Optimization with joint space reduction and extension induced by kinematic limits for redundant manipulators

This paper shows how to find the optimal trajectory with respect to a certain cost when a kinematically redundant manipulator is subject to joint kinematic limits. Relevant mathematical properties in accordance with our intuition are strictly enlightened to be useful for this seemingly easy albeit complicated problem. Key terms of ´reducibility´ and ´(weak) optimal extensionability´ are defined for further discussion of the solution. The final solution is basically achieved by the composition of applying the extended Jacobian method on each joint space where the joint coordinates saturated with the limits are eliminated. We would then be able to understand how the transition query is characterized and why the solution should be in that way. The simulation example would reveal how the solution gets links with our physical intuition and this would make us cope with unexpected algorithmic singularities