Predicting the Biomechanics of Walking - Human Motion Simulation: Predictive Dynamics - page 169

flexion of normal gait is around 60 degrees regardless of the subject's age and
gender. This constraint is expressed as

2 ε #

q knee 2 60

# ε;

t t midswing (7.34)

5

where ε is a small range of motion, i.e., ε 5 5 degrees.

7.9.3.2.3 Initial and final foot contacting position
Since the step length L is given, the foot initial and final contacting positions are
specified at the initial and final times to satisfy the step length constraint. It is
noted that the initial and final postures and velocities are determined by the opti-
mization process instead of specified from the experiments.

x i ð 0 Þ 5 x i ð 0 Þ;
x i ðTÞ 5 x i ðTÞ;

(7.35)

i AΩ

where x i is the specified initial and final contacting position, and ~ Ω is the set of

contacting points as specified in Table 7.2 .

7.10 Numerical discretization
In order for the numerical solver to calculate the design variables, the time
domain is discretized by using cubic B-spline curves, which are defined by a set
of control points P and time grid points (knots) t. A joint profile qðtÞ is parameter-
ized by using B-splines as follows:

X

m

qð t ; P Þ 5 B i ð t Þp i 0

# t # T

(7.36)

i 5 1

where B i ð t Þ 5 ft 0 ; ...;

t s g

are the basis functions, t is the knot vector, and

P p m g is the control points vector. With this representation, the control

points become the optimization variables (also called the design variables).
Since q , q , and q are functions of t and P, torque τ 5 τð t ; P Þ is an explicit
function of the knot vector and control points. Thus, the derivatives of a torque
with respect to the control points can be computed using the chain rule as

5 fp 1 ; ...;

q @ _ q @ €

@τ
@ P i 5 @τ q @ P i 1 @τ

q

P i 1 @τ

q q

(7.37)

@ @ @ _ @ @ € @ P i

Multiplicity is used in the knot vector at the end points of the time interval.
This property guarantees that the starting and ending joint angle values of a DOF
are exactly those corresponding to the first and last control point values. This
makes it easier to impose the symmetric posture constraints. The time-dependent
constraints are imposed not only at the knot points but also between the knots, so
that a very smooth motion is generated.