NASA Technical Reports Server (NTRS) 20000054886: A Comparison of Two Path Planners for Planetary Rovers

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by  NASA Technical Reports Server (NTRS)

A Comparison of Two Path Planners for Planetary Rovers
M. Tarokh t, Z- Shiller2 and S. Hayati 3
1 Robotics and Intelligent Systems Laboratory
San Diego State University, San Diego, CA 92182-7720
2 Dept of Mechanical and Aerospace Engineering
University of California, Los Angeles, CA 90095
3 Jet Propulsion Laboratory
California Institute of Technology, Pasadena, CA 91109
Abstract
many caseswhere abinary obstaclemodel hasresulted
The paper presents two path planners suitable for plan- inhaltedmotions,oftenleavingtheroverinan undesir-
etary rovers. The first is based on f_,zy description of
ablesituation[3].Recently severalpath plannershave
the terrain, and genetic algorithm tofind a traversable
been developed thatconsiderthe traversabilitoyf the
path in a rugged terrain. The second planner uses a
global optimization method with a cost f_nction that is terrain[4]-[7]T.erraintopology and simple vehicledy-
the path distance divided by the velocity limit obtained narnicsareconsideredin[4]togenerateglobaloptimal
from the consideration of the rover static and dynamic paths on generalterrain.In [5]the shortestfeasible
stability. A description of both methods is provided, path foroff-roadvehiclesiscomputed. A geneticalgo-
and the results of paths produced are given which show
rithm isused in[6]to synthesizepath from segments,
the effectiveness of the path planners in finding near
eachevaluatedforitsstaticstabilityand forsatisfying
optimal paths. The features of the methods and their
suitabilit_ and application .for rover path planning are certainmissiontasks.A recentlydeveloped planner[7]
comparea. usesfuzzylogicto characterizetheterraintraversabil-
ity,and thenfindstraversablepaths ina rockyterrain.
The purposeofthispaper istodiscusstwo pathplan-
nersforpossibleMars roverapplications.The firstal-
1 Introduction
gorithm isbased on fuzzy characterizationofthe ter-
Following the successful launch and deployment of rainroughness,and theuse ofageneticplannertoop-
Mars Sojourner rover, NASA has planned further rover timizea fitnessfunction. The second algorithmcon-
missions to Mars starting in 2001 "with Marie Curie, sidersconstraintsimposed by certainvehicledynam-
a rover similar to the Sojourner. Two additional rover icsand terraintopology to come up with an optimal
missions in 2003 and 2005 have been planned for in-situ path. The common featureofboth plannersisfinding
experiments, and another in 2007 for sample return to paths thatareoptimal in the senseof both distances
Earth. An important element for the success of these and traversabilitwyh,ere the latterquantifiestheease
missions is incorporating a reasonably high level of au- of traversalofthe terrain.These two algorithmsfind
tonomy in the rover so that it can traverse distances of paths thatresultinreduced roverenergy consumption
100 meters or more per communication cycle. In order and enableexploringlargerregionsoftheMartian ter-
to traverse these distances, it is necessary to delegate rain.
the motion planning task to the rover using the image
obtained from mast mounted cameras. The challenge 2 Genetic Path Planner
is then to use these images to perform on-board path The path planner starts by creating several random
planning.
paths between start and goal points on the terrain.
The existing path planners focus almost exclusively These initial paths in general go though rough or im-
on obstacle avoidance, treating obstacles as forbidden passable regions on the terrain, and must be improved.
regions and the rest of the terrain as free spaces [1]. This improvement is achieved by applying certain ge-
This binary environment is not appropriate for the netic operators to a randomly selected path from the
Martian terrain and a rover that can climb over some
population. Each genetic operator has a particular role
rocks [2] if such traversals result in more optimal routes. in bringing about a change in the path. For example,
In fact NASA's experience with Sojourner has revealed replace operator replaces an undesirable way-point (a
.way-point on a rough region), with a random and po-
Se and Pe are used to quantify the linguistic state-
tentially better way-point. The selection of particular
ments "h is He", "._ is Se" and "_ is Pa",
operator is based on the probability assigned to it. Af-
respectively. The fuzzy sets HA for the hight are
ter a genetic operation is performed, the quality of all
chosen as very low (Hi = VL), low (1"12 =-" LO),
paths are compared, and the worst path is eliminated
medium (H3 = ME), high (Ha = HI) and very high
from the population. The process of applying a ge-
(Ha =-"VH). The membership functions/_tt_ for these
netic operator to create a new path, and eliminating
fuzzy sets are standard triangular and have equal base
the worst path, is referred to as a generation. The pop.
width with a 25% overlap. The fuzzy sets associated
ulation goes through generations and is thus evolved.
with the rock size are tiny ($1 = TI), small (5'2 ---SM),
After each generation, the quality of the paths is either
medium ($3 = ME), large ($4 - LG) and extra large
improved or in the worst case remain unchanged. The
(8s =- XL), and are also triangular with 25% overlap.
evolution is continued until an acceptable path is found,
The fuzzy sets for roughness are very low (Pl = VL),
or until a preset number of generations are performed.
low (P2 = LO), medium (P3 = ME), high (P4 = HI)
2.1 Terrain Roughness
and very high (P5 = VH). The membership functions
Consider a terraindivided into a grid of regular #Pk for the roughness are designed to be triangular
squarecellswhose sizedepends on thedimension ofthe
with different base widths to give more weighting to
rover,and thedesiredresolutionofsurfacedescription. rougher terrains.
The roughness of a fiat obstacle free cell is assigned a The rule matrix implementing (1) is given in Figure
value of 0, and that of a rugged cell with large obsta- 1, and consists of 25 rules which are self-explanatory.
cles is assigned a value of 1. The measure of roughness Zadeh's compositional rule of inference, and center of
depends on a number of parameters as follows:
height defuzzification method is used to obtain the crisp
• Height of the tallest obstacle in the cell - The rough- value of the cell roughness p.
ness becomes smaller with a decrease in the rock height.
2.2 Path Representation
• Size or surface area of the cell occupied by obstacles
or rocks - If two cells have rocks of the same height, the A path is represented by a sequence of way-points
region with less rock occupied area is smoother and connecting the start to the goal. The way-points Wt,
thus has a lower roughness value. k = 1,2,-.-,m arespecifiedby their(xh,ye) coordi-
nateson the terrain.The generationand evolutionof
In addition to roughness, two path dependent quan-
tities, namely path slope and curvature, affect the dif- a path refersto the creationand modificationof the
ficulty of the traversal by a rover. These will be con- way-points. These way-points inturn specifythe ter-
sidered in Section 2.2. raincellsthatthe path traversesover. A ceilthat is
The most commonly used sensors for mobile robots locatedon a path,willbe referredtoasa pathcell,and
has two main attributesas follows:
are cameras and their associated image processing
hardware and software. Despite the availability of vi- • The roughnessp,of thecell,which providesinfor-
sion processing software, exact determination of the mation on theheights,sizesand concentrationofrocks
on a cell,asdescribedinSection2.1.
heights and sizes of rocks affecting roughness is not pos-
sible. These parameters can be found, at best, approxi- • The curvatureorjaggedness ofa path cellisob-
mately due to errors, misinterpretations and ambiguity tained using the information about the way-points.
involved in extracting information from images. It is Specifically, the curvature _h of the way-point Wh is
defined as
therefore essential to set the problem in a fuzzy and
approximate reasoning framework.
The height of the tallest rock in the cell under con- _e = d---Lk k = 1,2,3-.-,m (2)
De
sideration,-/r, and the size or surface area occupied by
rocks in this cell, s, are used to find the cell roughness where de is the perpendicular distance of W_ to the
p. The crisp values of h, s and p are fuzzified to obtain
line segment joining the previous way-point We-t to
the linguistic variables h, ] and _, respectively. The "if- the next way-point We+t, and Dt is the distance be-
then rule" of the following form is employed to obtain tween We-1 and Wk+t. Note that _ is a dimensionless
the fuzzy roughness,
quantity, and that 0 < _e < oo. Furthermore, (2) also
gives the curvature of the path cell that contains a way-
if h is file and _ is Se then _ is He (1) point.
It is noted from Section 2.1 that roughness is normal-
where/z/e, ._ and He, k = 1,2,..., v are the linguistic
ized and varies between 0 and 1. However, curvature
values associated with h, ] and _, respectively, and v
can have large values. In order to enable easy com-
is the number of linguistic values. The fuzzy sets H_,
parison between the two cell attributes, we normalize
curvatureasfollows:
appliedto way-points,and as a resultsofchanges in
C - t- e-_¢, (3)
way-points,thepath cellsare alsochanged. Note that
each time an operatorisapplied,a new path isgener-
where a is a constant whose role will be explained
ated.Ifthisnew path produces a path impedance that
shortly. Note that 0 < ¢i < 1 for all values of _i.
islowerthan the impedance ofany path inthepopu-
The above two quantities, namely roughness and cur-
lation,itisacceptedas a new member ofpopulation,
vature, which are attributes of path cells, are combined
and thepath with highestimpedance isdiscarded.
to define a cell impedance rh as follows
Cross-Over
1
_ = 5(p_+ ¢_) (4) This operatorrandomly selectstwo paths from the
population,sayPl and P_,and divideseachpath into
The ceil impedance varies between 0 and 1 and quan- two pathsegments about arandomly electedway-point.
tities the difficulty of the path cell traversal by a rover. Denoting these paths by PI -- (Pn,PI2) and P_ -_
Consequently, a path cell containing no rocks that is lo- (P2t, P2_), where Pij is the j-th segment of path i, then
cated on a straight path segment will have a minimum two new paths are formed as/31 _- (Pn, P2_) and/32
(P21, Pl2 ). --
impedance of 0. On the other hand a very rough cell on
a jagged path segment will have a maximum impedance Mutate
of 1. The constant a in (3) determines the weight given
to curvature relative to the roughness. Lower values of This operator randomly selects a path and a way-
a reduce the contribution of curvature to the overall cell point in this path. It then changes the z, y coordinates
impedance. It is noted that other path attributes such of the selected way-point with random values. Mutate
as slope can easily be included in the above formulation operator can produce a significant change in the path.
of the path impedance.
Replace
A cell with an impedance of more than a threshold
This operator is applied to an intraversable path. It
becomes intraversable. The value of the threshold is
replaces an intraversable way-point with one or more
chosen based On the mobility characteristics of the par-
way-points whose location and number are random. If
ticular rover being used. We identify a path as being
there are more than one intraversable way-points, one
traversable if every celis on the path is traversable, oth-
of them is selected randomly for replacement.
erwise the whole path becomes intraversable. In the ge-
Swap
netic evolutionary process, these two type of paths are
treated separately. Although, traversable paths have
The operator interchanges the locations of two ran-
priority over intraversable paths, the latter are not au-
domly selected way-points on a randomly selected path.
tomaticaUy discarded since they may prove to produce The swap operator can be applied to both traversable
good offsprings later on during the' evolutionary pro- and intraversable paths. It has the possibility of re-
cess. The path impedance is defined as the sum of moving or introducing a "zig-zag".
impedances of all cells on the path, that is
Smooth
' = (5) The role of this operator is to reduce sharp turns.
The way-point with the highest curvature, say Wk, is
k_
selected and two new way-points are inserted, one on
When a population of paths consisting of both a randomly selected cell between the way-points W__ z
travers_able .and intraversable paths are compared for and Wk and the other on a cell between Wk and Wk+l.
selection, any traversable path is given preference over After this insertion, the way-point W_ is removed. The
best (lowest r/) intraversable path. However, when the effect of this operation is the smoothing of a sharp turn.
population consists of only traversable paths or only in- This operator is only applied to traversable paths.
traversable paths, then the selection is based on lower Pull-out
valuesofr/.
2.3 Genetic Operators This operator is intended to pull out a path segment
from inside an intraversable region to its surrounding
In orderto evolvepaths from one generationto the
traversable region. Pull-out is more elaborate than the
next,severaloperatorshave been devised.Two ofthese
other operators, and details of its implementation is
operators,namely crossover and mutation, are com-
omitted here for the sake of brevity.
monly used in geneticalgorithms. Others are specif-
icallydesigned for the path planner. Operators are The probability of occurrence of an operator depends
on the role played by it in the evolution of paths. An
.adaptation scheme is devised to modify the probabili-
Reducing the v - w space to "a line reduces the B
ties based on the population diversity, and traversabil-
ity. For example, if most paths in the population are patch to a continuous curve that is guaranteed to stay
on the surface.
similar and have high impedances, mutation is given
3.2 Vehicle Model
higher probability and cross over is assigned a smaller
probability. This is due to the fact that in this'situa- At top speeds of I0 - 20 cm/s, the motion planning
tion cross over of intraversable paths also produce other problem for Mars Rover can be considered a kinematic
intraversable paths and a substantial change is needed problem. However, we do account for certain rover dy-
which is achieved by mutation.
namics for the purpose of quantifying traversability and
dynamic stability, with the premise that paths that axe
3 The Global Optimization Planner
traversable at a wide speed range are safer than those
This planner formulates the motion planning prob- that ate not.
lem as a three stage optimization. At the lowest level, a
The vehicle is modeled as a point mass, suspended
given path is evaluated for its traversability by comput-
above ground at the location of the vehicle's center of
ing the maximum speeds along the path at which the
mass. The height of the center of mass above ground
vehicle is dynamically stable. The second level consists
and the width between the wheels are used to evaluate
of a parameter optimization that selects a locally opti-
stability with respect to lateral tip over.
mal path in the neighborhood of an initial guess. The
The external forces acting on the vehicle consist of
third and highest level of the optimization selects the
the friction force F (the sum of all the horizontal tire
initial guesses for the local optimization. The global op-
forces), the normal force R (the sum of all normal tire
timization is based on a branch and bound search that
forces) applied by ground on the vehicle in the r direc-
prunes the initial set of all paths between the end points tion, and the gravity force.
to a small number of candidates for the local optimiza-
The equation of motion of the vehicle are written in
tion [4]. These candidates represent the most promis-
the vehicle fixed frame in terms of the tangential speed
ing regions, one of which contains the global optimal
.4and the tangential acceleration//[4]
path. Optimizing these paths with the local optimiza-
tion yields the best path, in addition to a number of
ft = mgkt + m_ (8)
good alternatives. These paths are not necessarily the
fq = mgkq + mlcnq_ 2 (9)
shortest, but they are traversable at the widest speed
range of all paths with similar or shorter lengths, as is R = mgkr+mtcn,$2 (10)
demonstrated inseveralexamples inthispaper.
.......... and fqarethecomponents ofthefrictionforce
3.1 Terrain and Path Representation tangent and normal to thepath,kt,kq and kr arethe
The terrainisrepresentedby a cubicB patch,which projectionoftheverticalunitvector,k,on therespec-
isa parametricsurfacemade ofa mesh ofcubicsplines. tiveaxisofthevehiclefixedcoordinateframe,and 1/s
A typicalpointp on asinglepatch inthreedimensional isthe path curvature,. The moment of the friction
spaceisa functionoftwo Parameters,v and w, : forcearound thecenterofmass isconsideredlaterwhen
we account forthetipover constraint.
P = VMRMTwT (6)
Equations (8)to (10)areused todetermine the fea-
where V = [v3,v2,v, 1], v = [0, 1], W = [w3,w2,w,i], siblespeedand accelerationforgivenlimitson thefric-
tionand normal forces.
w = [0, 1] M is the 4 x 4 matrix specifying the type
of spline used to construct the patch, and R is a 4 x 4 3.3 Dynamic Constraints
matrix of 16 control points.
Constraints between the vehicle and ground are con-
The control points of the patch axe generated by plac-
sidered to ensure vehicle dynamic stability along the
ing a dr_if6rm grid on the map-range data generated path.
from stereo images taken by the on-board mast cam-
Sliding Constraint
era. The resolution of this grid is chosen economically
at about half the rover size: roughly 20cm between The maximum frictionforceisa functionofthenor-
neighboring points. This ensures that obstacles the size realforceand the coefficientof frictionbetween the
wheels and ground:
of the rover and larger are depicted by the B-patch.
Smaller obstacles may be filtered out.
The path is represented by a smooth curve on the IFI_ _R (11)
surface, obtained by parameterizing v and w by a single
parameter u: Substituting(8)-(10)in(11),then solvingfor_ yields
constraintsoftheform [4]
c(u) = p(v(u), w(u)) = V(u)MRMrWr(u ) (7)
-g_,+ v_ _<_< -gk,- v_ (12)
• wh_re
The costfunctionforMars roveriscomputed by di-
A = a._4 + 2bs2 + c_> 0 (t3)
vidingthepath lengthby themaximum constantspeed
yields constraints on the feasible vehicle speed along thatdoes not crossthe velocitylimitsforthatpath.
the path. The feasible speed range is determined by the This costfunctionistheminimum motion timeat the
roots of (13). Only the positive roots are of interest. constantspeed alongthepath. Itquantifiesthecumu-
Contact Constraint lativeeffectsofpath distance,terraintopography, and
vehicledynamics. Italsofavorsregionswith high ve-
To ensurethatthevehicledoesnotloosecontactwith
locitylimits,which aretraversableatthe widestspeed
ground on rough terrain,thenormal forceR appliedon range.
the vehicleshould be positive._SettingR = 0 in (10),
The optimizationstartsby searchingfora setofbest
we obtain themaximum speed allowedby the contact
constraint: paths alonga uniform gridover the terrain,usingthe
Dryfus algorithm.These pathsarepruned by retaining
_ < __/-_'_k_. thebestpath ineach neighborhood, each representing
- v _:nr (14)
the neighborhood ofa potentiallocalminimum. Sub-
where nr is the projection of the path normal, n, on mittingthesepathstoa localoptimizationthatfurther
the surfacenormal, r. Equation (14)appliesonly for minimizes the costfunctionyieldsthe globaloptimal
the caseswhere path curvaturepointsoppositeto the path in addition to a set of good alternatives. This op-
directionofthesurfacenormal. Note thatthevelocity timization, admits paths that might go over obstacles if
limitisinfinitefora fiatterrain(n_= 0),and zerofor such a path isdynamically feasibleand itislesscostly
than goingaround.
a sharp verticalbump (_:n,= co),asexpected.
Tip--Over Constraint
4 Comparison of Results
The tip-overconstraintisobtainedby expressingthe
The two planners were tested on images obtained
limitingconditionbefore the vehicleisabout to tip-
from the JPL Mars Yard. The images were electroni-
overin terms of s,_/.The vehiclewillnot tip-overif
cally manipulated to make the terrain more challeng-
t[h6e]reactionforceand the lateralfrictionforcesatisfy ing by adding large rocks in the central region. A
monochrome version of the color image used for path
1,-<(Rb)2 (15) planning is shown in Fig. 2.
In the absence of stereo images, the apparent rock
Substituting(8)and (9)into(15)yieldsa constrainton height and size were determined from a single image
similarto (13). based on several assumptions on camera location and
Velocity Limit Curve geometry. The height is estimated by multiplying the
apparent height by a correction factor derived from per-
Plottingthe velocitylimitsdue tathedynamic con-
spective transformation. Similarly the size of a rock is
straints along the path formsthe velocity Hmit curve in
estimated from its apparent boundary by subjecting it
the phase plane s- _. It represents the upper bound for
to perspective transformation. The the number of pix-
vehicle speeds for which the dynamic constraints dis-
eis within the perspectively corrected boundary is then
cussed earlier are satisfied. The height of the velocity
found, giving the size (area) of the rock. A contour'map
limit represents a measure of safety and traversability: is then constructed on the basis of location, height and
a zero velocity limit implies static instability, whereas a size of each obstacle. The contour map of the Mars
nonzero but low velocity limit implies a stable but dan- terrain (Fig. 2) is shown in Fig. 3, where darker areas
gerous position along the path. Obviously, the higher correspond to higher elevations. This contour map was
the vel0dity Iinait, the wider the speed range that the used by both path planner.
vehicle can move along the path without sliding, tip-
For the genetic planner, the 512 x 512 pixel image
ping over, or flying off the ground.
representing a 10 square meter region was divided into
3.4 Global Search and Local Optimization 32 x 32 cells. The number of cells can be increased
The search for the optimal path follows the method for higher resolution, if required. The impedance of
each cell was determined using the method described
presented in [4]. It combines a grid search in the posi-
in Section 2.1. A population size of five paths was cho-
tion space with a local optimization to yield the global
optimal path for a variety of static and dynamic cost sen, and these paths went through the genetic evolu-
tion described in Section 2. The initial intraversable
functions, such as distance and motion time. This
approach eliminates the search in the 2n dimensional paths were quickly evolved into traversable paths, and
state-space without sacrificing global optimality. as the evolution continued these paths in turn changed
into shorter ones passing through less rock concentrated
Height
VL L0 ME H I VI:I
TI VL VL LO ME VH
SM VL VL LO ME VH
ME VL LO LO LO VH
(/3
LG VL LO ME LO VH
XL VL LO ME LO VH
Fig. 1. The fuzzyrule matrix. The entries are
terrain roughness
Fig. 3.
Three paths found by the genetic path planner
shownon the contour map (Smoothing not
applied to the paths)
l=,g2 A reconstructed Mars =mage
Fig. 4. Three paths found by the global optimization
planner shown on Conlour map
• areas and avoiding larger rocks. Near optimal paths
duced by both planners are generally longer than the
were usually found after 200 to 400 iterations (genera-
shortest paths between respective end points (Fig. 3
tions), thus good paths were found very quickly. Figure
and 4) but they seem to pass mostly through wider
3 shows three path generated by the genetic algorithm.
corridors and hence are safer.
Path I starts at the left part of the region near a rock
and the goal position is located to the right of the re- 5 Conclusions
gion at the base of a large rock. Path 2 starts at the The path planners described in this paper share the
lower left corner and has the same goal location as Path common attribute of attempting to optimize certain
1. Path 3starts at the upper left corner and has its goal performance indices. It has been shown through plan-
location in the lower center of the region. ning of paths for a simulated Mars terrain that both
The global planner uses the contour map directly, are capable of producing short paths that traverse over
and performs the optimization method described in smooth parts of the terrain and avoid areas with large
Section 2. Figure 4shows the paths found by the global rocks. While both planners perform some form of opti-
planner for the same start and goal locations as those mization, they are conceptually different. The genetic
used for the genetic planner. planner requires only an approximate description of the
Severalobservationsarenow made regardingthegen- terrain and operates on the basis of evolutionary pro-
erated paths. First,the geneticplannerproduces the tess and stochastic search to generate a near optimal
waypoints, and in Figure 3 thesewaypoints are con- path. The global planner incorporates certain kinemat-
nected by straightlinesegments. To obtain smoother ics and dynamics into the planning phase, and require
paths,thesewaypoints can be connectedby cubicpoly- more knowledge about the environment and the rover.
nomials or any othersuitableinterpolations.Itisalso The relative simplicity of the genetic planner and the
noted thatintheseruns a low weighting(ain(3))was benefit of incorporating kinematic/dynamic constraints
assignedto curvaturerelativeto thecellimpedance to of the global planner can be combined to achieve better
obtainshorterpaths. As a resultapath sometimes tra- results. For example, the genetic planner can quickly
versesoversmallrockstoachieveshorterpath lengths produce a number of paths based on imprecise terrain
(and path impedance). However, a closerexamination description and the global planner can then evaluate or
shows thatallpaths areinfacttraversableby therover modify these paths to take into consideration the rover
kinematic/dynamic constraints.
(inthiscase NASA's Rocky 7 rover[2]).The global
optimization planner produces smoother path due to 6 References
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