Simple closed-chain mechanism examples

Figure 6.2 Example of a Type 1 Simple Closed-Chain Mechanism...

The method outlined above is quite straightforward and illustrates some features of a parallel computational structure as discussed previously. Of course, the illustrated example represents a special case. We will now develop a similar approach fex a general simple closed-chain mechanism which, although the notation becomes a bit more complicated, also exhibits the parallel computational characteristics we expect. 

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. 

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. 

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