Optimality principles for model-based prediction of human gait☆
Introduction
Dynamic simulation of human gait is a well established research technique but has been mainly applied to track observed human movements (Zajac et al., 2003). Predictive simulations of gait, on the other hand, are based solely on an assumed optimality criterion, e.g. minimal energy, which is used to solve an optimal control problem. Such simulations can help uncover underlying principles of neuromuscular coordination and have potential applications in predicting patient responses to surgical interventions (e.g. a tendon transfer procedure), in the design of prosthetic and orthotic devices, or in the reconstruction of gait for dinosaurs (Sellers and Manning, 2007). Predictive simulation has not yet found widespread application because of its high computational cost (Anderson and Pandy, 2001). In addition, there is no generally accepted optimality criterion for human gait. Previous studies have only used a single optimality criterion and it is not clear how the choice of optimality criterion affects the results.
Energy consumption appears to play a role in the selection of overall gait characteristics, such as step length and cadence, as corroborated by many experimental studies (e.g. Bertram and Ruina, 2001, Ralston, 1976). This, however, does not necessarily imply that minimal energy also governs detailed features of gait such as joint angles and muscle recruitment. Other criteria such as muscle fatigue or peak joint loads might play a role.
In particular, the knee flexion observed in the weight acceptance phase of normal gait raises questions as it requires the activation of the large Quadriceps to prevent knee collapse and may, therefore, be inconsistent with minimal energy criteria. Indeed, stance phase knee flexion appears to be absent in several energy-based predictive gait simulations (Sellers et al., 2005, Nagano et al., 2005).
In this context, the objective of this paper is to shed some light into the effects of the cost function choice on the predicted kinematics and muscle recruitment patterns of gait. A series of predictive simulations of gait are performed utilizing a family of cost functions representative of a large range of performance criteria traditionally adopted in the literature.
Section snippets
Musculoskeletal model
The musculoskeletal dynamics model (Gerritsen et al., 1998, Hardin et al., 2004) consists of seven body segments (trunk, thighs, shanks, and feet) and has nine kinematic degrees of freedom. Eight muscle groups are included in each lower extremity: Iliopsoas, Glutei, Hamstrings, Rectus Femoris, Vasti, Gastrocnemius, Soleus, and Tibialis Anterior. Each muscle is represented by a 3-element Hill-type model, using the equations from McLean et al. (2003) and muscle properties from Gerritsen et al.
Results
Fig. 1, Fig. 2 show the results for the simulations using the muscle volume-based weighting factors, , corresponding to cost functions , , and , and for the simulations using a unitary weighting factors, , corresponding to cost functions , , and in Table 1, respectively. A comparison of the predicted kinematics on the left and of the ground contact forces on the right is shown for the four different exponents p (1, 2, 3 and 10) in the cost function, Eq. (9). As
Discussion
Different performance criteria led to substantially distinct gait predictions, Fig. 1, Fig. 2, Fig. 3, showing the importance of the choice of an appropriate cost function. Perhaps the most remarkable differences occur in knee flexion angle during initial and mid-stance. Cost functions , , and led to a maximal knee flexion of or higher in mid-stance, while cost functions , , and led to a straight-legged pattern, with only very slight knee flexion. As explained in Section
Conflict of interest statement
None declared.
Acknowledgments
This study is supported by the NIH Grant R01 EB006735 and NSF Grant BES 0302259.
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Parts of this work were first presented in January 2008 at the Workshop 3 — Biomechanics and Neural Control — Muscle, Limb and Brain, Mathematical Biosciences Institute, Columbus, OH.