Page 297 - Control Theory in Biomedical Engineering

270 Control theory in biomedical engineering

as swing and stance leg angles, respectively. The mathematical equations of
this model are presented as follows.
Eqs. (1), (2) are for the single support phase.

€
θ g=Lsin θðÞ ¼ 0 (1)
€ €
_ 2
ð
m=Mϕ m=M 1 cos ϕðÞÞθ m=Mθ sin ϕðÞ m=M sin θ ϕð Þ ¼ Tr
(2)
Tr is the swing hip torque assumed to be generated through a rotational
spring with stiffness, k, and equilibrium angle, ϕ 0 (Eq. 3).

Tr ¼ k ϕ ϕð 0 Þ (3)
The single support phase is terminated when:

ϕ 2θ ¼ 0 (4)
Double support phase is also represented by Eq. (5) as an event:
2 3 + 2 32 3 2 3
θ 1 0 0 0 θ 0
0 cos 2θðÞ
6
6 _ 7 6 7 _ 7 6 sin2θ 7
7Impulse
6 θ 7 6 00 7 θ 7 6 7
6
2 0 0
6 7 ¼ 6 76 7 + 6
4
4 ϕ 5 4 0 05 ϕ 5 4 5
_
ϕ 0 cos 2θðÞ 1 cos 2θðÞÞ 00 ϕ _ 1 cos 2θðÞ
ð
(5)
In Eq. (5),( ) and (+) indicate states values before and after foot-contact
condition. Also, Impulse is considered as the push-off impact amplitude.
In this model, the sum of two consecutive steps is considered as a stride.
2.2 Controller design
Impulse and k were considered as control parameters of the model
(Bahramian et al., 2019) (other model parameters values are mentioned in
Table 1). To clarify the effect of each parameter, first, values of Impulse¼0.4
and k¼4 were set. Then, after the system reached a steady state, the k value

Table 1 Model parameters values.
Parameter Value
M 50 kg
m 5kg
L 1m
g 9.8 m/s 2
0.04 rad
Φ 0