Acceleration of the End Effector

A similar difference expression may be developed for the end effector accelo -ation vector, x, as follows. The end effector velocity, it, is defined  

As was true for the joint accelerations, if the present state, driving actuator tffl ques and/or forces, and motion of the base are known, the open-chain term, Xopenf is known. Its value may also be determined using an tq pr(piate Direct Dynamics algorithm for ( n-chain manipulators. Because the joint positions are assumed known, the inverse operational space inertia matrix, A is defined. The efficient computation of A and its inverse was the primary topic of Ch ter 4. 

---

For every unknown component of the contact force vector, there is a corresponding known value of relative linear or angular acceleration of the end effector in (or along) the same direction which is a result of the constraint [33]. Because the acceleration of the contacted body is known, and the components of the relative acceleration are known in the constrained directions, the components of the absolute end effector acceleration in the constrained directions are also known. That is, fOT every component of the unknown h , th is a known component of g . Also, as with any joint , the components of force in the free directions, h, are known (or their relationship to the constrained components is known), and the relative accelerations in the free directions, g, are unknown. Examples of the two classes of contacts are now discussed. 

The reaction tcx ques/forces at the powered joints and the relative accelerations of the end effector in the free directions of the contact are also unknown, but their solution is not usually required. They may also be computed as desired. 

---