1. MEGN 536 – Computational Biomechanics Prof. Anthony J. Petrella

2. Inverse Kinematics in Musculoskeletal Analysis  Need to know joint angles that define model configuration at each increment in the motion  Markers used to track motion and match model to individual  We can then do dynamics calcs to find joint reactions / torques  Muscle forces and joint contact forces can also be estimated Ali, et al., CMBBE, 2013

3. Inverse Kinematics in Musculoskeletal Analysis Problems…  Model markers don’t perfectly match subject  Multiple sources of exp. error  Segments may have multiple markers  over-determined  Cannot solve for model config exactly (closed form)  Have to use alternate method, such as optimization Ali, et al., CMBBE, 2013

4. Optimization in Inverse Kinematics

5. Optimization Problem Statement  Find x that minimizes f(x), subject to A*x ≤ b and A eq *x = b eq x =design vector f(x) objective function A*x ≤ b inequality constraints  A eq *x = b eq equality constraints x1:xnx1:xn e.g. -F quad ≤ 0 e.g.  F x = 0 

6. Optimization Problem Statement  Note that the goal is to minimize the objective function in an absolute sense – not to make it small, but to make it as negative as possible  If you are solving a problem in which you wish to maximize a value (e.g., find the maximum volume of a box using a fixed amount of material for the sides of the box), then you simply minimize the negative of the objective function… f(x) = -f vol (x) = -(L * W * H)

7. Optimization Example  Given: the planex 1 + 2x 2 + 4x 3 = 7  Problem: find point on the plane closest to the origin design vector x = obj. function f(x)= dist = sqrt(x 1 2 + x 2 2 +x 3 2 ) ineq. constraints A*x ≤ b - none - eq. constraints x 1 + 2x 2 + 4x 3 = 7 (A eq = [1 2 4]; b eq = [7]) x1x2x3x1x2x3

8. Optimization Example - Solution

9. Optimization in MATLAB  Type ‘ doc fmincon ’ at the MATLAB command line  All of the inputs and outputs of fmincon are explained in the MATLAB documentation  To use fmincon you will need to write a MATLAB function and use function handles