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Grid Search Optimized SVM Method for Dish-like Underwater Robot
Attitude Prediction
Tian Wang, Xiufen Ye, Lei Wang College of Automation,
Harbin Engineering University, Harbin, 15001,China
E-mail: [email protected]
Heyi Li College of Underwater Acoustic Engineering,
Harbin Engineering University, Harbin, 15001,China
E-mail: [email protected]
AbstractThe control of dish-like underwater robot motion is
complex. It involves many kinds of influencing factors and its also
a nonlinear process. The model of attitude motion control is very
important for the accuracy control and self adapting predictive
control. For establishing the attitude motion model and predicting
the attitude, SVM algorithm was used to construct a MIMO identifier
in this paper. Moreover, in order to improve the effect of the
identification and prediction, the grid search method was adopted
to optimize the key parameter C and g in SVM. At last the effects
were contrasted with GA and PSO optimized SVM algorithm by the data
from the experiments in the pool, the results proved the
superiority of grid search method in both calculating time and
optimizing results. The results show the well performance of this
GS-SVM on the identification and prediction for the attitude of
dish-like underwater robot.
Keywords- Prediction; SVM; MIMO; Dish-like Underwater robot;
Grid Search
I. INTRODUCTION Underwater robot with the properties of flexible
free
degrees, non-liner and high tense coupling[1] requires the
high-accuracy dynamic model to achieve the optimum control.
Generally, the major methods for underwater robot modeling are
modeling by mechanism and system identification. Nevertheless,
considering the complexity of modeling by mechanism[2-3] and model
obtained by this method is not suitable for applying in the control
system directly, the model correction is needed by experiment
repeatedly. The traditional system identification ways such as the
Least Square method[4], the Gradient Calibration method, the
Maximum Likelihood method[5] cant attribute the demand of nonlinear
MIMO problem. The unsuitability of ANN[6] is resulted as the
network structure is hard to determine, Generalization ability is
poor, large amount samples is demanded.
By contrast, the Support Vector Machine(SVM) is proposed as a
modern technique[7] based on Statistical Learning Theory and
Structural Risk Minimization Inductive Principle to identify the
structure of model. The SVM is a kind of convex quadratic
optimization problem, it has simple structure, strong
generalization ability and the suitable solution of small samples
and non-liner high dimension problems, the global optimal solution
can be found quicker
and more accurate. The traditional SVM is mainly settle the
problem of SISO and is not suitable for underwater robot with MIMO,
close coupled, under-actuate, time variant and unstable complex
dynamic system. To face this issue, dish-like underwater robot
attitude MIMO identifier based on SVM is proposed, The kernel
function of this identifier is RBF .
As to improve the accuracy and reliability of SVM, the ultimate
model would have a decline in the accuracy, generalization ability
and cause over learning if SVM is under high study accuracy. To
solve this problem, the suitable balance factor C and kernel
functional parameter g are in significant position to get the best
result of identification. In this paper the grid search (GS) was
used as optimization of parameter. In the end, the comparison was
made with the results of GA and PSO optimized SVM.
II. INTRODUCTION AND ANALYSIS OF ROBOT This paper presented a
new type of underwater robot
structure in the condition of closed underwater robot profile.
And the dish hydrodynamic shape makes the robot a good hydrodynamic
performance and the ability of movement almost in any direction.
The robot is drove by 4 thrusters laid in vector in the control
surface and the included angle of control surface and inertial
coordinates by the moment caused by the change of relative position
of gravity center and buoyant center. The 2 slide mass is used to
change the relative position. With the results, the robot can move
in 5 DOF. The structure of robot is shown in Fig.1.
As shown in Fig.1, the coordinate systems O-XbYbZb are the
body-fixed coordinates on the dish underwater robot. Origin point O
is the center of the mass and geometric center. Slider block m1 and
m2 move separately at coordinate axes OXb1 and OZb2, and they make
change of the position of center of mass. The frame O-XvYvZv
parallels to the frame O-Xb1Yb1Zb1 and O-Xb2Yb2Zb2 , the Yb1 and Yb
are coincident. The distance of plane XbOYb to plane Xb1OYb1 or
plane Xb2OYb2 is bym . The thrusters are laid in vector and act on
the center of mass, and the sum of thrust is denoted as T , the sum
of moment caused by thrusters is denoted as MT. is the included
angle of thrusting line and axes OXb.
2012 Fifth International Joint Conference on Computational
Sciences and Optimization
978-0-7695-4690-2/12 $26.00 2012 IEEEDOI
10.1109/CSO.2012.189
839
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Figure 1. The structure diagram of robot.
Apparently, with the analyses above, the inputs of robot are the
force of 4 thrusters T1, T2, T3, T4 and the movement distance of
centroid adjust mass xb, zb. The outputs are the included angle , ,
of body-fixed coordinates and inertial coordinates. The depth is
negligible because the experiment is conducted in shallow water.
Underwater robots description of state space correspond to the
equation below
( , , , , , , )1 2 3 4( , , , , , , )1 2 3 4
x f x T T T T x zb by g x T T T T x zb b
==
(1)
The non-linear model after discretization is
( 1) ( ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ))1 1 2 3 4( 1)
( ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ))2 1 2 3 4( 1) ( ( ),
( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ))3 1 2 3 4
k f k k k T k T k T k T k x k z kb bk f k k k T k T k T k T k x
k z kb bk f k k k T k T k T k T k x k z kb b
+ =+ =+ =
(2)
III. THE DESIGN OF SVM IDENTIFIER Refer to the reference[10], a
SVM based MIMO dish-like
underwater robot attitude prediction system for the angles of
pitch, roll and yaw is proposed, the structure is shown in
Fig.2.
Figure 2. Prediction structure block diagram of SVM
identifier.
In Fig.2, u(k)=[T1, T2, T3, T4, xb(k), zb(k)] represents the 4
outputs of the thruster respectively at the k moment. xb(k), zb(k)
are the displacement distances of 2 attitude adjusted mass. v(k)=[
(k-1), (k-1), (k-1)] is the output at the k-1 moment .The outputs
of this identifier is (k+1), (k+1), (k+1), they are pitch, roll and
yaw angles.
First of all, to face the problem of linear regression,
regression function was estimated in the linear regression set
according to equation (3).
( )( , )f x x b = + (3)
Add to the given training sample set ( ) ( ) ( ){ }, , , , , ,1
1 2 2T x y x y x yn n=
Where lx X Ri = ,my Y Ri =
And n , l , m is the quantity of samples and the dimension of
the input and output.
Assuming ( ) ( )1,1
nR b y x bemp ini
= =
(4)
and b should be calculated in the condition of restriction based
on the minimum of empirical risk to satisfy the slack variable i ,
( )* 1, ,i ni = with the equation below
* *( , )1 1
n nF i ii i
= + = =
(5)
( )( )( )( ) *. .
*, 0, 1, ,
w x b yi i i
s t y w x bi i i
i ni i
+ + + + =
(6)
To build Lagrange function according to the equation (6) , that
is
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( ) ( ) ( )( )
* * * * *, , , , , , , ,( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1
1* * ( ) ( ) ( ) ( ) ( ) ( ) ( )
1
n nL b C y x bk k k i k i k k i k k k i ki i
nx b yi k k i k k i k k i k
i
= + + + = =
+ + + =
( ) ( )*( ) * * ( ) ( ) ( ) ( ) ( ) ( ) ( )2 1knC
cn k k k i k i k i k i ki + =
(7)
The solution of the equation (7) requires complex optimization,
which could be switched to the parameter C to
solve the problem of equation (8) with the restriction of
condition (9) the convex optimization.
( ) ( ) ( )( )( ) ( )1 * *
( ) ( ) ( ) ( ) ( ) ( )2* , 1min ,1 * *
( ) ( ) ( ) ( ) ( )1 1
Tnx x a am i k i k i k j k j k j k
i jn nk yi k i k i k i k i k
i i
==
= + + = =
(8)
( )* 0( ) ( )1. .*0 , , 1, , , 1, ,( ) ( ) ( )
ni k i k
is t
C i n k mi k i k k
=
= = =
(9)
( )i k , *( )i k calculated by the equation (8) (9) based on the
condition of Karush-Kuhn-tucker (KKT).The SVM regression estimation
equation is
( )( )*( )( ) ( )( ) ( ) ( ) ( )1n
f x x x bk ki k i k i k ki = + +
=
(10)
Aiming at the solution of the problem, kernel function was used
to make the non-linear data mapped to the high dimension space to
regression. Thus the equation can be drawn below
( )( ) ( )( )
( ) ( )
*min ,1 * *,2 , 1
* *1 1
TnK x xi i i j j j
i jn n
yi i i i ii i
= =
+ + = =
(11)
Where ( ),K x xi j is the kernel function and the regression
estimate function of MIMO-SVM conveyed that
( ) ( )*( ) 1n
f x K x x bi i ii
= + +=
i (12)
Corresponding that
( ) ( )( ) ( )
( ) ( )
*( )(1) (1) (1) (1) (1) (1)1
*( )(2) (2) (2) (2) (2) (2)1
*( )( ) ( ) ( ) ( ) ( ) ( )1
nf x K x x bi i i
in
f x K x x bi i ii
nf x K x x bm i m i m i m m m
i
= + + =
= + +=
= + + =
i
i
i
(13)
Where, the threshold value b is the constant as the equation
(14) .Then we can got the model by substitution of the support
vectors and the parameters.
( ) ( )1 * *, sgn( )
1 1
l nb y K x xk k i i i k k kl k i
= + = = (14)
The RBF is used here as equation (15)
( ) 2, expK x x g x xi i =
(15) The kernel function parameter 0g > , which could
approximate any non-liner relation easily.
IV. GRID SEARCH OPTIMIZATION
A. Grid Search The amount of support vectors, threshold b,
NZ-
Lagrangian multipliers can be obtained through identification.
The penalty factor C and RBF parameter g must be provided before
identification. They are the key factors of machine learning,
predictive ability and generalization ability.
This paper used grid search algorithm(GS). The GS is a
traditional digital plan method, its a kind of exhaustion method.
Theres no special requirement of objection function, it has a good
effect to solve constrained nonlinear extreme value problems. The
GS will mesh in the pre-set range, the grid intersections are
correspond to objective function values. Then the optimal value of
objective function will be searched in the feasible intersections
by rules, the parameters correspond to this optimal point will be
the optimal.
The advantages of the grid search method is that it can
simultaneously search multiple coupling parameters. This method is
the search for parameter combinations, therefore it avoids the
coupling of multiple solution problems that may arise between the
multi-parameter. So the optimal evaluation function value
corresponds to the optimal combination of parameters(C and g).
B. Optimization Routine The MSE of SVM output and samples is
used as
evaluation function. The optimal C and g can be obtained when
the MSE is minimum. Cross-validation is used to verify the MSE.
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Set the grid search region for C, g as (0,100) and (0,10).
To process the grid search. To draw the contour map and choose
the best
parameters accordingly.
V. EXPERIMENT
A. Experiment Environment The parameters of experiment pool is
17.3m x2m
x1.7m(length x wide x deep). And the major purpose of experiment
is to obtain the attitude angles of the robot (pitch, roll, yaw),
moreover to obtain the offset of the adjusted centroid mass, depth
and the outputs of thrusters. The thrust force was calculated by
the driving voltage of pump and the position of centroid adjusting
mass was reckoned by the steps of stepping motor after the power
initialization. The attitude angles were obtained by the AHRS (MTi,
made by Xsense, Holland) deployed in the robot. This refresh rate
of this AHRS is 120Hz, the accuracy of roll and pith
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a) GS-SVM Contrast Figure
b) GA-SVM Contrast figure
c) PSO-SVM Contrast figure
Figure 6. Contrast figure of SVM1 fitting data
VI. CONCLUSION In this paper, the characteristics of this novel
type of
underwater robot was analysed. To combine these characteristics,
the MIMO-SVM optimized by Grid Search Method was proposed. The
similarity of prediction and the experiment data could attain
99.15% at least with the computation speed of 1.5 s. These two
aspects of parameters were both better than that of GA and PSO
optimization. The model obtained by this GS-SVM attaining the
superiority of
high accuracy and instantaneity of prediction. This would
contribute a lot to the self-adaption control and prediction
control study.
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