Global finite-time inverse tracking control of robot manipulators

Robotics and Computer-Integrated Manufacturing 27 (2011) 550–557

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Robotics and Computer-Integrated Manufacturing journal homepage: www.elsevier.com/locate/rcim

Global finite-time inverse tracking control of robot manipulators Yuxin Su a,n, Chunhong Zheng b a b

School of Electro-Mechanical Engineering, Xidian University, Xi’an 710071, China School of Electronic Engineering, Xidian University, Xi’an 710071, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 30 April 2010 Received in revised form 22 September 2010 Accepted 28 September 2010 Available online 4 November 2010

This paper addresses the global finite-time tracking of robot manipulators. By replacing with the nonlinear exponential-like errors, the commonly used inverse dynamics control for robot manipulators is modified to produce global finite-time tracking. Using this method, the controlled robotic system is transformed into a nonlinear and decoupled one, and thus the tracking performance is very convenient to quantify. A Lyapunov-like argument along with finite-time stability analysis is employed to prove global finite-time stability. Simulations performed on a two degree-of-freedom (DOF) manipulator are provided to illustrate the effectiveness and the improved performance of the formulated algorithm. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Robot control Finite-time stability Tracking control Global stability Inverse dynamics control

1. Introduction Trajectory tracking of robot manipulators may be recognized as one of its basic aims in robot control and at the same time finds its main application in the robotic field. Several control schemes for global asymptotic tracking of robot manipulators can be found in the literature. For example, thanks to the uncertain parameters can be linearly parameterized with an unknown constant, the standard adaptive controls were developed [1,2]. Due to the high robustness with respect to uncertainties, various sliding mode controls were proposed [3–7]. Specifically, combining the high-order sliding mode technique and adaptive compensation, a fully decentralized adaptive control was formulated [6]. A very simple fully decentralized variable-structure nonlinear proportional plus derivative control was presented in [7]. By the mild assumption that the manipulators track a known periodic trajectory, various repetitive learning controls were developed (for example, see [8–10], and references therein). Another fundamental scheme for global asymptotic tracking of robot manipulators is the well-known inverse dynamics control proposed by Spong and Ortega [11]. In the case of exact knowledge of the robot parameters, inverse dynamics control yields linear and decoupled error dynamics such that the tracking performance is very convenient to quantify. Benefiting from this advantage, inverse dynamics control is widely used in robot control [12–14].

n

Corresponding author. Tel.: +86 29 8820 3040 E-mail addresses: [email protected] (Y. Su), [email protected] (C. Zheng). 0736-5845/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.rcim.2010.09.010

All of the mentioned controls produce global asymptotic tracking of robot manipulators, which implies that the system trajectories converge to the equilibrium as time goes to infinity. It is now known that finite-time stabilization of dynamical systems may give rise to fast transient and high-precision performances besides finite-time convergence to the equilibrium, and a lot of work has been done in the last several years [15–18]. Concerning the finite-time tracking robot manipulators, the knowledge of robot dynamics is required and the so-called terminal sliding mode (TSM) technique has dominated the scenario. In particular, Man et al. [19] presented a robust TSM control for rigid robotic manipulators. Tang [20] developed an improved TSM control. Feng et al. [21] addressed the singularity problem of the TSM control. To reduce the potential chattering of the commonly used TSM scheme, Parra-Vega et al. [22] formulated a dynamic sliding control for perfect tracking defined as the performance of zero tracking errors of position and force in finite-time. More recently, Yu et al. [23] proposed a continuous TSM tracking control for robot manipulators using a new form of terminal sliding mode. As an alternative, recently, Su [24] developed a finitetime proportional-derivative (PD) plus robot dynamics control. The closed-loop system is shown to be global finite-time stability with Lyapunov stability and finite-time stability theory. In this paper we modify the commonly used inverse dynamics control to achieve global finite-time tracking for robot manipulators. This is accomplished by replacing the feedback linear PD actions with exponential-like ones. The closed-loop system is shown to be global finite-time stability with Lyapunov’s direct method and finite-time stability theory. Similar to the conventional inverse dynamics control, the proposed finite-time inverse dynamics control has decoupled error dynamics, and thus the tracking performance is very convenient to quantify. Simulations performed on a two degree-of-freedom

Y. Su, C. Zheng / Robotics and Computer-Integrated Manufacturing 27 (2011) 550–557

(DOF) manipulator are included to demonstrate the effectiveness and improved performance of the proposed approach.

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4. Control design 4.1. Control formulation

2. Robot manipulator model and properties In the absence of disturbances, the dynamics of an n-DOF robot manipulator can be written as [12,13] _ q_ þ Dq_ þ gðqÞ ¼ t MðqÞq€ þCðq, qÞ

ð1Þ

n

_ q€ A R denote the link position, velocity and accelerawhere q, q, tion, respectively, M(q)ARn  n represents the inertia matrix, _ A Rnn denotes the centrifugal-Coriolis matrix, DARn  n Cðq, qÞ represents the diagonal matrix composed of damping friction coefficients for each joint, g(q)ARn is the gravitational force, and tARn denotes the torque input vector. Recalling that robot manipulators are being considered, the following property can be established [11–13]. Property 1. The matrix M(q) is symmetric and positive definite.

To aid the subsequent control design and analysis, we define the vector SigðUÞa A Rn as follows: a

a

ð6Þ SigðxÞa ¼ ½9x1 9 sgnðx1 Þ,    ,9xn 9 sgnðxn ÞT  T where x ¼ x1 ,    , xn A Rn , 0o a o1, and sgn(U) being the standard signum function. With this exponential-like function, the inverse dynamics control proposed by Spong and Ortega in [11] is modified as _ q_ þDq_ þgðqÞ _ a2  þ Cðq, qÞ t ¼ MðqÞ½q€ d Kp SigðeÞa1 Kd SigðeÞ

where Kp and Kd are positive definite constant diagonal proportional and derivative matrices, respectively, 0o a1 o1, and a2 ¼2a1/(a1 +1). Upon substituting (7) into (1), we have the following closedloop dynamics _ a2  ¼ 0 MðqÞ½e€ þKp SigðeÞa1 þ Kd SigðeÞ

3. Preliminaries

ð8Þ

From the positive definiteness of M(q) expressed in Property 1, we have M(q)a0. As a result, the decoupled error equation is given by

Some concepts of finite-time stability and stabilization of nonlinear systems, and the properties of homogeneous systems are reviewed, following the treatment in [15–17]. Consider the system

_ a2 ¼ 0 e€ þ Kp SigðeÞa1 þ Kd SigðeÞ

z_ ¼ f ðzÞ, f ð0Þ ¼ 0, zð0Þ ¼ z0 , z A Rn

4.2. Stability analysis

ð2Þ

ð7Þ

ð9Þ

n

with f:U0-R continuous on an open neighborhood U0 of the origin. Suppose that system (2) possesses unique solutions in forward time for all initial conditions. The equilibrium z ¼0 of system (2) is (locally) finite-time stable if it is Lyapunov stable and finite-time convergent in a neighborhood UCU0 of the origin. The finite-time convergence means the existence of a function T:U\{0}-(0, N), such that, 8z0AU CRn, the solution of (2) denoted by st(z0) with z0 as the initial condition is defined and st(z0)AU\{0} for tA(0, T(z0)), and limt-Tðz0 Þ st ðz0 Þ ¼ 0 with st(z0) for t 4T(z0). When U ¼Rn, we obtain the global finite-time stability. A scalar function V(z) is homogeneous of degree sAR with (r1,y,rn), ri 40, i¼1,y,n, if for any given e 40, Vðer1 z1 ,. . ., ern zn Þ ¼ es VðzÞ,

8z A Rn

ð3Þ T

A continuous vector field f(z)¼[f1(z), y, fn(z)] is homogeneous of degree kAR with r¼ (r1, y, rn), ri 40, i¼1, y, n, if for any given e 4 0, fi ðer1 z1 ,. . ., ern zn Þ ¼ ek þ ri fi ðzÞ,

i ¼ 1,. . .,n,8z A Rn

ð4Þ

System (2) is said to be homogeneous if f(z) is homogeneous. Some of the results on finite-time stability of a nonlinear system in [17] that will be used in this paper are summarized by the following two lemmas: Lemma 1. Suppose that system (2) is homogeneous of degree k o0 with respect to (r1, y, rn), ri 40, i¼1, y, n and z ¼0 is an asymptotically stable equilibrium. Then z ¼0 is a locally finitetime stable equilibrium of the system (2). Lemma 2. Global asymptotic stability and local finite-time stability of the closed-loop system imply global finite-time stability. Let Ck denote the set of k times continuously differentiable functions. We consider the global finite-time problem of controlling the system (1) towards any desired trajectory qd(t)AC2 with modified inverse dynamics control. We will quantify the control _ A Rn as follows: objective by defining the tracking errors eðtÞ, eðtÞ e ¼ qqd ,

_ q_ d e_ ¼ q

ð5Þ

Now we are in a position to state the following result. Theorem 1. Given the robot system (1), the modified inverse dynamics control input given by (7) ensures global finite-time tracking in the sense that _ ¼0 lim eðtÞ, eðtÞ ð10Þ t-Tðx0 Þ

where T(x0) is the convergent time which is a function of the initial _ T ÞT A R2n . state x0, and x0 ¼ xð0Þ ¼ ðeð0ÞT eð0Þ Proof. The proof proceeds in the following two steps. First, the global asymptotic stability is proven based on Lyapunov’s direct method. Second, the global finite-time stability is shown using Lemmas 1 and 2. (1) Global asymptotic stability To this end, we propose the following Lyapunov-like function candidate V¼

n 1 T 1 X a þ1 e_ e_ þ k 9e 9 1 2 a1 þ 1 i ¼ 1 pi i

ð11Þ

where kpi denotes the ith diagonal element of matrix Kp, and ei denotes the ith component of vector e defined by (5). It is straightforward that V defined by (11) is radially unbounded _ positive definite with respect to e, e. Differentiating V with respect to time, we have V_ ¼ e_ T e€ þ e_ T Kp SigðeÞa1

ð12Þ

Substituting e€ from (9) into (12), it follows that _ a2 V_ ¼ e_ T Kd SigðeÞ

ð13Þ

Hence, V(t) is a radially unbounded positive definite Lyapunov function whose time derivative V_ ðtÞ is negative semidefinite. In fact, V_  0 means e_  0. By LaSalle’s invariant principle [25], _ we have e(t)-0 and eðtÞ-0 as t-N for any initial state

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Y. Su, C. Zheng / Robotics and Computer-Integrated Manufacturing 27 (2011) 550–557

_ ðqð0Þ, qð0ÞÞ. Hence, we have the global asymptotic stability about the point ðe ¼ 0, e_ ¼ 0Þ. (2) Global finite-time stability Following the idea presented by Hong et al. [17], the local finitetime stability is proven first using Lemma 1. To this end, let x1 ¼e, _ and x ¼ ðxT1 xT2 ÞT . The decoupled error equation (9) is x2 ¼ x_ 1 ¼ e, ( x_ 1 ¼ x2 ð14Þ x_ 2 ¼ Kp Sigðx1 Þa1 Kd Sigðx2 Þa2

Clearly, x¼0 is the equilibrium of (14). Moreover, we have shown that x¼0 (i.e. ðe ¼ 0, e_ ¼ 0Þ) is the global asymptotic equilibrium from part (1). Furthermore, it can be easily verified that system (14) is homogeneous of degree k ¼ a1  1o0, (0o a1 o1) with respect to (r11, r12, y, r1n, r21, r22, y, r2n), in which r1i ¼r1 ¼2 and r2i ¼r2 ¼ a1 +1. Therefore, according to Lemma 1, we have the local finite-time stability of the closedloop system (14).

Position errors of first link Position errors [rad]

2 IDC FIDC

1.5 1 0.5 0 -0.5 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Position errors of second link Position errors [rad]

0.5

0

-0.5 IDC FIDC

-1 0

0.5

1

1.5

2

2.5 3 Time [sec]

3.5

4

4.5

5

Fig. 1. Position tracking error comparisons with the inverse dynamic control.

Velocity errors of first link Velocity errors [rad]

1 0 -1 -2 IDC FIDC

-3 -4 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Velocity errors of second link Velocity errors [rad]

3 IDC FIDC

2 1 0 -1 0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 2. Velocity tracking error comparisons with the inverse dynamic control.

4.5

5

Y. Su, C. Zheng / Robotics and Computer-Integrated Manufacturing 27 (2011) 550–557

Finally, by invoking Lemma 2, we get the global finite-time stability of (9). This completes the proof. &

5. Simulation illustrations Simulations on the two-DOF robot used in [26] were conducted to illustrate the effectiveness and improved performance of the proposed finite-time inverse dynamics control (FIDC). The entries

553

to model the robot manipulator are, respectively " # p1 þ2p2 cosðq2 Þ p3 þp2 cosðq2 Þ M¼ p4 p3 þp2 cosðq2 Þ " # p2 sinðq2 Þq_ 2 p2 sinðq2 Þðq_ 1 þ q_ 2 Þ C¼ 0 p2 sinðq2 Þq_ 1 " # p5 sinðq1 Þ þ p6 sinðq1 þ q2 Þ g¼ p6 sinðq1 þ q2 Þ

Input torques of the IDC Input torques [Nm]

500 Link 1

0

-500

-1000 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Input torques [Nm]

20 0 -20 -40 -60 Link 2

-80 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time [sec] Fig. 3. Input torques of the commonly used inverse dynamics control.

Input torques of the FIDC Input torques [Nm]

200 Link 1

0 -200 -400 -600 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

20 Input torques [Nm]

Link 2

0 -20 -40 -60 0

0.5

1

1.5

2

2.5 3 Time [sec]

3.5

4

Fig. 4. Input torques of the proposed finite-time inverse dynamics control.

4.5

5

ð15Þ

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Y. Su, C. Zheng / Robotics and Computer-Integrated Manufacturing 27 (2011) 550–557

with p1 ¼8.77, p2 ¼0.51, p3 ¼0.76, p4 ¼ 0.62, p5 ¼74.48, p6 ¼6.174. Here we assumed that there is no friction, i.e. D ¼diag(0,0). The parameters are given in SI units. The desired position trajectories for links 1 and 2 were selected as follows: 3

qd2 ðtÞÞ ¼ ð1:57sinð2tÞð1e0:05t Þ,

_ q_ þ Dq_ þ gðqÞ _ þ Cðq, qÞ t ¼ MðqÞ½q€ d Kp eKd e

3

1:2sinð3tÞð1e0:05t ÞÞT

ð16Þ

where Kp and Kd are the same as that in (7).

Position errors of first link Position errors [rad]

2 CTSM FIDC

1.5 1 0.5 0 -0.5 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Position errors of second link Position errors [rad]

0.5

0

-0.5 CTSM FIDC

-1 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time [sec] Fig. 5. Position tracking error comparisons with the continuous terminal sliding mode control.

Velocity errors of first link Velocity errors [rad]

1 0 -1 -2 CTSM FIDC

-3 -4 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Velocity errors of second link 3 Velocity errors [rad]

qd ðtÞ ¼ ðqd1 ðtÞ,

The sampling period was T¼1 ms. The initial conditions for both _ ¼ ð0,0ÞT . controls were q(0)¼ (2.0,–1.0)T and qð0Þ First we carried out the comparisons with the commonly used inverse dynamics control proposed by Spong and Ortega [11]. The inverse dynamics control (IDC) law is given by

CSTM FIDC

2 1 0 -1 0

0.5

1

1.5

2

2.5 Time [sec]

3

3.5

4

4.5

Fig. 6. Velocity tracking error comparisons with the continuous terminal sliding mode control.

5

ð17Þ

Y. Su, C. Zheng / Robotics and Computer-Integrated Manufacturing 27 (2011) 550–557

The gains of the proposed finite-time inverse dynamics control (FIDC) were chosen as: Kp ¼diag(50, 40), Kd ¼diag(25, 20), and a1 ¼0.5. For a fair comparison, the gains of the conventional inverse dynamics control (IDC) were chosen the same as those of the FIDC. Figs. 1 and 2 illustrate the position and velocity tracking errors. For a clear comparison, the requested input torques of the two controllers are shown in Figs. 3 and 4, respectively. It can be seen that the robot completed the motion successfully, and after a

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transient due to errors in initial condition, both the position and velocity tracking errors tend to zero. Furthermore, a faster response is achieved in comparison with the commonly used inverse dynamics control. Notice that the favorable results are obtained without an excessive input torques, which is very beneficial to avoid the actuator saturation. After that comparisons with the continuous terminal sliding mode (CTSM) control proposed by Yu et al. [23] were also conducted.

Inputs of the CTSM control Input torques [Nm]

200 0 -200 -400 Link 1

-600 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time [sec]

Input torques [Nm]

20 0 -20 -40 Link 2

-60 0

0.5

1

1.5

2

2.5 3 Time [sec]

3.5

4

4.5

5

Fig. 7. Input torques of the continuous terminal sliding mode control.

Position errors of first link Position errors [rad]

2 CFPD+ FIDC

1.5 1 0.5 0 -0.5 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Position errors of second link Position errors [rad]

0.5

0

-0.5 CFPD+ FIDC

-1 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time [sec] Fig. 8. Position tracking error comparisons s with the continuous finite-time PD plus control.

5

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Y. Su, C. Zheng / Robotics and Computer-Integrated Manufacturing 27 (2011) 550–557

The reasoning behind this comparison is that the proposed controller and the CTSM controller are all continuous finite-time tracking controls and they involve the model information in the control law formulation. The CTSM controller can be expressed as _ g s ¼ eþ bSigðeÞ _ q_ þ Dq_ þgðqÞ t ¼ MðqÞq€ d þCðq, qÞ 1 _ 2g þ k1 s þ k2 SigðsÞr  MðqÞ½b g1 SigðeÞ

ð18Þ

where b, g, k1 and k2 are constant positive definite diagonal matrices, 0o r o1, and 1o gi o2 and gi denotes the ith diagonal elements of the diagonal matrix g. According to the guidelines presented in [23], the gains of the CTSM controller were chosen as b ¼diag(0.5, 0.5), g ¼diag(1.5, 1.5), k1 ¼diag(25, 25), k2 ¼diag(20, 20), and r ¼ 0.2. Simulation results are shown in Figs. 5–7. It can be clearly seen that the proposed finite-time inverse dynamics control (FIDC) obtains a much faster

Velocity errors of first link Velocity errors [rad]

1 0 -1 -2 CFPD+ FIDC

-3 -4 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Velocity errors of second link Velocity errors [rad]

3 CFPD+ FIDC

2 1 0 -1 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time [sec] Fig. 9. Velocity tracking error comparisons with the continuous finite-time PD plus control.

Input torques of the CFPD+ control Input torques [Nm]

100 Link 1

50 0 -50 -100 -150 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Input torques [Nm]

150 Link 2

100 50 0 -50 0

0.5

1

1.5

2

2.5 3 Time [sec]

3.5

4

Fig. 10. Input torques of the continuous finite-time PD plus control.

4.5

5

Y. Su, C. Zheng / Robotics and Computer-Integrated Manufacturing 27 (2011) 550–557

transient over the CTSM controller. Moreover, the requested input torques of the proposed FIDC is much smoother than that of the CTSM, which is very helpful to avoid the chattering. Moreover, comparisons with the continuous finite-time PD plus robot dynamics (CFPD+ ) control proposed by Su in [24] were also carried out. The control law of CFPD+ is given by _ q_ d þ Dq_ d þgðqÞ _ a2 þ MðqÞq€ d þ Cðq, qÞ t ¼ Kp SigðeÞa1 Kd SigðeÞ

ð19Þ

where Kp, Kd, a1, a2 are the same as the ones in (7). They were chosen to be the same as those of the proposed finite-time inverse dynamics control (FIDC). The position and velocity tracking errors are illustrated in Figs. 8 and 9, respectively. The requested input torques of the CFPD+ control is shown in Fig. 10. It can be seen that the proposed FIDC also produces a much faster transient over the CFPD+ control. As a result, we can conclude that the proposed finite-time inverse dynamics control will give a faster transient and better tracking over the commonly used inverse dynamics control [11], the continuous terminal sliding mode (CTSM) control proposed by Yu et al. [23], and the continuous finite-time PD plus robot dynamics (CFPD + ) control developed by Su [24].

6. Conclusions We have modified the commonly used inverse dynamics control to give a global finite-time position and velocity tracking for robot manipulators. This is accomplished by replacing the linear position and velocity tracking errors with exponential-like ones. Lyapunov’s direct method and finite-time stability theory have been employed to show the global finite-time stability. Similar to the commonly used inverse dynamics control, the proposed finite-time inverse dynamics control has decoupled error dynamics, and thus, the tracking performance is very convenient to quantify. Simulations performed on a two-DOF robot manipulator demonstrate the effectiveness and improved performances of the proposed approach.

Acknowledgements The authors would like to thank the anonymous reviewers for their valuable comments. This work was supported in part by the National Natural Science Foundation of China under Grant 50675167, and in part by A Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant 200535, SRF for ROCS, SEM, the Fundamental Research Funds for the Central Universities and the State Key Laboratory of Robotics and System (HIT) under Grant SKLRS-2009-MS-04.

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