Biomedical Engineering Reference
In-Depth Information
@τ
2
@θ
1
5
2
m
2
L
1
l
2
θ
1
sin
θ
2
(4.94)
@τ
2
@θ
2
5
0
(4.95)
@τ
2
@θ
1
5
m
2
l
2
1
I
2
1
m
2
L
1
l
2
cos
θ
2
(4.96)
@τ
2
@θ
2
5
m
2
l
2
I
2
1
(4.97)
The gradients will become import in later chapters when the optimization for-
mulation is implemented.
4.7
Trajectory planning example
As an illustration of the use of recursive dynamics, we shall present an example
of trajectory planning. Trajectory planning is defined as creating motion for an
end-effector (e.g., a hand) from one point to another while avoiding collisions.
We shall use concepts from Chapters 3 on optimization and Chapter 4 on dynam-
ics to present a time-optimal trajectory planning for the two-link arm treated ear-
lier. We shall solve the optimization problem using the recursive Lagrangian
formulation.
The objective is to minimize total travel time
T
subjected to boundary condi-
tions and torque limits. The same problem has been examined by Dissanayake
et al. (1991) and
Wang et al. (2005)
using the closed-form equation of motion
without the gravity effects. The parameter optimization problem can then be
stated mathematically as follows: to compute design variables
, which are con-
trol points P and total travel time
T
, and to minimize
T
subject to the constraints
on boundary conditions and torque limits
x
Minimize
:
Tð
x
Þ
Such that
q
1
ð
0
Þ
5
0
:
0
;
q
2
ð
0
Þ
52
2
:
0
q
1
ðTÞ
5
1
:
0
;
q
2
ðTÞ
52
1
:
0
(4.98)
q
1
ð
0
Þ
5
_
_
q
2
ð
0
Þ
5
_
q
1
ðTÞ
5
_
q
2
ðTÞ
5
0
:
0
10
#
τð
x
Þ
#
10
2
where
T
is the final time that needs to be minimized. Gravity effects are neglected
in this time-optimal design problem, i.e.,
g
0.
We use the word profile to denote a quantity changing over time. The optimal
joint profiles are shown in
Figure 4.4
and the joint torque profiles are depicted in
Figure 4.5
.
5
Search WWH ::
Custom Search