Predictive Dynamics - Human Motion Simulation: Predictive Dynamics - page 106

Γ 3

x Front

heel

Γ 2

r 3

r 4

o z ZMP r 2

Γ 4 r 1

Rear
toe

Γ 1

FIGURE 5.7
Foot support polygon (top view).

5.7.1.4 Dynamic balance
The dynamic balance is achieved by enforcing the ZMP to remain within the foot
support polygon (FSP) as depicted in Figure 5.7 , where is a vector along the

boundary of FSP and r is the position vector from a vertex of the FSP to ZMP.
The ZMP constraint is mathematical expressed as follows:

Γ

ð r i 3 Γ i Þ n y # 0 ; i 1 ; ...; 4 (5.24)

5

where n y is the unit vector along the y axis.

5.7.1.5 Self-avoidance
Self-avoidance is considered in the current formulation to prevent penetration of
the arm in the body. A sphere-filling algorithm is used to formulate this constraint
as shown in Figure 5.8 , as follows:

dð q ; tÞ 2 r 1 2 r 2 $ 0 ; 0 t T (5.25)

# #

where r 1 is a constant radius to represent the wrist, and r 2 is another radius to rep-
resent the hip; d is the distance between wrist and hip.

5.7.2 Time-independent constraints
5.7.2.1 Symmetry conditions
The gait simulation starts from the left heel strike and ends with the right heel
strike. The initial and final postures and velocities should satisfy the symmetry
conditions to generate continuous walking motion. These conditions are expressed
as follows:

q L ð 0 Þ 2 q R ðTÞ 5 0 q L ð 0 Þ 2 _ _ q R ðTÞ 5 0

q Sx ð 0 Þ 2 q Sx ðTÞ 5 0 q Sx ð 0 Þ 2 _ _ q Sx ðTÞ 5 0

(5.26)

q Sy ð 0 Þ 1 q Sy ðTÞ 5 0 q Sy ð 0 Þ 1 _ _ q Sy ðTÞ 5 0

q Sz ð 0 Þ 1 q Sz ðTÞ 5 0 q Sz ð 0 Þ 1 _ _ q Sz ðTÞ 5 0