The interaction between surgical tools and soft tissue has nonlinear mechanical properties. In robot-assisted surgery, a pre-specified trajectory of tool positions (or applied forces) is desired for achieving a surgical treatment. Due to the computational complexity involved in solving the nonlinear optimal trajectory following problem, a tractable path is to find lower bounds on the optimal cost. The lower bound serves as a reference to evaluate the goodness of any specific controller. A mathematical derivation of a lower bound on the optimal cost of the dynamic optimization problem, using a linear matrix inequality (LMI) approach, is developed. Semi-definite programming is used to solve the LMIs. This approach is demonstrated for a previously studied soft tissue grasping control problem.
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