Equations of Motion and Joint Accelerations

The dynamic equations of motion for a single N degree-of-freedom chain may be written (as given in Chapter 3)  

The equations of motion for the chain may be used to partition the joint acceleration vector into the difference oT two tmns. Fa- a given position, the first term corresponds to the chain in an open, unconstrained configuration with the contact force removed, while the second term is a function of the contact 

Using Equation 5.1, we may solve for the vector of joint accelerations for the closed chain, q, as foUows  

Given the p sent state of the manipulator, q and q, the input joint actuator traques and forces, r, and the motion of the base, the open-chain term, ijopen. is completely defined. An appropiate Direct Dynamics algorithm for )en-chain manipulator may be used to detomine its value. With the same givoi information, the coefficient of f in the constrained term, G, is also known. The efficient computation of G was discussed in Chapter 4. 

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