Computing the COM from the COP in postural sway movements Pietro G. Morasso1,2, Gino Spada2 and Roberto Capra2 1 University of Genova - Dept. of Informatics, Systems and Telecommunication, Italy 2 Center of Bioengineering, Rehabilitation Hospital la Colletta, Arenzano, Italy

Abstract A method is described for computing the Center of Mass (COM) from empirical estimates of the Center of Pressure (COP) obtained by means of a force platform. The method is based on a biomechanical model of sway movements in quiet standing, according to which the horizontal acceleration of the COM is approximately proportional to the COM-COP difference. The equations are solved by approximating the solution with best fitting spline functions. The implications for movement control are discussed.

Introduction. In spite of its apparent simplicity, the nature of the control mechanisms that allow humans to stabilise postural sway is still an object of controversy. Sway is the persistent oscillation of the COM (Center Of Mass), the controlled variable, in the AP (Antero-Posterior) and ML (Medio-Lateral) planes. This variable is not measurable in a direct way, although is frequently confused with the COP (Center Of Pressure) that is typically measured by means of a force platform. The difference between the two trajectories is small but is significant and the study of its structure is an important method for addressing fundamental motor control problems. The most general way to compute the trajectory of the COM is to apply the definition (see, for example, Winter et al. 1998), i.e. measuring the COM of each body segment by means of stereophotogrammetric methods and then adding them up, weighted according to their mass. However, this is very cumbersome, especially if we think of clinical applications, and requires a very critical calibration. More practical methods are limited to force platform data. An empirical approach is based upon the fact that the two trajectories have similar shapes but the COP trajectory appears to be more "noisy"; thus, the computation can be seen as a type of "filtering" (Benda et al. 1994). A more formal approach (Shimba 1984, Zatsiorsky and King, 1998) takes into account that the second time derivative of the horizontal COM trajectory is proportional to the horizontal component of the ground reaction force: this force is measured by means of a fully equipped force platform and integrated twice, with a special technique for recovering the unknown initial conditions. Here we propose a new method that is based on a biomechanical model but does not require to measure the horizontal component of the ground reaction force and thus can operate with a much cheaper force platform. The model. In the following we only consider AP oscillations. However, the model and the algorithm can be applied to ML oscillations as well. Figure 1 shows the human “inverted pendulum” in v which we single out the ground reaction force f = ( f H , f V ) , the force of gravity mg, and the ankle torque τankle due to the muscles. For the foot we can write an equilibrium equation: f H δ + fV u + τ ankle = 0 (1)

1

Computing the COM from the COP

Human Movement Science 18, 759-67, 1999

where δ,u are, respectively, the vertical and horizontal displacements of the COP with respect to the ankle. For the sway movements of the rest of the body the following equations apply f V − mg = m&z& ≈ 0 f H = m&y& (2) &y& d τ ankle + mgy = ( Iθ& ) ≈ Iθ&& ≈ mh 2 k s = mhk s &y& dt h where m is the weight of the body excluding the feet, h is the distance of the COM from the ankle, I is its moment of inertia with respect to the ankle joint, and ks is a shape factor that depends on the distribution of mass in the body. In particular, ks=1 if all the mass were concentrated in the COM (in that case I=mh2) and ks=1.33 if it were distributed in a uniform way along a thin rod (in that case I=4/3mh2). For the human body ks is closer to the latter estimate than the former one. There are three approximations in the model, that are well satisfied for sway in quiet standing: (i) the vertical acceleration of the COM is negligible, (ii) the moment of inertia is constant, and (iii) the angular acceleration is proportional to the horizontal acceleration of the COM. We can now combine Eq. 1 and 2, obtaining the following sway equation: g &y& = ( y − u) (3) (hk s + δ ) or, more simply, &y& =

g ( y − u) he

(4)

having defined an "effective distance" he=ksh+δ. In particular, the equation says that the COM-COP difference is proportional to the horizontal component of the ground reaction force.

Fig. 1. Mechanics of quiet standing. 2

Computing the COM from the COP

Human Movement Science 18, 759-67, 1999

The algorithm. The problem is to integrate Eq. 4. Since we do not know the initial state, we cannot use standard numerical integration techniques that, in any case, have numerical instability problems when applied to unstable systems. However we know that the solution must be a smooth function of time and we can seek it within a suitable class of functions by means of a variational method. This requires to define a functional to be minimised, using the equation as a constraint. We chose a piece-wise polynomial representation of y(t) in terms of B-spline functions (De Boor, 1978) because it is linear_in_the_parameters. Since the system equation (4) is linear then the functional is quadratic and the solution can be provided by the LSE method. In order to have an algorithm whose performance is independent of the duration of the experiment, we adopted a moving-window paradigm, i.e. the LSE is iterated on shifted strings of data of duration ±Tw. The estimation step. Let m be the number of B-splines, n the number of samples in the time window, { p i } the set of unknown coefficients that weight the influence of each B-spline. The B-spline approximation of y(t) can be written as follows: m

y (t ) = ∑ Bi (t ) ⋅ pi

(5)

i =1

It must be m