Adaptive Sampling Based COLREGs-Compliant Obstacle Avoidance forAutonomous Surface Vehicles

Petr Svec1, Member, IEEE, Brual C. Shah1, Ivan R. Bertaska2, Wilhelm Klinger2, Armando J. Sinisterra2,Karl von Ellenrieder2, Member, IEEE, Manhar Dhanak2, and Satyandra K. Gupta3, Senior Member, IEEE

Abstract— Achieving persistent autonomy requires un-manned surface vehicles (USVs) to be able to deal with awide variety of complex planning situations. In this paper,we introduce a model-predictive, local trajectory planningalgorithm for USVs operating in congested and highly dynamictraffic. The planner generalizes the Velocity Obstacle conceptto systems with non-linear dynamics, nonholonomic constraints,and any form of low-level feedback controller. High planningperformance is achieved by searching in a resolution-adaptivespace of vehicle candidate motion goals for a motion goalthat optimizes the arrival time to a given global goal. Thealgorithm gradually identifies, with increasing resolution andfocused sampling, an approximate representation of spatio-temporal obstacle regions. The sampling of motion goals canbe constrained to ensure the International Regulations forPrevention of Collisions at Sea (COLREGs). We have carriedout experiments to demonstrate the practical feasibility of usingthe planner on a 14-foot (4.3 m) long catamaran USV operatingin a harbor-like environment.

I. INTRODUCTION

Achieving persistent autonomy requires unmanned surfacevehicles (USVs) [1] to be able to deal with a wide varietyof situations and tasks [2], [3], [4], [5], [6], [7], [8]. Forexample, the USV may have to transition from operatingin an open ocean to a crowded harbor. In these typesof environments, it will need to deal with a wide varietyof obstacles ranging from stationary piers to fast movingvessels. Other boats in the environment are likely to bedriven by humans who may exhibit different driving stylesranging from highly cautious to highly risky. The plannerhas to be capable of dealing with these different situationsappropriately by accounting for collision risks. It needs toutilize the knowledge of the USV’s maneuverability, enablefollowing the International Regulations for Prevention ofCollisions at Sea (COLREGs) [9], and ensure fast replanning.

In order to achieve long-term, persistent autonomy, it isimperative to utilize an accurate, system-identified modelof the vehicle dynamics and a state feedback control lawin trajectory planning. However, high-fidelity, local plan-ning that utilizes an expressive set of candidate maneuvers

1P. Svec and Brual C. Shah are with the Department of MechanicalEngineering, University of Maryland, College Park, MD 20742, USA{petrsvec,brual}@umd.edu

2I.R. Bertaska, W. Klinger, A.J. Sinisterra, K. Ellenrieder, and M.Dhanak are with the Department of Ocean & Mechanical Engineering,Florida Atlantic University, Dania Beach, FL 33004-3023, USA{ibertask,wklinger,asiniste,ellenrie,dhanak}@fau.edu

3S.K. Gupta is with the Department of Mechanical Engineering andInstitute for Systems Research, University of Maryland, College Park, MD20742, USA skgupta@umd.edu

Fig. 1: a) Expanded set of motion goals in classical, uniformsampling-based local planning and b) adaptive, non-uniformsampling-based local planning introduced in this paper.

generated using the model and corresponding controller iscomputationally demanding. This prevents the use of a shortsense-plan-control cycle to guarantee collision-free operationon unmanned systems.

Very recently, techniques have been developed that aimto address this challenge [10], [11], [12], [13] . However,they either make simplifying assumptions about the vehicle’sdynamics, do not have the capability to utilize advanced low-level control and rich sets of control actions in planning, ordo not consider constraints on control inputs.

In this paper, our goal is to address these assumptionsand still achieve high planning performance by searching ina resolution adaptive space of candidate motion goals (i.e.desired vehicle states). The developed planner generalizesthe Velocity Obstacle (VO) concept [14] to systems withnon-linear dynamics, nonholonomic constraints, and anylow-level motion control method. It computes a resolutionoptimal motion goal that optimizes the time to reach a pre-specified global goal, is guaranteed to be reachable by thevehicle with its specific control, and respects COLREGs, ifneeded. The computation is anytime in nature as it returnsthe best possible motion goal within a given time limit.

The search for a motion goal starts with a small, lowresolution, regularly sampled, bounded search space consist-ing of motion goals and state space trajectories that lead tothese goals in the vicinity of the vehicle’s state. The resolu-tion of this space is iteratively increased during the searchby expanding motion goals in the neighborhood of statespace trajectories that lead to motion goals with the currentknown minimum cost. In this way, the algorithm gradually

discovers, with increasing resolution and focused sampling,an approximate representation of spatio-temporal obstacleregions in the state space. The algorithm also ensures thedynamical feasibility of corresponding state trajectories andmaintains high replanning frequency through the use of ameta-model of the vehicle dynamics and low-level controller.

The sampling is based on the history of detected collisionsfrom previous sampling iterations. The aim is to minimizethe number of samples for collision testing during the searchby focusing on promising regions only. The motion goalsare sampled non-uniformly along trajectories based on thedifference in curvature between their consecutive segments,difference in surge speed, and the future projected poses ofdynamic obstacles. This saves computation, as a large num-ber of states do not have to be evaluated for collision. Equallyimportant, the algorithm decides the extent of the look-aheadfor each expanded sampled trajectory in the adaptive searchspace to save additional computational resources.

We have performed physical experiments with a 14-foot(4.3 m) long catamaran USV that has a non-linear dynamicsand utilizes backstepping control [15] to demonstrate thefeasibility of using the planner in a realistic physical environ-ment containing civilian vessels and static obstacle regions.

II. BACKGROUND AND RELATED WORK

In many recent applications, local obstacle avoidance isusually based on the Velocity Obstacle (VO) [14], InevitableCollision State (ICS) [16], or Dynamic Window (DW) [17]concepts. For example, in the marine robotics domain, VO-based, COLREGs-compliant local obstacle avoidance wasintroduced in [18] and integrated into NASA JPL’s CARA-CaS planning and control system. An extension of this workthat argues the need to consider the dynamics of the USVin planning of robust obstacle avoidance maneuvers waspresented in [3]. A thorough review of planners compliantwith COLREGs can be found in [19].

In the past, there were several attempts [10], [11], [12],[13] to incorporate vehicle kinematics and linearized dynam-ics into the computation of velocity obstacles. For example,Wilkie et al. introduced the concept of Generalized VelocityObstacle (GVO) [10] that was further extended to LQR-Obstacles [11] to incorporate a LQR controller into planning.Very recently, this algorithm was further extended in [13] totake reciprocal avoidance maneuvers of other agents into ac-count when computing velocity obstacles. Similarly, Berg etal. [12] proposed a VO-based, reciprocal collision avoidanceplanner that considers vehicle acceleration constraints.

The algorithm in this paper significantly extends theapproach published in [3] to carry out efficient planning inthe state space as opposed to the control space, similarly to[11]. This has the advantage that the planner can adapt thesearch space of all candidate maneuvers to environmentalconstraints (e.g., static obstacles) and navigation rules (e.g.,lane following while being COLREGs-compliant), and thenefficiently search in this constrained, well-posed space [20].Moreover, the low-level controller has higher flexibility inoptimizing its behavior as it takes each motion goal (specified

by the desired geometric pose and surge speed of the USV)as a higher-level control input [11], as opposed to only takingthe desired heading and speed objectives.

The latest work in non-uniform sampling based planningincludes [21]. The planning algorithm introduced in thispaper allows high performance planning through anytimesearch and adaptive, non-uniform sampling of state-spacetrajectories. It adaptively expands the boundaries of the ex-plored state space, increases its resolution during the search,and efficiently samples individual trajectories for collisionswith dynamic obstacles. The use of a meta-model of vehicle’sdynamics and control in planning and the above mentionednon-uniform sampling procedure allows the deployment ofthe planner in real-world unmanned systems.

III. PROBLEM FORMULATION

Given,(i.) the continuous state space X = Xη × Xψ × Xu × T in

which each state x = [ηT , ψ, u, t]T consists of USV’sposition η = [x, y]T ∈ Xη , heading ψ ∈ Xψ aboutthe z-axis in the North-East-Down (NED) coordinatesystem [22], surge speed u ∈ Xu, and the time t ∈ T ;

(ii.) the current USV’s state xU = [ηUT , ψU , uU , t]

T ∈ X ;(iii.) the bounded, discrete space of motion goals XG(xU) ⊂

XG in the USV’s body frame;(iv.) the meta-model m : Xu × XG × T → X of the USV’s

dynamics [3] and an associated low-level controller. Themodel takes uU , xG, and the time t ∈ T as inputsand returns a state x along a trajectory leading to xG.The USV’s dynamics is defined as xU = fU (xU,uh),where the thrust and moment are produced by modelactuators that take uh as the control input. This controlinput is determined by the controller hU (xU,uc, PU ),where uc = [ud, ψd]

T is the control action and PU isthe set of controller parameters;

(v.) the estimated states Xd = {xo,l|xo,l ∈ X}Ll=1

of dynamic obstacles and their geometric profiles{od(xo,l) ⊆ R2}Ll=1;

(vi.) the geometric profiles {os ⊆ R2}Kk=1 of static obstacles;(vii.) the global trajectory τG = {xτ,G,i}Ni=1 consisting of

waypoints to be followed;Compute,(i.) a high-level control input in the form of a motion goal

xG to track τG, while avoiding obstacles in COLREGs-compliant way and minimizing the travel time.

IV. APPROACH

In this section, we present a computationally efficient,local planning algorithm (see Alg. 1), which computestrajectories that respect the vehicle’s dynamic constraintsand COLREGs. The algorithm developed searches in thelocal space of dynamically reachable motion goals XG(i.e., desired vehicle’s states) that is incrementally expandedtowards a given global goal xτ,G,j ∈ τG with increasingresolution. The search terminates when either the allocatedcomputational time limit tc,T is reached, or all candidatemotion goals have been evaluated.

The motion goals are trapped in an underlying discretespace of dynamically feasible state space trajectories T ={τi|τi = {xi,j}j=Nj=1 ,xi,j ∈ X} emanating from the currentstate xU of the USV. The algorithm starts with motion goalsin an initial discrete set TI of sparsely sampled trajectories.Each trajectory in this initial set is expanded for the maxi-mum of te,T seconds. This set of trajectories is adaptivelysampled and expanded during the search to increase theresolution of the search in its most promising regions (seesubsection IV-C). New motion goals are also added to XGthrough adaptive sampling of individual trajectories basedon the difference in curvature between their consecutivesegments, difference in surge speed, and the future esti-mated poses of spatio-temporally distributed obstacles (seesubsection IV-D). The motion goals are evaluated using acost function defined in subsection IV-B. A new candidatemotion goal can be selected for execution if only the timet and distance d to reach the goal is equal or greater thanuser-specified thresholds th,T and dh,T , respectively. In orderto maximize the planning performance, the dynamics andcontrol of the USV are adaptively sampled and encapsulatedinto a meta-model as described in subsection IV-A.

Algorithm 1 FINDMOTIONGOAL(TI , O, tc,T , th,T , dh,T )

Require: The initial, low-resolution, uniformly sampled space ofstate trajectories TI ⊂ T ; geometric profiles O = {os}Kk=1 ∪{od(xo,l)}Ll=1 of static and dynamic obstacles, respectively; themaximum allowed computation time tc,T , and time th,T anddistance dh,T horizons, respectively.

Ensure: A motion goal xG,best that optimizes the time to reacha given global goal.

1: Q ← DETERMINENEXTSTATE(xG,i,1, TI) for all initial mo-tion goals xG,i,1 ∈ XG,I contained in TI . Here, the cost of themotion goals is computed using (1).

2: TU ← TI3: while Q 6= ∅ AND tc ≤ tc,T do4: xG,i,j ← Q.FIRST()5: if c(xG,i,j) < c(xG,best) and t ≥ th,T and d ≥ dh,T then6: xG,best ← xG,i,j

7: end if8: xG,i,j+k ← DETERMINENEXTSTATE(xG,i,j, TU )9: Compute cost c(xG,i,j+k) using (1).

10: Q← xG,i,j+k if not in collision with obstacles O.11: {Q, TU} ← SAMPLEMOTIONGOALS(T , TU , O)12: end while13: return xG,best

A. Meta-Model of USV14 Dynamics and Control

We precompute a meta-model m(uU ,xG, t) of theUSV14’s dynamics and control [3]. The meta-model is repre-sented as a 4D look-up table consisting of a discrete, finite setXG of motion goals and corresponding state space trajecto-ries T leading to these goals from the current USV14’s statexU. Each state space trajectory τi ∈ T is discretized into afinite number of motion goals (xG,i,1,xG,i,2, . . . ,xG,i,n),where the time difference between consecutive goals isdefined by the parameter ∆t.

We utilize a backstepping controller [15] to forwardsimulate the model of the USV14’s dynamics to generate

T . The meta-model is defined in the body frame of theUSV14. Hence, the only state variable that is varied duringthe computation of the meta-model is the initial USV’s surgespeed u in addition to a motion goal xG.

B. Cost Function

The cost of sampled motion goals is computed as follows:

c(xG) = ωtc + (1− ω)tg + ccol,soft + c¬COLREGs (1)

Here, tc is the time to come to xG from xU, tg =‖xG − xτ,G‖ is the heuristic estimate of the time to goto the global goal xτ,G of the trajectory τG, ω is theweight to tune the breadth the of search, c¬COLREGs isthe cost penalty for breaching COLREGs, and ccol,soft =(ccol,soft,T−ccol,hard,T )/dcol,soft,T dCPA+ccol,hard,T is thelinearly varied cost penalty for being inside a soft collisionregion around an obstacle. The upper bound of this costpenalty ccol,hard,T is incurred if the state lays at the boundaryof the obstacle, i.e., when dCPA = 0. The lower bound ofthis penalty ccol,soft,T is incurred if the state lays at theboundary of the soft collision region, i.e., when dCPA =dcol,soft,T . The penalty ccol,soft = 0 if dCPA > dcol,soft,T .It is important to use the soft collision region to compensatefor the imprecisions in the system identified USV model aswell as motion uncertainties that are not considered whencomputing the meta-model.

C. Adaptive sampling of state space trajectories

During the search, Alg. SAMPLEMOTIONGOALS gradu-ally increases the resolution of XG in the most promisingdirection and prioritizes the search. It samples in the set ofstate space trajectories T leading from xU to the boundaryof the search space, which results in TU ⊂ T . The terminalmotion goals XG,T ⊂ XG of these trajectories are uniformlydistributed on an arc with the central angle dα and the radiusdh (i.e., a distance horizon) from the USV (see Fig. 2).

The algorithm maintains a state space trajectoryτi,best ∈ TU that contains a motion goal xG,i,j with thecurrently known minimum cost. It determines neighboringgroups of expanded (even partially) state space trajectoriesTU,i−m and TU,i+n (vary in shape as well as speed) withterminal motion goals that are geometrically positionedto the left and to the right of the terminal motion goalxG,i,T of τi,best (given the current resolution of TU ),respectively. Given TU,i−m and TU,i+n, the algorithmdetermines trajectories T(2i−m)/2 and T(2i+n)/2 withterminal motion goals X(G,2i−m)/2,T and X(G,2i+n)/2,T

that are geometrically positioned in the middle betweenthe terminal motion goals of TU,i−m and τi,best, andτi,best and TU,i+n, respectively. In the final step, theinitial motion goals with the arrival time and distanceup to the thresholds th,T and dh,T , respectively, ofthe new trajectories are put onto the priority queue Qif they do not collide with obstacles. More precisely,XG,(2i−m)/2,j is computed in which each xG,(2i−m)/2,j ←DETERMINENEXTSTATE(xG,(2i−m)/2,j, TU ). Similarly,

Fig. 2: Adaptive sampling of the space of state trajectories.The expanded parts of the trajectories and motion goals aremarked as black.

Fig. 3: Adaptive sampling of a state space trajectory.

XG,(2i+n)/2,j is computed. The cost of each motion goal inboth the sets is computed using (1).

The time horizon of individual state space trajectories isdecided during the search (see IV-D). This allows the systemto find a balance between the extent of deliberation andreactivity in planning. In addition, if required, the new tra-jectories are not considered in future searches if they lead toa COLREGs-breach region (see [3] for a detailed descriptionof determination of COLREGs-compliant maneuvers).

D. Adaptive sampling of a state space trajectory

We also sample the space of motion goals XG alongpartially expanded state space trajectories TU leading fromxU to terminal motion goals XG,T as implemented by Alg.DETERMINENEXTSTATE. The motion goals of τi ∈ TUare sampled based on their spatio-temporal proximity toobstacles as well as the difference in heading of the vehicle atconsecutive goals. This saves computation time as the motiongoals of straight trajectory segments or the ones that will notlead to collisions are skipped during sampling (see Fig. 3)).

More precisely, the algorithm returns the next sampledmotion goal xG,i,j+k from the motion goal xG,i,j alonga state space trajectory τi when either 1) the delta timethreshold δt is exceeded when moving between the motiongoals, 2) the terminal motion goal xG,i,T is reached, 3) theglobal motion goal xτ,G,j is reached, or 4) the difference

in heading ψ of the vehicle at xG,i,j and the heading atxG,i,j+k exceeds the threshold δψ . For the last condition,we define a function n(xG,i,j) which returns the next motiongoal xG,i,j+k along the trajectory τi if δψ is exceeded. Werepresent this function as a look-up table and precompute italong with the meta-model.

In addition, a new, backtracked motion goal xG,i,p isdetermined when there is a collision, i.e., the minimumdistance dCPA,min at the closest point of approach (CPA)between the USV and time-projected obstacles along thetrajectory segment connecting xG,i,j and xG,i,j+k is belowa user-specified safety threshold.

In order to compute dCPA for a single obstacle, the timeto CPA, tCPA, is computed first. The algorithm computesthe quadratic discriminant D = (2(ηU,o ·vU,o))2− 4(vU,o ·vU,o)((ηU,o ·ηU,o)− (rU + ro)

2). Here, if D < 0, the USVdoes not collide with the obstacle and tCPA = −2(ηU,o ·vU,o)/(2(vU,o ·vU,o)). The time to CPA tCPA remains thesame for the case when the USV collides with the obstacle ata single point, i.e., D = 0. If D > 0, tCPA = min{t0, t1},where t0 and t1 are the roots of a quadratic equation with thediscriminant D. Here, vU,o = vU−vo, where vU and vo arevelocity vectors of the USV and obstacle, respectively, andthe term ηU,o = ηU−ηo. The USV and the obstacle intersectat any time in the future if only D ≥ 0 and tCPA ≥ 0.If tCPA < 0, then the distance between their trajectoriesincreases and CPA has already occurred at some time inthe past. In order to achieve fast collision detection, weapproximate the USV and the obstacle with a circle of aradius rU and ro, respectively. We approximate the staticobstacle region Os by circular regions with radius rs.

We then compute the minimum distance dCPA,min =mino∈OdCPA,o = ‖ηU(tCPA) − ηo(tCPA)‖ between CPAand obstacles from the currently sampled motion goal, whereO = Os ∪Od. The algorithm maintains the minimum valueof dCPA,min for each trajectory being expanded during thesearch. If dCPA,min < 0 and tCPA,min ∈ [0, tj+k−tj ], thenthe USV collides with the obstacle at some time tCPA,minin the future when transitioning between xG,i,j and xG,i,j+k

along τi. In that case, the algorithm backtracks to a motiongoal xG,i,p, where p ∈ [1, j + k] from which tCPA,min ≤tCPA,col. In this way, the newly backtracked motion goalcan be followed by the USV as the USV will have sufficientamount of time tCPA,col to execute avoidance maneuvers toavoid a collision if required.

V. EXPERIMENTAL RESULTS

We have performed a physical experiment to evaluate theplanner in a real-world surveillance task that involved theWAM-V USV14, a 14-foot long wave adaptive modularvessel, and a civilian vessel of the same length (see Fig. 4). Inthe experiment, the autonomous USV14 was directed to gothrough a sequence of predefined approach goals (1, 2, and 3in Fig. 5) in a harbor-like environment. While performing thistask, the USV14 had to avoid static obstacle regions and yieldto civilian vessels according to COLREGs. The experimentwas performed under the assumption that the USV14 has

(a)

(b)

Fig. 4: (a) The autonomous USV14; (b) The human-controlled vessel.

perfect knowledge of the locations of the civilian vesselsand static obstacle regions, which we marked using buoys.

The planning algorithm was initialized with 7 uniformlydistributed motion goals of state space trajectories in TI withthe initial execution time te,T = 1 s. The minimum planningtime and distance horizons to select a motion goal wereth,T = 7.5 s and dh,T = 5 m. The cost function parameterswere ω = 0.1, c¬COLREGs = 1000, ccol,soft,T = 100, andccol,hard,T = 1000. The resolution of XG of the precomputedmeta-model m (see Section IV-A) along x and y was 10m and 0.5 m/s for the surge speed of the initial and goalUSV’s states. The terminal goal states were defined on anarc with a radius dh = 100 m and the central angle ofdα = 270 degrees (see Fig. 2). The time difference betweenconsecutive sampled trajectory states was δt = 1 s andδψ = 5 degrees. In the current implementation, the completesense-plan-control cycle took 1.5 seconds on average. Theplanner was implemented in Matlab and converted to C++compiled code. The laptop used for planning was a LenovoThinkPad i5 with 2.5 Ghz and 8GB RAM.

The propulsion system used for the USV14 consists of twoelectric waterjets producing maximum combined total thrustof 205 N. The maximum speed of the USV14 throughoutthe test was kept at 2 m/s. The civilian vessel (see Fig.4b) was used to create “head-on” and “crossing from right”COLREGs situations during the course of the experiment.The vessel is actuated by an outboard 3.7 kW motor whichgives the maximum speed of 3.5 m/s.

We used MTi-G and MTI-G-700 IMU/GPS sensing unitsfor state estimation of the USV14 and the civilian vessel,respectively. These sensing units contain wide area augmen-tation system (WAAS) that enables global positioning (GPS).The position data is internally filtered to give accurate valuesof a boat’s position and speed. The estimated states of boththe boats are shared over RF communication channels usingthe LCM software architecture [23]. The planning systemreceives the combined package of the estimated states of boththe boats over one LCM channel, computes a new motiongoal based on the state estimates, and sends it to the USV14over a different LCM channel.

(a)

(b)

(c)

Fig. 5: An experiment carried out in a harbor environmentwith static obstacles and civilian vessels (GPS coordinatesare projected on top of the map data c©2014 Google).

Fig. 5a) depicts the USV14 has already visited first ap-proach goal and is currently approaching the second ap-proach goal while avoiding static obstacle regions on itsway. In order to ensure collision-free, dynamically feasibleglobal guidance, we used a 4D lattice-based, global trajectoryplanner that considers position, orientation, and speed of thevehicle during the search for the trajectory [3]. The currentapproach goal is changed to the next approach goal whenthe USV14 enters its acceptance region with 10 m radius.

The USV14 encountered a “head-on” situation (see Fig.5b)) with a civilian vessel after it reached its second approachgoal. In the absence of the civilian vessel, the USV14

would directly proceed to the third approach goal in thesouth west corner. Instead, it complied to COLREGs andyielded to the encountered civilian vessel by slightly turningright, while making sure it does not collide with the staticobstacle region. The avoidance maneuver was performed bythe local trajectory planner as described in Section IV. Theswitching between the 4D global trajectory planner and thelocal planner was decided based on the parameters of themaximum time tCPA,max = 50 s and the minimum distancedCPA,min = 60 m to CPA.

Finally, after the “head-on” avoidance maneuver, theUSV14 resumed its task and proceeded towards the thirdapproach goal. On its way, the USV14 encountered anothercivilian vessel in the “crossing-from-right” situation. Fig. 5c)depicts the USV14’s avoidance maneuver with respect to thecivilian vessel. The USV14 had to reduce its speed rapidlyonce it encountered the vessel and turn sharply to the rightto comply with COLREGs.

The use of the USV14’s dynamics and its low level controlin real-time trajectory planning contributed to the successof the designed experiment as the vehicle could quicklyand accurately predict the effect of its own maneuvers forCOLREGs-compliant obstacle avoidance.

VI. CONCLUSIONS AND FUTURE WORKWe have presented a local trajectory planning algorithm

that generalizes the Velocity Obstacle concept to systemswith non-linear dynamics and any low-level controller. Thealgorithm searches in the adaptive space of motion goals fora motion goal that optimizes the time of arrival to a giventarget state. The planner utilizes a meta-model of the USVdynamics and controller in order to make planning possible.The performed physical experiment demonstrated the feasi-bility of utilizing the planner for persistent USV’s operationsin a complex, harbor-like environment with civilian vessels.

In the future, we will further improve the performanceof the algorithm by also sampling along the surge speedand heading state space dimensions. We will also extend thealgorithm to better predict the future trajectories of obstaclevessels to improve the sampling. Finally, we will evaluatethe planner in more complex physical experiments.

ACKNOWLEDGMENTThis work was supported by the U.S. Office of Naval

Research under Grants N00014-11-1-0423 and N00014-12-1-0502, managed by R. Brizzolara and K. Cooper.

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