Postural feedback responses scale with biomechanical constraints in human standing

Abstract

We tested whether human postural responses can be described in terms of feedback control gains, and whether these gains are scaled by the central nervous system to accommodate biomechanical constraints. A feedback control model can describe postural responses for a wide range of perturbations, but biomechanical constraints—such as on the torque that can be exerted on the ground—make a single set of feedback gains inappropriate for all perturbations. To observe how postural responses change with perturbation magnitude, we applied fast, backward perturbations of magnitudes 3–15 cm to 13 healthy young volunteers (4 men, 9 women, aged 20–32 years). We used a 3-segment, sagittal-plane biomechanical model and a linear state feedback controller to reproduce the observed postural responses. Optimization was used to identify the best-fit feedback control gains for each trial. Results showed that trajectories of joint angles and joint torques were scaled with perturbation magnitude. This scaling occurred gradually, rather than abruptly changing at magnitudes where biomechanical constraints became active. Feedback gains were found to fit reasonably well with data (R ²=0.92) and to be multivariate and heterogenic in character, meaning that the torque produced at any joint is generally a function of motions not only at the same joint, but other joints as well. Hip gains increased and ankle gains decreased nearly linearly with perturbation magnitude, in accordance with biomechanical limitations on ground reaction torque. These results indicate that postural adjustments can be described as a single feedback control scheme, with scalable heterogenic gains that are adjusted according to biomechanical constraints.

Appendix

Biomechanical model and optimization of feedback parameters

The model’s equations of motion are of the form (Kuo and Zajac 1993):

$$M(\theta )\ddot{\theta } = V(\theta ,\dot{\theta }) + G(\theta ) + T + W(a) $$

(1)

where θ is a vector of joint angles, M is the mass matrix, V is a vector of velocity-dependent terms, G is a vector of gravity-dependent terms, T is a vector of joint torques, and W is a vector dependent on external perturbation magnitude a. The inertial parameters for the models were found by incorporating a series of 27 anthropometric measurements into a nonlinear regression model of the body segments (Yeadon and Morlock 1989). We used Eq. 1 in inverse dynamics computations to estimate joint torques (Kuo 1998), and also in the feedback control model (Fig. 2). We also developed a 4-segment model, including knee motion, to quantify the advantages of a more complex model (see Discussion).

These equations of motion were incorporated in a feedback control model of postural responses to perturbations of the support surface. We used a linear state feedback control (Barin 1989; Kuo 1995), producing joint torques as a function of the joint angles and velocities to stabilize the body (see Fig. 2). We assumed that the CNS can either directly sense or indirectly estimate the equivalent of the state information $x = {\left[ {\theta _{{{\text{ank}}}} \;\theta _{{{\text{hip}}}} \;\dot{\theta }_{{{\text{ank}}}} \;\dot{\theta }_{{{\text{hip}}}} } \right]}^{T}$. The control task was to produce joint torques $u{\mathop = \limits^\delta }T$ with the feedback law:

$$u = K{\left( {x - x_{{{\text{ref}}}} } \right)} $$

(2)

where K is the (2×4) feedback control gain matrix, and x _ref is the state corresponding to the upright reference position. The state equations, linearized about the upright vertical position, are of the form:

$$ \dot{x} = Ax + Bu + w $$

(3)

where A, and B are system matrices, and w describes perturbation. Combining Eqs. 2 and 3 yields the closed-loop system:

$$ \dot{x} = {\left( {A - BK} \right)}x + w $$

(4)

whose stability depends on the quantity (A--BK). For a given subject, A and B are relatively fixed, and the gain (K) is selected by the CNS. The selection of K effectively determines the postural response strategy. The movement resulting from a perturbation can be predicted from initial conditions and the perturbation magnitude (w), resulting in simulated state and torque trajectories over time, x _sim and u _sim, respectively.

We used optimization to describe the postural response strategy in terms of the feedback parameters. The objective was to minimize the sum-squared, normalized deviations of the model states x _sim from the experimental data x _exp and the model torques u _sim from the data u _exp:

$$J(K) = {\sum {\delta x^{T} Q\delta x + \delta u^{T} \delta u} } $$

(5)

where $\delta {\text{x}}{\mathop = \limits^\Delta }{\left( {x_{{\exp }} - x_{{{\text{sim}}}} } \right)}/{\left| {x_{{\exp }} } \right|},\;\delta {\text{u}}{\mathop = \limits^\Delta }{\left( {u_{{\exp }} - u_{{{\text{sim}}}} } \right)}/{\left| {u_{{\exp }} } \right|}$ and the summation occurs over samples of recorded data. The Q matrix was used to weight the relative contributions of errors in state and control, and was chosen to be Q=0.01I ^4×4 where I is the identity matrix. This places equal weighting on all states relative to each other, with the overall scaling factor of 0.01 chosen to place some weighting on matching experimentally-derived joint torques. One constraint was placed on the optimization, requiring a stable closed-loop system, i.e., eigenvalues of the system matrix having nonpositive real parts. Therefore, the constrained optimization problem is written mathematically as follows:

$$ {\mathop {\min }\limits_K }\;J(K)\;{\text{subject}}\;{\text{to}}\;\operatorname{Re} {\left\{ {{\text{eig}}{\left( {A - BK} \right)}} \right\}} \leqslant 0 $$

(6)

To perform the optimization, we used a sequential quadratic programming (SQP) algorithm, one of many algorithms that can minimize a quadratic objective subject to nonlinear constraints (Shittowski 1985). We repeated the optimization multiple times using random initial guesses for K, to check for local minima in the optimization. As a measure of the degree of fit, we calculated R ² for each fit using the same relative weightings of Eq. 5.