Single closed chain computations

In an attempt to circumvent the computational complexity of matrix inversion, some researchers are pursuing solutions for the joint accelerations which have a linear recursive form. The inversion of the in tia matrix is explicitly avoided. The resulting linear recursive algorithms have a reduced computational complexity which is 0(N). This is the second basic solution approach to the Direct Dynamics problem, and it has been rqjplied to serial open chains [3,7], single closed chains [22], and some more genoal multibody systems [4, 37]. It is believed that the structure of linear recursive algorithms may also facilitate their implementation on parallel computer systems. 

Although a single closed chain is a simple example of a closed-chain robotic mechanism, its real-time dynamic simulation is not trivial. The dynamics of the chain must be combined with the kinematic constraints which are imposed by the tip contact. In general, both the contact forces at the tip and the joint accelerations must be computed to completely solve the system. 

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. 

The qren-chain tom, 4open> turd the force vector coefficient, fl, are known from Stq) 1, and f is known from Step 2. The computational complexity of this step is 0 N). With this step, the Direct Dynamics problem has been completely solved for this single closed chain system. 

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. 

Tible 5.1 Computations for the O(N ) Single Closed Chain Dynamic Simulation Algtxithm... 

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... 

As in the single closed chain case, the open-chain terms, (qt)open and (Xik)open, are completely defined for each chain given the present state genial joint positions and rates, qt and qt, the applied graeral joint torques/forces in the free directions, n, and the motion of the supprat surface. Any appropriate open-chain Direct Dynamics algorithm may be used to calculate these terms. Because the general joint positions are known, fit and Aj are also defined. The efficient computation of fit and for a single serial-link chain was discussed in detail in Chapter 4. 

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 ... 

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. 

Given the computations required for each individual chain, the number of scalar operations needed to compute the spatial acceleration of the reference membo, ao, is given in Table 6.3. Equation 6.38 is used to obtain the solution, which requires 0(m) spatial additions and a single 6x6 symmetric linear system solution. Thus, the number of opmtions required for ao is a function only of m, the number of chains in the simple closed-chain mechanism. The example of three chains (m s 3) is given in the last two columns of this table. 

There are a number of mathematically simple problems involving single polymer chains which are basically exactly soluble by the standard methods. By the term exactly soluble we imply that a closed form analytical solution is available which may be evaluated, possibly with the use of computers, to any desired numerical accuracy. This category of problems includes the configurational statistics of polymers in ideal dilute solutions - (i.e., at the 0-point). (See Section II for a brief discussion of the 0-point.)... 

We have focused so far on single-chain surfactants with hydrocarbon chains, mostly with COOH or closely related head groups. Computer simulations have also been performed on a variety of other surfactants. We do not attempt here to exhanstively review all work, but describe some (hopefully) representative samples. 

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