Real-time visual feedback of COM and COP motion properties differentially modifies postural control structures

Abstract

The experiment was setup to investigate the control of human quiet standing through the manipulation of augmented visual information feedback of selective properties of the motion of two primary variables in postural control: center of pressure (COP) and center of mass (COM). Five properties of feedback information were contrasted to a no feedback dual-task (watching a movie) control condition to determine the impact of visual real-time feedback on the coordination of the joint motions in postural control in both static and dynamic one-leg standing postures. The feedback information included 2D COP or COM position and macro variables derived from the COP and COM motions, namely virtual time-to-contact (VTC) and the COP–COM coupling. The findings in the static condition showed that the VTC and COP–COM coupling feedback conditions decreased postural motion more than the 2D COP or COM positional information. These variables also induced larger sway amplitudes in the dynamic condition showing a more progressive search strategy in exploring the stability limits. Canonical correlation analysis (CCA) found that COP–COM coupling contributed less than the other feedback variables to the redundancy of the system reflected in the common variance between joint motions and properties of sway motion. The COP–COM coupling had the lowest weighting of the motion properties to redundancy under the feedback conditions but overall the qualitative pattern of the joint motion structures was preserved within the respective static and dynamic balance conditions.

Appendices

Appendix A—virtual time-to-contact (VTC) in 2D space

Input data for VTC (Slobounov et al. 1997) calculation in MATLAB was the 2D position of the COP or COM along with the instantaneous velocity and acceleration vectors, respectively. The 2D stability boundary was defined as the outside edge of the foot namely the base of support and was modeled by projecting the markers placed at the distal phalanges, fifth metatarsal, lateral malleolus heel and medial malleolus onto the ground and connecting them with line segments.

VTC (τ) at each time instant is the time the COM or COP would need to contact with the 2D stability boundary if it were to continue from the current position ($\overrightarrow {r} = \left[ {\mathop r\nolimits_{x} ,\mathop r\nolimits_{y} } \right]T$) with instantaneous initial velocity ($\overrightarrow {v} = \left[ {\mathop v\nolimits_{x} ,\mathop v\nolimits_{y} } \right]T$) and instantaneous constant acceleration ($\overrightarrow {a} = \left[ {\mathop a\nolimits_{x} ,\mathop a\nolimits_{y} } \right]T$).

Let (x _c , y _c ) denote the point on the stability boundary where the virtual trajectory intersects it for the first time. If the end points of the corresponding boundary line segment are (x ₁, y ₁) and (x ₂, y ₂), the slope (s) of the line connecting the two points is

$$ s = (\mathop y\nolimits_{2} - \mathop y\nolimits_{1} )/(\mathop x\nolimits_{2} - \mathop x\nolimits_{1} ) $$

(1)

Assuming constant slope in the differential segment between (x ₁, y ₁) and (x ₂, y ₂), the slope can also be computed as

$$ s = (\mathop y\nolimits_{c} - \mathop y\nolimits_{1} )/(\mathop x\nolimits_{c} - \mathop x\nolimits_{1} ) $$

(2)

Assuming a point mass model for the COM and constant acceleration, the point of virtual contact can be written as,

$$ \mathop x\nolimits_{c} (\tau ) = \mathop r\nolimits_{x} + \mathop v\nolimits_{x} \cdot \tau + \mathop a\nolimits_{x} \cdot \frac{{\mathop \tau \nolimits^{2} }}{2} $$

(3)

$$ \mathop y\nolimits_{c} (\tau ) = \mathop r\nolimits_{y} + \mathop v\nolimits_{y} \cdot \tau + \mathop a\nolimits_{y} \cdot \frac{{\mathop \tau \nolimits^{2} }}{2} $$

(4)

Substituting x _c and y _c from Eqs. 3–4 in 2, and equating it to 1, gives a quadratic equation in τ. VTC (τ) is the lowest positive solution of this quadratic equation. In the case where both velocity and acceleration were zero, VTC would be infinity.

Appendix B—canonical correlation analysis (CCA)

Canonical correlation analysis (CCA) is a general approach that can reveal the linear structure between two sets of variables (Brillinger 1975). CCA is based on simultaneous singular value (eigenvalue) decomposition of two multivariate datasets in such a way that the component scores associated with the first eigenvector of the first dataset has maximum correlation with the component scores associated with the first eigenvector of the second dataset. Given the first eigenvectors, the component scores associated with the second eigenvectors (which are orthogonal to the first eigenvectors) again have maximum correlation, etc.

Let Set 1 with p variables and n observations be represented by a n × p random variable X and set 2 with q variables and n observations by a n × q random variable Y. CCA creates d = min (rank(X), rank(Y)) pairs of n × 1 linear combinations (=component scores) U and V of the original variables from each set:

$$ U_{i} = Xa_{i} \quad {\text{and}}\quad V_{i} = Yb_{i} $$

(5)

where i = 1,…, d, a _i and b _i are p × 1 and q × 1 coefficient vectors. Let S be the total (p + q, p + q) dimensional variance–covariance matrix of X (set 1) and Y (set 2):

$$ S = \left[ {\begin{array}{*{20}c} {S_{11} } & {S_{12} } \\ {S_{21} } & {S_{22} } \\ \end{array} } \right] $$

(6)

Using singular value decomposition the eigenvalues in decreasing order and the corresponding eigenvectors of

$$ A_{p} = S_{11}^{ - 1} S_{12} S_{22}^{ - 1} S_{21} \quad {\text{and}}\quad A_{q} = S_{22}^{ - 1} S_{21} S_{11}^{ - 1} S_{12} $$

(7)

are obtained. The ith eigenvector of A _p constitutes the a _i coefficients and the ith eigenvector of A _q the b _i coefficients. The canonical correlations are derived from the first d eigenvalues λ _i . The canonical correlation r _i is the square root of λ _i . The eigenvalues of A _p and A _q are the same, and either one can be used to obtain the canonical correlation.

$$ r_{i} = \sqrt {\lambda_{i} } $$

(8)

CCA was performed using standardized data; therefore, S can be replaced by the correlation matrix ρ. A pair of component scores associated with the ith eigenvectors of the two sets is commonly termed the ith canonical function. The significance of each canonical function (pairs of U and V) was assessed using F-statistics.

The CCA cross-loadings are the bivariate correlations between each original variable and the component score of the other set. There are no general guidelines for distinguishing high versus low cross-loadings (Noble et al. 2004). Therefore, the interpretation of the cross-loadings is kept at a qualitative level. The CCA redundancy index of each set can be obtained by multiplying the average proportion of total variance by the squared canonical correlation coefficient.

The CCA redundancy index can also be obtained by averaging the squared cross-loadings. It quantifies the amount of variance represented by the component score associated with the ith eigenvector of set 1 that can be explained by the component score associated with the ith eigenvector of set 2 and vice versa. Similar to R ² in multiple regression it is the shared variance between the two sets.