Control of Standing

from Morasso  PG & Schieppati M, Journal of Neurophysiology (1999): 1622-1626

The Body Center of Mass (CoM) is at the top, balanced over the ankle joint by the moving force vector. The point at the bottom of the force vector is called the Center of Pressure (CoP).

Resolving moments, the equation of such a system is:

                       m.g.CoM - M_a + z =  I.a

where I = moment of inertia of the body about the ankle joint; a = angular acceleration about the ankle; m = body mass; g = acceleration due to gravity; M_a is the moment about the ankle; and z represents the perturbing moments arising from internal (arising from breathing motions etc.) and external forces.

In quiet standing, the moment at the ankle, M_a, must be equal to the ground reaction force (m.g) multiplied by its moment arm, the CoP:

                        M_a  =  m.g.CoP

and this must balance the center of mass (mg) multiplied by its moment arm (CoM), together with any inertial moments caused by acceleration of the CoM, and the perturbing moments:

                        m.g.CoP  = m.g.CoM - I.a + z

Since the angle of sway is usually small (< 8 degrees), a = a, the acceleration of the CoM. Furthermore, I = m.h²k_s, where k_s is a shape factor (1 if the CoM is concentrated at the CoM, and 1.33 if body mass is uniformly distributed along a rod-shape.

Therefore, we have:

                       m.g.CoP  = m.g.CoM - m.h².k_s.a + z

where h = height of body center of mass above the ankle. If we make h_e = h.k_s, then we have:

                        m.h.h_e.a = m.h.g(CoM - CoP) + z

and if we divide throughout by m.h_e, and making z' = z/(m.h_e), this becomes:

                        a - CoM.g/h_e = - CoP.g/h_e + z'

Converting to Laplace notation, the open-loop transfer function of this is:

                        CoM(s).s² - CoM(s).g/h_e = Z'(s) - CoP(s).g/h_e
or,
                       CoM(s) = (Z'(s) - CoP(s).g/h_e)/(s² - g/h_e)

Such systems are unstable if any of the roots of the denominator are positive:

for (s² - g/h_e):
                        Roots = +/- sqrt(g/h_e)
Thus, the system is unstable.

If we add a Proportional + Derivative controller, with gains of K_p and K_d, respectively, i.e:

                        CoP(s) = CoM(s).(K_p + sK_d)

we get the closed-loop transfer function,

                         CoM(s) = Z'(s)/(s² + s.g/h_eK_d + g(K_p - 1)/h_e)

The denominator this time is (s² + s.g/h_eK_d + g(K_p - 1)/h_e)

For this system, when the proportional gain, K_p > 1, both roots of the denominator must be negative, and the system will be stable, reacting to perturbations with a series of damped oscillations.