- Email: [email protected]
UNDERWATER ROBOT INTELLIGENT CONTROL BASED ON MULTILAYER NEURAL NETWORK
8th IFAC Conference on Control Applications in Marine Systems Rostock-Warnemünde, Germany September 15-17, 2010
UNDERWATER ROBOT INTELLIGENT CONTROL BASED ON MULTILAYER NEURAL NETWORK Alexander A. Dyda, Dmitry A. Os'kin
Laboratory of Nonlinear and Intelligent Control Systems, Department of Automatic and Information Systems, Maritime State University Vladivostok, Russia [email protected], [email protected]
Abstract: the paper is devoted to the design of the intelligent neuron network based control systems for underwater robot. New algorithm for intelligent controller learning is derived with usage of speed gradient method. Proposed systems provide the robot dynamics close to reference one. Results of simulation have confirmed the effectiveness of approach developed. Keywords: underwater robot, control, uncertain dynamics, multilayer neural network speed gradient method, .
1.
INTRODUCTION
Underwater robots (UR) promise great perspectives and have a widest scope of applications in the area of ocean exploration and exploitation. To provide exact movement along prescribed space trajectory, UR needs a high quality control system. It is well known that UR can be considered as multi-dimensional nonlinear and uncertain controllable object. Hence, the design procedure of UR control laws is difficult and complex problem. Modern control theory has derived a lot of methods and approaches to solve appropriate synthesis problems such as nonlinear feedback linearization, adaptive control (Fradkov, 1990), robust control, variable structure systems (Dyda, 2007) etc. However, most of mentioned methods of control systems synthesis use at least any information about structure of the UR mathematical model. The nature of interaction of a robot with water environment is so complicated that it is hardly possible to get exact detailed equations of UR movement. Possible way to overcome control laws synthesis problems can be found in the class of artificial intelligence systems, in particular, based on multi-layer neural networks (NN) (Narendra, Parthasaraty, 1990) (Yuh, 1990) In the paper the intelligent NN based control system for UR is designed. New learning algorithm for intelligent NN controller that uses speed gradient method is proposed. 2. UNDERWATER ROBOT MODEL UR mathematical model traditionally consists of differential equations of kinematics (1) q& 1 = J (q1 )q 2 and dynamics 978-3-902661-88-3/10/$20.00 © 2010 IFAC
179
D (q 1 )q& 2 + B( q 1 , q 2 )q 2 + G (q 1 , q 2 ) = U (2) where J the kinematical matrix; q1, q2 the vectors of generalized coordinates and body-fixed frame velocities of UR; U the control forces and torques vector; D the inertia matrix taking into account added masses of water; B the Coriolis – centripetal term matrix; G the vector of generalized gravity, buoyancy and nonlinear damping forces/torques.
Poor a priori knowledge of mathematical structure and parameters of matrices and vectors of the UR model can be compensated by intensive experimental research. As a rule, this way is expansive and takes a long time. One of perspective alternative approach is connected with usage of intelligent NN control 3. INTELLIGENT NN CONTROLLER AND LEARNING ALGORITHM DERIVATION
Our objective is synthesis of underwater robot NN controller to provide its movement along prescribed trajectory qd1(t), qd2(t). First we consider the control task with respect to velocities qd(t). Define error
e2 = q d2 − q 2 and introduce the function Q as measure of difference between desirable and real trajectories: 10.3182/20100915-3-DE-3008.00082
CAMS 2010 Rostock-Warnemünde, Germany, Sept 15-17, 2010
1 Q = e T2 De 2 2 Further we use the speed gradient method developed by (Fradkov, 1990). According th othe method, compute time derivative of Q:
& = e T De& + 1 e T D & Q 2 2 2 e2 2 As
q 2 = q d2 − e2 , one has
D(q1 )q& 2 = D(q1 )q& d 2 − D(q1 )e& 2 . Using expression of first term from dynamics equation, one can get the following:
D(q1 )e& 2 = D(q1 )q& d 2 + B(q1, q 2 )q d 2 − − B(q1, q 2 )e2 + G(q1, q 2 ) − U and time derivative of function Q can be written in the form
& = eT (D(q )q& + B(q , q )q − Q 2 1 d2 1 2 d2 1 & − B(q1 , q 2 )e 2 + G(q1 , q 2 ) − U) + e T2 D e2 . 2 After terms reorganization, one get
w 11 w w = 21 ... w H1
w 12 w 22 ... w H2
... w 1N ... w 2 N ... ... ... w HN
As result of nonlinear transformation f(.), hidden layer output vector can be written in the form
f1 ( w 1T x ) f ( w , x ) = ... , f H ( w TH x ) where wk denotes k-th raw of matrix w. By analogy, introduce matrix W which element Wli odenotes transform coefficient from i-th neuron of hidden l-th neuron of output layer. With defined NN parameters, the undwewater robot control signal (NN output) is computed as following:
U = y( W, w , x ) = Wf ( w , x ) Substitution of this control let us to get
& = eT (D(q )q& + B(q , q )q + G(q , q ) − Q 2 1 d2 1 2 d2 1 2
& = eT (D(q )q& + B(q , q )q + Q 2 1 d2 1 2 d2
1 & (q1)e2 = − U) − eT2B(q1, q2 )e2 + eT2D 2 eT2 (D(q1)q& d2 + B(q1, q2 )qd2 + G(q1, q2 ) −
+ G(q1, q 2 ) − Wf (w, x)).
1 & − U) + eT2 (D (q1) − B(q1, q2 )e2 ). 2 As known, the matrix in last term is skewsymmetric, hence, this term is equal to zero and we have simplified expression:
& = eT (D(q )q& + B(q , q )q + G(q , q ) − U). Q 2 1 d2 1 2 d2 1 2 We plan to implement intelligent UR control (Dyda, 2007) based on neural network. Without losing of generality of the approach, choose two-layer NN. Let hidden and output layers have H and m neurons appropriately (m is equal to dimension of e2). For the sake of simplicity, one supposes that only summing of weighted signals (without nonlinear transformation) is realized in output layer. Input vector has N coordinates. Define wij as transform coefficient for i-th input of j-th neuron of hidden layer. So these coefficients compose matrix 180
To derive NN learning algorithm, apply the speed gradient method. For this, compute partial derivatives of function Q time derivative with respect to adjustable NN parameters – matrices w and W. Direct differentiation gives
& ∂Q = −e 2 f T ( w, x ). ∂W It is easy to demonstrate that choosing of all activation functions in the usual form
f ( x ) = 1 /(1 + e − τ x ) imply property ∂ T T T f i ( w i x ) = f i ( w i x )[1 − f i ( w i x )] x j ∂ w ij
Introduce additional functions
CAMS 2010 Rostock-Warnemünde, Germany, Sept 15-17, 2010
[
ϕ i ( w i T x ) = f i ( w i T x )[1 − f i ( w i T x )]
q2 = vx
and matrix
vz
ωy
]
T
Dimensions of NN input (q2 and e2 ) and output (control forces and torque) are equal to 6 and 3.
Φ ( w , x ) = diag ( ϕ 1 ( w 1 T x )... ϕ H ( w H T x ))
Direct calculation gives
[
U = Fx
& ∂Q = − ΦW T e 2 x T ∂w As a final stage, we can write the NN learning algorithm in following form: W( k +1) := W ( k ) + γe 2f T (w , x ). w ( k +1) := w ( k ) + γΦWT e2 x T (γ is learning step, k is number of iteration). Now consider which components should be included in NN input vector. As NN controller is oriented to compensate an influence of appropriate matrix and vector functions, in common case the NN input vector must be composed of q1 ,q2 , e2 , qd2 and its time derivative. The NN learning procedure leads to reducing of function Q, consequently in ideal conditions, error e2 tends to zero and the UR movement follows to desirable trajectory
My
]
T
For the NN controller contaning 5 neurons in hidden layer results of simulation are given on Fig.1 - 6. Transient processes and control for taken nominal model are shown on Fig.1-2. 0.3
0.25
0.2
0.15
Vx, m/sec Vz, m/sec Wy, rad/sec
0.1
0.05
0
-0.05
0
10
20
30
40
50 t, sec
60
70
80
90
100
Fig.1. Transient processes (nominal model)
q 2 (t) → q d2 (t)
80
If UR trajectory is given by qd1(t), one can choose
70
q d 2 (t ) = J −1 (q1 )(q& d1 (t ) + k (q d1 (t ) − q1 ( t ))
Fz
Fx, H Fz, H My, Hm
60 50
(k is positive constant). As follows from kinematics equation,
40
q& 1 ( t ) → q& d1 ( t ) + k (q d1 ( t ) − q1 ( t ))
30
and
20
e& 1 ( t ) + ke1 ( t ) → 0
10
where
e1 ( t ) = q d1 ( t ) − q1 ( t )
0
10
20
30
40
50 t, sec
60
70
80
90
Fig.2. Control signals(nominal model)
Hence, UR follows to the planned trajectory qd1(t). (3.1), 4.
0
SIMULATION RESULTS OF INTELLIGENT NN CONTROLLER
To check the effectiveness of the approach, computer simulations have been carried. The UR model parameters were taken from Vector q2 consists of following components (linear and angular UR velocities): 181
Fig.3-4 present the same processes for tha case of reduced UR added masses.
100
CAMS 2010 Rostock-Warnemünde, Germany, Sept 15-17, 2010 0.3
0.3
0.25
0.25 0.2
0.2
0.15
Vx, m/sec Vz, m/sec Wy, rad/sec
0.1
0.15
0.05
Vx, m/sec Vz, m/sec Wy, rad/sec
0.1
0
0.05 -0.05
0
10
20
30
40
50 t, sec
60
70
80
90
100
0
Fig.3. Transient processes (reduced UR added masses)
-0.05
0
10
20
30
40
60
50
60
70
80
90
100
Fig.5. Transient processes (quadratic viscous friction)
Fx, H Fz, H My, Hm
40
50 t, sec
30 30
Fx, H Fz, H My, Hm
25 20
20
15
10
10 0
0
10
20
30
40
50 t, sec
60
70
80
90
100
Fig.4. Control signals (reduced UR added masses)
5 0 -5 -10
The exact description of hydrodynamic forces and torques is practically impossible. In the nominal model viscous friction was linear with respect to velocities. The effectiveness of the designed NN controller was proved and confirmed (Fig.5-6) for quadratic and cubic viscous friction forces (torques).
182
0
10
20
30
40
50 t, sec
60
70
80
90
Fig.6. Control signals (quadratic viscous friction) 5.
CONCLUSION
The approach to derive the learning algorithm for NN controller of underwater robot was proposed and studied. The numerical experiments have shown that high quality processes can be achieved on the basis of proposed intelligent NN control. The procedure of NN learning makes possible for UR control system to overcome parameter and, partially, structural uncertainties of object model. REFERENCES Dyda A.A. (2007) Adaptive and neural network control for complex dynamical objects. - Vladivostok, Dalnauka. – 149 p. (in Russian). Narendra K.S., Parthasaraty K. (1990) Identification and control of dynamical systems using neural
100
CAMS 2010 Rostock-Warnemünde, Germany, Sept 15-17, 2010
networks // IEEE Identification and Control of Dynamical System, Vol.1. № 1. 20, pp. 1475-1483. Ross A., Fossen T.and Johansen A. (2004) Identification of underwater vehicle hydrodynamic coefficients using free decay tests // Preprints of Int.Conf.CAMS-2004,. Ancona, Italy,2004. - pp.363368. Fradkov A.L. (1990) Adaptive control in large-scale systems.- M.: Nauka., (in Russian). Yuh Y. (1990) Modelling and control of underwater vehicles IEEE J. of Trans. Syst., Man, Cybern., vol.
183
UNDERWATER ROBOT INTELLIGENT CONTROL BASED ON MULTILAYER NEURAL NETWORK Alexander A. Dyda, Dmitry A. Os'kin
Laboratory of Nonlinear and Intelligent Control Systems, Department of Automatic and Information Systems, Maritime State University Vladivostok, Russia [email protected], [email protected]
Abstract: the paper is devoted to the design of the intelligent neuron network based control systems for underwater robot. New algorithm for intelligent controller learning is derived with usage of speed gradient method. Proposed systems provide the robot dynamics close to reference one. Results of simulation have confirmed the effectiveness of approach developed. Keywords: underwater robot, control, uncertain dynamics, multilayer neural network speed gradient method, .
1.
INTRODUCTION
Underwater robots (UR) promise great perspectives and have a widest scope of applications in the area of ocean exploration and exploitation. To provide exact movement along prescribed space trajectory, UR needs a high quality control system. It is well known that UR can be considered as multi-dimensional nonlinear and uncertain controllable object. Hence, the design procedure of UR control laws is difficult and complex problem. Modern control theory has derived a lot of methods and approaches to solve appropriate synthesis problems such as nonlinear feedback linearization, adaptive control (Fradkov, 1990), robust control, variable structure systems (Dyda, 2007) etc. However, most of mentioned methods of control systems synthesis use at least any information about structure of the UR mathematical model. The nature of interaction of a robot with water environment is so complicated that it is hardly possible to get exact detailed equations of UR movement. Possible way to overcome control laws synthesis problems can be found in the class of artificial intelligence systems, in particular, based on multi-layer neural networks (NN) (Narendra, Parthasaraty, 1990) (Yuh, 1990) In the paper the intelligent NN based control system for UR is designed. New learning algorithm for intelligent NN controller that uses speed gradient method is proposed. 2. UNDERWATER ROBOT MODEL UR mathematical model traditionally consists of differential equations of kinematics (1) q& 1 = J (q1 )q 2 and dynamics 978-3-902661-88-3/10/$20.00 © 2010 IFAC
179
D (q 1 )q& 2 + B( q 1 , q 2 )q 2 + G (q 1 , q 2 ) = U (2) where J the kinematical matrix; q1, q2 the vectors of generalized coordinates and body-fixed frame velocities of UR; U the control forces and torques vector; D the inertia matrix taking into account added masses of water; B the Coriolis – centripetal term matrix; G the vector of generalized gravity, buoyancy and nonlinear damping forces/torques.
Poor a priori knowledge of mathematical structure and parameters of matrices and vectors of the UR model can be compensated by intensive experimental research. As a rule, this way is expansive and takes a long time. One of perspective alternative approach is connected with usage of intelligent NN control 3. INTELLIGENT NN CONTROLLER AND LEARNING ALGORITHM DERIVATION
Our objective is synthesis of underwater robot NN controller to provide its movement along prescribed trajectory qd1(t), qd2(t). First we consider the control task with respect to velocities qd(t). Define error
e2 = q d2 − q 2 and introduce the function Q as measure of difference between desirable and real trajectories: 10.3182/20100915-3-DE-3008.00082
CAMS 2010 Rostock-Warnemünde, Germany, Sept 15-17, 2010
1 Q = e T2 De 2 2 Further we use the speed gradient method developed by (Fradkov, 1990). According th othe method, compute time derivative of Q:
& = e T De& + 1 e T D & Q 2 2 2 e2 2 As
q 2 = q d2 − e2 , one has
D(q1 )q& 2 = D(q1 )q& d 2 − D(q1 )e& 2 . Using expression of first term from dynamics equation, one can get the following:
D(q1 )e& 2 = D(q1 )q& d 2 + B(q1, q 2 )q d 2 − − B(q1, q 2 )e2 + G(q1, q 2 ) − U and time derivative of function Q can be written in the form
& = eT (D(q )q& + B(q , q )q − Q 2 1 d2 1 2 d2 1 & − B(q1 , q 2 )e 2 + G(q1 , q 2 ) − U) + e T2 D e2 . 2 After terms reorganization, one get
w 11 w w = 21 ... w H1
w 12 w 22 ... w H2
... w 1N ... w 2 N ... ... ... w HN
As result of nonlinear transformation f(.), hidden layer output vector can be written in the form
f1 ( w 1T x ) f ( w , x ) = ... , f H ( w TH x ) where wk denotes k-th raw of matrix w. By analogy, introduce matrix W which element Wli odenotes transform coefficient from i-th neuron of hidden l-th neuron of output layer. With defined NN parameters, the undwewater robot control signal (NN output) is computed as following:
U = y( W, w , x ) = Wf ( w , x ) Substitution of this control let us to get
& = eT (D(q )q& + B(q , q )q + G(q , q ) − Q 2 1 d2 1 2 d2 1 2
& = eT (D(q )q& + B(q , q )q + Q 2 1 d2 1 2 d2
1 & (q1)e2 = − U) − eT2B(q1, q2 )e2 + eT2D 2 eT2 (D(q1)q& d2 + B(q1, q2 )qd2 + G(q1, q2 ) −
+ G(q1, q 2 ) − Wf (w, x)).
1 & − U) + eT2 (D (q1) − B(q1, q2 )e2 ). 2 As known, the matrix in last term is skewsymmetric, hence, this term is equal to zero and we have simplified expression:
& = eT (D(q )q& + B(q , q )q + G(q , q ) − U). Q 2 1 d2 1 2 d2 1 2 We plan to implement intelligent UR control (Dyda, 2007) based on neural network. Without losing of generality of the approach, choose two-layer NN. Let hidden and output layers have H and m neurons appropriately (m is equal to dimension of e2). For the sake of simplicity, one supposes that only summing of weighted signals (without nonlinear transformation) is realized in output layer. Input vector has N coordinates. Define wij as transform coefficient for i-th input of j-th neuron of hidden layer. So these coefficients compose matrix 180
To derive NN learning algorithm, apply the speed gradient method. For this, compute partial derivatives of function Q time derivative with respect to adjustable NN parameters – matrices w and W. Direct differentiation gives
& ∂Q = −e 2 f T ( w, x ). ∂W It is easy to demonstrate that choosing of all activation functions in the usual form
f ( x ) = 1 /(1 + e − τ x ) imply property ∂ T T T f i ( w i x ) = f i ( w i x )[1 − f i ( w i x )] x j ∂ w ij
Introduce additional functions
CAMS 2010 Rostock-Warnemünde, Germany, Sept 15-17, 2010
[
ϕ i ( w i T x ) = f i ( w i T x )[1 − f i ( w i T x )]
q2 = vx
and matrix
vz
ωy
]
T
Dimensions of NN input (q2 and e2 ) and output (control forces and torque) are equal to 6 and 3.
Φ ( w , x ) = diag ( ϕ 1 ( w 1 T x )... ϕ H ( w H T x ))
Direct calculation gives
[
U = Fx
& ∂Q = − ΦW T e 2 x T ∂w As a final stage, we can write the NN learning algorithm in following form: W( k +1) := W ( k ) + γe 2f T (w , x ). w ( k +1) := w ( k ) + γΦWT e2 x T (γ is learning step, k is number of iteration). Now consider which components should be included in NN input vector. As NN controller is oriented to compensate an influence of appropriate matrix and vector functions, in common case the NN input vector must be composed of q1 ,q2 , e2 , qd2 and its time derivative. The NN learning procedure leads to reducing of function Q, consequently in ideal conditions, error e2 tends to zero and the UR movement follows to desirable trajectory
My
]
T
For the NN controller contaning 5 neurons in hidden layer results of simulation are given on Fig.1 - 6. Transient processes and control for taken nominal model are shown on Fig.1-2. 0.3
0.25
0.2
0.15
Vx, m/sec Vz, m/sec Wy, rad/sec
0.1
0.05
0
-0.05
0
10
20
30
40
50 t, sec
60
70
80
90
100
Fig.1. Transient processes (nominal model)
q 2 (t) → q d2 (t)
80
If UR trajectory is given by qd1(t), one can choose
70
q d 2 (t ) = J −1 (q1 )(q& d1 (t ) + k (q d1 (t ) − q1 ( t ))
Fz
Fx, H Fz, H My, Hm
60 50
(k is positive constant). As follows from kinematics equation,
40
q& 1 ( t ) → q& d1 ( t ) + k (q d1 ( t ) − q1 ( t ))
30
and
20
e& 1 ( t ) + ke1 ( t ) → 0
10
where
e1 ( t ) = q d1 ( t ) − q1 ( t )
0
10
20
30
40
50 t, sec
60
70
80
90
Fig.2. Control signals(nominal model)
Hence, UR follows to the planned trajectory qd1(t). (3.1), 4.
0
SIMULATION RESULTS OF INTELLIGENT NN CONTROLLER
To check the effectiveness of the approach, computer simulations have been carried. The UR model parameters were taken from Vector q2 consists of following components (linear and angular UR velocities): 181
Fig.3-4 present the same processes for tha case of reduced UR added masses.
100
CAMS 2010 Rostock-Warnemünde, Germany, Sept 15-17, 2010 0.3
0.3
0.25
0.25 0.2
0.2
0.15
Vx, m/sec Vz, m/sec Wy, rad/sec
0.1
0.15
0.05
Vx, m/sec Vz, m/sec Wy, rad/sec
0.1
0
0.05 -0.05
0
10
20
30
40
50 t, sec
60
70
80
90
100
0
Fig.3. Transient processes (reduced UR added masses)
-0.05
0
10
20
30
40
60
50
60
70
80
90
100
Fig.5. Transient processes (quadratic viscous friction)
Fx, H Fz, H My, Hm
40
50 t, sec
30 30
Fx, H Fz, H My, Hm
25 20
20
15
10
10 0
0
10
20
30
40
50 t, sec
60
70
80
90
100
Fig.4. Control signals (reduced UR added masses)
5 0 -5 -10
The exact description of hydrodynamic forces and torques is practically impossible. In the nominal model viscous friction was linear with respect to velocities. The effectiveness of the designed NN controller was proved and confirmed (Fig.5-6) for quadratic and cubic viscous friction forces (torques).
182
0
10
20
30
40
50 t, sec
60
70
80
90
Fig.6. Control signals (quadratic viscous friction) 5.
CONCLUSION
The approach to derive the learning algorithm for NN controller of underwater robot was proposed and studied. The numerical experiments have shown that high quality processes can be achieved on the basis of proposed intelligent NN control. The procedure of NN learning makes possible for UR control system to overcome parameter and, partially, structural uncertainties of object model. REFERENCES Dyda A.A. (2007) Adaptive and neural network control for complex dynamical objects. - Vladivostok, Dalnauka. – 149 p. (in Russian). Narendra K.S., Parthasaraty K. (1990) Identification and control of dynamical systems using neural
100
CAMS 2010 Rostock-Warnemünde, Germany, Sept 15-17, 2010
networks // IEEE Identification and Control of Dynamical System, Vol.1. № 1. 20, pp. 1475-1483. Ross A., Fossen T.and Johansen A. (2004) Identification of underwater vehicle hydrodynamic coefficients using free decay tests // Preprints of Int.Conf.CAMS-2004,. Ancona, Italy,2004. - pp.363368. Fradkov A.L. (1990) Adaptive control in large-scale systems.- M.: Nauka., (in Russian). Yuh Y. (1990) Modelling and control of underwater vehicles IEEE J. of Trans. Syst., Man, Cybern., vol.
183