Coordinate Constraint Influences Static Optimization Results

[Alexander Bruno]

Hi Everyone,

I'm trying to understand how the presence of a coordinate coupled constraint in a model factors into the static optimization analysis. I've been using the Lumbar 210 spine model, which uses a coordinate coupled constraint to specify the amount of rotation of each intervertebral joint as a function of total trunk flexion. However, if this constraint is removed, and rotations at each intervertebral joint are specified independently (in the motion file for instance), the model appears weaker during static optimization (larger muscle forces are required to achieve equilibrium).

So it appears that the constraint is adding strength or stability to the model since for the same body position muscle forces are different. This makes some intuitive sense since the constraint reduces the model degrees of freedom, but I was hoping someone could explain to me the precise way in which the constraint is being factored into the static optimization analysis, or point me to some reference/documentation that does, so that I can attempt to quantify this strengthening effect.

Ultimately, I'm interested in estimating vertebral loading, and although the constraint makes specifying the model kinematics convenient, I really want to determine if it's appropriate to have from a mechanics perspective.

Thanks for any advice on this issue,
Alex

[Ton van den Bogert]

Dear Alex,

Your question intrigued me because I also use coordinate coupler constraints in my models.

I did a simple example (on paper), with a two-link inverted pendulum, stabilized by two 1-joint muscles against an external horizontal load at the top. In the 2-DOF case (no coordinate coupling), you can easily solve the two muscle forces (no static optimization needed) and this solution is independent of muscle properties. In the 1-DOF case (with coordinate coupling), you can use virtual work or Lagrange method to get the static equilibrium equation. This is one equation defining a relationship between the external force and the two unknown muscle forces. It is easily shown that the original solution (from the 2-DOF case) is still a solution for the 1-DOF model. But it may not be the optimal solution.

So your intuition was right, the reduction of DOFs allows other solutions, possibly with lower muscle forces because the load can shift to the muscles with larger moment arms.

I suppose this means that you have to be careful with static optimization in models that have coordinate coupling. Coordinate coupling acts as if there are kinematic elements (such as gears) that couple the joint rotations. In reality, these gears do not exist and the muscles are responsible for this coupling. The optimal solution from the coupled model would not necessarily create an equilibrium posture in the spine if you removed the kinematic coupling.

Ton van den Bogert

[jimmy d]

Hi Alex,

Just wanted to add that coordinate coupled constraints account for the moment between bodies during a simulation. Static Optimization works by distributing the required joint moment across the muscles of that joint. If the coordinate coupled constraint is taking up the moment, then there is no need for static optimization to use the muscles. So the behavior you are seeing is to be expected.

Hope that, with Tons excellent response, helps.

cheers,
-james

[Alexander Bruno]

Thank you Ton and James! That cleared some things up.

-Alex