Generality is the main advantage rate control methods enjoy over closed-form methods. With little (if any) modification, rate control methods will solve the inverse kinematics problem for robots with a wide range of geometries. Computational efficiency has historically been a concern with rate control methods, but modern computing hardware greatly reduces this concern. Closed-form approaches are computationally very efficient. They also provide insight into the underlying physical phenomena often associated with ill-conditioned equations and singularities. The closed-form analysis is often quite complex and specific to a particular geometry. The outstanding work developing closed-form approaches diminished this concern by deriving a body of available solutions covering most robot
geometries.
Constraint Tracking - Constraint tracking uses the equality
constraints on the position and orientation of the robot's
End EFfector (EEF) to reduce the solution space of the optimization problem by six (three position and three orientation). If the robot has two arms, then constraint tracking will reduce the solution space by twelve. Figure 1. depicts the geometry of the
transformations. To satisfy the constraints, the transformation associated with the robot's n joints must equal the above transformation.
Tracking the equality constraints requires extracting from T a transform associated with only six joint displacements. The extraction procedure follows a standard implementation.
Inverting the transform T using inverse kinematics generates six joint displacements to satisfy the six equality constraints.
Simulated Annealing - The operator assist interface must solve a global optimization problem. Table 1. lists some options for finding global optima. These options include: a "shotgun" approach tracking gradients from different starting places, simulated annealing based on models of the physical annealing process, genetic algorithms based on models of biological genetics, brute force exhaustive evaluation, and the Monte Carlo based on randomness and statistics. All of these methods will solve global optimization problems. The difficulty lies in the need for interactive response (a few seconds) from the configuration advisor. In an optimization with seventeen DOF, none of these methods would have interactive response on available computer hardware. With constraint tracking, all of the methods except brute force will have interactive response in a configuration advisor application. This section discusses an implementation of the simulated annealing method. Even in complex environments with multiple obstacles and competing performance criteria, the implementation has proven reliable.
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