Adaptive Neural-Network Control for Redundant ... - Semantic Scholar
Adaptive Neural-Network Control for Redundant Nonholonomic Mobile Modular Manipulators Yangmin Li1 , Yugang Liu1 , and Shaoze Yan2 1
2
Faculty of Science and Technology, University of Macau, Av. Padre Tomas Pereira S.J., Taipa, Macao S.A.R., P.R. China {ymli,ya27401}@umac.mo Department of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, P.R. China [email protected]
Abstract. This paper discusses the trajectory following issue for redundant nonholonomic mobile modular manipulators. Dynamic model is established and an adaptive neural-network controller is developed to control the end-effector to follow a desired spacial trajectory. The proposed algorithm doesn’t need any priori dynamics and provides a new solution for stabilization of redundant robotic selfmotions. Simulation results for a real robot demonstrate the proposed algorithm is effective.
1
Introduction
A nonholonomic mobile modular manipulator can be defined as a kind of robot integrating a N -degree of freedom (DOF) modular manipulator together with a M -wheeled nonholonomic mobile platform. If the integrated structure has more DOFs than required, it is called a redundant one. This integration extends the workspace of the entire robot drastically. However, the nonholonomic constraints, the interactive motions, as well as the self-motions make the trajectory following task difficult to realize. Neural networks (NNs) with characteristics of not needing exact priori dynamic parameters and universal approximators are being widely used for robotic control. In related research work, back-propagation (BP) NN was used for vibration control of a 9-DOF redundant modular manipulator [1]. A multi-layer NN controller, which did not need off-line learning, was designed to control rigid robotic arms [2]. A fuzzyGaussian NN controller was proposed for trajectory tracking control of mobile robots [3]. A dual NN was presented for the bi-criteria kinematic control of redundant manipulators [4]. A sliding mode adaptive NN controller was devised for control of mobile modular manipulators [5]. In this paper, the dynamic model is established in Section 2. An adaptive NN controller (ANNC) is designed in task space in Section 3. Simulations are carried out in Section 4. Finally, some conclusions are given in Section 5.
2
Dynamic Modeling
In this paper, a 3-wheeled nonholonomic mobile platform is studied as shown in Fig. 1(a), and only end-effector positions x = [px py pz ]T are concerned. The moJ. Wang, X. Liao, and Z. Yi (Eds.): ISNN 2005, LNCS 3498, pp. 271–276, 2005. c Springer-Verlag Berlin Heidelberg 2005
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Yangmin Li, Yugang Liu, and Shaoze Yan
bile platform is supposed to just move on a horizontal plane. The coordinate system can be defined as follows: an arbitrary inertial base frame OB XB YB ZB is fixed on the motion plane, while a frame Om Xm Ym Zm is attached to the mobile platform. The parameters can be observed in Fig. 1(b) in details.
YB
φL
Ym
Driving wheel G
φm
dm
Driving wheel
Modular manipulator Xm
Om ( xm , ym )
rf lG
φR
XB
OB
(a) A real robot
(b) Platform motion
Fig. 1. A real mobile modular manipulator and its motion on a plane
T Define ξ = xm ym φm φL φR , then the nonholonomic velocity constraints can be described as follows, [6] rf · Cm /2 rf · Cm /2 dm Cm Sm − 2 −rf 0 rf · Sm /2 rf · Sm /2 dm Cm Sm · rf /dm = 0 (1) −rf /dm 0 −r f 2 1 0 −Sm Cm 0 0 0 0 1 In short A (ξ) · S (ξ) = 0. T T Define q = φL φR q1 · · · qN , and ζ = ξ T q1 · · · qN , then the Jacobian matrix can be derived by
∂x ∂x S (ξ) 05×N J = T · S¯ (ξ) = T · (2) 0N ×2 IN ×N ∂ζ ∂ζ The constrained dynamics can be determined by Eqn. 3, see [5] in details. H · ζ¨ + V · ζ˙ + G = B · τ + J T · Fext + C · λ
(3)
Where H, V and G denote the inertial matrix, the centripetal and coriolis matrix, T T and the gravitational force vector. B = 0n×3 In×n , C = A (ξ) 03×N , τ = T T x y z τL τR τ1 · · · τN , Fext = Fext Fext Fext is an external-force vector. From Eqn. 2 x˙ = J · q, ˙ ζ˙ = S¯ · q. ˙ (4) Solving Eqn. 4 and its derivative, yields ¯ † x˙ + S¯ In − J † J q˙s ζ˙ = SJ † ¯ † J˙ J † x˙ + In − J † J q˙s ¨ + In − J † J q¨s + S¯˙ − SJ ζ¨ = S¯ J x
(5)
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−1 Where J † = J T · J · J T is the Moore-Penrose generalized inverse of J; q˙s ∈ n † is an arbitrary joint velocity vector; J · x˙ is a least-square solution; In − J † · J · q˙s ∈ ℵ (J), the null space of J, is a homogeneous solution. Define Jℵ ∈ n×(n−m) as a matrix with all its columns being the normalized bases
T of ℵ (J), JE† = J † Jℵ , x˙ ℵ = JℵT · q˙s , and xE = xT xTℵ . Substituting † T Eqn. 5 into 3, and left multiplying J · S¯T , yields E
¯ ·x ¯ = τ¯ H ¨E + V¯ · x˙ E + G (6) † T T T ¯ = J Where H · S¯ · H · S¯ · JE† , V¯ = JE† · S¯T · H · S¯˙ − S¯ · J † · J˙ + V · S¯ , E ¯ = J † T · S¯T · G, τ¯ = J † T · S¯T · B · τ + J T · Fext ; J † T · S¯T · C · λ = 0. G E
E
E
¯ · r ≥ 0. Remark 1. For any r ∈ , r · H ˙ ¯ − 2V¯ · r = 0 . Remark 2. For any r ∈ n , rT · H n
T
−1 T Remark 3. If J is full rank, JE† is invertible, and JE = JE† = J
Jℵ
T
.
¯ V¯ , G ¯ are all bounded as long as the Jacobian J is full rank. Remark 4. H,
3
Controller Design
Let xd , x˙ d and x ¨d be the desired trajectory, velocity and acceleration in task space. The desired self-motions can be used to fulfil a secondary task. In this paper, the system is assumed to be far away from singularity, physical limits, obstacles. So, q˙sd and q¨sd and can be selected for the optimization problem of: min q˙T · q˙ subject to x˙ = J · q. ˙ Then, xℵd , x˙ ℵd and x ¨ℵd can be determined. Let xEd = [ xTd | xTℵd ]T , then the error system can be defined by e (t) = xE (t) − xEd (t) , x˙ r (t) = x˙ Ed (t) − Λ · e (t) , r (t) = x˙ E (t) − x˙ r (t) . (7) Where r (t) is the tracking error measure; Λ ∈ n×n > 0 is a constant matrix. Substituting Eqn. 7 into 6, yields ¯ · r˙ (t) + V¯ · r (t) + H ¯ ·x ¯ = τ¯ H ¨r + V¯ · x˙ r + G
(8)
It is verified that a multilayer perceptron (MLP) trained with BP algorithm can approximate any continuous multivariate functions to any desired degree of accuracy, provided that sufficient hidden neurons are available [7]. To ensure rapid convergence, multiple-input single-output (MISO) MLPs with only one hidden layer are applied in this paper as shown in Fig. 2(a). Output of this MLP is: N Nh i fN N (x, wih , who , θh , θo ) = xi · wihji + θhj · whoj + θo (9) ϕ j=1
i=1
Where ϕ (◦) is the activation function named hyperbolic tangent function; Ni and Nh represent neuron numbers; x is the inputs; wih , who , θh and θo denote weights and
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Yangmin Li, Yugang Liu, and Shaoze Yan
thresholds accordingly; the subscript ”i, h and o” represents the input, hidden and output layer respectively. T ¯ ·x ¯ Define xin = [ ζ T q˙T xTEd x˙ TEd x ¨TEd ] , h (xin ) = H ¨r + V¯ · x˙ r + G. From Remark 4 and Eqn. 7, all the elements hk (k = 1, 2, · · · n) are bounded as long as J keeps full rank. Then, they can be approximated by MISO NNs, hk (xin ) = hN N k (xin ) + k (xin )
(10)
Where hN N k = fN N (xin , wkih , wkho , θkh , θko ); k is the approximated error. ˆ Let w ˆkih , w ˆkho , θˆkh and θko be estimates of wkih , wkho , θkh and θko respectively, ˆ ˆkih , w ˆkho , θˆkh , θˆko , then the Taylor series expansions of define hN N k = fN N xin , w hN N k can be derived hN N k around ˆ N Nh i 2 2 ˆNNk ˜ h ˜hN N k = ∂ hˆ N N k · w ˜ + O w ˜ · θko + O θ˜ko + ∂ ∂θ kihji kihji ∂wkihji ko j=1 i=1 (11) N 2 2 h ∂ hˆ N N k ˆNNk ˜ ∂h ˜ ˜khoj + ∂θkhj · θkhj + O w ˜khoj + O θkhj + ∂wkhoj · w j=1
˜ N N k = hN N k − h ˆN N k, w Where h ˜kihji = wkihji − w ˆ ,w ˜ =w −w ˆkhoj , 2 khoj 2 khoj 2 2 kihji ˜ ˜ ˆ ˜ ˆ ˜khoj , O θkhj and O θ˜ko θkhj = θkhj − θkhj , and θko = θko − θko ; O w ˜kihji , O w are higher-order terms. The ANNC is given by Eqn. 12, and the control system is shown in Fig. 2(b). t T −1 T ˆ ¯ τ = S ·B ·JE · hN N −KP ·r−KI · r (t) dt−K ·sgn (r) −J T ·Fext (12) 0
ˆ N N ∈ n forms the adaptive NN term; KP , KI ∈ n×n are proportional and Where h integral gain matrices of the PID controller; K = diag k1 k2 · · · kn is the gain matrix of the robust term, and its elements are selected as follows:
N
Ni h
2 2 2 2
(13) O w ˜kihji + O w ˜khoj + O θ˜khj + O θ˜ko + k kk ≥
j=1
i=1
Substituting Eqn. 12 into 8, and considering 6 at the same time, yields t ˜NN + = 0 ¯ ¯ H · r˙ + V · r + KP · r + KI · r (t) dt + K · sgn (r) + h 0
θ h1
ζ
Wih
q xEd
Who
θh2
x1
θo
xℵd
yNN θh ( N
h
−1)
xd
xℵd
x Ni
xd
θ hN
xℵd
h
xd
Input layer
Hidden layer
M u x
xEd xEd
M u x M u x M u x
(
f NN xin , wˆ ih , wˆ ho , θˆh ,θˆo
xin
θˆ
x1 yNN
xNi
θˆo
NN
Σ _
xEd +
Λ
Σ
(
in
ih
ho
,θˆ , θˆ
)
)
) )
h
o
∂ wˆ ih
∂f NN xin , wˆ ih , wˆ ho , θˆh , θˆo
)
∂ θˆh ∂ θˆo
Adaptive Law
xE + _
( ( x , wˆ , wˆ
∂f NN xin , wˆih , wˆ ho ,θˆh ,θˆo ∂ wˆ ho ∂f NN
h
xEd
xEd
(
∂ f NN xin , wˆ ih , wˆ ho , θˆh ,θˆo
wˆ ih wˆ ho
xE ζ q e
+
xr
_ +
Kε
Σ xE
r
KP ∫
_ _
Σ _
KI
S
T
−1
⋅ B ⋅ JET
+ _
Σ Robot
JT
(a) A MISO NN
xE Fext
Output layer
(b) An adaptive NN controller
Fig. 2. A MISO NN and an adaptive NN controller
(14)
Adaptive Neural-Network Control
275
Theorem 1. If KP > 0, KIT = KI > 0, the closed-loop system in Eqn. 14 is asymptotically stable under the adaptation laws given by Eqn. 15. The error signals are convergent with time, i.e., e (t) , e˙ (t) → 0, as t → +∞. ˆNNk ∂h w ˆ˙ kihji = −Γwkihji · rk · ∂w , kihji ˆNNk ∂h ˙ w ˆkhoj = −Γwkhoj · rk · ∂wkhoj ,
ˆNNk ˙ h θˆkhj = −Γθkhj · rk · ∂∂θ , khj ˆNNk ˙ ∂h ˆ θko = −Γθko · rk · ∂θko .
(15)
Where Γwkihji , Γwkhoj , Γθkhj , and Γθko are positive constants. The terms with partial differentiation can be derived from Eqn. 10, details will not be listed here. Proof. Considering the following nonnegative Lyapunov candidate: T ¯ · r + 1 · t r (t) dt · KI · t r (t) dt VS = 12 · rT · H 2 0 0 ! N 2
N n 2 i h 2 ˜2 θ w ˜ w ˜ θ˜ko kihji khoj khj 1 + 2· + Γw + Γθ ≥0 + Γθ Γw k=1
j=1
i=1
kihji
khoj
khj
ko
The time derivative of Lyapunov candidate is n ˙ ¯ · r˙ + KI · t r (t) dt + 1 · rT · H ¯˙ · r + 1 · θ˜ko ·θ˜ko V˙ S = rT · H 2 2 Γθ 0 k=1 ! ko
N Ni n ˙ h ˙ kihji ˙ khoj θ˜ w ˜ kihji ·w ˜ w ˜ ·w ˜ ·θ˜ khj + 12 · + khj + khoj Γw Γw Γθ k=1
j=1
i=1
kihji
khoj
(16)
(17)
khj
˙ ˙ ˙ ˙ Notice that w ˜˙ kihji = −w ˆ˙ kihji , w ˜˙ khoj = −w ˆ˙ khj , θ˜khj = −θˆkhj , θ˜ko = −θˆko . Substituting Eqn. 12 into 15, then substituting the result together with Eqns. 14,16 into 18, and considering Remark 2 at the same time, yields V˙ S ≤ −rT · KP · r ≤ 0
(18)
Therefore VS is a Lyapunov function, iff r = 0, VS and V˙ S equal to zero. According to LaSalle’s theorem, the system is asymptotically stable and r → 0 as t → +∞. Define p = {x (t) ∈ n : x p < ∞} the p − norm space. From Eqns. 16 and 18, r (t) ∈ 2 . According to Eqns. 7, e (t) ∈ 2 ∩ ∞ , e˙ (t) ∈ 2 , and e (t) → 0, as t → +∞. It is obvious that the higher-order terms in Eqn. 11 are bounded, so K ∈ ∞ . Then, from Eqn. 14, r˙ (t) ∈ ∞ . Since r (t) ∈ 2 and r˙ (t) ∈ ∞ , r (t) → 0 as t → +∞, which is followed by e˙ (t) → 0. End of proof.
4
Simulation Results
The simulation is performed on a real robot as shown in Fig. 1(a), in which N = 3 and n = 5. Simulation time is 10 seconds and the end-effector is to follow the desired trajectory in Fig. 3(a). The gain matrices and constants are: Nh = 200, KP = diag {50}, KI = diag {10}, Γwkihji = Γwkhoj = Γθkhj = Γθko = 0.01, K = diag {50}, Λ = diag {2.0}. To examine the disturbance suppression ability of the ANNC, an exz = 1N is applied to the end-effector along the direction Zm at the time ternal force Fext instant t = 2s.
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Yangmin Li, Yugang Liu, and Shaoze Yan
The desired and the controlled locus are shown in Fig. 3(a). Figure 3(b) shows the self-motion velocities. The tracking position and velocity errors are given by Fig. 3(c)– 3(d). From these figures, we can see that the ANNC is effective to control the endeffector to follow a desired spacial trajectory.
1 1.5
3
1
2
1 pz (m)
Desired 0 −1
•
xℵ 1
0
Tracing position erros, m
Self−motion velocities, m/s
Controlled
−0.5
−1
x•
ℵ2
Tracking velocity errors, m/s
0.5 2
0.5 0 ez
ex
−0.5
ey −1
−1.5
−2
−1.5
1
e•z
0 e•x
−1
e•
y
−2 −3
5 0 px (m)
−5 0 −5
5
p (m) y
(a) Spacial locus
−2
0
2
4
6
8
Time, s
(b) Self-motions
10
−2
0
2
4
6
8
Time, s
(c) Position errors
10
−4
0
2
4
6
8
10
Time, s
(d) Velocity errors
Fig. 3. Simulation results
5
Conclusions
Dynamic model is established and an ANNC is developed for a general mobile modular manipulator. The proposed controller does not need precise knowledge of dynamic parameters in advance and can suppress bounded external disturbances effectively. The torque instability problem caused by self-motions of the redundant robot can be solved by the presented control algorithm. The simulation is carried out on a real redundant nonholonomic mobile modular manipulator, which has verified the effectiveness of the dynamic modeling method and the controller design method.
References 1. Li, Y., Liu, Y., Liu, X. and Peng, Z.: Parameters Identification and Vibration Control in Modular Manipulators. IEEE/ASME Trans. Mechatronics. 9 (2004) 700–705 2. Lewis, F., Yegildirek, A. and Liu, K.: Multilayer neural-net robot controller with guaranteed tracking performance. IEEE Trans. on Neural Networks. 7 (1996) 388–399 3. Watanabe, K., Tang, J., Nakamura, M., Koga, S. and Fukuda, T.: A Fuzzy-Gaussian Neural Network and Its Application to Mobile Robot Control. IEEE Trans. on Control Systems Technology. 4 (1996) 193–199 4. Zhang, Y. N., Wang, J. and Xu, Y. S.: A Dual Neural Network for Bi-Criteria Kinematic Control of Redundant Manipulators. IEEE Trans. on Robotics and Automation. 18 (2002) 923–931 5. Li, Y. and Liu, Y.: Sliding Mode Adaptive Neural-Network Control for Nonholonomic Mobile Modular Manipulators. The 2nd Int. Conf. on Autonomous Robots and Agents(ICARA04), New Zealand (2004) 95–100 6. de Wit, C. C., Siciliano, B. and Bastin, G.: Theory of Robot Control. Springer-Verlag London Limited (1996) 7. Haykin, S.: Neural Networks: A Comprehensive Foundation. 2nd edition. Prentice-Hall, Inc. (1999)
2
Faculty of Science and Technology, University of Macau, Av. Padre Tomas Pereira S.J., Taipa, Macao S.A.R., P.R. China {ymli,ya27401}@umac.mo Department of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, P.R. China [email protected]
Abstract. This paper discusses the trajectory following issue for redundant nonholonomic mobile modular manipulators. Dynamic model is established and an adaptive neural-network controller is developed to control the end-effector to follow a desired spacial trajectory. The proposed algorithm doesn’t need any priori dynamics and provides a new solution for stabilization of redundant robotic selfmotions. Simulation results for a real robot demonstrate the proposed algorithm is effective.
1
Introduction
A nonholonomic mobile modular manipulator can be defined as a kind of robot integrating a N -degree of freedom (DOF) modular manipulator together with a M -wheeled nonholonomic mobile platform. If the integrated structure has more DOFs than required, it is called a redundant one. This integration extends the workspace of the entire robot drastically. However, the nonholonomic constraints, the interactive motions, as well as the self-motions make the trajectory following task difficult to realize. Neural networks (NNs) with characteristics of not needing exact priori dynamic parameters and universal approximators are being widely used for robotic control. In related research work, back-propagation (BP) NN was used for vibration control of a 9-DOF redundant modular manipulator [1]. A multi-layer NN controller, which did not need off-line learning, was designed to control rigid robotic arms [2]. A fuzzyGaussian NN controller was proposed for trajectory tracking control of mobile robots [3]. A dual NN was presented for the bi-criteria kinematic control of redundant manipulators [4]. A sliding mode adaptive NN controller was devised for control of mobile modular manipulators [5]. In this paper, the dynamic model is established in Section 2. An adaptive NN controller (ANNC) is designed in task space in Section 3. Simulations are carried out in Section 4. Finally, some conclusions are given in Section 5.
2
Dynamic Modeling
In this paper, a 3-wheeled nonholonomic mobile platform is studied as shown in Fig. 1(a), and only end-effector positions x = [px py pz ]T are concerned. The moJ. Wang, X. Liao, and Z. Yi (Eds.): ISNN 2005, LNCS 3498, pp. 271–276, 2005. c Springer-Verlag Berlin Heidelberg 2005
272
Yangmin Li, Yugang Liu, and Shaoze Yan
bile platform is supposed to just move on a horizontal plane. The coordinate system can be defined as follows: an arbitrary inertial base frame OB XB YB ZB is fixed on the motion plane, while a frame Om Xm Ym Zm is attached to the mobile platform. The parameters can be observed in Fig. 1(b) in details.
YB
φL
Ym
Driving wheel G
φm
dm
Driving wheel
Modular manipulator Xm
Om ( xm , ym )
rf lG
φR
XB
OB
(a) A real robot
(b) Platform motion
Fig. 1. A real mobile modular manipulator and its motion on a plane
T Define ξ = xm ym φm φL φR , then the nonholonomic velocity constraints can be described as follows, [6] rf · Cm /2 rf · Cm /2 dm Cm Sm − 2 −rf 0 rf · Sm /2 rf · Sm /2 dm Cm Sm · rf /dm = 0 (1) −rf /dm 0 −r f 2 1 0 −Sm Cm 0 0 0 0 1 In short A (ξ) · S (ξ) = 0. T T Define q = φL φR q1 · · · qN , and ζ = ξ T q1 · · · qN , then the Jacobian matrix can be derived by
∂x ∂x S (ξ) 05×N J = T · S¯ (ξ) = T · (2) 0N ×2 IN ×N ∂ζ ∂ζ The constrained dynamics can be determined by Eqn. 3, see [5] in details. H · ζ¨ + V · ζ˙ + G = B · τ + J T · Fext + C · λ
(3)
Where H, V and G denote the inertial matrix, the centripetal and coriolis matrix, T T and the gravitational force vector. B = 0n×3 In×n , C = A (ξ) 03×N , τ = T T x y z τL τR τ1 · · · τN , Fext = Fext Fext Fext is an external-force vector. From Eqn. 2 x˙ = J · q, ˙ ζ˙ = S¯ · q. ˙ (4) Solving Eqn. 4 and its derivative, yields ¯ † x˙ + S¯ In − J † J q˙s ζ˙ = SJ † ¯ † J˙ J † x˙ + In − J † J q˙s ¨ + In − J † J q¨s + S¯˙ − SJ ζ¨ = S¯ J x
(5)
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−1 Where J † = J T · J · J T is the Moore-Penrose generalized inverse of J; q˙s ∈ n † is an arbitrary joint velocity vector; J · x˙ is a least-square solution; In − J † · J · q˙s ∈ ℵ (J), the null space of J, is a homogeneous solution. Define Jℵ ∈ n×(n−m) as a matrix with all its columns being the normalized bases
T of ℵ (J), JE† = J † Jℵ , x˙ ℵ = JℵT · q˙s , and xE = xT xTℵ . Substituting † T Eqn. 5 into 3, and left multiplying J · S¯T , yields E
¯ ·x ¯ = τ¯ H ¨E + V¯ · x˙ E + G (6) † T T T ¯ = J Where H · S¯ · H · S¯ · JE† , V¯ = JE† · S¯T · H · S¯˙ − S¯ · J † · J˙ + V · S¯ , E ¯ = J † T · S¯T · G, τ¯ = J † T · S¯T · B · τ + J T · Fext ; J † T · S¯T · C · λ = 0. G E
E
E
¯ · r ≥ 0. Remark 1. For any r ∈ , r · H ˙ ¯ − 2V¯ · r = 0 . Remark 2. For any r ∈ n , rT · H n
T
−1 T Remark 3. If J is full rank, JE† is invertible, and JE = JE† = J
Jℵ
T
.
¯ V¯ , G ¯ are all bounded as long as the Jacobian J is full rank. Remark 4. H,
3
Controller Design
Let xd , x˙ d and x ¨d be the desired trajectory, velocity and acceleration in task space. The desired self-motions can be used to fulfil a secondary task. In this paper, the system is assumed to be far away from singularity, physical limits, obstacles. So, q˙sd and q¨sd and can be selected for the optimization problem of: min q˙T · q˙ subject to x˙ = J · q. ˙ Then, xℵd , x˙ ℵd and x ¨ℵd can be determined. Let xEd = [ xTd | xTℵd ]T , then the error system can be defined by e (t) = xE (t) − xEd (t) , x˙ r (t) = x˙ Ed (t) − Λ · e (t) , r (t) = x˙ E (t) − x˙ r (t) . (7) Where r (t) is the tracking error measure; Λ ∈ n×n > 0 is a constant matrix. Substituting Eqn. 7 into 6, yields ¯ · r˙ (t) + V¯ · r (t) + H ¯ ·x ¯ = τ¯ H ¨r + V¯ · x˙ r + G
(8)
It is verified that a multilayer perceptron (MLP) trained with BP algorithm can approximate any continuous multivariate functions to any desired degree of accuracy, provided that sufficient hidden neurons are available [7]. To ensure rapid convergence, multiple-input single-output (MISO) MLPs with only one hidden layer are applied in this paper as shown in Fig. 2(a). Output of this MLP is: N Nh i fN N (x, wih , who , θh , θo ) = xi · wihji + θhj · whoj + θo (9) ϕ j=1
i=1
Where ϕ (◦) is the activation function named hyperbolic tangent function; Ni and Nh represent neuron numbers; x is the inputs; wih , who , θh and θo denote weights and
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Yangmin Li, Yugang Liu, and Shaoze Yan
thresholds accordingly; the subscript ”i, h and o” represents the input, hidden and output layer respectively. T ¯ ·x ¯ Define xin = [ ζ T q˙T xTEd x˙ TEd x ¨TEd ] , h (xin ) = H ¨r + V¯ · x˙ r + G. From Remark 4 and Eqn. 7, all the elements hk (k = 1, 2, · · · n) are bounded as long as J keeps full rank. Then, they can be approximated by MISO NNs, hk (xin ) = hN N k (xin ) + k (xin )
(10)
Where hN N k = fN N (xin , wkih , wkho , θkh , θko ); k is the approximated error. ˆ Let w ˆkih , w ˆkho , θˆkh and θko be estimates of wkih , wkho , θkh and θko respectively, ˆ ˆkih , w ˆkho , θˆkh , θˆko , then the Taylor series expansions of define hN N k = fN N xin , w hN N k can be derived hN N k around ˆ N Nh i 2 2 ˆNNk ˜ h ˜hN N k = ∂ hˆ N N k · w ˜ + O w ˜ · θko + O θ˜ko + ∂ ∂θ kihji kihji ∂wkihji ko j=1 i=1 (11) N 2 2 h ∂ hˆ N N k ˆNNk ˜ ∂h ˜ ˜khoj + ∂θkhj · θkhj + O w ˜khoj + O θkhj + ∂wkhoj · w j=1
˜ N N k = hN N k − h ˆN N k, w Where h ˜kihji = wkihji − w ˆ ,w ˜ =w −w ˆkhoj , 2 khoj 2 khoj 2 2 kihji ˜ ˜ ˆ ˜ ˆ ˜khoj , O θkhj and O θ˜ko θkhj = θkhj − θkhj , and θko = θko − θko ; O w ˜kihji , O w are higher-order terms. The ANNC is given by Eqn. 12, and the control system is shown in Fig. 2(b). t T −1 T ˆ ¯ τ = S ·B ·JE · hN N −KP ·r−KI · r (t) dt−K ·sgn (r) −J T ·Fext (12) 0
ˆ N N ∈ n forms the adaptive NN term; KP , KI ∈ n×n are proportional and Where h integral gain matrices of the PID controller; K = diag k1 k2 · · · kn is the gain matrix of the robust term, and its elements are selected as follows:
N
Ni h
2 2 2 2
(13) O w ˜kihji + O w ˜khoj + O θ˜khj + O θ˜ko + k kk ≥
j=1
i=1
Substituting Eqn. 12 into 8, and considering 6 at the same time, yields t ˜NN + = 0 ¯ ¯ H · r˙ + V · r + KP · r + KI · r (t) dt + K · sgn (r) + h 0
θ h1
ζ
Wih
q xEd
Who
θh2
x1
θo
xℵd
yNN θh ( N
h
−1)
xd
xℵd
x Ni
xd
θ hN
xℵd
h
xd
Input layer
Hidden layer
M u x
xEd xEd
M u x M u x M u x
(
f NN xin , wˆ ih , wˆ ho , θˆh ,θˆo
xin
θˆ
x1 yNN
xNi
θˆo
NN
Σ _
xEd +
Λ
Σ
(
in
ih
ho
,θˆ , θˆ
)
)
) )
h
o
∂ wˆ ih
∂f NN xin , wˆ ih , wˆ ho , θˆh , θˆo
)
∂ θˆh ∂ θˆo
Adaptive Law
xE + _
( ( x , wˆ , wˆ
∂f NN xin , wˆih , wˆ ho ,θˆh ,θˆo ∂ wˆ ho ∂f NN
h
xEd
xEd
(
∂ f NN xin , wˆ ih , wˆ ho , θˆh ,θˆo
wˆ ih wˆ ho
xE ζ q e
+
xr
_ +
Kε
Σ xE
r
KP ∫
_ _
Σ _
KI
S
T
−1
⋅ B ⋅ JET
+ _
Σ Robot
JT
(a) A MISO NN
xE Fext
Output layer
(b) An adaptive NN controller
Fig. 2. A MISO NN and an adaptive NN controller
(14)
Adaptive Neural-Network Control
275
Theorem 1. If KP > 0, KIT = KI > 0, the closed-loop system in Eqn. 14 is asymptotically stable under the adaptation laws given by Eqn. 15. The error signals are convergent with time, i.e., e (t) , e˙ (t) → 0, as t → +∞. ˆNNk ∂h w ˆ˙ kihji = −Γwkihji · rk · ∂w , kihji ˆNNk ∂h ˙ w ˆkhoj = −Γwkhoj · rk · ∂wkhoj ,
ˆNNk ˙ h θˆkhj = −Γθkhj · rk · ∂∂θ , khj ˆNNk ˙ ∂h ˆ θko = −Γθko · rk · ∂θko .
(15)
Where Γwkihji , Γwkhoj , Γθkhj , and Γθko are positive constants. The terms with partial differentiation can be derived from Eqn. 10, details will not be listed here. Proof. Considering the following nonnegative Lyapunov candidate: T ¯ · r + 1 · t r (t) dt · KI · t r (t) dt VS = 12 · rT · H 2 0 0 ! N 2
N n 2 i h 2 ˜2 θ w ˜ w ˜ θ˜ko kihji khoj khj 1 + 2· + Γw + Γθ ≥0 + Γθ Γw k=1
j=1
i=1
kihji
khoj
khj
ko
The time derivative of Lyapunov candidate is n ˙ ¯ · r˙ + KI · t r (t) dt + 1 · rT · H ¯˙ · r + 1 · θ˜ko ·θ˜ko V˙ S = rT · H 2 2 Γθ 0 k=1 ! ko
N Ni n ˙ h ˙ kihji ˙ khoj θ˜ w ˜ kihji ·w ˜ w ˜ ·w ˜ ·θ˜ khj + 12 · + khj + khoj Γw Γw Γθ k=1
j=1
i=1
kihji
khoj
(16)
(17)
khj
˙ ˙ ˙ ˙ Notice that w ˜˙ kihji = −w ˆ˙ kihji , w ˜˙ khoj = −w ˆ˙ khj , θ˜khj = −θˆkhj , θ˜ko = −θˆko . Substituting Eqn. 12 into 15, then substituting the result together with Eqns. 14,16 into 18, and considering Remark 2 at the same time, yields V˙ S ≤ −rT · KP · r ≤ 0
(18)
Therefore VS is a Lyapunov function, iff r = 0, VS and V˙ S equal to zero. According to LaSalle’s theorem, the system is asymptotically stable and r → 0 as t → +∞. Define p = {x (t) ∈ n : x p < ∞} the p − norm space. From Eqns. 16 and 18, r (t) ∈ 2 . According to Eqns. 7, e (t) ∈ 2 ∩ ∞ , e˙ (t) ∈ 2 , and e (t) → 0, as t → +∞. It is obvious that the higher-order terms in Eqn. 11 are bounded, so K ∈ ∞ . Then, from Eqn. 14, r˙ (t) ∈ ∞ . Since r (t) ∈ 2 and r˙ (t) ∈ ∞ , r (t) → 0 as t → +∞, which is followed by e˙ (t) → 0. End of proof.
4
Simulation Results
The simulation is performed on a real robot as shown in Fig. 1(a), in which N = 3 and n = 5. Simulation time is 10 seconds and the end-effector is to follow the desired trajectory in Fig. 3(a). The gain matrices and constants are: Nh = 200, KP = diag {50}, KI = diag {10}, Γwkihji = Γwkhoj = Γθkhj = Γθko = 0.01, K = diag {50}, Λ = diag {2.0}. To examine the disturbance suppression ability of the ANNC, an exz = 1N is applied to the end-effector along the direction Zm at the time ternal force Fext instant t = 2s.
276
Yangmin Li, Yugang Liu, and Shaoze Yan
The desired and the controlled locus are shown in Fig. 3(a). Figure 3(b) shows the self-motion velocities. The tracking position and velocity errors are given by Fig. 3(c)– 3(d). From these figures, we can see that the ANNC is effective to control the endeffector to follow a desired spacial trajectory.
1 1.5
3
1
2
1 pz (m)
Desired 0 −1
•
xℵ 1
0
Tracing position erros, m
Self−motion velocities, m/s
Controlled
−0.5
−1
x•
ℵ2
Tracking velocity errors, m/s
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ey −1
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(c) Position errors
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(d) Velocity errors
Fig. 3. Simulation results
5
Conclusions
Dynamic model is established and an ANNC is developed for a general mobile modular manipulator. The proposed controller does not need precise knowledge of dynamic parameters in advance and can suppress bounded external disturbances effectively. The torque instability problem caused by self-motions of the redundant robot can be solved by the presented control algorithm. The simulation is carried out on a real redundant nonholonomic mobile modular manipulator, which has verified the effectiveness of the dynamic modeling method and the controller design method.
References 1. Li, Y., Liu, Y., Liu, X. and Peng, Z.: Parameters Identification and Vibration Control in Modular Manipulators. IEEE/ASME Trans. Mechatronics. 9 (2004) 700–705 2. Lewis, F., Yegildirek, A. and Liu, K.: Multilayer neural-net robot controller with guaranteed tracking performance. IEEE Trans. on Neural Networks. 7 (1996) 388–399 3. Watanabe, K., Tang, J., Nakamura, M., Koga, S. and Fukuda, T.: A Fuzzy-Gaussian Neural Network and Its Application to Mobile Robot Control. IEEE Trans. on Control Systems Technology. 4 (1996) 193–199 4. Zhang, Y. N., Wang, J. and Xu, Y. S.: A Dual Neural Network for Bi-Criteria Kinematic Control of Redundant Manipulators. IEEE Trans. on Robotics and Automation. 18 (2002) 923–931 5. Li, Y. and Liu, Y.: Sliding Mode Adaptive Neural-Network Control for Nonholonomic Mobile Modular Manipulators. The 2nd Int. Conf. on Autonomous Robots and Agents(ICARA04), New Zealand (2004) 95–100 6. de Wit, C. C., Siciliano, B. and Bastin, G.: Theory of Robot Control. Springer-Verlag London Limited (1996) 7. Haykin, S.: Neural Networks: A Comprehensive Foundation. 2nd edition. Prentice-Hall, Inc. (1999)
