Open Journal of Discrete Mathematics, 2011, 1, 139-152
doi:10.4236/ojdm.2011.13018 Published Online October 2011 (http://www.SciRP.org/journal/ojdm)
Copyright © 2011 SciRes. OJDM
Optimal Trajectory of Underwater Manipulator Using
Adjoint Variable Method for Reducing Drag
Kazunori Shinohara
JAXAS Engineering Digital Innovation Center, Japan Aerospace Exploration Agency, Sagamihara City, Japan
E-mail: shinohara@06.alumni.u-tokyo.ac.jp
Received June 28, 2011; revised July 30, 2011; accepted August 16, 2011
Abstract
In order to decrease the fluid drag on an underwater robot manipulator, an optimal trajectory method based
on the variational method is presented. By introducing the adjoint variables, which are Lagrange multipliers,
we formulate a Lagrange function under certain constraints related to the target angle, target angular velocity,
and dynamic equation of the robot manipulator. The state equation (the partial differentiation of the La-
grange function with respect to the state variables), adjoint equation (the partial differentiation of the La-
grange function with respect to the adjoint variables), and sensitivity equation (the partial differentiation of
the Lagrange function with respect to torques) can be derived from the stationary conditions of the Lagrange
function. Using the state equation, we can calculate the state variables (angles, angular velocities, and angu-
lar acceleration) at every time step in the forward time direction. These state variables are stored as data at
every time step. Next, by using the adjoint equation, we can calculate the adjoint variables by using these
state variables at every time step in the backward time direction. These adjoint variables are stored as data at
every time step. Third, the sensitivity equation is calculated by using both the state variables and the adjoint
variables. Finally, the optimal trajectory of the manipulator is obtained using the sensitivities. The proposed
method is applied to the problem of two-link manipulators. It can obtain the optimal drag reduction trajectory
of the manipulator under the constraints mentioned above.
Keywords: Robot Manipulator Dynamics, Optimal Trajectory, Adjoint Variable Method, Euler-Lagrange
Equation, Fluid Drag Force, Calculus of Variations
1. Introduction
Presently, we are facing serious environmental problems
such as global warming and abnormal climatic condi-
tions, which are closely related to the ocean. Therefore,
the establishment of ocean study technology is extremely
important. Since the 1990s, researchers have investigated
the development of underwater robot manipulators for
oceanic studies [1-6].
In an extreme environment such as the abyssal ocean,
it is difficult to supply energy to manipulators. However,
because of fluid tractions, the energy consumption of an
underwater manipulator is greater than that of a manipu-
lator in air. In order to reduce the energy consumption, it
is important to determine the optimal trajectory to reduce
the drag on the manipulator.
Optimal time control for a manipulator trajectory was
studied in the 1970s [7,8]. Kahn and Roth first presented
an optimal time control method based on kinematic dy-
namics [9]. Vukobratovic and Kiranski presented an op-
timal time control method based on dynamic program-
ming [10]. Townsend et al. presented optimal control by
approximating a function [11]. Lee et al. presented the
formulation of a genetic algorithm based on trajectory
planning [12]. Constantinescu et al. presented a method
for determining smooth and time-constrained optimal path
trajectories for a robot manipulator [13]. These studies
were carried out with the objective of constructing an
optimal time trajectory for a manipulator, from its initial
position to the target position. On the other hand, Eiji
presented a method for determining the minimum energy
trajectory of an underwater manipulator [14]. Uno et al.
presented a minimum torque change model [15]
A method for drag reduction control has not been de-
veloped thus far. A marine robot has energy limitations
during its operations. Therefore, drag reduction control
K. SHINOHARA
140
under an extreme environment is crucial for low energy
consumption.
In this study, we propose an optimal trajectory method
for reducing the drag on the manipulator. As the ma-
nipulator moves from its initial position to the target po-
sition, the fluid generates an external force on the ma-
nipulator. A method based on the variational principle is
developed to determine the optimal trajectory to reduce
the drag. This method is called the adjoint variable me-
thod. The adjoint variable method is based on a varia-
tional method. By introducing Lagrange multipliers
called adjoint variables, we transform the constrained
optimization of the cost function into the unconstrained
optimization of the Lagrange function. The cost function
is defined as the fluid drag on the manipulator. The La-
grange function is formulated under the constraints of the
robot manipulator dynamics. The stationary conditions
(the state equation, adjoint equation, and sensitivity
equation) are derived from the Lagrange function. An
algorithm is developed on the basis of the stationary
conditions. First, the state variables (the angle, angular
velocity, and angular accelerations) are calculated by
using the state equation in the forward time direction and
stored as data at every time step. Next, by using the state
variables at every time step, we calculate the adjoint
variables by using the adjoint equation in the backward
time direction. Finally, the sensitivity (gradient) is cal-
culated at every time step, and the time history of the
joint torques is determined.
Using this optimal trajectory algorithm developed in
three phases (state analysis, adjoint analysis, and sensi-
tivity analysis), we resolve the problem of the two-link
manipulator. The effectiveness of the algorithm is then
verified by comparing it with the optimal time control
methods described in the literature.
2. Theory
2.1. Variable
In this paper, the two-dimensional motion of a manipu-
lator with respect to the x-y plane is considered, as shown
in the Figure 1. The links are arranged in the shape of a
circular cylinder. The manipulator consists of two links
that are connected by joints. The coordinates at each
joint are defined. The joints and links are numbered from
the base to the tip.
The angles with respect to joint i are defined as
1, 2qi
i (1)
The angular velocities with respect to joint i are de-
fined as:
1, 2qi
i
(2)
1
q
2
q
0
x
0
y
0
z
1
y1
x
2
y2
x
1
r
1
l
2
r
2
l
Joint 2
Joint 1
Link 2
Link 1
Figure 1. Two-link manipulator.
The angular accelerations with respect to joint i are
defined as
1, 2qi
i
 (3)
In this study, the variables obtained from Eq.(1) to
Equation (3) are called state variables (q = (1212
12
)). The penalty parameters with respect to the angle
and angular velocity are defined as
,,,,qqqq

,qq
 
1,2,3,4i
i (4)
Variables with respect to the angular accelerations
() are defined as
12
,qq
 
1, 2i
i (5)
The variables obtained from Equation (5) are called
adjoint variables (λ=(λ1, λ2)). The torques with respect to
joint i are defined as
1, 2i
i (6)
The masses with respect to link i are defined as
1, 2mi
i (7)
The diameters with respect to link i are defined as
1, 2di
i (8)
The lengths with respect to link i are defined as
1, 2li
i (9)
The lengths from the centroid of a link to joint i are
defined as
1, 2ri
i (10)
The drag coefficients with respect to link i are defined
as
1, 2Ci
i (11)
The density of the water is defined as
(12)
Copyright © 2011 SciRes. OJDM
K. SHINOHARA 141
The translational veloci
de
(13)
The sign function sgn(x) is def
ties with respect to link i are
fined as
1, 2vi
i
ined as

0
sgn0 if0
1if 0
xx
x
1 ifx

(14)
In order to derive the dynamics of an underwater robot
m
em
order to minimize the cost function under certain con-
anipulator, the external force exerted by the fluid drag
needs to be added to the robot manipulator dynamics.
The fluid drag on an object is proportional to the square
of the object’s speed [16]. The fluid drag always has a
positive value in the calculation of the robot manipulator
dynamics. With respect to the motion direction of the
manipulator, the fluid drag acts in the opposite direction
in a real environment. Therefore, the sign of the fluid
drag direction has to be determined according to the mo-
tion of the manipulator. The motion direction of the ma-
nipulator can be identified by the sign of the translational
velocities.
.2. Probl2
In
straints, a Lagrange function is formulated by introduc-
ing the adjoint variables. The input data are the time his-
tories of the torques (τ1, τ2) from the start time 0 to the
end time t. The optimal trajectory is searched for in the
set of inputs. After determining the values of τ1 and τ2,
the angle, angular velocities, and angular accelerations
are determined using the state equation from start time 0
to the end time t. The tip of the underwater manipulator
moves from the initial position to the target position. The
objective of this study is to reduce the fluid drag by the
2D trajectory of the manipulator. The cost function is
defined as
12
J
DD N (15)
(16)





 
2
1122 3142
11 11
0
22 22
0
22
3142
d0
d0
t
t
Nqaqb qcqd
qtqa qa
qtqbq b
qcq d




 


 
22 2
 


32
11111
sgn
6
DCdlqq
1


(17)
22
2
22
2ll
q llqq




2
22
22112 11212
2
2
2
2
23
3
3
D Cdllll
lq



(18)
where the parameters a, b, c, and d in Equation (16) are
constants. The first and second terms in Equation (16)
are the constraints with respect to the target angle a of
joint 1 and target angle b of joint 2, respectively. The
third and fourth terms in Equation (16) are the con-
straints with respect to the target angular velocity c of
joint 1 and target angular velocity d of joint 2, respec-
tively. Equations (17) and (18) represent the fluid drag
on link 1 and link 2, respectively (see Appendix A). In
order to simplify the formulation of the adjoint variable
method in this study, it is assumed that the inequality
11
lq
21 20lq q
 is satisfied at all times. The La-
grange function is defined as
12 1122
DD NFFLJF

  (19)
where equations F and F co
12 nstitute the state equation.
This state equation represents the robot manip
quation of the Lagrange function
int variable λ is called the state
ulator dy-
namics. In this study, a weak formulation is applied to
the Lagrange function by the time integration of the state
equation. The Lagrange function can also be formulated
by a strong formulation that satisfies the equation at
every time step.
2.3. State Equation
The partial differential e
ith respect to the adjow
equation:
1
111
d00
LL L
F
 
 
dt

 
 
(20)

 

2
222
d00
d
LL L
F
t

 
 
 
 

 

Equations (20)-(21) represent the robot d
nipulator as
(21)
ynamic ma-

,Mqq Cqqfq

 (22)
The parameters M, C, f, and τ represent the i
trix, vector of the coriolis and centrifugal
fo
nertia ma-
forces, drag
rce, and joint torque, respectively. By using the pa-
rameters in Appendix B, Eqaution (22) is written as
5621 131
3
22214 2
A
AAq A
A
AB qA


 





 (23)
Equation F is defined as
Copyright © 2011 SciRes. OJDM
K. SHINOHARA
142
23 25214 213
11
11
36213614
11
22
11
1
2
6
2
11
0
0
22
25
A
ABAAAB
AA
Fq
A
Fq
A
A
 
 
 AAAA AA
AA
BA
AA
A
A
AA



  
 
 
 
 


 


 






(24)
where parameters A1 A14, B1, and B2 are discussed in
Appendix B and Appendix C.
2.4. Adjoint Equations
uler-Lagrange equations
erived from the stationary condition of the Lagrange
ons are derived as
The adjoint equations are the E
d
function. The adjoint equati
11
d0
d
LL
qtq







(25)
22
d0
d
LL
qtq







(26)
The time derivation of the adjoint variable λ1 is de-
rived from Equation (25) (see Appendix D).

111 3141516
210 2725216 217
1
11
2
2710621765216
2
11
22
2
tq aq cBqBA
A
ABAABrABA
AA



(27)
AA A AAAABrA
AA
 
  






The time derivation of the adjoint variable λ2 is de-
rived from Equaiton (26).

22242 518
212 25218 21819
1
BAABrABA A
1
2122 181965218
2
1
2
tq bq dBA
A
A
A AAA ABrA
A
 
 




The condition of the end time tf is derived from the
partial differential equation of the Lagrange function
with respect to the state variables as


(28)
 
12
0, 0
ff
Lt Lt
qq



 (29)
01,
if
ti
 2 (30)
2.5. Sensitivity Equations
rtial differential equation of the Lagrange function
with respect to the torque τ is calle
on:
The pa
d the sensitivity equa-
ti

12 22
dBA
LL




 1,( )00
k
L
Gt

 

 
11 1 1
dtA
  

(31)


12 26
2,( )
22 12
d00
dk
AA
LL L
Gt
tA

 
 
 

 
 
 

(32)
where the subscript (k) represents an iteration, as sh
in the Figure 2. The time histories of the torques are
iteratively modified from the time histories of the initial
torques. The algorithm determines the optimal trajectory
by minimizing the Lagrange function. Finally, Gi,(k)
own
(t) (I
= 1, 2) reaches zero if the subscript (k) represents a suffi-
cient number of iterations.
2.6. Steepest Descent Method
The time histories of the torques are modified by the
gradient as
Using Equation (29), we obtain the following equa-
tions:






1, 11,1,kk k
ttG
t (33)


tt


 

2, 12,2,kk k
G
t
(34)
er to robustly converge to the optimal trajec-
tory and to avoid numerical vibration and di
this study, the parameter α is set to 0.1.
3.
The data of the
every time step are stored in the PC
econd phase, using the state variable at
The value of the coefficient α should be sufficiently
small in ord
vergence. In
Algorithm
The algorithm is shown in the Figure 2. In the first phase,
the state variable (q) is calculated from the start time to
he end time in the forward direction. t
state variable at
emory. In the sm
every time step, we can calculate the adjoint variables (λ)
from the end time to the start time in the backward direc-
tion. The adjoint variables are also stored at every time
step. In the third phase, by using the state variable and
adjoint variable, we obtain sensitivity G, which is the
gradient of the Lagrange function, at every time step.
The time histories of the torques are modified by using
the steepest descent method. Finally, the results are visu-
alized in the case where the position of the manipulator
Copyright © 2011 SciRes. OJDM
K. SHINOHARA
Copyright © 2011 SciRes. OJDM
143
0(
0(
qcondtionsInitial:
)
)
Convergence?
condtionsInitial:
)0(
)0(
λ
:
)(
)(
1
n
k
:
)(
)(
,2
n
k
variablesAdjoint
n
k
n
k:,)(
)(
,2
)(
)(
,1

Start time
Visualization by Gnuplot
YES
NO
YES
NO
n=n+1
n=n-1
Torques
n
k
i:
)(
)(
,
Time derivation of adjoint variables
Time derivation of adjointvariables
Anglesqq n
k
n
k:, )(
)(,
2
)(
)(,
1
Velocitiesqq n
k
n
k:, )(
)(
,2
)(
)(
,1
onsAcceleratiqq n
k
n
k:,)(
)(
,2
)(
)(
,1
Start time
End time
ySensitivitG n
k
i:
)(
)(
,
Sensitivity
equations
E qs.(31) -
(32)
Steepest
decent
method
Eqs.(33)-
(34)
State
equations
Adjoint
equations
Table 1
Eq. (24)
Obtained
by Runge -
Kutta
method
Eq.(30)
End time
Eq.(27)
Eq.(28)
Obtained
by Runge -
Kutta
method
k=k+1
Figure 2. Algorithm.
almost agrees with the target position. In the case where
the position of the manipulator does not reach the target
position, this algorithm returns to the first phase.
method
r reducing the drag on a manipulator. Using this
rag reduction trajectory can be obtained.
he optimal trajectory obtained by the drag reduction
optimal trajectory obtained by the time optimal control
method described in the literature [17] (see Appendix E).
e time optimal control method, the manipulator
requires a minimum amount of time to move from the
Using th
4. Results
In Section 2, we formulated the adjoint variable
fo
method, the d
T
control method can be verified by comparing it with the
initial position to the target position.
4.1. Calculation Conditions
Two trajectories for manipulators located at different
initial positions are calculated using both the time opti-
mal control and the drag reduction control. These two
K. SHINOHARA
144
are summarized in
able 1.
origin (0, 0). The
aximum and minimum values for both torque 1 and
37, B2 = 0.3033, and B3 = 0.1482 in the litera-
tu
gures show angle 1 and angle 2, respec-
vely. Angle 1 (q) constantly converges to the target
s away from
e target angle. After that, by rapidly closing to the tar-
itial a
calculation conditions are defined as case 1 and case 2.
The initial and objective positions
T
The edge of link 1 is fixed at the
m
torque 2 are set to ±30 (N) and ±10 (N), respectively. The
initial parameters are summarized in Table 1. The solver
is applied to the Runge-Kutta method. The time span is
set to 0.001. The time histories of the initial torques, τi,(k)
(I = 1,2), are constantly defined as zero. The parameters
B1 = 4.53
re [17] are used in the manipulator computing model
(see Appendix B). The density ρ is set to 1.0. The drag
coefficients, C1 in link 1 and C2 in link 2, with respect to
the circular cylinder are each set to 0.1. The lengths, L1
of link 1 and L2 of link 2, are each set to 1.0. The diame-
ters of the cylinder, d1 and d2, with respect to link 1 and
link 2 are each set to 0.01. The penalty parameters, ε1 ~ ε4,
are set to 5.
4.2. Computational Results in Case 1
The trajectories for case 1 are shown in the Figure 3
(time optimal control) and the Figure 4 (drag reduction
control). The end time is 0.81. The time histories of the
angles are shown in the Figure 5. The black and red
lines in the fi
ti 1
angle. During the first 0.4 s, angle 2 (q2) i
th
get angle, the time optimal trajectory using the inertia
force is created. As shown in the Figure 7, the angular
velocities (12
,qq

) almost become zero at end time tf and
satisfy the constraint condition. In link 1 of the manipu-
lator, the angular velocity 1
q
increases monotonically
during approximately the first 0.4 s. After that, the angu-
lar velocity 1
q
decreases monotonically in order to sat-
isfy 10q
. In link 2 of the manipulator, the angular
velocity 20q
from 0 s to 0.4 s. After that, the angular
Table 1. Innd target conditions in case 1 and case 2.
Parameter Initial condition Target condition
1
q π/3 0.0
2
q 0.0 0.0
1
q
0.0 0.0
Case 1
0.0 0.0
Case 2
0.0 0.0
2
q
1
q π/3 0.0
2
q π/6 × 5 0.0
1
q
0.0 0.0
2
q
–2
–1.8
–1.6
–1.4
–1.2
–1
–0.8
–0.6
–0.4
–0.2
0
00.511.522.5
Meter (m)
Meter(m)
Figure 3. Trajectory of two-link manipulator for time op-
timal control in case 1.
–2
–1.8
–1.6
–1.4
–1.2
Meter(
–1
–0.8
–0.6
–0.4
–0.2
0
00.511.522.5
Meter (m)
m)
Figure 4. Trajectory of two-link manipulator for drag re-
duction control in case 1.
Figure 5. Time history of angles for time optimal control in
case 1.
velocity , and it decreases in order to satisfy the
constraint, 20q
2
q0
me
at end time tf. The time history of the
torques for the ti optimal control is shown in the Fig-
ure 9. The maximum torque of +30 (Nm) acts on joint 1
during approximately the first 0.4 s. After that, in order
Copyright © 2011 SciRes. OJDM
K. SHINOHARA 145
to meet the end constraint condition, the inverse
maximum torque of –30 (Nm) The maxi-
mum torque of –10 (Nm) acts 0.0 s to
0.15 s. After that, the inverse maximue of +10
(Nm) acts on joint 2 during 0.19 s - . Again, the
maximum torque of –10 (Nm) acts on
The drag reduction trajectory and te history of
angles are shown in Figure 4 and Fi pectively.
The end time tf is 1.08 s. The angle s a constant
value of zero. By avoiding any extrment of the
manipulator, a trajectory from sition to the
end position is created for minimum
The time history of the angular velshown in
the Figure 8. Equaiton (40) is angular ve-
locity increases monotonicallyproximately
The angular velocity to zero
10q
,
acts on joint 1.
on joint 2 from
um torq
0.50 s
joint 2.
he tim
gure 6, res
q2 ha
a move
the initial po
drag reduction.
ocity is
satisfied. The
during ap
1
remains close
1
q
y.
the first 0.4 s. After that, in order to satisfy the constraint
condition, 0q
, angular velocity q
decreases mono-
nicall 1
to 2
q
by adjusting torque 2 at joint 2.
Figure 6. Time history of angles for drag reduction control
in case 1.
Figure 8. Time history of angular velocities for drag reduc-
ion controt
l in case 1.
Figure 9. Time history of torques for time optimal control
in case 1.
The time history of the torques for the drag reduction
control is shown in Figure 10. The maximum torque of
+30 (Nm) acts on joint 1 from 0.0 s to 0.4 s. After that,
by loading the inverse torque, torque 1 is adjusted such
that 10q
0
. In the case of torque 2, in order to satisfy
2
q
and 20q
positive torque and the negati
, alternative torques using both the
ve torque act on joint 2.
4.3. Computational Results in Case 2
For case 2, as shown in Table 1, the trajectory of the
manipulator obtained by the time optimal control is
shown in the Figure 11. The end time is 0.71 s. The tim
nverges to the target angle. During
e first 0.2 s, angle 2 (q2) is away from the target angle.
After that, by rapidly closing to the target angle, a time
optimal trajectory is created using the pullback force
e
histories of the angles are shown in the Figure 13. The
angle q1 constantly co
th
Figure 7. Time history of angular velocities for time optimal
control in case 1.
Copyright © 2011 SciRes. OJDM
K. SHINOHARA
146
obtained by the inertia force.
The time histories of the angular velocities are shown
in the Figure 15. In link 1, the absolute value of the an-
gular velocity increases monotonically during ap-
proximately th0.35 s.
After that, the absolute value of angular velocity
decreases monotonically in order to satisfy
1
q
e first
1
q
10q
the first
.
. A
gular velocity becomes during
s. After that, eter
The time t
trajectory are shown in the Figure 17. The maximum
torques of +30 (Nm) act on joint 1 during approximately
the first 0.35 s. After that, the inverse torques act on joint
1 so that
The cl torque 2 acts on joint 2 during the firs
. Again, the clockwise torque takes effect after
.45
el
the coion, a
tra
otion of the manipulator, the manipulator is prevented
target angle.
ink 2 of the manipulator moves in a straight path with
n-
0.18
al
2
q
param
e histories of t
20q
turns to
o
2
q
h20q
rques for the time-optim
10q
.
ockwise t
0.1 s. The counterclockwise torque takes effect from 0.1
s to 0.45 s
0 s.
For case 2, as shown in Table 1, the trajectory of the
manipulator obtained by the drag reduction control and
the time history of angles for drag reduction control in
case 2 are shown in the Figure 12 and the Figure 14,
respectivy. The end time is 1.08 s. In order to reduce
st functsmall angle for q2 is selected at every
time step. To prevent the generation of drag by any ex
m
from swinging link 2 with respect to the
L
respect to the target position.
The time histories of the angular velocities are shown
in the Figure 16. Equation (40) is satisfied. The absolute
value of the velocities, 1
q
, increases monotonically.
After that, the absolute value of the velocities, 1
q
, de-
creases monotonically in order to satisfy 10q
. Angu-
lar velocity 2
q
increases monotonically. After that, 2
q
decreases monotonically in order to satisfy 20q
.
Figure 10. Time history of torques for drag reduction con-
trol in case 1.
Mer
Meter (m)
et
(
m
)
0 0.5 1 1.5 2 2.
5
1
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
0.8
Figure 11. Trajectory of two-link manipulator for te op-
timal control in case 2.
im
Meter (m)
Meter (m)
0 0.5
1 1.5
2
2.5
0.8
0.6
0.4
0.2
0
–0.2
–0.4
Figure 12. Trajectory of two-link manipulator for drag-
reduction control in case 2.
Figure 13. Time history of angles for time optimal control
in case 2.
Copyright © 2011 SciRes. OJDM
147
K. SHINOHARA
Figure 14. Time history of angles for drag reduction control
in case 2.
Figure 15. Time history of angular velocities for time opti-
mal control in case 2.
Figure 16. Time history of angular velocities for drag re-
duction con
Figure 17. Time history of torques for time optimal control
in case 2.
Figure 18. Time history of torques for drag reduction con-
trol in case 2.
ure 18. The clockwise torque 1 takes effect during ap-
proximately the first 0.25 s. After 0.25 s, the counter-
clockwise torque takes effect. The clockwise and coun-
terclockwise torques act alternately. The clockwise torque
acts on joint 2 during the first 0.2 s. The counterclock-
wise torque takes effect from 0.2 s to 0.4 s. After ap-
proximately 0.43 s, the alternative torque acts on joint 2.
5. Conclusions
We proposed the adjoint variable method for obtaining
an optimal trajectory in order to decrease the fluid drag
on a manipulator when the manipulator moves from its
initial position to the target position. By considering hy-
drodynamic effects, we formulated the dynamics of an
s the fluid drag. By introducing the adjoint
The time histories of the torques are shown in the Fig-
underwater robot manipulator. The cost function was
defined a
trol in case 2.
Copyright © 2011 SciRes. OJDM
K. SHINOHARA
Copyright © 2011 SciRes. OJDM
148
ariable, we formulated the Lagrange function under
certain constraints, which consisted of the target angle,
target angular velocities, and robot manipulator dynam-
ics equations. The gradient of the Lagrange function with
respect to the torque was derived from the stationary
condition. The algorithm was developed on the basis of
the adjoint variable method. This algorithm can be suffi-
ciently converged to the optimum value under the c
straints, and it can determine an optimal trajectory to
reduce the fluid drag under the constraints.
Simulation results showed that the performance ca
enhanced for the control of an underwater manipulato
By using this approach, we can use the motion to reduce
fluid drag. It may be important to note that significant
performance enhancement is achieved by the motion o
es
v
on-
n be
r.
f
the manipulator.
. Referenc6
[1] T. L. McLain, S. M. Rock, and M. J. Lee, “Experiments
in the Coordinated Control of an Underwater Arm/Vehi-
cle System,” Autonomous Robots, Vol. 3, No. 2-3, 1996,
pp. 213-232. doi:10.1007/BF00141156
[2] K. N. Leabourne and S. M. Rock, “Model Development
of an Underwater Manipulator for Coordinated Arm-Ve-
hicle Control,” Proceedings of the OCEANS 98 Confer-
ence, Nice France, No. 2, 1998, pp. 941-946.
[3] J. Yuh, S. Zhao and P. M. Lee, “Application of Adaptive
erver Control to an Underwater Manipu-
onal Conference on Robotics and Auto-
Disturbance Obs
lator,” Internati
mation, Vol. 4, 2001, pp. 3244-3249.
[4] S. Sagara, T. Tanikawa, M. Tamura and R. Katoh, “Ex-
periments on a Floating Underwater Robot with a Two-
Link Manipulator,” Artificial Life and Robotics, Vol. 5,
No. 4, 2001, pp. 215-219. doi:10.1007/BF02481505
[5] G. R. Vossoughi, A. Meghdari and H. Borhan, “Dynamic
Modeling and Robust Control of an Underwater ROV
Equipped with a Robotic Manipulator Arm,” 2004 Japan
USA Symposium on Flexible Automation, Denver USA,
2004.
[6] K. Ioi and K. Itoh, “Modelling and Simulation of an Un-
derwater Manipulator,” Advanced Robotics, Vol. 4, No. 4,
1989, pp. 303-317. doi:10.1163/156855390X00152
[7] M. L. Nagurka and V. Yen “Optimal Design of Robotic
Manipulator Trajectories: A Nonlinear Programming
echnical Report CMU-RI-TR-87-12, the
te, Carnegie Mellon University, April,
ia, Philadelphia,
nd Control,
Approach,” T
Robotics Institu
1987.
[8] M. Žefran, “Review of the Literature on Time-Optimal
Control of Robotic Manipulators,” Technical Report MS-
CIS-94-30, University of Pennsylvan
1994.
[9] M. E. Kahn and B. Roth, “The Near Minimum-Time
Control of Open-Loop Articulated Kinematic Chains,”
Journal of Dynamic Systems, Measurement a
Vol. 93, No. 3, 1971, pp. 164-172.
doi:10.1115/1.3426492
[10] M. Vukobratović and M. Kirćanski, “A Method for Op-
timal Synthesis of Manipulation Robot Trajectories,”
Journal of Dynamic Systems, Measurement, and Control,
Vol. 104, No. 2, 1982, pp. 188-193.
doi:10.1115/1.3139695
[11] M. A. Townsend, “Optimal Trajectories and Controls for
pulator with Acceleration Parameterization,” Jour-
and
Systems of Coupled Rigid Bodies with Application of
Biped Locomotion,” Thesis (Ph.D.), University of Wis-
consin Madison, Madison, 1971.
[12] Y. D. Lee and B. H. Lee, “Genetic Trajectory Planner for
a Mani
nal of Universal Computer Science, Vol. 3, No. 9, 1997,
pp. 1056-1073.
[13] D. Constantinescu and E. A. Croft, “Smooth
Time-Optimal Trajectory Planning for Industrial Ma-
nipulators Along Specified Paths,” Journal of Robotic
Systems, Vol. 17, No. 5, 2000, pp. 233-249.
doi:10.1002/(SICI)1097-4563(200005)17:5<233::AID-R
OB1>3.0.CO;2-Y
[14] E. Shintaku, “Minimum Energy Trajectory for an Un-
derwater Manipulator and Its Simple Planning Method by
Using a Genetic Algorithm,” Advanced Robotics, Vol. 13,
No. 6-13, 1999, pp. 115-138.
[15] Y. Uno, M. Kawato and R. Suzuki, “Formation and Con-
trol of Optimal Trajectory in Human Multijoint Arm
Movement,” Biological cybernetics, Vol. 61, No. 2, 1989,
pp. 89-101. doi:10.1007/BF00204593
[16] J. Saleh, “Fluid Flow Handbook,” McGraw Hill, New
York, 2002.
[17] H. Kobayashi et al., “The Robot Control Actually,” The
Society of Instrument and Control Engineers, Tokyo,
1997.
K. SHINOHARA 149
the coordinates (x0,y0,z0)
0
Appendix A: Fluid Drag
The underwater manipulator receives a reactive force
when it moves to the target position. In this section, the
fluid force on the manipulator is derived. In this study,
the link is defined as a circular cylinder. The transla-
tional velocity with respect to
(=(0,0,0)) of link 1 is given by
1
1011 11
1
00 0
00 0
00
x
x
q
q
 
 
 
 
 
 
vvωx (35)
here the variable v0 represents the velocity on the
round. In this study, the base is fixed at v0 = 0. The
ariable ω1 is the angular velocity vector in link 1. The
anslational velocity in link 2 is given by
w
g
v
tr

12
11 212
0
0
2
00
00
00
21 22 11
x
lq

 

vvωx
qq
 


 
 

lqxqq







(36)
Using Equation (35), we obtain the fluid drag in link 1
as


12
1101111 1
0
sgnd
2
l
Cdxq x
 
vωx
122
111 111
0
0
sgn()d
2
l
Cdxqxqx
1
0


(37)
The function
11
sgn
x
q
always becomes the function
1
sgn q
because x1 > 0. Therefore,
11
sgn xq
1
sgn q
The integration in Equation (37) is not related
to the function
1
sgn q
. Thus, Equation (37) becomes
 
32 32
111 1111111
11
00
1sgn sgn
23 6
00
0
0
CdlqqCdl qq
D

 
 
 
 
 
 
 






 
D
(38)
The fluid drag in link 2 is (see next page)
In this study, in order to simplify the formulation of
the adjoint variable method, it is assumed to satisfy the
inequality as
112 120lql qq

 (40)
n (39) bec
Appen
The variables used in this study can be defined as fol-
lows:
2
Thus, Equatioomes (see next page)
dix B: Definition of Variables
222
11223
cos
A
BB BBq (42)
 







2
2
2
221121211212 2
0
2
22 2
2 21111122122
0
0
sgn d
2
0
0
2sgn
2
l
l
Cdlqx qqlqx qqx
Cdlqlqq qxq qx


112 122
d
lqx qqx
22
2
21 22112122
0
sgn d
2
l
Cdlqx qqx
 
0
 








 
 

vωx
(39)
 

222
2222
2
2221121212 12
2
33
l
Cdllllqlllqq2
2
2
sgnd3
ll
Cdlqxqqxq

22 122112 122
0
2
0
2
0


 





0
 
vωx
22
0
D





D
(41)
 
Copyright © 2011 SciRes. OJDM
K. SHINOHARA
150
2223
cos
A
BB q  (43)
2
2
331
sin
A
Bq q (44)
2
2
4312232
2cos cos
A
Bqqq Bqq
 (45)
2
2
5312232
2sin sin
A
Bqqq Bqq
 (46)
261 3
2cos
A
BB q (47)
2732
2sin
A
Bq q (48)
283
sin
A
Bq (49)
293
cos
A
Bq (50)
103 12
sin
A
Bq q (51)
2
113 12
cos
A
Bq q (52)
in

123 122
2s
A
Bq qq
 (53)
22
134 1525126 17 1 282
cos 2
A
BqBr BlqBqBqqBq 

(54)
2
145 261
2
7 1 28 2
A
Br BqB
qqBq

(55)

15 5 2
sin 22
6 17 1 282
A
BqB qBqqBq 

(56)
166 17 2
2
A
Bq Bq

(57)

1741525 12
2cos

6 172
2
A
BqBr Blq

Bq Bq
(58)
187 182
2
A
Bq Bq
(59)
195251 c2
os
A
Br Blq (60)
where the variables B1 B8 are defined as follows:
 
2
21
ml
112
111
22
11
1112 2221
16 12
zz zz
zzg
BI I
Imr
dl
m





(61)
22
2
2 2221
zzg
I mrml 
22
222
22
dl
mrmmrml
 


16 12

22
22
22
22 22222
16 12
zz zzg
dl
BI Imrmmr

 


2
1
(62)
322
Bmrl (63)
3
41111
sgn( )
6
BCdlq
 (64)
52
2
BCd
 22
l (65)
2
22
6112
3
l
Blll




(66)
2
721
2
3
Blll




2
(67)
2
2
83
l
B
(68)
Appendix C: Robot Manipulator Dynamics
The translational acceleration, a1, of the barycenter in
link 1 is as
1111 11
2
11
11
111
000
00000
00
rr
rq
qqq
 


 

 
 


 


 
 



 
aωrωωr
11
0
rq
(69)
The translational acceleration a2 of the barycenter in
link 2 is given by
2
21122222
2
22 11
12
2
12 12
sin cos 000
0010
00
000
0
qq lq
qq
r
qq qq

22 11 2
cossin 00qq lqr

0



 


 








 








 
 
aReωrωωr


2
2
1121222 12
2
1121122 12
cos sin
sin cos
0
lqqlqqr qq
lqqlqqr qq







 
 

(70)
The vect
with respect to the tip o
sents the rotation matri
coordinates (x1,y1,z1). The translational forces f1 and f2
with respect to l
2
or ei represents the angular acceleration vector
f link i. The matrix 1
R repre-
x from coordinates (x2,y2,z2) to
2
ink 1 and link 2 are given by
222
m
fa D (71)
1
11 221
m
1
aRfD (72) f
1
where the matrix represents the rotation matrix
from the coord
(x2,y2,z2).
1
2
R
inates (x1,y1,z1) to the coordinates
22
1sin cos 0qq
22
2
cossin 0
00
qq
R (73)
In this study, the buoyancy and gravity are ignored. By
equilibrant moment of the relation between link 1 and
link 2, it is derived as
Copyright © 2011 SciRes. OJDM
K. SHINOHARA 151
2
(74)
where the first and second terms on the right-
are

222222 2
 InIaωωfr
hand side

2
212212
00 00
x
zz
I
22 2
00
0 0
00
y
I
I
qq Iqq









 
I


α (75)

2
2222
122 12
0000
00 00
00 0
x
y
z
I
I
qqI qq




 







 
ωIω
0
0
(76)
where the matrix I2 represents the inertia tensor of lin
The vector α2 represents the acceleration of link 2.
equilibrant moment at joint 1 is derived as
(77)
where the matrix I1 represents the inertia tensor of link 1.
Th
The adjoint equations are derived. The v
are
k 2.
The



11
122112211
1111 1
 

nRnfr Rfrl
Iαω Iω
e vector α1 represents the acceleration of link 1. With
respect to the z axis, moments n1 of the joint 1 and n2 of
the joint 2 are defined as torques τ1 and τ2. Equation (24)
is thus derived.
Appendix D. Derivation of Adjoint Equations
ariables 1
q, 2
q,
1
q
, 2
q
, 1
q
 , and 2
q
 independent of each other. The
partial differential equation of the Lagrange function
with respect to q
is given by
1

113141
0
1
210 2725216 217
1
516
11
1d 22
2
t
Ltq aq cBqBA
q
AA BA ABrA BA
A



 




(78)
11
2710621765216
2
2
AA
AAA AAAABrA
A



The partial differential equation of the Lagrange func-
tion with respect to q
 is given by
1
1
1
L
q

 (79)
The partial differential equation of the Lagrange func-
tion with respect to is given by
2
q

2242
0
2
1d 2
t
Ltq bq dBA
q


212 25218 21819
1
1
2 122 181965218
2
1
BAABrABA A
A
AAAA AABrA
A








(80)
The partial differential equation of the Lagrange func-
tion with respect to 2
q
 is given by
2
2
L
518
q
 (81)
Appendix E. Derivation of Adjoint Equations
. The parameters are defined as follows:
The time optimal control method is described in the lit-
erature [17]. The Hamiltonian formulation is applied in
time optimal control. Using a Legendre transformation,
the Hamiltonian formulation is derived from the Lagran-
ian formulationg
11
x
q
(82)
21
x
q
(83)
31
x
q
(8)
42
4
x
q
(85)
The state equation is defined as
3
11
4
22
23 2522
12
111
33
44
25 366
2
12
111
d
d
x
xS
x
xS
AA BABA
AAA
xS
t
xS
AA AAA
A
AAA




























(86)
The cost function with respect to the time optimal con-
trol is given by
00
d1d
tt
J
Lt Nt NtN
 
 (87)
where the constraint N represe
and Equation (16). Using the a
3, 4), the Hamiltonian is defined as
nts the constraint condition
djoint variables pi (i = 1, 2,
T
H
L
pS (88)
where L represents 1 of the integrand in Equation (87).
The adjoint equation is given by

TT
T
d
d
H
L
t


 




ppS
xx
(89)
The boundary condition of the adjoint equ
given by
ation is
 


T
f
f
f
Nt
pt t



x
x (90)

Copyright © 2011 SciRes. OJDM
K. SHINOHARA
Copyright © 2011 SciRes. OJDM
152
The Hamiltonian Equation (89) is
13 24 334
1
H
pxpxpS pS (91) 1
H
1
d
d0p


(92)
tx

Using Equation (89), the adjoint equation is given by
2325 8936 89
24 38 211
3
1
2
AAA AA
pp p
A
  
25
2458386112
2A
AAAAAAAA
 
22
11
2898 2898
31 32
22
11
11
d
22
tA A
BAAA AAAA
pp
AA
AA





 
 
 
 
2
dAA BA AA
BAAA AA



4
2
1
A
289 8
41
2
1
1
2AAA A
p
A
A

689
42
2
1
2AAA H
p
A






2
x

(93)
27 21027 610
4
AA AA
31 3
113
22
d
d
BA AA
H
pp p
tA
 
p
 
Ax
(94)
212 26
AA BA
21
2 22
4
14
d
d
BA BA
H
42 3
1
pp p
tA
 p



Ax

(95)
The constraint condition at the end time is given by
 
111
2
ff
ptxt a
 (96)
 
222
2
ff
ptxt b
 (97)
0)
(98)
The gradient is given by
2
B
Hpp
2
34
111
34
21
A
 
333
2(
ff
ptxtc
 
 
444
2(0)
ff
ptxt d
 (99)
6
2
1
A
A
A
HA
pp
A
A





(100)

