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We get and can once again implement the position level constraint on the velocity level by requesting , where is a position stabilization constant. This lets us implement the position level linear motor by using the corresponding velocity level linear motor. 2 3-DOF angular motor Let’s first present a 3-DOF velocity level angular motor that directly controls relative angular and in the world coordinate frame, both in terms of the rotation direction velocity of and speed. This motor is an angular analog to the 3-DOF linear velocity motor presented earlier and hence we can simply define the three dimensional motor velocity measure (t) as .

34 rad (20 deg) with the nonlinear controller for the two links (see Table 2 and Table 3). The hybrid controller proposed deals well with the flexibility of the link as the deflection is lessened (see Figure 5 and Figure 10). However, results obtained with the nonlinear control alone are slightly better. 036 m with the nonlinear control. 011 m with the nonlinear control We notice also from Figure 5 and Figure 10, the appearance of vibrations with the hybrid control. However, their amplitude is lessened.

We have already seen this formulation for the first three rows and so only need to consider . We have 0 and thus , 0 and , 22 Motion Control . Note that 0 equals the relative angular velocity (rotation speed) about the axis and so the 0 explicitly prohibits relative body rotation about velocity-level constraint . In turn, the two constrained bodies can only rotate about and . The unstabilized acceleration level formulation of the constraint is obtained by differentiating the velocity level formulation with respect to time.