Surrogate Model Creation

The time varying quantities muscle-tendon lengths ($l^{MT}$), muscle-tendon velocities ($v^{MT}$), and moment arms (r) are calculated using polynomial functions of the joint angles and velocities that share common coefficients. For muscles that span a single degree of freedom (DOF) and for the selection of a cubic polynomial, the muscle-tendon length is approximated using the following equation:

$$l^{MT}(t) = b_0 + b_1 \theta + b_2 \theta ^2 + b_3 \theta ^3$$

where $\theta$ is joint angle and $b_0$ through $b_3$ are constant coefficients. Muscle-tendon velocity can then be calculated using the first derivative with respect to time of muscle-tendon length.

$$v^{MT}(t) = \frac {dl^{MT}} {dt} = b_1 \.\theta + 2 b_2 \theta \.\theta + 3 b_3 \theta ^2 \.\theta$$

where $\.\theta$ is joint angular velocity. Similarly, the muscle-tendon moment arm can be calculated from muscle-tendon length using a relationship from An et al.:

$$r(t) = - \frac {\partial l^{MT}} {\partial \theta} = -b_1 -2b_2 \theta -3b_3 \theta ^2$$

The negative sign in this expression is needed for consistency with the OpenSim musculoskeletal modeling environment, where a positive joint moment causes a positive change in joint angle. For muscles that span two DOFs and for the selection of a cubic polynomial, these equations are extended as follows:

$$l^{MT}(t) = b_0 + b_1 \theta_1 + b_2 \theta_2 + b_3 \theta_1 \theta_2 + b_4\theta_1^2 + b_5\theta_2^2 + b_6 \theta_1^2 \theta_2 + b_7 \theta_1 \theta_2^2 + b_8 \theta_1^3 + b_9 \theta_2^3$$

$$v^{MT}(t) = b_1\.\theta_1 + b_2\.\theta_2 + b_3(\.\theta_1 \theta_2 + \theta_1 \.\theta_2) + 2b_4\theta_1\.\theta_1 + 2b_5\theta_2\.\theta_2 + ...$$

$$b_6(2\theta_1\.\theta_1\theta_2 + \theta_1^2\.\theta_2) + b_7(\.\theta_1\theta_2^2 + 2\theta_1\theta_2\.\theta_2) + 3b_8\theta_1^2\.\theta_1 + 3b_9\theta_2^2\.\theta_2$$

$$r_1 = - \frac {\partial l^{MT}} {\partial \theta_1} = -b_1 -b_3\theta_2 -2b_4 \theta_1 -2b_6\theta_1\theta_2 - b_7\theta_2^2 -3b_8\theta_1^2$$

$$r_2 = - \frac {\partial l^{MT}} {\partial \theta_2} = -b_2 -b_3\theta_1 -2b_5 \theta_2 -b_6\theta_1^2 -2b_7\theta_1\theta_2 -3b_9\theta_2^2$$

For muscles that span three, four or more degrees of freedom, the equations for muscle-tendon length, muscle-tendon velocities, and moment arms are extended in a similar manner by adding terms corresponding to the additional joint angles and velocities.