Predictive Dynamics - Human Motion Simulation: Predictive Dynamics - page 99

two categories: active forces and passive forces. Active forces include inertia,
gravity, and applied external forces and moments. Passive forces are the ground
reaction forces (GRF). The ZMP position and the GRF are calculated from the
equations of motion by using a two-step algorithm: the resultant active forces and
the ZMP location are calculated in the first step, and the GRF are calculated in
the second step. The basic idea of this algorithm is to obtain the GRF from the
resultant active forces by imposing the overall equilibrium of the digital human
model at ZMP. This algorithm was first introduced in Xiang et al. (2007) to simu-
late 3D human gait, and details were presented elsewhere ( Xiang, 2008; Xiang
et al., 2009b ). It is outlined below but will be expanded upon in the next chapter.
The following calculations are performed at each time instant:
Step 1: Calculation of resultant active forces and ZMP

o )at
the origin in the global coordinate system (o-xyz), excluding GRF, are
calculated from equations of motion using inverse dynamics.
1.2. The ZMP position is calculated from its definition using the global resultant
active forces and assuming the feet to be on the level ground, as follows:

o , F

1.1. Given q , q , _ q for each DOF, the global resultant active forces (M €

M z
F y ;

M x
F y

z zmp 5 2 y zmp 5 0 ; x zmp 5

(5.4)

.
1.3. After obtaining the ZMP position, the resultant active forces at ZMP (M

o

M x M y M z

T o

F x F y F z

T

where M and F

5 5

zmp ,

zmp ) are computed using the equilibrium conditions as follows:

F

zmp o o

M

5 M

r zmp 3 F

o zmp o

1 ; F

5 F

(5.5)

where o r zmp is the ZMP position in the global coordinate system
obtained from Equation (5.4) .
Step 2: Calculation of GRF

The value and location of GRF are calculated from the equilibrium between
the resultant active forces and passive forces at the ZMP:

GRF zmp GRF zmp o o

M

1 M 5 0 ; F 1 F 5 0 ;

r GRF 2 r zmp 5 0 (5.6)

All active forces (gravity, inertia and applied external forces) and passive
forces (GRF) are applied to the entire human body model, and the equations of
motion are used to obtain the real joint torques. In this case, the global forces are
zero and the equilibrium of the system is satisfied automatically.
There are two prominent features to calculate GRF in the foregoing two-step
algorithm:

1. GRF are explicitly calculated from joint kinematics; and differentiation of
GRF with respect to the system kinematics is easily achieved.