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Robust adaptive tracking control for nonholonomic mobile manipulator with uncertainties Jinzhu Peng n, Jie Yu, Jie Wang School of Electrical Engineering, Zhengzhou University, Zhengzhou, Henan 450001, China
art ic l e i nf o
a b s t r a c t
Article history: Received 6 December 2013 Received in revised form 10 May 2014 Accepted 15 May 2014 This paper was recommended for publication by Rickey Dubay
In this paper, mobile manipulator is divided into two subsystems, that is, nonholonomic mobile platform subsystem and holonomic manipulator subsystem. First, the kinematic controller of the mobile platform is derived to obtain a desired velocity. Second, regarding the coupling between the two subsystems as disturbances, Lyapunov functions of the two subsystems are designed respectively. Third, a robust adaptive tracking controller is proposed to deal with the unknown upper bounds of parameter uncertainties and disturbances. According to the Lyapunov stability theory, the derived robust adaptive controller guarantees global stability of the closed-loop system, and the tracking errors and adaptive coefficient errors are all bounded. Finally, simulation results show that the proposed robust adaptive tracking controller for nonholonomic mobile manipulator is effective and has good tracking capacity. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Mobile manipulator Robust adaptive control Tracking control Nonholonomic system
1. Introduction Tracking control for multi-joint robotic manipulators and mobile robots always is a challenging problem and has been given a lot of attention in the control field. Many powerful methodologies have been applied on robotic manipulators or mobile robots to achieve good tracking performances. Robust adaptive control method combines the advantages of adaptive control and robust control methods [1–4], which have been widely used to control the robotic manipulators and mobile robots or other electromechanical systems. Slotine and Li [5] introduced a robust fixed gain based on the basic of adaptive control for controlling the manipulator, which enhanced the robustness of passive structure adaptive controller. Su and Leung [6] introduced an estimation algorithm of unknown parameters' upper bound based on the robust control structure, which can effectively reduce the robust gain conservatism. However, only parameter uncertainties were considered in the above works. By combining robust adaptive control and fuzzy logic control, Gueaieb et al. [7] proposed a decentralized robust adaptive fuzzy control strategy, which was specially for the parametric and nonparametric uncertainties, and the stability of system was proven by using the Lyapunov stability theory. González-Vázquez et al. [8] introduced a class of PD-type robust controllers for robotic manipulators by using the theory of singularly perturbed systems.
n
Corresponding author. Tel./fax: þ 86 371 67783113. E-mail address: [email protected] (J. Peng).
Tomei [9] proposed a robust adaptive controller, which can maintain a high tracking accuracy and adjust system transient quality discretionarily under the circumstance that the unknown parameters of the system and the external interference coexist. However, the upper bound of system parameters should be known in the above works. Aiming at the system uncertainties, Wang et al. [10] proposed a robust adaptive tracking control for robotic manipulator, where the upper bound of system parameters was assumed to be unknown. However, the above works mainly focused on controlling for robotic manipulators or mobile robots only. A mobile manipulator is a robotic manipulator mounted on a mobile platform, with the function of mobile and operation. It not only possesses the flexible function of manipulator's operation but also has mobile robot's extensity in workspace. The mobile manipulator shows its nonholonomic characteristics since the mobile platform is a typical nonholonomic system. In general, the tracking control methods of mobile manipulator are mainly divided into two categories: one is called centralized control strategy, the other is called decentralized control strategy [11]. In the centralized control strategy, the mobile platform and robotic manipulator are regarded as a whole. Seraji [12] established a unified dynamic model of mobile manipulators, the idea of configuration control was proposed. However, the control method that was based on the kinematics was difficult to implement in practice. Tan and Xi [13,14] established a dynamic model of mobile manipulators, and a hybrid force/position control method was proposed. In [15], the dynamic coupling was compensated by linearizing the dynamic model of mobile manipulators, however,
http://dx.doi.org/10.1016/j.isatra.2014.05.012 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Peng J, et al. Robust adaptive tracking control for nonholonomic mobile manipulator with uncertainties. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.012i
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the decoupling matrix was required to be full rank, that is, the initial states of system were restricted. Consider the dynamic coupling, Dong [16] designed a robust controller which is based on the Lyapunov theory to achieve system stable, however, the system structure was complicated due to the regression matrix was used. In the decentralized control strategy, the mobile manipulator is divided into nonholonomic mobile platform subsystem and holonomic manipulator subsystem. Evangelos and John [17] used computed torque control theory to implement the tracking control of mobile manipulator, which is subject to external disturbances. Liu and Lewis [18] took manipulator and mobile platform as two independent systems, the controllers for each system were designed, and the system dynamic couplings were regarded as external disturbances. Lin and Goldenberg [19] designed tracking controllers for mobile manipulator based on neural networks, which were used to estimate the system dynamic coupling and uncertainties online. Whereafter, they proposed a robust damping control algorithm as well [20], the control algorithm only needs a few control parameters. However, the above investigations cannot handle the varying parameter uncertainties well, because the adaptive parameters identification is generally not included in these methods. Wang and Wang [21] established the dynamic model of nonholonomic mobile manipulator based on Screw theory and designed a robust adaptive fuzzy controller of mobile manipulator. Andaluz et al. [22] proposed a kinematic controller of mobile manipulator with uncertainties. The controller was able to solve the problem, not only the point stabilization and the trajectory tracking, but also the path following. Shojaei et al. [23] proposed an input–output model of mobile manipulator, where a tracking controller was designed by using the dynamic surface control technique. Then, according to adaptive robust technique, the influence of parametric and nonparametric about the uncertainties in the mobile manipulators model was compensated. However, the upper bound of system parameters should be known from the above works. In this paper, the decentralized control strategy is studied. The mobile manipulator is divided into nonholonomic mobile platform subsystem and holonomic manipulator subsystem. The kinematic controller of the mobile platform is derived. And considering that the dynamic model, Lyapunov functions of the two subsystems are derived, the couplings between the two subsystems are regarded as external disturbances. Then, the dynamic parameters and external disturbances are assumed to be unknown, a robust adaptive controller is proposed to achieve the closed-loop system stability. The proposed controller can eliminate interference of dynamic uncertainties and external disturbances as well. Simulation results show that the proposed robust adaptive controller is effective for controlling the mobile manipulator. The rest of this paper is organized as follows. In Section 2, the mathematic models of nonholonomic mobile platform and holonomic manipulator are addressed respectively. In Section 3, the Lyapunov function for mobile platform is designed, which consists of the design of kinematic controller and Lyapunov function based on dynamic model. In Section 4, the design of Lyapunov function for manipulator is drawn. The design of robust adaptive control scheme is given in Section 5, as well as the robust stability is analysed. The simulation results are given in Section 6, and the conclusions are drawn in Section 7.
2. Model of nonholonomic mobile manipulator A mobile manipulator system is depicted in Fig. 1, where a twolink manipulator is mounted on the centre C of a mobile platform. The two rear wheels of mobile platform are driven independently, and the two links of manipulator are also driven by motors
Fig. 1. System of a mobile manipulator.
independently. In addition, the manipulator is generally considered to be a holonomic system, while the mobile platform is subject to nonholonomic constraint, the mechanical system of mobile manipulator can be expressed as _ q_ þ Fðq; qÞ _ þ τd ¼ BðqÞτ AT ðqÞλ MðqÞq€ þ Cðq; qÞ
ð1Þ
The nonholonomic constraint can be expressed as AðqÞq_ ¼ 0
ð2Þ
_ q€ A Rp are the state vectors of the mobile manipulator where q; q; system, representing position, velocity and acceleration vectors respectively, MðqÞ A Rpp is the symmetric, positive definite inertia _ A Rpp represents the vector of centripetal and matrix, Cðq; qÞ _ A Rpp represents the gravity and Coriolis forces term; Fðq; qÞ p friction term, τd A R is the vector of unknown bounded external disturbances, BðqÞ A Rpðp rÞ is the input transformation matrix, τ A Rp r is the input torque, AðqÞ A Rrp is the constraint matrix, λ A Rr is the constraint force. Let q ¼ ½qTv qTr T , where qv A Rm represents the position and direction of mobile platform, qr A Rn represents the link position of manipulator, and p ¼ m þ n. Since the nonholonomic characteristics of mobile manipulator is caused by the movement of mobile platform, Eq. (2) can be simplified as follows: Av ðqv Þq_ v ¼ 0
ð3Þ
where Av ðqv Þ A Rmp is the constraint matrix of mobile platform. Therefore, Eq. (1) can be rewritten as " #" # " #" # " # q_ v q€ v M 11 M 12 C 11 C 12 F1 þ þ q_ r q€ r M 21 M 22 C 21 C 22 F2 " # " # " # τd1 Bv ðqv Þτv AT ðq Þλ ¼ v v þ ð4Þ τd2 τr 0 where τv A Rm r is the control torque of mobile platform, τr A Rn is the control torque of manipulator, M11 and M22 represent the inertia matrices of mobile platform and manipulator respectively, M 12 q€ r and M 21 q€ v represent the interaction inertia between the manipulator and mobile platform, and C 12 q_ r and C 21 q_ v also represent the interaction centripetal and Coriolis forces between the two subsystems. 2.1. Subsystem of mobile platform Select a full rank matrix Sðqv Þ ¼ ½s1 ðqv Þ; …; sm r ðqv Þ A Rmðm rÞ to be a basis of null space Av ðqv Þ. Then, we have ST ðqv ÞATv ðqv Þ ¼ 0
ð5Þ
Please cite this article as: Peng J, et al. Robust adaptive tracking control for nonholonomic mobile manipulator with uncertainties. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.012i
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There exists an auxiliary input vector νðtÞ A Rm r , and satisfy q_ v ¼ Sðqv ÞνðtÞ
ð6Þ
From Eq. (4), the dynamic equation of mobile platform subsystem can be expressed as M 11 q€ v þ M 12 q€ r þ C 11 q_ v þC 12 q_ r þF 1 þ τd1 ¼ Bv τv ATv ðqv Þλ
ð7Þ
Introducing Eq. (5), the constraint forces term ATv ðqv Þλ can be eliminated by multiplying both sides of Eq. (7) by ST . Then, substituting Eq. (6) into Eq. (7) yields M 11 ν_ þ C 11 ν þ τ d1 þ f 1 ¼ τ v
ð8Þ
M 11 ¼ ST M 11 S; C 11 ¼ ST M 11 S_ þ ST C 11 S; τ d1 ¼ ST τd1 ; τ v ¼
where
S Bv τv and T
f 1 ¼ ST ðM 12 q€ r þ C 12 q_ r þ F 1 Þ
ð9Þ
Remark 1. Eq. (9) contains the gravity, the static and dynamic friction term F1 of mobile platform subsystem, and the dynamic coupling term ðM 12 q€ r þC 12 q_ r Þ that is caused by the manipulator. 2.2. Subsystem of manipulator From Eq. (4), the dynamic equation of manipulator subsystem can be expressed as M 22 q€ r þ C 22 q_ r þ τd2 þf 2 ¼ τr
ð10Þ
where f 2 ¼ M 21 q€ v þ C 21 q_ v þF 2
ð11Þ
Remark 2. Eq. (11) contains the gravity, the static and dynamic friction term F2 of manipulator subsystem, and the dynamic coupling term ðM 21 q€ v þC 21 q_ v Þ that is caused by the mobile platform. Property 1. The inertia matrices M11 and M22 are symmetric positive definite, and for arbitrary x A Rn satisfy _ 11 2C 11 gx ¼ 0; x T fM
_ 22 2C 22 gx ¼ 0 xT fM
Property 2. The inertia matrices M12 and M21 are positive definite, and M 12 ¼ M T21 ;
J C ij J ¼ C ijb J q_ J ; i; j ¼ 1; 2
where Cijb are positive constants, i,j¼1,2. Assumption 1. Friction terms and disturbances are bounded, that is,
3
3. Design of Lyapunov function for mobile platform 3.1. Design of kinematic controller For the two-wheel driven mobile platform, its kinematic model can be expressed as Eq. (6), that is, 3 2 3 2 x_ cos φ d sin φ 7 v 6 y_ 7 6 d cos φ 5 ð12Þ 4 5 ¼ 4 sin φ ω φ_ 0 1 where qv ¼ ½x y φT represents the current pose of mobile platform, which is shown in Fig. 1, (x,y) is the coordinate of point C in the coordinate system XOY; φ is the direction angle when the mobile platform rotates around the X-axis anticlockwise; d is the distance between the point C of mobile platform and the axis midpoints G of two driven wheels, v and ω represent the linear and angular velocities of mobile platform, respectively. Suppose that the mobile platform is required to follow a reference trajectory, with position and velocity are qvd ¼ ½xr yr φr T
and νr ¼ ½vr ωr T respectively, then the tracking errors can be given as 32 2 3 2 3 xr x e1 cos φ sin φ 0 76 y y 7 6e 7 6 ð13Þ 4 2 5 ¼ 4 sin φ cos φ 0 54 r 5 φr φ e3 0 0 1
Theorem 1. The velocity control law νd is designed as # " k1 e1 þ vr cos e3 v νd ¼ ¼ ωr þ k2 e2 vr þ k3 vr sin e3 ω
ð14Þ
where k1 4 0; k2 4 0; k3 4 0 are controller gains. Then, the velocity control law νd may achieve theoretical stability with respect to the reference trajectory. Select a Lyapunov function as follows: 1 1 cos e3 V 0 ¼ ðe21 þe22 Þ þ 2 k2
ð15Þ
It has been proven that the kinematic system of mobile robot, which consisted of Eqs. (12) and (14), is closed-loop stable [18]. Remark 3. Eq. (14) is only designed specially based on the kinematic model. However, the velocity νd cannot be generated directly by the motors. Instead, the motor provides a control torque to the wheels, which will result in an actual velocity νd . Therefore, it is necessary to design the torques for mobile platform based on the dynamic model. 3.2. Design of Lyapunov function based on dynamics Suppose the desired velocity of mobile platform to be νd , which is given by Eq. (14). Define the velocity tracking errors as follows: z ¼ ν νd
ð16Þ
J F 1 J r ξ0 þ ξ1 J q_ J ;
J τ d1 J r τ D1 ;
where ν is the actual velocity of mobile platform. After differentiating Eq. (16) in time, multiplying by M 11 , and substituting Eq. (8), the following expression can be obtained:
J F 2 J r ξ2 þ ξ3 J q_ J ;
J τ d2 J r τ D2
M 11 z_ þ C 11 z þ f 1 þM 11 ν_ d þ C 11 νd ¼ τ v
ð17Þ
Consider a Lyapunov function as follows: where ξ0 ; ξ1 , ξ2 ; ξ3 , τD1 and τD2 are positive constants.
V 1 ¼ 12 zT M 11 z
Assumption 2. The full rank matrix S is bounded, that is,
Taking the derivative of Eq. (18) with respect to time and substituting Eq. (17) yield
J S J r ζs
_ z ¼ zT ðτ τ f M ν_ þ C ν Þ V_ 1 ¼ zT M 11 z_ þ 12 zT M v 11 11 d 11 d d1 1
where
ζs is a positive constant.
ð18Þ
ð19Þ
where Property 1 was utilized.
Please cite this article as: Peng J, et al. Robust adaptive tracking control for nonholonomic mobile manipulator with uncertainties. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.012i
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4
Substituting Eqs. (29) and (30) into Eq. (28), we can obtain
4. Design of Lyapunov function for manipulator Suppose that qrd ; q_ rd and q€ rd represent the desired link position, velocity and acceleration vectors of manipulator respectively. Then, the link position and velocity tacking errors can be expressed as e ¼ qr qrd ;
e_ ¼ q_ r q_ rd
ð20Þ
Define a filter tracking error as follows: r ¼ e þ ke_
ð21Þ
where k is a positive definite diagonal matrix. After differentiating Eq. (21) in time and multiplying by M22, we can obtain _ ¼ τr þ ðM 22 k C 22 Þðr keÞ þ f 2 þ τd2 M 22 r_ ¼ M 22 ðq€ rd q€ r þ keÞ
ð22Þ
Redefine f2 given by Eq. (11) as f 2 ¼ M 21 q€ v þ C 21 q_ v þ M 22 q€ rd þ C 22 q_ rd þ F 2
ð23Þ
V 2 ¼ 12 r T M 22 r
ð24Þ
Taking the derivative of Eq. (24) with respect to time and substituting Eq. (22) yield _ 22 r ¼ r V_ 2 ¼ r T M 22 r_ þ 12 r T M
τ r þ M 22 kr ðC 22 M22 kÞke þ f 2 þ τd2
zT f 1 zT τd1 þr T f 2 þ r T τd2 þ Ξ
ð31Þ
where d dt þ r T ½ðM 21 SÞðz_ þ ν_ d Þ þ ðM 21 S_ þ C 21 SÞðz þ νd Þ
Ξ ¼ frT MT12 ðSzÞg þ zT ST ðM12 r_ þ ðC 12 M12 Þðr keÞÞ ¼ ðSzÞT ½ C 12 ke M 12 kðr keÞ þ r T ðM T12 Sν_ d þ M T12 S_ νd þ C 21 Sνd Þ
ð32Þ
Substituting Eq. (32) into Eq. (31), we can obtain V_ 3 ¼ zT ðτ v M 11 ν_ d C 11 νd Þ þ r T ½ τr þ M 22 kr ðM 22 k C 22 keÞ þ ðSzÞT ½ C 12 ke M 12 kðr keÞ þ r T ðM T12 Sν_ d þ M T12 S_ νd þ C 21 Sνd Þ zT f 1 zT τd1 þr T f 2 þ r T τd2 ¼ zT fτ v M 11 ν_ d C 11 νd ST ½C 12 ke þ M 12 kðr keÞ f 1 τd1 g
Consider a Lyapunov function as follows:
T
V_ 3 ¼ zT ðτ v M 11 ν_ d C 11 νd Þ þ r T ½ τr þ M 22 kr ðM 22 k C 22 keÞ
ð25Þ where Property 1 was utilized.
þ r T f τr þ M 22 kr ðM 22 k C 22 keÞ þ M T12 Sν_ d þ M T12 S_ νd þ C 21 Sνd þ f 2 þ τd2 g
ð33Þ
For the first item of the right hand side of Eq. (33), introducing f 1 , we can obtain zT fτ v M 11 ν_ d C 11 νd ST ½C 12 ke þ M 12 kðr keÞ f 1 τ d1 g ¼ zT fτ v M 11 ν_ d ST ½M 12 q€ rd þ M12 kðr keÞg þ zT f C 11 νd ST ½C 12 ðke þ q_ rd Þ þ F 1 τd1 g r zT fτ v M 11 ν_ d ST ½M 12 q€ rd þ M 12 kðr keÞg þ J z J fC 11b J q_ J J νd J þ ζ s ðC 12b J q_ J J ke þ q_ rd J þ ξ0 þ ξ1 J q_ J Þ þ τ D1 g
5. Robust adaptive controller for nonholonomic mobile manipulator and stability analysis
¼ zT fτ v M 11 ν_ d ST ½M 12 q€ rd þ M12 kðr keÞg þ J z J Θ1 Ψ 1 T
Because of the couplings between mobile platform subsystem and manipulator subsystem, we have to analyse the stability of whole mobile manipulator. According to the Lyapunov functions of mobile platform and manipulator in the previous sections, a robust adaptive controller of whole mobile manipulator system will be designed and its stability will be also analysed in this section. Suppose the desired trajectory of mobile platform to be ðqvd ; νd ; ν_ d Þ and the desired trajectory of manipulator to be ðqrd ; q_ rd ; q€ rd Þ. For the mobile manipulator system, the Lyapunov function can be given as # T " M 11 M 12 Sz 1 Sz ð26Þ V3 ¼ M 21 M 22 2 r r Eq. (26) can be expanded as
ΘT1 ¼ ½C 11b ; ζ s C 12b ; ζ s ξ1 ; ζ s ξ0 þ τD1
ð35Þ
Ψ T1 ¼ ½jq_ J J νd J ; J q_ J J ke þ q_ rd J ; J q_ J ; 1
ð36Þ
For the second item of the right hand side of Eq. (33), introducing f 2 , we can obtain r T f τ r þ M 22 kr ðM22 k C 22 keÞ: þ M T12 Sν_ d þ M T12 S_ νd þ C 21 Sνd þ f 2 þ τd2 g
¼ r T f τr þ M 22 ½kðr keÞ þ q€ rd þ M T12 Sν_ d þ M T12 S_ νd g þ r T ½C 22 ðke þ q_ rd Þ þ C 21 Sνd þ F 2 þ τd2
r r T f τ r þ M 22 ½kðr keÞ þ q€ rd þ M T12 ðSν_ d þ S_ νd Þg þ J r J ðC 22b J q_ J J ke þ q_ rd J þ C 21b ζ s J q_ J J νd J þ ξ2 þ ξ3 J q_ J þ τ D2 Þ T
¼ V 1 þV 2 r T M T12 ðSzÞ
ð27Þ
Taking the derivative of Eq. (27) with respect to time substituting Eqs. (19) and (25) yield o dn T T r M 12 ðSzÞ ¼ zT ðτ v τ d1 f 1 M 11 ν_ d þC 11 νd Þ V_ 3 ¼ V_ 1 þ V_ 2 dt þ r T τr þ M 22 kr ðC 22 M 22 kÞke þf 2 þ τd2
where Property 2, Assumptions 1 and 2 were utilized, and
¼ r T f τ r þ M 22 ½kðr keÞ þ q€ rd þ M T12 ðSν_ d þ S_ νd Þg þ J r J Θ2 Ψ 2
V 3 ¼ 12 zT ðST M 11 SÞz r T M T12 ðSzÞ þ 12 r T M 22 r
d T T fr M 12 ðSzÞg dt
ð28Þ
ð34Þ
ð37Þ
where Property 2, Assumptions 1 and 2 were utilized, and
ΘT2 ¼ ½C 22b ; ζ s C 21b ; ξ3 ; ξ2 þ τD2
ð38Þ
Ψ T2 ¼ ½jq_ J J ke þ q_ rd J ; J q_ J J νd J ; J q_ J ; 1
ð39Þ
Substituting Eqs. (34) and (37) into Eq. (33) yields V_ 3 rzT fτ v M 11 ν_ d ST ½M 12 q€ rd þ M 12 kðr keÞg þr T f τr þ M 22 ½kðr keÞ þ q€ rd þ M T12 ðSν_ d þ S_ νd Þg þ J z J Θ1 Ψ 1 þ J r J Θ2 Ψ 2 T
T
ð40Þ
Rearrange f1 given by Eq. (9), f 1 ¼ ST ðM12 q€ r þ C 12 q_ r þ F 1 Þ ¼ ST ½M 12 r_ þ ðC 12 M 12 kÞðr keÞ þ f 1
ð29Þ
where f 1 ¼ ST ðM 12 q€ rd þ C 12 q_ rd þ F 1 Þ. In the same way, rearrange f2 given by Eq. (23), f 2 ¼ M 21 q€ v þ C 21 q_ v þ M 22 q€ rd þ C 22 q_ rd þ F 2 ¼ ðM 21 SÞðz_ þ ν_ Þ þ ðM 21 S_ þC 21 SÞðz þ ν Þ þ f d
where f 2 ¼ M 22 q€ rd þC 22 q_ rd þ F 2 .
d
2
ð30Þ
Remark 4. If the upper bounds of model errors and coefficient matrices Θ1 and Θ2, which are shown in Eqs. (35) and (38) respectively, could be known entirely, one can easily design an effective robust controller with sliding dynamics. However, in practice, the entire portion of the upper bounds are usually unknown. If the upper bounds were selected as relatively large values, it will cause chattering too large, even if the system is unstable.
Please cite this article as: Peng J, et al. Robust adaptive tracking control for nonholonomic mobile manipulator with uncertainties. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.012i
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The main objective of this paper is to suppose the upper bounds to be unknown, the following theorem can be given as follows, where the upper bounds will be estimated by using an adaptive algorithm. Theorem 2. The kinematic model of mobile platform is given by Eq. (12) and the dynamic model of the mobile manipulator is given by Eq. (4), if the kinematic controller is designed as Eq. (14) and the robust controller with adaptive mechanism is designed as follows:
5
Substituting Eqs. (51) and (52) into Eq. (50) yields ~ Tμ Θ ~T ^ ^ V_ r zT κ 1 z r T κ 2 r Θ 1 1 1 Θ 2 μ2 Θ 2 Since 1 2ð
Θ~ þ ΘÞT μðΘ~ þ ΘÞ Z 0
ð54Þ
Expanding Eq. (54), we can obtain 1 2
Θ~ μΘ~ þ Θ~ μΘ þ 12 ΘμΘ Z0 T
T
ð55Þ
τ v ¼ τ v0 κ 1 z þ ϕ1
ð41Þ
^ Θ, we get ~ ¼Θ And using Θ
τr ¼ τr0 þ κ 2 r ϕ2
ð42Þ
Θ~ μΘ^ Z 12 ðΘ~ μΘ~ ΘT μΘÞ
where κ1 and κ2 are the controller gains, ϕ1 and ϕ2 are the robust terms, which are used for compensating the parameter uncertainties and external disturbances, and
ð53Þ
T
T
ð56Þ
Substituting Eq. (56) into Eq. (53) yields T ~ Tμ Θ ~ V_ r zT κ 1 z r T κ 2 r 12 ðΘ 1 1 1 Θ1 μ1 Θ1 Þ
τ v0 ¼ M 11 ν_ d ST M12 ½q€ rd þ kðr keÞ
ð43Þ
τr0 ¼ M22 ½kðr keÞ þ q€ rd þ MT12 ðSν_ d þ S_ νd Þ
ð44Þ
T ~ Tμ Θ ~ 12 ðΘ 2 2 2 Θ2 μ2 Θ2 Þ þ δ1 ðΘ1 ; J z J Þ þ δ2 ðΘ2 ; J r J Þ n o T ~ Tμ Θ ~ 1 ~ ~ ¼ zT κ 1 z þr T κ 2 r þ 12 Θ 1 1 1 þ 2 Θ 2 μ2 Θ 2 n o T T þ 12 Θ1 μ1 Θ1 þ Θ2 μ2 Θ2
ð45Þ
r 12 λmin ðQ Þ‖x‖2 þ ε
The robust terms
ϕ1 and ϕ2 can be designed as follows:
T
ϕ1 ¼ Θ^ 1 Ψ 1 signðzÞ T
ϕ2 ¼ Θ^ 2 Ψ 2 signðrÞ
ð46Þ
^ and Θ ^ represent the estimate values of coefficient matrices where Θ 1 2 Θ1 and Θ2 respectively. ^ and Θ ^ can be designed as The adaptive law of Θ 1 2 _T Θ^ 1 ¼ Γ 1 1 ðΨ 1 J z J μ1 Θ^ 1 Þ _
T
Θ^ 2 ¼ Γ 2 1 ðΨ 2 J r J μ2 Θ^ 2 Þ
T
ð49Þ
^ T Ψ signðzÞg þ J z J ΘT Ψ r zT f κ 1 z Θ 1 1 1 1 ^ T Ψ signðrÞg þ J r J ΘT Ψ þ r T tf κ 2 r Θ 2 2 2 2 ð50Þ
Introducing Eq. (47), we can obtain
0
0 0
μ1
0
0
μ2
7 7 7; 7 5
~ ;Θ ~ T ; x ¼ ½Sz; r; Θ 1 2
¼ 12
λmax ðμÞ‖Θ‖
2
T
1 2
λmin ðPÞ‖x‖2 r V r 12 xT Px r 12 λmax ðPÞ‖x‖2
where 2 M 11 6 M 21 6 P¼6 4 0 0
M 12 M 22
0 0
0
Γ1
0
0
T
ð58Þ
3 0 0 7 7 7 0 5
Γ2
From Eqs. (57) and (58), we get
λ ðQ Þ Vðx; tÞ þ ε V_ ðx; tÞ r min λmax ðPÞ r ρVðx; tÞ þ ε
ð59Þ
where ρ ¼ λmin ðQ Þ=λmax ðPÞ. According to the Lyapunov stability theory, the closed-loop system is achieved to be stable. Moreover, for arbitrary xðt 0 Þ, as long as t 4t 0 , we have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2ε J xðtÞ J r ‖xðt 0 Þ‖2 e ρt þ ð1 e ρt Þ ð60Þ λmin ðPÞ λmin ðPÞρ Since S has an upper bound, which is given by Assumption 2, z is also bounded. That is, the tracking errors z and r, the adaptive coefficient ~ and Θ ~ are all bounded. matrix errors Θ 1 2
~ TΓ Θ ~_ ^ T Ψ signðzÞ þ J z J ΘT Ψ þ Θ z Θ 1 1 1 1 1 1 1 T
^ T Ψ þ J z J ΘT Ψ þ Θ ~ T ðΨ J z J μ Θ ^ ¼ JzJΘ 1 1 1Þ 1 1 1 1 1 ð51Þ
In the same way, introducing Eq. (48), we can obtain
6. Simulation examples The mobile manipulator which is shown in Fig. 1 is utilized to demonstrate the effectiveness of the proposed robust adaptive control method. The nonholonomic constraint is given as
~_ ~ TΓ Θ ^ T Ψ signðrÞ þ J r J ΘT Ψ þ Θ r T Θ 2 2 2 2 2 2 2 ~ Tμ Θ ^ r Θ 2 2 2
2κ 2 0
3
where κ 1 ¼ S T κ 1 S 1 , μ ¼ ½μT1 μT2 T ; Θ ¼ ½Θ1 Θ2 T , λmin ðAÞ and λmax ðAÞ represent the minimum and maximum eigenvalue values of matrix A respectively. From Eq. (49), we can obtain
~ TΓ Θ ~ TΓ Θ ~_ þ Θ ~_ V_ ¼ V_ 3 þ Θ 1 2 1 1 2 2
~ Tμ Θ ^ r Θ 1 1 1
0
ð48Þ
~ ¼Θ ^ Θ ; Θ ~ ¼Θ ^ Θ are the estimation errors of where Θ 1 1 1 2 2 2 coefficient matrices. Taking the derivative of Eq. (49) with respect to time substituting Eqs. (41)–(46) yield
~ TΓ Θ ~ TΓ Θ ~_ þ Θ ~_ þΘ 1 2 1 1 2 2
0
ε
Proof. Define a Lyapunov function candidate as follows: T
0
0
ð47Þ
where Γ1 and Γ2 are the positive definite constant diagonal adaptive gain matrices, μ1 and μ2 are the positive definite constant diagonal matrices. Then, the closed-loop system, which is composed of Eqs. (4), (12), (14) and (41)–(48), is stable. Moreover, all the variables of the closedloop system are bounded.
~ Γ Θ ~ ~ 1 ~ V ¼ V 3 þ 12 Θ 1 1 1 þ 2 Θ2 Γ2 Θ2
where 2 2κ 1 6 6 0 Q ¼6 6 0 4
ð57Þ
ð52Þ
_ ¼0 y_ c cos φ x_ c sin φ dφ
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Then, the constraint matrix of nonholonomic system corresponding to Eq. (3) can be obtained as Av ðqv Þ ¼ ½ sin φ cos φ d
x_ r ¼ vr cos φr ;
Select Sðqv Þ as a basis in null space Av ðqv Þ, that is, 2r 3 r r r 2 cos φ þ Rd sin φ 2 cos φ Rd sin φ r r r 6 r 7 Sðqv Þ ¼ 4 2 sin φ Rd cos φ 2 sin φ þ Rd cos φ 5 r Rr R The corresponding parameters of the dynamic model of mobile manipulator, which is shown in Eq. (4), are given as 2 3 2 φ m012 þ 2I w rsin 2Iw sin r2φ cos φ 2m12 d sin φ 2 6 7 2 7 2I sin φ cos φ φ M 11 ¼ 6 m012 þ 2Iw cos 2m12 d cos φ 5; 4 w r2 r2
2m12 d sin φ 2m12 d cos φ " T I 12 0 0 I 12 T M 12 ¼ M 21 ¼ ; M 22 ¼ 0 0 0 0 2 6 C 11 ¼ 6 4
_ sin φ cos φ 2I w φ r2 _ cos 2φ Iw φ r2
_ 2φ Iw φ cos r2
2Iw φ
_ sin φ cos φ r2
_ cos φ m12 dφ _ sin φ m12 dφ T 0 0 0 0 0 C 12 ¼ C T21 ¼ ; C 22 ¼ ; 0 0 0 0 0
F 1 ¼ ½0 0 0T ; 2 cos φ 16 B11 ¼ 4 sin φ r R
0 I2
For the mobile platform subsystem, consider the kinematic controller, the reference trajectory is chosen as qvd ¼ ½xr ; yr ; φr T , then y_ r ¼ vr sin φr ;
φ_ r ¼ ωr
where the reference linear velocity is chosen as vr ¼ 0:5 m=s and the reference angular velocity is chosen as ωr ¼ 0 rad=s. The reference initial position of mobile platform is qvd ð0Þ ¼ ½2; 2; 451T , while the actual initial position is qv ð0Þ ¼ ½3; 1; 01T . The kinematic controller is shown as Eq. (14), where the gains are set to be k1 ¼ 10; k2 ¼ 5; k3 ¼ 4.
mat 33
# ;
_ cos φ m12 dφ
3
7 _ sin φ 7 m12 dφ 5; 0
F 2 ¼ ½0 m2 gl2 sin θ2 T ; cos φ
3
7 sin φ 5 R 2
where m012 ¼ m0 þ m1 þ m2 , m12 ¼ m1 þ m2 , mat 33 ¼ I 012 þ m12 d þ2I w R2 =r 2 , I 012 ¼ I 0 þ I 1 þ I 2 , I 12 ¼ I 1 þ I 2 , m0 is the mass of mobile platform, r is the drive wheel radius of mobile platform; 2R is the distance between the two drive wheels; Iw is the moment inertia of platform which rotates around central axis, I0 is the moment inertia of the drive wheel which rotates around axis, m1 and m2 represent the mass of link 1 and link 2 of manipulator respectively, l1 and l2 represent the length of link 1 and link 2 respectively, I1 and I2 represent the moment inertia of link 1 and link 2 respectively. The mobile manipulator parameters used for the simulation are shown in Table 1, where the nominal values are used to calculate the controller functions τ v0 and τr 0 to design the robust adaptive controller and the actual values are used to test the robustness of the controller. The external disturbances are assumed to be τd1 ¼ ½ sin t 3 cos t 2 sin tT and τ d2 ¼ ½ sin t cos tT . The simulation sampling steps length is given by 0.01 s. Table 1 Simulation parameters. Parameters (unit)
Nominal values
Actual values
m0 ðkgÞ m1 ðkgÞ m2 ðkgÞ l1 ðmÞ l2 ðmÞ r ðmÞ 2R ðmÞ d ðmÞ I w ðkg m2 Þ I 0 ðkg m2 Þ I 1 ðkg m2 Þ I 2 ðkg m2 Þ
40 6 4 1.1 0.8 0.1 0.3 0.3 2 2 1 1
50 8 2 þ 2 sin 0:5t 1.2 1.2 0.1 0.3 0.3 2 2 þ cos 0:5t 1 1 þ sin 0:5t
Fig. 2. Computed torque control for mobile manipulator. (a) Tracking trajectory of mobile platform on the x–y plane. (b) Tracking errors of mobile platform. (c) Tracking trajectory of link 1. (d) Tracking trajectory of link 2. (e) Tracking errors of two-link manipulators.
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Fig. 4. Robust adaptive control for mobile manipulator. (a) Tracking trajectory of mobile platform on the x–y plane. (b) Tracking errors of mobile platform. (c) Tracking trajectory of link 1. (d) Tracking trajectory of link 2. (e) Tracking errors of two-link manipulators.
Fig. 3. Robust control for mobile manipulator. (a) Tracking trajectory of mobile platform on the x–y plane. (b) Tracking errors of mobile platform. (c) Tracking trajectory of link 1. (d) Tracking trajectory of link 2. (e) Tracking errors of two-link manipulators.
For the manipulator subsystem, choosing k ¼ diagf5; 5g in Eq. (21), the desired trajectory of the manipulator is chosen as qrd ¼ ½ sin t cos tT ;
q_ rd ¼ ½ cos t sin tT ; q€ rd ¼ ½ sin t cos tT
The initial position is qr ð0Þ ¼ ½0:5 0:5T rad, the initial velocity is q_ r ð0Þ ¼ ½0 0T rad=s, and the initial acceleration is q€ r ð0Þ ¼ ½0 0T rad=s2 . For the purpose of comparison, the mobile manipulators are firstly controlled by the computed torque control method [17]. The tracking trajectory on x–y plane of mobile platform is shown in Fig. 2(a). The tracking errors in x and y directions of mobile platform are shown in Fig. 2(b). The tracking trajectories of link 1 and link 2 are shown in
Please cite this article as: Peng J, et al. Robust adaptive tracking control for nonholonomic mobile manipulator with uncertainties. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.012i
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Fig. 2(c) and (d) respectively. The tracking errors of manipulator are shown in Fig. 2(e). From Fig. 2, it can be seen that the computed torque control method cannot drive the mobile manipulator to reach the desired positions and steady-state tracking errors exist due to the existence of parameter uncertainties and external disturbances. Secondly, the mobile manipulators are controlled by the robust control method without adaptive mechanism [20]. The tracking trajectory on the x–y plane of mobile platform is shown in Fig. 3(a). The tracking errors in x and y directions of mobile platform are shown in Fig. 3(b). The tracking trajectories of link 1 and link 2 are shown in Fig. 3(c) and (d) respectively. The tracking errors of manipulator are shown in Fig. 3(e). From Fig. 3, it can be seen that the steady-state tracking errors still exist by using the robust control method without adaptive mechanism due to the existence of varying parameter uncertainties and external disturbances. Finally, the proposed robust adaptive control method is utilized to control the mobile manipulator. The controller is given by Eqs. (41)– (48), where the controller gains are designed as κ 1 ¼ diagf50; 50g, κ 2 ¼ diagf150; 150g to guarantee the tracking errors to be convergent to zero quickly, the adaptive gains are designed as Γ 1 ¼ Γ 2 ¼ diagf5; 5; 5; 5g to guarantee the adaptive errors to be convergent to ^ and Θ ^ are given as Θ ^ ð0Þ ¼ zero quickly, the initial values of Θ 1 2 1 T ^ Θ 2 ð0Þ ¼ ½0 0 0 0 , the constants are given as μ1 ¼ μ2 ¼ diagf0:5; 0:5; 0:5; 0:5g. The tracking trajectory on the x–y plane of mobile platform is shown in Fig. 4(a). The tracking errors in x and y directions of mobile platform are shown in Fig. 4(b). The tracking trajectories of link 1 and link 2 are shown in Fig. 4(c) and (d). The tracking errors of manipulator are shown in Fig. 4(e). From Fig. 4, the tracking errors significantly decrease using the proposed robust adaptive controller in comparison to the computed torque control method and robust control method without adaptive mechanism. The proposed robust adaptive controller can achieve a better performance in the presence of these varying parameter uncertainties and disturbances. Fig. 5 shows the con^ and Θ ^ , it can vergence procedure of the adaptive coefficients Θ 1 2 be seen that the adaptive coefficients are convergent.
^ 1 ; (b) Θ ^ 2. Fig. 5. Adaptive coefficients. (a) Θ
7. Conclusions This paper presents a robust adaptive controller for mobile manipulator under the system parameter uncertainties and external disturbances, where the mobile manipulator is divided into nonholonomic mobile platform subsystem and holonomic manipulators subsystem. By considering the kinematic controller of mobile platform, Lyapunov functions of the two subsystems are designed, the coupling between the two subsystems is regarded as disturbances. Then, aiming at the unknown upper bound of system parameters, corresponding robust adaptive tracking controller is proposed. The proposed control method demonstrated robust and effective control performance on mobile manipulator having uncertainties with good disturbance rejection. As a future work, the proposed adaptive robust controller will be proven experimentally and will be also applied to other kinds of electromechanical systems.
Acknowledgement The authors would like to acknowledge the funding received from the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20124101120001), China Postdoctoral Science Foundation (No. 2013M541992), Henan Provincial Postdoctoral Science Foundation (No. 2013073), National Natural Science Foundation of China (Nos. 61104022 and 61374128) and Key Project for Science and Technology of the Education Department of Henan Province (No. 14A413009) to conduct this research investigation. References [1] Tao G. On robust adaptive control of robot manipulators. Automatica 1992;28(4): 803–7. [2] Slotine JJE, Li W. Composite adaptive control of robot manipulators. Automatica 1989;25(4):509–19. [3] Whitcomb LL, Rizzi AA, Koditschek DE. Comparative experiments with new adaptive controller for robot arms. IEEE Trans Robot Autom 1993;9(1):59–70. [4] Khoshnood AM, Moradi HM. Robust adaptive vibration control of a flexible structure. ISA Trans 2014 〈http://dx.doi.org/10.1016/j.isatra.2014.03.004i〉. [5] Slotine JJE, Li WP. Adaptive manipulator control: a case study. IEEE Trans Autom Control 1998;33(11):995–1003. [6] Su CY, Leung TP. A sliding mode controller with bound estimation for robot manipulators. IEEE Trans Robot Autom 1993;9(2):208–14. [7] Gueaieb W, Karray R, Sharhan SA. A robust adaptive fuzzy position/force control scheme for cooperative manipulators. IEEE Trans Control Syst Technol 2003;11(4):516–28. [8] González-Vázquez S, Moreno-Valenzuela J. Time-scale separation of a class of robust PD-type tracking controllers for robot manipulators. ISA Trans 2013;52:418–28. [9] Tomei P. Robust adaptive control of robots with arbitrary transient performance and disturbance attenuation. IEEE Trans Autom Control 1999;44(3): 654–8. [10] Wang YN, Peng JZ, Sun W, Yu HS, Zhang H. Robust adaptive tracking control of robotic system with uncertainties. J Control Theory Appl 2008;6(3):281–6. [11] Song ZS, Yi JQ, Zhao DB. Survey of the control for mobile manipulators. Robot 2003;25(5):465–9. [12] Seraji H. Motion control of mobile manipulators. In: Proceedings of the IEEE/RSJ international conference on intelligent robots and systems, 1993. p. 2056–63. [13] Tan JD, Xi N. Unified model approach for planning and control of mobile manipulators. In: Proceedings of the international conference on robotics and automation, 2001. p. 3145–52. [14] Tan JD, Xi N, Wang Y. Hybrid force/position control of redundant mobile manipulators. In: The 15th IFAC triennial world congress, 2002. [15] Yamamoto Y, Yun XP. Effect of the dynamic interaction on coordinated control of mobile manipulators. IEEE Trans Robot Autom 1996;12(5):816–24. [16] Dong W, Xu W. On robust control of mobile manipulators. Control Theory Appl 2002;19(3):345–8. [17] Evangelos P, John P. Planning and model-based control for mobile manipulators. In: Proceedings of the IEEE/RSJ international conference on intelligent robots and systems, 2000. p. 1810–5. [18] Liu K, Lewis FL. Decentralized continuous robust controller for mobile robots. In: Proceedings of the IEEE international conference on robotics and automation, 1990. p. 1822–7.
Please cite this article as: Peng J, et al. Robust adaptive tracking control for nonholonomic mobile manipulator with uncertainties. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.012i
J. Peng et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ [19] Lin S, Goldenberg AA. Neural-network control of mobile manipulators. IEEE Trans Neural Netw 2001;12(5):1121–33. [20] Lin S, Goldenberg AA. Robust damping control of mobile manipulators. IEEE Trans Syst Man Cybern Part B 2002;32(1):126–32. [21] Wang HQ, Wang QL. Robust and adaptive fuzzy control for mobile manipulator. Control Decision 2010;25(3):461–5.
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[22] Andaluz V, Roberti F, Toibero JM, Carelli R. Adaptive unified motion control of mobile manipulators. Control Eng Pract 2012;20(12):1337–52. [23] Shojaei K, Shahri AM. Output feedback tracking control of uncertain nonholonomic wheeled mobile robots: a dynamic surface control approach. IET Control Theory Appl 2012;6(2):216–28.
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Robust adaptive tracking control for nonholonomic mobile manipulator with uncertainties Jinzhu Peng n, Jie Yu, Jie Wang School of Electrical Engineering, Zhengzhou University, Zhengzhou, Henan 450001, China
art ic l e i nf o
a b s t r a c t
Article history: Received 6 December 2013 Received in revised form 10 May 2014 Accepted 15 May 2014 This paper was recommended for publication by Rickey Dubay
In this paper, mobile manipulator is divided into two subsystems, that is, nonholonomic mobile platform subsystem and holonomic manipulator subsystem. First, the kinematic controller of the mobile platform is derived to obtain a desired velocity. Second, regarding the coupling between the two subsystems as disturbances, Lyapunov functions of the two subsystems are designed respectively. Third, a robust adaptive tracking controller is proposed to deal with the unknown upper bounds of parameter uncertainties and disturbances. According to the Lyapunov stability theory, the derived robust adaptive controller guarantees global stability of the closed-loop system, and the tracking errors and adaptive coefficient errors are all bounded. Finally, simulation results show that the proposed robust adaptive tracking controller for nonholonomic mobile manipulator is effective and has good tracking capacity. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Mobile manipulator Robust adaptive control Tracking control Nonholonomic system
1. Introduction Tracking control for multi-joint robotic manipulators and mobile robots always is a challenging problem and has been given a lot of attention in the control field. Many powerful methodologies have been applied on robotic manipulators or mobile robots to achieve good tracking performances. Robust adaptive control method combines the advantages of adaptive control and robust control methods [1–4], which have been widely used to control the robotic manipulators and mobile robots or other electromechanical systems. Slotine and Li [5] introduced a robust fixed gain based on the basic of adaptive control for controlling the manipulator, which enhanced the robustness of passive structure adaptive controller. Su and Leung [6] introduced an estimation algorithm of unknown parameters' upper bound based on the robust control structure, which can effectively reduce the robust gain conservatism. However, only parameter uncertainties were considered in the above works. By combining robust adaptive control and fuzzy logic control, Gueaieb et al. [7] proposed a decentralized robust adaptive fuzzy control strategy, which was specially for the parametric and nonparametric uncertainties, and the stability of system was proven by using the Lyapunov stability theory. González-Vázquez et al. [8] introduced a class of PD-type robust controllers for robotic manipulators by using the theory of singularly perturbed systems.
n
Corresponding author. Tel./fax: þ 86 371 67783113. E-mail address: [email protected] (J. Peng).
Tomei [9] proposed a robust adaptive controller, which can maintain a high tracking accuracy and adjust system transient quality discretionarily under the circumstance that the unknown parameters of the system and the external interference coexist. However, the upper bound of system parameters should be known in the above works. Aiming at the system uncertainties, Wang et al. [10] proposed a robust adaptive tracking control for robotic manipulator, where the upper bound of system parameters was assumed to be unknown. However, the above works mainly focused on controlling for robotic manipulators or mobile robots only. A mobile manipulator is a robotic manipulator mounted on a mobile platform, with the function of mobile and operation. It not only possesses the flexible function of manipulator's operation but also has mobile robot's extensity in workspace. The mobile manipulator shows its nonholonomic characteristics since the mobile platform is a typical nonholonomic system. In general, the tracking control methods of mobile manipulator are mainly divided into two categories: one is called centralized control strategy, the other is called decentralized control strategy [11]. In the centralized control strategy, the mobile platform and robotic manipulator are regarded as a whole. Seraji [12] established a unified dynamic model of mobile manipulators, the idea of configuration control was proposed. However, the control method that was based on the kinematics was difficult to implement in practice. Tan and Xi [13,14] established a dynamic model of mobile manipulators, and a hybrid force/position control method was proposed. In [15], the dynamic coupling was compensated by linearizing the dynamic model of mobile manipulators, however,
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the decoupling matrix was required to be full rank, that is, the initial states of system were restricted. Consider the dynamic coupling, Dong [16] designed a robust controller which is based on the Lyapunov theory to achieve system stable, however, the system structure was complicated due to the regression matrix was used. In the decentralized control strategy, the mobile manipulator is divided into nonholonomic mobile platform subsystem and holonomic manipulator subsystem. Evangelos and John [17] used computed torque control theory to implement the tracking control of mobile manipulator, which is subject to external disturbances. Liu and Lewis [18] took manipulator and mobile platform as two independent systems, the controllers for each system were designed, and the system dynamic couplings were regarded as external disturbances. Lin and Goldenberg [19] designed tracking controllers for mobile manipulator based on neural networks, which were used to estimate the system dynamic coupling and uncertainties online. Whereafter, they proposed a robust damping control algorithm as well [20], the control algorithm only needs a few control parameters. However, the above investigations cannot handle the varying parameter uncertainties well, because the adaptive parameters identification is generally not included in these methods. Wang and Wang [21] established the dynamic model of nonholonomic mobile manipulator based on Screw theory and designed a robust adaptive fuzzy controller of mobile manipulator. Andaluz et al. [22] proposed a kinematic controller of mobile manipulator with uncertainties. The controller was able to solve the problem, not only the point stabilization and the trajectory tracking, but also the path following. Shojaei et al. [23] proposed an input–output model of mobile manipulator, where a tracking controller was designed by using the dynamic surface control technique. Then, according to adaptive robust technique, the influence of parametric and nonparametric about the uncertainties in the mobile manipulators model was compensated. However, the upper bound of system parameters should be known from the above works. In this paper, the decentralized control strategy is studied. The mobile manipulator is divided into nonholonomic mobile platform subsystem and holonomic manipulator subsystem. The kinematic controller of the mobile platform is derived. And considering that the dynamic model, Lyapunov functions of the two subsystems are derived, the couplings between the two subsystems are regarded as external disturbances. Then, the dynamic parameters and external disturbances are assumed to be unknown, a robust adaptive controller is proposed to achieve the closed-loop system stability. The proposed controller can eliminate interference of dynamic uncertainties and external disturbances as well. Simulation results show that the proposed robust adaptive controller is effective for controlling the mobile manipulator. The rest of this paper is organized as follows. In Section 2, the mathematic models of nonholonomic mobile platform and holonomic manipulator are addressed respectively. In Section 3, the Lyapunov function for mobile platform is designed, which consists of the design of kinematic controller and Lyapunov function based on dynamic model. In Section 4, the design of Lyapunov function for manipulator is drawn. The design of robust adaptive control scheme is given in Section 5, as well as the robust stability is analysed. The simulation results are given in Section 6, and the conclusions are drawn in Section 7.
2. Model of nonholonomic mobile manipulator A mobile manipulator system is depicted in Fig. 1, where a twolink manipulator is mounted on the centre C of a mobile platform. The two rear wheels of mobile platform are driven independently, and the two links of manipulator are also driven by motors
Fig. 1. System of a mobile manipulator.
independently. In addition, the manipulator is generally considered to be a holonomic system, while the mobile platform is subject to nonholonomic constraint, the mechanical system of mobile manipulator can be expressed as _ q_ þ Fðq; qÞ _ þ τd ¼ BðqÞτ AT ðqÞλ MðqÞq€ þ Cðq; qÞ
ð1Þ
The nonholonomic constraint can be expressed as AðqÞq_ ¼ 0
ð2Þ
_ q€ A Rp are the state vectors of the mobile manipulator where q; q; system, representing position, velocity and acceleration vectors respectively, MðqÞ A Rpp is the symmetric, positive definite inertia _ A Rpp represents the vector of centripetal and matrix, Cðq; qÞ _ A Rpp represents the gravity and Coriolis forces term; Fðq; qÞ p friction term, τd A R is the vector of unknown bounded external disturbances, BðqÞ A Rpðp rÞ is the input transformation matrix, τ A Rp r is the input torque, AðqÞ A Rrp is the constraint matrix, λ A Rr is the constraint force. Let q ¼ ½qTv qTr T , where qv A Rm represents the position and direction of mobile platform, qr A Rn represents the link position of manipulator, and p ¼ m þ n. Since the nonholonomic characteristics of mobile manipulator is caused by the movement of mobile platform, Eq. (2) can be simplified as follows: Av ðqv Þq_ v ¼ 0
ð3Þ
where Av ðqv Þ A Rmp is the constraint matrix of mobile platform. Therefore, Eq. (1) can be rewritten as " #" # " #" # " # q_ v q€ v M 11 M 12 C 11 C 12 F1 þ þ q_ r q€ r M 21 M 22 C 21 C 22 F2 " # " # " # τd1 Bv ðqv Þτv AT ðq Þλ ¼ v v þ ð4Þ τd2 τr 0 where τv A Rm r is the control torque of mobile platform, τr A Rn is the control torque of manipulator, M11 and M22 represent the inertia matrices of mobile platform and manipulator respectively, M 12 q€ r and M 21 q€ v represent the interaction inertia between the manipulator and mobile platform, and C 12 q_ r and C 21 q_ v also represent the interaction centripetal and Coriolis forces between the two subsystems. 2.1. Subsystem of mobile platform Select a full rank matrix Sðqv Þ ¼ ½s1 ðqv Þ; …; sm r ðqv Þ A Rmðm rÞ to be a basis of null space Av ðqv Þ. Then, we have ST ðqv ÞATv ðqv Þ ¼ 0
ð5Þ
Please cite this article as: Peng J, et al. Robust adaptive tracking control for nonholonomic mobile manipulator with uncertainties. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.012i
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There exists an auxiliary input vector νðtÞ A Rm r , and satisfy q_ v ¼ Sðqv ÞνðtÞ
ð6Þ
From Eq. (4), the dynamic equation of mobile platform subsystem can be expressed as M 11 q€ v þ M 12 q€ r þ C 11 q_ v þC 12 q_ r þF 1 þ τd1 ¼ Bv τv ATv ðqv Þλ
ð7Þ
Introducing Eq. (5), the constraint forces term ATv ðqv Þλ can be eliminated by multiplying both sides of Eq. (7) by ST . Then, substituting Eq. (6) into Eq. (7) yields M 11 ν_ þ C 11 ν þ τ d1 þ f 1 ¼ τ v
ð8Þ
M 11 ¼ ST M 11 S; C 11 ¼ ST M 11 S_ þ ST C 11 S; τ d1 ¼ ST τd1 ; τ v ¼
where
S Bv τv and T
f 1 ¼ ST ðM 12 q€ r þ C 12 q_ r þ F 1 Þ
ð9Þ
Remark 1. Eq. (9) contains the gravity, the static and dynamic friction term F1 of mobile platform subsystem, and the dynamic coupling term ðM 12 q€ r þC 12 q_ r Þ that is caused by the manipulator. 2.2. Subsystem of manipulator From Eq. (4), the dynamic equation of manipulator subsystem can be expressed as M 22 q€ r þ C 22 q_ r þ τd2 þf 2 ¼ τr
ð10Þ
where f 2 ¼ M 21 q€ v þ C 21 q_ v þF 2
ð11Þ
Remark 2. Eq. (11) contains the gravity, the static and dynamic friction term F2 of manipulator subsystem, and the dynamic coupling term ðM 21 q€ v þC 21 q_ v Þ that is caused by the mobile platform. Property 1. The inertia matrices M11 and M22 are symmetric positive definite, and for arbitrary x A Rn satisfy _ 11 2C 11 gx ¼ 0; x T fM
_ 22 2C 22 gx ¼ 0 xT fM
Property 2. The inertia matrices M12 and M21 are positive definite, and M 12 ¼ M T21 ;
J C ij J ¼ C ijb J q_ J ; i; j ¼ 1; 2
where Cijb are positive constants, i,j¼1,2. Assumption 1. Friction terms and disturbances are bounded, that is,
3
3. Design of Lyapunov function for mobile platform 3.1. Design of kinematic controller For the two-wheel driven mobile platform, its kinematic model can be expressed as Eq. (6), that is, 3 2 3 2 x_ cos φ d sin φ 7 v 6 y_ 7 6 d cos φ 5 ð12Þ 4 5 ¼ 4 sin φ ω φ_ 0 1 where qv ¼ ½x y φT represents the current pose of mobile platform, which is shown in Fig. 1, (x,y) is the coordinate of point C in the coordinate system XOY; φ is the direction angle when the mobile platform rotates around the X-axis anticlockwise; d is the distance between the point C of mobile platform and the axis midpoints G of two driven wheels, v and ω represent the linear and angular velocities of mobile platform, respectively. Suppose that the mobile platform is required to follow a reference trajectory, with position and velocity are qvd ¼ ½xr yr φr T
and νr ¼ ½vr ωr T respectively, then the tracking errors can be given as 32 2 3 2 3 xr x e1 cos φ sin φ 0 76 y y 7 6e 7 6 ð13Þ 4 2 5 ¼ 4 sin φ cos φ 0 54 r 5 φr φ e3 0 0 1
Theorem 1. The velocity control law νd is designed as # " k1 e1 þ vr cos e3 v νd ¼ ¼ ωr þ k2 e2 vr þ k3 vr sin e3 ω
ð14Þ
where k1 4 0; k2 4 0; k3 4 0 are controller gains. Then, the velocity control law νd may achieve theoretical stability with respect to the reference trajectory. Select a Lyapunov function as follows: 1 1 cos e3 V 0 ¼ ðe21 þe22 Þ þ 2 k2
ð15Þ
It has been proven that the kinematic system of mobile robot, which consisted of Eqs. (12) and (14), is closed-loop stable [18]. Remark 3. Eq. (14) is only designed specially based on the kinematic model. However, the velocity νd cannot be generated directly by the motors. Instead, the motor provides a control torque to the wheels, which will result in an actual velocity νd . Therefore, it is necessary to design the torques for mobile platform based on the dynamic model. 3.2. Design of Lyapunov function based on dynamics Suppose the desired velocity of mobile platform to be νd , which is given by Eq. (14). Define the velocity tracking errors as follows: z ¼ ν νd
ð16Þ
J F 1 J r ξ0 þ ξ1 J q_ J ;
J τ d1 J r τ D1 ;
where ν is the actual velocity of mobile platform. After differentiating Eq. (16) in time, multiplying by M 11 , and substituting Eq. (8), the following expression can be obtained:
J F 2 J r ξ2 þ ξ3 J q_ J ;
J τ d2 J r τ D2
M 11 z_ þ C 11 z þ f 1 þM 11 ν_ d þ C 11 νd ¼ τ v
ð17Þ
Consider a Lyapunov function as follows: where ξ0 ; ξ1 , ξ2 ; ξ3 , τD1 and τD2 are positive constants.
V 1 ¼ 12 zT M 11 z
Assumption 2. The full rank matrix S is bounded, that is,
Taking the derivative of Eq. (18) with respect to time and substituting Eq. (17) yield
J S J r ζs
_ z ¼ zT ðτ τ f M ν_ þ C ν Þ V_ 1 ¼ zT M 11 z_ þ 12 zT M v 11 11 d 11 d d1 1
where
ζs is a positive constant.
ð18Þ
ð19Þ
where Property 1 was utilized.
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4
Substituting Eqs. (29) and (30) into Eq. (28), we can obtain
4. Design of Lyapunov function for manipulator Suppose that qrd ; q_ rd and q€ rd represent the desired link position, velocity and acceleration vectors of manipulator respectively. Then, the link position and velocity tacking errors can be expressed as e ¼ qr qrd ;
e_ ¼ q_ r q_ rd
ð20Þ
Define a filter tracking error as follows: r ¼ e þ ke_
ð21Þ
where k is a positive definite diagonal matrix. After differentiating Eq. (21) in time and multiplying by M22, we can obtain _ ¼ τr þ ðM 22 k C 22 Þðr keÞ þ f 2 þ τd2 M 22 r_ ¼ M 22 ðq€ rd q€ r þ keÞ
ð22Þ
Redefine f2 given by Eq. (11) as f 2 ¼ M 21 q€ v þ C 21 q_ v þ M 22 q€ rd þ C 22 q_ rd þ F 2
ð23Þ
V 2 ¼ 12 r T M 22 r
ð24Þ
Taking the derivative of Eq. (24) with respect to time and substituting Eq. (22) yield _ 22 r ¼ r V_ 2 ¼ r T M 22 r_ þ 12 r T M
τ r þ M 22 kr ðC 22 M22 kÞke þ f 2 þ τd2
zT f 1 zT τd1 þr T f 2 þ r T τd2 þ Ξ
ð31Þ
where d dt þ r T ½ðM 21 SÞðz_ þ ν_ d Þ þ ðM 21 S_ þ C 21 SÞðz þ νd Þ
Ξ ¼ frT MT12 ðSzÞg þ zT ST ðM12 r_ þ ðC 12 M12 Þðr keÞÞ ¼ ðSzÞT ½ C 12 ke M 12 kðr keÞ þ r T ðM T12 Sν_ d þ M T12 S_ νd þ C 21 Sνd Þ
ð32Þ
Substituting Eq. (32) into Eq. (31), we can obtain V_ 3 ¼ zT ðτ v M 11 ν_ d C 11 νd Þ þ r T ½ τr þ M 22 kr ðM 22 k C 22 keÞ þ ðSzÞT ½ C 12 ke M 12 kðr keÞ þ r T ðM T12 Sν_ d þ M T12 S_ νd þ C 21 Sνd Þ zT f 1 zT τd1 þr T f 2 þ r T τd2 ¼ zT fτ v M 11 ν_ d C 11 νd ST ½C 12 ke þ M 12 kðr keÞ f 1 τd1 g
Consider a Lyapunov function as follows:
T
V_ 3 ¼ zT ðτ v M 11 ν_ d C 11 νd Þ þ r T ½ τr þ M 22 kr ðM 22 k C 22 keÞ
ð25Þ where Property 1 was utilized.
þ r T f τr þ M 22 kr ðM 22 k C 22 keÞ þ M T12 Sν_ d þ M T12 S_ νd þ C 21 Sνd þ f 2 þ τd2 g
ð33Þ
For the first item of the right hand side of Eq. (33), introducing f 1 , we can obtain zT fτ v M 11 ν_ d C 11 νd ST ½C 12 ke þ M 12 kðr keÞ f 1 τ d1 g ¼ zT fτ v M 11 ν_ d ST ½M 12 q€ rd þ M12 kðr keÞg þ zT f C 11 νd ST ½C 12 ðke þ q_ rd Þ þ F 1 τd1 g r zT fτ v M 11 ν_ d ST ½M 12 q€ rd þ M 12 kðr keÞg þ J z J fC 11b J q_ J J νd J þ ζ s ðC 12b J q_ J J ke þ q_ rd J þ ξ0 þ ξ1 J q_ J Þ þ τ D1 g
5. Robust adaptive controller for nonholonomic mobile manipulator and stability analysis
¼ zT fτ v M 11 ν_ d ST ½M 12 q€ rd þ M12 kðr keÞg þ J z J Θ1 Ψ 1 T
Because of the couplings between mobile platform subsystem and manipulator subsystem, we have to analyse the stability of whole mobile manipulator. According to the Lyapunov functions of mobile platform and manipulator in the previous sections, a robust adaptive controller of whole mobile manipulator system will be designed and its stability will be also analysed in this section. Suppose the desired trajectory of mobile platform to be ðqvd ; νd ; ν_ d Þ and the desired trajectory of manipulator to be ðqrd ; q_ rd ; q€ rd Þ. For the mobile manipulator system, the Lyapunov function can be given as # T " M 11 M 12 Sz 1 Sz ð26Þ V3 ¼ M 21 M 22 2 r r Eq. (26) can be expanded as
ΘT1 ¼ ½C 11b ; ζ s C 12b ; ζ s ξ1 ; ζ s ξ0 þ τD1
ð35Þ
Ψ T1 ¼ ½jq_ J J νd J ; J q_ J J ke þ q_ rd J ; J q_ J ; 1
ð36Þ
For the second item of the right hand side of Eq. (33), introducing f 2 , we can obtain r T f τ r þ M 22 kr ðM22 k C 22 keÞ: þ M T12 Sν_ d þ M T12 S_ νd þ C 21 Sνd þ f 2 þ τd2 g
¼ r T f τr þ M 22 ½kðr keÞ þ q€ rd þ M T12 Sν_ d þ M T12 S_ νd g þ r T ½C 22 ðke þ q_ rd Þ þ C 21 Sνd þ F 2 þ τd2
r r T f τ r þ M 22 ½kðr keÞ þ q€ rd þ M T12 ðSν_ d þ S_ νd Þg þ J r J ðC 22b J q_ J J ke þ q_ rd J þ C 21b ζ s J q_ J J νd J þ ξ2 þ ξ3 J q_ J þ τ D2 Þ T
¼ V 1 þV 2 r T M T12 ðSzÞ
ð27Þ
Taking the derivative of Eq. (27) with respect to time substituting Eqs. (19) and (25) yield o dn T T r M 12 ðSzÞ ¼ zT ðτ v τ d1 f 1 M 11 ν_ d þC 11 νd Þ V_ 3 ¼ V_ 1 þ V_ 2 dt þ r T τr þ M 22 kr ðC 22 M 22 kÞke þf 2 þ τd2
where Property 2, Assumptions 1 and 2 were utilized, and
¼ r T f τ r þ M 22 ½kðr keÞ þ q€ rd þ M T12 ðSν_ d þ S_ νd Þg þ J r J Θ2 Ψ 2
V 3 ¼ 12 zT ðST M 11 SÞz r T M T12 ðSzÞ þ 12 r T M 22 r
d T T fr M 12 ðSzÞg dt
ð28Þ
ð34Þ
ð37Þ
where Property 2, Assumptions 1 and 2 were utilized, and
ΘT2 ¼ ½C 22b ; ζ s C 21b ; ξ3 ; ξ2 þ τD2
ð38Þ
Ψ T2 ¼ ½jq_ J J ke þ q_ rd J ; J q_ J J νd J ; J q_ J ; 1
ð39Þ
Substituting Eqs. (34) and (37) into Eq. (33) yields V_ 3 rzT fτ v M 11 ν_ d ST ½M 12 q€ rd þ M 12 kðr keÞg þr T f τr þ M 22 ½kðr keÞ þ q€ rd þ M T12 ðSν_ d þ S_ νd Þg þ J z J Θ1 Ψ 1 þ J r J Θ2 Ψ 2 T
T
ð40Þ
Rearrange f1 given by Eq. (9), f 1 ¼ ST ðM12 q€ r þ C 12 q_ r þ F 1 Þ ¼ ST ½M 12 r_ þ ðC 12 M 12 kÞðr keÞ þ f 1
ð29Þ
where f 1 ¼ ST ðM 12 q€ rd þ C 12 q_ rd þ F 1 Þ. In the same way, rearrange f2 given by Eq. (23), f 2 ¼ M 21 q€ v þ C 21 q_ v þ M 22 q€ rd þ C 22 q_ rd þ F 2 ¼ ðM 21 SÞðz_ þ ν_ Þ þ ðM 21 S_ þC 21 SÞðz þ ν Þ þ f d
where f 2 ¼ M 22 q€ rd þC 22 q_ rd þ F 2 .
d
2
ð30Þ
Remark 4. If the upper bounds of model errors and coefficient matrices Θ1 and Θ2, which are shown in Eqs. (35) and (38) respectively, could be known entirely, one can easily design an effective robust controller with sliding dynamics. However, in practice, the entire portion of the upper bounds are usually unknown. If the upper bounds were selected as relatively large values, it will cause chattering too large, even if the system is unstable.
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The main objective of this paper is to suppose the upper bounds to be unknown, the following theorem can be given as follows, where the upper bounds will be estimated by using an adaptive algorithm. Theorem 2. The kinematic model of mobile platform is given by Eq. (12) and the dynamic model of the mobile manipulator is given by Eq. (4), if the kinematic controller is designed as Eq. (14) and the robust controller with adaptive mechanism is designed as follows:
5
Substituting Eqs. (51) and (52) into Eq. (50) yields ~ Tμ Θ ~T ^ ^ V_ r zT κ 1 z r T κ 2 r Θ 1 1 1 Θ 2 μ2 Θ 2 Since 1 2ð
Θ~ þ ΘÞT μðΘ~ þ ΘÞ Z 0
ð54Þ
Expanding Eq. (54), we can obtain 1 2
Θ~ μΘ~ þ Θ~ μΘ þ 12 ΘμΘ Z0 T
T
ð55Þ
τ v ¼ τ v0 κ 1 z þ ϕ1
ð41Þ
^ Θ, we get ~ ¼Θ And using Θ
τr ¼ τr0 þ κ 2 r ϕ2
ð42Þ
Θ~ μΘ^ Z 12 ðΘ~ μΘ~ ΘT μΘÞ
where κ1 and κ2 are the controller gains, ϕ1 and ϕ2 are the robust terms, which are used for compensating the parameter uncertainties and external disturbances, and
ð53Þ
T
T
ð56Þ
Substituting Eq. (56) into Eq. (53) yields T ~ Tμ Θ ~ V_ r zT κ 1 z r T κ 2 r 12 ðΘ 1 1 1 Θ1 μ1 Θ1 Þ
τ v0 ¼ M 11 ν_ d ST M12 ½q€ rd þ kðr keÞ
ð43Þ
τr0 ¼ M22 ½kðr keÞ þ q€ rd þ MT12 ðSν_ d þ S_ νd Þ
ð44Þ
T ~ Tμ Θ ~ 12 ðΘ 2 2 2 Θ2 μ2 Θ2 Þ þ δ1 ðΘ1 ; J z J Þ þ δ2 ðΘ2 ; J r J Þ n o T ~ Tμ Θ ~ 1 ~ ~ ¼ zT κ 1 z þr T κ 2 r þ 12 Θ 1 1 1 þ 2 Θ 2 μ2 Θ 2 n o T T þ 12 Θ1 μ1 Θ1 þ Θ2 μ2 Θ2
ð45Þ
r 12 λmin ðQ Þ‖x‖2 þ ε
The robust terms
ϕ1 and ϕ2 can be designed as follows:
T
ϕ1 ¼ Θ^ 1 Ψ 1 signðzÞ T
ϕ2 ¼ Θ^ 2 Ψ 2 signðrÞ
ð46Þ
^ and Θ ^ represent the estimate values of coefficient matrices where Θ 1 2 Θ1 and Θ2 respectively. ^ and Θ ^ can be designed as The adaptive law of Θ 1 2 _T Θ^ 1 ¼ Γ 1 1 ðΨ 1 J z J μ1 Θ^ 1 Þ _
T
Θ^ 2 ¼ Γ 2 1 ðΨ 2 J r J μ2 Θ^ 2 Þ
T
ð49Þ
^ T Ψ signðzÞg þ J z J ΘT Ψ r zT f κ 1 z Θ 1 1 1 1 ^ T Ψ signðrÞg þ J r J ΘT Ψ þ r T tf κ 2 r Θ 2 2 2 2 ð50Þ
Introducing Eq. (47), we can obtain
0
0 0
μ1
0
0
μ2
7 7 7; 7 5
~ ;Θ ~ T ; x ¼ ½Sz; r; Θ 1 2
¼ 12
λmax ðμÞ‖Θ‖
2
T
1 2
λmin ðPÞ‖x‖2 r V r 12 xT Px r 12 λmax ðPÞ‖x‖2
where 2 M 11 6 M 21 6 P¼6 4 0 0
M 12 M 22
0 0
0
Γ1
0
0
T
ð58Þ
3 0 0 7 7 7 0 5
Γ2
From Eqs. (57) and (58), we get
λ ðQ Þ Vðx; tÞ þ ε V_ ðx; tÞ r min λmax ðPÞ r ρVðx; tÞ þ ε
ð59Þ
where ρ ¼ λmin ðQ Þ=λmax ðPÞ. According to the Lyapunov stability theory, the closed-loop system is achieved to be stable. Moreover, for arbitrary xðt 0 Þ, as long as t 4t 0 , we have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2ε J xðtÞ J r ‖xðt 0 Þ‖2 e ρt þ ð1 e ρt Þ ð60Þ λmin ðPÞ λmin ðPÞρ Since S has an upper bound, which is given by Assumption 2, z is also bounded. That is, the tracking errors z and r, the adaptive coefficient ~ and Θ ~ are all bounded. matrix errors Θ 1 2
~ TΓ Θ ~_ ^ T Ψ signðzÞ þ J z J ΘT Ψ þ Θ z Θ 1 1 1 1 1 1 1 T
^ T Ψ þ J z J ΘT Ψ þ Θ ~ T ðΨ J z J μ Θ ^ ¼ JzJΘ 1 1 1Þ 1 1 1 1 1 ð51Þ
In the same way, introducing Eq. (48), we can obtain
6. Simulation examples The mobile manipulator which is shown in Fig. 1 is utilized to demonstrate the effectiveness of the proposed robust adaptive control method. The nonholonomic constraint is given as
~_ ~ TΓ Θ ^ T Ψ signðrÞ þ J r J ΘT Ψ þ Θ r T Θ 2 2 2 2 2 2 2 ~ Tμ Θ ^ r Θ 2 2 2
2κ 2 0
3
where κ 1 ¼ S T κ 1 S 1 , μ ¼ ½μT1 μT2 T ; Θ ¼ ½Θ1 Θ2 T , λmin ðAÞ and λmax ðAÞ represent the minimum and maximum eigenvalue values of matrix A respectively. From Eq. (49), we can obtain
~ TΓ Θ ~ TΓ Θ ~_ þ Θ ~_ V_ ¼ V_ 3 þ Θ 1 2 1 1 2 2
~ Tμ Θ ^ r Θ 1 1 1
0
ð48Þ
~ ¼Θ ^ Θ ; Θ ~ ¼Θ ^ Θ are the estimation errors of where Θ 1 1 1 2 2 2 coefficient matrices. Taking the derivative of Eq. (49) with respect to time substituting Eqs. (41)–(46) yield
~ TΓ Θ ~ TΓ Θ ~_ þ Θ ~_ þΘ 1 2 1 1 2 2
0
ε
Proof. Define a Lyapunov function candidate as follows: T
0
0
ð47Þ
where Γ1 and Γ2 are the positive definite constant diagonal adaptive gain matrices, μ1 and μ2 are the positive definite constant diagonal matrices. Then, the closed-loop system, which is composed of Eqs. (4), (12), (14) and (41)–(48), is stable. Moreover, all the variables of the closedloop system are bounded.
~ Γ Θ ~ ~ 1 ~ V ¼ V 3 þ 12 Θ 1 1 1 þ 2 Θ2 Γ2 Θ2
where 2 2κ 1 6 6 0 Q ¼6 6 0 4
ð57Þ
ð52Þ
_ ¼0 y_ c cos φ x_ c sin φ dφ
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Then, the constraint matrix of nonholonomic system corresponding to Eq. (3) can be obtained as Av ðqv Þ ¼ ½ sin φ cos φ d
x_ r ¼ vr cos φr ;
Select Sðqv Þ as a basis in null space Av ðqv Þ, that is, 2r 3 r r r 2 cos φ þ Rd sin φ 2 cos φ Rd sin φ r r r 6 r 7 Sðqv Þ ¼ 4 2 sin φ Rd cos φ 2 sin φ þ Rd cos φ 5 r Rr R The corresponding parameters of the dynamic model of mobile manipulator, which is shown in Eq. (4), are given as 2 3 2 φ m012 þ 2I w rsin 2Iw sin r2φ cos φ 2m12 d sin φ 2 6 7 2 7 2I sin φ cos φ φ M 11 ¼ 6 m012 þ 2Iw cos 2m12 d cos φ 5; 4 w r2 r2
2m12 d sin φ 2m12 d cos φ " T I 12 0 0 I 12 T M 12 ¼ M 21 ¼ ; M 22 ¼ 0 0 0 0 2 6 C 11 ¼ 6 4
_ sin φ cos φ 2I w φ r2 _ cos 2φ Iw φ r2
_ 2φ Iw φ cos r2
2Iw φ
_ sin φ cos φ r2
_ cos φ m12 dφ _ sin φ m12 dφ T 0 0 0 0 0 C 12 ¼ C T21 ¼ ; C 22 ¼ ; 0 0 0 0 0
F 1 ¼ ½0 0 0T ; 2 cos φ 16 B11 ¼ 4 sin φ r R
0 I2
For the mobile platform subsystem, consider the kinematic controller, the reference trajectory is chosen as qvd ¼ ½xr ; yr ; φr T , then y_ r ¼ vr sin φr ;
φ_ r ¼ ωr
where the reference linear velocity is chosen as vr ¼ 0:5 m=s and the reference angular velocity is chosen as ωr ¼ 0 rad=s. The reference initial position of mobile platform is qvd ð0Þ ¼ ½2; 2; 451T , while the actual initial position is qv ð0Þ ¼ ½3; 1; 01T . The kinematic controller is shown as Eq. (14), where the gains are set to be k1 ¼ 10; k2 ¼ 5; k3 ¼ 4.
mat 33
# ;
_ cos φ m12 dφ
3
7 _ sin φ 7 m12 dφ 5; 0
F 2 ¼ ½0 m2 gl2 sin θ2 T ; cos φ
3
7 sin φ 5 R 2
where m012 ¼ m0 þ m1 þ m2 , m12 ¼ m1 þ m2 , mat 33 ¼ I 012 þ m12 d þ2I w R2 =r 2 , I 012 ¼ I 0 þ I 1 þ I 2 , I 12 ¼ I 1 þ I 2 , m0 is the mass of mobile platform, r is the drive wheel radius of mobile platform; 2R is the distance between the two drive wheels; Iw is the moment inertia of platform which rotates around central axis, I0 is the moment inertia of the drive wheel which rotates around axis, m1 and m2 represent the mass of link 1 and link 2 of manipulator respectively, l1 and l2 represent the length of link 1 and link 2 respectively, I1 and I2 represent the moment inertia of link 1 and link 2 respectively. The mobile manipulator parameters used for the simulation are shown in Table 1, where the nominal values are used to calculate the controller functions τ v0 and τr 0 to design the robust adaptive controller and the actual values are used to test the robustness of the controller. The external disturbances are assumed to be τd1 ¼ ½ sin t 3 cos t 2 sin tT and τ d2 ¼ ½ sin t cos tT . The simulation sampling steps length is given by 0.01 s. Table 1 Simulation parameters. Parameters (unit)
Nominal values
Actual values
m0 ðkgÞ m1 ðkgÞ m2 ðkgÞ l1 ðmÞ l2 ðmÞ r ðmÞ 2R ðmÞ d ðmÞ I w ðkg m2 Þ I 0 ðkg m2 Þ I 1 ðkg m2 Þ I 2 ðkg m2 Þ
40 6 4 1.1 0.8 0.1 0.3 0.3 2 2 1 1
50 8 2 þ 2 sin 0:5t 1.2 1.2 0.1 0.3 0.3 2 2 þ cos 0:5t 1 1 þ sin 0:5t
Fig. 2. Computed torque control for mobile manipulator. (a) Tracking trajectory of mobile platform on the x–y plane. (b) Tracking errors of mobile platform. (c) Tracking trajectory of link 1. (d) Tracking trajectory of link 2. (e) Tracking errors of two-link manipulators.
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Fig. 4. Robust adaptive control for mobile manipulator. (a) Tracking trajectory of mobile platform on the x–y plane. (b) Tracking errors of mobile platform. (c) Tracking trajectory of link 1. (d) Tracking trajectory of link 2. (e) Tracking errors of two-link manipulators.
Fig. 3. Robust control for mobile manipulator. (a) Tracking trajectory of mobile platform on the x–y plane. (b) Tracking errors of mobile platform. (c) Tracking trajectory of link 1. (d) Tracking trajectory of link 2. (e) Tracking errors of two-link manipulators.
For the manipulator subsystem, choosing k ¼ diagf5; 5g in Eq. (21), the desired trajectory of the manipulator is chosen as qrd ¼ ½ sin t cos tT ;
q_ rd ¼ ½ cos t sin tT ; q€ rd ¼ ½ sin t cos tT
The initial position is qr ð0Þ ¼ ½0:5 0:5T rad, the initial velocity is q_ r ð0Þ ¼ ½0 0T rad=s, and the initial acceleration is q€ r ð0Þ ¼ ½0 0T rad=s2 . For the purpose of comparison, the mobile manipulators are firstly controlled by the computed torque control method [17]. The tracking trajectory on x–y plane of mobile platform is shown in Fig. 2(a). The tracking errors in x and y directions of mobile platform are shown in Fig. 2(b). The tracking trajectories of link 1 and link 2 are shown in
Please cite this article as: Peng J, et al. Robust adaptive tracking control for nonholonomic mobile manipulator with uncertainties. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.012i
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Fig. 2(c) and (d) respectively. The tracking errors of manipulator are shown in Fig. 2(e). From Fig. 2, it can be seen that the computed torque control method cannot drive the mobile manipulator to reach the desired positions and steady-state tracking errors exist due to the existence of parameter uncertainties and external disturbances. Secondly, the mobile manipulators are controlled by the robust control method without adaptive mechanism [20]. The tracking trajectory on the x–y plane of mobile platform is shown in Fig. 3(a). The tracking errors in x and y directions of mobile platform are shown in Fig. 3(b). The tracking trajectories of link 1 and link 2 are shown in Fig. 3(c) and (d) respectively. The tracking errors of manipulator are shown in Fig. 3(e). From Fig. 3, it can be seen that the steady-state tracking errors still exist by using the robust control method without adaptive mechanism due to the existence of varying parameter uncertainties and external disturbances. Finally, the proposed robust adaptive control method is utilized to control the mobile manipulator. The controller is given by Eqs. (41)– (48), where the controller gains are designed as κ 1 ¼ diagf50; 50g, κ 2 ¼ diagf150; 150g to guarantee the tracking errors to be convergent to zero quickly, the adaptive gains are designed as Γ 1 ¼ Γ 2 ¼ diagf5; 5; 5; 5g to guarantee the adaptive errors to be convergent to ^ and Θ ^ are given as Θ ^ ð0Þ ¼ zero quickly, the initial values of Θ 1 2 1 T ^ Θ 2 ð0Þ ¼ ½0 0 0 0 , the constants are given as μ1 ¼ μ2 ¼ diagf0:5; 0:5; 0:5; 0:5g. The tracking trajectory on the x–y plane of mobile platform is shown in Fig. 4(a). The tracking errors in x and y directions of mobile platform are shown in Fig. 4(b). The tracking trajectories of link 1 and link 2 are shown in Fig. 4(c) and (d). The tracking errors of manipulator are shown in Fig. 4(e). From Fig. 4, the tracking errors significantly decrease using the proposed robust adaptive controller in comparison to the computed torque control method and robust control method without adaptive mechanism. The proposed robust adaptive controller can achieve a better performance in the presence of these varying parameter uncertainties and disturbances. Fig. 5 shows the con^ and Θ ^ , it can vergence procedure of the adaptive coefficients Θ 1 2 be seen that the adaptive coefficients are convergent.
^ 1 ; (b) Θ ^ 2. Fig. 5. Adaptive coefficients. (a) Θ
7. Conclusions This paper presents a robust adaptive controller for mobile manipulator under the system parameter uncertainties and external disturbances, where the mobile manipulator is divided into nonholonomic mobile platform subsystem and holonomic manipulators subsystem. By considering the kinematic controller of mobile platform, Lyapunov functions of the two subsystems are designed, the coupling between the two subsystems is regarded as disturbances. Then, aiming at the unknown upper bound of system parameters, corresponding robust adaptive tracking controller is proposed. The proposed control method demonstrated robust and effective control performance on mobile manipulator having uncertainties with good disturbance rejection. As a future work, the proposed adaptive robust controller will be proven experimentally and will be also applied to other kinds of electromechanical systems.
Acknowledgement The authors would like to acknowledge the funding received from the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20124101120001), China Postdoctoral Science Foundation (No. 2013M541992), Henan Provincial Postdoctoral Science Foundation (No. 2013073), National Natural Science Foundation of China (Nos. 61104022 and 61374128) and Key Project for Science and Technology of the Education Department of Henan Province (No. 14A413009) to conduct this research investigation. References [1] Tao G. On robust adaptive control of robot manipulators. Automatica 1992;28(4): 803–7. [2] Slotine JJE, Li W. Composite adaptive control of robot manipulators. Automatica 1989;25(4):509–19. [3] Whitcomb LL, Rizzi AA, Koditschek DE. Comparative experiments with new adaptive controller for robot arms. IEEE Trans Robot Autom 1993;9(1):59–70. [4] Khoshnood AM, Moradi HM. Robust adaptive vibration control of a flexible structure. ISA Trans 2014 〈http://dx.doi.org/10.1016/j.isatra.2014.03.004i〉. [5] Slotine JJE, Li WP. Adaptive manipulator control: a case study. IEEE Trans Autom Control 1998;33(11):995–1003. [6] Su CY, Leung TP. A sliding mode controller with bound estimation for robot manipulators. IEEE Trans Robot Autom 1993;9(2):208–14. [7] Gueaieb W, Karray R, Sharhan SA. A robust adaptive fuzzy position/force control scheme for cooperative manipulators. IEEE Trans Control Syst Technol 2003;11(4):516–28. [8] González-Vázquez S, Moreno-Valenzuela J. Time-scale separation of a class of robust PD-type tracking controllers for robot manipulators. ISA Trans 2013;52:418–28. [9] Tomei P. Robust adaptive control of robots with arbitrary transient performance and disturbance attenuation. IEEE Trans Autom Control 1999;44(3): 654–8. [10] Wang YN, Peng JZ, Sun W, Yu HS, Zhang H. Robust adaptive tracking control of robotic system with uncertainties. J Control Theory Appl 2008;6(3):281–6. [11] Song ZS, Yi JQ, Zhao DB. Survey of the control for mobile manipulators. Robot 2003;25(5):465–9. [12] Seraji H. Motion control of mobile manipulators. In: Proceedings of the IEEE/RSJ international conference on intelligent robots and systems, 1993. p. 2056–63. [13] Tan JD, Xi N. Unified model approach for planning and control of mobile manipulators. In: Proceedings of the international conference on robotics and automation, 2001. p. 3145–52. [14] Tan JD, Xi N, Wang Y. Hybrid force/position control of redundant mobile manipulators. In: The 15th IFAC triennial world congress, 2002. [15] Yamamoto Y, Yun XP. Effect of the dynamic interaction on coordinated control of mobile manipulators. IEEE Trans Robot Autom 1996;12(5):816–24. [16] Dong W, Xu W. On robust control of mobile manipulators. Control Theory Appl 2002;19(3):345–8. [17] Evangelos P, John P. Planning and model-based control for mobile manipulators. In: Proceedings of the IEEE/RSJ international conference on intelligent robots and systems, 2000. p. 1810–5. [18] Liu K, Lewis FL. Decentralized continuous robust controller for mobile robots. In: Proceedings of the IEEE international conference on robotics and automation, 1990. p. 1822–7.
Please cite this article as: Peng J, et al. Robust adaptive tracking control for nonholonomic mobile manipulator with uncertainties. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.012i
J. Peng et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ [19] Lin S, Goldenberg AA. Neural-network control of mobile manipulators. IEEE Trans Neural Netw 2001;12(5):1121–33. [20] Lin S, Goldenberg AA. Robust damping control of mobile manipulators. IEEE Trans Syst Man Cybern Part B 2002;32(1):126–32. [21] Wang HQ, Wang QL. Robust and adaptive fuzzy control for mobile manipulator. Control Decision 2010;25(3):461–5.
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[22] Andaluz V, Roberti F, Toibero JM, Carelli R. Adaptive unified motion control of mobile manipulators. Control Eng Pract 2012;20(12):1337–52. [23] Shojaei K, Shahri AM. Output feedback tracking control of uncertain nonholonomic wheeled mobile robots: a dynamic surface control approach. IET Control Theory Appl 2012;6(2):216–28.
Please cite this article as: Peng J, et al. Robust adaptive tracking control for nonholonomic mobile manipulator with uncertainties. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.012i
