The Rover Analysis Modeling and Simulation (ROAMS) algorithm is to solve the kinematics of a wheeled mo- bile robot (rover) traversing on a rocky terrain. The rover is constructed using a “rocker-bogey-differential” type suspension and steering system as shown in the figure. By exploring the mechanical symmetry and the wheeled-terrain contact characteristics on a rough terrain profile, we developed a novel algorithm to carry out the rover’s configuration, including the vehicle’s wheels, steering and suspension linkages, and the position and orientation of the chassis. Because of its efficient and reliable numerical results, the ROAMS algorithm is well suited for the real-time simulation test bed, e.g., a simulation software system, of the mobile robotic vehicles in the planetary surface exploration missions. Currently, it is used to support the development of simulation and operation tools for the Mars Exploration Rover (MER) in the Mars ‘03 mission.

Rocky-7, a small robotic vehicle, is designed to traverse rocky terrain. The algorithm described in the text solves the inverse kinematic problem of the rover/terrain system during a computational simulation of such a traversal.

One of the most important properties of the algorithm is the robust computational result, which yields a fast prediction of the rover’s position and orientation in an uncertain environment. The ROAMS algorithm treats the underlying mathematical model as an inverse-kinematics problem, and carries out the solutions using the computational techniques for constrained optimization. In this framework, the objective functions comprise three conditions: (1) rover’s internal differential mechanisms, (2) wheel-terrain contact, and (3) rover’s driving motion. The resultant nonlinear equations can be derived, and their solution can be computed by a straightforward application of numerical methods. However, the standard approaches to this nonlinear system of equations suffer from the instability and inefficiency in the numerical solutions due to the rough terrain profile, which is often modeled as a piecewise smooth digital elevation map (DEM). Moreover, the modeling of the physical limits of the linkages (e.g., the bumper-stops) can be problematic to the solution, as well as the case that all wheels may not be in contact with the ground at all times. All these modeling and numerical difficulties are resolved in the novel algorithm that achieves the real-time simulation of the rover traversal on a rocky terrain.

In this algorithm, we implemented a Newton-type iterative method that handles the non-smooth wheel-terrain contact equations using global searching and relaxation techniques. The solution also takes account for the bump-stopper devices, which have been modeled as the joint limits at the corresponding linkages, and maintains a fast convergent rate. Regular Newton-type iteration requires the smoothness of the equations to ensure a fast convergence. This prerequisite of a robust convergence is violated since the roughness of the terrain has been embedded in the contact equations. When the rocker or bogey linkages reach their limits, an abruption of the iteration can induce unpredictable solution of the configuration. To overcome these numerical difficulties, we applied a weight factor to the residual of each contact equation. During the iterations, the weight factor for a given wheel can be made to approach zero to relax the contact condition. Whenever the wheel leaves the ground, its corresponding weight factor is set to zero for a total relaxation of this wheel-terrain contact. The re-scaling of the weight factors is coupled with the global search algorithm, which can detect the joint limits (associated with each of the wheel-terrain contacts) for locking the joints, and can sample small perturbations around the contacting locations to determine the occurrence of a separation of the wheel and the ground.

The step-selection strategy used in the global search is a backtracking line search algorithm that monitors the progress of the iteration. For a smooth terrain profile, the iterative solution generated by the Newton method converges very rapidly to a local minimum of the nonlinear equations. However, the rate of convergence can be tremendously decreased when a non-smooth terrain profile appears. Special cares have to be taken to maintain robust and efficient solution in the case of non-smooth terrain profile. Although the problem in hand is ill-posed (i.e., it is well known that the Newton method cannot treat non-smooth equations), we developed a heuristic solution to ease the computational difficulty in the iterations. In practice, the wheel-terrain contact is treated as a non-penetrative type, which is not a realistic portrait of the nature of the wheel-terrain interaction. Therefore, a search direction to the wheel-terrain contact may not be in-line with the normal direction of the terrain (at the contact location), instead, it could be anywhere along the perimeter of the wheel. The heuristic solution leads to modeling the wheel-terrain contact equation as the distance constraint between the wheel center and the terrain profile. As shown in all the preliminary testing cases, the modification of contact equations yields a much-improved convergence in non-smooth terrain profiles, and allows the modified Newton iteration to overcome many local irregularities.

This work was done by Jeng Yen of Cal-tech for NASA’s Jet Propulsion Laboratory.


This Brief includes a Technical Support Package (TSP).

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Motion Control Tech Briefs Magazine

This article first appeared in the June, 2002 issue of Motion Control Tech Briefs Magazine.

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