Physiological LQR Design for Postural Control Coordination of Sit-to-Stand Movement

Abstract

The neurophysiological mechanisms involved in postural stabilization are not well understood. Active and passive mechanisms at muscle and spinal levels as well as visual and vestibular processes are known to contribute toward postural stabilization and coordination of voluntary movement. The motivation for this research is to use a modeling–simulation framework to achieve two aims: (a) to ascertain viability of a physiologically motivated optimal controller design in the maintenance of posture and coordination of voluntary movement and (b) to study the relative contribution from active (feedforward) and passive (feedback) mechanisms in the execution of said movement. We employ a multi-segment sagittal model built on anatomical proportions with three degrees of freedom, including rotation at the ankle, knee, and hip joints. The behavior of the biomechanical model is controlled by an optimal linear quadratic regulator whose state and control weights are derived from physiological considerations. Representative postural and voluntary movements are simulated to illustrate the analysis–synthesis framework of biomechanical movement. Our analytical and simulation results support an active–passive model of postural stabilization and movement coordination. Besides expanding our understanding of the physiological stabilization processes in the body, the insight gained from this study promotes awareness of the existence of optimizing controllers in the central nervous system.

Appendix

The inertial, Coriolis, and gravitational matrices appearing in the nonlinear dynamic model (1) are given as [18]:

$$ D = \left[ {\begin{array}{*{20}c} {d_{11} } & {d_{12} \cos \left( {\theta_{1} - \theta_{2} } \right)} & {d_{13} \cos \left( {\theta_{1} - \theta_{3} } \right)} \\ {d_{12} \cos \left( {\theta_{1} - \theta_{2} } \right)} & {d_{22} } & {d_{23} \cos \left( {\theta_{2} - \theta_{3} } \right)} \\ {d_{13} \cos \left( {\theta_{1} - \theta_{3} } \right)} & {d_{23} \cos \left( {\theta_{2} - \theta_{3} } \right)} & {d_{33} } \\ \end{array} } \right] $$

(A1)

$$ H = \left[ {\begin{array}{*{20}c} 0 & {d_{12} \theta_{5} \sin \left( {\theta_{1} - \theta_{2} } \right)} & {d_{13} \theta_{6} \sin \left( {\theta_{1} - \theta_{3} } \right)} \\ { - d_{12} \theta_{4} \sin \left( {\theta_{1} - \theta_{2} } \right)} & 0 & {d_{23} \theta_{6} \sin \left( {\theta_{2} - \theta_{3} } \right)} \\ { - d_{13} \theta_{4} \sin \left( {\theta_{1} - \theta_{3} } \right)} & { - d_{23} \theta_{6} \sin \left( {\theta_{2} - \theta_{3} } \right)} & 0 \\ \end{array} } \right] $$

(A2)

$$ G = g\left[ {\begin{array}{*{20}c} {f_{1} \cos \theta_{1} } & {f_{2} \cos \theta_{2} } & {f_{3} \cos \theta_{3} } \\ \end{array} } \right] $$

(A3)

$$ x_{COP} = l_{f} - a + \frac{{\tau_{1} + bF_{x} - cm_{f} g}}{{F_{y} }}. $$

(A4)