Simple closed-chain mechanism reference member

A brief review of previous work related to the dynamic simulation of simple closed-chain mechanisms is given in the second section of this chapter. A model for simple closed-chain mechanisms is described in the third section, and the basic problem statement is also discussed. The dynamic equations of motion for the individual chains and the common reference member are summarized in the fourth section. 

In [31], Oh and Orin extend the basic method of Orin and McGhee [33] to include simple closed-chain mechanisms with m chains of N links each. The dynamic equations of motion for each chain are combined with the net face and moment equations for the reference membo and the kinematic constraint equations at the chain tips to form a large system of linear algebraic equations. The system unknowns are the joint accelerations for all the chains, the constraint fcwces applied to the reference memba, and the spatial acceleration of the reference member, lb find the Joint accelerations, this system must be solved as a whole via standard elimination techniques. Although this approach is sbmghtforward, its computational complexity of 0(m N ) is high. 

The structure of a simple closed-chain mechanism is characterized by m actuated chains which support a single common reference member [31]. The supporting chains are assumed to be serial-link chains, firee of intmial closed loops. Therefore, the removal of the reference membo breaks all closed Io( in the system. Each chain may have an arbitrary numbo of links and degrees of freedom. The... 

As was mentioned biefly in Chapt 1, th arc two basic types of simple closed-chain mechanisms called Type 0 and lype 1, respectively [31]. These two types are defined based on the nature of the interactions which occur between the links of the chain and the reference member or support surface. For both types, the support surface acts as the base of each chain. We will refa to the link which interacts with the support surface as link 1, and the link which intmcts with the reference member will be called the last link or end effector (link N). The far end of link N is the tip of the chain. 

More specifically, in a simple closed-chain mechanism, the m actuated chains act on the reference member in parallel, and their motion is coupled with that of the reference member. If the reference member is removed, the chains may function independently. Computationally, the physical removal of the reference member corresponds to solving for the forces which it ex on each chain. Once these forces are known, the system is equivalent to a group of independent chains with known tip forces. The general joint accelerations may then be computed for each chain separately. Given enough processors (at least one per chain), the computations for each chain may be carried out simultaneously. 

We may now solve for ao ftom this linear system of algebraic equations using any linear system solver. Note that the characteristic system mabix is only 6x6 and represents the effective operational space inotia of the complete simple closed-chain mechanism as seen by the reference member. 

The triplication of Equation 6.42 to ev actuated chain in the simple closed-chain mechanism results in a complete solution to the Direct Dynamics problem for this robotic system. The next state positions and velocities may be computed by integrating the appropriate quantities for each chain and the reference member. As discussed in Chapter 5, small amounts of negative position and rate feedback may be employed to countra t the drift which is a result of the integration process. 

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. 

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