Dynamic coupling switching control incorporating Support Vector Machines for wheeled mobile manipulators with hybrid joints

Automatica 46 (2010) 832–842

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Dynamic coupling switching control incorporating Support Vector Machines for wheeled mobile manipulators with hybrid jointsI Z. Li a,∗ , Y. Kang b a

Department of Automation, Shanghai Jiao Tong University, Shanghai, 200240, China

b

Department of Automation, The University of Science and Technology of China, Hefei, China

article

info

Article history: Received 6 May 2009 Received in revised form 13 October 2009 Accepted 8 February 2010 Available online 17 March 2010 Keywords: Wheeled mobile manipulators Under-actuated joints Zero dynamics

abstract The hybrid joints can be switched to either active (actuated) or passive (under-actuated) mode as needed (Li, Ming, Xi, & Shimojo, 2006), in this paper, dynamic coupling switching control incorporating Support Vector Machines (SVMs) is developed for wheeled mobile manipulators with hybrid joints. The hybrid actuated robot dynamics is a mixed under-actuated and actuated model. In order to approximate the high dimension unmodelled dynamics, the SVMs matrix and its operator are proposed. Considering the joint switching as an event, the event driven switching control strategy is used to ensure that the system outputs track the given bounded reference signals within a small neighborhood of zero, and guarantee semi-global uniform boundedness of all closed loop signals, and the switch stability. The effectiveness of the proposed controls is verified through extensive simulations. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction The hybrid joint shown in Fig. 1 was proposed in Li, Ming, Xi, and Shimojo (2006), which is with one clutch, when the clutch is released, the link is free, and the passive link is directly controlled by the dynamic coupling of mobile manipulators, and when it is on, the joint is actuated by the motor. The robot with hybrid joints is called hybrid actuated robot. One of the advantages of hybrid actuated robots is that they may consume less energy than their fully actuated ones. For example, hyper-redundant robots, such as snake-like robots or multi-legged mobile robots, need large redundancy for dexterity and specific task completion, under-actuation allows a more compact design and simpler control and communication schemes. The hybrid actuated robot concept is also useful for the reliability or faulttolerant design of fully actuated robots working in hazardous areas or with dangerous materials. If any of the joint actuators of such a device fails, one degree of freedom of the system would be lost. It is important, in these cases, that the passive (failed) joint can still be controlled via the dynamic coupling by the active joints, so the

I This work is supported by Shanghai Pujiang Program under Grant No. 08PJ1407000 and Natural Science Foundation of China under Grant Nos. 60804003, 60704007, 60935001 and the Science and Technological Fund of Anhui Province for Outstanding Youth (08040106841). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Yong-Yan Cao under the direction of Editor Toshiharu Sugie. ∗ Corresponding author. Tel.: +86 21 34204616. E-mail address: [email protected] (Z. Li).

0005-1098/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2010.02.019

system can still make use of all of its degrees of freedom originally planned. Hybrid actuated mobile manipulator is the robot manipulator consisting of hybrid joints mounted on a wheeled mobile robot, which first appeared in Li, Ming, Xi, Gu, and Shimojo (2005). Hybrid actuated mobile manipulators are different from full actuated mobile manipulators in Dong (2002), Li, Ge, and Ming (2007), Lin and Goldenberg (2001), due to simultaneously integrating both kinematic constraints and dynamic constraints, on the other hand, it is more complex than the mobile wheeled inverted pendulums (Li & Luo, 2009) or pendulums (Zhang & Tarn, 2002), whose dynamic balances in the vertical plane are achieved due to the existent gravity. For these reasons, increasing effort need to be made towards control design that guarantee stability and robustness for hybrid actuated mobile manipulators with the consideration of joint switching. These systems are intrinsically nonlinear and their dynamics will be described by nonlinear differential equations. The hybrid joint is with the characteristic of under-actuated joints (Arai & Tanie, 1998; Bergerman, Lee, & Xu, 1995; Spong, 1995). The hybrid joints in the free mode, which can rotate freely, can be indirectly driven by the effect of the dynamic coupling between the active and the passive joints. The zero torque at the hybrid joint results in a second-order nonholonomic constraint. The effective control of second-order nonholonomic systems could reduce the weight, energy consumption and cost. For example, the coordination of multi-manipulators using passive joints was proposed in Liu, Xu, and Bergerman (1999), Tinos, Terra, and Ishihara (2006) to decrease the undesired internal forces. Safe human–robot interaction approach was proposed in Li et al. (2005), where the hybrid joints are used to secure the safety of human.

Z. Li, Y. Kang / Automatica 46 (2010) 832–842

833

Nomenclature q

The generalized coordinates for the mobile manipulators with q = [qTv , qTa , qTh ]T ∈ Rn ; qv The generalized coordinates for the mobile platform with qv = [x, y, θ]T ∈ Rnv ; x, y The position coordinates of the mid-point of the two driving wheels in the fixed frame; θ The heading angle in motion relative to the x-axis of the fixed frame; qa The coordinates of the active joints with qa ∈ Rna ; qh Denoting the coordinates of the hybrid joints with qh ∈ Rnh ; D(q) The symmetric positive definite inertia matrix with D(q) ∈ Rn×n ; Dv , Da , Dh The inertia matrices for the mobile platform, the active links and the hybrid links respectively; Dv a , Dv h , Dah The coupling inertia matrices of the mobile platform, the active links and the hybrid links; C (q, q˙ ) The Centripetal and Coriolis torques with C (q, q˙ ) ∈ Rn×n ; Cv , Ca , Ch The Centripetal and Coriolis torques for the mobile platform, the active links and the hybrid links respectively; Cv a , Cv h , Cah The coupling Centripetal and Coriolis torques of the mobile platform, the active links and the hybrid links; G(q) The gravitational torque vector with G(q) = [GTv , GTa , GTh ]T ∈ Rn ; Gv , Ga , Gh The gravitational torque vectors for the mobile platform, the active links and the hybrid links respectively; d(t ) The external disturbances with d(t ) = [dTv , dTa , dTh ]T ∈ Rn ; dv , da , dh The external disturbances on the mobile platform, the active links and the hybrid links respectively; B(q) The known full rank input transformation matrix with B(q) ∈ Rn×m ; τ The control inputs with τ = [τvT , τaT , τhT ]T ∈ Rm ; τv The input vector associated with the left driven wheel and the right driven wheel respectively; τa The control input vectors for the active joints of the manipulator; τh The control input vectors for the hybrid joints of the manipulator; f The external force on the mobile manipulators with f ∈ Rl ; Jv The kinematic constraint matrix related to nonholonomic constraints with Jv ∈ Rnv ×l ; λn The associated Lagrangian multipliers with the generalized nonholonomic constraints with λn ∈ Rl . Switched control system has recently received much attention due to its applicability to many physical systems exhibiting switching in nature. In Liu, Zhang, and Wang (2007), a type-2 fuzzy switching control system was proposed for a biped robot, which includes switched nonlinear system modeling, type-2 fuzzy control system design, and a type-2 fuzzy modeling algorithm. In Tan, Xi, and Wang (2004), the robot workspace is partitioned into subspaces based on the singular configurations of the robot. Switching between continuous controllers is involved when the robot travels across the subspaces. With the hybrid controller, the robot can work at the vicinity of singular configurations, but also can go through and stay at the singular configurations.

Fig. 1. The hybrid joint.

In Tarn, Wu, Xi, and Isidori (1996), sensor based control using positive acceleration feedback with a switching control strategy is developed for robot impact control and force regulation. The hybrid joint is a typical switching system, which has not been investigated until now. In the switching control for hybrid joints, the control can be switched among several controllers, and each controller is designed for a specific nominal mode of the hybrid joint. Since the hybrid joint in the under-actuated mode is mainly actuated by the dynamic coupling between the joints, which depends on the dynamic parameters, and is subject to errors if there are uncertainties on the values of these parameters. However, in Arai and Tanie (1998), Liu et al. (1999), Luca and Oriolo (2002), Tan et al. (2004), Tarn et al. (1996), Tinos et al. (2006), the precise knowledge of the dynamic models is required to be known in order to achieve satisfactory performance. While in most cases it is difficult to obtain an accurate dynamic model especially for the under-actuated system. Joint friction and compliance are difficult to model; the tip weight/mass in transporting tasks is usually unknown; and the inertia of the entire manipulator assembly is usually inaccurate because of the lack of experimental methods to determine it. Therefore, it is impossible to control them with simple control schemes if the dynamics is unknown beforehand, because the dynamics uncertainty causes the unknown coupling between the joints. Recently, Least Squares Support Vector Machine (LS-SVM) as an attractive variation of the standard Support Vector Machines has been proposed for solving nonlinear function estimation problems (Suykens, Vandewalle, & Moor, 2001). Standard SVM formulation is modified in the sense of ridge regression. LS-SVM takes equality instead of inequality constraints of SVM in the problem formulation. Therefore, one solves a linear system instead of a QP problem, so LS-SVM is easy to be trained. Although several SVM learning approaches have been proposed in Wang, Li, and Bi (2004), Zhang, Wang, Zhang, and Cai (2006) for modeling nonlinear system and SVM-based controls have been investigated in Lu, Song, Hua, and Sun (2007) and Xu and Chen (2004) for nonlinear system, those works assume beforehand the approximation using SVM is within a compact set, but for practical control systems, the tracking errors could sweep the whole space rather than a compact set. Therefore, the approximation using SVMs is invalid and can not be applied. Moreover, these works lack the explicit stability proof of the closed-loop system using SVM approaches. Finally, these works is only for SISO system, where the output of SVMs is a scalar, and can not applied to the MIMO system. In this paper, we consider switching control incorporating SVMs for hybrid actuated mobile manipulators in the presence of parametric and functional uncertainties in the dynamics. We propose a state-dependent switched controller for motion control

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Z. Li, Y. Kang / Automatica 46 (2010) 832–842

of mobile manipulator with joint switching. Moreover, SVMs matrix and its operator for approximating the MIMO nonlinear unmodelled dynamics are proposed. The switching signal is assumed to generate by a event, which leads to two different switching controls. In each stage the system reaches a subspace of the whole space, and converges to the desired trajectory. We prove the convergence of the closed-loop system to the equilibrium point for every stage. The main contributions of this paper lie in: (i) dynamic coupling switching control incorporating Support Vector Machines (SVMs) is proposed for mobile manipulators with hybrid actuated joints for every switching stage; (ii) SVMs matrix and its operator are proposed for approximating the unmodelled dynamics of mobile manipulators, and closeloop stability approximation stability is proven; (iii) the input-to-state stability properties of its zero dynamics are used to derive bounds on tracking errors when the hybrid joint is under-actuated; and the switching stability for the hybrid joints is presented.

Lemma 2.1 (Ge, Hang, Lee, & Zhang, 2001). Let e = H (s)r with H (s) representing an (n × m)-dimensional strictly proper exponentially stable transfer function, r T and e denoting its input and T output, n m ˙ respectively. Then r ∈ Lm L implies that e , e ∈ L Ln∞ , e is ∞ 2 2 continuous, and e → 0 as t → ∞. If, in addition, r → 0 as t → ∞, then e˙ → 0. Lemma 2.2. For a, b ∈ R, and a + b > 0, the following inequality holds: a+b

(1)

2.1. The dynamics of wheeled mobile manipulators The dynamics of an n-DOF mobile manipulator can be expressed as the following: D(q)¨q + C (q, q˙ )˙q + G(q) + d(t ) = B(q)τ + f Dv Dav Dhv

Dv a Da Dha

Gv Ga , Gh

Dv h Dah , Dh

#

" # G(q) =

B(q)τ = τvT



(3)

The effect of the constraints can be viewed as a restriction of the dynamics on the manifold Ωn as Ωn = {(qv , q˙ v )|Jv q˙ v = 0}. Assume that the annihilator of the co-distribution spanned by the covector fields JvT1 (qv ), . . . , JvTl (qv ) is an (nv − l)-dimensional smooth nonsingular distribution ∆ on Rnv . This distribution ∆ is spanned by a set of (nv − l) smooth and linearly independent vector fields S1 (qv ), . . . , Snv −l (qv ), i.e., ∆ = span{S1 (qv ), . . . , Snv −l (qv )}, which satisfy, in local coordinates, the following relation (Su & Stepanenko, 1994) S T (qv )JvT (qv ) = 0

(4) nv ×(nv −l)

where S (qv ) = [S1 (qv ), . . . , Snv −l (qv )] ∈ R . Note that S T S is of full rank. Constraints (3) implies the existence of vector η˙ ∈ Rnv −l , such that (5)

Considering (5) and its derivative, the dynamics of mobile manipulator can be expressed as

D (ζ )ζ¨ + C (ζ , ζ˙ )ζ˙ + G(ζ ) + d1 (t ) = U

C (q, q˙ ) = dv da , dh

" # d(t ) =

τaT

τhT

T

Cv Cav Chv

"



Cv a Ca Cha

Cv h Cah Ch

#



JvT λn  f = 0  0

.

Remark 2.1. The definition of symbols are listed in the Nomenclature. Assumption 2.1. The wheeled mobile manipulator is subject to known nonholonomic constraints.



S T Dv S  D (ζ ) = Dav S Dhv S



T

S Gv

Assumption 2.2. For simplification, only one hybrid joint is within the mobile manipulator under study.

S T Dv S˙ + S T Cv S C (ζ , ζ˙ ) =  Dav S˙ + Cav S Dhv S˙ + Chv S S T τv

T

ζ = qa , qh



H dv d1 (t ) =  da  , dh



S T Cv a Ca Cha

S T Cv h Cah  , Ch





U =  τa  .

τh

Remark 2.4. In this paper, we choose ζ˙ = [υ, ω, q˙ Ta , q˙ Th ]T , and η˙ = [υ, ω]T , where υ is the forward velocity of the mobile platform; and ω is the rotation velocity of the mobile platform. Assumption 2.3. For simplicity, the disturbance d1 is bounded. 2.3. Physical properties for switching modes 2.3.1. Actuated hybrid joints Assume that the hybrid joint qh = ζ3 is known, we select the group of the actuated joints ζ1 = qa from all actuated joints including the mobile platform such that the dimension of ζ1 and ζ3 are same, and the left actuated variables are grouped as ζ2 . Partitioning (6) in quantities related to the active joints, the hybrid joints, and the remaining joints of the mobile manipulators as D11 D21 D31

D (ζ ) =

D12 D22 D32 C1 C2 C3

" # C (ζ , ζ˙ )ζ˙ = G1

" # Remark 2.3. Although we discuss one hybrid joint, the approach we propose could be extended to multiple hybrid joints easily.

S T Dv h Dah  , Dh



G(ζ ) =  Ga  ,



" # η



S T Dv a Da Dha



" Remark 2.2. In actual implementation, we can adopt methods of producing enough friction between the wheels of the mobile platform and the ground such that the assumption of nonholonomic constraints holds.

(6)

where

(2)

where D(q) =

Jv q˙ v = 0.

Gh

≤ a.

"

The vehicle subjected to nonholonomic constraints can be expressed as

q˙ v = S (qv )η. ˙

2. System description

ab

2.2. Reduced system

G = G2 , G3

D13 D23 D33



#

C11 ζ˙1 + C12 ζ˙2 + C13 ζ˙3



  = C21 ζ˙1 + C22 ζ˙2 + C23 ζ˙3  C31 ζ˙1 + C32 ζ˙2 + C33 ζ˙3 " # " # d11

P = d12 , d13

u1

U = u2 . u3

(7)

Z. Li, Y. Kang / Automatica 46 (2010) 832–842

Considering the property of the above mechanical system, we list the following properties (Ge, Lee, & Harris, 1998) for the active hybrid joints: Property 2.1. The inertia matrix D (ζ ) is symmetric and positive definite, and bounded.

˙ − 2C is skew-symmetric. Property 2.2. The matrix D ˙ are Remark 2.5 (Ge et al., 1998). Since D and therefore D ˙ − 2C symmetric matrices, the skew-symmetry of the matrix D ˙ = C + CT . can also be seen from the fact that D 2.3.2. Unactuated hybrid joints The hybrid joint is under-actuated, i.e. u3 = 0. In order to make ζ3 controllable, we assume that matrices D13 and D31 are full rank 1 and it is obvious that D− 11 exists. However, if D13 and D31 are not full rank, while D23 and D32 are full rank, which means that ζ3 will be coupled with ζ2 , we only need to exchange ζ1 with the vector ζ2 . In this paper, for simplification, we assume D13 = D31 are full rank and every element in the vector D13 is more than zero. After some simple manipulations, we can obtain three dynamics as D11 ζ¨1 = u1 − C1 − G1 − d11 − D12 ζ¨2 − D13 ζ¨3

(8)

1 −1 ¨ ¨ (D22 − D21 D− 11 D12 )ζ2 + (D23 − D21 D11 D13 )ζ3

−1

1 −1 − D21 D− 11 d11 = u2 − D21 D11 u1 1 −1 ¨ ¨ (D32 − D31 D− 11 D12 )ζ2 + (D33 − D31 D11 D13 )ζ3

−1

(10) −1

Let A = D22 − D21 D11 D12 , B = D23 − D21 D11 D13 , J = D32 − 1 −1 −1 ˙ D31 D− 11 D12 , L = D33 − D31 D11 D13 , E = (C22 − D21 D11 C12 )ζ2 + − 1 −1 (C23 − D21 D11 C13 )ζ˙3 , F = (C32 − D31 D11 C12 )ζ˙2 + (C33 − 1 −1 −1 ˙ ˙ D31 D− 11 C13 )ζ3 , H = (C21 − D21 D11 C11 )ζ1 + G2 + d12 − D21 D11 G1 − −1 −1 − D21 D11 d11 , K = (C31 − D31 D11 C11 )ζ˙1 + G3 + d13 − D31 D111 G1 − 1 D31 D− 11 d11 . Then, we can rewrite (8)–(10) as D11 ζ¨1 = u1 − C1 − G1 − d11 − D12 ζ¨2 − D13 ζ¨3

(11)

Aζ¨2 + B ζ¨3 + E + H = −D21 D11 u1 + u2

(12)

1 J ζ¨2 + Lζ¨3 + F + K = −D31 D− 11 u1 .

(13)

−1

Let ξ = [ζ , ζ ] become

T T , 2

considering (6) and (7), the Eqs. (12) and (13)

D1 (ζ )ξ¨ + C1 (ζ , ζ˙ )ξ˙ + P1 = B1 U1

(14)

where



L D1 (ζ ) = B



J , A

 

K P1 = , H

 1 C − D31 D− 11 C13 ˙ C1 (ζ , ζ ) = 33 −1 C23 − D21 D11 C13  T T U1 = −u1 uT2 .

1 D31 D− 11 B1 = −1 D21 D11



1 C32 − D31 D− 11 C12 −1 C22 − D21 D11 C12



0 , I



1 −1 −1 −1 ¨ ¨ ζ¨1 = −(D− 31 J + D11 D12 )ζ2 − (D31 L + D11 D13 )ζ3 −1 −1 − D31 (F + K ) − D11 (C1 + G1 + d11 )

(15)

ξ¨ = −D1−1 C1 (ζ , ζ˙ )ξ˙ − D1−1 P1 + D1−1 B1 U1

(16)

where D31 is the inverse of D31 .

Remark 2.6. In this paper, since only one hybrid joint is consid1 ered, D31 ∈ R > 0, and D− 11 are positive, it is obvious that the eigenvalues of the inertia matrix B1 are positive. If D31 ∈ R < 0, we can design U1 = [uT1 , uT2 ]T such that the eigenvalues of the inertia matrix B1 are positive. However, it is easy to guarantee D31 > 0 by designing the proper physical parameters when the robot is manufactured. Assumption 2.4. If the hybrid joint is under-actuated, then u3 = 0, we assume that nv + na − l > nh such that the under-actuated joint can be controlled. For example, If nh = 1, that is, there exist one hybrid joint, even if na = 0, and since nv = 2, the system still can be controlled. Remark 2.7. There exist the minimum and maximum eigenvalues λmin (B1 ) and λmax (B1 ), such that ∀x ∈ R(n−l−nh ) , xT λmin (B1 )Ix ≤ xT B1 x ≤ xT λmax (B1 )Ix with the identity matrix I.

For the hybrid joints, we give the following assumptions for the actuated and passive modes, respectively, Assumption 3.1. Since the hybrid joint as a mechanical system, the response time for the joint switch is about α second, i.e., 50 ms, therefore, the switching frequency is no more than 1/α Hz, for example, 20 Hz, such that the hybrid joint can be controlled. Assumption 3.2 (Actuated Hybrid Joints, Chang & Chen, 2000; Li et al., 2007). The desired trajectories ζ1d (t ), ζ2d (t ), ζ3d (t ) and their time derivatives up to the 3rd order are continuously differentiable and bounded for all t ≥ 0. For the switching joints in the actuated mode, we could design the controllers that ensure the tracking errors for the variables ζ1 , ζ2 , ζ3 from any (ζj (0), ζ˙j (0)) ∈ Ω , where j = 1, 2, 3, ζj , ζ˙j converge to a manifold Ωad specified as

Ωad = {(ζj , ζ˙j )||ζj − ζjd | ≤ j1 , |ζ˙j − ζ˙jd | ≤ j2 }

(17)

where ji > 0, i = 1, 2. Ideally, ji should be the threshold of measurable noise. At the same time, all the closed loop signals are to be kept bounded. Assumption 3.3 (Under-actuated Hybrid Joints, Chang & Chen, 2000; Li et al., 2007). The desired trajectories ζ2d (t ), ζ3d (t ) and their time derivatives up to the 3rd order are continuously differentiable and bounded for all t ≥ 0. The control objective for the system with the unactuated hybrid joint is to design, if possible, controllers that ensure the tracking errors for the variables ζ2 , ζ3 from any (ζ2 (0), ζ3 (0), ζ˙2 (0), ζ˙3 (0)) ∈ Ω , ζ2 , ζ˙2 , ζ3 , ζ˙3 converge to a manifold Ωud specified as Ω where

Considering (11) and (14), we have

−1

Property 2.5. The eigenvalues of the inertia matrix B1 are positive.

3. Control objectives for the hybrid joints

− D31 D11 d11 = −D31 D11 u1 .

T 3

˙ 1 − 2C1 is skew-symmetric. Property 2.4. The matrix D

(9)

1 −1 + C3 + G3 + d13 − D31 D− 11 C1 − D31 D11 G1

−1

Property 2.3. The inertia matrix D1 is symmetric and positive definite.

Assumption 2.5. The known positive parameters b1 and b2 satisfy 0 < b1 ≤ λmin (B1 ) and b2 ≥ λmax (B1 ), that is, xT b1 Ix ≤ xT λmin (B1 )Ix, and xT b2 Ix ≥ xT λmax (B1 )Ix.

−1

+ C2 + G2 + d12 − D21 D11 C1 − D21 D11 G1

−1

835

Ωud = {(ζk , ζ˙k )||ζk − ζkd | ≤ k1 , |ζ˙k − ζ˙kd | ≤ k2 }

(18)

where ki > 0, i = 1, 2, k = 2, 3. Ideally, ki should be the threshold of measurable noise. At the same time, all the closed loop signals are to be kept bounded. In the following, we can analyze and design the control for each subsystem. For clarity, define the

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Z. Li, Y. Kang / Automatica 46 (2010) 832–842

tracking errors and the filtered tracking errors as ej = ζj − ζjd

(19)

rj = e˙ j + Λj ej

(20)

where Λj is positive definite, j = 1, 2, 3. Then, based on Lemma 2.1, to study the stability of ej and e˙ j , we only need to study the properties of rj . In addition, the following computable signals are defined:

ζ˙jr = ζ˙jd − Λj ej

(21)

ζ¨jr = ζ¨jd − Λj e˙ j .

(22)

4. Support Vector Machine 4.1. LS-SVM For Support Vector Machine regression, the basic idea is to map the data to a higher-dimensional feature space H (Reproducing kernel Hilbert space), via a nonlinear mapping φ , and then to do the linear regression in this space. Therefore given a training set of l training samples (x1 , y1 ), . . . , (xl , yl ) ∈ Rn × R, we introduce a nonlinear mapping φ(·) : Rn −→ H ∈ Rh , which maps the training samples to a new data set (φ1 (x), y1 ), . . . , (φl (x), yl ). In  insensitive Support Vector Regression the goal is to estimate the following function fˆ (x) =< w,

φ(x) > +b;

ω ∈ Rh , b ∈ R

(23)

where w and b are the coefficients, which are estimated by the risk function R = min

w,b,E

1 2

kwk2 + c

l 1X

2 i=1

! Ei2 ,

s.t. yi − fˆ (xi ) = Ei

(24)

where l is the number of the training samples and the constant c > 0 measure the trade-off between complexity and losses. Construct a Lagrangian to solve the optimization problem:

( L=

maxa minw,b

1 2

l 1 X

wT w + c 2

i=1

)

l

−

Ei2

X

ai {yi − [w φ(xi ) + b] − Ei } . T

(25)

i=1

According to Karush–Kuhn–Tucker optimization condition, we can seek the optimal solution and transform this optimization problem into a matrix function as:

0 1    .. . 

1 K (x1 , x1 ) +

1

 = 0

.. .

1 c

K (xl , x1 ) y1

...

yl

...

1

...

K (x1 , xl )

..

.. .

.

... T

K (xl , xl ) +

1

   b  a1      ..   .  al

c

Linear : K (xi , xj ) = xi · xTj ;

kx i − x j k 2σ 2

The obtained nonlinear model is:

ai K (x, xi ) + b.

(26)

i=1

In this paper, the kernel function is the Gaussian kernel function. 4.2. On-line training with time window In order to have a proper performance of LS-SVM, we need to select as many samples as possible for training, however, the dimension of SVM will greatly increase in the process of on-line training. Based on the aim of designing a control which depends on the current state of the nonlinear dynamic system, the training data collected earlier might not suit for real time system, the large data set might lead to time consuming calculation. Therefore, sliding time window is constructed by l with selected sampling time interval, then sample data is collected orderly from current to past. Moreover, a new data sample is collected while the oldest data being dropped. We assume that the nearest data can more properly describe the feature of the system than the oldest data. Theorem 4.1. For any given continuous real functions f (x) on a compact set U ∈ Rn and arbitrary ε > 0, there exists an LS-SVM approximation fˆ (x) formed by (26) such that supx∈U |fˆ (x) − f (x)| < ε. Proof. See Hammer and Gersmann (2003); Wang, Chen, and Chen (2004).  Theorem 4.2. For any given continuous real functions f (x) on a compact set, for a large enough length of slide time window l combined with properly selected sampling time interval and any given  > 0, there exists an LS-SVM approximation function fˆ (x) formed by (26) such that supt ∈[T ,T −l+1] |fˆ (x) − f (x)| ≤ ε . Proof. Let {a1 , a2 , . . . , al } be the estimated weights of LSSVM by minimizing L(·) given by (25). Let the input data be (xT −l+1 , yT −l+1 ), . . . , (xT −1 , yT −1 ), (xT , yT ), where, T denotes the current time. It is easy to know that the length l and the selected sampling time interval determine the size of data samples. From (25), we know the regularization constant c determines the tradeoff between the empirical risk and the model complexity. From Theorem 4.1 and Vapnik (1998), when the time interval is properly selected and l is sufficiently large, we can obtain that for any given  > 0, there exists an LS-SVM approximation function fˆ (x) formed by (26) such that supt ∈[T ,T −l+1] |fˆ (x) − f (x)| ≤ ε .  4.3. SVM matrix operators The above description is the traditional definition for LS-SVM, which is used to approximate a scalar in R, not a vector in Rm . If we need to approximate a vector ∈ Rm , then we propose the LSSVM matrix operator approach for the vector approximation. The SVM row vector bAc and bAcT are defined in the following way:

bAc = ba1 , a2 , . . . , an c bAc = b , T

aT1

aT2

,...,

aTn

(27)

c.

(28)

The definition of SVM matrix is listed as follows

Polynomial : K (xi , xj ) = (xi · xTj + c )d ; Gaussian : K (xi , xi ) = exp −

l X



where K (xi , xj ) = φ(xi )T φ(xj ), i, j = 1, . . . , l, it is a kernel function, which satisfies the Mercer’s theorem. For K (xi , xj ), we have the following typical choice



fˆ (x) =

a11 a21

a12 a22

.. .

··· ··· .. .

an1

an2

···

 2



.

bAc =   .. .

  b a1 c ba2 c   ..   =  ..  . . . ann b an c

a1n a2n 



(29)

Z. Li, Y. Kang / Automatica 46 (2010) 832–842

The transpose of a SVM matrix is defined as



aT11 aT21

aT12 aT22

  bAT c =   ..  .

aTn1

···

.. .

··· .. .

aTn2

···

aT1n aT2n

From (20), we obtain the error dynamics as e˙ = −Λe + r



 T  ba c   1T   ba2 c  ..  . ..  =  .   .  baTn c aT

f11 (x)  f21 (x)

Fˆ (x) =  

f12 (x) f22 (x)

(30)

X˙ 1 =

f1n (x) f2n (x) 

ban1 c bΨn1 c

ban2 c bΨn2 c T

···

eT3 T ,

banm c bΨnm c T

   −Λ e˙ = r˙ 0

 

I

−D

−1





0 e + U1 r D −1

C

(39)

which can be described by brief form X˙ 1 = A1 (ζ , ζ˙ )X1 + B1 (ζ )U1 with A1 (ζ , ζ˙ ) =

R



 = [bAcT ⊗ bΨ c] ..  . fm1 (x) fm2 (x) · · · fmn (x)   T ba11 c bΨ11 c ba12 cT bΨ12 c · · · ba1n cT bΨ1n c ba21 cT bΨ21 c ba22 cT bΨ22 c · · · ba2n cT bΨ2n c    =  (31) ..   . T

where e = [ , , ] Λ = diag[Λ1 , Λ2 , Λ3 ], and r = [r1T , r2T , r3T ]T . We could build up the following augmented system eT2

as

nn

··· ···

(38) eT1

It can be seen that the transpose of a SVM matrix also transpose its elementary vectors locally. To avoid any possible confusion, [∗] is used to denote an ordinary matrix, and b∗c to denote a SVM matrix explicitly. For any nonlinear matrix function F (x) ∈ Rn×m , the corresponding SVM matrix operator, denoted by ⊗ is defined as below



837

2(n−l)×1

−Λ

h

0

(40)

I −D −1 C

i

h

, B1 (ζ ) =

0 D −1

i

, and X1 ∈

.

Lemma 5.1. Consider the dynamics described by (6). Given a weighted matrix Q = Q T > 0, if there exists a symmetric positive definite matrix P1 = P1T > 0 satisfying the following algebraic Riccatilike equation 1 T ˙ P1 A1 + AT1 P1 − P1 B1 R− 1 B1 P1 + P1 + Q = 0

(41)

−1 hfor a gaini matrix R1 = Q22 and positive definite matrix Q = Q11 Q12 T , and Q12 + Q12 < 0, P1 = diag[K1 , D ]. The feedback QT Q 22

12

control

where

bajk cT bΨjk c =

l X

ajki Kik (x, xi ) + b.

(32)

i=1

By letting m = 1, we can see that a vector emulator can be expressed as a SVM product of two SVM vectors.

   ba1 cT bΨ1 c F1 (x) T  ba2 c bΨ2 c  F2 (x)   = [bAcT ⊗ bΨ c] =  Fˆ (x) =   . . ..  .    . . T Fn (x) ban c bΨn c.

sup

t ∈[T ,T −l+1]

kεk ≤ εm

V1 =

˙1 = V =

(34)

(42)

1 2

X1T P1 X1 .

(43)

1 2 1 2

(X˙ 1T P1 X1 + X1T P˙1 X1 + X1T P1 X˙ 1 ) 1

(X1T AT1 (ζ , ζ˙ ) + UT1 BT1 (ζ ))P1 X1 + X1T P˙1 X1 2

1

+ X1T (P1 A1 (ζ , ζ˙ )X1 + P1 B1 (ζ )U1 ) 2

=

5. Dynamic coupling switching control using SVMs In reality, dynamics of the system cannot be exactly known. In addition, external disturbances may also affect the performance of the system. In this section, we take both factors into consideration to develop an LS-SVM based control to deal with uncertainties as well as external disturbances.

1 2

X1T AT1 (ζ , ζ˙ )P1 X1 +

1 2

UT1 BT1 (ζ )P1 X1 +

1

1

2

2

1 2

X1T P˙ 1 X1

+ X1T P1 A1 (ζ , ζ˙ )X1 + X1T P1 B1 (ζ )U1 =

1 2

X1T (AT1 (ζ , ζ˙ )P1 + P˙ 1 + P1 A1 (ζ , ζ˙ ))X1

+ X1T P1 B1 (ζ )U1 .

(44)

Consider (41) and (42), we have

5.1. Actuated hybrid joint Since ζ˙j = ζ˙jr + rj , ζ¨j = ζ¨jr + r˙j , where j = 1, 2, 3, the Eq. (6) becomes (35)

T ¨T ¨T T where r = [r1T , r2T , r3T ]T , ζ¨r = [ζ¨1r , ζ2r , ζ3r ] . Define new control input as

U1 = U − ∆1

2

Proof. Let us choose a Lyapunov function as

where εm > 0 is the given LS-SVM approximation error.

D r˙ + C r = −D ζ¨r − C ζ˙r − G − d1 + U

2

Taking the time derivative of V1 along the dynamics (39), we obtain

From Theorem 4.2, we can define the LS-SVM approximation matrix as

kFˆ (x) − F (x)k ≤ ε,

1

guarantees that all the variables of the close loop system are bounded and the tracking performance is achieved.



(33)

1

1 T −1 U1 = − R− 1 B1 P1 X1 = − R1 r

(36)

 1 ˙V1 = − X1T Q11 T 2

Q12



Q12 X ≤ 0. Q22 1

(45)

Therefore, using the feedback control (42) results in the controller nonlinear system X˙ 1 =



A1 (ζ , ζ˙ ) −

1 2

1 T B1 (ζ )R− 1 B1 (ζ )P1

 X1

(46)

where U1 is auxiliary control inputs to be optimized later, and ∆1 = D ζ¨r + C ζ˙r + G + d1 , therefore, the Eq. (35) becomes

being globally exponentially stable about the origin in R2 , that is, e and r is bounded. Let

D r˙ + C r = U1 .

ˆ svm (ζ , ζ˙ ) + u ∆1 = U

(37)

(47)

838

Z. Li, Y. Kang / Automatica 46 (2010) 832–842

where Uˆ sv m (ζ , ζ˙ ) denotes SVM estimation from (33) with on-line LS-SVM method, due to the approximation property of SVM and Theorem 4.2, such that

ˆ svm (ζ , ζ˙ )k ≤ sup{|u |} kU∗ (ζ , ζ˙ ) − U

(48)

Z ∈Ω

where u is bounded error by Theorem 4.2. We select the vector of ˆ svm (ζ , ζ˙ )) for the on-line the sample pairs as (x, y) = ((ζ , ζ˙ ), U time window, where Uˆ sv m (ζ , ζ˙ ) = baj cT bΨj c from (33). 

˙ 1 ≤ −ν V1 +δ(t ). Thus, r converges to a set Therefore, we arrive at V containing the origin with a rate at least at fast as e−ν t . Integrating both sides of (56) gives V1 (t ) − V1 (0) ≤ −

(49)

where τr1 is a robustifying vector defined later. The state space description (40) can be given by

ˆ svm + τr1 − ∆1 ). X˙ 1 = A1 (ζ , ζ˙ )X1 + B1 (ζ )(U1 + U

(50)

Considering the feedback control (42), we have

  1 T ˙X1 = A1 − 1 B1 R− ˆ 1 B1 P1 X1 + B1 (Usv m + τr1 − ∆1 ).

(51)

2

Theorem 5.1. Consider U1 provided by (42) with the robust term given by

τr1 = −

2 r εm

(52)

kr kεm + δ(t )

with the positive εm defined in (34) and r defined by (20), δ(t ) is a time R t varying positive function converging to zero as t → ∞, such that δ(ω)dω = a < ∞ with bounded constant a, then X1 converges 0 to a set containing the origin with a rate at least at fast as e−ν t with ν = λmin (Q )/λmax (P1 ) > 0. Proof. Considering (43), we have

˙1 = V



X1T

P1 X˙ 1 +

1 2

P˙ 1



X1 .

(53)

1 2

1 T X1T P1 B1 R− 1 B1 P1 X1 +

1 2

(54)

Using X1T P1 A1 X1 = 21 X1T (AT1 P1 + P1 A1 )X1 , integrating (41), the time derivative of Lyapunov function becomes 1 ˙ 1 = − X1T QX1 + X1T P1 B1 (Uˆ svm + τr1 − ∆1 ). V 2

= 

diag[K1 , D ] 0



0

D

D −1

 −1 T

T

from (39), X1T P1 B1

(55)

=

eT



rT



T

= r , consider Lemma 2.2, we have

1 2

2

 kX1 k2 λmin (Q ) ds ≤ V1 (0) − V1 (t ) + a

(58)

which leads to X1 ∈ L2 . From r = e˙ + Λe, it can be obtained that e, e˙ ∈ L∞ . As we have established e, e˙ ∈ L∞ , from Assumption 3.2, we conclude that ζ˙ , ζ˙r ∈ L∞ .  5.2. Unactuated hybrid joint 5.2.1. ζ2 and ζ3 -subsystems Since ξ˙ = ξ˙r + r, ξ¨ = ξ¨r + r˙ , the Eq. (14) become

D1 r˙ + C1 r = −D1 ξ¨r − C1 ξ˙r − P1 + B1 U

(59)

T ¨T T where r = [r3T , r2T ]T , ξ¨r = [ζ¨3r , ζ2r ] . Define new control input as

U = U2 + ∆2

(60)

where U2 is auxiliary control inputs to be optimized later, and ∆2 = B1−1 (D1 ξ¨r + C1 ξ˙r + P1 ), therefore, the close loop system (59) becomes

D1 r˙ + C1 r = B1 U2 .

(61)

From (20), we obtain the error dynamics as e˙ = −Λe + r

(62)

where e = [eT3 , eT2 ]T , Λ = diag[Λ3 , Λ2 ], and r = [r3T , r2T ]T . We could build up the following augmented system as

   ˙X2 = e˙ = −Λ r˙ 0

 

I

−D1−1 C1

with A2 =

h

−Λ 0

I −1

i

−D1 C1





0 e + U2 r D1−1 B1

(63)

(64)

h

0 −1

i

, B2 = D B , and X2 ∈ R2(n−l−nh )×1 . 1 1

Lemma 5.2. Consider the dynamics described by (63). Given a weighted matrix Q = QT > 0, if there exists a symmetric positive definite matrix P2 = P2T > 0 satisfying the following algebraic Riccatilike equation 1 T ˙ P2 A2 + AT2 P2 − P2 B2 R− 2 B2 P2 + P2 + Q = 0

−1

for a gain matrix R2

h

1 ˙ 1 ≤ − kX1 k2 λmin (Q ) + krkε + X1T PBτr1 V 2 1 krk2 εm2 ≤ − kX1 k2 λmin (Q ) + krkεm − 2 krkεm + δ(t ) 1 krkεm δ(t ) 2 ≤ − kX1 k λmin (Q ) + 2 krkεm + δ(t )

≤ − kX1 k2 λmin (Q ) + δ(t ).

0

1

X˙ 2 = A2 (ζ , ζ˙ )X2 + B2 (ζ )U2

X1T P˙ 1 X1

+ X1T P1 B1 (Uˆ svm + τr1 − ∆1 ).

Since B1

(57)

which can be described by brief form

Introducing (51) into (53) yields

˙ 1 = X1T P1 A1 X1 − V

 kX1 k λmin (Q ) ds + a. 2

Thus V1 is bounded, which implies that X1 ∈ L∞ . From (57), we have

Remark 5.1. From Lemma 5.1, we know r, e are bounded uniformly in time, therefore, Theorem 4.2 can be applied.

ˆ svm + U1 + τr1 U=U

1 2

0

Z t

Therefore, the external torques are given as

Z t

Q11 T Q12

Q12 Q22

(65)

= Q22 and positive definite matrix Q =

i

T , and Q12 + Q12 < 0, P2 = diag[K2 , D1 ]. The feedback

control 1

1

2

2

1 T −1 U2 = − R − 2 B2 P2 X2 = − R2 r

(66)

guarantees that all the variables of the close loop system are bounded and the tracking performance is achieved. (56)

Proof. Similar as the proof of Lemma 5.1.



Z. Li, Y. Kang / Automatica 46 (2010) 832–842

ˆ svm (ζ , ζ˙ ) + u , where U ˆ svm (ζ , ζ˙ ) denotes SVM Let ∆2 = U estimation from (33) with on-line LS-SVM method, due to the approximation property of SVM and Theorem 4.2, such that ˆ svm (ζ , ζ˙ )k ≤ sup{|u |} k∆2 − U

(67)

Z ∈Ω

where u is bounded error by Theorem 4.2. We select the vector ˆ svm (ζ , ζ˙ )), and of the sample pairs as (x, y) = ((ζ , ζ˙ ), U ˆ svm (ζ , ζ˙ ) = baj cT bΨj c. U Remark 5.2. From Lemma 5.2, we know r, e are bounded uniformly in time, therefore, Theorem 4.2 can be applied.

˙ 2 ≤ −ν V2 + δ(t ). Thus, r converges to a Therefore, we arrive at V set containing the origin with a rate at least at fast as e−ν t . Integrating both sides of the above equation gives V2 (t ) − V2 (0) ≤ −

ˆ svm + U2 + τr2 U=U

(68)

where τr2 is a robustifying vector defined later. The state space description (64) can be given by

ˆ svm + τr2 − ∆2 ). X˙ 2 = A2 (ζ , ζ˙ )X2 + B2 (ζ )(U2 + U

(69)

Considering the feedback control (66), we have X˙ 2 =

 A2 −

1 2

1 T B2 R− 2 B2 P2



ˆ svm + τr2 − ∆2 ). X2 + B2 (U

(70)

Theorem 5.2. Consider the U2 provided by (66) with the robust term given by 2 rb22 εm

1

τr2 = −

(71)

b1 kr kb2 εm + δ(t )

with the positive εm defined in (34) and r defined by (20), δ(t ) is a time R t varying positive function converging to zero as t → ∞, such that δ(ω)dω = a < ∞ with bounded constant a, then X2 converges 0 to a set containing the origin with a rate at least at fast as e−ν t with ν = λmin (Q)/λmax (P2 ) > 0. Proof. Let us choose a Lyapunov function as

V2 =

1 2

X2T P2 X2

(72)

we have

  1 ˙V2 = X2T P2 X˙ 2 + P˙2 X2 .

(73)

2

Introducing (70) into (73) yields

˙ 2 = X2T P2 A2 X2 − V

1 2

(74)

2

Using X2T P2 A2 X2 = 21 X2T (AT2 P2 + P2 A2 )X2 , integrating (65), the time derivative of Lyapunov function becomes (75)

2

X2T P2 B2



T

T



= 0 D1 B1 from (63), = e r  T −1 T diag[K2 , D1 ] 0 D1 B1 = r B1 , consider Lemma 2.2, we have Since B2

1

˙ 2 ≤ − kX2 k2 λmin (Q) + kr kkB1 kε + X2T P2 B2 τr2 V 2 1 kr k2 b22 εm2 ≤ − kX2 k2 λmin (Q) + kr kb2 εm − 2 kr kb2 εm + δ(t ) 1 kr kb2 εm δ(t ) 2 ≤ − kX2 k λmin (Q) + 2 kr kb2 εm + δ(t ) 1

≤ − kX2 k λmin (Q) + δ(t ). 2

2

(77)

Thus V2 is bounded, which implies that X2 ∈ L∞ . From (77), we have

Z t

1 2

 kX2 k2 λmin (Q) ds ≤ V2 (0) − V2 (t ) + a

(78)

which leads to X2 ∈ L2 . From r = e˙ + Λe, it can be obtained that e, e˙ ∈ L∞ . As we have established e, e˙ ∈ L∞ , from Assumption 3.2, we conclude that ξ˙ , ξ˙r ∈ L∞ .  5.2.2. ζ1 -subsystem Finally, for system (8)–(10) under control laws (68), apparently, the ζ1 -subsystem (8) can be rewritten as

ϕ˙ = f (ν, ϕ, U)

(79)

where ϕ = [ζ1T , ζ˙1T ]T , ν = [r T , r˙ T ]T , U = [uT1 , uT2 ]T . Assumption 5.1. From (9) and (10), the reference signal satisfies Assumption 3.3, and the following functions are Lipschitz in γ , i.e., there exists Lipschitz positive constants Lγ and Lf such that

kC1 + G1 + d1 k ≤ L1γ kγ k + L1f

(80)

kF + K k ≤ L2γ kγ k + L2f

(81)

moreover, from the stability analysis of ζ2 and ζ3 subsystems, γ converges to a small neighborhood of γd = [ζ3d , ζ2d , ζ˙3d , ζ˙2d ]T . Remark 5.3. Under the stability of ζ2 and ζ3 subsystems, and considering (18), let kγ − γd k ≤ ς1 , it is easy to obtain kγ k ≤

kγd k + ς1 , and similarly, let µ = [ζ¨3 , ζ¨2 ]T , and µd = [ζ¨3d , ζ¨2d ]T , kµk ≤ kµd k + ς2 , where ς1 and ς2 are small bounded errors. Lemma 5.3. The ζ1 -subsystem (8), if ζ2 -subsystem and ζ3 -subsystem are stable, is global asymptotically stable, too.

˙ 3 = V˙ 2 + tanh(ζ˙1 )ζ¨1 V 1 = V˙ 2 + tanh(ζ˙1 )D− 11 (u1 − C1 − G1 − d11

(83)

From (13), we have 1 ¨ ¨ u1 = −D11 D− 31 (J ζ2 + Lζ3 + F + K ).

Integrating (84) into (83), we have 1 ¨ ¨ ˙ 3 = V˙ 2 + tanh(ζ˙1 )[−D− V 31 (J ζ2 + Lζ3 + F + K ) 1 ¨ ¨ + D− 11 (−C1 − G1 − d11 − D12 ζ2 − D13 ζ3 )] " #T   1 −1 tanh(ζ˙1 )D− ζ¨2 31 J + D11 D12 = V˙ 2 − −1 −1 ˙ ζ¨3 tanh(ζ1 )D31 L + D11 D13

(76)

(82)

Differentiating (82) along (8) gives

− D12 ζ¨2 − D13 ζ¨3 ).

1 ˙ 2 = − X2T QX2 + X2T P2 B2 (Uˆ svm + τr2 − ∆2 ). V

T

2

 kX2 k λmin (Q) ds + a. 2

V3 = V2 + ln(cosh(ζ˙1 )).

+ X2T P˙2 X2 + X2T P2 B2 (Uˆ svm + τr2 − ∆2 ).

−1

1

Proof. From (8)–(10), we choose the following function as

1 T X2T P2 B2 R− 2 B2 P2 X2

1



Z t 0

0

Therefore, the external torques are given as

839

1 − tanh(ζ˙1 )D− 31 (F + K ) 1 − tanh(ζ˙1 )D− 11 (C1 + G1 + d11 ).

(84)

840

Z. Li, Y. Kang / Automatica 46 (2010) 832–842

1 Since k tanh(ζ˙2 )k ≤ 1, D− 31 , D12 and D13 are all bounded. Let

T



tanh(ζ˙1 )D−311 J + D−111 D12 1 −1

tanh(ζ˙ )D−1 L + D−1 D

≤ %1 , and kD− 11 k ≤ %2 , kD31 k ≤ 1 31

11 13 %3 , where %1 , %2 and %3 are bounded constants. Considering

Assumption 5.1 and Remark 5.3, we have

switching. Similarly, if the switching joint is switched from the passive mode to the active mode, although the joint torque is added, since the motion of the system is continuous because of the inertia, that is 1V ≤ 0, the motion of the system is also stable.  7. Simulations

1

˙ 3 ≤ − kX2 k2 λmin (Q) + δ(t ) V 2 + %1 (kµd k + ς2 ) + %2 (L1γ (kγd k + ς1 ) + L1f ) + %3 (L2γ (kγd k + ς2 ) + L2f ).

(85)

Let ρ1 = δ(t ) + %1 (kµd k + ς2 ) + %2 (L1γ (kγd k + ς1 ) + L1f ) + %3 (L2γ (kγd k + ς2 ) + L2f ) and it is apparently bounded positive, we

˙ 3 ≤ 0, when kX2 k ≥ have V

q

2ρ1 , λmin (Q)

we can choose the proper Q

such that the r can be arbitrary small. Therefore, we can obtain the internal dynamics is stable with respect to the output ζ˙1 . Therefore, the ζ1 -subsystem (8) is global asymptotically stable.  Theorem 5.3. Consider the system (8)–(10) with Assumption 3.3, under the action of control laws (68). For compact set Ω , where (ζ2 (0), ζ3 (0), ζ˙2 (0), ζ˙3 (0)) ∈ Ω , the tracking errors r converges to the compact sets Ω defined in (18), and all the signals in the closed loop system are bounded. Proof. From the results (78), it is clear that the tracking errors r2 , r3 converges to the compact sets Ω defined by (18). From Lemma 2.1, we can know e2 , e˙ 2 , e3 , e˙ 3 are also bounded. From the boundedness of ζ2d , ζ3d in Assumption 3.3, we know that ζ2 , ζ3 are bounded. Since ζ˙2d , ζ˙3d are also bounded, it follows that ζ˙2 , ζ˙3 are bounded. From Lemma 5.3, we know that the ζ1 -subsystem (8) is stable, and ζ1 , ζ˙1 are bounded. This completes the proof.  6. Switching stability For the system switching stability between the actuated and under-actuated mode, we give the following theorem as Theorem 6.1. Consider the system (6) with the actuated mode (7) and the under-actuated mode (8)–(10), if the system is both stable before and after the switching phase using the control laws (49) and (68). Assume that there exists no external impacts during the switching, the system is also stable during the switching phase. Proof. Since V1 and V2 is decreasing from Theorems 5.1 and 5.2, we know the system is stable no matter the hybrid joint is either actuated or under-actuated. In the preceding, we have shown that the Lyapunov function non-increasing during the switching. Let + 1 ˙− 1 ˙+ ˙ ˙− ˙ ˙ ˙+ ˙ V− 12 = 2 (ζ −ζ )D (ζ −ζ ) and V12 = 2 (ζ −ζ )D (ζ −ζ ) denote

the Lyapunov function before and after the switching, and ζ˙ + and ζ˙ − represent the post- and pre-switch velocities, respectively. The Lyapunov function change during the switching can be simplified as follows:

1V = V+ − V−

=

1 2

1

(ζ˙ + − ζ˙ )D (ζ˙ + − ζ˙ ) − (ζ˙ − − ζ˙ )D (ζ˙ − − ζ˙ ). 2

(86)

There is no external impact during the switching, which means that there are no extra energy injected into the system. Since the inertia properties of the switching joint and link exists, during the switching joint, if the switching joint is switched from the active mode to the passive mode without considering the friction, the motion of the link should be continuous, that is, ζ˙ + = ζ˙ − = ζ˙ . Therefore, during the switching, the Lyapunov function is nonincreasing. If considering the friction, the Lyapunov function is decreasing, that is, 1V ≤ 0, the motion is stable during the

Let us consider a mobile wheeled hybrid-actuated manipulators shown in Fig. 2. The following variables have been chosen to describe the vehicle (see also Fig. 2): τl , τr : the torques of two wheels; τ1 : the torques of the hybrid joint; θl , θr : the rotation angle of the left wheel and the right wheel of the mobile platform; v : the forward velocity of the mobile platform; θ : the direction angle of the mobile platform; ω: the rotation velocity of the mobile platform, and ω = θ˙ ; θ1 : the joint angle of the hybrid link; m1 , m2 : the mass of links of the manipulator; Iz1 , Iz2 : the inertia moment of the link 1 and the link 2; l1 , l2 : the link length of the manipulator; r: the radius of the wheels; 2l: the distance of the wheels; d: the distance between the manipulator and the driving center of the mobile base; mp : the mass of the mobile platform; Ip : the inertia moment of the mobile platform; Iw : the inertia moment of each wheel; mw : the mass of each wheel; g: gravity acceleration. The mobile manipulator is subjected to the following constraint: x˙ cos θ − y˙ sin θ = 0. Using Lagrangian approach, we can obtain the standard form with q = [θl , θr , θ1 ]T , ζ˙ = [ζ˙1 , ζ˙2 , ζ˙3 ]T = [υ, θ˙ , θ˙1 ]T , then we could obtain M (q)¨q + C (q, q˙ )˙q + G(q) = B(q)τ + J T f where m11 m21 m31

" M (q) =

m12 m22 m32

m13 m23 , m33

#

" C (q, q˙ ) =

c11 c21 c31

c12 c22 c32

c13 c23 , c33

#

m11 = p0 − p1 cos θ1 + p2 + p3 sin θ1 , m12 = q0 + p1 cos θ1 − p2 , m13 = q1 sin θ1 − q2 cos θ1 + q23 + q4 , m21 = q0 + p1 cos θ1 − p2 , m22 = p0 − p1 cos θ1 + p2 − p3 sin θ1 , m23 = q1 sin θ1 + q2 cos θ1 − q3 − q4 , m31 = q1 sin θ1 − q2 cos θ1 + q3 + q4 , m32 = q1 sin θ1 + q2 cos θ1 − q3 − q4 , m33 = Iz1 + Iz2 + m2 l22 , p0 = 1 (mp + m1 + m2 + 2mw )r 2 + 41 (Ip + 2I w+ m1 d2 + m2 d2 + 2mw l2 )r 2 + 4 (Iz1 + Iz2 )r 2 /4, p1 = m2 l2 dr 2 /2, p2 = m2 l22 r 2 /4, p3 = m2 l2 r 2 /2, q0 = (mp + m1 + m2 + 2mw )r 2 /4 − 14 (Ip + 2Iw + m1 d2 + m2 d2 + 2mw l2 )r 2 − (Iz1 + Iz2 )r 2 /4, q1 = m2 l2 r /2, q2 = m2 l2 dr /2, q3 = m2 l2 r 2 /2, q4 = (Iz1 + Iz2 )r /2, c11 = (p1 sin θ1 + p3 cos θ1 )θ˙1 /2, c12 = −p1 sin θ1 θ˙1 , c13 = (p1 sin θ1 +p3 cos θ1 )θ˙r /2−p3 sin θ1 θ˙l + (q1 cos θ1 + q2 sin θ1 )θ˙1 , c21 = −p1 sin θ1 θ˙1 /2, c22 = (p1 sin θ1 − p3 cos θ1 )θ˙1 /2, c23 = −p1 sin θ1 θ˙r /2+(p1 sin θ1 −p3 cos θ1 )θ˙l /2+ (q1 cos θ1 − q2 sin θ1 )θ˙1 , c31 = −(p1 sin θ1 + p3 cos θ1 )θ˙r /2 + p1 sin θ1 θ˙l /2, c32 = p1 sin θ1 θ˙r /2 + (p3 cos θ1 − p1 sin θ1 )θ˙l /2, c33 = 0, G(q) = 0, B(q) = diag[1]. In the simulation, we assume the parameters p0 = 6.0 kg m2 , p1 = 1.0 kg m2 , p2 = 0.5 kg m2 , p3 = 1.0 kg m2 , p4 = 2.0 kg m2 , q0 = 4.0 kg m2 , q1 = 1.0 kg m2 , q2 = 1.0 kg m2 , q3 = 1.0 kg m2 , q4 = 0.5 kg m2 , d = 1.0m, r = 0.5m.The disturbances from environments on the system are introduced as 0.1 sin(t ), 0.1 sin(t ) and 0.1 sin(t ) to the simulation model. The desired trajectories are chosen as ζ1 = 0 rad and θd = 0.0 rad, ζ2 = 0.5t m and vd = 0.5 m/s, θ1d = 0 rad, and the initial value is [0.0, 0.1, 0.1, −0.2, 0.1, π /18]T . The design parameters of the LS-SVM controller for full actuated joints are: T Q11 = diag[20, 20, 20], Q12 = Q21 = diag[−20, −20, −20], Q22 = diag[100, 100, 100], Λ = diag[1, 1, 1], K1 = diag[10, 10, 10], 200 training data are sampled for the sliding window during [0.0, 5.0] s as the phase I. The training data pairs (x, y) are constructed using x = (θ˙ , υ, θ˙1 ) and y = Uˆ sv m , and U = [u1 , u2 , u3 ]T , U = 1 ˆ − 12 R− 1 r + Usv m +τr1 . We use the radial basis function (RBF) kernels

Z. Li, Y. Kang / Automatica 46 (2010) 832–842

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Fig. 4. The profiles of ζ2 = θ and ω. Fig. 2. The mobile hybrid-actuated manipulator in the simulation.

Fig. 5. Tracking the desired position of θ1 . Fig. 3. Tracking the forward velocity v and the corresponding ζ1 .

for SVM, the parameters of RBF kernels are δ1 = δ2 = δ3 = 0.001, the parameters of LS-SVM are selected as c1 = c2 = c3 = 100, εm = 0.02. From t = [5.0, 10.0] s as the phase II, the hybrid joint is switched to passive mode, the parameters of the LS-SVM controller T = for under-actuated joints are: Q11 = diag[20, 20], Q12 = Q21 diag[−20, −20], Q22 = diag[100, 100], Λ = diag[1, 1], K2 = diag[10, 10]. 200 training data are sampled for the sliding window during [5, 10] s. The training data pairs (x, y) are constructed using 1 x = (υ, θ˙1 ) and y = Uˆ sv m , and U = [u1 , u2 ]T , U = − 12 R− 2 r +

Uˆ sv m + τr2 . The parameters of RBF kernels are δ1 = δ2 = 0.001, the parameters of LS-SVM are selected as c1 = c2 = 100, εm = 0.02, b1 = 1.0, b2 = 10.0. From t = [10.0, 15.0] s as the phase III, the hybrid joint is switched to the original mode. The used controller is the same as the the phase I. The trajectory profiles for ζ˙1 and ζ1 are shown in Fig. 3. The positions tracking and the corresponding velocity profiles for the ζ2 and ζ3 are shown in Figs. 4 and 5. During the whole switching, the switching points happen on t = 5.0 s and t = 10 s, we can see the positions and velocities are continuous and converge to the desired values, which are stable and bounded during the switching. The input torques of the preswitch and post-switch are shown in Fig. 6. Finally, the trajectory is shown in Fig. 7. Since the initial values for the dynamics are assumed to be unknown in the simulations, from these figures, even if the nominal parameters of the system are uncertain, and the initial disturbances boundedness from the environment are unknown, we can get good performance by the proposed switching control. 8. Conclusion In this paper, adaptive dynamic coupling switching control designs are carried out for dynamic balance and tracking of desired trajectories of mobile manipulators with hybrid actuated joints in the presence of unmodelled dynamics, or parametric/functional

Fig. 6. Input torques of the pre-switch and post-switch.

Fig. 7. The produced trajectory.

uncertainties. The controllers are mathematically shown to guarantee semi-globally uniformly bounded stability, and the steady state compact sets to which the closed loop error signals converge are derived. The size of compact sets can be made small through appropriate choice of control design parameters. Simulation results demonstrate that the system is able to track reference signals satisfactorily, with all closed loop signals uniformly bounded.

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Z. Li received the Dr. Eng. Degree in mechatronics, Shanghai Jiao Tong University, PR China, in 2002. From 2003 to 2005, he was a postdoctoral fellow in Department of Mechanical Engineering and Intelligent systems, The University of Electro-Communications, Tokyo, Japan. From 2005 to 2006, he was a research fellow in the Department of Electrical and Computer Engineering, National University of Singapore, and Nanyang Technological University, Singapore. Currently, he is an associate professor in the Department of Automation, Shanghai Jiao Tong University, PR China. Dr. Li is IEEE Senior Member and his current research interests are the adaptive/robust control, mobile manipulator, nonholonomic system, etc.

Y. Kang received the Dr. Eng. Degree in Control Theory and Control Engineering, University of Science and Technology of China, PR China, in 2005. From 2005 to 2007, he was a postdoctoral fellow in Academy of Mathematics and Systems Science, Chinese Academy of Sciences, PR China. Currently, he is Associate Professor in the Department of Automation, University of Science and Technology of China, PR China. Dr. Kang’s current research interests are in the adaptive/robust control, variable structure control, mobile manipulator, Markovian jump systems, etc.