Optimal Path-Planning with Random Breakdowns

Marissa Gee1,2, Alexander Vladimirsky2,3

Abstract— We propose a model for path-planning based ona single performance metric that accurately accounts for thethe potential (spatially inhomogeneous) cost of breakdowns andrepairs. These random breakdowns (or system faults) happenat a known, spatially inhomogeneous rate. Our model includesbreakdowns of two types: total, which halt all movement until anin-place repair is completed, and partial, after which movementcontinues in a damaged state toward a repair depot. We usethe framework of piecewise-deterministic Markov processes todescribe the optimal policy for all starting locations. We alsointroduce an efficient numerical method that uses hybrid value-policy iterations to solve the resulting system of Hamilton-Jacobi-Bellman PDEs. Our method is illustrated through a seriesof computational experiments that highlight the dependence ofoptimal policies on the rate and type of breakdowns, with oneof them based on Martian terrain data near Jezero Crater.

I. INTRODUCTION

Robustness is one of the central challenges in path-planningfor autonomous vehicles. Even when the dynamics and nav-igation environment are fully known, there is a questionof how to account for the possibility of a partial or totalbreakdown. Much of the existing literature on planetary roversmodels the undesirability of such events heuristically. Thespeed or cost functions might be modified ad hoc to reflectthe risk [1], or terrain obstacles might be classified andessentially excluded from the environment before the actualtrajectory planning [2]. A more rigorous approach could bebased on multiobjective path-planning [3], recovering thePareto Frontier to capture all possible tradeoffs between theprimary optimization criterion (e.g., the time to target whenfully functional) and the risk of a breakdown along theway. This, however, still ignores the possibility of repeatedbreakdowns. More importantly, all of these approaches ignorethat the actual consequences for the mission depend notjust on the fact of a random breakdown but also on itstype and location. The primary contribution of this paperis a model that addresses these limitations rigorously andsystematically in the framework of piecewise-deterministicMarkov processes (PDMPs) [4].

We consider an autonomous robot attempting to reach atarget while subject to random breakdowns, whose (location-dependent) probability is known in advance. After a totalbreakdown, the robot cannot move, and must pay for anin-place repair. After a partial breakdown, it can continuetraveling, but must pass through a repair depot on its wayto the target. Until a depot is reached, the damaged robot isessentially solving a different optimization problem since it

*Supported by the NSF DMS (awards 1645643, 1738010, and 2111522).1mag433@cornell.edu2Center for Applied Mathematics, Cornell University3Department of Mathematics, Cornell University

might have a reduced speed or control authority, a restrictedset of directions of motion, or a lower energy efficiency. Wemodel these different modes (fully functional and damaged)using PDMPs. A PDMP is a stochastic model where at anypoint in time the system is in one of finitely many modes. Thesystem switches between these modes stochastically at knownrates, while each mode specifies its own deterministic dynam-ics and running cost. Recently, PDMPs have been applied topath planning problems with changing environments, wherethe modes represent environmental states such as changingwind direction [5], [6]. We take a similar approach here, butuse the modes to model the status of the robot itself.

The range of path-planning methods used in the roboticsand optimal control literature is truly broad (see [7]-[11] forsome examples). Our approach is based on dynamic program-ming in continuous state and time: we obtain globally optimaltrajectories by solving Hamilton-Jacobi-Bellman PDEs. Thisis a popular framework (e.g., [12]-[14]), particularly suitablewhen the state space is low-dimensional. But we note that ourmain ideas are also suitable for modifying popular discretestate path-planning methods in higher dimensions [9], [10].

We limit our discussion to a simplified isotropic modelof robot dynamics, primarily to streamline the exposition.However, our framework is quite general and can be extendedto the case of anisotropic dynamics or to more realistic,curvature-constrained models [15]. We start by describingthe problem statement and the structure of the governingPDEs for three path-planning scenarios in Section II. We thenpresent a novel iterative numerical method for solving thesePDEs (Section III) and results of computational experiments(Section IV). Most of our test problems use synthetic datato illustrate the effects of system parameters on optimaltrajectories, but the last example is more realistic and is basedon terrain data for a region of Mars near Jezero crater [16].We conclude by discussing future extensions in Section V.

II. GENERAL SETTING

A. Classical Path Planning

We consider a robot that obeys the isotropic dynamics

y(s) = a(s)f(y(s)), y(0) = x, (1)

throughout some bounded domain Ω, where a(s) : R → S1

is the chosen direction of motion and f is the speed oftravel. Henceforth, we use x to refer to a generic point inΩ or a trajectory’s initial condition and y(s) to encode thedependence of the robot’s position on time s. We seek thepolicy a(·) that minimizes the cumulative cost

J(x,a(·)) =

∫ T

0

K(y(s))ds+ q(y(T )). (2)

arX

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109.

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[m

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OC

] 1

0 N

ov 2

021

Here K is a running cost and q is a terminal cost. While Tcould be set a priori, our focus is on exit-time problems: letG ⊂ Ω be a finite set of points representing the target, thenT = infs|y(s) ∈ G.

The value function u(x) is defined to encode the optimalcost-to-go from each point in Ω:

u(x) = infa(·)∈A

J(x,a(·)), (3)

where A is the set of measurable functions from R to S1.Classical arguments from control theory [17] show that umust be the viscosity solution of the Eikonal PDE (suppress-ing the dependence on x for clarity):

|∇u|f = K, x ∈ Ω u = q, x ∈ G, (4)

for which there are several well-known efficient numer-ical solvers; e.g., [18]-[21]. Once u is computed, theoptimal policy can be recovered by setting a∗(s) =−∇u(y(s))/|∇u(y(s))|. In the special case that K = 1 andq = 0, the control a∗(s) is time-optimal.

B. Simplified Model: Total Breakdowns Only

Our first extension introduces the possibility of total break-downs. We assume that breakdowns happen instantaneouslyand at random times. We view the time until the nextbreakdown as an exponential random variable with rate λ > 0,except we allow λ(x) to vary in space, making the risk ofbreakdown trajectory-dependent. Let R(x) be the in-placerepair cost the robot must pay before continuing. The expectedcost associated with a trajectory y(s) is then given by

J(x,a(·)) =

∫ T

0

K(y(s)) + λ(y(s))R(y(s))ds (5)

(where we now assume q = 0 at the target). While we modelR as being paid instantaneously, we will see later that we canchoose R to capture the time taken for a repair (assuming therobot does not accrue running cost while broken down). Inthat case, s becomes the time spent moving, not the total timetaken. The structure of (5) is identical to that of (2), and thusthe same arguments show that the value function u solves

|∇u|f = K + λR, x ∈ Ω u = 0, x ∈ G. (6)

We now describe a particular model for R. Suppose thedomain contains a finite number of repair depots at D =x1, ..., xM ⊂ Ω, and that a repair vehicle must travel froma depot to the broken down robot to repair it. If the repairvehicle has its own speed fR(x) and running cost KR(x) andminimizes its own travel cost, we can write

R = uR +RF , (7)|∇uR|fR = KR, x ∈ Ω uR = RD, x ∈ D (8)

where RD(x) and RF (x) are depot and breakdown location-dependent repair costs, respectively. This structure providesgreat flexibility in modeling the cost of a total breakdown,while still allowing R to be precomputed throughout thedomain. For example, if we set KR = 1, RF to be the timefor an in-place repair, and RD to be the pre-dispatch waitingtime at each depot, then R captures the total time spent broken

down. Alternatively, if the repair vehicle charges a rate C forits time, we can scale RF by C, KR by 2C (to account for thereturn trip), and set RD = 0 (assuming we are not chargedfor waiting time) to model the amount paid for a repair.

C. Full Model: Total and Partial Breakdowns

Now, we allow for two types of breakdown: total, as definedabove, and partial, where the robot can continue moving afterthe breakdown (possibly with a lower f or higher K or λ).As before, a total breakdown immobilizes the robot until itis fixed in place. Thus, we now have two distinct modes inwhich the robot can travel and must decide on a trajectory:mode 1, when it is fully functional, and mode 2, when it isdamaged and moving toward a repair depot. We model thisas a PDMP, where each mode has its own value functionand they are coupled due to switching via breakdowns andrepairs. If we first assume u2 (the value function in mode 2)is known, then the expected cost in mode 1 is

J1(x,a(·)) = E

[∫ T1

0

K1(y(s)) + λ1(y(s))R(y(s))ds

+1[Tb<TG]u2(y(Tb))

]. (9)

where Tb is the time at which the first partial breakdown oc-curs, TG = infs|y(s) ∈ G is the time the robot would reachthe target if no breakdown occurred, and T1 = minTb, TG.The last term in (9) is only nonzero when a partial breakdownoccurs before the robot reaches the goal, in which case therobot switches to mode 2. If partial breakdowns occur at someknown rate φ(x), then this is an example of a randomly-terminated finite-horizon control problem, as outlined in [22].

We can similarly define J2, assuming that u1 is known. Inmode 2, the robot returns to mode 1 by being repaired at adepot or having a total breakdown and paying R. Let TB bethe time of the first total breakdown, TD = infs|y(s) ∈ Dbe the time the robot would reach the depot if no breakdownsoccurred, and T2 = minTB, TD, then

J2(x,a(·)) = E

[∫ T2

0

K2(y(s))ds

+1[TB<TD](R(y(TB)) + u1(y(TB)) (10)

+1[TB≥TD](RD(y(TD)) + u1(y(TD))

].

Here the last two terms again encode the optimal cost-to-go whenever a mode switch occurs. Following the derivationpresented in [6], we arrive at the following system of coupledPDEs for the value functions:

|∇u1|f1 = K1 + λ1R+ φ(u2 − u1) x ∈ Ω (11)u1 = 0 x ∈ G

|∇u2|f2 = K2 + λ2(R+ u1 − u2) x ∈ Ω (12)u2 = RD + u1 x ∈ D.

The inclusion of partial breakdowns significantly compli-cates the model. Previously it was possible to sequentially

solve for R and then u, but due to the coupling of u1 and u2that strategy is no longer possible in general. One exceptionis when the only depot is located at the target and thereare no total breakdowns in mode 2 (λ2 = 0). In this case,the boundary condition for u2 is known, since plugging inu1 = 0 on G gives u2 = RD on D, and thus the PDEsbecome decoupled. We can also recover simpler models asspecial cases of (11) and (12). First, setting λ1 = λ2 =φ = 0 recovers the classical path-planning problem withoutbreakdowns. If we instead set λ2 = φ = 0, we obtain thesimplified model from the previous subsection. Finally, settingλ1 = λ2 = 0 corresponds to an additional model with onlypartial breakdowns, which we examine in Examples 3 and 4.

III. NUMERICS

A. Discretized Equations

The main computational challenge in solving (11) and (12)is the coupling between u1 and u2: without it, the PDEs couldbe solved entirely using existing numerical techniques. Still,we will make use of multiple existing methods, so we start byreviewing them and later address solving the coupled systems.We will use the discretization (x, y) ≈ (xi, yj), where weassume for convenience that the distances between gridpoints,∆x and ∆y, are constant.

When solving Eikonal PDEs, we use the Fast MarchingMethod (FMM) [18], which relies on having a causal dis-cretization of the PDE. We use one-sided finite differenceapproximations of the derivatives, given by

Dij±xu =

±ui±1,j ∓ uij

∆x, Dij

±yu =±ui,j±1 ∓ uij

∆y,

(13)to define the upwind difference operator in the x-direction

Dijx u = minDij+xu,−D

ij−xu, 0 (14)

and its equivalent for the y-direction. Thus, for the Eikonalequation, the discretized u must satisfy√(

Dijx u)2

+(Dijy u

)2=Kij

f ij. (15)

at each point in the discretized domain. We can solve for uby propagating the boundary values as outlined in [18].

We also require solvers for a class of uncertain horizonproblems described in [22], which extends the FMM toproblems with random termination and known terminal cost.Equations (11) and (12) fit this framework if we assume thateither u2 or u1, respectively, are known. If that were the case,we would arrive at the following discretized equations:√(

Dijx u1)2

+(Dijy u1

)2(16)

=Kij

1 + λij1 Rij

f ij1− φij

f ij1(uij1 − u

ij2 )√(

Dijx u2)2

+(Dijy u2

)2(17)

=Kij

2

f ij2− λij2f ij2

(uij2 − (uij1 +Rij))

Algorithm 1: Solve equations (11) and (12) via value-policy iteration.Input: tol, termination thresholdInput: ρ < 1, policy evaluation threshold

1 Solve Eq. (8) for uR using FMM2 Set R = uR +RF

3 Set n = 0, d(0) = tol + 1, δ = d(n)

4 Initialize u(0)1 and u(0)2 with overestimates5 while d(n) > tol do6 while d(n) > ρ · δ do7 Initialize u(n+1)

2 = u(n)1 on D

8 Solve Eq. (12) for u(n+1)2 using modified

FMM with u1 = u(n)1

9 Solve Eq. (11) for u(n+1)1 using modified

FMM with u2 = u(n+1)2

10 Set n = n+ 1, dn = ‖u(n)1 − u(n−1)1 ‖∞11 end12 Solve Eqs. (19) and (20) for r1 and r213 Set δ = ‖u(n)1 − r1‖∞, u(n)1 = r1, u(n)2 = r214 end

and each value function could be found using an existingcausal method, assuming the other were known.

B. Value Iterations

To take advantage of the methods presented above, we takean iterative approach similar to standard value iteration. Ateach iteration, we first freeze the value of u2 and use an exist-ing method to solve (11) for u1. We then use the new versionof u1 to solve (12) for u2 in the same way. We alternateupdates in this manner until the change between iterations, δ,falls below a specified tolerance tol. The convergence can beproved by interpreting this algorithm as a (mode-by-mode)Gauss-Seidel relaxation of standard value iterations [23].

To start the iterative process, we need to initialize u2using some overestimates, which we obtain by posing simpler“pessimistic” problems. For all pessimistic problems, weassume that the robot does not leave mode 2 after a totalbreakdown, even after paying R — thus, the only couplingbetween u1 and u2 is on D. This provides an overestimateif f2 ≤ f1, K2 ≥ K1, and λ2 ≥ λ1. To obtain a pessimisticversion of u1 at every xm ∈ D (and thus an overestimateof the boundary conditions for (12)), we replace fk, φ, andKk + λkR by their worst values and assume that the vehiclemoves along a straight line from xm to the closest point inG, returning to xm along the same line in case of a partialbreakdown. This makes the problem essentially 1D, makingit easy to solve analytically using a system of two coupledlinear ODEs. We then initialize u2 throughout Ω by solving

|∇u2|f2 = K2 + λ2R, x ∈ Ω u2 = RD + u1, x ∈ D(18)

where u1 is the overestimate produced above.

(a) (b)

Fig. 1: Example 1. Radially symmetric value functions with G = D = (0.5, 0.5).(a) Value functions for φ = 5, λ1 = 0.5, λ2 = 1.5, f1 = 1, f2 = 0.2, andfR = 0.1. (b) L∞ error between analytic and numeric solutions, with value iterationtolerance and slope −1 line for reference. Error exhibits first-order convergence.

C. Acceleration via Value-Policy Iteration

One shortcoming of value iterations is that δ quicklybecomes small, even when the current value function is farfrom the solution. Thus, pure value iterations are slow toconverge, especially for poor initializations of the system. Toremedy this, we extend to PDMPs the method of combinedvalue iterations and policy evaluations, outlined in [24] forsingle mode problems. Policy evaluation is commonly foundas a step in policy iteration, a popular dynamic programmingtechnique, and involves fixing the control a(s) and computingthe value function exactly for that fixed, suboptimal policy.

Fixing a1(s) and a2(s) in equations (11) and (12) we get

(a1 · ∇r1)f1 = K1 + λ1R+ φ(r2 − r1) x ∈ Ω (19)r1 = 0 x ∈ G

(a2 · ∇r2)f2 = K2 + λ2(R+ r1 − r2) x ∈ Ω (20)r2 = RD + r1 x ∈ D,

a system of coupled linear PDEs that can be discretized usingfinite differences and solved efficiently using any large-scalelinear solver. The combined value-policy iteration algorithmfor two coupled value functions is outlined in Algorithm 1.

IV. NUMERICAL RESULTS

For all examples except the first, the value functions arerepresented on a 501×501 grid on [0, 1] × [0, 1]. For allexamples, we set tol = 2/(∆x + ∆y), K = 1 in bothmodes, RF = 1, RD = 0, and use spatially homogenousenvironmental parameters f1, f2, fR, λ1, λ2, and φ unlessotherwise specified. 1

A. Example 1: Convergence of Iterative Scheme

We start with a model for which the solution can becomputed analytically: one target and one depot, both at thesame location. The 2D solution is radially symmetric about G,and can be obtained by solving a system of ODEs analytically.Figure 1a shows the value functions as functions of distanceto the target. Figure 1b shows the first-order convergence ofthe iterative scheme under grid refinement.

1Code for all examples is available at https://github.com/eikonal-equation/Random_Breakdowns

(a) (b)

(c) (d)

Fig. 2: For all figures: gold stars are targets, red pentagons are depots, grey dots aresample starting locations, red Xs are potential breakdown locations, and green dotsare potential in-place repairs. Black trajectories are optimal in mode 1 and red dottedtrajectories are optimal in mode 2. Example 2:G = D = (0.9, 0.5). (a) The partialbreakdown rate. (b) Value function for mode 1 and sample trajectories for modes 1 and2 when λ1 = λ2 = 0. (c) Value function and sample trajectories for mode 2, whenλ1 = 1 and λ2 = 3. The optimal trajectories are orthogonal to the level sets of u2.(d) Value function and sample trajectories for mode 1 when λ1 = 1 and λ2 = 3.

B. Example 2: Inhomogenous Partial Breakdown Rate

We now consider an inhomogenous partial breakdown rate,shown in Figure 2a. We set f1 = 1, and use f2 = 0.2f1 andfR = 0.1f1 to enforce slower speeds for the damaged robotand the repair vehicle. Figure 2b shows the mode 1 valuefunction in the absence of total breakdowns. The optimaltrajectories clearly avoid areas with high values of φ. Withouttotal breakdowns (λ1 = λ2 = 0), there is no need to computeuR and u2 is radially symmetric about G, so it is optimal totravel in a straight line to the target in mode 2.

This changes once we introduce a chance of total break-downs. Figures 2c and 2d show the value functions for aconstant chance of total breakdown (λ1 = 1, λ2 = 3). Now,the optimal trajectories in mode 2 are no longer straight lines.Instead, the robot avoids areas where φ is high so as not to becaught there if it is repaired in-place after a total breakdown.Thus, the introduction of even a constant value of λ2 candisrupt the radial symmetry of u2 about the target when φ isinhomogenous.

C. Example 3: Changing Partial Breakdown Rate

We now examine how changes in φ affect the optimalpaths. We set λ1 = λ2 = 0, and use the same speed valuesas Example 2 but with three different depots (Figure 3a).Since G

⋂D = ∅, a damaged robot must travel to a depot

rather than directly to the target. This example illustratesthe tradeoffs associated with extending the planned mode 1trajectories. Longer trajectories can remain closer to D, whichreduces the expected time spent traveling in mode 2, but alsoincreases the chance that a breakdown occurs at all.

(a) (b)

Fig. 3: Example 3, G = (0.9, 0.9) and D =(0.3, 0.1), (0.35, 0.55), (0.8, 0.6). (a) Mode 2 value function with φ = 3 andλ1 = λ2 = 0. (b) Mode 1 value function for the same parameters. Sample optimaltrajectories in mode 1 for φ = 0 (yellow), φ = 3 (orange & orthogonal to the shownlevel sets of u1), and φ = 5 (red) starting at x = (0.1, 0.1).

Figure 3b shows the impact of changing φ on the optimaltrajectories in mode 1. With no chance of breakdowns (φ =0), the robot would follow the yellow straight line to G. Forrelatively low values of φ > 0, it is optimal to extend thepath (orange) in order to stay close to D and decrease thetime spent traveling in mode 2. However, as φ increases,breakdowns are more frequent and the robot may have totravel in mode 2, even after a recent repair. Eventually, it isno longer worth extending the path (red) to be near everyrepair depot, and it becomes optimal to prioritize reachingthe best depot (the one with the lowest expected cost-to-go)as quickly as possible.

D. Example 4: Real-world Terrain

As our last example, we consider path-planning for a Marsrover. We examine a region of Mars near Jezero crater, whichwas the landing site of NASA’s Perseverance rover, and thushas extensively characterized terrain. All data was accessedvia JMARS, a Mars GIS [16]. We base the rover’s speed onthe slope of the terrain and measure it in units of meters/sol (asol is a Martian day). Current Mars rovers can navigate slopesof up to approximately σ = 20, and travel more slowlyon steeper slopes due to increased wheel slippage [25]. Forsimplicity we assume that the rover travels at a small butnonzero speed fmin = 1m/sol for any slope above σ, andfollowing [26] we set fmax = 200m/sol, which yields

f1(x) =

fmax − fmax−fmin

σ σ(x) σ ≤ σfmin σ > σ

(21)

where σ(x) is the slope. We assume that σ and fmax aredecreased by a factor of 2 in mode 2. Figures 4a and 4bshow f1 and f2 for the region we consider.

We do not allow total breakdowns (λ1 = λ2 = 0), sincerepairs are not feasible for a Mars rover. However, we doallow for partial breakdowns, and set D = G, ensuringthat the rover will continue to travel to the target in mode2. The breakdown rate is modeled as proportional to theterrain roughness, ρ(x). We compute ρ as the root-mean-square height, a standard measure in geology [27] that hasalso been used to characterize traversability for Martian rovers

in the past [1]. We model the breakdown rate as

φ(x) =ρ(x)2

5000(22)

and it is shown in Figure 4c. This scaling is largely arbitrary,and can be modified based on the rover’s capabilities.

Figures 4d and 4e show u2 and u1 and sample optimaltrajectories for Example 4. In mode 1, the rover is ableto traverse the crater’s rim by avoiding the most steeplysloped portions. If a breakdown occurs before it has finishedclimbing, it may become optimal to take a much longerroute that avoids the steep slopes of the rim. However, ifa breakdown occurs after the rover has navigated most ofthe treacherous terrain, it is optimal to climb down the slopetowards the target, though with an altered path that avoidssteeper slopes. Figure 4f shows the elevation of the regionof interest, a single mode 1 optimal trajectory, two samplebreakdown locations, and subsequent mode 2 paths.

V. CONCLUSION

We presented a piecewise-deterministic framework for op-timal path planning with two types of random breakdown,total and partial, and showed that each leads to structurallydifferent PDE systems for the value functions. We alsopresented an efficient numerical method for solving thesesystems via value-policy iterations, and verified the first-orderaccuracy of our approximate solutions experimentally. Ourtest problems illustrated the effect that the breakdown type,breakdown rate, and depot/goal placement have on optimaltrajectories. Finally, we computed time-optimal trajectoriesthat account for random partial breakdowns in an environmentbased on Martian terrain data.

One obvious extension of our model is to allow the robotto voluntarily terminate mode 2, paying for the full in-place repair before a total breakdown actually occurs. Thiswould in effect replace equation (12) with a quasi-variationalinequality [22]. Another natural extension is to consideranisotropic problems, allowing for control-dependent K, f ,φ, and λ, and taking advantage of the numerical methodsin [19] or [28]. Extensions could also focus on switchingrates, either by allowing them to be time-dependent, or byadding more modes to represent additional levels of damage.In addition, it will be useful to look beyond the expectedcost, maximizing the probability that the cumulative cost fallsbelow any specified threshold [6]. Multiobjective extensionsare yet another possibility, with hard or chance constraints,similar to [29], imposed on additional criteria (e.g., directlyconstraining the probability of a full breakdown on Mars).

ACKNOWLEDGMENT

The authors would like to thank Lars Grune for his adviceon value-policy iteration and Elliot Cartee, whose work onmodeling environmental crime inspired this project.

REFERENCES

[1] D. B. Gennery, “Traversability analysis and path planning for a plane-tary rover,” Autonomous Robots, vol. 6, no. 2, pp. 131–146, 1999.

[2] H. Seraji, “Traversability index: A new concept for planetary rovers,”in Proc. 1999 IEEE ICRA, vol. 3. IEEE, 1999, pp. 2006–2013.

(a) (b) (c)

(d) (e) (f)

Fig. 4: Example 4, G = D = (0.1, 0.1). (a) Speed of rover in mode 2. (b) Speed of rover in mode 1. (c) Partial breakdown rate. (d) Value function and sample trajectoriesfor mode 2. (e) Value function and sample trajectories for mode 1. (f) Elevation and sample optimal trajectories in modes 1 and 2.

[3] A. Kumar and A. Vladimirsky, “An efficient method for multiobjectiveoptimal control and optimal control subject to integral constraints,” J.Comput. Math., vol. 28, pp. 517–551, 2010.

[4] M. H. Davis, “Piecewise-deterministic Markov processes: A generalclass of non-diffusion stochastic models,” J. Roy. Statistical Soc.: SeriesB (Methodological), vol. 46, no. 3, pp. 353–376, 1984.

[5] Z. Shen and A. Vladimirsky, Piecewise-Deterministic Optimal PathPlanning. [Online]. Available: http://arxiv.org/abs/1512.08734

[6] E. Cartee, A. Farah, A. Nellis, J. Van Hook, and A. Vladimirsky, “Quan-tifying and managing uncertainty in piecewise-deterministic Markovprocesses,” preprint: https://arxiv.org/abs/2008.00555, 2020.

[7] E. W. Dijkstra, “A note on two problems in connection with graphs,”Numerische Mathematik, vol. 1, pp. 269–271, 1959.

[8] B. Chazelle, “Approximation and decomposition of shapes,” Algorith-mic and Geometric Aspects of Robotics, vol. 1, pp. 145–185, 1985.

[9] L. E. Kavraki, P. Svestka, J.-C. Latombe, and M. H. Overmars, “Prob-abilistic roadmaps for path planning in high-dimensional configurationspaces,” IEEE Trans. Robot. Autom., vol. 12, no. 4, pp. 566–580, 1996.

[10] S. M. LaValle, “Rapidly-exploring random trees: A new tool for pathplanning,” 1998.

[11] K. Alton and I. M. Mitchell, “Optimal path planning under defferentnorms in continuous state spaces,” in Proc. 2006 IEEE ICRA. IEEE,2006, pp. 866–872.

[12] S. Garrido, D. Alvarez, and L. Moreno, “Path planning for Mars roversusing the Fast Marching Method,” in Robot 2015: Second IberianRobot. Conf., L. P. Reis, A. P. Moreira, P. U. Lima, L. Montano, andV. Munoz-Martinez, Eds. Springer, 2016, pp. 93–105.

[13] E. Cartee and A. Vladimirsky, “Control-theoretic models of environ-mental crime,” SIAM J. Appl. Math., vol. 80, no. 3, pp. 1441–1466,2020.

[14] C. Parkinson, D. Arnold, A. Bertozzi, and S. Osher, “A model foroptimal human navigation with stochastic effects,” SIAM J. Appl. Math.,vol. 80, no. 4, pp. 1862–1881, 2020.

[15] R. Takei and R. Tsai, “Optimal trajectories of curvature constrainedmotion in the Hamilton-Jacobi formulation,” J. Sci. Comput., vol. 54,no. 2–3, p. 622–644, Feb. 2013.

[16] P. Christensen, E. Engle, S. Anwar, S. Dickenshied, D. Noss, N. Gore-lick, and M. Weiss-Malik, “JMARS - a planetary GIS,” in AGU FallMeeting Abstracts, 2009, pp. IN22A–06.

[17] M. G. Crandall and P.-L. Lions, “Viscosity solutions of Hamilton-Jacobiequations,” Trans. Amer. Math. Soc., vol. 277, no. 1, pp. 1–42, 1983.

[18] J. A. Sethian, “A fast marching level set method for monotonicallyadvancing fronts,” Proc. Nat. Acad. Sci., vol. 93, no. 4, pp. 1591–1595,February 1996.

[19] Y.-H. R. Tsai, L.-T. Cheng, S. Osher, and H.-K. Zhao, “Fast sweepingalgorithms for a class of Hamilton–Jacobi equations,” SIAM J. Numer.Anal., vol. 41, no. 2, pp. 673–694, 2003.

[20] A. Chacon and A. Vladimirsky, “Fast two-scale methods for Eikonalequations,” SIAM J. Sci. Comput., vol. 34, no. 2, pp. A547–A578, 2012.

[21] ——, “A parallel two-scale method for eikonal equations,” SIAM J. Sci.Comput., vol. 37, no. 1, pp. A156–A180, 2015.

[22] J. Andrews and A. Vladimirsky, “Deterministic control of randomly-terminated processes,” Interfaces and Free Boundaries, vol. 16, no. 1,pp. 1–40, 2014.

[23] D. P. Bertsekas, Dynamic Programming and Optimal Control, 3rd ed.Boston, MA: Athena Scientific, 2001, vol. I and II.

[24] L. Grune and W. Semmler, “Using dynamic programming with adaptivegrid scheme for optimal control problems in economics,” J. Econ. Dyn.Control, vol. 28, no. 12, pp. 2427 – 2456, 2004.

[25] M. Heverly, J. Matthews, J. Lin, D. Fuller, M. Maimone, J. Biesiadecki,and J. Leichty, “Traverse performance characterization for the MarsScience Laboratory rover,” Journal of Field Robotics, vol. 30, no. 6,pp. 835–846, 2013.

[26] M. Ono, B. Rothrock, E. Almeida, A. Ansar, R. Otero, A. Huertas, andM. Heverly, “Data-driven surface traversability analysis for Mars 2020landing site selection,” in 2016 IEEE Aerosp. Conf. IEEE, 2016, pp.1–12.

[27] M. K. Shepard, B. A. Campbell, M. H. Bulmer, T. G. Farr, L. R. Gaddis,and J. J. Plaut, “The roughness of natural terrain: A planetary andremote sensing perspective,” Journal of Geophysical Research: Planets,vol. 106, no. E12, pp. 32 777–32 795, 2001.

[28] J. A. Sethian and A. Vladimirsky, “Ordered Upwind Methods for StaticHamilton-Jacobi Equations,” Proc. Nat. Acad. Sci., vol. 98, no. 20, pp.11 069–11 074, 2001.

[29] M. Ono, M. Pavone, Y. Kuwata, and J. Balaram, “Chance-constraineddynamic programming with application to risk-aware robotic spaceexploration,” Autonomous Robots, vol. 39, no. 4, pp. 555–571, 2015.