TRAJECTORY CONTROL FOR AUTONOMOUS PLANETARY ROVERS

Jan Filip1, Martin Azkarate2, and Gianfranco Visentin2

1 Czech Technical University in Prague, Faculty of Electrical Engineering, Czech Republic,E-mail:jan.filip2@gmail.com

2 Automation and Robotics Section, European Space Agency, ESA, Noordwijk, The Netherlands

ABSTRACT

In this paper, a modified pure-pursuit path following al-gorithm is presented. The proposed modification uti-lizes piecewise linear path representation and reductionof lookahead distance based on tracking error. A keysafety requirement in application to planetary rovers isto keep the rover within a safety corridor of the path.The main shortcoming of the original algorithm, cuttingcorners during turns, is addressed through the proposedmodifications.

Presented simulation results show a reduction of pathtracking error of up to 15% compared to the original al-gorithm while respecting the safety corridor.

Key words: Path following, trajectory control, pure-pursuit, planetary rover locomotion control.

1. INTRODUCTION

Path following is a critical control system for automatedoperation of remote planetary missions. With large time-delays in extraterrestrial exploration scenarios, directcontrol of the rover is not feasible. Therefore, the rovermust be capable of executing higher-level commands,such as reaching a given goal position, to carry out thescience interest of the mission safely and efficiently.

A path is comprised of ordered points in the opera-tional Cartesian space (called waypoints) which the robotshould visit. The task of path following consists of steer-ing the vehicle along the path to the goal position. Aparticularly important aspect of path following in plan-etary exploration is the safety of the rover. The path isusually provided by a higher-level layer of the navigationsubsystem, either from an automated path planner or bya human operator, and can be assumed traversable up tocertain lateral deviation from it. This gives rise to the no-tion of nominal path and its safety corridor. The nominalpath is given as piecewise-linear interconnection of thewaypoints, where each line connecting two consecutivewaypoints defines a path segment. The safety corridorof each segment is bounded by two lines parallel to the

path segment line in a fixed distance. Tracking error isdefined as the lateral deviation of the rover position fromthe nearest segment line.

The purpose of this paper is to introduce a modifiedgeometry-based path following algorithm. The algorithmbuilds upon the pure-pursuit follower of [Cou92], whichis a popular path following solution, often considered asa reference. The pure-pursuit algorithm was used for tra-jectory control of cars by successful teams both in theDARPA Grand [UAB+05] and Urban Challenges [PH08,BIS09], as well as for planetary rover trajectory con-trol [HRC+06]. A comprehensive summary of the algo-rithm’s applications can be found in [MMM16].

The introduced c-pursuit, standing for conservative pur-suit, modifies the existing algorithm to add a guarantee ofpath following bounded within a predefined safety cor-ridor along the path, which is a critical safety featurefor planetary exploration missions. The paper presentsanalysis, simulation results and experimental verificationwith rover prototypes in Mars-like terrain. Additionally,we present design guidelines for adjusting the controllerfor similar platforms with respect to their geometric con-straints. The proposed controller is tuned using a singleparameter with an intuitive geometric interpretation, al-lowing for a quick adoption to similar platforms.

2. METHODOLOGY

This section describes the used methodology, namely therepresentation of the path, the model of the rover, the con-cepts of geometry-based path following and the metricsused in the evaluation of path following results.

2.1. Path Representation

We selected to represent the path as a polyline, a series ofline segments interconnecting the waypoints. This repre-sentation allows to define the path either by user-selectedpoints or using grid-based planner nodes directly, with-out any additional processing needed. This format is

user-intuitive and highly efficient in terms of computa-tions. The step of generating a smooth interpolating tra-jectory, which is commonly done in other path controlalgorithms, is not necessary. Instead, the path followingalgorithm utilizes the safety corridor along the path. Thevehicle is allowed to deviate from the nominal path to ex-ecute turns, resulting in smooth transitions to the consec-utive path segments while respecting the bounds of thesafety corridor. The path following error at each con-trol step is evaluated efficiently using the vehicle distancefrom the current segment line. However, the interpola-tion step might still be used if the waypoints are selectedcoarsely. Then, the path represents the piecewise-lineardiscretization of the smooth curve.

2.2. Kinematic Model

The proposed path following algorithm is aimed mainlyat planetary exploration rovers. In this application, thenominal forward velocity is relatively low, on the orderof units of cm/s. Therefore, the dynamics of the motionare negligible, and the problem can be represented usingthe kinematic equations only.

A typical locomotion architecture of planetary roversconsidered is the triple-boogie, which commonly has sixindependently driven wheels. The front and rear wheelscan be steered independently, enabling the rover to ex-ecute both Ackermann turns and turns on the spot. Toexecute a spot turn, the rover must stop first to align itswheels. While spot-turning capability is advantageous inmany scenarios, such as narrow passages or dead ends,Ackermann turns are more efficient for general purposepath tracking as the rover can execute the path withoutstopping, adjusting the steering angles of wheels contin-uously.

The rover is represented using a body-fixed frame at-tached to its kinematic center, aligned with the axis ofthe middle pair of wheels. Since the rover is moving on asurface, only two positional coordinates are independent;hence the xr = [x, y] coordinates are used, determin-ing the position of the rover projected onto the horizontalplane. The heading of the rover θ is given as the angle ofrotation of the body-fixed frame with respect to the worldreference frame. The selection of reference frames andthe steering geometry is illustrated by Fig. 1.

The Ackermann-steering geometry of the rover can besimplified by assuming a 2D kinematic bicycle modelwith an equivalent turn radius rturn. Simplification is doneby representing each pair of wheels by a single wheel lo-cated in the middle, with the steering angle correspondingto the radius of turn which the rover center is executing.Moreover, the rear pair of the wheels is steered symmet-rically as the front pair, and so it can be omitted in thekinematic model. This way, the steering geometry of thesix wheeled triple-boogie is equivalently represented us-ing the well-known kinematic bicycle model.

While executing a circular arc, each wheel is aligned tan-

xw (m)0 0.5 1 1.5

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Figure 1: Reference frames and Ackermann steering ge-ometry of the rover

gentially to a concentric circle (with respect to the cen-ter of the turn) passing through its contact point. Duringpath following, the steering angle is varied. For simplifi-cation, the dynamics of the steering mechanisms are notmodeled, assuming that the response of the steering isfast compared to the variation of steering angle. Suchassumption is valid for this application due to the low ve-locity of the motion execution.

2.3. Geometric-based Following

There are two main path following approaches for vehi-cles with nonholonomic constraints: the geometric-basedand control theory based followers. The proposed algo-rithm for path following is based on the geometric ap-proach and is a modification of the original pure-pursuitfollower introduced in [Cou92]. This method uses ge-ometric relationships between the vehicle and the path,resulting in control law for the path following problem.

A survey comparing various control theory and geome-try based approaches is presented in [Sni09], from whichthe pure-pursuit algorithm comes as the most promisingsolution for slow-moving vehicles in terms of trackingperformance and robustness.

Pure-pursuit type path followers are position-based andvelocity-independent. The controller has no requirementson the path smoothness and is relatively robust to tran-sient disturbances. The disadvantages arising from theneglected system dynamics become significant only atspeeds several orders of magnitude higher than those con-sidered in this application. Therefore, due to slow ve-locity of the rover (on the order of cm/s) and relativelyfast response of the steering mechanism, the shortcom-ings can be neglected, making the pure-pursuit followera viable solution.

The pure-pursuit algorithm solves the path followingproblem of an Ackermann-steered vehicle, producing theradius of the turn (or equivalently the turn curvature) as

the input into the locomotion control subsystem. Themain steps of the algorithm are

1. Determine the position of the vehicle xr with respectto the nominal path.

2. Determine a lookahead point xlh ahead on the path.

3. Calculate the radius rturn required to steer the vehiclefrom xr to xlh.

This approach uses the notion of lookahead point andlookahead distance. The lookahead point is a virtualpoint located on the path ahead of the vehicle. The ve-hicle is instantaneously steered toward this virtual point.As the vehicle moves, the lookahead point also shifts for-ward, hence the name pure-pursuit. The lookahead pointis uniquely determined by the path and the current po-sition. A single scalar parameter is used to parametrizehow far ahead is the lookahead point placed: the looka-head distance.

2.4. Performance evaluation

The immediate path tracking error is evaluated as the dis-tance from the line of the current path segment. Theperformance of different algorithms or parameters settingwas evaluated using two criteria. Firstly, it is required thatthe vehicle stays within the safety corridor, which has abinary outcome. Secondly, the root mean square track-ing error is evaluated to account for both the mean andthe variability of the error, penalizing larger deviationsmore. Lastly, the improvement of the proposed algorithmis evaluated as

I = 100eRMS,A − eRMS,B

eRMS,A(%), (2.1)

where eRMS is the RMS of path tracking error of the orig-inal (A) resp. the proposed (B) algorithm.

3. PROPOSED ALGORITHM

This section describes the modifications to the originalpure-pursuit algorithm described in [Cou92, Sni09].

3.1. Position along the path

To determine the vehicle position and progress along thepath in each iteration, the current n-th path segment (con-sisting of waypoints wn−1,wn) is kept in memory, aswell as the precalculated unit line vector sn of n-th seg-ment

dn = ‖wn −wn−1‖2 , (3.1)

sn =wn −wn−1

dn. (3.2)

The position with respect to the path is determined as per-pendicular projection xi of the current position xr into thepath segment line

ks = (xr −wn−1) · (sn), (3.3)xi = wn−1 + kssn. (3.4)

To determine the progress along the segment and the tran-sition into the subsequent one, distance ks is comparedwith current segment length dn. The distance from thecurrent path segment line determines the tracking error ε

ε = ‖xr − xi‖2 = |s⊥n ·xr|. (3.5)

The polyline path representation allows for a simple andcomputationally efficient evaluation of the tracking error.

3.2. Adaptive lookahead distance

Given the position xr, the location of the lookaheadpoint xlh is determined uniquely and parametrized by thelookahead distance dlh only. Our approach to lookaheaddistance adaptation was inspired by [Kel97], where thedistance was increased by the lateral deviation ε from thepath, in order to slow down the convergence and preventovershooting. The aim was to use the pure-pursuit algo-rithm for vehicles with non-negligible motion dynamics.Lookahead distance increasing was applied in [G+05] fora fast-moving all-wheel-drive military vehicle.

However, in the planetary robotics application, the rovervelocity is low and fast convergence with the nominalpath is the main path following objective. Therefore,lookahead distance reduction was investigated. In ourapproach, the lookahead distance modification is gener-alized to unify both schemes

dlh,rd = dlh − kε ε, (3.6)

where dlh is the initial setting of the lookahead distanceand kε is the gain of the deviation penalization. In thispaper, reduction (kε > 0) of the lookahead distance isfurther investigated, in order to prevent cutting cornersand to guarantee corridor-bounded path following.

With lower values of lookahead distance, the vehicle issteered more towards the current position on the nominalpath rather than towards the path heading, speeding upthe convergence with the path. However, this can also re-sult in steering angle saturation and path intersection at alarge heading error. While this approach may not be suit-able for car-like vehicles with non-negligible motion dy-namics, it is well suited for rovers capable of point-turns.In the rare cases of sharp turns where the commandedarc curvature exceeds significantly the steering limits, apoint turn can be commanded to realign the rover with theheading towards the lookahead point and the path. Thisprevents overshooting, improves the convergence withthe nominal path and helps satisfy the corridor-boundedtracking at the cost of stopping to align the wheels forspot turn. While this maneuver is useful in practice, itis not further developed in the theoretical analysis of thealgorithm.

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W (m)

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xy

dlh

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xr

wn-1

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xlh

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Figure 2: Lookahead point xlh calculation at position xrwith dlh = 0.6 m, kε = 1 and ε = 0.15 m

3.3. Lookahead point search

The original algorithm searched the lookahead point xlhas a point satisfying

‖xlh − xr‖2 = dlh (3.7)

and maximizing the progress along the path, by evaluat-ing intersections of a circle with the path. Although thisapproach was applied previously in a planetary roboticsapplication [HRC+06], it has a disadvantageous tendencyto cut corners while executing sharper turns (angle be-tween segment lines γ > 45◦), which is a safety concern.

In the proposed adaptive lookahead approach, the looka-head point is found by moving distance dlh,rd of Eq. (3.6)along the path from the projected position xi. If thedistance from xi to the end of the segment is not suffi-cient, following segment(s) are used, summing the dis-tance along the segment lines until dlh,rd is reached. Thecalculation of lookahead point is illustrated in Fig. 2 withdlh = 0.6 m and ε = 0.15 m. Yellow arrow showsthe vector of the desired heading towards the lookaheadpoint. For the given heading, this translates to a circu-lar arc rturn drawn in brown color. As the vehicle wouldmove and the lookahead point would shift along the path,this control scheme would result in the trajectory shownin gray, which gradually converges back towards the pathwhile progressing along it.

3.4. Turn Radius Calculation

Using the current position and the lookahead point, turnradius is calculated. The radius is determined by inter-connecting points xr,xlh with a circular arc. The centerof the circle is located on the vehicles yb-axis so that thearc is tangential to the current vehicle heading θ. Thecalculation in the body frame is given as

xlh,b = R>z,θ(xlh − xr), (3.8)

rturn =x2lh,b + y2lh,b

2ylh,b, (3.9)

where R>z,θ denotes the rotation from world to body-fixed reference frame. In the original algorithm, the com-manded arc curvature κ = 1

rturnis proportional to the

the lateral error between the current vehicle heading vec-tor and the lookahead point ylh,b, since the numeratorof Eq. (3.9) was the constant lookahead distance. There-fore, the pure-pursuit path follower has the properties of aP controller in terms of ylh,b → κ, as discussed in [Sni09],with the P gain being inversely proportional to the looka-head distance dlh. In the proposed algorithm, the sameapplies, only the P gain is no longer constant, but adjustedwith the tracking error ε.

4. ANALYSIS

The assignment of lookahead point to each position canbe visualized as a vector field, where the local directioncorresponds to the direction vector to the assigned looka-head point – recalling the yellow arrow of Fig. 2. Theactual motion direction of the vehicle may be differentas it is heading dependent. However, if the vehicle isaligned with the vector field, its instantaneous motioncorresponds to the heading vector. Moreover, the vehi-cle is gradually aligned with the field as a result of thecontrol. Therefore, the vector field represents the controlstrategy and can locally approximate the resulting trajec-tory.

Fig. 3a resp. 3b shows the behaviour of the original pure-pursuit algorithm for increasing angle of turn γ. Whilethe approaches are initially similar, the difference occursat the transition to the next segment. The original ap-proach commands the vehicle to cut the corner, directingit outside of the safety corridor, which is not acceptable inour mission scenario. Decreasing the constant dlh wouldreduce this problem, but at the cost of making the trackingresponse too sensitive to small heading changes, causingoscillatory tracking response.

On the other hand, the proposed modification maintainsthe rover trajectory at all positions inside of the safetycorridor, as shown in Fig. 3c. The vector field stream-lines do not exit the safety corridor, and the vehicle isnever commanded outside of it. The commanded direc-tion towards the lookahead point is directed inwards thesafety corridor up to

γ ≤ 90◦ and dlh ≤3

2dcorr, (4.1)

which is in the limit case shown using arrows in Fig. 3c,decomposing the distances in the same way as in Fig. 2.For this case, the direction to lookahead point is incidentwith the safety corridor edge. The proposed modifica-tion satisfies the requirement of corridor-bounded track-ing while the original algorithm does not give such guar-antee.

5. SIMULATION

The path following algorithms were compared in simula-tion using kinematic bicycle model implemented in Mat-

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(a) original pure-pursuit algorithm, γ = 45◦ between segments

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(b) original pure-pursuit algorithm, γ = 90◦ between segments

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(c) proposed modification, γ = 90◦ between segments

Figure 3: Vector fields of direction vector to lookaheadpoint xlh from all possible xr inside the safety corridor

lab. As a benchmark, paths consisting of five segmentswith following angles γ were selected

γ = γ [1 −1 −1 1] . (5.1)

The simulation parameters are summarized in Tab. 1. Anexample of path with γ = 90◦ is shown in Fig. 4a, em-ulating an obstacle avoiding maneuver. The proposed al-gorithm follows the nominal path with lower deviationand successfully maintains the safety corridor. On theother hand, a violation of corridor bounds occurs twicein the case of the original algorithm with the same dlhlookahead distance setting. The path tracking error ε isdepicted in Fig. 4b, indicating a lower mean error andlower peak deviation of the proposed algorithm. The twomaxima of the original algorithm correspond to the cutcorners. The same experiment was repeated for differ-ent values of γ and the results are summarized in Tab. 2.Overall, the performance difference becomes significantas the angle between segments increases above γ > 45◦.Using the modified approach, corridor-bounded trackingis satisfied, path tracking error is reduced, and computa-tional performance of the algorithm is improved. Theseresults are in correspondence with the analysis of Fig. 3.

Table 1: Simulation parameters

Parameter Set using Value

Min. turn radius rmin given 0.6 m

Lookahead distance dlh32rmin 0.9 m

Safety corridor width dcorr23dlh 0.6 m

Deviation penalization kd – 1Path segment length – 2 m

0 1 2 3 4 5 6x (m)

0

0.5

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)

Simulation of path following algorithms

xy

xy

xy

xy

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(a) simulated trajectories

0 20 40 60 80 100 120 140 160time (s)

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ral d

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originalmodified

(b) lateral error from the closest path segment

Figure 4: Comparison of path tracking algorithms, simu-lated on path with 90◦ between consecutive segments

Table 2: Simulation results

γ(◦) Mean ε (mm) RMS ε (mm) Improvedorig. mod. orig. mod. (%)

30 29.30 27.74 37.96 35.47 6.5545 43.79 40.36 56.63 50.72 10.4360 58.23 52.39 75.23 64.50 14.2690 86.19 79.06 111.39 93.87 15.73

6. TESTING

The proposed path following algorithm was integratedinto the software of two rover platforms and tested inMars-like terrain conditions. The implementation ofproposed path following algorithm is available as opensource in the repository [F+16].

Figure 5: ExoMars Test Rover (ExoTeR) platform in theterrain of Planetary Robotics Laboratory, ESTEC

Figure 6: Measured trajectory of path following experi-ments with ExoTeR platform in PRL using planner-basedpath

6.1. Planner-based paths

The first experiment was conducted with the ExoTeRplatform, which is shown in Fig. 5, in the terrain of thePlanetary Robotics Laboratory in ESTEC. The path wasgenerated onboard using AD* planner operating on thetraversability map of the terrain, thus planning the pathto avoid obstacles and excessive slopes. Vicon motiontracking system was used for rover localization during theexperiment.

The results are depicted in Fig. 6, overlapping the pathand the trajectory with the traversability map of the ter-rain from the top view. The modified algorithm managedto steer the rover along the nominal path in the terrain ofmoderate difficulty, maintaining the deviation from thenominal within the safety corridor ±20 cm. Moreover,the deviation from the nominal path was below 15 cmduring the whole experiment, and the goal positions werereached successfully. This experiment demonstrates theperformance of the proposed algorithm and integrationwith a grid-based planner outputting the path as a list ofnodes. In the presented application, the path can be used

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10 20 30 40 50y

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Outdoor field test with HDPR rover using DGPS localization

rock

crater

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goal

Figure 7: Measured trajectory of outdoor path follow-ing experiments with HDPR platform using user-definedGPS waypoints

directly with no need for further processing.

6.2. User-defined waypoints

A path consisting of ordered waypoints interconnected bystraight lines offers a user-intuitive interface and can beconveniently defined manually. Moreover, the notion ofthe safety corridor is easy to interpret, visualize and givesthe operator a clear indicator of the safety of a candidatepath.

Path following of user-defined GPS waypoints wasdemonstrated using HDPR platform [BH+16] in outdoorenvironment. HDPR platform is a full-sized rover. Dur-ing the experiment, its velocity was set to 30 cm s−1 andthe minimum turn radius restricted to 1 m. The exper-iment was conducted at an outdoor test site with Mars-analogous terrain in the Netherlands, around the craterlocated at 52.215930N, 4.426796E, the same test site asdepicted in [BH+16]. Test results are shown in Fig. 7.The rover was commanded to traverse the terrain aroundthe crater using several user-defined GPS waypoints. Al-though the waypoints were coarsely spaced, forming anon-smooth path, the traverse was successful. The devi-ation from the nominal path was minimal with the onlyexception of turns. Turns were executed successfully,maintaining the safety corridor. Overall, the algorithmbehavior was reliable and predictable, making it a highlyuseful tool in automating the field tests with rovers.

7. TUNING GUIDELINES

This section presents the tuning guidelines of the pro-posed algorithm, based both on simulation analysis andlaboratory tests, to provide tips for adaptation of the al-gorithm on similar locomotion platforms. If Ackermannsteering geometry is applicable, the minimum radius ofturn is the performance constraint, and other controllerparameters can be derived from its value.

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lh = 0.45m, = 22°

dlh

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xr

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permissable area

Figure 8: Minimum turn radius as the limitation on usablelookahead distance value

Fig. 8 shows the impact of the lookahead point Eu-clidean distance from the vehicle for minimum turn ra-dius rturn,min

.= 0.6 m. The sector (the right-hand side

from the black circles) for which an executable arc existsshrinks rapidly with decreasing distance. In order to trackpaths varying in direction, the algorithm must allow a suf-ficient ∆θ ≈ ylh,b

xlh,bso that saturation of the commanded

turn radius is mostly avoided. Therefore, a lookaheaddistance equivalent to the minimum turn radius is recom-mended as a good starting point for controller tuning. Itallows a change of heading to the lookahead point up to30◦ before reaching turn radius saturation. For lower val-ues the response is too aggressive, causing oscillations.

Based on our simulations and experimental results, itis advisable to increase the lookahead distance slightlyabove the minimum setting, in the range of

dlh ∈ 〈rturn,min, 1.5 rturn,min〉. (7.1)

Increasing the lookahead distance slightly improves themargin before reaching the steering command saturation.Using a setting above the minimum is advisable since theeffective lookahead distance is reduced using Eq. (3.6).Staying within the linear range of the steering commandavoids overshooting and improves the path tracking be-haviour.

On the other hand, the minimum safety corridor, whichthe algorithm can guarantee, scales with the selectedlookahead distance. Based on the result of Eq. (4.1)and Fig. 3c, the width of the corridor should be set to

dcorridor = 23dlh, (7.2)

which allows path tracking error up to ± 13dlh. This set-

ting guarantees the corridor-bounded tracking under no-slip assumption.

8. CONCLUSION

In this paper, a modification to the popular path follow-ing pure-pursuit algorithm was proposed. The modifica-tion utilizes piecewise-linear path representation for effi-cient lookahead point and tracking error calculations. Ascheme for reducing the lookahead distance based on the

current tracking error was developed. Using these twonovel ideas, the executed trajectory is bounded within aguaranteed safety corridor along the path. The results ofthe modified algorithm were successfully demonstratedboth in simulations and in laboratory experiments withtwo rover platforms in Mars-analogous terrain.

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