Page 117 - Handbook of Biomechatronics

Model-Based Control of Biomechatronic Systems 113

Accordingly, in the simplified driver model, two representative muscles, one
flexor and one extensor, were used to actuate each arm segment to represent
the muscles producing clockwise and counterclockwise torques. A model
inspired from the popular Hill muscle model (Mehrabi et al., 2014;
Thelen, 2003) was used to simulate the muscle contraction dynamics.
The Hill muscle model consists of a contractile element (CE) and a parallel
elastic (PE) element in series with a series elastic (SE) element. In this study,
the SE dynamics representing the tendon were neglected because the
steering motion is relatively slow and the amount of energy transfer to ten-
dons is small. Therefore, the muscle model was reduced to the CE element,
and the muscle force (F TM ) was computed as follows:

F TM ¼ F max ð ð (15)
0 F PE t, L M Þ + F CE t, a, L M , V M Þ cos α p
where F CE represents the active force of the muscle and L M , V M , α p , and
max
F 0 are the muscle length, contraction velocity, pennation angle, and max-
imum isometric muscle force, respectively. The muscle activation level (a)
represents the number of active motor units in the muscle (between 0% and
100%), and since the SE element was removed, the pennation angle for all
muscles was assumed to be zero. The force generated by F CE can be sepa-
rated into force-length and force-velocity relations scaled by the muscle acti-
vation command (a):

F CE ¼ atðÞF L ð t, L M ÞF V ð t, a, L M , V M Þ (16)
CE CE
L V
where the force-length (F CE ) and force-velocity (F CE ) relations are:
2
L M
L L opt 1 =γ
F ¼ e M (17)
CE
8
V M max
> + AV
> max opt M
V L
>
> M M
>
> V M < 0
>
V M
> max
+ AV
>
>
> max opt M
< V M L M A f
F V ¼ len (18)
CE V M BF
> max max
> + ACV
> max opt M
V L
>
>
> M M
> V M > 0
V M B
>
>
> max
> + ACV
: max opt M
V
M L A f
M
max
where γ, A, B, and C are shape factors, V M is the maximum fiber velocity,
opt
L M is the optimal length of fiber at which F CE is a maximum, and F len is the
max