APPROXIMATION OF ATTAINABLE LANDING AREA OF A MOON LANDER BY REACHABILITY ANALYSIS

DLR – German Aerospace CentreInstitute of Space SystemsGuidance, Navigation and Control Dept.Robert-Hooke-Str. 728359, Bremen, Germany

17th International Conference on Hybrid Systems: Computation and Control (HSCC 2014)15th-17th April 2014, Berlin, Germany

Method

Introduction

Discretization of Optimal Control Problem

References: [1] T. Oehlschlägel, S. Theil, H. Krüger, M. Knauer, J. Tietjen, C. Büskens, Optimal Guidance and Control of Lunar Landers with Non-throttable Main Engine, Advances in Aerospace Guidance, Navigation and Control, pp. 451-463, 2011

[2] R.Baier, M. Gerdts, I. Xausa, Approximation of Reachable Sets Using Optimal Control Algorithms, Numerical Algebra; Control and Optimization, Vol. 3, 2013

[3] M. Sagliano, S. Theil, Hybrid Jacobian Computation for Fast Optimal Trajectories Generation, AIAA Guidance, Navigation and Control(GNC) Conference, Boston, August 19-22,2013

Results

Problem Statement

Yunus Emre ArslantasYunus.Arslantas@dlr.de

Thimo OehlschlägelThimo.Oehlschlaegel@dlr.de

Marco SaglianoMarco.Sagliano@dlr.de

Stephan TheilStephan.Theil@dlr.de

Claus BraxmaierClaus.Braxmaier@dlr.de

Determination of AttainableLanding Area by Forward

Reachability

Equations of motion of the moon lander are taken from [1]. The vector of states and control inputs are defined as follows:

Initial condition and terminal conditions:

DCA Representation

Reference Frame: Downrange-Crossrange-Altitude (DCA)

0 100 200 300 400

0

100

200

Jacobian Structure

0 100 200 300 400

0

100

200

Numerical Jacobian

0 100 200 300 400

0

100

200

PseudoSpectral Jacobian

0 100 200 300 400

0

100

200

Analytical Jacobian

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1f-LGR

LG

LGRSchematic diagram of Legendre points

Acknowledgements: This research is supported by DLR (German Aerospace Center) and DAAD (German Academic

Exchange Service) Research Fellowship Programme.

Developments in space technology have paved the way for more challenging missions which require advancedguidance and control algorithms for safely and autonomously landing on celestial bodies.

Instant determination of hazards, automatic guidance during landing maneuvers and likelihood maximization ofa safe landing are of paramount importance.

Reachability analysis is used to obtain attainable landing areas for the final phase of interplanetary spacemissions given initial conditions, admissible control inputs and landing constraints.

: Downrange Rate : Downrange : Pitch

: Altitude Rate : Altitude : Yaw

: Crossrange Rate : Crossrange : Mass

: RCS Thrusters

: Pitch Rate

: Yaw Rate

Non-uniform collocation points toavoid the Runge phenomenon

0 20 40 60 80 100 120 140 160 180 200-12

-10

-8

-6

-4

-2

0

Number of Nodes

log 10

of

infin

ity n

orm

inte

rpol

atio

n er

ror

Interpolation error comparison using LGR,f-LGR,LG discretization points

LGR

f-LGRLG

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2Interpolation of a function with Lagrange Polynomials using Legendre-Gauss-Radau Polynomial roots

y

x

y = 11+ 50x2

InterpolationNodes

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Interpolation of a function with Lagrange Polynomials using Legendre-Gauss-Radau Polynomial roots

y

x

y = 11+ 50x2

InterpolationNodes

: Number of discretization points : Number of control Inputs

: Number of states : Number of constraints

Generic Mission Profile for Lunar Lander

Exponentialconvergence w.r.t. number ofcollocation points

Lagrange polynomials forinterpolation ofstates,control inputs

Jacobian of theassociated NLP isexpressed as sum of 3 different contributors. Resulting sparse is usedby commercial off-the-shelf SQP solver.

In this study, we apply an optimal-control-based algorithm for approximating nonconvex reachable sets of nonlinearsystems [2].

Discretize region of interest

Find optimal control law that steers the system from the initial condition to the target state

s.t. a.e. in

a.e. in

OCP is transcribed into NLP by SPARTAN (SHEFEX-3 Pseudospectral Algorithm for Reentry Trajectory ANalysis) [3].

-500 -400 -300 -200 -100 0 100 200 300 400 500-500

-400

-300

-200

-100

0

100

200

300

400

500

Downrange(m)

Cro

ssra

nge(

m)

Reachable Landing SiteforLunarLanderattf = 35 -Grid

Grid

Reachable

Approximate the reachable set with an error of discretization step by solving following optimal control problem (OCP) for each grid points

Discretization of State Space

Reachable Set at tf=35 (s)-Propellant Cost-

Reachable Landing Tunnel tf <35 (s)-Propellant Cost-

Reachable Set with free final time -Propellant Cost-

Reachable Set with free final time -Time Cost-

Attainable landing area and propellant&time cost map ofassociated region is computed using reachability analysis

Jacobian structure of associated NLP

Interpolation of a function with Lagrange polynomials using LGR roots

Schematic diagram of Legendre points and Interpolation Error

Condition for successful landing: