OPTEC - K.U.Leuven Center of Excellence: Optimization in Engineering

"(Bio-)Mechanical Engineering Applications of Convex Optimization"
Bram Demeulenaere (K.U. Leuven, Dept. Mechanical Eng. PMA)

Convex optimization problems constitute a class of nonlinear optimization problems for which every local optimum is also a global optimum. As a result, the global optimum can be found very efficiently using dedicated algorithms. Optimization problems belonging to this class are, for instance, linear programs (LPs), second-order cone programs (SOCPs) and semidefinite programs (SDPs). In this talk we introduce three mechanical engineering applications, and show how convex optimization techniques can be applied to them.
The first application is counterweight balancing, that is, the problem of determining counterweights for a linkage, such that it exerts smaller forces and moments on its supporting frame. It is shown that determining the optimal counterweight parameters can be formulated as an SDP for spatial mechanisms and an SOCP for planar mechanisms.
As a second application, we show that generating (dynamically) optimal motion trajectories, parametrized using B-splines, can be formulated as an LP, provided that one limits oneself to certain classes of constraints and goal functions. It is shown that the resulting optimization framework is nevertheless very flexible and capable of tackling real-life examples. Quite interestingly, the presented framework behaves like a spline knot-optimization algorithm with fast and guaranteed global convergence.
As a third, bio-mechanical application, we consider dynamic musculoskeletal analysis, that is, the problem of determining the muscle forces that underly some experimentally observed human motion. It is shown that this challenging, large-scale, nonconvex optimization problem can be solved in an efficient manner by using convex optimization techniques. That is, an approximate, convex program is formulated and solved, in order to provide a hot-start for the exact, nonconvex program. The key element in this approximation is a (global) linearization of muscle physiology, based on techniques from experimental system identification. This approach is applied to the study of muscle forces during gait.
The results presented comprise joint work with Goele Pipeleers, Myriam Verschuure, Erwin Aertbelien, Jan Swevers, Joris De Schutter (all affiliated with the Mech. Eng. Dept., PMA Division) Pieter Spaepen (Mech. Eng. Dept., BMGO Division), Ilse Jonkers (Dept. of Kinesiology) and Lieven Vandenberghe (University of California Los Angeles -- Electrical Engineering Dept.).