Coordination of muscle torques stabilizes upright standing posture: an UCM analysis

Abstract

The control of upright stance is commonly explained on the basis of the single inverted pendulum model (ankle strategy) or the double inverted pendulum model (combination of ankle and hip strategy). Kinematic analysis using the uncontrolled manifold (UCM) approach suggests, however, that stability in upright standing results from coordinated movement of multiple joints. This is based on evidence that postural sway induces more variance in joint configurations that leave the body position in space invariant than in joint configurations that move the body in space. But does this UCM structure of kinematic variance truly reflect coordination at the level of the neural control strategy or could it result from passive biomechanical factors? To address this question, we applied the UCM approach at the level of muscle torques rather than joint angles. Participants stood on the floor or on a narrow base of support. We estimated torques at the ankle, knee, and hip joints using a model of the body dynamics. We then partitioned the joint torques into contributions from net, motion-dependent, gravitational, and generalized muscle torques. A UCM analysis of the structure of variance of the muscle torque revealed that postural sway induced substantially more variance in directions in muscle torque space that leave the Center of Mass (COM) force invariant than in directions that affect the force acting on the COM. This difference decreased when we decorrelated the muscle torque data by randomizing across time. Our findings show that the UCM structure of variance exists at the level of muscle torques and is thus not merely a by-product of biomechanical coupling. Because muscle torques reflect neural control signals more directly than joint angles do, our results suggest that the control strategy for upright stance involves the task-specific coordination of multiple degrees of freedom.

Appendix: The relationship between forces at the COM and joint torques

Let x be a point on the body given by the position of the COM in reference configuration.

The equation of motion of the full body is given by

$$M\left( \theta \right)\ddot{\theta } + C\left( {\theta ,\dot{\theta }} \right)\dot{\theta } + G\left( \theta \right) = \tau$$

(1)

where M(θ) is the inertia or mass matrix of the full dynamical equations of motion, $C(\theta ,\dot{\theta })\dot{\theta }$ are the centrifugal and Coriolis joint torques, G(θ) is gravity torque vector, and τ is the vector of muscle torques. J is the kinematic Jacobian relating displacement of the COM position x to changes in joint angles.

$$\partial x = J \cdot \partial \theta$$

(2.1)

$$\Rightarrow J = \frac{\partial x}{\partial \theta }$$

(2.2)

In a given body configuration θ, the relationship between joint torques τ.

And the force F that these ⇒torques exert on the COM is linear and given by

$$F = \bar{J}^{\text{T}} \tau$$

(3)

The general solution of this equation is

$$\tau = J^{\text{T}} F + \left[ {I - J^{\text{T}} \bar{J}^{\text{T}} } \right]\tau_{0}$$

(4)

where τ ₀ is and arbitrary joint torque vector. Together with Eq. 1 we get

$$\left[ {I - J^{\text{T}} \bar{J}^{\text{T}} } \right]\tau_{0} = M\ddot{\theta } + C + G$$

(5)

In the dynamic case with gravity, torques $[I - J^{\text{T}} \bar{J}^{\text{T}} ]\tau_{0}$ that do not affect the endpoint force in Eq. 4 must satisfy the following dynamical constraint.

$$JM^{ - 1} \left[ {I - J^{\text{T}} \bar{J}^{\text{T}} } \right]\tau_{0} = 0$$

(6)

We solve the Eq. 6 for $\bar{J}^{\text{T}}$

$$JM^{ - 1} \tau_{0} - JM^{ - 1} J^{\text{T}} \bar{J}^{\text{T}} \tau_{0} = 0$$

$$JM^{ - 1} J^{\text{T}} \bar{J}^{\text{T}} = JM^{ - 1}$$

$$\bar{J}^{\text{T}} = \left[ { JM^{ - 1} J^{\text{T}} } \right]^{ - 1} JM^{ - 1}$$

(7)

$$\bar{J} = M^{ - 1} J^{\text{T}} \left( {JM^{ - 1} J^{\text{T}} } \right)^{ - 1}$$

(8)