Predicting the Biomechanics of Walking - Human Motion Simulation: Predictive Dynamics - page 160

To calculate ZMP, the inertia J i and angular acceleration θ i must be evaluated
in the global coordinates; however, they have been defined in the local coordi-
nates associated to link i in the DH method. The transformation to the global
coordinates is tedious and time-consuming. Therefore, some researchers simply
ignore these terms in ZMP calculation.
To overcome this difficulty, an alternative method is developed to calculate
ZMP based on global equilibrium condition, or equivalently, the forces and
moments in the virtual branch obtained from the equations of motion. The basic
idea is to use the resultant of the active forces and moments to calculate ZMP
directly instead of evaluating them link by link as in Equations (7.13).
Given state variables ( q j , q j ) for each joint, apply active forces to the

mechanical system, excluding GRF. The calculated generalized forces (

q j , and _ €

τ 1 , τ 2 , τ 3 ,

τ 4 , τ 6 ) in the virtual branch from the equations of motion are in fact the resul-

tant active forces and moments. After obtaining the resultant of the active forces,
we can use them to calculate ZMP using the following three steps:
1. Calculate the resultant of the active forces and moments at the pelvis in the
inertial reference frame;
2. Transfer these forces and moments to the origin of the inertial reference frame
(o- xyz );
3. Calculate the ZMP from its definition.

τ 5 ,

We present more details of these three steps below.

7.7.1 Global forces at the pelvis
We denote the forces at the pelvis by F p 5 ½ F x F y F z

T .

τ 6 ) at the pelvis are defined
in the local coordinates ( z 5 , z 6 ). Because of the global rotational movements
( q 4 , q 5 ), the forces at the pelvis no longer align with the global Cartesian coordi-
nates (o- xyz ), as shown in Figure 7.8 . Since ZMP is defined in the global
Cartesian coordinates, we need to recover

τ 5 ,

The direction of the resultant active moments (

the resultant active moments

5 ½ M x M y M z

M p

T at the pelvis in the global Cartesian coordinates. This is
accomplished using the following equilibrium equation:

2 3 2 3 2 3

M z
M x
M y

cos ðz 4 ; xÞ cos ðz 4 ; yÞ cos ðz 4 ; zÞ τ 4

τ 5
τ 6

4 5 4 5 1 4 5 5

cos ðz 5 ; xÞ cos ðz 5 ; yÞ cos ðz 5 ; zÞ 0 (7.14)

cos ðz 6 ; xÞ cos ðz 6 ; yÞ cos ðz 6 ; zÞ

where τ 6 are resultant moments along the DH local axes associated with

their DOFs; cos ðz 4 ;

τ 4 , τ 5 ,

xÞ 5 0, cos ðz 4 ; yÞ 5 0, and cos ðz 4 ; zÞ 5 1 because the first rota-

tional joint is aligned with global z - axis.
The resultant active forces F p

T at the pelvis are obtained by
considering the equilibrium between two sets of forces as follows:
F x 1 τ 2 5

5 ½ F x F y F z

F y 1 τ 3 5 F z 1 τ 1 5

0 ; 0 ; 0 (7.15)