1. Prof. Anthony Petrella Musculoskeletal Modeling & Inverse Dynamics MEGN 536 – Computational Biomechanics

2. MSM: Computing Muscle Forces  With ID problem for arm curl…  Six equations (3 per body):  Six unknowns: F Ex, F Ey, M E, F Ox F Oy, M O  Determinate system  Add muscle forces…  Same eq’s + more unknowns  Indeterminate system  Need optimization shoulder hand elbow

3. MSM: Computing Muscle Forces  Reaction moment (joint torque) is created by muscle forces spanning the joint  Reaction force is created by resultant of muscle forces and contact force Inverse Dynamics Muscle Forces & Contact 

4. MSM: Biarticular Muscles  Works the same way as uniarticular case  Consider all muscle forces that cross the joint Inverse Dynamics Muscle Forces & Contact 

5. MSM: Muscle Moment Arms Moment arm is perpendicular… …but cross product takes care of this automatically

6. MSM: Muscle Recruitment Problem  Muscle forces make MSM system indeterminate, also referred to as underdetermined system…  For example, several knee extensor muscles contribute to extensor moment… no unique solution  How to decide distribution of force among “parallel” muscles? This is referred to as the muscle recruitment problem  Something like… min  (a i )  strong muscles do work first  How about… min max (a i )  extreme sharing across all m’s  Most common… min  (a i p = 2 or 3 )  a reasonable balance, there is sharing among muscles, but strong still work a bit more  AnyBody default… min  (a i 3 )

7. Muscle Recruitment: Optimization Problem  Design vector: {a i } = vector of muscle activations  Objective function: G =  (a i 3 )  Recall… forces expressed as F = a * Strength, where activation restricted to range [0,1]  Lower bound (0): muscle forces cannot be negative  Upper bound (1): muscles have finite strength   Constraints: 1.0 ≤ a i ≤ 1 (can be two inequality constraints or LB + UB) 2.Muscle forces must also satisfy moment equation(s)… (equality constraint(s))

8. Muscle Recruitment: Optimization Problem   Constraints: Muscle forces must also satisfy moment equation(s)… (equality constraint(s))  If you want to use Aeq and beq, you simply rearrange the constraint equations to be linear in x… for example…

9. Muscle Recruitment: Optimization Problem   Constraints: Muscle forces must also satisfy moment equation(s)… (equality constraint(s))  If you wish to use the rxf() function for the moments, you’ll need to implement a non-linear constraint function (which is easy and the better way to do it)

10. Muscle Recruitment: Optimization Problem  Non-linear constraints… from the MATLAB Help: