Fig 1.
Single shooting framework for dynamic optimization.
We trained a planar, musculoskeletal model actuated by 18 Hill-type musculotendon actuators to walk by optimizing the parameters of a gait controller based on an objective function that sought to minimize metabolic cost, avoid falling and injury, and stabilize the head. The gait controller computed muscle excitations, u(t), for a musculoskeletal model to generate a forward simulation. Sensory feedback, based on the model’s muscle and joint states, was used in a feedback loop with the gait controller. The objective function quantified the performance of each simulation, and a Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) optimization method updated the values of the variables in the optimization problem.
Fig 2.
A planar musculoskeletal model for walking.
The musculoskeletal model had nine degrees of freedom (dof). A 3-dof planar joint between the pelvis and ground was the base joint. Each hip and ankle was a 1-dof pin joint, and each knee was a 1-dof coupled joint. The model’s 18 musclulotendon actuators (red lines) represented the 9 major uniarticular and biarticular muscle groups per leg that drive sagittal plane motion: gluteus maximus (GMAX), biarticular hamstrings (HAMS), iliopsoas (ILPSO), rectus femoris (RF), vasti (VAS), biceps femoris short head (BFSH), gastrocnemius (GAS), soleus (SOL), and tibialis anterior (TA). A compliant contact model was used to generate forces between the spheres at the heel and toes of the feet and the ground plane.
Fig 3.
The gait controller used a combination of a state machine and low-level control laws to determine excitations.
The state machine had three states in stance (i.e., early stance (ES), mid-stance (MS), and pre-swing (PS)) and two states in swing (i.e., swing (S) and landing preparation (LP)), and it determined when low-level control laws were active. Low-level control laws included constant signals (C), feedback terms based on muscle length (L), muscle velocity (V), and muscle force (F), and proportional-derivative control (PD) based on the pelvis tilt angle. Positive and negative feedback are denoted by (+) and (-), respectively. All feedback laws based on muscle states acted upon the same muscle, except for a negative force feedback from the soleus to the tibialis anterior.
Fig 4.
Kinematics and kinetics of simulated walking over a range of speeds.
Seven prescribed speeds between 0.50 m/s (blue) and 2.00 m/s (red) at intervals of 0.25 m/s were analyzed. Joint angles (left column) and joint moments (middle column) are plotted for the hip (top row), knee (middle row), and ankle (bottom row). Positive joint angles indicate flexion, and positive joint moments indicate extension. Ground reaction forces (GRFs) (right column) are shown for both horizontal (top row) and vertical (middle row) directions. With exceptions at the knee joint, we observed trends that occur in the experimental data of Schwartz et al. [58], including greater joint angle ranges, joint moments, and ground reaction forces at higher speeds.
Fig 5.
Spatiotemporal and metabolic parameters over a range of speeds.
Simulated (dots) and experimental (lines) data are plotted for percent stance phase (left), step length (middle left), cadence (middle right), and cost of transport (right). Experimental data for the left three panels represent the mean ± 2 standard deviations from Schwartz et al. [58]. In the right panel, the magnitude and characteristic shape of the cost of transport bowl were similar between the simulated data and three experimental data sets from Ralston, Martin et al., and Browning et al. [59–61].
Fig 6.
Kinematics and kinetics of gait at self-selected speed.
Simulated (black line) kinematics and kinetics are compared to experimental data (gray area) collected by Schwartz et al. [58]. Joint angles (left column) and joint moments (middle column) are plotted for the hip (top row), knee (middle row), and ankle (bottom row). Ground reaction forces (right column) are shown for both horizontal (top row) and vertical (middle row) directions. Positive angles indicate flexion, and positive moments indicate extension. Note that the experimental data for hip flexion angle was shifted by 11.6° to account for the difference in pelvis orientation definitions between the experimental data set and the musculoskeletal model.
Table 1.
Similarity metrics between simulated and experimental data for self-selected gait.
Fig 7.
Muscle activity during gait at self-selected speed.
Simulated muscle activations (black) are compared to on-off timings estimated from experimental electromyograms (EMG) reported by Perry and Burnfield (gray) [63]. Because the muscles in the model represent groups of muscles, we show on-off timing of multiple muscles for some comparisons (e.g., for HAMS, experimental data represent the long head of the biceps femoris, semitendinosus, and semimembranosus). S3 Table contains more details about these comparisons.
Fig 8.
Z-scores of key kinematic and kinetic measures of gait for all deficit cases.
Z-scores were computed using previous experimental data of individuals walking at a self-selected speed from Schwartz et al. [58]. All 18 cases (dots) are plotted for each measure. A black dot indicates any case within 2 SDs of normal (gray band). A blue or orange dot indicates either a weakness or contracture case, respectively, which is outside of 2 SDs. All of these cases are labeled with the affected muscle or muscles: soleus (SOL), gastrocnemius (GAS), or both (PF); and severity level: mild (Mild), moderate (Mod), or severe (Sev). One other case is labeled (see Peak Plantarflexion Moment, PF Mod*) as it was a clear outlier compared to other deficit cases. The count above each line (e.g., n = 13) indicates the number of unlabeled dots.
Fig 9.
Ankle kinematics and kinetics for moderate and severe PF weakness.
Simulated ankle kinematics and kinetics for unimpaired walking (black lines) and walking with PF weakness (blue lines) are compared to experimental data of unimpaired individuals (gray area) [58]. Ankle dorsiflexion (top) and ankle plantarflexion moment (bottom) are plotted for cases of moderate (left) and severe (right) PF weakness.
Fig 10.
Ankle kinematics and kinetics for severe contracture.
Simulated ankle kinematics and kinetics for unimpaired walking (black lines) and walking with severe contracture (orange lines) are compared to experimental data of unimpaired individuals (gray area) [58]. Ankle dorsiflexion (top) and ankle plantarflexion moment (bottom) are plotted for cases of SOL (left), GAS (middle), and PF (right) contracture.
Fig 11.
Hip and knee kinematics for severe contracture.
Simulated hip and knee kinematics for unimpaired walking (black lines) and walking with severe contracture (orange lines) are compared to experimental data collected from unimpaired individuals (gray area) [58]. Hip flexion (top) and knee flexion (bottom) are plotted for cases of SOL (left), GAS (middle), and PF (right) contracture.