Prooxdings of the 1993 IEEE/RSJ International Conference on Intelligent Robots and Systems Yokohama, Japan July 2630,1993
On Modeling and Motion Planning of Planetary Vehicles F. BEN AMAR Ph. BIDAUD and F. BEN OUEZDOU Laboratoire de Robotique de Paris 10-12, Avenue de 1’Europe 78 140 Velizy Villacoublay FRANCE e-mail:
[email protected] 1 >
-8f vehicle motions over considers the kinemati physics of the interaction between the wheels and the groundi obtained by integration o ics. This system is exploi that the vehicle remains
1
4
>f sentation of path. We choose to control the extrema of the steer angle along the path, which affects vehicle stability and also is limited by technological construction. The smoothness of the steer angle profile appears in energy consumption criterion. Vehicle heading values at sub-goal points are assumed to be not prescribed by the mission. These values will be optimized by the planner. Paths must satisfy kinematic and static constraints which relates respectively to the steering capacity of the vehicle and its traction performance.
-rl...p 2 World Relpresentation
Introduction
This paper presents a general framework for the analysis of wheeled vehicles performances. This framework takes into consideration: the mechanical behaviour of the articulated locomotion mechanism, terrain geometry and some basic terramechanics aspects of ground-wheel interactions. Vehicle navigation on uneven terrain has been addressed in a limited number of works. Some of them concern path planning for vehicle considered as a point [4] [5] or considered as a single body with fixed contact points in the platform reference frame [3]. The locomotion mechanism considered in this paper, as an example, is a four wheeled vehicle with an articulated front axle: steering and suspension joints. The kinematic and static models derived take account of the geometry properties of the terrain. We use a continuous representation of the terrain geometry which is defined by Bspline patches. T h e kinestatic developement used resorts in particular to a formalism used by [l]for analysing multifinger motion coordination and by [2] for deriving control laws of an articulated track robot on 3-dimensional terrain. The static indeterminacy is solved by assuming a symetrical behaviour i n the transversal direction [7]and by means of optimization of wheel-terrain contact forces. The smoothness of a trajectory relates to the smoothness of its curvature profile. Clothoids and spiral cubics have a great interest for trajectory generation because they provide a simple curvature profile, but they are difficult to compute because no closed-form expression of their coordinates is available. In o u r Hiiiiuliit,or, we gciicrtite cubic spline curves joining a set of a given sub-goal points which moreover offer a flexible repre-
Figure 1: Vi.ew of
2.1
the graphic simulator
Locomotion! Mechanism of the Vehicle
The locomotion mechanism of the vehicle (fig.1 2 and 3) considered to illustrate this work is a two-axles vehicle with four driven conventional w:heels. The front axle is connected to the platform by revolute pairs with orthogonal axis. One for steering(@, the other one ( 7 ) is to maintain a permanent contact of wheels with the ground. This last one could be actuated to provide a better obstacle negociation. However, this increase the degree of static indeterminacy and the complexity of optimal load distribution procedure.
0-7803-0823-9/93/$3.0 (C) 1993 IEEE 1381
Figure 3: 2-D instantaneous kinematic. From figure 2 and for planar motion, we have: Figure 2: Schematic view of locomotion mechanism
4=8+a
2.2
where: 4 is path tangent, 8 is the steering angle and CY is the heading angle of the platform. If k ( s ) is the curvature along the path,(s is the curve lenght) and 1 is the distance between the front and rear axle, we can write:
Terrain Geometry and Physics Representation
111 the study of the performance of off-road vehicles, the characterization of the terrain and the geometry of the surface are required. In fact, the geometric properties of the terrain are of importance for the study of the ride quality and obstacle crossing capabilities of the vehicle. Wheras, the mechanical properties of the terrain are the most important factors affecting the tractive performance of the vehicle. The use of a continuous surface representation allows to compute the normal and tangent vectors a t wheel-terrain contact point directly, as required for kinestatic analysis. The surface is defined by a second order continuous B-spline patches. Each patch is bi-cubic spline. T h e control points are specified using a computer graphic interface (fig.1 and 4). The selection of the method for characterizing terrain phys‘ics properties depends on the method adopted for modeling the interaction between the wheel and the terrain. Cone Index(C1) obtained by cone penetrometer technique has been widely used in mobility studies of agricultural and military vehicle [6].
3
d4 k(s) = dt
Then, the necessary steering law to guide the front point of this vehicle over the prescribed path defined by k ( s ) must satisfy: d8 - k(s) - da = k ( s ) - sin(8) _ ds
ds
1
(4)
This equation(4) is resolved by the use of Range-Kutta a fourth order method. For technology and stability reasons, the steer angle 8 has a limiting value.
Path Representation
section G .
ds 1 da sin(8) dt
_ -- -ds
The prescribed path is expressed on a medium plane which is defined to be the best fit-plane through the successives sub-goal points. Simulation results (fig. 4) shows that the projection of the executable path in this plane keeps close to the prescribed one, since the vehicle do not cross high (or low) altitude areas. In fact, we consider that terrain elevation is about the vehicle size since vehicle action field is more than I O g h times this last one. Given a pair of points, we can generate cubic polynomial curves(z(u), y(u)) joining these two points, with U E [0,1]. The first derivative vectors ( ~ ‘ ( u )~, ‘ ( z L at ) ) initial and final poii11.s arc prcscribetl hy the planner. This will be discussed iii
(1)
Then, wheel angular velocities can be easily deduced along the curve, as function of i,8 , l and wheelbase (see fig.3). It should be mentionned t h a t the system has a partitioned mobility (0). It has two degrees of freedom on a flat surface, with a pure rolling condition. But the vehicle pratform connectivity with the ground is equal to one.
4 3-D Kinematic Model The aim of this section is t o compute, by means of integration of kinematic variables, geometric parameters required for the static analysis of vehicle stability. Initial conditions are then assumed to be known. If we consider for instance that the vehicle start on a flat area, the geometric parameters(rol1, pitch and elevation of the vehicle body and roll of suspension joint
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y) are initially equal to zero. The location of point contact on wheel circumference is computed as the same for platform geometric parameters. This has a great interest for analyzing wheeLground contact forces. Given a velocity of the steering joint(6) and for a wheel(wi), as computed in the previous section, we can compute the complete platform twist relative to the ground(V, 0) and the other joint velocities(wj ,7)(fig.2 and 3). The developement of this section resorts in particular to a formalism used by [I] [2]. This formalism allows to express differential kinematic equations in a matrix form, but in general it does not provide direct or inverse models, since some equations are linearly dependants. For each contact point C;, we define a local frame Ri = (Ci,b i , ti, ni) where: 0
0
ni is the oriented normal to the contact tangent plane. ( b i , l i ) defines the contact tangent plane, with 2; the unit vector through wheel plane and 6i the binormal vector.
The kinematic equation of the opened sub-chain joining the contact Cl, for example, to the platform is expressed by the following vectorial equation: (see fig.2)
vc1 = V
+ Q A RCi + W ~ ( Z UA PiC1) ~ +
~ ( ZA OR C i )
+ f(z7A R e i )
(6)
Where: 0
ucl is the velocity of contact point C1 relative to the ground,
0
) is the complete twist of the platform whose (components in platform reference frame ( R ,i, k) are 1
v,,vz
Z w l l , Ze
1
Q,,
R,, Qz
and 2, are unit vectors along joint axis.
A permanent contact on the ground: vC,.ni = 0 , N o longitudinal slip: v c , .ti = 0 ,
No lateral slip: uc,.bi = 0, Therefore, we obtain for each contact C; 3 equations corresponding to the previous roiling contact constraints, and then the kinematic of the system can be expressed by the following matrix form equation (see appendix A):
L'
15
25
20
30
35
1
Kinematic constraints relative to the i-th contact are:
0
10
Figure 4: Horizontal projection of the executable path(so1id) and the prescribed one(dashed)
j,
(V.
5
( E)
=2q
If61 # 62 and b3 # bg, the system has a moblity equal to zero with pure rolling assumption, because equation system (Eq.7) has 12 independant equiltions with 12 kinematic parameters. We can notice that for plane motion equations vc,.bl = 0 and vc2.62 = 0 are equivalents, as vc1.C1C2 = vc,.C1C2 and ClC2 = b l = 62. We assume that it is available even for b l N 62. Then one equation per axle can be removed, that means that we permit i.he correspondant contact to slip in the transversal direction to the wheel. In our study case, two equations are removed. Therefore, by given two joint velocities, for example Y = ( 6 , w l ) ' as computed in the previous section, the direct lcineniatic model derived from (Eq.7) can be written in the foffowing form:
A X = BY
(7)
where: where: 0
are internal joint velocities: ( w l ? w 2 > W3r w 4
i
A is a (10,lO) regulixr matrix,
B is a ( 1 0 4 matrix,
7)' i 0
L is called locomotion matrix (6,12),
X is the unknown kinematic parameters and is equal for this example to
J is the jacobian matrix (12,6).
(vSl
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v,
1
I 0=1
0Z 1 W21
w41
7)'
By solving Eq.8, we deduce directly 3-D trajectory of the front point of the vehicle, Euler geometry parameters of the vehicle body, passive joint parameters and the location of contact points required for the next iteration. This method has been validated, as roll, pitch of the platform and suspension joint parameter become zero, when we reach flat area, as shown in fig.5 corresponding to the path plotted in fig.4.
0.4 .,
0.3
................... i...... .............................................
...
............ :: ~
I
!
i
..l
;
the vehicle body. Therefore, the system has a high degree of static indeterminacy, since it uses a redundant actuation due to actuator redundancy. On the one hand, longitudinal forces (driving or braking forces) are controled directly by wheel torques. Then, they could be optimized. To obtain a transversal symetrical traction, longitudinal forces f, on same axle are supposed to be equal. This is expressed by:
fi
f; = f; f; = f;
...........
Optimization of traction of front and rear wheels must considers normal forces fi. For this reason, we impose equality of longitudinal traction coefficient between front and rear wheels, which can be expressed by:
0.2
0. I
0
f; + f: - f; + f; f,' + f,2 - f: + f,"
-0.I
-0.2 -0.3
0
10
20
30
40
50
60
Covered Distance
Figure 5: The corresponding roll(so1id curve), pitch(dashed) and suspension joint(dotted)
This equation can not be included in the matrix form of equations 9 and 10, because it has a nonlinear form. However, it could be linearized for each position, since normal forces do not vary rapidly. On the other hand, it is difficult to make a priori assumptions on lateral forces, because they can not be controled. For symetry reasons, it is assumed that lateral components are equals for wheels mounted on the same axle. This relation was established in [7], resulting from what he called "zero-interaction assumption" and is expressed by:
3-D Static Model
5
At the average speed of the planetary rover (about O.lm/s) [7], the inertiel forces can be neglected. The purpose of this section is to present a quasi-static model to predict contact forces and the applied wheel torques. Exploiting duality between wrench and twist sub-spaces, static equilibrium equat8ionsof the system writes [l] [2]:
(6)
LXf
=
J'X,
= 'I
(9)
(10)
where: 0
0
L and J are respectively the locomotion and the jacobian matrices defined in the previous section,
X, = (f:, fi ,fj)' is the contact forces vector expressed in local contact frames,
0
0
is the resulting external wrench acting on the (vehicle)body,
I? is joint torques vector (zero if not actuated-joint).
This formalism assumes a point contact between the wheel and the ground (no moment components) and only accounts for reversible effects of contact forces. Rolling resistance bue I,o terrain compaction is considered as an irreversible effort and then added to the resulting external wrench acting on
f: = f: f: = f:
(15)
Then equations 11, 12, 13 (after linearization) and 14 are added to the matrix form equation 9 and 10, and contact forces and wheel torques are then deduced. Wheel-terrain interaction model Since most wheeled off-road vehicle are equipped with pneumatic tyres, available methods for predicting wheel-terrain forces deal with elastic wheels. Moreover, a tyre with a high inflation pressure operating on a weak soil behave as a rigid wheel. As the wheel-terrain interaction is very complex, theoretical methods require numerous data, sometimes not realistic or difficult to obtain. Therefore empirical approach, used for predicting performances of agricultural and military vehicles [6], provides simple models which are suitable for our purpose. Based on a dimensional analysis of the interaction problem, this approach supplies a dimensionless terms characterizing the contact. These parameters depend basically on wheel geometry (radius, width, section height), terrain cone index and the applied vertical load. Then force components (rolling resistance, driving force, braking force, lateral force) are plotted as function of this dimensionless terms. Readers could report to [8] for a review of principal empirical laws for predicting wheel-terrain forces. When both lateral and longitudinal forces exist simultaneously, there is a limited value of total force that can be developed by the wheel. This is assumed to be defined by the
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friction ellipse [9] shown in fig.6. Friction ellipse does not account for irreversible forces, as rolling resistance due to terrain compaction. Let p i and p i be the maximum friction coefficients respectively in the lateral and in the longitudinal direction (see appendix B). Their expressions are given in [8] as function of the dimensionless mobility index. Maximum force condition, expressed by:
must be satisfied for each i contact. We notice that this relation expresses also the vehicle stability condition, since the friction limit decreases with the normal component of the contact force which is related to the quasistatic stability.
t 1
brakine
lateral force
I
driving
The proposed method is based on a deformation of a nominal path joining sub-goal points. So, we assume that heading values of the vehicle body at these points are not prescribed by the mission. The first derivative to the trajectory a t these points could be discontinuous, since the front axle can rotates freely when the vehicle is at rest. First, a cubic spline curve ( r ( u ) , y ( u ) )joining this set of points is generating. Heading value a t sub-goal point dP,,/du are initially given parallel to (P,-,Pn+l) line (fig.7). The deformation of this nominal path is obtained by varying heading values a t sub-goal point. Then a set of paths, which keep close the nominal one, is scaned and is evaluated by the simulation system described in previous sections, Feasible paths must satisfy steering conditions (Eq.5) and contact force constraint for each wheel (Eq.16). By mean of this method, some performance indexes as covered distance, energy consumption and risk factor, can be compared for several paths. Figure 8 shows an example with 5 points. The minimal risk path is the one who keeps the farthest from friction limit.
\ loneitudi
;Specified sub-goal points
Figure 6: Friction ellipse describing maximum force.
6
Path Planning
Shortest feasible path
The least risk path
Figure 8: study case with 3 sub-goal points.
7
Conclusion
Figure 7: Definition of the nominal curve
A general framework for path planning of wheeled vehicles As mentionned in section 3, the path is specified by a set of sub-goal points which will be executed autonomously. These points are partially specified by the operator and corresponds to interest point of the mission. The remaining points are specified to guide the path search. Since configuration parameters of the system (z,y, a,0) are numerous and are consthained by kinematic conditions, methods based on graph srnrch are difficiilt t,o apply. Difficulties concern also the tlclintion of the cost function.
has been proposed. The mechanical behaviour of articulated locomotion mechanism and its mechanical interaction with its environment are conslidered. This simulator system is exploited to search paths clonsidering the steering capability of the vehicle and its traction performances. In this study, terrain geometry is represented by a continuous functions and its physics is characterized, by region, by a single parameter. Some optimal paths corresponding to different criterions are generated. The choice of them depends on many factors, such
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are: mission orders, self-recovery ability of the system and t,errain knowledge incertainty. The proposed method, does not search path in the whole Cartesian space, but scans paths which keep close a nominal one. It fails when an important detour or some manoeuvres are needed. In this case, intermediate point have to be introduced, nevertheless by an heuristic. This system will be used to define some basic rules for crossing some specific regions or negociating of some basic obstacles. I n t.he future, this work wijl be extended to account for more realistic contact geometry considering wheel width and terrain discontinuities.
References Z. Li, P. Hsu and S. Sastry. Grasping and Coordinated Manipulation by a multzfingered Robot Hand, Int. J. of Robotics Research, Vol. 8, No. 4, August 1989.
Appendix B: Empirical laws for contact forces prediction Mobility number M (dimensionless term) characterizing the contact is defined as:
F.X. Potel, F. Richard and P. Tournassoud. Design and execution control of locomotion plans, Proc. of the Int. Conference on Advanced Robotics, Pisa, 1991.
T. Simeon. Motion Planning for a Non-holonomic Mob& Robot on 3-dimensional Terrains, IEEE Int. Workshop on Intelligent Robots and Systems, Osaka, 1991.
where:
(31:Cone Index given by cone penetrometer
Z. Shiller and J.C. Chen. Optimal Motion Planning of Autonomous Vehicles in Three Dimensional Terrains, Proc. of IEEE Int. Conference on Robotics an Automation, Cincinnata, 1990.
b: wheel width
D. Caw and A. Meystel. Minimum-Time Navigaiion of an Unmanned Mobile Robot in a 2-1/20 World with Obstacles, Proc. of IEEE Int. Conference on Robotics an
h: wheel section height
d: unloaded wheel diameter 6 : vertical wheel deflection
This number is related to contact force components by:
Automation, San Francisco, 1986.
R
- = 0.287/M+ 0.049
J.Y. Wong. Terramechanics and Off-Road Vehicles,
f*
pp20-42, Elsevier, 1989.
fY
K.J. Waldron, V. Kumar and A. Burkat. An Actively Coordinated Moobzlzty System for a Planetary Rover, Proc.
py kPY
Int. Conf. on Advanced Robotics, 1987.
= 0.796-0.92/M
= 4.838 + 0.287M
Jx -
C.W. Plackett. A Review of Force Prediction Methods f o r 08-Road Wheels, J . of Agricultural Engineering Research, No. 31, 1985.
D.A. Crolla. Off-Road Vehicle Dynamics, Vehicle System Dynamics, No.10, 1981.
= py(l - e - k i )
f*
pz(l
- e-BC)
fi
p2
= 0.89- 0.14M
Bpx = 2.18+ 0.38M
where: R: Rolling resistance due to terrain compaction
i: longitudinal slip ratio
Appendix A: Computing L and J matrices
c: side slip angle
For sub-chain 1:
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