KINETICS: FORCES AND MOMENTS OF FORCE - Biomechanics and Motor Control of Human Movement - page 130

decelerate it and reverse its direction. Note that all reversals of direction of
COM coincide with an overshoot of the COP signal.

5.2.9 Kinematics and Kinetics of the Inverted Pendulum Model
All human movement (except in space flights) is done in a gravitational
environment, and, therefore, posture and balance are continuous tasks that
must be accomplished. In normal daily activity at home, at work, and in our
sports and recreation, we must maintain a safe posture and balance. The base
of support can vary from one foot (running) to a four-point support (football),
and it is essential that the COM remain within that base of support or move
safely between the two feet if it lies temporarily outside the base of support
(as it does in running and during the single-support phase of walking). There
is a common model that allows us to analyze the dynamics of balance: the
inverted pendulum model, which relates the trajectories of the COP and COM.
As was seen in Section 5.2.8, the position of the COP relative to the COM
decides the direction of the angular acceleration of the inverted pendulum. A
full biomechanical analysis of the inverted pendulum model in both sagittal
and frontal planes has been presented by Winter et al. (1998). In the sagittal
plane, assuming that the body swayed about the ankles, it was shown that:

I S C OM x
Wh

COP x − COM x =− (5.9)

where: I s = moment of inertia of body about ankles in sagittal plane

C OM x = forward acceleration of COM

W = body weight above ankles

h = height of COM above ankles

In the frontal plane, the balance equation is virtually the same:

I f C OM z
Wh

COP z − COM z =− (5.10)

where: I f = moment of inertia of body about ankles in frontal plane

C OM z = medial/lateral acceleration of COM

The major difference between Equations (5.9) and (5.10) is in the muscle
groups that control the COP. In Equation (5.9), COP x M a /R , where M a is

the sum of the right and left plantarflexor moments and R is the total vertical
reaction force at the ankles. In Equation (5.10), COP z

=

= M t /R , where M t =

M al + M hr. M al with M ar are the left and right frontal ankle

moments, while M hl and M hr are the left and right frontal plane hip moments.
Thus, frontal plane balance is controlled by four torque motors, one at each

M ar + M hl +