Single closed chain

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. 

EFFICIENT DYNAMIC SIMULATION OF A SINGLE CLOSED CHAIN... 

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 this chapter, an efficient serial algorithm for the dynamic simulation of a single closed chain is developed. The algorithm is valid fw a manipulator with... 

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. 

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. 

One of the first Direct Dynamics algorithms for single closed-chain robotic mechanisms is presented by Orin and McGhee in [33]. This algorithm is based on the in a matrix invasion aj roach. The dynamic equations of motion for the chain are augmented with kinematic constraint equations at the tip of the... 

Dynamic Simulation Algorithm for a Single Closed Chain... 

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 problem, we will begin our analysis with the dynamic equations of motion for the entire simple closed-chain system. First, we will consider the dynamic equations for each supporting chain, and then we will formulate an appropriate dynamic equation for the reference member al ie. 

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. 

If (nc)jb is the number of degrees of constraint for the genoal joint at the tip of chain k, then Mt is the (nc)k x (ne)t transformation matrix which solves for h. The basic solution procedure for h, evoi with ao unknown, is identical to the i proach discussed in Section S.6.2 for a single closed chain. Thus, this general solution is still valid for a chain in a singular position or a chain with less than six (xiginal degrees of freedom. The solution procedure requires 0[(ne) ] scalar opo ons. 

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 ... 

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