Design of a Waypoint-tracking Controller for
a Biomimetic-autonomous Underwater Vehicle
.Jenhwa
Guo,
Jing-F& Tsai, Fomg-Chen Chiu, Sheng-Wen Cheng, Ye-Sheng Ho
Dept. of Engineering Science & Ocean Engineering, National Taiwan University73 Chou-Shan Road, Taipei, Taiwan 106 R.O.C. [email protected]
Abshacf- We develop a control system for the waypoint-tracking of a biomimetic-autonomous underwater vehicle (BAUV). The BAUV swims forward by oscillating its body and caudal fin and turns by slanting its body and caudal fin to the side of turning direction. Because of the undulatory
motion of BAUV, we take moving averages of swimming
velocity and heading error as feedback signal to control the velocity and angular velocity of BAUV. The control algorithm
utilizes the oscillating frequency to control the forward speed,
and a body-spline offset parameter to control the yawing rate.
We verify the effectiveness of control algorithms by experiments. Finally, we discuss the stability of the control
system based on a Lyapunov function. Fig. 1 Photograph of the testhed vehicle
I. INTRODUCTION
AUVs have great potential for applications in many undersea missions. Traditionally, rotary propellers driven by electric motors power AUVs. But the low efficiency of the
small diameter propellers coupled with the large fraction of Fig. 2 Definitions of coordinate systems the hull volume induced positioning, turning and hovering
problems. Recently, the integration of engineering and biology provide us a new direction to solve these problems. Fish have great ability to perform precise hovering and agile turning. The shape of fish is suitable for swimming in the water. So we imitate the shape and the motion modes of the
fish to develop a prototype AUV. In our previous work, we had established an optimal body-spline for the BAUV to swim forward [I]. In this paper, we combine a turning mode with the forward-swimming mode to design a control system for BAUV’s waypoint tracking.
A review of fish swimming modes for aquatic locomotion is presented in [2]. So far, the focus of our BAUV development focused on the control of body andor caudal fin movements. In the near future, we will include the median andor paired fins movements to the BAUV system. The turning performance of fish robots is discussed in [3].
Chiu et al. did simulations on undulatoxy locomotion of a flexible slender body [4]. Chiu analyzed the dynamic characteristics of a BAUV [ 5 ] . Guo et al. presented a method to coordinate body segments and paired fins for the BAUV’s motion control and guidance [6].
Section I1 introduces the optimal body-wave equation
of BAUV for the forward swimming and the turning. Experimental measurements of BAUV’s turning parameter were included in Section 11. In Section 111, we divided the control system into two levels. The local control level controls the motion of BAUV in the vehicle-fixed
coordinate. The global control level controls the BAUV to track waypoints in the space-fixed coordinate. The stNChue of the feedback control system is presented in Section 111. Section 1V presents experimental results of the waypoint tracking system. Finally, the concluding remarks and future works are given in section V.
11. FORWARD AND TURNING MOTION Figure 1 is the outlook of the testbed BAUV. We defme three coordinate systems as: space-fixed coordinate system 0 -
X
Y , body-fixed coordinate systemo - x y and segment-fixed coordinate system 6i - i ; j i shown as in Fig. 2. Each coordinate can be transferred to another by using the relationship of the position and angle between two coordinates.
From observations of fish [8,9], the bodily motion can be considered as traveling waves that increase in amplitude from the nose to the tail. A specific form of traveling wave suggested by Lighthill [9] was
f i ( x , t ) = f2(x)sin(kx+ a t ) . ( I - e - - ) ( 1 ) where
f i ( x ) : amplitude envelope; here we define 2
f 2 ( x ) = c,x
+
c2xX : displacement along main axis
k = 271 I
A
:body wave number1 :body wave length
c, :coefficient of linear wave amplitude envelope
c2 :coefficient of quadratic wave amplitude envelope
o = 2nf = 2n / T : frequency
T : period of the body wave
T,
: period of the initial undulating delay cycleEquation (1) expresses a sinusoidal wave traveling from right to left (i.e. from x = 0 to x = -L ) within the bounds of a second-order ( c I ~ + c ; x 2 ) amplitude envelope. The exponential term defining the initial delay when the body s t m s to undulate is for the convenience of conducting experiments. The slope of the body is y , ( x , < ) = 8 y / a x , the angle between the body and the x-axis is 0.- tan-' y., . From
[I], we found that the optimal parameters for our BAUV to swim forward efficiently is c, =-0.075,' c2 =0.017,
Though real fish turn skillfully using not only tail fin but also pectoral fins or ventral fins, our researches on the
BAUV are focused on the tail fin. The turning mode of BAUV is defined as in Fig. 3.
As we know, fish swim by oscillation pattern. They
swing their fms and bodies to generate the propulsion force and control the heading direction. Using this tnming mode,
the BAUV keeps swinging and slanting to one side during
tuming. We considered it as the most fundamental and important tuming mode. One can easily control the turning diameter and forward velocity by this mode.
Since there is no involvement of side fins, it is necessaw to keep the swimming speed by oscillating the tail fin, or the BAUV may stop during the turning. We then rewrite the body wave as follow
a
=3.522, T 4.y ( x , t ) = k , . r ; ( . J ) + k 2 . f i ( X ) (2) where
y ( x , l ) : transverse displacement of body
k, : scale factor of the oscillating amplitude
k2 : offset parameter, lk2
I
5 1In (2),. f ; (.,I) is the optimal body-wave equation.
kz controls the magnitude of the.center of oscillation away from the nominal position along the BAUV's body. The
parameter kz controls the offset of BAUV's body spline
while the BAUV is turning. For our BAUV, the amplitude
of oscillation is restrained by its propulsion mechanism. The parameter k, guarantees that joint angles do not exceed their limits.
.
1 'Swine the iail 10 th: !kit \
. 1'
TlillC
Fig.3 Turning mode of the BAUV
I +
forward direction maximum offset maximum d i s p l a c e m e n t k,=-1, . k2'=0 k,=l 1 Fig. 4 Definitions.ofk2
full oscillating amplitude
implitude decreased due to contramts (of mechanism
minimum amplitude
1 Fig.5 Definition of kl
The parameter
k,
also keeps the BUAV swimming forward.Their defmitions are illustrated in Fig.4 and Fig.5,
respectively.
To investigate the characteristics of the BAUV's
turning mode, we chose two sets of parameter: Set 1 is
c, =-0.075, c2 =0.017, 1 =3.522. Set 2 is cI =-0.094,
c2 =0.031, A: =6.51. The set 1 data is the optimal body
spline obtained from free running experiments using the testbed B A W [I]. The variables are period ( T ), offset ( k , ) and forward velocity (v). The BAUV is swimming forward
using its full oscillating amplitude ( k , = I ) .
The model setup is shown in Fig. 6. The BAUV was
bung on a three-degree-of-freedom force gauge, which was fixed on a carriage cart above a water tank. The carriage cart
provided constant velocity to the model. The force gauge measured the force in the x and JJ directions together with
the moment in the
z
-direction.Figures 7 , s and 9 are typical data ofthe experimental results. In this experiment, we set
T
= 4 ~ offset = 0 , Y = 0 . FromFig.10, we observed that the set 1 is a better forward-swimming mode in most of the periods. Figure 11 shows that the set 2 has better yawing moment under the same offset parameter compared to set 1. This is due to the fact that the oscillating envelope of the set 2 is larger than the set 1. Also in Fig.11, we observed that the moment in the z -direction increased with the value of the offset, parameter. The BAUV tums faster with larger offset. In Fig.12, the BAUV had a variable initial velocity, and the higher the initial velocity, the BAUV will tum faster under the same offset. Figures 13 and 14 are results from free running trials using the testbed BAUV in a water tank. The BAUV reached a steady forward speed, then an offset was applied to the body spline. We measured the relationship between the offset values and the yaw rates, sideslip velocities, respectively. It is shown that the offset parameter of the body spline can be used as a direct control factor to modify the yaw rate in a proportional fashion. The offset and the sideslip velocity do not exhibit a proportional relationship.
Fig. 6 Model setup for the force and moment measurement
F O ~ ! " " d l ~ l a " , . , , . , , . ' , ,
.
, , , , I ? O IO 10 10 $0 sa 70 10 I O am./., .,yJFig. 7 The force in the
x
-direction. . .. ..
.
I W . . . . .
, , , , , , , ,
m.,.,
Fig. 9 The moment in the z -direction
+Sei .
~-
2-
100.00 3 .E
50.00a
0.00 M-
X 3 4 5 6 Period, T(SI
Fig. 10 Force in the x -direction and oscillation period
Set I . 1'=4 ~ o f f s c r = l l 6
-0- off~cl=ll3
, c a i ~ i a g veloiily, Y ( i d s )
Fig. 12 Moment in the z -direction with carriage velocity
Sei I t T = 4
-
-13- T=5.
g
0" 0.8 ,757 0.6 ,625 0.4-
-
y
0.2 5 0 F-
I16 I 13 olfwFig. 13 The relation between offset and angular velocity
Set I +TA -D- T=5 P r -8
'
0 1 16 113 offmFig. 14 Offset and the velocity in the
y
-direction.111. CONTROLLER DESIGN
In this section, we develop a waypoint-tracking control system for the BAUV. Basing on the forward swimming and turning modes we determined in the previous section, by alternating the amplitude and the frequency, we can control the forward velocity of the BAUV. By control the offset parameter, we can control the angular velocity of the tuming motion. Because the BAUV moves in an undulating fashion, we take the average of forward speeds, as well as heading errors during the motion period as feedback to control the forward velocity and heading of the BAUV.
Unlike traditional underwater vehicles, BAUVs do not have direct control devices such as thrusters to control the states of BAUVs. The joint angles of BAUV determine the bodily motion, and they move by the interaction forces between the BAUV and the surrounding fluid. The uncertainty of hydrodynamic forces and limitations due to the power supply and mechanical design of the propulsion mechanism make it difficult to precisely control the BAUVs' motion.
In order to overcome the difficulties, we divide the control system to two levels. The local control level generates the oscillation for swimming forward. The global control level controls the performance of waypoint tracking. At the global control level, we let the BAUV track waypoints by evaluate the k ? . We applied the so-called
"line-of-sight" guidance scheme in the horizontal plane. Control error, e , is dzfined as the angle between the waypoint and the BAUV in the global coordinate minus the heading angle of the BAUV. Feedback signals are the average of errors in one swimming period. The averaged error is defined as
(3) where
F
: averaged errore : angular difference between the waypoint and BAUV's
T : period of body-wave
We take the sum of averaged error and its differentiation and divide the sum by a parameter k ,
.
k, is responsible for the sensitivity of error correction. The offsetk , is evaluated by the following equations
heading
.where
: coefficient of
-
e : differentiation of the averaged error
k, : coefficient of sensitivity
To show that by using (3) and (4), we can control the BAUV to track waypoints, the following proof is provided.
In Fig. 15, the feedback system is shown in its block diagram form. The following definitions are used
.
e=v</-v
( 5 )e =
e,,
- Jq ( 6 )I
w
Fig.15 Control system block diagram
where y is the yaw angle in the 0 -
X
Y , q is the yaw ratein the 0 - x y , and J = I is the transformation from 0 - x y to 0 - X Y . N represents the transformation from the offset to the yawing rate discussed in the section 11. N is approximately a linear function.
1 . .2,
s = - e + - e
T T (9)
We define a Lyapunov function:
v = - s
s2 (10)
I T
The time derivative of the Lyapunov function is
whcre k, is positive, then
From the above derivations, we can prove that the system is stable in the Lyapunov sense. The fact that the yaw rate,
p
, is linearly dependent on k, provides stability to the controlled behaviors of the BAUV.IV EXPERIMENTS
The BAUV has five components: head, caudal part, tail fin, and two pectoral fins. The head segment, tail fin, and pectoral fins are rigid, and the caudal part is supported by a rigid link and some flexible materials. A Doppler sonar is
set in the abdominal part of the BAUV for sensing the direction, velocity, and water depth. We focus on body and
caudal fin movements. Pectoral fins are not concerned now, hut they will he studied in the near future.
Two cases of waypoint-tracking experiments are illustrated here. In the following experimental setups, case 1 involves results of tracking a single target point. Case 2 provides three waypoints for the BAUV to swim around. These points are at (4,0.5) m, (4,2) m, and (2,2)
m.
When the distance between the position of BAUV and the target is less than 0.5 m, this waypoint is considered being reached.We set the parameters as e, =-0.075, c, =0.017, ,I=3.522, T =4sec., and the scale factor k, started to decrease at k,=O.5. That is, the BAUV will decrease its speed for the cases where large offset k2 is required. The
variables to he tumed are Ae and k , . In case 1, Ae=2 and
k, =90. Figures 16-26 are outputs of the control system of tracking single waypoints.
For the case 2, the parameters are the same as in the case 1, and the variables to he turned are ,le and k, . Figure 27 shows the results of tracking three waypoints arranged in a way such that large turning maneuvers are required.
I 1
lime (I)
, , I . . , . . , .
.
. . . . . , . . . I 0 1 0 2@ 10 .o %a 60 10 ~ 80 time ( 5 )Fig. 21 The error differentiation
-q o + -< time ,I)
Fig. 2;: The offset k2
a
mi
' 1 IIO 20 24 40 YI f4 IO U time (I)
O O s r
.
. . ,4
(0 io io h 1. .b ;oA
time(s)Fig. 26 The velocity in the y -direction
, , . . . , . , , , , , . , , . , . , , . , ) . , . , . , , . , , . , . , . , 0.6 I 3 . 5 I *.I I I 5 4 4.5 I x(m)
Fig. 27 The tracking of three waypoints
It was observed that the parameter k, affects the sensitivity of the control system. If k, is too small, the
system response tends to he unstable. The time derivative of
error helps to improve the stability when the value of k, is large. But, if the value of k, is small, the time derivative of
error makes the system less stable instead.
V. CONCLUDING REMARKS
To implement waypoint-tracking capability, we propose a way of combining the forward-swimming mode and the turning mode of a biomimetic-autonomous underwater vehicle (BAUV) for the motion control. By altemating the amplitude and frequency of body-wave equation, we can control the forward speed of BAUV. By control an offset parameter of the body spline, we can
control the yaw rate directly. Averages of forward speeds and heading errors during the motion period are served as feedback to control the velocity and heading of BAUV. Limits in joint angles of the swimming mechanism were considered in the controller design. We have demonstrated by experiments that our waypoint-tracking controller provides satisfactory tracking capability for the BAUV. The performance of the control system can be adjusted by choosing different control parameters.
To improve the performance of the control system, environmental information can be used to better the
precision of navigation system. More sensors such as vision camera or forward-looking sonar are necessary. To decrease the roll and pitch motions, the balance between buoyancy and weight in every body section shall be considered in the future. The recoil motion decreases swimming efficiency, pectoral fins must he considered in the BAUV control system in the future.
ACKNOWLEDGMENT
The authors would like to thank the National Science Council of the R.O.C. for financially supporting this research under Contract No. NSC90-2611-002-030.
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