A general contact model with smooth forces for optimal trajectory generation for multibody systems

This dissertation presents a general contact model for generating joint trajectories for multibody systems using techniques from optimal control. Many joint trajectories of interest involve contact; however, the dynamics of the system change whenever it contacts another object. Either different dynamics models must be used for different contact states, or one dynamics model can be used throughout the motion with a model to calculate the contact forces acting on the system. Because switching models does not allow for modeling impact vibration, sliding, or chattering, one model is used throughout our motions. A contact model is developed to use with any system and environmental obstacles that can be represented as the union of convex shapes. This model produces continuous, continuously differentiable (C 1) contact forces to make possible the analytic gradient of the cost function and not impede convergence when solving the optimal control problem.; This research produces a compliant contact model in which penetration distance and velocity are used with a virtual spring-damper system to calculate the normal and friction forces. Smoothing functions are used to make the contact forces C1 and to make the friction force change smoothly close to zero. The friction model is of the modified Karnopp variety, and includes viscous and Coulomb effects. To make the contact model general, the directions for the contact forces were defined in terms of the near points between the robot and the obstacle. Gilbert et al.'s distance algorithm is used to formulate an optimization for which the near points are the solution, and use the derivative of the first order necessary conditions to obtain the requisite derivatives. Analytic expressions for the distance and contact force derivatives, and the contact model itself, are the main contributions of this research. This contact model extends the scope of problems for which the optimal control trajectory problem can be solved reliably to systems that experience contact and collisions. We demonstrate with several novel examples that our contact model enables the solution of previously unsolved problems in the optimal control of multibody systems.