Simple closed-chain mechanism algorithm

As mentioned above, more efficient algorithms are needed to make real-time dynamic simulation a reality. This need is particularly great fw robotic systems with multiple chains and closed kinematic loops. Thus, a fundamental goal of this book is the development of better and more efficient algorithms for the dynamic simulation of multiple chain robotic systems. In particular, solutions to the Direct Dynamics problem fw simple closed-chain mechanisms are investigated. 

In this chapter, an efficient soial algorithm for the dynamic simulation of simple closed-chain mechanisms is derived. The development of this algorithm... 

While considerable effort has been spent studying the simulation problem for single closed chains, fewer results are available for more complex multiple chain robotic systems. Existing algorithms for simple closed-chain mechanisms are, in general, difficult to apply and/or computationally inefficient Some relevant... 

Dynamic Simulation Algorithm for Simple Closed-Chain Mechanisms... 

In developing an efficient algwithm for the dynamic simulation of simple closed-chain mechanisms, we are naturally led to consider the relationship between the physical structure of the robotic system and the computational structure of the desired algorithm. Intuitively, it seems tqyparent that the structural parallelism present in a simple closed-chain mechanism should lead to computational parallelism in the solution of the Direct Dynamics problem for that mechanism. 

With ao known, we may also solve explicitly for the spatial tip force fit, jk = 1,..., m, using Equation 6.12. Thus, the motion of the refnence membo and the spatial force exerted at the tip of each chain are completely defined, and the simple closed-chain mechanism is effectively decoupled. Each manipulator may now be treated as an independent chain with a known spatial tip force. The joint accelerations for each chain may be computed separately using an r pro xiate Direct Dynamics algorithm and then integrated to obtain the next state. 

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 ... 

In the develc ment of the simulation algorithm in the previous section, the objective was to decouple the simple closed-chain mechanism by computing the spatial force vectors exerted by the chains on the reference membo. The spatial tip forces computed in that algorithm are real, measurable forces, associated with the general jdnts which connect the reference membo and each chain tip. Once these forces are known, the chains are effectively decoupled from the refnence member, and the general joint accelerations may be computed for each chain separately. 

The coefficient matrix of ao in Equation 6.51 rq>resoits the effective simple closed-chain mechanism as seen by the reference member at the origin of its own coordinate system. The operational space inertia of the reference member is just its spatial inertia matrix, lo. Note that the operational space inertias of the augmented chains (acting in parallel on the reference member) add in a simple sum. This is a general rule for inertia matrices. For actuated chains connected in series, the combination rule is not as simple. In this case, extended versions of the recursive algorithms of Chapter 4 may be applied. 

In this secdon, we will consid the computadonal requirements of the dynanuc simuladon algorithm for simple closed-chain mechanisms presented in Secdon 6.S. First, the number of scalar q>eradons required fw each chain of the mechanism will be tabulated, followed by the number of operadons required to compute the spadal acceleration of the reference member. The computadonal complexity of the complete algorithm will then be discussed. The parallel implementation of this algorithm will also be consid ed. 

In this chapter, a general and efficient dynamic simulation algorithm for simple closed-chain mechanisms was derived. The algorithm is tq>plicable to both lype... 

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. 

The book may be organized into two parts. Part one addresses the efficient computation of the joint space and operational space inotia matrices. Four algorithms are presented for the computation of each inertia matrix. Part two of the book presents the dynamic simidation algorithms which are develt red for single closed chains and simple closed-chain mechanisms, respectively. 

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