Post
by Ton van den Bogert » Wed Nov 21, 2012 1:49 pm
I do not use Opensim for inverse dynamics but the problem seems somewhat fundamental and not limited to Opensim. We are working on inverse dynamic analysis of the shoulder mechanism. There is a closed kinematic chain (thorax-clavicle-scapula-thorax) and therefore more unknown joint moments than equations of motion. The way we are dealing with this is to do a combined inverse dynamic analysis and static optimization in one step, rather than sequentially.
The equations of motion are written for the open kinematic chain, and the reaction loads at the loop closure are treated as unknown loads, along with the joint moments. So there is a vector T that contains all these unknown loads. The equations of motion are:
M(q)*qdd + c(q,qd) + B(q)*T = 0
(I use Autolev, which gives me symbolic expressions for M, c, and B)
(d and dd stand for "dot'" and "dotdot", the first and second derivatives with respect to time)
B is a matrix with more columns than rows, so we can't solve T. The classical example would be a planar 4-bar linkage with one grounded link and all joints actuated. The open chain has 3 DOF, so 3 equations of motion, but T has 6 elements: 4 joint torques and X and Y force at the loop closure.
In a musculoskeletal model, T contains joint moments which are related to muscle forces F: T = R(q)*F. R is a matrix with all moment arms. The rows in R that correspond to the reaction loads in T will simply be all zeros. Now we can formulate the static optimization problem:
minimize a cost function f(F)
subject to linear constraints A*F = b, where
A = B(q)*R(q)
b = -M(q)*qdd - c(q,qd)
We're just starting on this, so I can't report results yet but I believe that this is a good and easy way to do it. I would be interested in Sherm's comments.
Alternatively, you could obtain the closed-loop equation of motion, so the reaction loads would not appear at all. You still would have more unknowns than equations of motion. For instance, the closed 4-bar linkage has 1 DOF but 4 unknown driving torques. For the shoulder, we did not want to do it that way because we need to know some of the reaction loads (and because it is much easier to get the open loop equations of motion). Some static optimization approaches include the constraint that the force between scapula and thorax (the loop closure force) must be compressive and normal to the sliding surface (van der Helm, J Biomech 1994). So we needed to bring this force into the equations anyway.
One aspect that is not quite elegant is that we estimate body pose q with an inverse kinematic analysis that does not enforce the kinematic constraint at the loop joint. But if the data is good anyway, that should be OK. In ergonomics, you are probably doing forward kinematics so you will need a way to generate consistent kinematics for the closed loop system. Opensim may be able to do the inverse or forward kinematics with full consideration of the closed loop mechanism.
If your model is torque-driven rather than muscle-driven, it can be done the same way but you then need to define a torque-based cost function f(T). The reaction loads are linearly related to the joint torques that are responsible for the movement (via the equation of motion), but they should be excluded from f(T) because there is no physiological cost. In Daniel's simple 1-segment example, the solution of this static optimization problem would be zero joint torque which is exactly what I would expect a human to do in this situation.
Apologies for the long post, but this is a good opportunity for me to get some early feedback on what we are doing. It may also help solve Daniel's problem or be useful to the Opensim developer team.
Ton van den Bogert