Dynamic Simulation Algorithm for a Single Closed Chain

6 Dynamic Simulation Algorithm for a Single Closed Chain 

We now present an efficient serial algorithm for the dynamic simulation of a single closed chain which is based on the concq ts developed in the previous three sections. For this simulation problem, we assume that the following information is given for the chain or contacted body, as indicated  

In the solution of this closed-chain simulation problem, the following unloiown quantities are desired  

The reaction tcx ques/forces at the powered joints and the relative accelerations of the end effector in the free directions of the contact are also unknown, but their solution is not usually required. They may also be computed as desired. 

The new simulation algorithm developed hoe may be presented as a series of four computational steps. These steps will be referred to as follows  

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In the sixth section, the complete dynamic simulation algorithm for a single closed chain is presented as a series of four computational steps. Each step is explained in detail, particularly the step which computes the unknown contact forces and moments. The integration of the joint rates and accelerations to obtain the next state positions and rates is also briefly discussed. The computational requirements of both versions of the simulation algorithm are tabulated and compared in the seventh section of this chapter. 

In this section, the computational requirements of the dynamic simulation algorithm for a single closed chain are summarized and discussed. The number of required scalar operations is tabulated for each step, with the exception of the integration step. The q)erations required for integration are usually not included in the overall computational complexity of a simulation algorithm. 

Like the dynamic simulation algorithm fw a single closed chain, the algorithm developed here for simple closed-chain mechanisms may also be presented as a series of steps. In this case, five computational steps are required, and they are as follows ... 

A review of previous work related to the dynamic simulation of single closed chains is given in the second section of this chapter. The next three sections discuss several steps in the development of the simulation algorithm. In particular, in the third section, the equations of motion for a single chain are used to partition the joint acceleration vector into two terms, one known and one unknown. The unknown term is a function of the contact forces and moments at the tip. The end effector acceloation vector is partitioned in a similar way in the fourth section, making use of the operational space inertia matrix. In the fifth section, two classes of contacts are defined which may be used to model interactions between the end effector and other rigid bodies. Specific examples are provided. 

The purpose of this book is to present computationally efficient algorithms for the dynamic simulation of closed-chain robotic systems. In particular, the simulation of single closed chains and simple closed-chain mechanisms (such as multilegged vehicles or dexterous hands) is investi ted in detail. In conjunction with the simulation algorithms, efficient algorithms are also derived for the computation of the joint space and operational space inntia matrices of a manipulator. These two inertial quantities are important factors in a variety of robotics applications, including both simulation and control. 

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