Direct Dynamics for Single Closed Chains

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 

In [8], Featherstone introduces a similar approach which combines the N dynamic equations for the chain with n simultaneous quadratic equations in the unknown contact forces. The main difficulty with this proach is the solution of the quadratic equations, which have, in general, 2 = roots. Even afta the unknown contact forces have been found, an N x N inverse is still required to solve for the joint accelerations. Although this tq proach is quite genoal, it is computationally intensive, and the quadratic equations are not easily dealt with. 

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