Simple closed-chain mechanism dynamic equations

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. 

Each chain in the simple closed-chain mechanism is governed by the dynamic equations of motion fw a single chain. These are ... 

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. 

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. 

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