Theoretical three-and four-axis gimbal robot wrists

In high-performance flight simulations, a four-axis gimbal system allows all possible rotations with acceptable gimbal angle rates while it avoids the so-callled 'gimbal lock' that occurs when gimbal rotational axes are colinear. In this paper, pertinent equations (including quaternions) are assembled for a hypothetical robot wrist, functionally equivalent to this four-axis gimbal system, and also for a true three-axis gimbal robot wrist. These equations are used to simulate the rotation of a robot hand by the robot wrist in response to operator rotational velocity commands to the robot hand. Near gimbal lock (wrist singularity), excessive rotational rates occur. Scaling the rates, which is necessary for the three-gimbal robot wrist to prevent rate limiting, introduces an undesirable time delay in the robot hand rotation with respect to the commanded rotation. However, the merit of the four-gimbal robot wrist is that the fourth gimbal angle keeps the robot wrist away from the singularity so that the robot hand moves exactly as commanded. It appears that in a 'worst-type' maneuver of the robot hand, the fourth gimbal angle can be defined so that none of the gimbal angle rates exceed about twice the commanded rates.