3
BACKGROUND
3.1
MODELING
The dynamics between the applied force to the linear position of each joint can be represented by the transfer function
q
(
s
) =
1
M
t
s
2
F
(
s
)
(3.1)
where
q
(
s
) =
L
[
q
(
t
)]
is the Laplace of the joint position
q
(
t
)
,
F
(
s
)
is the Laplace of the applied linear force, and
M
t
is the total mass being moved by the motor (i.e., both pre-load and payload).
3.2
POSITION CONTROLLER DESIGN
The joint positions on the Hexapod system is controlled using a standard proportional-integral-derivative (PID) con-
trol. The PID control used has following structure:
F
(
t
) =
k
p
(
q
d
(
t
)
−
q
(
t
)) +
k
i
Z
q
d
(
t
)
−
q
(
t
)
dt
+
k
d
( ˙
q
d
(
t
)
−
˙
q
(
t
))
(3.2)
where
k
p
is the proportional control gain,
k
i
is the integral gain,
k
d
is the derivative control gain,
q
d
(
t
)
∈ <
6
is the
setpoint or reference joint position (for all six joints),
q
(
t
)
∈ <
6
is the measured joint position (for all six joints), and
F
(
t
)
is the force (i.e., control effort). The block diagram of the control is given in Figure 3.1.
Figure 3.1: Block diagram of Hexapod position control.
In software implementation, the controller force is converted into motor current using
τ
m
=
F
r
I
m
=
τ
m
k
t
Thus the controller force is divided by the lead-screw radius,
r
, to get the torque and this is multiplied by the current
constant,
k
t
, to obtain the necessary motor current.
HEXAPOD Laboratory Guide
v 1.3
