Recursive Dynamics - Human Motion Simulation: Predictive Dynamics - page 80

The backward recursive dynamics sensitivity equations are implemented as
follows:
Input state variables q

; _ q ; € q and final condition D ½n 1 5 0, E ½n 1 5 0,

1 1

F ½n 0 and use the stored information calculated from forward

recursive sensitivity algorithm.

1 5 0, G ½n 1 5

1 1

Do i 1

ð 1 Þ Calculate and store D i ;

n ;

5

E i ; F i ; G i

ð 2 Þ Calculate and store τ i

(4.54)

Do j i

Calculate and store

1 ;

5

2 A i =@

@ q i @ q j

End Do

Do k n

ð a Þ Calculate and store

1 ;

5

@ D i =@ q k ;@ D i =@ _ q k ;@ D i =@ € q k

ð b Þ Calculate and store @ E i =@ q k

ð c Þ Calculate and store @ F i =@ q k (4.55)

ð d Þ Calculate and store @τ i =@ q k ;@τ i =@ _ q k ;@τ i =@ € q k

EndDo
EndDo

4.5.6 Joint profile discretization
A joint profile qðtÞ is parameterized by using uniform B-splines as follows:

X

m

qð t ; P Þ 5 B i ð t Þp i 0 t T (4.56)

# #

i 5 1

where B i ð t Þ are the basis functions, t 5 ft 0 ; ...; t s g is the knot vector, and

P 5 fp 1 ; ...; p m g is the control point vector. With this representation, the control

points become the optimization variables (also called the design variables).
B-spline interpolation has many important properties, such as continuity, differen-
tiability, and local control. These properties, especially differentiability and local
control, make B-splines competent to represent joint angle trajectories, which
require smoothness and flexibility ( Wang et al., 2007 ).
The B-spline basis functions are uniquely determined by knot vector t, which
is evenly spaced on the time interval ½ 0 T with time step Δ t , as follows:

T
s ;

t i 1 1 5 t i 1 Δ t ; Δ t 5 i 5 0 ; ...; s 2 1 (4.57)

where s is the number of discretized segments.
Note that q , q , and _ q are calculated as functions of t and P; therefore torque €

τ 5 τð t ; P Þ is an explicit function of the knot vector and control points from the