Table 1.
Summary of the ten predictive simulations.
The task simulated is either sprinting or marathon running. Motor coordination is always a decision variable, whereas body segment factors and muscle volume scaling factors are either set to the values of a model or also optimized. The model name describes which model is used or results from the simulation. We finally mention which figure in the results section displays results from each simulation.
Fig 1.
Our differentiable musculoskeletal simulator generates the derivatives of the state variables given the state variables (muscle activations am, torque actuator activations aT, tendon forces Ft, generalized positions q and velocities ) and the decision variables (skeleton segment scaling factors ps, muscle volume scaling factors
, muscle excitations em, torque actuator excitations eT). This is achieved by evaluating a set of dynamics equations: activation dynamics, torque actuator dynamics, muscle dynamics, and skeleton dynamics. Evaluating muscle and skeleton dynamics depends on the outputs of musculoskeletal geometry computations (i.e., muscle-tendon lengths lmt and velocities
and muscle moment-arm matrices R) and on the scaled muscle parameters (pm,scaled). Since the scaling of the skeleton and muscle volumes are decision variables, we formulated musculoskeletal geometry computation, muscle parameter scaling and skeleton dynamics as a differentiable function of these decision variables. The dotted boxes indicate the parts of the simulator where we turned non-differentiable computation used in OpenSim and Falisse et al. [39] into differentiable computation. Tendon forces are mapped to joint muscle torques (τm) by the moment-arm matrix (R). Torque actuator activations are scaled to torque actuator torques (τT) by a scaling factor of 150 [10]. A contact function (fcontact) based on the Hunt-Crossley contact model gives the generalized forces resulting from contact (fc).
Fig 2.
Sprinting speed (A), marathon lower limb energy cost (B), height (C), mass (D) for the generic model and models with optimal body-segment dimensions for sprinting and marathon running. The dots and names in (C) and (D) represent the seven all-time fastest male 100m and marathon runners. In (E) the scaling of individual body segments are shown for the three dimensions: length, width, and depth. The scaling factors are normalized to the generic model. In (F) the figure shows the joint torque capacity of the sagittal plane degrees of freedom for the different models normalized by the capacity of the generic model.
Fig 3.
Sprinting speed (A) and marathon lower limb energy cost (B) for the generic model and models after optimal ‘strength training’ for sprinting and marathon running. (C) shows the joint torque capacity of the sagittal degrees of freedom for the different models normalized by the capacity of the generic model. (D) shows normalized muscle maximal isometric force for the different models organized per degree of freedom.
Fig 4.
Joint torques for the sagittal-plane degrees of freedom across the sprinting cycle for the right leg.
FLEX indicates flexion moment, EXT indicates extension moment, DF indicates dorsiflexion moment, PF indicates plantarflexion moment.