Fast, Safe, Propellant-Efficient Spacecraft Motion...

Fast Safe Propellant-Efficient Spacecraft Motion PlanningUnder ClohessyndashWiltshirendashHill Dynamics

Joseph A Stareklowast Edward Schmerlingdagger and Gabriel D Maherdagger

Stanford University Stanford California 94305

Brent W BarbeeDagger

NASA Goddard Space Flight Center Greenbelt Maryland 20771

and

Marco Pavonesect

Stanford University Stanford California 94305

DOI 1025141G000324

This paper presents a sampling-based motion planning algorithm for real-time and propellant-optimized

autonomous spacecraft trajectory generation in near-circular orbits Specifically this paper leverages recent

algorithmic advances in the field of robot motion planning to the problem of impulsively actuated propellant-

optimized rendezvous and proximity operations under the ClohessyndashWiltshirendashHill dynamics model The approach

calls upon a modified version of the FMT algorithm to grow a set of feasible trajectories over a deterministic low-

dispersion set of sample points covering the free state space To enforce safety the tree is only grown over the subset of

actively safe samples from which there exists a feasible one-burn collision-avoidance maneuver that can safely

circularize the spacecraft orbit along its coasting arcunder a given set of potential thruster failuresKey features of the

proposed algorithm include 1) theoretical guarantees in terms of trajectory safety and performance 2) amenability to

real-time implementation and 3) generality in the sense that a large class of constraints can be handled directly As a

result the proposed algorithm offers the potential for widespread application ranging from on-orbit satellite

servicing to orbital debris removal and autonomous inspection missions

I Introduction

R EAL-time guidance for spacecraft proximity operations near

circular orbits is an inherently challenging task particularly for

onboard applications for which computational capabilities are limited

Fortunately for the unconstrained case many effective real-time

solutions have been developed (eg state transition matrix

manipulation [1] Lambert targeting [2] glideslope methods [3]

safety ellipses [4] and others [5]) However the difficulty of real-time

processing increases when there is a need to operate near other objects

andor incorporate some notion of propellant optimality or control-

effort minimization In such cases care is needed to efficiently handle

collision-avoidance plume-impingement sensor line-of-sight and

other complex guidance constraints which are often nonconvex and

may depend on time and amixture of state and control variables State-

of-the-art techniques for collision-free spacecraft proximity operations

(both with and without optimality guarantees) include artificial

potential function guidance [67] nonlinear trajectory optimization

with [89] or without [10] convexification line-of-sight or approach

corridor constraints [11ndash13] relative separation techniques [14]Keep-

Out Zone (KOZ) constraints with mixed-integer programming [15]

and kinodynamic motion planning algorithms [16ndash19]

Requiring hard assurances of mission safety with respect to a wide

variety and number of potential failure modes [20] provides an

additional challenge Often the concept of passive safety (safetycertifications on zero-control-effort failure trajectories) over a finitehorizon is employed to account for the possibility of control failuresthough this potentially neglects mission-saving opportunities andfails to certify safety for all time A less conservative alternative thatmore readily adapts to infinite horizons aswewill see is to use activesafety in the form of positively invariant set constraints For instance[11] enforces infinite-horizon active safety for a spacecraft byrequiring each terminal state to lie on a collision-free orbit of equalperiod to the target Reference [17] achieves a similar effect by onlyplanning between waypoints that lie on circular orbits (a morerestrictive constraint) Likewise [21] requires an autonomousspacecraft to maintain access to at least one safe forced equilibriumpoint nearby Finally [22] devises the Safe and Robust ModelPredictive Control algorithm which employs invariant feedbacktubes about a nominal trajectory (which guarantee resolvability)together with positively-invariant sets (taken about referencesafety states) designed to be available at all times over the planninghorizonThe objective of this paper is to design an automated approach to

actively-safe spacecraft trajectory optimization for rendezvous andproximity operations near circular orbits which we model usingClohessyndashWiltshirendashHill (CWH) dynamics Our approach leveragesrecent advances from the field of robot motion planning particularlysampling-based motion planning [23] Several decades of research inthe robotics community have shown that sampling-based planningalgorithms (dubbed ldquoplannersrdquo throughout this paper) show promisefor tightly constrained high-dimensional optimal control problemssuch as the one considered in this paper Sampling-based motionplanning essentially breaks down a complex trajectory controlproblem into a series ofmany local relaxed two-point boundary valueproblems (2PBVPs or steering problems) between intermediatewaypoints (called samples) which are later evaluated a posteriori forconstraint satisfaction and efficiently strung together into a graph(ie a tree or roadmap) Bymoving complex constraints like obstacleavoidance or plume impingement into a posteriori evaluation we candecouple trajectory generation from constraint checking a fact weexploit to achieve real-time capability Critically this approachavoids the explicit construction of the unconstrained state space acomputationally prohibitive task for complex planning problems In

Received 4 December 2015 revision received 15 June 2016 accepted forpublication 18 June 2016 published online 23 August 2016 Copyright copy2016 by the American Institute of Aeronautics and Astronautics Inc Allrights reserved Copies of this paper may be made for personal and internaluse on condition that the copier pay the per-copy fee to the CopyrightClearance Center (CCC) All requests for copying and permission to reprintshould be submitted to CCC at wwwcopyrightcom employ the ISSN 0731-5090 (print) or 1533-3884 (online) to initiate your request

Graduate Student Aeronautics andAstronautics 496 LomitaMall Room009

daggerGraduate Student Institute for Computational and MathematicalEngineering 475 Via Ortega

DaggerAerospace Engineer Navigation and Mission Design Branch Mail Code595 8800 Greenbelt Road Senior Member AIAA

sectAssistant Professor Aeronautics and Astronautics 496 Lomita MallRoom 261 Member AIAA

Article in Advance 1

JOURNAL OF GUIDANCE CONTROL AND DYNAMICS

this way sampling-based algorithms can address a large variety of

constraints and provide significant computational benefits withrespect to traditional optimal control methods and mixed-integer

programming [23] Furthermore through a property called as-

ymptotic optimality (AO) sampling-based algorithms can bedesigned to provide guarantees of optimality in the limit that

the number of samples taken approaches infinity This makessampling-based planners a strong choice for the problem of space-

craft controlAlthough the aforementioned works [16ndash19] on sampling-based

planning for spacecraft proximity operations have addressed several

components of the safety-constrained optimal CWH autonomous

rendezvous problem few have addressed the aspect of real-timeimplementability in conjunction with both a 2-norm propellant-cost

metric and active trajectory safety with respect to control failures

This paper seeks to fill this gap The paperrsquos central theme is arigorous proof of asymptotic optimality for a particular sampling-

based planner namely a modified version of the FMT algorithm[24] under impulsive CWH spacecraft dynamics with hard safety

constraints First a description of the problem scenario is provided in

Sec II along with a formal definition of the sum-of-2-norms costmetric that we employ as a proxy for propellant consumption

Section III then follows with a thorough discussion of chasertargetvehicle safety defining precisely how abort trajectories may be

designed under CWH dynamics to deterministically avoid for all

future times an ellipsoidal region about the CWH frame origin undera prescribed set of control failures Next we proceed in Sec IV to our

proposed approach employing the modified FMT algorithm The

section begins with presentation of a conservative approximation tothe propellant-cost reachability set which characterizes the set of

states that are nearby a given initial state in terms of propellant useThese sets bounded by unions of ellipsoidal balls are then used to

show that the modified FMT algorithm maintains its (asymptotic)

optimalitywhen applied toCWHdynamics under our propellant-costmetric From there in Sec V the paper presents two techniques for

improving motion planning solutions 1) an analytical technique that

can be called both during planning and postprocessing to merge Δvvectors between any pair of concatenated graph edges and 2) a

continuous trajectory smoothing algorithm which can reduce themagnitude of Δv vectors by relaxing the implicit constraint to pass

through sample points while still maintaining solution feasibilityPut together these tools yield a flexible real-time framework for

near-circular orbit spacecraft guidance that handles a wide variety

of (possibly nonconvex) state time and control constraints and

provides deterministic guarantees on abort safety and solution quality(propellant cost) The methodology is demonstrated in Sec VI on a

single-chaser single-target scenario simulating a near-field low-Earth-orbit (LEO) approach with hard constraints on the total

maneuver duration relative positioning (including KOZ and antennainterference constraints) thruster plume-impingement avoidanceindividual and net Δv-vector magnitudes and a two-fault thrusterstuck-off failure tolerance Trades are then conducted studyingthe effects of the sample count and a propellant cost threshold onthe performance of FMT (both with and without trajectorysmoothing)Preliminary versions of this paper appear in [2526] This extended

and revised work introduces the following as additional contri-butions 1) a more detailed presentation of the FMT optimalityproof 2) improved trajectory smoothing and 3) a six-dimensional (3degree-of-freedom) numerical example demonstrating nonplanarLEO rendezvous

II Problem Formulation

Consider the problem of a chaser spacecraft seeking to maneuvertoward a single target moving in a well-defined circular orbit (see

Fig 1a) Define the state space X sub Rd as a d-dimensional region inthe targetrsquos local vertical local horizontal (LVLH) frame and let theobstacle region or Xobs be the set of states in X that result in missionfailure (eg states outside of an approach corridor or in collisionwiththe target) Let the free space orX free be the complement ofXobs Asseen in Fig 1b let xinit represent the chaserrsquos initial state relative tothe target and let xgoal isin Xgoal be a goal state inside goal region

X goal Finally define a state trajectory (or simply ldquotrajectoryrdquo) as a

piecewise-continuous function of time xt R rarr X and let Σrepresent the set of all state trajectories Every state trajectory isimplicitly generated by a control trajectory ut R rarr U where U isthe set of controls through the system dynamics _x fx u twhere f is the state transition function A state trajectory is called afeasible solution to the planning problem (X free tinit xinit xgoal) if1) it satisfies the boundary conditions xtinit xinit and xtfinal xgoal for some time tfinal gt tinit 2) it is collision free that is xt isinX free for all t isin tinit tfinal and 3) it obeys all other trajectoryconstraints The optimal motion planning problem can then bedefined as followsDefinition 1 (Optimal Planning Problem) Given a planning

problem (X free tinit xinit xgoal) and a cost functional J Σ times U times R rarrRge0 find a feasible trajectory x

twith associated control trajectoryut and time span t tinit tfinal for tfinal isin tinitinfin such thatJxmiddot umiddot t minfJxmiddot umiddot t j xt andut are feasiblegIf no such trajectory exists report failureTailoring Definition 1 to impulsively actuated propellant-optimal

motion planning near circular orbits (using a control-effort costfunctional J that considers only ut and the final time tfinal denotedas Jut tfinal) the optimal spacecraft motion planning problemmay be formulated as

(Cross-track)radial

(In track)

(Normal to orbital plane)out of plane

Attractor

Reference line

Target

Chaser

a) Schematic of CWH dynamics which modelsrelative guidance near a target in circular orbit

b) A representative motion planning query betweenfeasible states xinit and xgoal

Fig 1 Illustration of the CWH planning scenario

2 Article in Advance STAREK ETAL

Given initial state xinittinit goal regionXgoal free spaceX free

minimizeuttfinal

JuttfinalZ

tfinal

tinit

kutk2 dtXNi1

kΔvik2

subject to xtinitxinit initial condition

xtfinalisinXgoal terminal condition

_xtfxtutt system dynamics

xtisinX free for all tisin tinittfinal obstacle avoidance

gxtuttle0

hxtutt0for all tisin tinittfinal other constraints

exist safe xCAMτ τgtt for all xt active safety (1)

where tinit and tfinal are the initial and final times and xCAMτ refersto an infinite-horizon collision-avoidance maneuver (CAM) Note

we restrict our attention to impulsive control laws ut PNi1 Δviδt minus τi where δmiddot denotes the Dirac delta function

which models finite sequences of instantaneous translational burns

Δvi fired at discrete times τi (note that the number of burns N is not

fixed a priori) Although it is possible to consider all control laws it is

both theoretically and computationally simpler to optimize over a

finite-dimensional search space of Δv vectors furthermore these

represent the most common forms of propulsion systems used on

orbit (including high-impulse cold-gas and liquid bipropellant

thrusters) and they can (at least in theory) approximate continuous

control trajectories in the limit in which N rarr infin

We now elaborate on the objective function and each constraint

in turn

A Cost Functional

A critical component of our problem is the choice of cost

functional Consistent with [27] we define our cost as theL1-norm of

the lp-norm of the control The best choice for p ge 1 depends on thepropulsion system geometry and on the frame within which ut P

Ni1 Δviδt minus τi in J is resolved Unfortunately minimizing the

propellant exactly involves resolving vectors Δvi into the spacecraftbody-fixed frame requiring the attitude q to be included in our state

x To avoid this a common standard is to employ p 2 so that eachΔvi is as short as possible allocating the commandedΔvi to thrustersin a separate control allocation step (conducted later once the attitude

is known see Sec IIE) Although this moves propellant mini-

mization online it greatly simplifies the guidance problem in a

practical way without neglecting attitude Because the cost of Δvallocation can only grow from the need to satisfy torque constraints or

impulse bounds (eg necessitating counteropposing thrusters to

achieve the same netΔv vector) we are in effect minimizing the best-

case unconstrained propellant use of the spacecraft As wewill show

in our numerical experiments however this does not detract

significantly from the technique the coupling of J with p 2 to theactual propellant use through theminimum control-effort thrusterΔvallocation problem seems to promote low propellant-cost solutions

Hence J serves as a good proxy to propellant use with the added

benefit of independence from propulsion system geometry

B Boundary Conditions

Planning is assumed to begin at a known initial state xinit andtime tinit and end at a single goal state x

Tgoal δrTgoal δvTgoal (exact

convergence Xgoal fxgoalg) where δrgoal is the goal position and

δvgoal is the goal velocity During numerical experiments however

we sometimes permit termination at any state of which the position

and velocity liewithin Euclidean ballsBδrgoal ϵr andBδvgoal ϵvrespectively (inexact convergenceXgoalBrgoalϵrtimesBvgoalϵv)where the notation Br ϵ fx isin X jkr minus xk le ϵg denotes a ball

with center r and radius ϵ

C System Dynamics

Because spacecraft proximity operations incorporate significant

drift spatially dependent external forces and changes on fast time

scales any realistic solution must obey dynamic constraints we

cannot assume straight-line trajectories In this paper we employ the

classical CWH equations [2829] for impulsive linearized motion

about a circular reference orbit at radius rref about an inverse-square-law gravitational attractor with parameter μ This model provides a

first-order approximation to a chaser spacecraftrsquos motion relative to a

rotating target-centered coordinate system (see Fig 1) The linearized

equations of motion for this scenario as resolved in the LVLH frame

of the target are given by

δ x minus 3n2refδx minus 2nrefδ _y Fδx

m(2a)

δ y 2nrefδ _x Fδy

m(2b)

δz n2refδz Fδz

m(2c)

where nref μr3ref

qis the orbital frequency (mean motion) of the

reference spacecraft orbit m is the spacecraft mass F Fδx Fδy FδzT is some applied force and (δx δy δz) and (δ _x δ _yδ_z) represent the cross-track (radial) in-track and out-of-plane

relative position and velocity vectors respectively The CWHmodel

is used often especially for short-duration rendezvous and proxi-

mity operations in LEO and for leaderndashfollower formation flight

dynamicsDefining the state x as δx δy δz δ _x δ _y δ_zT and the control u as

the applied force per unit mass 1mF the CWH equations can be

described by the linear time-invariant (LTI) system

_x fxu t AxBu (3)

where the dynamics matrix A and input matrix B are given by

A

266666666664

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

3n2ref 0 0 0 2nref 0

0 0 0 minus2nref 0 0

0 0 minusn2ref 0 0 0

377777777775B

266666666664

0 0 0

0 0 0

0 0 0

1 0 0

0 1 0

0 0 1

377777777775As for any LTI system there exists a unique solution to Eq (3) given

initial condition xt0 and integrable input ut that can be ex-

pressed using superposition and the convolution integral as xteAtminust0xt0int t

t0eAtminusτBuτdτ for any time t ge t0 The expression

Φt τ ≜ eAtminusτ is called the state transition matrix which impor-

tantly provides an analytical mechanism for computing state tra-

jectories that we rely on heavily in simulations Note throughout this

work we sometimes represent Φt τ as Φ when its arguments are

understoodWe now specialize our solution to the case ofN impulsive velocity

changes at times t0 le τi le tf for i isin 1 N inwhich caseuτPNi1Δviδτminusτi where δy f1wherey 0 or 0otherwiseg

signifies the Dirac-delta distribution Substituting for Φ and uτ

xt Φt t0xt0 Z

t

t0

Φt τBXNi1

Δviδτ minus τidτ

Φt t0xt0 XNi1

Zt

t0

Φt τBΔviδτ minus τi dτ

Article in Advance STAREK ETAL 3

where on the second linewe used the linearity of the integral operator

By the sifting property of δ denoting Nt P

Ni1 1τi le t as the

number of burns applied from t0 up to time t we have for all times

t ge t0 the following expression for the impulsive solution to Eq (3)

xt Φt t0xt0 XNt

i1

Φt τiBΔvi (4a)

Φt t0xt0 Φt τ1B Φt τNtB |z

≜Φvtfτigi

2664Δv1

ΔvNt

3775|z

≜ΔV

(4b)

Φt t0xt0 Φvt fτigiΔV (4c)

Throughout this paper the notations ΔV for the stacked Δv vector

andΦvt fτigi for the aggregated impulse state transition matrix (or

simply Φv for short when the parameters t and fτigi are clear)

implicitly imply only those burns i occurring before time t

D Obstacle Avoidance

Obstacle avoidance is imposed by requiring that the solution xtstay within X free (or equivalently outside of Xobs) typically a

difficult nonconvex constraint For CWH proximity operationsXobs

might include positions in collision with a nearby object position

velocity pairs outside of a given approach corridor etc In our

numerical experiments to prevent the chaser from interfering with

the target we assumeXobs comprises an ellipsoidal KOZ centered at

the origin and a conical nadir-pointing region that approximates the

targetrsquos antenna beam patternNote that according to the definition of X free this also requires

xt to stay within X (CWH dynamics do not guarantee that state

trajectories will lie inside X despite the fact that their endpoints do)

Although not always necessary in practice if X marks the extent of

reliable sensor readings or the boundary inside which linearity

assumptions hold then this can be useful to enforce

E Other Trajectory Constraints

Many other types of constraints may be included to encode

additional restrictions on state and control trajectories which we

represent here by a set of inequality constraints g and equality

constraints h (note that g and h denote vector functions) To illustrate

the flexibility of sampling-based planning we encode the following

into constraints g (for brevity we omit their exact representation

which is a straightforward exercise in vector geometry)

Tplanmax le tfinal minus tinit le Tplanmax plan duration bounds

Δvi isin Uxτi for all i 1 N control feasibility[kisin1 K

PikΔvk βplume Hplume cap Starget empty for all i 1 N plume impingement avoidance

Here 0 le Tplanmin lt Tplanmax represent minimum and maximummotion plan durations Uxτi is the admissible control setcorresponding to state xτiPik is the exhaust plume emanating fromthruster k of the chaser spacecraft while executing burnΔvi at time τiand Starget is the target spacecraft circumscribing sphereWemotivateeach constraint in turn

1 Plan Duration Bounds

Plan duration bounds facilitate the inclusion of rendezvous

windows based on the epoch of the chaser at xinittinit as determined

by illumination requirements ground communication opportunities

or mission timing restrictions for example Tplanmax can also ensure

that the errors incurred by linearization which growwith time do not

exceed acceptable tolerances

2 Control Feasibility

Control set constraints are intended to encapsulate limitations on

control authority imposed by propulsive actuators and their

geometric distribution about the spacecraft For example given the

maximum burn magnitude 0 lt Δvmax the constraint

kΔvik2 le Δvmax for all i 1 N (5)

might represent an upper bound on the achievable impulses of a

gimbaled thruster system that is free to direct thrust in all directions

In our case we use Uxτi to represent all commanded net Δvvectors that 1) satisfy Eq (5) and also 2) can be successfully allocated

to thrusters along trajectory xt at time τi according to a simple

minimum-control-effort thruster allocation problem (a straight-

forward linear program [30]) To keep the paper self-contained we

repeat the problem here and in our own notation LetΔvijbf andMijbfbe the desired netΔv andmoment vectors at burn time τi resolved inthe body-fixed frame according to attitude qτi (we henceforth dropthe bar for clarity) Note the attitude qτimust either be included in

the state xτi or be derived from it as we assume here by imposing

(along nominal trajectories) a nadir-pointing attitude profile for the

chaser spacecraft Let Δvik kΔvikk2 be the Δv magnitude

allocated to thruster k which generates an impulse in direction Δvikat position ρik from the spacecraft center-of-mass (both are constant

vectors if resolved in the body-fixed frame) Finally to account for

the possibility of on or off thrusters let ηik be equal to 1 if thruster k isavailable for burn i or zero otherwise Then the minimum-effort

control allocation problem can be represented as

Given on-of flagsηik thruster positions ρik thruster axesΔvik

commandedΔv-vectorΔviandcommanded

momentvectorMi

minimizeΔvik

XKk1

Δvik

subject toXKk1

ΔvikηikΔvik Δvi netΔv-vector allocation

XKk1

ρik timesΔviηikΔvik Mi net moment allocation

Δvmink leΔvik leΔvmaxk thrusterΔvbounds (6)

where Δvmink and Δvmaxk represent minimum- and maximum-impulse limits on thruster k (due to actuator limitations minimumimpulse bits pulse-width constraints or maximum on-timerestrictions for example) Note that by combining the minimizationof commanded Δv-vector lengths kΔvik with minimum-effortallocation Δvik to thrusters k in Eq (6) the previous formulation


corresponds directly to minimum-propellant consumption (by theTsiolkovsky rocket equation subject to our thrust bounds and nettorque constraints) see also Sec IIA In this work we setMi 0 toenforce torque-free burns and minimize disturbances to our assumedattitude trajectory qtNote that we do not consider aminimum-norm constraint in Eq (5)

forΔvi as it is not necessary and would significantly complicate thetheoretical characterization of our proposed planning algorithmsprovided in Sec IV As discussed in Sec IIA kΔvik is only a proxyfor the true propellant cost computed from the thrust allocationproblem [Eq (6)]

3 Plume Impingement

Impingement of thruster exhaust on neighboring spacecraft canlead to dire consequences including destabilizing effects on attitudecaused by exhaust gas pressure degradation of sensitive opticalequipment and solar arrays and unexpected thermal loading [3132]To account for this during guidance we first generate representativeexhaust plumes at the locations of each thruster firing For burn ioccurring at time τi a right circular cone with axis minusΔvik half-angleβplume and height Hplume is projected from each active thruster k(ηik 1) the allocated thrust Δvik of which is nonzero asdetermined by Eq (6) Intersections are then checked with the targetspacecraft circumscribing sphere Starget which is used as a simpleconservative approximation to the exact target geometry For anillustration of the process refer to Fig 2

4 Other Constraints

Other constraints may easily be added Solar array shadowingpointing constraints approach corridors and so forth all fit within theframework and may be represented as additional inequality orequality constraints For more we refer the interested reader to [33]

F Active Safety

An additional feature we include in our work is the concept ofactive safety in which we require the target spacecraft to maintain a

feasible CAM to a safe higher or lower circular orbit from every pointalong its solution trajectory in the event that any mission-threateningcontrol degradations take place such as stuck-off thrusters (as inFig 3) This reflects our previous work [25] and is detailed morethoroughly in Sec III

III Vehicle Safety

In this section we devise a general strategy for handling the activesafety constraints introduced in Eq (1) and Sec IIF which we use toguarantee solution safety under potential control failuresSpecifically we examine how to ensure in real-time that safe aborttrajectories are always available to the spacecraft up to a givennumber of thruster stuck-off failures As will be motivated the ideabehind our approach is to couple positively invariant set safetyconstraints with escape trajectory generation and embed them intothe sampling routines of deterministic sampling-based motionplanners We prioritize active safety measures in this section (whichallow actuated CAMs over passive safety guarantees (which shut offall thrusters and restrict the system to zero control) in order to broadenthe search space for abort trajectories Because of the propellant-limited nature of many spacecraft proximity operations missionsemphasis is placed on finding minimum-Δv escape maneuvers inorder to improve mission reattempt opportunities In many ways weemulate the rendezvous design process taken byBarbee et al [34] butnumerically optimize abort propellant consumption and removemuch of its reliance on user intuition by automating the satisfaction ofsafety constraintsConsistent with the notions proposed by Schouwenaars et al [35]

Fehse [36] Sec 412 and Fraichard [37] our general definition forvehicle safety is taken to be the followingDefinition 2 (vehicle safety) A vehicle state is safe if and only if

there exists under the worst-possible environment and failureconditions a collision-free dynamically feasible trajectory satis-fying the constraints that navigates the vehicle to a set of states inwhich it can remain indefinitelyNote ldquoindefinitelyrdquo (or ldquosufficiently longrdquo for all practical pur-

poses under the accuracy of the dynamics model) is a criticalcomponent of the definition Trajectories without infinite-horizonsafety guarantees can ultimately violate constraints [11] therebyposing a risk that can defeat the purpose of using a hard guarantee inthe first place For this reason we impose safety constraints over aninfinite-horizon (or as we will show using invariant sets aneffectively infinite horizon)Consider the scenario described in Sec II for a spacecraft with

nominal state trajectory xt isin X and control trajectory ut isinUxt evolving over time t in time spanT tinitinfin LetT fail sube Trepresent the set of potential failure times we wish to certify (forinstance a set of prescribed burn times fτig the final approach phaseT approach or the entire maneuver span T ) When a failure occurscontrol authority is lost through a reduction in actuator functionalitynegatively impacting system controllability Let U failx sub Uxrepresent the new control set wherewe assume that 0 isin U fail for all x(ie we assume that no actuation is always a feasible option)Mission safety is commonly imposed in two different ways([36] Sec 44)1) For all tfail isin T fail ensure that xCAMt satisfies Definition 2

with uCAMt 0 for all t ge tfail (called passive safety) For aspacecraft this means its coasting arc from the point of failure mustbe safe for all future time (though practically this is imposed only overa finite horizon)2) For all tfail isin T fail and failure modes U fail devise actuated

collision-avoidancemaneuvers xCAMt that satisfy Definition 2withuCAMt isin U fail for all t ge tfail where uCAMt is not necessarilyrestricted to 0 (called active safety)See Fig 4a for an illustration In much of the literature only

passive safety is considered out of a need for tractability (to avoidverification over a combinatorial explosion of failure modepossibilities) and to capture the common case in which controlauthority is lost completely Although considerably simpler to

Fig 2 Illustration of exhaust plume impingement from thruster firings

a) Thruster allocation without stuck-off failures

b) The same allocation problemwith both upper-right thrusters stuck off

Fig 3 Changes to torque-free control allocation in response to thrusterfailures


implement this approach potentially neglects many mission-saving

control policies

A Active Safety Using Positively-Invariant Sets

Instead of ensuring safety for all future times t ge tfail it is more

practical to consider finite-time abort maneuvers starting at xtfailthat terminate inside a safe positively invariant set X invariant If the

maneuver is safe and the invariant set is safe for all time then vehicle

safety is assured

Definition 3 (positively invariant set) A set X invariant is positively

invariant with respect to the autonomous system _xCAM fxCAM ifand only if xCAMtfail isin X invariant implies xCAMt isin X invariant for

all t ge tfailSee Fig 4b This yields the following definition for finite-time

verification of trajectory safety

Definition 4 (finite-time trajectory safety verification) For all

tfail isin T fail and for all U failxtfail sub Uxtfail there exists

fut t ge tfailg isin U failxtfail and Th gt tfail such that xt is feasiblefor all tfail le t le Th and xTh isin X invariant sube X free

Here Th is some finite safety horizon time Although in principle

any safe positively invariant set X invariant is acceptable not just any

will do in practice in real-world scenarios unstable trajectories

caused by model uncertainties could cause state divergence toward

configurations of which the safety has not been verified Hence care

must be taken to use only stable positively-invariant sets

Combining Definition 4 with our constraints in Eq (1) from

Sec II spacecraft trajectory safety after a failure at xtfail xfail canbe expressed in its full generality as the following optimization

problem in decision variables Th isin tfailinfin xCAMt and uCAMtfor t isin tfail Th

Given failure state xfailtfail failure control setU failxfail the free space X free

a safe stable invariant setX invariant and a fixed number of impulsesN

minimizeuCAMtisinUfailxfail

ThxCAMt

JxCAMt uCAMt t Z

Th

tfail

kuCAMtk2 dt XNi1

kΔvCAMik2

subject to _xCAMt fxCAMt uCAMt t system dynamics

xCAMtfail xfail initial condition

xCAMTh isin X invariant safe termination

xCAMt isin X free for all t isin tfail Th obstacle avoidance

gxCAM uCAM t le 0

hxCAM uCAM t 0for all t isin tfail Th other constraints (7)

This is identical to Eq (1) except that now under failure mode

U failxfail we abandon the attempt to terminate at a goal state inXgoal

and instead replace it with a constraint to terminate at a safe stable

positively invariant set X invariant We additionally neglect any timing

constraints encoded in g as we are no longer concerned with our

original rendezvous Typically any feasible solution is sought

following a failure in which case one may use J 1 However toenhance the possibility of mission recovery we assume the same

minimum-propellant cost functional as before but with the exception

that here aswewillmotivateweusea single-burn strategywithN 1

B Fault-Tolerant Safety Strategy

The difficulty of solving the finite-time trajectory safety problem lies

in the fact that a feasible solution must be found for all possible failure

times (typically assumed to be any time during the mission) as well as

for all possible failures To illustrate for an F-fault tolerant spacecraftwith K control components (thrusters momentum wheels control

moment gyroscopes etc) each of which we each model as either

operational or failed this yields a total of Nfail P

Ff0

Kf

P

Ff0

KKminusff possible optimization problems that must be solved for

every time tfail along the nominal trajectory By any standard this is

intractable and hence explains why so often passive safety guarantees

are selected (requiring only one control configuration check instead of

Nfail since we prescribe uCAM 0 which must lie in U fail given our

assumption this is analogous to setting f K withF ≜ K) One ideafor simplifying this problemwhile still satisfying safety [the constraints

of Eq (7)] consists of the following strategyDefinition 5 (fault-tolerant active safety strategy) As a

conservative solution to the optimization problem in Eq (7) it is

sufficient (but not necessary) to implement the following procedure1) From each xtfail prescribe a CAM policy ΠCAM that gives a

horizon time Th and escape control sequence uCAM ΠCAMxtfaildesigned to automatically satisfy uCAMτ sub U for all tfail le τ le Th

and xTh isin X invariant

2) For each failure mode U failxtfail sub Uxtfail up to toleranceF determine if the control law is feasible that is see if uCAM ΠCAMxtfail sub U fail for the particular failure in question

This effectively removes decision variables uCAM from Eq (7)

allowing simple numerical integration for the satisfaction of the

Passive Abort

Active Abort

Failure

a) Comparing passive and active abort safety followingthe occurrence of a failure

b) A positively invariant set within which statetrajectories stay once entered

Fig 4 Illustrations of various vehicle abort safety concepts


dynamic constraints and a straightforward a posteriori verification ofthe other trajectory constraints (inclusion in X free and satisfaction ofconstraintsg andh) This checks if the prescribedCAM guaranteed toprovide a safe escape route can actually be accomplished in the givenfailure situation The approach is conservative due to the fact that thecontrol law is imposed and not derived however the advantage is agreatly simplified optimal control problem with difficult-to-handleconstraints relegated toaposteriori checks exactly identical to thewaythat steering trajectories are derived and verified during the planningprocess of sampling-based planning algorithms Note that formaldefinitions of safety require that this be satisfied for all possible failuremodes of the spacecraft we do not avoid the combinatorial explosionof Nfail However each instance of problem Eq (7) is greatlysimplified and with F typically at most 3 the problem remainstractable The difficult part then lies in computingΠCAM but this caneasily be generated offline Hence the strategy should work well forvehicles with difficult nonconvex objective functions and constraintsas is precisely the case for CWH proximity operationsNote that it is always possible to reduce this approach to the (more-

conservative) definition of passive safety that has traditionally beenseen in the literature by choosing some finite horizon Th and settinguCAM ΠCAMxtfail 0 for all potential failure times tfail isin T fail

C Safety in CWH Dynamics

We now specialize these ideas to proximity operations underimpulsive CWH dynamics Because manymissions require stringentavoidance (before the final approach and docking phase forexample) it is quite common for aKOZXKOZ typically ellipsoidal inshape to be defined about the target in the CWH frame Throughoutits approach the chaser must certify that it will not enter this KOZunder any circumstance up to a specified thruster fault tolerance Fwhere here faults imply zero-output (stuck-off) thruster failures PerDefinition 4 this necessitates a search for a safe invariant set forfinite-time escape along with as outlined by Definition 5 thedefinition of an escape policy ΠCAM which we describe next

1 CAM Policy

We now have all the tools we need to formulate an active abortpolicy for spacecraft maneuvering under CWH dynamics Recallfrom Definition 4 that for mission safety following a failure we mustfind a terminal state in an invariant set X invariant entirely containedwithin the free state space X free To that end we choose for X invariant

the set of circularized orbits of which the planar projections lieoutside of the radial band spanned by the KOZ The reasons wechoose this particular set for abort termination are threefold circularorbits are 1) stable (assuming Keplerian motion which is reasonableeven under perturbations because the chaser and target are perturbedtogether and it is their relative state differences that matter)2) accessible (given the proximity of the chaser to the targetrsquos circularorbit) and 3) passively safe (once reached provided there is no

intersection with the KOZ) In the planar case this set of safe

circularized orbits can fortunately be identified by inspection As

shown in Fig 5 the set of orbital radii spanning theKOZare excluded

in order to prevent an eventual collisionwith theKOZellipsoid either

in the short term or after nearly one full synodic period In the event of

an unrecoverable failure or an abort scenario taking longer than one

synodic period to resolve circularization within this region would

jeopardize the target a violation of Definition 2 Such a region is

called a zero-thrust region of inevitable collision (RIC) which we

denote asX ric as without additional intervention a collision with the

KOZ is imminent and certain To summarize this mathematically

XKOZ fxjxTEx le 1g (8)

where E diagρminus2δx ρminus2δy ρminus2δz 0 0 0 with ρi representing the

ellipsoidal KOZ semi-axis in the i-th LVLH frame axis direction

X ric xjjδxj lt ρδx δ _x 0 δ _y minus

3

2nrefδx

sup XKOZ (9)

X invariant xjjδxj ge ρδx δ _x 0 δ _y minus

3

2nrefδx

X c

ric (10)

In short our CAM policy to safely escape from a state x at which

the spacecraft arrives (possibly under failures) at time tfail as

visualized in Fig 6 consists of the following1) Coast from xtfail to some new Th gt tfail such that xCAMTminus

h lies at a position in X invariant2) Circularize the (in-plane) orbit at xCAMTh such that

xCAMTh isin X invariant

3) Coast along the neworbit (horizontal drift along the in-track axisin the CWH relative frame) in X invariant until allowed to continue themission (eg after approval from ground operators)

a) Safe circularization burn zones invariant for planar

b) Inertial view of the radial band spanned by theKOZ that defines the unsafe RIC

CWH dynamics

Fig 5 Visualizing the safe and unsafe circularization regions used by the CAM safety policy

CoastingArc

Circularized Orbit

Fig 6 Examples of safe abort CAMs xCAM following failures


2 Optimal CAM Circularization

In the event of a thruster failure at state xtfail that requires anemergency CAM the time Th gt tfail at which to attempt a circulari-zation maneuver after coasting from xtfail becomes a degree offreedom As we intend to maximize the recovery chances of thechaser after a failure we choose Th so as to minimize the cost of thecircularization burnΔvcirc the magnitude of which we denoteΔvcircDetails on the approach which can be solved analytically can befound in Appendix A

3 CAM Policy Feasibility

Once the circularization time Th is determined feasibility of theescape trajectory under every possible failure configuration at xtfailmust be assessed in order to declare a particular CAM as actively safeTo show this the constraints of Eq (7) must be evaluated under everycombination of stuck-off thrusters (up to fault tolerance F) with theexception ofKOZ avoidance as this is embedded into the CAMdesignprocess How quickly thismay be done depends on howmany of theseconstraints may be considered static (unchanging ie independent oftfail in the LVLH frame of reference) or time varying (otherwise)

Fortunately most practical mission constraints are static (ieimposed in advance by mission planners) allowing CAM trajectoryfeasibility verification to bemoved offline For example consideringour particular constraints in Sec IIE if we can assume that the targetremains enclosed within its KOZ near the origin and that it maintainsa fixed attitude profile in the LVLH frame then obstacle and antennalobe avoidance constraints become time invariant (independent of thearrival time tfail) If we further assume the attitude qt of the chaser isspecified as a function of xt then control allocation feasibility andplume-impingement constraints become verifiable offline as wellBetter still because of their time independence we need only toevaluate the safety of arriving at each failure state xfail once thismeans the active safety of a particular state can be cached a fact wewill make extensive use of in the design of our planning algorithm

Some constraints on the other hand cannot be defined a prioriThesemust be evaluated online once the time tfail and current environ-ment are known which can be expensive due to the combinatorial ex-plosion of thruster failure combinations In such cases theseconstraints can instead be conservatively approximated by equivalentstatic constraints or otherwise omitted from online guidance until aftera nominal guidance plan has been completely determined (called lazyevaluation) These strategies can help ensure active safety whilemaintaining real-time capability

IV Planning Algorithm and TheoreticalCharacterization

With the proximity operations scenario established we are now inposition to describe our approach As previously described the

constraints that must be satisfied in Eq (1) are diverse complex anddifficult to satisfy numerically In this section we propose a guidancealgorithm to solve this problem followed by a detailed proof of itsoptimality with regard to the sum-of-2-norms propellant-cost metric

J under impulsive CWH dynamics As will be seen the proof relies

on an understanding of 1) the steering connections between sampled

points assuming no obstacles or other trajectory constraints and 2) the

nearest-neighbors or reachable states from a given state We hence

start by characterizing these two concepts in Secs IVA and IVB

respectively We then proceed to the algorithm presentation (Sec IV

C) and its theoretical characterization (Sec IVD) before closingwith

a description of two smoothing techniques for rapidly reducing the

costs of sampling-based solution trajectories for systems with

impulsive actuation (Sec V)

A State Interconnections Steering Problem

For sample-to-sample interconnections we consider the un-

constrained minimal-propellant 2PBVP or steering problem

between an initial state x0 and a final state xf under CWH dynamics

Solutions to these steering problems provide the local building

blocks from which we construct solutions to the more complicated

problem formulation in Eq (1) Steering solutions serve two main

purposes1) They represent a class of short-horizon controlled trajectories

that are filtered online for constraint satisfaction and efficientlystrung together into a state-space spanning graph (ie a tree orroadmap)

2) The costs of steering trajectories are used to inform the graphconstruction process by identifying the unconstrained nearest neigh-bors as edge candidates

Because these problems can be expressed independently of the

arrival time t0 (as will be shown) our solution algorithm does not

need to solve these problems online the solutions between every

pair of samples can be precomputed and stored before receiving a

motion query Hence the 2PBVP presented here need not be solved

quickly However we mention techniques for speedups due to the

reliance of our smoothing algorithm (Algorithm 2) on a fast solution

methodSubstituting our boundary conditions into Eq (4) evaluating at

t tf and rearranging we seek a stacked burn vector ΔV such that

Φvtf fτigiΔV xf minusΦtf t0x0 (11)

for some numberN of burn times τi isin t0 tf Formulating this as an

optimal control problem that minimizes our sum-of-2-norms cost

functional (as a proxy for the actual propellant consumption as

described in Sec IIA) we wish to solve

Given initial state x0 final state xf burn magnitude boundΔvmax

and maneuver duration boundTmax

minimizeΔviτi tfN

XNi1

kΔvik2

subject to Φvtf fτigiΔV xf minusΦtf t0x0 dynamics=boundary conditions

0 le tf minus t0 le Tmax maneuver duration bounds

t0 le τi le tf for burns i burn time bounds

kΔvik2 le Δvmax for burns i burn magnitude bounds (12)

Notice that this is a relaxed version of the original problempresented as Eq (1) with only its boundary conditions dynamicconstraints and control norm bound As it stands because of thenonlinearity of the dynamics with respect to τi tf andN Eq (12) is


nonconvex and inherently difficult to solve However we can make

the problem tractable if we make a few assumptions Given that we

plan to string many steering trajectories together to form our overall

solution let us ensure they represent the most primitive building

blocks possible such that their concatenation will adequately rep-resent any arbitrary trajectory Set N 2 (the smallest number of

burns required to transfer between any pair of arbitrary states as it

makesΦvtf fτigi square) and select burn times τ1 t0 and τ2 tf (which automatically satisfy our burn time bounds) This leaves

Δv1 isin Rd∕2 (an intercept burn applied just after x0 at time t0)Δv2 isin Rd∕2 (a rendezvous burn applied just before xf at time tf) andtf as our only remaining decisionvariables If we conduct a search for

tf isin t0 t0 Tmax the relaxed 2PBVP can nowbe solved iterativelyas a relatively simple bounded one-dimensional nonlinear

minimization problem where at each iteration one computes

ΔVtf Φminus1v tf ft0 tfgxf minusΦtf t0x0

where the argument tf is shown for ΔV to highlight its dependence

By uniqueness of the matrix inverse (provided Φminus1v is nonsingular

discussed in what follows) we need only check that the resulting

impulsesΔvitf satisfy the magnitude bound to declare the solutionto an iteration feasible Notice that becauseΦ andΦminus1

v depend only

on the difference between tf and t0 we can equivalently search overmaneuver durations T tf minus t0 isin 0 Tmax instead solving the

following relaxation of Eq (12)

Given∶ initial state x0 final state xf burn magnitude boundΔvmax

and maneuver duration boundTmax lt2π

nref

minimizeTisin0Tmax

X2i1

kΔvik2

subject to ΔV Φminus1v T f0 Tgxf minusΦT 0x0 dynamics∕boundary conditions


This dependence on the maneuver duration T only (and not on the

time t0 at which we arrive at x0) turns out to be indispensable for

precomputation as it allows steering trajectories to be generated and

stored offline Observe however that our steering solution ΔVrequires Φv to be invertible ie that tf minus τ1 minus tf minus τ2 tf minus t0 T avoids singular values (including zero orbital period

multiples and other values longer than one period [38]) and we

ensure this by enforcingTmax to be shorter than one period To handle

the remaining case of T 0 note a solution exists if and only if x0and xf differ in velocity only in such instances we take the solutionto be Δv2 set as this velocity difference (with Δv1 0)

B Neighborhoods Cost Reachability Sets

Armed with a steering solution we can now define and identify

state neighborhoods This idea is captured by the concept of

reachability sets In keeping with Eq (13) since ΔV depends onlyon the trajectory endpoints xf and x0 we henceforth refer to the costof a steering trajectory by the notation Jx0 xf We then define the

forward reachability set from a given state x0 as followsDefinition 6 (forward reachable set) The forward reachable setR

from state x0 is the set of all states xf that can be reached from x0 witha cost Jx0 xf below a given cost threshold J ie

Rx0 J ≜ fxf isin X jJx0 xf lt Jg

Recall fromEq (13) in Sec IVA that the steering cost may bewritten

as

Jx0 xf kΔv1k kΔv2k kS1ΔVk kS2ΔVk (14)

whereS1 Id∕2timesd∕2 0d∕2timesd∕2S2 0d∕2timesd∕2 Id∕2timesd∕2 andΔV is

given by

ΔVx0 xf Δv1Δv2

Φminus1

v tf ft0 tfgxf minusΦtf t0x0

The cost function Jx0 xf is difficult to gain insight into directlyhowever as we shall see we can work with its bounds much more

easilyLemma 1 (fuel burn cost bounds) For the cost function in Eq (14)

the following bounds hold

kΔVk le Jx0 xf le2

pkΔVk

Proof For the upper bound note that by the Cauchyndash

Schwarz inequality we have J kΔv1k middot 1 kΔv2k middot 1 lekΔv1k2 kΔv2k2

pmiddot

12 12

p That is J le

2

p kΔVk Similarly

for the lower bound note that J kΔv1k kΔv2k2

pge

kΔv1k2 kΔv2k2p

kΔVk

Now observe that kΔVkxfminusΦtft0x0TGminus1xfminusΦtft0x0

q where Gminus1 ΦminusT

v Φminus1v

This is the expression for an ellipsoid Exf resolved in the LVLH

frame with matrix Gminus1 and center Φtf t0x0 (the state T tf minus t0time units ahead of x0 along its coasting arc) Combined with

Lemma 1 we see that for a fixedmaneuver timeT and propellant cost

threshold J the spacecraft at x0 can reach all states inside an area

underapproximated by an ellipsoid with matrix Gminus1∕ J2 and

overapproximated by an ellipsoid of matrix2

pGminus1∕ J2 The forward

reachable set for impulsiveCWHdynamics under our propellant-costmetric is therefore bounded by the union over all maneuver times ofthese under- and overapproximating ellipsoidal sets respectively Tobetter visualize this see Fig 7 for a geometric interpretation

C Motion Planning Modified FMT Algorithm

Wenowhave all the tools we need to adapt sampling-basedmotionplanning algorithms to optimal vehicle guidance under impulsiveCWHdynamics as represented by Eq (1) Sampling-based planningessentially breaks down a continuous trajectory optimizationproblem into a series of relaxed local steering problems (as inSec IVA) between intermediate waypoints (called samples) beforepiecing them together to form a global solution to the originalproblem This framework can yield significant computationalbenefits if 1) the relaxed subproblems are simple enough and 2) the aposteriori evaluation of trajectory constraints is fast compared to asingle solution of the full-scale problem Furthermore providedsamples are sufficiently dense in the free state-space X free and graphexploration is spatially symmetric sampling-based planners canclosely approximate global optima without fear of convergence tolocal minima Althoughmany candidate planners could be used herewe rely on the AO FMT algorithm for its efficiency (see [24] fordetails on the advantages of FMT over its state-of-the-art counter-parts) and its compatibilitywith deterministic (as opposed to random)batch sampling [39] a key benefit that leads to a number of algo-rithmic simplifications (including the use of offline knowledge)


The FMT algorithm tailored to our application is presented as

Algorithm 1 (we shall henceforth refer to our modified version of

FMT as simply FMT for brevity) Like its path-planning variantour modified FMT efficiently expands a tree of feasible trajectories

from an initial state xinit to a goal state xgoal around nearby obstaclesIt begins by taking a set of samples distributed in the free state space

X free using the SAMPLEFREE routine which restricts state sampling toactively safe feasible states (which lie outside ofXobs and have access

to a safe CAM as described in Sec IIIC) In our implementation we

assume samples are taken using theHalton sequence [40] though anydeterministic low-dispersion sampling sequence may be used [39]

After selecting xinit first for further expansion as the minimum cost-to-come node z the algorithm then proceeds to look at reachable

samples or neighbors (samples that can be reached with less than agiven propellant cost threshold J as described in the previous

subsection) and attempts to connect those with the cheapest cost-to-

come back to the tree (using Steer) The cost threshold J is a freeparameter of which the value can have a significant effect on

performance see Theorem 2 for a theoretical characterization andSec VI for a representative numerical trade study Those trajectories

satisfying the constraints of Eq (1) as determined by CollisionFreeare saved As feasible connections are made the algorithm relies on

adding and removing nodes (savedwaypoint states) from three sets a

set of unexplored samples Vunvisited not yet connected to the tree afrontier Vopen of nodes likely to make efficient connections to

unexplored neighbors and an interior Vclosed of nodes that are nolonger useful for exploring the state space X More details on FMT

can be found in its original work [24]To make FMT amenable to a real-time implementation we

consider an onlinendashoffline approach that relegates as much compu-

tation as possible to a preprocessing phase To be specific the sam-

ple set S (Line 2) nearest-neighbor sets (used in Lines 5 and 6)and steering trajectory solutions (Line 7) may be entirely pre-processed assuming the planning problem satisfies the followingconditions1) The state space X is known a priori as is typical for most LEO

missions (notewe do not impose this on the obstacle spaceXobs sub X which must generally be identified online using onboard sensorsupon arrival to X )2) Steering solutions are independent of sample arrival times t0 as

we show in Sec IVAHere Item1allows samples tobe precomputedwhile Item2enables

steering trajectories to be stored onboard or uplinked from the groundup to the spacecraft since their values remain relevant regardless of thetimes at which the spacecraft actually follows them during the missionThis leaves only constraint checking graph construction and termi-nation checks as parts of the online phase greatly improving the onlinerun time and leaving the more intensive work to offline resources forwhich running time is less important This breakdown into online andoffline components (inspired by [41]) is a valuable technique forimbuing kinodynamicmotion planning problems with real-time onlinesolvability using fast batch planners like FMT

D Theoretical Characterization

It remains to show that FMT provides similar asymptoticoptimality guarantees under the sum-of-2-norms propellant-costmetric and impulsive CWH dynamics (which enter into Algorithm 1under Lines 6 and 7) as has already been shown for kinematic(straight-line path-planning) problems [24] For sampling-basedalgorithms asymptotic optimality refers to the property that as thenumber of samples n rarr infin the cost of the trajectory (or path)returned by the planner approaches that of the optimal cost Here a

Algorithm 1 The fast marching tree algorithm (FMT) Computes a minimal-costtrajectory from an initial state xt0 xinit to a target state xgoal through a fixed number

n of samples S

1) Add xinit to the root of the tree T as a member of the frontier set Vopen

2) Generate samples Slarr SAMPLEFREE (X n t0) and add them to the unexplored set Vunvisited

3) Set the minimum cost-to-come node in the frontier set as zlarrxinit4) while true do5) for each neighbor x of z in Vunvisited do6) Find the neighbor xmin in Vopen of cheapest cost-to-go to x7) Compute the trajectory between them as xt ut tlarrSteerxmin x (see Sec IVA)8) if COLLISIONFREE xt ut t then9) Add the trajectory from xmin to x to tree T10) Remove all x from the unexplored set Vunvisited

11) Add any new connections x to the frontier Vopen

12) Remove z from the frontier Vopen and add it to Vclosed

13) if Vopen is empty then14) return failure15) Reassign z as the node in Vopen with smallest cost-to-come from the root (xinit)16) if z is in the goal region Xgoal then17) return success and the unique trajectory from the root (xinit) to z

a) The set of reachable positions δrf withinduration Tmax and propellant cost Δυmax

b) The set of reachable velocities δvf within duration Tmax and propellant cost Δυmax

Fig 7 Bounds on reachability sets from initial state xt0 under propellant-cost threshold J


proof is presented showing asymptotic optimality for the planningalgorithm and problem setup used in this paper We note that whileCWH dynamics are the primary focus of this work the followingproof methodology extends to any general linear system controlledby a finite sequence of impulsive actuations the fixed-duration2-impulse steering problem of which is uniquely determined (eg awide array of second-order control systems)The proof proceeds analogously to [24] by showing that it is

always possible to construct an approximate path using points in Sthat closely follows the optimal path Similarly to [24] are great wewill make use here of a concept called the l2 dispersion of a set ofpoints which places upper bounds on how far away a point inX canbe from its nearest point in S as measured by the l2-normDefinition 7 (l2 dispersion) For a finite nonempty set S of points

in a d-dimensional compact Euclidean subspace X with positiveLebesgue measure its l2 dispersion DS is defined as

DS ≜ supxisinX

minsisinS

ks minus xk

supfR gt 0jexist x isin X withBx R cap S emptygwhere Bx R is a Euclidean ball with radius R centered at state xWe also require a means for quantifying the deviation that small

endpoint perturbations can bring about in the two-impulse steeringcontrol This result is necessary to ensure that the particularplacement of the points of S is immaterial only its low-dispersionproperty mattersLemma 2 (steering with perturbed endpoints) For a given steering

trajectory xt with initial time t0 and final time tf let x0 ≔ xt0xf ≔ xtf T ≔ tf minus t0 and J ≔ Jx0 xf Consider now the per-turbed steering trajectory ~xt between perturbed start and endpoints~x0x0δx0 and ~xfxfδxf and its corresponding cost J ~x0 ~xfCase 1 (T 0) There exists a perturbation center δxc (consisting of

only a position shift) with kδxck OJ2 such that if kδx0k leηJ3 and kδxf minus δxck le ηJ3 then J ~x0 ~xfleJ14ηOJand the spatial deviation of the perturbed trajectory ~xt fromxt is OJ

Case 2 (T gt 0) If kδx0k le ηJ3 and kδxfk le ηJ3 thenJ ~x0 ~xf le J1OηJ2Tminus1 and the spatial deviation of theperturbed trajectory ~xt from xt is OJ

Proof For the proof see Appendix B

We are now in a position to prove that the cost of the trajectoryreturned by FMT approaches that of an optimal trajectory as thenumber of samples n rarr infin The proof proceeds in two steps Firstwe establish that there is a sequence of waypoints in S that are placedclosely along the optimal path and approximately evenly spaced incost Then we show that the existence of these waypoints guaranteesthat FMT finds a path with a cost close to that of the optimal costThe theorem and proof combine elements from Theorem 1 in [24]and Theorem IV6 from [42]Definition 8 (strong δ clearance) A trajectory xt is said to have

strong δ clearance if for some δ gt 0 and all t the Euclidean distancebetween xt and any point in Xobs is greater than δTheorem 1 (existence of waypoints near an optimal path) Let xt

be a feasible trajectory for the motion planning problem Eq (1) with

strong δ clearance let ut PNi1 Δvi middot δt minus τi be its asso-

ciated control trajectory and let J be its cost Furthermore letS cup fxinitg be a set of n isin N points from X free with dispersion

DS le γnminus1∕d Let ϵ gt 0 and choose J 4γnminus1∕d∕ϵ1∕3 Thenprovided that n is sufficiently large there exists a sequence of points

fykgKk0 yk isin S such that Jyk yk1 le J the cost of the path ytmade by joining all of the steering trajectories between yk and yk1 isP

Kminus1k0 Jyk yk1 le 1 ϵJ and yt is itself strong δ∕2 clearProof We first note that if J 0 then we can pick y0 xt0

and y1 xtf as the only points in fykg and the result is trivialThus assume that J gt 0 Construct a sequence of times ftkgKk0 andcorresponding points xk xtk spaced along xt in cost intervalsof J∕2 We admit a slight abuse of notation here in that xτi mayrepresent a statewith any velocity along the length of the impulseΔvi to be precise pick x0 xinit t0 0 and for k 1 2 definejk minfjjPj

i1 kΔvi k gt kJ2g and select tk and xk as

tk τjk

xk limtrarrtminus

k

xt kJ

2minusXjkminus1i1

kΔvi kB

ΔvikΔvi k

Let K dJe∕ J∕2 and set tK tf xK xtf Since the

trajectory xt to be approximated is fixed for sufficiently small J(equivalently sufficiently large n) we may ensure that the controlapplied between each xk and xk1 occurs only at the endpoints In

particular this may be accomplished by choosing n large enough so

that J lt minikΔvi k In the limit J rarr 0 the vast majority of the

two-impulse steering connections between successive xk will be

zero-time maneuvers (arranged along the length of each burn Δvi )with only N positive-time maneuvers spanning the regions of xtbetween burns By considering this regime of n we note thatapplying two-impulse steering between successive xk (which

otherwise may only approximate the performance of amore complexcontrol scheme) requires cost no greater than that of x itself along

that step ie J∕2We now inductively define a sequence of points fxkgKk0 by

x0 x0 and for each k gt 01) If tk tkminus1 pick x

k xk δxck xkminus1 minus xkminus1 where δxck

comes from Lemma 2 for zero-time approximate steering betweenxkminus1 and xk subject to perturbations of size ϵJ

32) Otherwise if tk gt tkminus1 pick xk xk xkminus1 minus xkminus1 The

reason for defining these xk is that the process of approximatingeachΔvi by a sequence of small burns necessarily incurs some short-term position drift Since δxck O J2 for each k and sinceK O Jminus1 the maximum accumulated difference satisfiesmaxkkxk minus xkk O JFor each k consider the Euclidean ball centered at xk with radius

γnminus1d ie let Bk ≔ Bxk γnminus

1d By Definition 7 and our restriction

on S eachBk contains at least one point from S Hence for everyBk

we can pick awaypoint yk such that yk isin Bk cap S Then kyk minus xkk leγnminus

1d ϵ J∕23∕8 for all k and thus by Lemma 2 (with η ϵ∕8) we

have that

Jyk yk1 leJ

2

1 ϵ

2O J

le

J

21 ϵ

for sufficiently large n In applying Lemma 2 to Case 2 for k such thattk1 gt tk we note that the T

minus1 term is mitigated by the fact that thereis only a finite number of burn times τi along xt Thus for eachsuch k tk1 minus tk ge minjtj1 minus tj gt 0 so in every case we have

Jyk yk1 le J∕21 ϵ That is each segment connecting yk toyk1 approximates the cost of the corresponding xk to x

k1 segment

of xt up to a multiplicative factor of ϵ and thus

XKminus1k0

Jyk yk1 le 1 ϵJ

Finally to establish that yt the trajectory formed by steeringthrough the yk in succession has sufficient obstacle clearance we

note that its distance from xt is bounded by maxkkxk minus xkk O J plus the deviation bound from Definition 7 again O J Forsufficiently large n the total distance O J will be bounded by δ∕2and thus yt will have strong δ∕2 clearance

We now prove that FMT is asymptotically optimal in the numberof points n provided the conditions required in Theorem 1 hold notethe proof is heavily based on Theorem VI1 from [43]Theorem 2 (asymptotic performance of FMT) Let xt be a

feasible trajectory satisfying Eq (1) with strong δ clearance and costJ LetS cup fx0g be a set of n isin N samples fromX free with dispersionDS le γnminus1∕d Finally let Jn denote the cost of the path returned byFMT with n points in S while using a cost threshold Jn ωnminus1∕3d and J o1 [That is Jn asymptotically dominates


nminus1∕3d and is asymptotically dominated by 1] Thenlimnrarrinfin Jn le JProof Let ϵ gt 0 Pick n sufficiently large so that δ∕2 ge J ge

4γnminus1∕d∕ϵ1∕3 such that Theorem 1 holds That is there exists a

sequence of waypoints fykgKk0 approximating xt such that the

trajectory yt created by sequentially steering through the yk is

strong δ∕2 clear the connection costs of which satisfy Jyk yk1 leJ and the total cost of which satisfies

PKminus1k0 Jyk yk1 le 1 ϵJ

We show that FMT recovers a path with cost at least as good as ytthat is we show that limnrarrinfinJn le JConsider running FMT to completion and for each yk let cyk

denote the cost-to-come of yk in the generated graph (with valueinfin ifyk is not connected) We show by induction that

mincym Jn leXmminus1

k0

Jyk yk1 (15)

for all m isin 1 K For the base case m 1 we note by theinitialization of FMT in Line 1 of Algorithm 1 that xinit is in Vopentherefore by the design of FMT (per Lines 5ndash9) every possiblefeasible connection ismade between the first waypoint y0 xinit andits neighbors Since Jy0 y1 le J and the edge y0 y1 is collisionfree it is also in the FMT graph Then cy1 Jy0 y1 Nowassuming that Eq (15) holds for m minus 1 one of the followingstatements holds1) Jn le

Pmminus2k0 Jyk yk1

2) cymminus1 leP

mminus2k0 Jyk yk1 and FMT ends before

considering ym3) cymminus1 le

Pmminus2k0 Jyk yk1 and ymminus1 isin Vopen when ym is

first considered4) cymminus1 le

Pmminus2k0 Jyk yk1 and ymminus1 isin= Vopen when ym is

first consideredWe now show for each case that our inductive hypothesis holds

Case 1 Jn leP

mminus2k0 Jyk yk1 le

Pmminus1k0 Jyk yk1

Case 2 Since at every step FMT considers the node that is theendpoint of the path with the lowest cost if FMT ends beforeconsidering ym we have

Jn le cym le cymminus1 Jymminus1 ym leXmminus1

k0

Jyk yk1

Case 3 Since the neighborhood of ym is collision free by theclearance property of y and since ymminus1 is a possible parentcandidate for connection ym will be added to the FMT tree assoon as it is considered with cym le cymminus1 Jymminus1 ym leP

mminus1k0 Jyk yk1

Case 4 When ym is considered it means there is a node z isin Vopen

(with minimum cost to come through the FMT tree) andym isin Rz J Then cymleczJzym Since cymminus1 ltinfin ymminus1 must be added to the tree by the time FMT terminatesConsider the path from xinit to ymminus1 in the final FMT tree andlet w be the last vertex along this path which is in Vopen at the

time when ym is considered If ym isin Rw J iew is a parentcandidate for connection then

cym le cw Jw ymle cw Jw ymminus1 Jymminus1 ymle cymminus1 Jymminus1 ym

leXmminus1

k0

Jyk yk1

Otherwise if ym isin= Rw J then Jw ym gt J and

cym le cz Jz ymle cw J

le cw Jw ymle cw Jw ymminus1 Jymminus1 ymle cymminus1 Jymminus1 ym

leXmminus1

k0

Jyk yk1

where we used the fact that w is on the path of ymminus1 to establishcw Jw ymminus1 le cymminus1 Thus by induction Eq (15) holdsfor all m Taking m K we finally have that Jn le cyK leP

Kminus1k0 Jyk yk1 le 1 ϵJ as desiredRemark 1 (asymptotic optimality of FMT) If the planning

problem at hand admits an optimal solution that does not itself havestrong δ clearance but is arbitrarily approximable both pointwise andin cost by trajectories with strong clearance (see [43] for additionaldiscussion on why such an assumption is reasonable) thenTheorem 2 implies the asymptotic optimality of FMT

V Trajectory Smoothing

Because of the discreteness caused by using a finite number ofsamples sampling-based solutions will necessarily be approxima-tions to true optima In an effort to compensate for this limitation weoffer in this section two techniques to improve the quality of solutionsreturned by our planner from Sec IVC We first describe a straight-forwardmethod for reducing the sum ofΔv-vectormagnitudes alongconcatenated sequences of edge trajectories that can also be used toimprove the search for propellant-efficient trajectories in the feasiblestate spaceX freeWe then followwith a fast postprocessing algorithmfor further reducing the propellant cost after a solution has beenreportedThe first technique removes unnecessary Δv vectors that occur

when joining subtrajectories (edges) in the planning graph Considermerging two edges at a nodewith position δrt and velocity δvt asin Fig 8a A naive concatenation would retain both Δv2tminus (therendezvous burn added to the incoming velocity vtminus) and Δv1t

a) Smoothing during graph construction(merges Δ vectors at edge endpoints)

b) Smoothing during postprocessing (see algorithm 2)υ

Fig 8 Improving sampling-based solutions under minimal-propellant impulsive dynamics


(the intercept burn used to achieve the outgoing velocity vt)individually within the combined control trajectory Yet becausethese impulses occur at the same time a more realistic approach

should merge them into a single net Δv vector Δvitminus By the

triangle inequality we have that

kΔvnettminusk kΔv2tminus Δv1tk le kΔv2tminusk kΔv1tk

Hence merging edges in this way guarantees Δv savings for

solution trajectories under our propellant-cost metric Furthermoreincorporating netΔv into the cost to come during graph construction

can make the exploration of the search space more efficient the cost

to come cz for a given node z would then reflect the cost torendezvous with z from xinit through a series of intermediate

intercepts rather than a series of rendezvous maneuvers (as atrajectory designer might normally expect) Note on the other hand

that these new net Δv vectors may be larger than either individual

burn which may violate control constraints control feasibility tests(allocation feasibility to thrusters plume impingement etc) must

thus be reevaluated for each new impulse Furthermore observe thatthe node velocity δvt is skipped altogether at edge endpoints whentwo edges as in Fig 8a are merged in this fashion This can be

problematic for the actively safe abort CAM (see Sec IIIC) fromstates along the incoming edge which relies on rendezvousing with

the endpoint x δrt δvt exactly before executing a one-burncircularization maneuver To compensate for this care must be taken

to ensure that the burn Δv2tminus that is eliminated during merging is

appropriately appended to the front of the escape control trajectoryand verified for all possible failure configurations Hence we see the

price of smoothing in this way is 1) reevaluating control-dependentconstraints at edge endpoints before accepting smoothing and 2) that

our original one-burn policy now requires an extra burn which may

not be desirable in some applicationsThe second technique attempts to reduce the solution cost by

adjusting the magnitudes of Δv vectors in the trajectory returned byFMT [denoted by xnt with associated stacked impulse vector

ΔVn] By relaxing FMTrsquos constraint to pass through state samples

strong cost improvements may be gained The main idea is to deformour low-cost feasible solution xnt as much as possible toward the

unconstrained minimum-propellant solution xt between xinit andxgoal as determined by the 2PBVP [Eq (12)] solution from Sec IVA

[in other words use a homotopic transformation from xnt to xt]However a naive attempt to solve Eq (12) in its full generality wouldbe too time consuming to be useful and would threaten the real-time

capability of our approach Assuming our sampling-based trajectoryis near optimal (or at least in a low-cost solution homotopy) we can

relax Eq (12) by keeping the number of burnsN end time tf ≔ tfinaland burn times τi fixed from our planning solution and solve for an

approximate unconstrained minimum-propellant solution ΔVdagger with

associated state trajectory xdaggert via

minimizeΔvi

XNi1

kΔvik2

subject to Φvtfinal fτigiΔV xgoal minusΦtfinal tinitxinitdynamics=boundary conditions

kΔvik2 le Δvmax for all burns i

burn magnitude bounds (16)

(see Sec IIC for definitions) It can be shown that Eq (16) is a

second-order cone program and is hence quickly solved using

standard convex solvers As the following theorem shows explicitly

we can safely deform the trajectory xnt toward xdaggert without

violating our dynamics and boundary conditions if we use a convex

combination of our two control trajectories ΔVn and ΔVdagger This

follows from the principle of superposition given that the CWH

equations are LTI and the fact that both solutions already satisfy the

boundary conditionsTheorem 3 (dynamic feasibility of CWH trajectory smoothing)

Suppose xnt and xdaggert with respective control vectors ΔVn and

ΔVdagger are two state trajectories satisfying the steering problemEq (11)

between states xinit and xgoal Then the trajectory xt generated by

the convex combination of ΔVn and ΔVdagger is itself a convex

combination of xnt and xdaggert and hence also satisfies Eq (11)Proof Let ΔV αΔVn 1 minus αΔVdagger for some value α isin 0 1

From our dynamics equation

xt Φt tinitxinit Φvt fτigiΔV α 1 minus αΦt tinitxinit Φvt fτigiαΔVn 1 minus αΔVdagger αΦt tinitxinit Φvt fτigiΔVn 1 minus αΦt tinitx0 Φvt fτigiΔVdagger

αxnt 1 minus αxdaggert

which is a convex combination as required Substituting t tinit ort tgoal we see that xt satisfies the boundary conditions given thatxnt and xdaggert doWe take advantage of this fact for trajectory smoothing Our

algorithm reported as Algorithm 2 and illustrated in Fig 8b

computes the approximate unconstrained minimum- propellant

solution xdaggert and returns it (if feasible) or otherwise conducts a

bisection line search on α returning a convex combination of our

original planning solution xnt and xdaggert that comes as close to xdaggert

Algorithm 2 Trajectory smoothing algorithm for impulsive CWH dynamicsGiven a trajectory xnt t isin tinittgoal between initial and goal states xinit and xgoalsatisfying Eq (1) with impulses ΔVn applied at times fτigi returns another feasible

trajectory with reduced propellant cost

1) Initialize the smoothed trajectory xsmootht as xnt with ΔVsmooth ΔVn

2) Compute the unconstrained optimal control vector ΔVdagger by solving Eq (16)3) Compute the unconstrained optimal state trajectory xdaggert using Eq (4) (See Sec IIC)4) Initialize weight α and its lower and upper bounds as αlarr1 αllarr0 αularr15) while true do6) xtlarr1 minus αxnt αxdaggert7) ΔVlarr1 minus αΔVn αΔVdagger

8) if COLLISIONFREE (xtΔV t) then9) αllarrα10) Save the smoothed trajectory xsmootht as xt and control ΔVsmooth as ΔV11) else12) αularrα13) if αu minus αl is less than tolerance δαmin isin 0 1 then14) break15) αlarrαl αu∕216) return the smoothed trajectory xsmootht with ΔVsmooth


as possible without violating trajectory constraints Note becauseΔVn lies in the feasible set of Eq (16) the algorithm can only improvethe final propellant cost By design Algorithm 2 is geared towardreducing our original solution propellant cost as quickly as possiblewhile maintaining feasibility the most expensive computationalcomponents are the calculation of ΔVdagger and constraint checking(consistent with our sampling-based algorithm) Fortunately thenumber of constraint checks is limited by the maximum number ofiterations dlog2 1

δαmine 1 given tolerance δαmin isin 0 1 As an

added bonus for strictly time-constrained applications that require asolution in a fixed amount of time the algorithm can be easilymodified to return the αl-weighted trajectory xsmootht when timeruns out as the feasibility of this trajectory is maintained as analgorithm invariant

VI Numerical Experiments

Consider the two scenarios shown in Fig 9 modeling both planarand nonplanar near-field approaches of a chaser spacecraft in closeproximitypara to a target moving on a circular LEO trajectory (as inFig 1)We imagine the chaser which starts in a circular orbit of lowerradius must be repositioned through a sequence of prespecifiedCWH waypoints (eg required for equipment checks surveyingetc) to a coplanar position located radially above the target arrivingwith zero relative velocity in preparation for a final radial (R-bar)approach Throughout the maneuver as described in detail in Sec IIthe chaser must avoid entering the elliptic target KOZ enforce a hardtwo-fault tolerance to stuck-off thruster failures and otherwise avoidinterfering with the target This includes avoiding the targetrsquos nadir-

pointing communication lobes (represented by truncated half-cones)and preventing exhaust plume impingement on its surfaces For

context we use the Landsat-7 [44] spacecraft and orbit as a reference([45] Sec 32) (see Figs 10 and 11)

A Simulation Setup

Before proceeding to the results we first outline some key featuresof our setup Taking the prescribed query waypoints one at a time as

individual goal points xgoal we solve the given scenario as a series ofmotion planning problems (or subplans) linked together into an

overall solution calling FMT from Sec IVC once for each subplanFor this multiplan problem we take the solution cost to be the sum of

individual subplan costs (when using trajectory smoothing theendpoints between two plans are merged identically to two edges

within a plan as described in Sec V)As our steering controller from Sec IVA is attitude independent

states x isin Rd are either xT δx δy δ _x δ _y with d 4 (planarcase) or xT δx δy δz δ _x δ _y δ_z with d 6 (nonplanar case)

This omission of the attitude q from the state is achieved byemploying an attitude policy (assuming a stabilizing attitude

controller) which produces qt from the state trajectory xt Forillustration purposes a simple nadir-pointing attitude profile is

chosen during nominal guidance representing a mission requiring

constant communication with the ground for actively safe abort weassume a simple turnndashburnndashturn policy which orients the closest-

available thruster under each failure mode as quickly as possible intothe direction required for circularization (see Sec IIIC)Given the hyperrectangular shape of the state space we call upon

the deterministic low-dispersion d-dimensional Halton sequence

[40] to sample positions and velocities To improve samplingdensities each subplan uses its own sample space defined around its

respective initial and goal waypoints with some arbitrary threshold

a) Planar motion planning query b) 3-dimensional motion planning queryFig 9 Illustrations of the planar and 3D motion plan queries in the LVLH frame

Fig 10 Target spacecraft geometry and orbital scenario used in numerical experiments

paraClose proximity in this context implies that any higher-order terms of thelinearized relative dynamics are negligible eg within a few percent of thetarget orbit mean radius


space added around them Additionally extra samples ngoal are takeninside each waypoint ball to facilitate convergenceFinally we make note of three additional implementation details

First for clarity we list all relevant simulation parameters in Table 1Second all position-related constraint checks regard the chaserspacecraft as a point at its center of mass with all other obstaclesartificially inflated by the radius of its circumscribing sphere Thirdand finally all trajectory constraint checking is implemented bypointwise evaluation with a fixed time-step resolution Δt using theanalytic state transition equations Eq (4) together with steeringsolutions from Sec IVA to propagate graph edges for speed the linesegments between points are excluded Except near very sharpobstacle corners this approximation is generally not a problem inpractice (obstacles can always be inflated further to account for this)To improve performance each obstacle primitive (ellipsoid right-circular cone hypercube etc) employs hierarchical collisionchecking using hyperspherical andor hyperrectangular boundingvolumes to quickly prune points from consideration

B Planar Motion Planning Solution

A representative solution to the posed planning scenario bothwithand without the trajectory smoothing algorithm (Algorithm 2) isshown in Fig 12 As shown the planner successfully finds safetrajectories within each subplan which are afterward linked to forman overall solution The state space of the first subplan shown at thebottom is essentially unconstrained as the chaser at this point is toofar away from the target for plume impingement to come into play

This means every edge connection attempted here is added so thefirst subplan illustrates well a discrete subset of the reachable statesaround xinit and the unrestrained growth of FMT As the secondsubplan is reached the effects of the KOZ position constraints comein to play and we see edges begin to take more leftward loops Insubplans 3 and 4 plume impingement begins to play a role Finally insubplan 5 at the top where it becomes very cheap to move betweenstates (as the spacecraft can simply coast to the right for free) we seethe initial state connecting to nearly every sample in the subspaceresulting in a straight shot to the final goal As is evident straight-linepath planning would not approximate these trajectories wellparticularly near coasting arcs along which our dynamics allow thespacecraft to transfer for freeTo understand the smoothing process examine Fig 13 Here we

see how the discrete trajectory sequence from our sampling-basedalgorithm may be smoothly and continuously deformed toward theunconstrained minimal-propellant trajectory until it meets trajectoryconstraints (as outlined in Sec V) if these constraints happen to beinactive then the exact minimal-propellant trajectory is returned as

Table 1 List of parameters used during near-circular orbitrendezvous simulations

Parameter Value

Chaser plume half-angle βplume 10 degChaser plume height Hplume 16 mChaser thruster fault tolerance F 2Cost threshold J 01ndash04 m∕sDimension d 4 (planar) 6

(nonplanar)Goal sample count ngoal 004nGoal position tolerance ϵr 3ndash8 mGoal velocity tolerance ϵv 01ndash05 m∕sMax allocated thruster Δv magnitude Δvmaxk infin m∕sMax commanded Δv-vector magnitude kΔvikΔvmax

infin m∕s

Max plan duration Tplanmax infin sMin plan duration Tplanmin 0 sMax steering maneuver duration Tmax 01 middot 2π∕nrefMin steering maneuver duration Tmin 0 sSample count n 50ndash400 per planSimulation time step Δt 00005 middot 2π∕nrefSmoothing tolerance δαmin 001Target antenna lobe height 75 mTarget antenna beamwidth 60degTarget KOZ semi-axes ρδx ρδy ρδz 35 50 15 m

a) Chaser spacecraft b) Target spacecraft

Fig 11 Models of the chaser and target together with their circumscribing spheres

Fig 12 Representative planar motion planning solution using theFMT algorithm (Algorithm 1) with n 2000 (400 per subplan)J 03 m∕s and relaxed waypoint convergence (Soln Solution TrajTrajectory)


Fig 13a shows This computational approach is generally quite fastassuming a well-implemented convex solver is used as will be seenin the results of the next subsectionThe net Δv costs of the two reported trajectories in this example

come to 0835 (unsmoothed) and 0811 m∕s (smoothed) Comparethis to 0641 m∕s the cost of the unconstrained direct solution thatintercepts each of the goal waypoints on its way to rendezvousingwithxgoal (this trajectory exits the state space along the positive in-trackdirection a violation of our proposed mission hence its costrepresents an underapproximation to the true optimal cost J of theconstrained problem) This suggests that our solutions are quite closeto the constrained optimum and certainly on the right order ofmagnitude Particularlywith the addition of smoothing at lower samplecounts the approach appears to be a viable one for spacecraft planningIf we compare the net Δv costs to the actual measured propellant

consumption given by the sum total of all allocated thruster Δvmagnitudes [which equal 106 (unsmoothed) and 101 m∕s(smoothed)] we find increases of 270 and 245 as expected oursum-of-2-norms propellant-cost metric underapproximates the truepropellant cost For point masses with isotropic control authority(eg a steerable or gimbaled thruster that is able to point freely in anydirection) our cost metric would be exact Although inexact for ourdistributed attitude-dependent propulsion system (see Fig 11a) it isclearly a reasonable proxy for allocated propellant use returningvalues on the same order of magnitude Although we cannot make astrong statement about our proximity to the propellant-optimalsolutionwithout directly optimizing over thrusterΔv allocations oursolution clearly seems to promote low propellant consumption

C Nonplanar Motion Planning Solution

For the nonplanar case representative smoothed and unsmoothedFMT solutions can be found in Fig 14 Here the spacecraft isrequired to move out of plane to survey the target from above beforereaching the final goal position located radially above the target Thefirst subplan involves a long reroute around the conical regionspanned by the targetrsquos communication lobes Because the chaserbegins in a coplanar circular orbit at xinit most steering trajectoriesrequire a fairly large cost to maneuver out of plane to the firstwaypoint Consequently relatively few edges that both lie in thereachable set of xinit and safely avoid the large conical obstacles areadded As we progress to the second and third subplans thecorresponding trees become denser (more steering trajectories areboth safe and within our cost threshold J) as the state space becomesmore open Compared with the planar case the extra degree offreedom associated with the out-of-plane dimension appears to allowmore edges ahead of the target in the in-track direction than beforelikely because now the exhaust plumes generated by the chaser are

well out of plane from the target spacecraft Hence the space-craft smoothly and tightly curls around the ellipsoidal KOZ tothe goalThe netΔv costs for this example come to 0611 (unsmoothed) and

0422 m∕s (smoothed) Counterintuitively these costs are on thesame order of magnitude and slightly cheaper than the planar casethe added freedom given by the out-of-plane dimension appears tooutweigh the high costs typically associated with inclination changesand out-of-plane motion These cost values correspond to totalthruster Δv allocation costs of 0893 and 0620 m∕s respectivelyincreases of 46 and 47 above their counterpart cost metric valuesAgain our cost metric appears to be a reasonable proxy for actualpropellant use

D Performance Evaluation

To evaluate the performance of our approach an assessment ofsolution quality is necessary as a function of planning parametersie the number of samples n taken and the reachability set costthreshold J As proven in Sec IVD the solution cost will eventuallyreduce to the optimal value as we increase the sample size nAlternatively we can attempt to reduce the cost by increasing the costthreshold J used for nearest-neighbor identification so that moreconnections are explored Both however work at the expense of therunning time To understand the effects of these changes on qualityespecially at finite sample counts for which the asymptoticguarantees of FMT do not necessarily imply cost improvements wemeasure the cost vs computation time for the planar planningscenario parameterized over several values of n and JResults are reported in Figs 15 and 16 Here we call FMT once

each for a series of sample countcost threshold pairs plotting thetotal cost of successful runs at their respective run times asmeasured by wall clock time (CVXGEN and CVX [46] disciplinedconvex programming solvers are used to implement Δv allocationand trajectory smoothing respectively) Note only the onlinecomponents of each FMT call ie graph searchconstructionconstraint checking and termination evaluations constitute the runtimes reported everything else may either be stored onboard beforemission launch or otherwise computed offline on ground computersand later uplinked to the spacecraft See Sec IVC for detailsSamples are stored as a d times n array while intersample steeringcontrolsΔvi and times τi are precomputed asn times n arrays ofd∕2 times Nand N times 1 array elements respectively Steering state and attitude

Fig 13 Visualizing trajectory smoothing (Algorithm 2) for the solution shown in Fig 12

All simulations are implemented in MATLABreg 2012b and run on a PCoperated byWindows 10 clocked at 400GHz and equipped with 320 GB ofRAM


trajectories xt and qt on the other hand are generated online

through Eq (4) and our nadir-pointing attitude policy respectively

This reduces memory requirements though nothing precludes them

from being generated and stored offline as well to save additional

computation time

Figure 15 reports the effects on the solution cost from varying the

cost threshold J while keeping n fixed As described in Sec IVB

increasing J implies a larger reachability set size and hence increases

the number of candidate neighbors evaluated during graph construc-

tion Generally this gives a cost improvement at the expense of extra

processing although there are exceptions as in Fig 15a at Jasymp03 m∕s Likely this arises from a single new neighbor (connected at

the expense of another since FMT only adds one edge per

neighborhood) that readjusts the entire graph subtree ultimately

increasing the cost of exact termination at the goal Indeed we see

that this does not occur where inexact convergence is permitted

given the same sample distribution

We can also vary the sample count n while holding J constant

From Figs 15a and 15b we select J 022 and 03 m∕srespectively for each of the two cases (the values that suggest the best

solution cost per unit of run time) Repeating the simulation for

varying sample count values we obtain Fig 16 Note the general

downward trend as the run time increases (corresponding to larger

sample counts) a classical tradeoff in sampling-based planning

However there is bumpiness Similar to before this is likely due to

new connections previously unavailable at lower sample counts that

Fig 14 Representative nonplanarmotion planning solution using the FMT algorithm (Algorithm1)withn 900 (300 per subplan) J 04 m∕s andrelaxed waypoint convergence

a) Exact waypoint convergence (n = 2000) b) Inexact waypoint convergence (n = 2000)

Fig 15 Algorithm performance for the given LEO proximity operations scenario as a function of varying cost threshold ( J isin 0204) with n heldconstant

b) Inexact waypoint convergence (J = 03 ms)a) Exact waypoint convergence (J = 022 ms)Fig 16 Algorithm performance for the given LEO proximity operations scenario as a function of varying sample count (n isin 6502000) with J heldconstant


cause a slightly different graph with an unlucky jump in thepropellant costAs the figures show the utility of trajectory smoothing is clearly

affected by the fidelity of the planning simulation In each trajectorysmoothing yields a much larger improvement in cost at modestincreases in computation time when we require exact waypointconvergence It provides little improvement on the other hand whenwe relax these waypoint tolerances FMT (with goal regionsampling) seems to return trajectories with costs much closer to theoptimum in such cases making the additional overhead of smoothingless favorable This conclusion is likely highly problem dependentthese tools must always be tested and tuned to the particularapplicationNote that the overall run times for each simulation are on the order

of 1ndash5 s including smoothing This clearly indicates that FMT canreturn high-quality solutions in real time for spacecraft proximityoperations Although run on a computer currently unavailable tospacecraft we hope that our examples serve as a reasonable proof ofconcept we expect that with a more efficient coding language andimplementation our approach would be competitive on spacecrafthardware

VII Conclusions

A technique has been presented for automating minimum-propellant guidance during near-circular orbit proximity operationsenabling the computation of near-optimal collision-free trajectoriesin real time (on the order of 1ndash5 s for our numerical examples) Theapproach uses amodified version of the FMTsampling-basedmotionplanning algorithm to approximate the solution to minimal-propellantguidance under impulsiveClohessyndashWiltshirendashHill (CWH)dynamicsOurmethod begins by discretizing the feasible space of Eq (1) throughstate-space sampling in the CWH local-vertical local-horizontal(LVLH) frame Next state samples and their cost reachability setswhich we have shown comprise sets bounded by unions of ellipsoidstaken over the steering maneuver duration are precomputed offline

and stored onboard the spacecraft together with all pairwise steeringsolutions Finally FMT is called online to efficiently construct a treeof trajectories through the feasible state space toward a goal regionreturning a solution that satisfies a broad range of trajectory constraints(eg plume impingement and obstacle avoidance control allocationfeasibility etc) or else reporting failure If desired trajectorysmoothing using the techniques outlined in Sec V can be employed toreduce the solution propellant costThe key breakthrough of our solution for autonomous spacecraft

guidance is its judicious distribution of computations in essenceonlywhatmust be computed onboard such as collision checking andgraph construction is computed online everything else includingthe most intensive computations are relegated to the ground wherecomputational effort and execution time are less critical Further-more only minimal information (steering problem control trajec-tories costs and nearest-neighbor sets) requires storage onboard thespacecraft Although we have illustrated through simulations theability to tackle a particular minimum-propellant LEO homingmaneuver problem it should be noted that the methodology appliesequally well to other objectives such as the minimum-time problemand can be generalized to other dynamic models and environments

The approach is flexible enough to handle nonconvexity and mixedstatendashcontrolndashtime constraints without compromising real-timeimplementability so long as constraint function evaluation isrelatively efficient In short the proposed approach appears to beuseful for automating the mission planning process for spacecraftproximity operations enabling real-time computation of low-costtrajectoriesThe proposed guidance framework for impulsively actuated

spacecraft offers several avenues for future research For examplethough nothing in the methodology forbids it outside of computa-tional limitations it would be interesting to revisit the problem withattitude states included in the state (instead of assuming an attitudeprofile) This would allow attitude constraints into the planningprocess (eg enforcing the line of sight keeping solar panelsoriented towards the sun maintaining a communication link with theground etc) Also of interest are other actively safe policies that relaxthe need to circularize escape orbits (potentially costly in terms ofpropellant use) or thatmesh better with trajectory smoothing withoutthe need to add compensating impulses (see Sec V) Extensions todynamic obstacles (such as debris or maneuvering spacecraft whichare unfixed in the LVLH frame) elliptical target orbits higher-ordergravitation or dynamics under relative orbital elements alsorepresent key research areas useful for extending applicability tomore general maneuvers Finally memory and run time performanceevaluations of our algorithms on spacelike hardware are needed toassess our methodrsquos true practical benefit to spacecraft planning

Appendix A Optimal Circularization Under ImpulsiveCWH Dynamics

As detailed in Sec IIIC1 a vital component of our CAMpolicy isthe generation of one-burn minimal-propellant transfers to circularorbits located radially above or below the target Assuming we needto abort from some state xtfail xfail the problem we wish tosolve in order to assure safety (per Definition 4 and as seen in Fig 6)is

Given∶ failure state xfail andCAM uCAMtfail le t lt Tminush ≜ 0 uCAMTh ≜ ΔvcircxTh

minimizeTh

Δv2circTh

subject to xCAMtfail xfail initial condition

xCAMTh isin X invariant invariant set termination

_xCAMt fxCAMt 0 t for all tfail le t le Th system dynamics

xCAMt isin= XKOZ for all tfail le t le Th KOZcollision avoidance

Because of the analytical descriptions of state transitions as givenby Eq (4) it is a straightforward task to express the decision variableTh invariant set constraint and objective function analytically interms of θt nreft minus tfail the polar angle of the target spacecraftThe problem is therefore one dimensional in terms of θ We canreduce the invariant set termination constraint to an invariant setpositioning constraint if we ensure the spacecraft ends up at a position

inside X invariant and circularize the orbit since xθcircxθminuscirc

h0

ΔvcircθcirciisinX invariant Denote θcirc nrefTh minus tfail

as the target anomaly at which we enforce circularization Nowsuppose the failure state xt lies outside of theKOZ (otherwise thereexists no safe CAM and we conclude that xfail is unsafe) We candefine a new variable θmin nreft minus tfail and integrate the coastingdynamics forward from time tfail until the chaser touches theboundary of the KOZ (θmax θminuscollision) or until we have reached onefull orbit (θmax θmin 2π) such that between these two boundsthe CAM trajectory satisfies the dynamics and contains only thecoasting segment outside of the KOZ Replacing the dynamics andcollision-avoidance constraints with the bounds on θ as a boxconstraint the problem becomes


minimizeθcirc

Δv2circθcirc

subject to θmin le θcirc le θmax theta bounds

δx2θminuscirc ge ρ2δx invariant set positioning

Restricting our search range to θ isin θmin θmax this is a functionof one variable and one constraint Solving by the method ofLagrange multipliers we seek to minimize the Lagrangian L Δv2circ λgcirc where gcircθ ρ2δx minus δx2θminuscirc There are two

cases to considerCase 1 is the inactive invariant set positioning constraint We set

λ 0 such thatL Δv2circ Candidate optimizers θmust satisfynablaθLθ 0 Taking the gradient of L

nablaθL partΔv2circpartθ

3

43nrefδxfail 2δ _yfail2 minus

3

4δ _x2fail n2refδz

2fail minus δ_z2fail

sin 2θ

3

2δ _xfail3nrefδxfail 2δ _yfailminus 2nrefδ_zfailδzfail

cos 2θ

and setting nablaθLθ 0 we find that

tan 2θ minus3∕2δ _xfail3nrefδxfail2δ _yfailminus2nrefδ_zfailδzfail3∕43nrefδxfail2δ _yfail2minus 3∕4δ _x2failn2refδz

2failminusδ_z2fail

Denote the set of candidate solutions that satisfy Case 1 by Θ1Case 2 is the active invariant set positioning constraint Here the

chaser attempts to circularize its orbit at the boundary of thezero-thrust RIC shown in Fig 5a The positioning constraint isactive and therefore gcircθ ρ2δx minus δx2θminuscirc 0 This isequivalent to finding where the coasting trajectory fromxtfail xfail crosses δxθ ρδx for θ isin θmin θmax Thiscan be achieved using standard root-finding algorithms Denotethe set of candidate solutions that satisfy Case 2 by Θ2

For the solution to theminimal-cost circularization burn the globaloptimizer θ either lies on the boundary of the box constraint at anunconstrained optimum (θ isin Θ1) or at the boundary of the zero-thrust RIC (θ isin Θ2) all of which are economically obtained througheither numerical integration or a root-finding solver Therefore theminimal-cost circularization burn time T

h satisfies

θ nrefTh minus tfail argmin

θisinfθminθmaxgcupΘ1cupΘ

2

Δv2circθ

If no solution exists (which can happen if and only if xfail startsinside the KOZ) there is no safe circularization CAM and wetherefore declare xfail unsafe Otherwise the CAM is saved for futuretrajectory feasibility verification under all failure modes of interest(see Sec IIIC3) This forms the basis for our actively safe samplingroutine in Algorithm 1 as described in detail in Sec IVC

Appendix B Intermediate Results for FMTOptimality Proof

We report here two useful lemmas concerning bounds on thetrajectory costs between samples which are used throughout theasymptotic optimality proof for FMT in Sec IV We begin with alemma bounding the sizes of the minimum and maximum eigen-

values ofG useful for bounding reachable volumes from x0We thenprove Lemma 2 which forms the basis of our asymptotic optimalityanalysis for FMT Here Φtf t0 eAT is the state transitionmatrix T tf minus t0 is the maneuver duration and G is the N 2impulse Gramian matrix

GT ΦvΦminus1v eATB B eATB B T (B1)

where Φvt fτigi is the aggregate Δv transition matrix corres-

ponding to burn times fτigi ft0 tfgLemma 3 (bounds on Gramian eigenvalues) Let Tmax be less

than one orbital period for the system dynamics of Sec IIC and let

GT be defined as in Eq (B1) Then there exist constants Mmin

Mmax gt 0 such that λminGT ge MminT2 and λmaxGT le Mmax

for all T isin 0 TmaxProof We bound the maximum eigenvalue of G through

norm considerations yielding λmaxGT le keATkB kBk2 leekAkTmax 12 and takeMmax ekAkTmax 12 As long as Tmax is

less than one orbital period GT only approaches singularity near

T 0 [38] Explicitly Taylor expanding GT about T 0 reveals

that λminGT T2∕2OT3 for small T and thus λminGT ΩT2 for all T isin 0 Tmax

Reiteration of Lemma 2 (steering with perturbed endpoints) For a

given steering trajectory xtwith initial time t0 and final time tf letx0 ≔ xt0 xf ≔ xtf T ≔ tf minus t0 and J∶ Jx0 xf Considernow the perturbed steering trajectory ~xt between perturbed start andend points ~x0 x0 δx0 and ~xf xf δxf and its correspondingcost J ~x0 ~xfCase 1 (T 0) There exists a perturbation center δxc (consisting

of only a position shift) with kδxck OJ2 such that

if kδxck le ηJ3 and kδxf minus δxfk le ηJ3 then J ~x0 ~xf leJ1 4ηOJ and the spatial deviation of the perturbedtrajectory ~xt is OJ

Case 2 (T gt 0) If kδx0k le ηJ3 and kδxfk le ηJ3 then

J ~x0 ~xf le J1OηJ2Tminus1 and the spatial deviation of the

perturbed trajectory ~xt from xt is OJProof For bounding the perturbed cost and J ~x0 ~xf we consider

the two cases separatelyCase 1 (T 0) Here two-impulse steering degenerates to a single-

impulse Δv that is xf x0 BΔv with kΔvk J To aid inthe ensuing analysis denote the position and velocity com-ponents of states x rT vT T as r I 0x and v 0 IxSince T 0 we have rf r0 and vf v0 Δv We pick theperturbed steering duration ~T J2 (which will provide anupper bound on the optimal steering cost) and expand thesteering system [Eq (4)] for small time ~T as

rf δrf r0 δr0 ~Tv0 δv0 fΔv1 O ~T2 (B2)

vf δvf v0 δv0 fΔv1 fΔv2 ~TA21r0 δr0A22v0 δv0 fΔv1 O ~T2 (B3)

where A213n2ref 0 0

0 0 0

0 0 minusn2ref

and A22

0 2nref 0

minus2nref 0 0

0 0 0

Solving Eq (B2) for fΔv1 to first order we find fΔv1~Tminus1δrfminusδr0minusv0minusδv0O ~T By selecting δxc ~TvT0 0T T[note that kδxck J2kv0k OJ2] and supposing that

kδx0k le ηJ3 and kδxf minus δxck le ηJ3 we have that

kfΔv1k le Jminus2kδx0k kδxf minus δxck kδx0k OJ2 2ηJ OJ2

Now solving Eq (B3) for fΔv2 Δv δvf minus δv0 minus fΔv1 OJ2 and taking norm of both sides

kfΔv2k le kΔvk kδx0k kδxf minus δxck 2ηJ OJ2le J 2ηJ OJ2

Therefore the perturbed cost satisfies

J ~x0 ~xf le kfΔv1k kfΔv2k le J1 4ηOJ


Case 2 (T gt 0) We pick ~T T to compute an upper bound on theperturbed cost Applying the explicit form of the steeringcontrolΔV [see Eq (13)] along with the norm bound kΦminus1

v k λminGminus1∕2 le Mminus1∕2

min Tminus1 from Lemma 3 we have

J ~x0 ~xf le kΦminus1v tf ft0 tfg ~xf minusΦtf t0 ~x0k

le kΦminus1v xf minusΦx0k kΦminus1

v δxfk kΦminus1v Φδx0k

le J Mminus1∕2min Tminus1kδxfk M

minus1∕2min Tminus1ekAkTmaxkδx0k

le J1OηJ2Tminus1

In both cases the deviation of the perturbed steering trajectory~xt from its closest point on the original trajectory is bounded(quite conservatively) by the maximum propagation of thedifference in initial conditions that is the initial disturbance δx0plus the difference in intercept burns fΔv1 minus Δv1 over themaximum maneuver duration Tmax Thus

k ~xt minus xtk le ekAkTmaxkδx0k kfΔv1k kΔv1kle ekAkTmaxηJ3 2J oJ OJ

where we have used kΔv1k le J and kfΔv1k le J ~x0 ~xf leJ oJ from our previous arguments

Acknowledgments

This work was supported by an Early Career Faculty grant fromNASArsquos Space Technology Research Grants Program (grant numberNNX12AQ43G)

References

[1] Dannemiller D P ldquoMulti-Maneuver Clohessy-Wiltshire TargetingrdquoAAS Astrodynamics Specialist Conference American AstronomicalSoc Girdwood AK July 2011 pp 1ndash15

[2] Kriz J ldquoA Uniform Solution of the Lambert Problemrdquo Celestial

Mechanics Vol 14 No 4 Dec 1976 pp 509ndash513[3] Hablani H B Tapper M L and Dana Bashian D J ldquoGuidance and

Relative Navigation for Autonomous Rendezvous in a Circular OrbitrdquoJournal of Guidance Control and Dynamics Vol 25 No 3 2002pp 553ndash562doi10251424916

[4] Gaylor D E and Barbee B W ldquoAlgorithms for Safe SpacecraftProximityOperationsrdquoAAS Space FlightMechanicsMeeting Vol 127American Astronomical Soc Sedona AZ Jan 2007 pp 132ndash152

[5] Develle M Xu Y Pham K and Chen G ldquoFast Relative GuidanceApproach for Autonomous Rendezvous and Docking ControlrdquoProceedings of SPIE Vol 8044 Soc of Photo-Optical InstrumentationEngineers Orlando FL April 2011 Paper 80440F

[6] Lopez I and McInnes C R ldquoAutonomous Rendezvous usingArtificial Potential Function Guidancerdquo Journal of Guidance Controland Dynamics Vol 18 No 2 March 1995 pp 237ndash241doi102514321375

[7] Muntildeoz J D and Fitz Coy N G ldquoRapid Path-Planning Options forAutonomous Proximity Operations of Spacecraftrdquo AAS AstrodynamicsSpecialist Conference American Astronomical Soc Toronto CanadaAug 2010 pp 1ndash24

[8] Accedilıkmeşe B and Blackmore L ldquoLossless Convexification of a Classof Optimal Control Problems with Non-Convex Control ConstraintsrdquoAutomatica Vol 47 No 2 Feb 2011 pp 341ndash347doi101016jautomatica201010037

[9] Harris M W and Accedilıkmeşe B ldquoLossless Convexification of Non-Convex Optimal Control Problems for State Constrained LinearSystemsrdquo Automatica Vol 50 No 9 2014 pp 2304ndash2311doi101016jautomatica201406008

[10] Martinson N ldquoObstacle Avoidance Guidance and Control Algorithmsfor SpacecraftManeuversrdquoAIAAConference onGuidanceNavigation

and Control Vol 5672 AIAA Reston VA Aug 2009 pp 1ndash9[11] Breger L and How J P ldquoSafe Trajectories for Autonomous

Rendezvous of Spacecraftrdquo Journal of Guidance Control and

Dynamics Vol 31 No 5 2008 pp 1478ndash1489doi102514129590

[12] Vazquez R Gavilan F and Camacho E F ldquoTrajectory Planning forSpacecraft Rendezvous with OnOff Thrustersrdquo International

Federation of Automatic Control Vol 18 Elsevier Milan ItalyAug 2011 pp 8473ndash8478

[13] Lu P and Liu X ldquoAutonomous Trajectory Planning for Rendezvousand Proximity Operations by Conic Optimizationrdquo Journal of

Guidance Control and Dynamics Vol 36 No 2 March 2013pp 375ndash389doi102514158436

[14] Luo Y-Z Liang L-B Wang H and Tang G-J ldquoQuantitativePerformance for Spacecraft Rendezvous Trajectory Safetyrdquo Journal ofGuidance Control andDynamics Vol 34 No 4 July 2011 pp 1264ndash1269doi102514152041

[15] Richards A Schouwenaars T How J P and Feron E ldquoSpacecraftTrajectory Planning with Avoidance Constraints Using Mixed-IntegerLinear Programmingrdquo Journal of Guidance Control and DynamicsVol 25 No 4 2002 pp 755ndash764doi10251424943

[16] LaValle S M and Kuffner J J ldquoRandomized KinodynamicPlanningrdquo International Journal of Robotics Research Vol 20 No 52001 pp 378ndash400doi10117702783640122067453

[17] Frazzoli E ldquoQuasi-Random Algorithms for Real-Time SpacecraftMotion Planning and Coordinationrdquo Acta Astronautica Vol 53Nos 4ndash10 Aug 2003 pp 485ndash495doi101016S0094-5765(03)80009-7

[18] Phillips J M Kavraki L E and Bedrossian N ldquoSpacecraftRendezvous and Docking with Real-Time Randomized OptimizationrdquoAIAA Conference on Guidance Navigation and Control AIAAReston VA Aug 2003 pp 1ndash11

[19] Kobilarov M and Pellegrino S ldquoTrajectory Planning for CubeSatShort-Time-Scale Proximity Operationsrdquo Journal of Guidance

Control and Dynamics Vol 37 No 2 March 2014 pp 566ndash579doi102514160289

[20] Haught M and Duncan G ldquoModeling Common Cause Failures ofThrusters on ISSVisitingVehiclesrdquoNASA JSC-CN-30604 June 2014httpntrsnasagovsearchjspR=20140004797 [retrieved Dec 2015]

[21] Weiss A BaldwinM Petersen C Erwin R S andKolmanovsky IV ldquoSpacecraft Constrained Maneuver Planning for Moving DebrisAvoidance Using Positively Invariant Constraint Admissible SetsrdquoAmerican Control Conference IEEE Publ Piscataway NJ June 2013pp 4802ndash4807

[22] Carson J M Accedilikmeşe B Murray R M and MacMartin D G ldquoARobust Model Predictive Control Algorithm with a Reactive SafetyModerdquo Automatica Vol 49 No 5 May 2013 pp 1251ndash1260doi101016jautomatica201302025

[23] Lavalle S Planning Algorithms Cambridge Univ Press CambridgeEngland UK 2006 pp 1ndash842

[24] Janson L Schmerling E Clark A and Pavone M ldquoFast MarchingTree A Fast Marching Sampling-Based Method for Optimal MotionPlanning in Many Dimensionsrdquo International Journal of Robotics

Research Vol 34 No 7 2015 pp 883ndash921doi1011770278364915577958

[25] Starek J A Barbee B W and Pavone M ldquoA Sampling-BasedApproach to Spacecraft Autonomous Maneuvering with SafetySpecificationsrdquo American Astronomical Society Guidance Navigation

and Control Conference Vol 154 American Astronomical SocBreckenridge CO Feb 2015 pp 1ndash13

[26] Starek J A Schmerling E Maher G D Barbee B W and PavoneM ldquoReal-Time Propellant-Optimized Spacecraft Motion Planningunder Clohessy-Wiltshire-Hill Dynamicsrdquo IEEE Aerospace

Conference IEEE Publ Piscataway NJ March 2016 pp 1ndash16[27] Ross I M ldquoHow to Find Minimum-Fuel Controllersrdquo AIAA

Conference onGuidance Navigation andControl AAAARestonVAAug 2004 pp 1ndash10

[28] Clohessy W H and Wiltshire R S ldquoTerminal Guidance System forSatellite Rendezvousrdquo Journal of the Aerospace Sciences Vol 27No 9 Sept 1960 pp 653ndash658doi10251488704

[29] Hill G W ldquoResearches in the Lunar Theoryrdquo JSTOR American

Journal of Mathematics Vol 1 No 1 1878 pp 5ndash26doi1023072369430

[30] Bodson M ldquoEvaluation of Optimization Methods for ControlAllocationrdquo Journal of Guidance Control and Dynamics Vol 25No 4 July 2002 pp 703ndash711doi10251424937

[31] Dettleff G ldquoPlume Flow and Plume Impingement in SpaceTechnologyrdquo Progress in Aerospace Sciences Vol 28 No 1 1991pp 1ndash71doi1010160376-0421(91)90008-R


[32] Goodman J L ldquoHistory of Space Shuttle Rendezvous and ProximityOperationsrdquo Journal of Spacecraft and Rockets Vol 43 No 5Sept 2006 pp 944ndash959doi102514119653

[33] Starek J A Accedilıkmeşe B Nesnas I A D and Pavone MldquoSpacecraft Autonomy Challenges for Next Generation SpaceMissionsrdquo Advances in Control System Technology for Aerospace

Applications edited byFeron EVol 460LectureNotes inControl andInformation Sciences SpringerndashVerlag New York Sept 2015 pp 1ndash48 Chap 1doi101007978-3-662-47694-9_1

[34] Barbee B W Carpenter J R Heatwole S Markley F L MoreauM Naasz B J and Van Eepoel J ldquoA Guidance and NavigationStrategy for Rendezvous and Proximity Operations with aNoncooperative Spacecraft in Geosynchronous Orbitrdquo Journal of the

Aerospace Sciences Vol 58 No 3 July 2011 pp 389ndash408[35] Schouwenaars T How J P andFeron E ldquoDecentralizedCooperative

Trajectory Planning of Multiple Aircraft with Hard Safety GuaranteesrdquoAIAA Conference on Guidance Navigation and Control AIAAReston VA Aug 2004 pp 1ndash14

[36] Fehse W Automated Rendezvous and Docking of Spacecraft Vol 16Cambridge Univ Press Cambridge England UK 2003 pp 1ndash494

[37] Fraichard T ldquoA Short Paper About Motion Safetyrdquo Proceedings of

IEEEConference onRobotics andAutomation IEEEPubl PiscatawayNJ April 2007 pp 1140ndash1145

[38] Alfriend K Vadali S R Gurfil P How J and Breger L SpacecraftFormation Flying Dynamics Control and Navigation Vol 2Butterworth-Heinemann Burlington MA Nov 2009 pp 1ndash402

[39] Janson L Ichter B and Pavone M ldquoDeterministic Sampling-BasedMotion Planning Optimality Complexity and Performancerdquo 17th

International Symposium of Robotic Research (ISRR) SpringerSept 2015 p 16 httpwebstanfordedu~pavonepapersJansonIchtereaISRR15pdf

[40] Halton J H ldquoOn the Efficiency of Certain Quasirandom Sequences ofPoints in Evaluating Multidimensional Integralsrdquo Numerische

Mathematik Vol 2 No 1 1960 pp 84ndash90doi101007BF01386213

[41] Allen R and Pavone M ldquoA Real-Time Framework for KinodynamicPlanning with Application to Quadrotor Obstacle Avoidancerdquo AIAA

Conference on Guidance Navigation and Control AIAA Reston VAJan 2016 pp 1ndash18

[42] Schmerling E Janson L and Pavone M ldquoOptimal Sampling-BasedMotion Planning under Differential Constraints the Drift Case withLinearAffineDynamicsrdquoProceedings of IEEEConference onDecisionand Control IEEE Publ Piscataway NJ Dec 2015 pp 2574ndash2581doi101109CDC20157402604

[43] Schmerling E Janson L and Pavone M ldquoOptimal Sampling-BasedMotion Planning Under Differential Constraints The Driftless CaserdquoProceedings of IEEE Conference on Robotics and Automation IEEEPubl Piscataway NJ May 2015 pp 2368ndash2375

[44] Landsat 7 Science Data Users Handbook NASA March 2011 httplandsathandbookgsfcnasa govpdfsLandSat7_Handbookhtml

[45] Goward S N Masek J G Williams D L Irons J R andThompson R J ldquoThe Landsat 7 Mission Terrestrial Research andApplications for the 21st Centuryrdquo Remote Sensing of EnvironmentVol 78 Nos 1ndash2 2001 pp 3ndash12doi101016S0034-4257(01)00262-0

[46] Grant M and Boyd S ldquoCVX Matlab Software for DisciplinedConvex Programmingrdquo Ver 21 March 2014 httpcvxrcomcvxciting [retrieved June 2015]


this way sampling-based algorithms can address a large variety of

constraints and provide significant computational benefits withrespect to traditional optimal control methods and mixed-integer

programming [23] Furthermore through a property called as-

ymptotic optimality (AO) sampling-based algorithms can bedesigned to provide guarantees of optimality in the limit that

the number of samples taken approaches infinity This makessampling-based planners a strong choice for the problem of space-

craft controlAlthough the aforementioned works [16ndash19] on sampling-based

planning for spacecraft proximity operations have addressed several

components of the safety-constrained optimal CWH autonomous

rendezvous problem few have addressed the aspect of real-timeimplementability in conjunction with both a 2-norm propellant-cost

metric and active trajectory safety with respect to control failures

This paper seeks to fill this gap The paperrsquos central theme is arigorous proof of asymptotic optimality for a particular sampling-

based planner namely a modified version of the FMT algorithm[24] under impulsive CWH spacecraft dynamics with hard safety

constraints First a description of the problem scenario is provided in

Sec II along with a formal definition of the sum-of-2-norms costmetric that we employ as a proxy for propellant consumption

Section III then follows with a thorough discussion of chasertargetvehicle safety defining precisely how abort trajectories may be

designed under CWH dynamics to deterministically avoid for all

future times an ellipsoidal region about the CWH frame origin undera prescribed set of control failures Next we proceed in Sec IV to our

proposed approach employing the modified FMT algorithm The

section begins with presentation of a conservative approximation tothe propellant-cost reachability set which characterizes the set of

states that are nearby a given initial state in terms of propellant useThese sets bounded by unions of ellipsoidal balls are then used to

show that the modified FMT algorithm maintains its (asymptotic)

optimalitywhen applied toCWHdynamics under our propellant-costmetric From there in Sec V the paper presents two techniques for

improving motion planning solutions 1) an analytical technique that

can be called both during planning and postprocessing to merge Δvvectors between any pair of concatenated graph edges and 2) a

continuous trajectory smoothing algorithm which can reduce themagnitude of Δv vectors by relaxing the implicit constraint to pass

through sample points while still maintaining solution feasibilityPut together these tools yield a flexible real-time framework for

near-circular orbit spacecraft guidance that handles a wide variety

of (possibly nonconvex) state time and control constraints and

provides deterministic guarantees on abort safety and solution quality(propellant cost) The methodology is demonstrated in Sec VI on a

single-chaser single-target scenario simulating a near-field low-Earth-orbit (LEO) approach with hard constraints on the total

maneuver duration relative positioning (including KOZ and antennainterference constraints) thruster plume-impingement avoidanceindividual and net Δv-vector magnitudes and a two-fault thrusterstuck-off failure tolerance Trades are then conducted studyingthe effects of the sample count and a propellant cost threshold onthe performance of FMT (both with and without trajectorysmoothing)Preliminary versions of this paper appear in [2526] This extended

and revised work introduces the following as additional contri-butions 1) a more detailed presentation of the FMT optimalityproof 2) improved trajectory smoothing and 3) a six-dimensional (3degree-of-freedom) numerical example demonstrating nonplanarLEO rendezvous

II Problem Formulation

Consider the problem of a chaser spacecraft seeking to maneuvertoward a single target moving in a well-defined circular orbit (see

Fig 1a) Define the state space X sub Rd as a d-dimensional region inthe targetrsquos local vertical local horizontal (LVLH) frame and let theobstacle region or Xobs be the set of states in X that result in missionfailure (eg states outside of an approach corridor or in collisionwiththe target) Let the free space orX free be the complement ofXobs Asseen in Fig 1b let xinit represent the chaserrsquos initial state relative tothe target and let xgoal isin Xgoal be a goal state inside goal region

X goal Finally define a state trajectory (or simply ldquotrajectoryrdquo) as a

piecewise-continuous function of time xt R rarr X and let Σrepresent the set of all state trajectories Every state trajectory isimplicitly generated by a control trajectory ut R rarr U where U isthe set of controls through the system dynamics _x fx u twhere f is the state transition function A state trajectory is called afeasible solution to the planning problem (X free tinit xinit xgoal) if1) it satisfies the boundary conditions xtinit xinit and xtfinal xgoal for some time tfinal gt tinit 2) it is collision free that is xt isinX free for all t isin tinit tfinal and 3) it obeys all other trajectoryconstraints The optimal motion planning problem can then bedefined as followsDefinition 1 (Optimal Planning Problem) Given a planning

problem (X free tinit xinit xgoal) and a cost functional J Σ times U times R rarrRge0 find a feasible trajectory x

twith associated control trajectoryut and time span t tinit tfinal for tfinal isin tinitinfin such thatJxmiddot umiddot t minfJxmiddot umiddot t j xt andut are feasiblegIf no such trajectory exists report failureTailoring Definition 1 to impulsively actuated propellant-optimal

motion planning near circular orbits (using a control-effort costfunctional J that considers only ut and the final time tfinal denotedas Jut tfinal) the optimal spacecraft motion planning problemmay be formulated as

(Cross-track)radial

(In track)

(Normal to orbital plane)out of plane

Attractor

Reference line

Target

Chaser

a) Schematic of CWH dynamics which modelsrelative guidance near a target in circular orbit

b) A representative motion planning query betweenfeasible states xinit and xgoal

Fig 1 Illustration of the CWH planning scenario



minimizeuttfinal

JuttfinalZ

tfinal

tinit

kutk2 dtXNi1

kΔvik2





gxtuttle0















in turn

A Cost Functional

























C System Dynamics












m(2a)


m(2b)

δz n2refδz Fδz

m(2c)

where nref μr3ref









_x fxu t AxBu (3)


A

266666666664

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1




377777777775B

266666666664

0 0 0

0 0 0

0 0 0

1 0 0

0 1 0

0 0 1












xt Φt t0xt0 Z

t

t0

Φt τBXNi1


Φt t0xt0 XNi1

Zt

t0








xt Φt t0xt0 XNt

i1

Φt τiBΔvi (4a)


≜Φvtfτigi

2664Δv1

ΔvNt

3775|z

≜ΔV

(4b)





























































momentvectorMi

minimizeΔvik

XKk1

Δvik

subject toXKk1


XKk1







3 Plume Impingement


4 Other Constraints


F Active Safety



III Vehicle Safety















control policies





safety is assured




















ThxCAMt

JxCAMt uCAMt t Z

Th

tfail

kuCAMtk2 dt XNi1

kΔvCAMik2





gxCAM uCAM t le 0


















Ff0

Kf

P

Ff0















Passive Abort

Active Abort

Failure









1 CAM Policy


















3

2nrefδx

sup XKOZ (9)


3

2nrefδx

X c

ric (10)









CWH dynamics


CoastingArc

Circularized Orbit
















































minimizeΔviτi tfN

XNi1

kΔvik2






















v depend only





nref

minimizeTisin0Tmax

X2i1

kΔvik2


















as



given by

ΔVx0 xf Δv1Δv2

Φminus1





kΔVk le Jx0 xf le2

pkΔVk



pmiddot

12 12

p That is J le

2

p kΔVk Similarly


pge

kΔv1k2 kΔv2k2p

kΔVk



v Φminus1v






































n of samples S














DS ≜ supxisinX

minsisinS

ks minus xk

















tk τjk

xk limtrarrtminus

k

xt kJ

2minusXjkminus1i1

kΔvi kB

ΔvikΔvi k



















have that

Jyk yk1 leJ

2

1 ϵ

2O J

le

J

21 ϵ




k1 segment


XKminus1k0

Jyk yk1 le 1 ϵJ















k0

Jyk yk1 (15)


Pmminus2k0 Jyk yk1

2) cymminus1 leP







Case 1 Jn leP


Pmminus1k0 Jyk yk1



k0

Jyk yk1


mminus1k0 Jyk yk1





leXmminus1

k0

Jyk yk1




leXmminus1

k0

Jyk yk1







































minimizeΔvi

XNi1

kΔvik2








































A Simulation Setup


























Parameter Value



infin m∕s























computation time




























VII Conclusions









minimizeTh

Δv2circTh







h0




minimizeθcirc

Δv2circθcirc







3


3



sin 2θ

3


cos 2θ



2failminusδ_z2fail




h satisfies



2

Δv2circθ



























where A213n2ref 0 0

0 0 0

0 0 minusn2ref

and A22

0 2nref 0

minus2nref 0 0

0 0 0

















le J1OηJ2Tminus1




Acknowledgments


References


































































minimizeuttfinal

JuttfinalZ

tfinal

tinit

kutk2 dtXNi1

kΔvik2





gxtuttle0















in turn

A Cost Functional

























C System Dynamics












m(2a)


m(2b)

δz n2refδz Fδz

m(2c)

where nref μr3ref









_x fxu t AxBu (3)


A

266666666664

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1




377777777775B

266666666664

0 0 0

0 0 0

0 0 0

1 0 0

0 1 0

0 0 1












xt Φt t0xt0 Z

t

t0

Φt τBXNi1


Φt t0xt0 XNi1

Zt

t0








xt Φt t0xt0 XNt

i1

Φt τiBΔvi (4a)


≜Φvtfτigi

2664Δv1

ΔvNt

3775|z

≜ΔV

(4b)





























































momentvectorMi

minimizeΔvik

XKk1

Δvik

subject toXKk1


XKk1







3 Plume Impingement


4 Other Constraints


F Active Safety



III Vehicle Safety















control policies





safety is assured




















ThxCAMt

JxCAMt uCAMt t Z

Th

tfail

kuCAMtk2 dt XNi1

kΔvCAMik2





gxCAM uCAM t le 0


















Ff0

Kf

P

Ff0















Passive Abort

Active Abort

Failure









1 CAM Policy


















3

2nrefδx

sup XKOZ (9)


3

2nrefδx

X c

ric (10)









CWH dynamics


CoastingArc

Circularized Orbit
















































minimizeΔviτi tfN

XNi1

kΔvik2






















v depend only





nref

minimizeTisin0Tmax

X2i1

kΔvik2


















as



given by

ΔVx0 xf Δv1Δv2

Φminus1





kΔVk le Jx0 xf le2

pkΔVk



pmiddot

12 12

p That is J le

2

p kΔVk Similarly


pge

kΔv1k2 kΔv2k2p

kΔVk



v Φminus1v






































n of samples S














DS ≜ supxisinX

minsisinS

ks minus xk

















tk τjk

xk limtrarrtminus

k

xt kJ

2minusXjkminus1i1

kΔvi kB

ΔvikΔvi k



















have that

Jyk yk1 leJ

2

1 ϵ

2O J

le

J

21 ϵ




k1 segment


XKminus1k0

Jyk yk1 le 1 ϵJ















k0

Jyk yk1 (15)


Pmminus2k0 Jyk yk1

2) cymminus1 leP







Case 1 Jn leP


Pmminus1k0 Jyk yk1



k0

Jyk yk1


mminus1k0 Jyk yk1





leXmminus1

k0

Jyk yk1




leXmminus1

k0

Jyk yk1







































minimizeΔvi

XNi1

kΔvik2








































A Simulation Setup


























Parameter Value



infin m∕s























computation time




























VII Conclusions









minimizeTh

Δv2circTh







h0




minimizeθcirc

Δv2circθcirc







3


3



sin 2θ

3


cos 2θ



2failminusδ_z2fail




h satisfies



2

Δv2circθ



























where A213n2ref 0 0

0 0 0

0 0 minusn2ref

and A22

0 2nref 0

minus2nref 0 0

0 0 0

















le J1OηJ2Tminus1




Acknowledgments


References






































































xt Φt t0xt0 XNt

i1

Φt τiBΔvi (4a)


≜Φvtfτigi

2664Δv1

ΔvNt

3775|z

≜ΔV

(4b)





























































momentvectorMi

minimizeΔvik

XKk1

Δvik

subject toXKk1


XKk1







3 Plume Impingement


4 Other Constraints


F Active Safety



III Vehicle Safety















control policies





safety is assured




















ThxCAMt

JxCAMt uCAMt t Z

Th

tfail

kuCAMtk2 dt XNi1

kΔvCAMik2





gxCAM uCAM t le 0


















Ff0

Kf

P

Ff0















Passive Abort

Active Abort

Failure









1 CAM Policy


















3

2nrefδx

sup XKOZ (9)


3

2nrefδx

X c

ric (10)









CWH dynamics


CoastingArc

Circularized Orbit
















































minimizeΔviτi tfN

XNi1

kΔvik2






















v depend only





nref

minimizeTisin0Tmax

X2i1

kΔvik2


















as



given by

ΔVx0 xf Δv1Δv2

Φminus1





kΔVk le Jx0 xf le2

pkΔVk



pmiddot

12 12

p That is J le

2

p kΔVk Similarly


pge

kΔv1k2 kΔv2k2p

kΔVk



v Φminus1v






































n of samples S














DS ≜ supxisinX

minsisinS

ks minus xk

















tk τjk

xk limtrarrtminus

k

xt kJ

2minusXjkminus1i1

kΔvi kB

ΔvikΔvi k



















have that

Jyk yk1 leJ

2

1 ϵ

2O J

le

J

21 ϵ




k1 segment


XKminus1k0

Jyk yk1 le 1 ϵJ















k0

Jyk yk1 (15)


Pmminus2k0 Jyk yk1

2) cymminus1 leP







Case 1 Jn leP


Pmminus1k0 Jyk yk1



k0

Jyk yk1


mminus1k0 Jyk yk1





leXmminus1

k0

Jyk yk1




leXmminus1

k0

Jyk yk1







































minimizeΔvi

XNi1

kΔvik2








































A Simulation Setup


























Parameter Value



infin m∕s























computation time




























VII Conclusions









minimizeTh

Δv2circTh







h0




minimizeθcirc

Δv2circθcirc







3


3



sin 2θ

3


cos 2θ



2failminusδ_z2fail




h satisfies



2

Δv2circθ



























where A213n2ref 0 0

0 0 0

0 0 minusn2ref

and A22

0 2nref 0

minus2nref 0 0

0 0 0

















le J1OηJ2Tminus1




Acknowledgments


References



































































3 Plume Impingement


4 Other Constraints


F Active Safety



III Vehicle Safety















control policies





safety is assured




















ThxCAMt

JxCAMt uCAMt t Z

Th

tfail

kuCAMtk2 dt XNi1

kΔvCAMik2





gxCAM uCAM t le 0


















Ff0

Kf

P

Ff0















Passive Abort

Active Abort

Failure









1 CAM Policy


















3

2nrefδx

sup XKOZ (9)


3

2nrefδx

X c

ric (10)









CWH dynamics


CoastingArc

Circularized Orbit
















































minimizeΔviτi tfN

XNi1

kΔvik2






















v depend only





nref

minimizeTisin0Tmax

X2i1

kΔvik2


















as



given by

ΔVx0 xf Δv1Δv2

Φminus1





kΔVk le Jx0 xf le2

pkΔVk



pmiddot

12 12

p That is J le

2

p kΔVk Similarly


pge

kΔv1k2 kΔv2k2p

kΔVk



v Φminus1v






































n of samples S














DS ≜ supxisinX

minsisinS

ks minus xk

















tk τjk

xk limtrarrtminus

k

xt kJ

2minusXjkminus1i1

kΔvi kB

ΔvikΔvi k



















have that

Jyk yk1 leJ

2

1 ϵ

2O J

le

J

21 ϵ




k1 segment


XKminus1k0

Jyk yk1 le 1 ϵJ















k0

Jyk yk1 (15)


Pmminus2k0 Jyk yk1

2) cymminus1 leP







Case 1 Jn leP


Pmminus1k0 Jyk yk1



k0

Jyk yk1


mminus1k0 Jyk yk1





leXmminus1

k0

Jyk yk1




leXmminus1

k0

Jyk yk1







































minimizeΔvi

XNi1

kΔvik2








































A Simulation Setup


























Parameter Value



infin m∕s























computation time




























VII Conclusions









minimizeTh

Δv2circTh







h0




minimizeθcirc

Δv2circθcirc







3


3



sin 2θ

3


cos 2θ



2failminusδ_z2fail




h satisfies



2

Δv2circθ



























where A213n2ref 0 0

0 0 0

0 0 minusn2ref

and A22

0 2nref 0

minus2nref 0 0

0 0 0

















le J1OηJ2Tminus1




Acknowledgments


References


































































control policies





safety is assured




















ThxCAMt

JxCAMt uCAMt t Z

Th

tfail

kuCAMtk2 dt XNi1

kΔvCAMik2





gxCAM uCAM t le 0


















Ff0

Kf

P

Ff0















Passive Abort

Active Abort

Failure









1 CAM Policy


















3

2nrefδx

sup XKOZ (9)


3

2nrefδx

X c

ric (10)









CWH dynamics


CoastingArc

Circularized Orbit
















































minimizeΔviτi tfN

XNi1

kΔvik2






















v depend only





nref

minimizeTisin0Tmax

X2i1

kΔvik2


















as



given by

ΔVx0 xf Δv1Δv2

Φminus1





kΔVk le Jx0 xf le2

pkΔVk



pmiddot

12 12

p That is J le

2

p kΔVk Similarly


pge

kΔv1k2 kΔv2k2p

kΔVk



v Φminus1v






































n of samples S














DS ≜ supxisinX

minsisinS

ks minus xk

















tk τjk

xk limtrarrtminus

k

xt kJ

2minusXjkminus1i1

kΔvi kB

ΔvikΔvi k



















have that

Jyk yk1 leJ

2

1 ϵ

2O J

le

J

21 ϵ




k1 segment


XKminus1k0

Jyk yk1 le 1 ϵJ















k0

Jyk yk1 (15)


Pmminus2k0 Jyk yk1

2) cymminus1 leP







Case 1 Jn leP


Pmminus1k0 Jyk yk1



k0

Jyk yk1


mminus1k0 Jyk yk1





leXmminus1

k0

Jyk yk1




leXmminus1

k0

Jyk yk1







































minimizeΔvi

XNi1

kΔvik2








































A Simulation Setup


























Parameter Value



infin m∕s























computation time




























VII Conclusions









minimizeTh

Δv2circTh







h0




minimizeθcirc

Δv2circθcirc







3


3



sin 2θ

3


cos 2θ



2failminusδ_z2fail




h satisfies



2

Δv2circθ



























where A213n2ref 0 0

0 0 0

0 0 minusn2ref

and A22

0 2nref 0

minus2nref 0 0

0 0 0

















le J1OηJ2Tminus1




Acknowledgments


References





































































1 CAM Policy


















3

2nrefδx

sup XKOZ (9)


3

2nrefδx

X c

ric (10)









CWH dynamics


CoastingArc

Circularized Orbit
















































minimizeΔviτi tfN

XNi1

kΔvik2






















v depend only





nref

minimizeTisin0Tmax

X2i1

kΔvik2


















as



given by

ΔVx0 xf Δv1Δv2

Φminus1





kΔVk le Jx0 xf le2

pkΔVk



pmiddot

12 12

p That is J le

2

p kΔVk Similarly


pge

kΔv1k2 kΔv2k2p

kΔVk



v Φminus1v






































n of samples S














DS ≜ supxisinX

minsisinS

ks minus xk

















tk τjk

xk limtrarrtminus

k

xt kJ

2minusXjkminus1i1

kΔvi kB

ΔvikΔvi k



















have that

Jyk yk1 leJ

2

1 ϵ

2O J

le

J

21 ϵ




k1 segment


XKminus1k0

Jyk yk1 le 1 ϵJ















k0

Jyk yk1 (15)


Pmminus2k0 Jyk yk1

2) cymminus1 leP







Case 1 Jn leP


Pmminus1k0 Jyk yk1



k0

Jyk yk1


mminus1k0 Jyk yk1





leXmminus1

k0

Jyk yk1




leXmminus1

k0

Jyk yk1







































minimizeΔvi

XNi1

kΔvik2








































A Simulation Setup


























Parameter Value



infin m∕s























computation time




























VII Conclusions









minimizeTh

Δv2circTh







h0




minimizeθcirc

Δv2circθcirc







3


3



sin 2θ

3


cos 2θ



2failminusδ_z2fail




h satisfies



2

Δv2circθ



























where A213n2ref 0 0

0 0 0

0 0 minusn2ref

and A22

0 2nref 0

minus2nref 0 0

0 0 0

















le J1OηJ2Tminus1




Acknowledgments


References














































































































minimizeΔviτi tfN

XNi1

kΔvik2






















v depend only





nref

minimizeTisin0Tmax

X2i1

kΔvik2


















as



given by

ΔVx0 xf Δv1Δv2

Φminus1





kΔVk le Jx0 xf le2

pkΔVk



pmiddot

12 12

p That is J le

2

p kΔVk Similarly


pge

kΔv1k2 kΔv2k2p

kΔVk



v Φminus1v






































n of samples S














DS ≜ supxisinX

minsisinS

ks minus xk

















tk τjk

xk limtrarrtminus

k

xt kJ

2minusXjkminus1i1

kΔvi kB

ΔvikΔvi k



















have that

Jyk yk1 leJ

2

1 ϵ

2O J

le

J

21 ϵ




k1 segment


XKminus1k0

Jyk yk1 le 1 ϵJ















k0

Jyk yk1 (15)


Pmminus2k0 Jyk yk1

2) cymminus1 leP







Case 1 Jn leP


Pmminus1k0 Jyk yk1



k0

Jyk yk1


mminus1k0 Jyk yk1





leXmminus1

k0

Jyk yk1




leXmminus1

k0

Jyk yk1







































minimizeΔvi

XNi1

kΔvik2








































A Simulation Setup


























Parameter Value



infin m∕s























computation time




























VII Conclusions









minimizeTh

Δv2circTh







h0




minimizeθcirc

Δv2circθcirc







3


3



sin 2θ

3


cos 2θ



2failminusδ_z2fail




h satisfies



2

Δv2circθ



























where A213n2ref 0 0

0 0 0

0 0 minusn2ref

and A22

0 2nref 0

minus2nref 0 0

0 0 0

















le J1OηJ2Tminus1




Acknowledgments


References
















































































v depend only





nref

minimizeTisin0Tmax

X2i1

kΔvik2


















as



given by

ΔVx0 xf Δv1Δv2

Φminus1





kΔVk le Jx0 xf le2

pkΔVk



pmiddot

12 12

p That is J le

2

p kΔVk Similarly


pge

kΔv1k2 kΔv2k2p

kΔVk



v Φminus1v






































n of samples S














DS ≜ supxisinX

minsisinS

ks minus xk

















tk τjk

xk limtrarrtminus

k

xt kJ

2minusXjkminus1i1

kΔvi kB

ΔvikΔvi k



















have that

Jyk yk1 leJ

2

1 ϵ

2O J

le

J

21 ϵ




k1 segment


XKminus1k0

Jyk yk1 le 1 ϵJ















k0

Jyk yk1 (15)


Pmminus2k0 Jyk yk1

2) cymminus1 leP







Case 1 Jn leP


Pmminus1k0 Jyk yk1



k0

Jyk yk1


mminus1k0 Jyk yk1





leXmminus1

k0

Jyk yk1




leXmminus1

k0

Jyk yk1







































minimizeΔvi

XNi1

kΔvik2








































A Simulation Setup


























Parameter Value



infin m∕s























computation time




























VII Conclusions









minimizeTh

Δv2circTh







h0




minimizeθcirc

Δv2circθcirc







3


3



sin 2θ

3


cos 2θ



2failminusδ_z2fail




h satisfies



2

Δv2circθ



























where A213n2ref 0 0

0 0 0

0 0 minusn2ref

and A22

0 2nref 0

minus2nref 0 0

0 0 0

















le J1OηJ2Tminus1




Acknowledgments


References



























































































n of samples S














DS ≜ supxisinX

minsisinS

ks minus xk

















tk τjk

xk limtrarrtminus

k

xt kJ

2minusXjkminus1i1

kΔvi kB

ΔvikΔvi k



















have that

Jyk yk1 leJ

2

1 ϵ

2O J

le

J

21 ϵ




k1 segment


XKminus1k0

Jyk yk1 le 1 ϵJ















k0

Jyk yk1 (15)


Pmminus2k0 Jyk yk1

2) cymminus1 leP







Case 1 Jn leP


Pmminus1k0 Jyk yk1



k0

Jyk yk1


mminus1k0 Jyk yk1





leXmminus1

k0

Jyk yk1




leXmminus1

k0

Jyk yk1







































minimizeΔvi

XNi1

kΔvik2








































A Simulation Setup


























Parameter Value



infin m∕s























computation time




























VII Conclusions









minimizeTh

Δv2circTh







h0




minimizeθcirc

Δv2circθcirc







3


3



sin 2θ

3


cos 2θ



2failminusδ_z2fail




h satisfies



2

Δv2circθ



























where A213n2ref 0 0

0 0 0

0 0 minusn2ref

and A22

0 2nref 0

minus2nref 0 0

0 0 0

















le J1OηJ2Tminus1




Acknowledgments


References




































































DS ≜ supxisinX

minsisinS

ks minus xk

















tk τjk

xk limtrarrtminus

k

xt kJ

2minusXjkminus1i1

kΔvi kB

ΔvikΔvi k



















have that

Jyk yk1 leJ

2

1 ϵ

2O J

le

J

21 ϵ




k1 segment


XKminus1k0

Jyk yk1 le 1 ϵJ















k0

Jyk yk1 (15)


Pmminus2k0 Jyk yk1

2) cymminus1 leP







Case 1 Jn leP


Pmminus1k0 Jyk yk1



k0

Jyk yk1


mminus1k0 Jyk yk1





leXmminus1

k0

Jyk yk1




leXmminus1

k0

Jyk yk1







































minimizeΔvi

XNi1

kΔvik2








































A Simulation Setup


























Parameter Value



infin m∕s























computation time




























VII Conclusions









minimizeTh

Δv2circTh







h0




minimizeθcirc

Δv2circθcirc







3


3



sin 2θ

3


cos 2θ



2failminusδ_z2fail




h satisfies



2

Δv2circθ



























where A213n2ref 0 0

0 0 0

0 0 minusn2ref

and A22

0 2nref 0

minus2nref 0 0

0 0 0

















le J1OηJ2Tminus1




Acknowledgments


References










































































k0

Jyk yk1 (15)


Pmminus2k0 Jyk yk1

2) cymminus1 leP







Case 1 Jn leP


Pmminus1k0 Jyk yk1



k0

Jyk yk1


mminus1k0 Jyk yk1





leXmminus1

k0

Jyk yk1




leXmminus1

k0

Jyk yk1







































minimizeΔvi

XNi1

kΔvik2








































A Simulation Setup


























Parameter Value



infin m∕s























computation time




























VII Conclusions









minimizeTh

Δv2circTh







h0




minimizeθcirc

Δv2circθcirc







3


3



sin 2θ

3


cos 2θ



2failminusδ_z2fail




h satisfies



2

Δv2circθ



























where A213n2ref 0 0

0 0 0

0 0 minusn2ref

and A22

0 2nref 0

minus2nref 0 0

0 0 0

















le J1OηJ2Tminus1




Acknowledgments


References





























































































minimizeΔvi

XNi1

kΔvik2








































A Simulation Setup


























Parameter Value



infin m∕s























computation time




























VII Conclusions









minimizeTh

Δv2circTh







h0




minimizeθcirc

Δv2circθcirc







3


3



sin 2θ

3


cos 2θ



2failminusδ_z2fail




h satisfies



2

Δv2circθ



























where A213n2ref 0 0

0 0 0

0 0 minusn2ref

and A22

0 2nref 0

minus2nref 0 0

0 0 0

















le J1OηJ2Tminus1




Acknowledgments


References








































































A Simulation Setup


























Parameter Value



infin m∕s























computation time




























VII Conclusions









minimizeTh

Δv2circTh







h0




minimizeθcirc

Δv2circθcirc







3


3



sin 2θ

3


cos 2θ



2failminusδ_z2fail




h satisfies



2

Δv2circθ



























where A213n2ref 0 0

0 0 0

0 0 minusn2ref

and A22

0 2nref 0

minus2nref 0 0

0 0 0

















le J1OηJ2Tminus1




Acknowledgments


References








































































Parameter Value



infin m∕s























computation time




























VII Conclusions









minimizeTh

Δv2circTh







h0




minimizeθcirc

Δv2circθcirc







3


3



sin 2θ

3


cos 2θ



2failminusδ_z2fail




h satisfies



2

Δv2circθ



























where A213n2ref 0 0

0 0 0

0 0 minusn2ref

and A22

0 2nref 0

minus2nref 0 0

0 0 0

















le J1OηJ2Tminus1




Acknowledgments


References


















































































computation time




























VII Conclusions









minimizeTh

Δv2circTh







h0




minimizeθcirc

Δv2circθcirc







3


3



sin 2θ

3


cos 2θ



2failminusδ_z2fail




h satisfies



2

Δv2circθ



























where A213n2ref 0 0

0 0 0

0 0 minusn2ref

and A22

0 2nref 0

minus2nref 0 0

0 0 0

















le J1OηJ2Tminus1




Acknowledgments


References




































































VII Conclusions









minimizeTh

Δv2circTh







h0




minimizeθcirc

Δv2circθcirc







3


3



sin 2θ

3


cos 2θ



2failminusδ_z2fail




h satisfies



2

Δv2circθ



























where A213n2ref 0 0

0 0 0

0 0 minusn2ref

and A22

0 2nref 0

minus2nref 0 0

0 0 0

















le J1OηJ2Tminus1




Acknowledgments


References

































































minimizeθcirc

Δv2circθcirc







3


3



sin 2θ

3


cos 2θ



2failminusδ_z2fail




h satisfies



2

Δv2circθ



























where A213n2ref 0 0

0 0 0

0 0 minusn2ref

and A22

0 2nref 0

minus2nref 0 0

0 0 0

















le J1OηJ2Tminus1




Acknowledgments


References









































































le J1OηJ2Tminus1




Acknowledgments


References

































































Fast, Safe, Propellant-Efficient Spacecraft Motion...

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Transcript of Fast, Safe, Propellant-Efficient Spacecraft Motion...