Sources of Error in a Simulation of Rigid Parts on a Vibrating Rigid Plate

Abstract

We present a simulation study of an important rigid-body contact problem. The system in question is composed of a rigid plate and a single rigid body (or particle). The plate follows a prescribed periodic motion of small amplitude and high frequency, such that the net force applied to the part appears to be from a time-independent, position-dependent velocity field in the plane of the plate. Theoretical results obtained by Vose et al. were found to be in good agreement with simulation results obtained with the Stewart–Trinkle time-stepping method. In addition, simulations were found to agree with the qualitative experimental results of Vose et al. After such verification of the simulation method, additional numerical studies were done that would have been impossible to carry out analytically. Specifically, we were able to demonstrate the convergence of the method with decreasing step size (as predicted theoretically by Stewart). Further analytical and numerical studies will be carried out in the future to develop and select robust simulation methods that best satisfy the speed and accuracy requirements of different applications. With the accuracy of our time-stepper verified for this system, we were able to study the inverse problem of designing new plate motions to generate a desired part motion. This is done through an optimization framework, where a simulation of the part interacting with the plate (including the full dynamics of the system) is performed, and based on the results of the simulation the motion of the plate is modified. The learned (by simulation) plate motion was experimentally run on the device, and without any tuning (of the simulation parameters or device parameters) our learned plate motion produced the desired part motion.

A direct method for trajectory optimization of rigid bodies through contact

“…In [5], Berard et al used the LCP formulation of contact to design trajectories of single body on a vibrating plate. Here, they optimize over a small set of parameters describing the oscillating behavior of the plate and the approach can best be described as a shooting method.…”

Per-Contact Iteration Method for Solving Contact Dynamics

“…Developing very accurate models is an active research area [1], [2] but the focus of this paper is to develop a very fast contact solver which uses less approximations than the existing simulators. There are three fundamental principles that are commonly used to model rigid body contacts: the Signorini condition, the Coulomb's friction cone constraint, and the maximum dissipation principle.…”

The application of particle filtering to grasping acquisition with visual occlusion and tactile sensing

“…Strictly speaking, a Kalman filter or extended Kalman filter could be used, but only after removing the contact terms from the dynamic model and adding noise. This would mean eliminating complementarity conditions (2) entirely and all but the first equation in equation (1). This single equation left is Newton's second law with all contact forces treated as disturbances in p app .…”

“…The resulting dynamic model can be formulated as a mixed nonlinear complementarity problem (mixed NCP) and solved by the path solver [3]. This mixed NCP could be linearized by approximating the quadratic cones with polyhedral cones, but previous results show that the NCP model is more accurate and can be solved just as quickly as the linearized model [2].…”