Step 3. Closed-Chain Joint Accelerations

The formulation of the simulation algorithm described here makes this third stq particularly straightforward. With the contact force vector, f, completely defined, the vector of closed-chain joint acceloations, q, may be found using Equation 5.6, which is repeated here for convenience  

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. 

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In the second step, the dynamic equations of the end effector are combined with the contact model to determine the unknown components of the contact force vector. The computations required for this second step differ slightly for the two classes of contacts discussed in Section 5.5, but the basic concq>tual approach is the same in either case. Once the contact face vecto is completely defined, a full solution for the closed-chain joint accelerations may be found from the corresponding equations of motion. This is the third step. In the fourth and final step, the joint accelerations and rates are integrated to find the next state joint rates and positions. The next four subsections explain each of these four steps in some detail. 

The spatial fcare vecux, f, exited by each chain on the reference member, and the closed-chain joint accelerations for the chain, q, are calculated in Steps 3 and 4 of the simulation algoithm, respectively. The appropriate equations are given in Table 6.1. The q>erations required to calculate these vectors complete the table of computations. The specific number of operations required for the special case of TV = 6 and ne = 3 are also provided in Table 6.2. This value of Tie might correspond to a hard point contact between the manipulator tip and a constraining body or surface when the tip is not slipping. 

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. 

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