Improved Formulations for Real-Time Dynamics

Abstract

The general purpose dynamic formulations described in Chapter 5 are simple and efficient, but they are not suitable for real time dynamic simulation. Real time performance requires faster formulations. These can be developed by taking into account the systemÕs kinematic configuration or topology. In the last two decades, a big effort has been dedicated to developing very efficient dynamic formulations for serial robots or manipulators. These formulations have been extended later on to general open and closed chain configurations. In the first section of this chapter, a survey of two of the most efficient available formulations that need O(N 3) and O(N) arithmetic operations is presented. It follows, a detailed description of a formulation based on velocity transformations that can be parallelized at body level. In the next section, a description of how the penalty formulation can be used for improved performance is also included. Finally, two complex examples: a model of the human body and a heavy truck, with theoretical count of arithmetic floating-point operations, and some numerical results are presented. The methods and results presented in this chapter are contributions coming mainly from JimŽnez (1993), and Avello et al. (1993). 8.1 Survey of Improved Dynamic Formulations Most of the improvements in multibody dynamic formulations that have been developed in the last 25 years come from the robotics field. This field was very active in the 60Õs and 70Õs with scientists trying to solve very hard problems of simulation and control with the limited computational resources available at that time. Anyone familiar with robot control algorithms based on dynamic models knows the very competitive race that took place in order to decrease the number of arithmetic operations required for the inverse dynamics in serial robots. This research was quite important and led to the solution of the inverse dynamic problem two or three hundred times per second with a DEC-PDP 11 processor. The recursive Newton-Euler formulation seems to have been the winner of the race