The control of planetary rovers, which are high performance mobile robots that move on deformable rough terrain, is a challenging problem. Taking lateral skid into account, this paper presents a rough terrain model and nonholonomic kinematics model for planetary rovers. An approach is proposed in which the reference path is generated according to the planned path by combining look-ahead distance and path updating distance on the basis of the carrot following method. A path-following strategy for wheeled planetary exploration robots incorporating slip compensation is designed. Simulation results of a four-wheeled robot on deformable rough terrain verify that it can be controlled to follow a planned path with good precision, despite the fact that the wheels will obviously skid and slip.
Wheeled mobile robots (WMRs) are typical nonholonomic systems and they have attracted the attention of many researchers as they do not satisfy Brockett’s necessary condition [
Despite the rich results that have been obtained in studies of WMRs that applied ideal assumptions, new control problems arose with the development of WMRs for deployment in challenging terrain, such as the planetary exploration rovers. The Mars exploration rovers, Sojourner, Spirit, and Opportunity of the USA have greatly enhanced our knowledge horizon [
In addition to the longitudinal slip and lateral skid of wheels moving on rough and deformable terrain, the redundant control of different wheels is another challenging problem. The current planetary exploration rovers have four or six independently driving wheels, and the four wheels at the corners are independently steering wheels, as shown in Figure
Flight rover Spirit [
The above-mentioned research mainly concerns the redundant control of planetary rovers, with the objective of improving their traversing performance by coordinating the velocity or torques of the driving wheels, while the path-following problem is solved by coordinating the velocity or position of the different steering wheels. A steering maneuver strategy for a four-wheeled rover tested on lunar soil regolith simulant was investigated under different steering angles using both dynamics simulation and experiments [
This study focuses on the path-following problem of a planetary rover on deformable and very rough terrain, that is, when the wheels and the vehicle body are not at the same slope and their local orientation coordinates differ. A nonholonomic kinematics model of planetary rovers traversing deformable rough terrain and a control strategy that coordinates the different steering wheels to realize path-following control of planetary rovers on challenging terrain are presented. The control algorithm is verified using a high-fidelity simulation platform [
For the sake of simplicity, studies in the literature often assume that wheel-terrain interaction occurs at a single point beneath the center of the wheel. This simplification will, however, lead to large errors when a WMR traverses over deformable rough terrain. On the one hand, the contact area between the wheel and the soil is large enough to need to be considered; on the other hand, it is determined by the geometry of the terrain rather than by the point beneath the wheel’s center. The local coordinates of the contact areas and the wheels should be calculated, as they are indispensable for the kinematics modeling of WMRs on rough and deformable terrain. For instance, the direction of a wheel’s velocity is approximately parallel to the contact surface rather than the horizontal plane.
Figure
Contact area of a wheel moving on rough and deformable terrain [
The equation of the inclined plane
Line
Figure
The vector direction of
The transformation matrix from
where
In Figure
Coordinates and slope angles of a wheel moving on the inclined plane.
When a virtual rover is being controlled in a numerical simulation, the coordinates of its wheel’s center
As the terrain is rough and deformable, the wheels experience longitudinal slip and lateral skid and all the coordinates of the wheels and of the rover’s vehicle are different in terms of not only position but also orientation. The nonholonomic kinematics model, which includes the properties of the terrain, vehicle, and wheels, constitutes the basis of path following. A model of a six-wheeled planetary rover is shown in Figure
Model of a six-wheeled planetary rover.
Lateral velocity that is perpendicular to the longitudinal direction of the vehicle body exists when a rover is moving on rough and deformable terrain. There is an included angle between the longitudinal velocity and the actual velocity of the vehicle, which is called the side skid angle and denoted by
Let
Let
In order to demonstrate the calculation of
Coordinates of four-wheeled planetary rover.
Front view
Top view
According to [
In (
By substituting (
Let
Parameter
The velocity of the
Substituting (
By substituting (
A rover’s optimal path in challenging terrain can be planned on a digital elevation map (DEM) [
Position coordinates and yaw angles of the planned path.
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0 | 0 | 1 | 2 | 3 | 4 | 4 | 5 | 5 |
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0 |
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A planned path in DEM.
The rover’s reference path is updated in real time based on the carrot following method according to the path planned in DEM. Let If there is only one forward intersection point, it is the goal point, as shown in Figure If there are several forward intersection points, the most forward one is the goal point, as shown in Figure If there is no forward intersection point and the shortest distance from the rover’s center to the planned path is larger than If there is no forward intersection point and the entire forward path is within the look-ahead circle, the destination point is the goal point, as shown in Figure
Classifications of goal points.
The look-ahead distance,
Influence of
Paths,
Paths,
Paths,
Desired yaw angle
For the DEM to show the small dimensional obstacles or craters that the rover has to overcome, the resolution of the DEM should be comparable to the dimension of the wheels. However,
Influence of
Paths,
Paths,
Paths,
Desired yaw angle
For the rover to follow the reference path and to compensate for the lateral skid, path-following strategy should be studied.
Figure
Illustration of path-following control.
When following the reference path, (1) the yaw angle of the rover,
In order to realize the control objective by coordinating the angular velocities and steering angles of the wheels, the relationship between them should be analyzed. On deformable rough terrain, the motion in the vertical direction, that is, the motion along the
The differences between the control of WMRs on flat, hard terrain and on deformable rough terrain involve the longitudinal slip and lateral skid of wheels and the nonholonomic kinematics models. All the factors except for the longitudinal slip are well considered in this study. The longitudinal slip is characterized by a variable named slip ratio:
The remaining problem is how to coordinate the steering angles of the wheels in order to realize the path-following objective. The proportional-integral-derivative (PID) control algorithm is used to generate the angular velocity of the rover’s body:
The angular velocity of the rover’s body should follow the yaw angle and compensate for lateral skid. Such an angular velocity is realized by the steering motion of the wheels. The forward wheels and the rear wheels can play different roles by using different PID parameters. For example, the forward wheels compensate mainly for the skid and the rear wheels mainly follow the yaw angle. Variables
Given the desired velocity of the rover’s body,
The state variables of the
The angular velocity of the suspension joints,
Substituting the above desired wheel velocity into (
In order to maneuver the wheels to achieve the steering angle,
Wheel’s steering control strategy for path following.
The El-Dorado II rover and the parameters of Toyoura sand were applied in the simulation, and the DEM of rough terrain was generated using MATLAB [
The rear wheels were controlled to follow the yaw angle; the PID parameters in (
Figure
Path-following results of simulation of the El-Dorado II rover.
Snapshots of the El-Dorado II rover
Trajectory of wheels
Trajectory of rover’s center
Roll and pitch angles of the rover’s body and angles of rocker joints
The curves in Figure
Simulation results for the wheels of El-Dorado II rover.
Slip ratios of all the wheels
Drawbar pull generated by all the wheels
Slope angles of the forward right (
This paper presents a path-following control method for wheeled planetary exploration robots moving on deformable, rough terrain. The modeling of a WMR on such challenging terrain includes a geometric model of the wheel-terrain contact area and a nonholonomic kinematics model. The coordinate systems of the rover’s body and of the wheels are different, and their transformation matrix to the inertia coordinate system can be described using the derived equations presented in this paper. In order to follow the path planned in DEM, the reference path should be updated in real time. By combining a longer look-ahead distance and a shorter path updating distance, the DEM information can be reflected very well and the actual path is smoothed to decrease the steering motion of the rover. The path following of a WMR is primarily achieved by controlling the angular steering velocity of the rover’s body, which in turn is realized by coordinating the position of the steering wheels. The path-following strategy for a WMR moving on deformable and rough terrain is designed. Different PID parameters can ensure that the forward and rear wheels play different roles in terms of following the yaw angle and compensating for the lateral skid. The four-wheeled El-Dorado II rover is used in a simulation experiment, and the effectiveness of the path-following strategy in deformable and rough terrain is verified.
The authors have declared no conflict of interests.
This study was supported by National Natural Science Foundation of China (Grant nos. 61005080, 61370033, and 51275106), National Basic Research Program of China (Grant no. 2013CB035502), Program for New Century Excellent Talents in University of Ministry of Education of China (NCET-10-0055), Special Postdoctoral Foundation of China (Grant no. 201104405), and the “111” Project (B07018).