Finite-time tracking control for robot manipulators with actuator saturation

Robotics and Computer-Integrated Manufacturing 30 (2014) 91–98

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

Finite-time tracking control for robot manipulators with actuator saturation Yuxin Su n,1, Jan Swevers Department of Mechanical Engineering, Division PMA, KU Leuven, Celestijnenlaan 300B, B-3001 Leuven-Heverlee, Belgium

art ic l e i nf o

a b s t r a c t

Article history: Received 8 March 2013 Received in revised form 10 September 2013 Accepted 12 September 2013 Available online 16 October 2013

This paper addresses the finite-time tracking of robot manipulators in the presence of actuator saturation. The commonly-used proportional-derivative (PD) plus dynamics compensation (PD þ) scheme is extended by replacing the linear errors in the PD þ scheme with saturated non-smooth but continuous exponential-like ones. Advantages of the proposed controller include semi-global finite-time tracking stability featuring faster transient and high-precision performances and the ability to ensure that actuator constraints are not violated. This is accomplished by selecting control gains a priori, removing the possibility of actuator failure due to excessive torque input levels. Lyapunov's direct method and finite-time stability are employed to prove semi-global finite-time tracking. Simulations performed on a three degree-of-freedom (DOF) manipulator are provided to illustrate the effectiveness and the improved performance of the formulated algorithm. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Actuator saturation Finite-time stability Geometric homogeneity Robot control Tracking control

1. Introduction One of the basic functionalities of robot manipulators is trajectory tracking. Several control schemes to implement tracking tasks of robot manipulators can be found in the literature [1–4]. While these control schemes are elegant and intuitively appealing, there is an implicit assumption in the development of these schemes that the robot system actuators are able to provide any requested torque. It is known that if the controller requests more torque than the actuators can supply, degraded or unpredictable motion and thermal or mechanical failure may result [5–7]. Recognizing these difficulties, several solutions that take into account actuator constraints during robot trajectory tracking have been proposed. For example, Loria and Nijmeijer [8] first addressed the semi-global asymptotic tracking of Euler-Lagrange systems with saturated position feedback PD plus feedforward dynamics (PDþ) scheme. Aguinga-Ruiz et al. [9] proposed state feedback saturated PDþ control for global asymptotic stability with a restrictive and limitative condition that the inherent damping friction coefficient on each joints is larger than the upper boundedness of the desired trajectories. Lefeber and Nijmeijer [10] combined a bounded regulation controller with a local asymptotically stable tracking controller and achieved global asymptotic tracking. Dixon et al. [11] formulated

n

Corresponding author. E-mail addresses: [email protected], [email protected] (Y. Su), [email protected] (J. Swevers). 1 He is on leave from the School of Electro-Mechanical Engineering, Xidian University, Xi'an, China. 0736-5845/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.rcim.2013.09.005

a saturated output feedback PDþ scheme and saturated adaptive control and obtained semi-global asymptotically tracking. MorenoValenzuela et al. [12] incorporated gains scaling of the argument of the hyperbolic tangents into the saturated PDþ control and showed the local exponential stability. Recently, Su and Zheng [13] developed a simple decentralized saturated repetitive learning controller for semi-global asymptotic tracking of robot manipulators. However, these bounded controllers only achieve asymptotic tracking stability implying that the system trajectories converge to the equilibrium as time goes to infinity. It is 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 [14–18]. This observation is supported by a review of literature which yields different approaches to address finite-time tracking of robot manipulators, which implies that the tracking errors become and remain zero within a finite time. Specifically, Man et al. [19] presented a robust terminal sliding mode (TSM) control for rigid robotic manipulators. Tang [20] developed an improved TSM control to implement global finite-time tracking. Feng et al. [21] addressed the singularity problem of the TSM control. Yu et al. [22] proposed a continuous TSM controller by using a new form of terminal sliding mode. Jin et al. [23] presented a practical nonsingular TSM tracking control by using time-delay estimation technique. Zhao et al. [24] developed a new terminal sliding mode control approach for robotic manipulators based on finite-time stability theory and differential inequality principle. To cope with model uncertainty, Barambones and Etxebarria [25] incorporated adaptive control into TSM for high precision tracking of uncertain

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Y. Su, J. Swevers / Robotics and Computer-Integrated Manufacturing 30 (2014) 91–98

robot manipulators. Parra-Vega and Hirzinger [26] proposed a dynamic sliding controller to implement the perfect tracking in finite time. Liu and Zhang [27] proposed a neural network-based robust finite-time controller with the consideration of actuator dynamics. On the other hand, Su [28] and Su and Zheng [29] formulated an alternative continuous finite-time tracking control for robot manipulators by using geometric homogeneity technique. Although these finite-time tracking schemes give a faster transient and higher precision, the desired favourable performances relies heavily on the assumption that the robot system actuators are able to provide any requested torque. This paper proposes a saturated finite-time tracking controller for robot manipulators with the consideration of actuator constraints. This is accomplished by replacing the linear errors in the commonly used PD þ with saturated non-smooth exponential-like ones. Lyapunov stability and finite-time stability theory are employed to prove semi-global finite-time tracking stability. The practical implications are that the actuators can be appropriately sized without an ad hoc saturation scheme to protect the actuator. Simulations are presented to demonstrate the improved performance of the proposed approach. Throughout this paper, the norm pffiffiffiffiffiffiffi of a vector x A ℜn is defined as J x J ¼ xT x and that of a matrix A is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the corresponding induced norm J A J ¼ ðAT AÞM , where AM

2.2. Fundamental facts Some concepts of finite-time stability of nonlinear systems are reviewed following the approach of Bhat and Bernstein [14] and Hong et al. [15]. Consider the system x_ ¼ f ðxÞ;

f ð0Þ ¼ 0;

x A ℜn

ð7Þ

with f : U 0 -ℜn continuous on an open neighborhood U 0 of the origin. Suppose that system (7) possesses unique solutions in forward time for all initial conditions. The equilibrium x ¼ 0 of system (7) is (locally) finite-time stable if it is Lyapunov stable and finite-time convergent in a neighborhood U  U 0 of the origin. The finite-time convergence means the existence of a function T : U n f0g-ð0; 1Þ, such that, 8 x0 A U  ℜn , the solution of (7) denoted by st ðx0 Þ with x0 as the initial condition and st ðx0 Þ A U n f0g for t A ½0; Tðx0 ÞÞ, and limt-Tðx0 Þ st ðx0 Þ ¼ 0 with st ðx0 Þ for t 4 Tðx0 Þ. When U ¼ ℜn , we obtain the global finite-time stability. A scalar function VðxÞ is homogeneous of degree κ A ℜ with ðr 1 ; ⋯; r n Þ; r i 40; i ¼ 1; ⋯; n, if for any given ε 4 0, Vðεr1 x1 ; …; εrn xn Þ ¼ εκ VðxÞ;

8 x A ℜn :

ð8Þ T

A continuous vector field f ðxÞ ¼ ½f 1 ðxÞ; ⋯; f n ðxÞ is homogeneous of degree κ A ℜ with r ¼ ðr 1 ; ⋯; r n Þ, if for any given ε 4 0,

denotes the maximum eigenvalue of the matrix A.

f i ðεr1 x1 ; ⋯; εrn xn Þ ¼ εκ þ ri f i ðxÞ; i ¼ 1; ⋯; n; 8 x A ℜn :

2. Preliminaries

Some of the results on finite-time stability of a nonlinear system in [15] that will be used in this paper are summarized by the following two lemmas.

2.1. Robot model and properties The dynamics of a rigid revolute joint robot manipulator can adequately be described as [4] MðqÞq€ þCðq; q_ Þq_ þ Dq_ þ gðqÞ ¼ τ;

ð1Þ

where q; q_ and q€ A ℜn denote the link position, velocity, and acceleration, respectively, MðqÞ A ℜnn is the symmetric inertia _ A ℜnn denotes the centrifugal-Coriolis matrix, matrix, Cðq; qÞ D A ℜnn represents the matrix composed of damping friction coefficients, gðqÞ A ℜn denotes the influence of gravity, and τ A ℜn is the input torque. The following properties of robot manipulator dynamics (1) have been established [4].

Lemma 1. Consider the following system x_ ¼ f ðxÞ þ f^ ðxÞ; f ð0Þ ¼ 0; f^ ð0Þ ¼ 0; x A ℜn ;

f^ i ðεr1 x1 ; ⋯; εrn xn Þ ¼ 0; xκ þ r i ε-0 lim

m1 J ζ J 2 r ζ MðqÞζ rm2 J ζ J 2 ;

3.1. Control objective

ð2Þ

where m1 and m2 are known positive constants. Property 2. There are positive constants dM and c2 such that _ J r c2 J q_ J ; 8 q; q_ A ℜn : ð3Þ J D J r dM ; J Cðq; qÞ

i ¼ 1; ⋯; n; 8 x a 0:

ð11Þ

Lemma 2. Semi-global asymptotic stability and local finite-time stability imply semi-global finite-time stability. 3. Control design

8 q; ζ A ℜn ;

ð10Þ

where f ðxÞ is a continuous homogeneous vector field of degree κ o0 with respect to ðr 1 ; ⋯; r n Þ. Assume that x ¼ 0 is an asymptotically stable equilibrium of the system x_ ¼ f ðxÞ. Then x ¼ 0 is a locally finite-time stable equilibrium of the system (10) if

Property 1. The inertia matrix MðqÞ is symmetric positive definite and satisfies the following inequality: T

ð9Þ

Let qd A ℜn be any reference trajectory for system (1) that is continuously differentiable up to its second derivative such that J q_ d J rV M ;

J q€ d J r AM :

ð12Þ

We also assume that each joint actuator has a maximum torque

τi; max that satisfies

_ Property 3. The matrix MðqÞ 2Cðq; q_ Þ is skew-symmetric, i.e. _ ζ T ðMðqÞ  2Cðq; q_ ÞÞζ ¼ 0;

8 q; q_ ; ζ A ℜn ;

ð4Þ

ð13Þ

with

which implies that _ _ MðqÞ ¼ Cðq; q_ Þ þ C T ðq; qÞ;

τi; max 4T i;M

8 q; q_ ; ζ A ℜn :

ð5Þ

T i;M ¼ m2 AM þ c2 V 2M þ dM V M þ κ gi

ð14Þ

The output tracking errors are defined as Property 4. The vector gðqÞ is bounded for all q A ℜn , i.e., there exist finite constants κ gi Z0 such that sup jg i ðqÞj r κ gi ;

q A ℜn

i ¼ 1; ⋯; n;

where g i ðqÞ denotes the ith component of gðqÞ.

ð6Þ

e ¼ q  qd ; e_ ¼ q_  q_ d

ð15Þ

Our objective is to design a saturated control input τðtÞ such that lim eðtÞ; e_ ðtÞ ¼ 0 while respecting the actuator constraints t-Tðx0 Þ

jτi j r τi; max ;

ð16Þ

Y. Su, J. Swevers / Robotics and Computer-Integrated Manufacturing 30 (2014) 91–98

where Tðx0 Þ is the finite convergence time which is a function of the initial values of the system state x0 , x0 ¼ ðeð0ÞT e_ ð0ÞT ÞT A ℜ2n , and τi denotes the torque of the ith actuator. 3.2. Control formulation First we introduce the following saturated function ( α jxj sgnðxÞ; jxj o1 ; sðxÞα ¼ sgnðxÞ; jxj Z1

ð17Þ

where 0 o α o 1, and sgnð U Þ denotes the standard sign function. It can easily be shown that the following expression holds xsðxÞα Z x2 ;

8 x A ð0; 1:

ð18Þ

To facilitate the subsequent design and analysis, we define a saturated vector SatðξÞα A ℜn as follows: SatðξÞα ¼ ½sðξ1 Þα ; ⋯; sðξn Þα T ;

ξ ¼ ½ξ1 ; ⋯; ξn T :

ð19Þ

Now the saturated finite-time tracking (SFT) control law is proposed as follows:

τ ¼  K p SatðeÞα1  K d Satðe_ Þα2 þ MðqÞq€ d þ Cðq; q_ d Þq_ d þ Dq_ d þ gðqÞ; ð20Þ where K p and K d are positive definite constant diagonal matrices, respectively, 0 o α1 o 1, and α2 ¼ ð2α1 Þ=ðα1 þ 1Þ.

with Z e Z n K p SatðsÞα1 ds ¼ ∑ i¼1

0

ei

93

kpi sðsÞα1 ds;

ð26Þ

0

where ei denotes the ith component of vector e. It can be shown that (26) satisfies Z e K p SatðsÞα1 ds 4 0; 8 ea 0 A ℜn ;

ð27Þ

0

because K p is a diagonal positive definite matrix, SatðsÞα1 js ¼ 0 ¼ 0, and SatðsÞα1 is an increasing function with respect to s. Therefore, this term is positive definite with respect to e. As a result, we can conclude that the proposed Lyapunov function is a radially unbounded, globally positive definite function for all eðtÞ; e_ ðtÞ. Differentiating V with respect to time, we have 1 _ V_ ¼ e_ T MðqÞ e_ þ e_ T MðqÞe€ þ e_ T K p SatðeÞα1 : 2

ð28Þ

Substituting MðqÞe€ from (23) into (28), and using (4) of Property 3, it follows that V_ ¼  e_ T De_  e_ T K d Satðe_ Þα2  e_ T Cðq; q_ d Þe_ :

ð29Þ

Remark 1. The control effort given by (20) is upper bounded by

In light of (3) of Property 2 and (12), the last term of the righthand-side of (29) can be upper bounded by

jτi jr kpi þkdi þ m2 AM þc2 V 2M þ dM V M þ κ gi ;

e_ T Cðq; q_ d Þe_ r c2 V M J e_ J 2

ð21Þ

where kpi and kdi denote the ith diagonal elements of control gain matrices K p and K d , respectively. Based on this fact, the actuator constraints expressed in (16) are satisfied by selecting the control gains according to the following inequalities kpi þ kdi r τi; max  T i;M :

ð22Þ

Substituting (20) into (1), the closed-loop dynamics equal _ þ Cðq; q_ d Þe_ þ De_ þ K p SatðeÞα1 þ K d Satðe_ Þα2 ¼ 0: MðqÞe€ þ ½Cðq; qÞ

ð23Þ

3.3. Stability analysis Theorem 1. Given the robot dynamics of (1) under the input constraints (16), the proposed SFT control ensures semi-global finite-time tracking, provided that the control gains are selected to fulfill (22) and the derivative control gain K d is selected to satisfy the following sufficient condition K dm 4 c2 V M ;

ð24Þ

where K dm indicates the smallest gain of the matrix K d , and c2 and V M are positive constants defined by (3) and (12), respectively. (

ð30Þ

Suppose J e_ J r 1, from inequality (18), we have e_ T K d Satðe_ Þα2 Z K dm e_ T e_ ;

8 ‖e_ ‖ r 1:

ð31Þ

Upon substituting (30) and (31) into (29), yields V_ r  e_ T De_  ðK dm  c2 V M Þ J e_ J 2 ;

8 J e_ J r 1:

ð32Þ

From condition (24) given in Theorem 1, we can conclude that V_ r 0; 8 J e_ J r1. Since V_  0 means e_  0, by LaSalle's invariance theorem [30], we have eðtÞ-0 and e_ -0, as t-1 for a neighborhood of the equilibrium state. Furthermore, from (24) and (32), we can see that the region of attraction can be increased arbitrarily by increasing the gain K d . Hence, we have the semi-global asymptotic stability about the point ðe ¼ 0; e_ ¼ 0Þ. Step 2: Semi-global finite-time tracking stability Following the idea presented in [15,28], the semi-global finitetime tracking stability is proven using Lemmas 1 and 2. To this end, let x1 ¼ e, x2 ¼ x_ 1 ¼ e_ , and x ¼ ðxT1 xT2 ÞT . The equation of the closed-loop system is

x_ 1 ¼ x2 : x_ 2 ¼  M  1 ðx1 þ qd Þ½ðCðx1 þ qd ; x2 Þ þ Cðx1 þqd ; q_ d ÞÞx2 þ Dx2 þ K p Satðx1 Þα1 þ K d Satðx2 Þα2 

Proof. The proof proceeds in two steps. First, semi-global asymptotic tracking stability is proven based on Lyapunov's direct method. Second, local finite-time tracking stability is shown using Lemma 1 and then Lemma 2 is involved to conclude the result. Step 1: Semi-global asymptotic tracking stability We propose the following Lyapunov function candidate Z e 1 V ¼ e_ T MðqÞe_ þ K p SatðsÞα1 ds 2 0

ð33Þ

Clearly, x ¼ 0 is the equilibrium of (33). But, the system (33) is not homogeneous. To use Lemma 1, we rewrite (33) as follows: ( x_ 1 ¼ x2 ð34Þ x_ ¼ M  1 ðq Þ½K p Satðx Þα1 þ K Satðx Þα2  þ f^ ðxÞ 2

1

d

d

2

2

with

ð25Þ

f^ 2 ¼  M  1 ðx1 þ qd Þ½ðCðx1 þqd ; x2 Þ þ Cðx1 þ qd ; q_ d ÞÞx2 ~ 1 ; q Þ½K p Satðx1 Þα1 þ K d Satðx2 Þα2 ; þDx2   Mðx d

ð35Þ

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Y. Su, J. Swevers / Robotics and Computer-Integrated Manufacturing 30 (2014) 91–98

~ 1 ; q Þ ¼ M  1 ðx1 þ q Þ  M  1 ðq Þ: Mðx d d d

ð36Þ

It is easy to verify that, 8 J e_ J r1, the following system ( x_ 1 ¼ x2 x_ 2 ¼  M  1 ðqd Þ½K p Satðx1 Þα1 þ K d Satðx2 Þα2 

q3 q2

ð37Þ

is homogeneous of degree κ ¼ α1  1 o 0 with respect to ðr 11 ; r 12 ; …; r 1n ; r 21 ; r 22 ; …; r 2n Þ with r 1i ¼ r 1 ¼ 2 and r 2i ¼ r 2 ¼ α1 þ 1. Note also that f ð0Þ ¼ 0 and f^ ð0Þ ¼ 0 from (34) and (35), respectively. Now consider the system (37) and take a nonnegative Lyapunov function candidate as follows: Z x1 1 V n ¼ xT2 Mðqd Þx2 þ K p SatðsÞα1 ds ð38Þ 2 0 with Z x1 Z n K p SatðsÞα1 ds ¼ ∑ i¼1

0

x1i

kpi sðsÞα1 ds;

q1

Fig. 1. The three-DOF robot.

ð39Þ

0

Lemma 2, we get the semi-global finite-time stability of (33) (i.e. (23)). This completes the proof.

where x1i denotes the ith component of vector x1 . After taking the time derivative of (38) along (37), we have 1 _ T _ V_ n jð37Þ ¼ x_ T1 K p Satðx1 Þα1 þ xT2 Mðq d Þx2 þ x2 Mðqd Þx2 : 2

ð40Þ

Upon substituting (37) into (40), it follows that 1 _ α2 T V_ n jð37Þ ¼ xT2 Mðq d Þx2  x2 K d Satðx2 Þ : 2

ð41Þ

4. An illustrative example Simulations on a three-DOF robot were conducted to illustrate the improved performance of the proposed SFT control. The used three-DOF robot is illustrated in Fig. 1. The robot dynamics are given in Appendix. The desired trajectories were selected as follows:

In light of (5) of Properties 3 and (3) of Property 2, we have

qd ðtÞ ¼ ð2 sin ðtÞ; cos ðtÞ; 0:5 sin ð2tÞÞT ðradÞ:

1 T _ 1 x Mðqd Þx2 ¼ xT2 ½Cðqd ; q_ d Þ þ C T ðqd ; q_ d Þx2 rc2 V M xT2 x2 : 2 2 2

The actuator constraints were set as τmax ¼ ½50; 100; 70 Nm. Inserting the system parameters and (48) into (2), (3), (6) and (12), respectively, we get the upper bounds required to determine the gains of the proposed controller

ð42Þ

Following the logical progression found in the proof of Step 1, we can show the asymptotic stability of (37). Next we will use Lemma 1 to show the local finite-time stability of the closed-loop system (33) (i.e. (23)). To this end, first note that, since M  1 ðx1 þqd Þ and Cðx1 þ qd ; x2 Þ are smooth [15,28] and κ o 0, we have lim

M  1 ðεr1 x1 þ qd Þ½ðCðεr1 x1 þqd ;

ε-0

εr2 x2 Þþ Cðεr1 x1 þqd ; q_ d ÞÞεr2 x2 þDεr2 x2  εκ þ r2

¼  M  1 ðqd ÞðCðqd ; 0Þ þ Cðqd ; q_ d Þ þ DÞx2 lim ε  κ ¼ 0: ε-0

ð43Þ

Upon applying the mean value theorem to each entry of ~ 1 ; q Þ, it follows that [15,28] Mðx d ~ εr1 x1 ; q Þ ¼ M  1 ðεr1 x1 þ q Þ  M  1 ðq Þ ¼ οðεr1 Þ: Mð d d d

ð44Þ

Thus, we have lim 

~ εr1 x1 ; q Þ½K p Satðεr1 x1 Þα1 þ K d Satðεr2 x2 Þα2  Mð d

εκ þ r 2

ε-0

¼  lim οðεr1  κ  r2 Þ½K p Satðεr1 x1 Þα1 þ K d Satðεr2 x2 Þα2  ¼ 0 ε-0

ð45Þ

lim

f^ 2 ðεr1 x1 ; εr2 x2 Þ

ε-0

εκ þ r 2

M 2 ¼ 8:9; C 2 ¼ 3:5; dM ¼ 0; κ g1 ¼ 0; κ g2 ¼ 40; κ g3 ¼ 12:2; V M ¼ 1:3949; AM ¼ 2:2219:

¼ 0:

ð46Þ

Therefore, according to Lemma 1, we have the local finite-time stability of the closed-loop system (33). Finally, by invoking

ð49Þ

With these bounds, T i; M defined by (14) are determined as T max ¼ ½26; 66; 38T Nm. We have a margin to select the control gains of the proposed SFT control. The SP-SD plus dynamics compensation (SP-SD þ) control developed by Aguinga-Ruiz et al. [9] is selected for comparison. The control law is given by

τ ¼  s2 ðK 2 eÞ  s1 ðK 1 e_ Þ þ MðqÞq€ d þ Cðq; q_ d Þq_ d þ Dq_ d þ gðqÞ;

ð50Þ

where K 1 and K 2 are positive definite diagonal matrices, and the saturation vector sj A ℜn ; j ¼ 1; 2 are defined by sj ðξÞ ¼ ½sj1 ðξ1 Þ; ⋯; sjn ðξn ÞT

ð51Þ

and the generalized saturation functions sji ðxÞ is given by   8 xþL >  Lji þ ðM ji  Lji Þtanh Mji  Lji ji ; x o  Lji > > < jxj r Lji sji ðxÞ ¼ x;   > > > L þ ðM  L Þtanh x  Lji ; : x4L ji

Note that in the derivations of (43) and (45) we have used the facts that  κ ¼ 1  α1 4 0, r 1  κ  r 2 ¼ 2ð1  α1 Þ 4 0 for 0 o α1 o 1. As a result, for any fixed x ¼ ðxT1 xT2 ÞT A ℜ2n , we get

ð48Þ T

ji

ji

M ji  Lji

ð53Þ

ji

with Lji o M ji ; i; j A Pf1; 2g  f1; 2g are design parameter, and tanhð UÞ being the standard hyperbolic tangent function. The sampling period was T ¼ 1 ms. The initial conditions were set as qð0Þ ¼ ð1:5; 1; 2ÞT ðradÞ and qð0Þ ¼ ð0; 0; 0ÞT ðradÞ. The gains of the proposed SFT control were selected according to input constraints (16) and the sufficient condition (24), by trial-anderror until a good tracking was obtained. The control gains of the proposed SFT control were determined as α1 ¼ 0:5, K p ¼ diagð17; 23; 16Þ, and K d ¼ diagð5; 6; 14Þ. According to the guidelines

Y. Su, J. Swevers / Robotics and Computer-Integrated Manufacturing 30 (2014) 91–98

95

Position eror [rad]

Position tracking errors of first link 2

0

-2

proposed SFT SP-SD+ 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Position error [rad]

Position tracking errors of second link 1 0 proposed SFT SP-SD+

-1 -2

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Position error [rad]

Position tracking errors of third link 2 proposed SFT SP-SD+

1 0 -1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time [sec] Fig. 2. Position tracking errors.

Velocity error [rad/sec]

Velocity tracking errors of first link 5

0 proposed SFT SP-SD+ -5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Velocity error [rad/sec]

Velocity tracking errors of second link 5 proposed SFT SP-SD+ 0

-5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Velocity error [rad/sec]

Velocity tracking errors of third link 2

0

-2

-4

proposed SFT SP-SD+ 0

0.5

1

1.5

2

2.5

3

Time [sec] Fig. 3. Velocity tracking errors.

3.5

4

4.5

5

96

Y. Su, J. Swevers / Robotics and Computer-Integrated Manufacturing 30 (2014) 91–98

Torque [Nm]

Torque [Nm]

Torque [Nm]

Input torques of the SFT control 20 0 link 2 -20

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

50 link 2

0 -50

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

50 0 link 3 -50

0

0.5

1

1.5

2

2.5 3 Time [sec]

3.5

4

4.5

5

Fig. 4. Input torques of the proposed SFT control.

Torque [Nm]

Torque [Nm]

Torque [Nm]

Input torques of the SP-SD+ control 50 0 link 1 -50

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

50 link 2 0 -50

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

50 0 link 3 -50

0

0.5

1

1.5

2

2.5 3 Time [sec]

3.5

4

4.5

5

Fig. 5. Input torques of the SP-SD þ control.

presented by Aguinga-Ruiz et al. [9], the gains of the SP-SD þ control were chosen as K 1 ¼ diagð5; 8; 10Þ, K 2 ¼ diagð500; 800; 1000Þ, M 11 ¼ 20, M 12 ¼ 30, M 13 ¼ 29, M 21 ¼ 4, M 22 ¼ 3, M 23 ¼ 3, and Lji ¼ 0:9M ji ; i; jA f1; 2g  f1; 2g. Figs. 2 and 3 illustrate the position and velocity tracking errors. After a transient due to errors in initial condition, both the position and velocity tracking errors tend to zero for both controllers. Moreover, the proposed SFT control gives a faster transient, especially for the second link. The requested input torques are shown in Figs. 4 and 5. Both requested input torques remain uniformly within the stated torque constraints.

of robot manipulators with actuator saturation. This is accomplished by replacing the feedback linear errors with the saturated exponential-like ones. Semi-global finite-time tracking stability is proven with Lyapunov's direct method and finite-time stability. The developed approach is an improvement upon the saturated robot tracking controller, and also solves the semi-global finitetime tracking problem for a large class of nonlinear systems with actuator saturation. Simulations demonstrate the improved performance of the proposed approach. The practical implications are that the actuators can be appropriately sized without an ad hoc saturation scheme to protect the actuator.

5. Conclusions

Acknowledgments

We have extended the conventional PD plus feedforward dynamics compensation to ensure semi-global finite-time tracking

Yuxin Su is funded by a senior post-doctoral fellowship of the KU Leuven Research council (SF/10/008).

Y. Su, J. Swevers / Robotics and Computer-Integrated Manufacturing 30 (2014) 91–98

97

c33 ¼ p7 sin ð2ðq2 þ q3 Þq_ 2 þ ðp7 sin ð2ðq2 þ q3 ÞÞ  2p11 2 sin ð2q3 ÞÞq_ 3

Appendix A

ð55Þ This appendix presents the dynamics of the three-DOF robot. The different elements of the robot manipulator model (1) are, respectively q ¼ ½q1 ; q2 ; q3 T 2 m11 m12 6 MðqÞ ¼ 4 m12 m12 m13 m13

D ¼ diagð0; 0; 0Þ; 2

m13

3 0 6 p cos ðq þ q Þ  ðp þ p Þ cos ðq Þ 7 gðqÞ ¼ 4 13 2 3 13 14 2 5 p13 ð cos ðq2 þ q3 Þ

ð56Þ

3

m13 7 5 m33

ð57Þ

with p1 ¼ 0:7437, p2 ¼ 0:5150, p3 ¼ 0:5200, p4 ¼ 4:0, p5 ¼ 0:31, p6 ¼ 0:4, p7 ¼ 0:1922, p8 ¼ 0:7322, p9 ¼ 0:5175, p10 ¼ 0:248, p11 ¼ 0:0025, p12 ¼ 0:32, p13 ¼ 12:152, and p14 ¼ 15:68 (in SI units).

m11 ¼ p1 þ 2p2 cos ðq2  q3 Þ2  2p3 ð cos ðq2  q3 Þ2 1Þ þ p4 ðp5 cos ðq2 þ q3 Þ þ p6 cos ðq2 Þ2 þ p7 cos ð2q2 Þ m12 ¼ p8 þ 2p2 cos ðq2  q3 Þ2  2p3 ð cos ðq2  q3 Þ2 1Þ þ p4 ðp5 cos ðq2 þ q3 Þ þ p6 cos ðq2 Þ2 þ p7 cos ð2q2 Þ m13 ¼ 2p9 cos ðq2 Þ þ p7 ð1 þ cos ð2ðq2 þ q3 ÞÞÞþ p10 ð cos ðq3 Þ þ cos ð2q2 þ q3 ÞÞ  2p11 cos ðq2  2q3 Þ m33 ¼ 2p3 þ 2p7 cos ðq2 þ q3 Þ2 4p11 cos ðq3 Þ; 2

c11 6 Cðq; q_ Þ ¼ 4 c21 c31

c12

c13

3

c22

c23 7 5

c32

0

c11 ¼  ðp7 ð sin ð2q2 Þ þ sin ð2ðq2 þq3 ÞÞÞ þ 2p10 sin ð2q2 þ q3 Þ  2p11 sin ð2ðq2  q3 Þ þ p12 sin ð2q2 ÞÞq_ 2  ðp4 p5 sin ðq2 þ q3 Þðp5 cos ðq2 þ q3 Þ þ p6 cos ðq2 Þ þ 2p11 sin ð2ðq2  q3 ÞÞÞq_ 3 c12 ¼  ðp7 ð sin ð2q2 Þ þ sin ð2ðq2 þq3 ÞÞÞ þ p12 sin ð2q2 Þ  2p11 sin ð2ðq2  q3 ÞÞÞq_ 1  ð2p7 ð sin ð2q2 Þ þ sin ð2ðq2 þ q3 ÞÞÞ þ 4p10 sin ð2q2 þq3 Þ  4p11 sin ð2ðq2  q3 ÞÞ þ 2p12 sin ð2q2 ÞÞq_ 2  ð2p7 sin ð2ðq2 þ q3 ÞÞ þ p9 sin ðq2 Þ þ p11 ð2 sin ð2ðq2 q3 ÞÞ  sin ðq2  2q3 ÞÞÞq_ 3 c13 ¼  ðp4 p5 sin ðq2 þ q3 Þðp5 cos ðq2 þ q3 Þ þ p6 cos ðq2 ÞÞ þ 2p11 sin ð2ðq2  q3 ÞÞÞq_ 1  ðp7 sin ð2ðq2 þ q3 ÞÞ þ p9 sin ðq2 Þ þ p10 ð sin ðq3 Þ þ 2 sin ð2q2 þ q3 ÞÞ þ p11 ð sin ð2ðq2  q3 Þ  sin ðq2  2q3 ÞÞÞq_ 2  ð2p7 sin ð2ðq2 þ q3 ÞÞ þ p10 ð sin ðq3 Þ þ 2 sin ð2q2 þ q3 ÞÞ þ 2p11 sin ðq2  2q3 ÞÞq_ 3 c21 ¼ ðp7 ð sin ð2ðq2 Þ þ sin ð2ðq2 þ q3 ÞÞÞ þ2p10 sin ð2q2 þ q3 Þ  2p11 sin ð2ðq2  q3 ÞÞ þ p12 sin ð2q2 ÞÞq_ 1 þ ðp9 sin ðq2 Þ  p10 sin ðq3 Þ  p11 ð2 sin ð2ðq2 q3 Þ þ sin ðq2  2q3 ÞÞÞq_ 3 c22 ¼  ðp7 ð sin ð2ðq2 Þ þ sin ð2ðq2 þ q3 ÞÞÞ þ 2p10 sin ð2q2 þ q3 Þ  2p11 sin ð2ðq2  q3 ÞÞ þ p12 sin ð2q2 ÞÞq_ 2 þ ðp7 sin ð2ðq2 þ q3 Þ þ p10 ð sin ðq3 Þ þ2 sin ð2q2 þq3 ÞÞ þ 2p11 sin ð2ðq2  q3 ÞÞÞq_ 3 c23 ¼ ðp9 sin ðq2 Þ  p10 sin ðq3 Þ  p11 ð2 sin ð2ðq2  q3 Þ þ sin ðq2  2q3 ÞÞÞq_ 1 þ ðp4 p5 sin ðq2 þ q3 Þðp5 cos ðq2 þ q3 Þ þ p6 cos ðq2 ÞÞ þ 2p11 sin ð2ðq2  q3 ÞÞq_ 2 þ ðp7 sin ð2ðq2 þ q3 Þ þ 2p10 ð sin ðq3 Þ þ sin ð2q2 þq3 ÞÞ þ 2p11 sin ðq2  2q3 ÞÞq_ 3 c31 ¼ ðp4 p5 sin ðq2 þq3 Þðp5 cos ðq2 þq3 Þ þ p6 cos ðq2 ÞÞ þ 2p11 sin ð2ðq2  q3 ÞÞÞq_ 1 þ ðp9 sin ðq2 Þ  p10 sin ðq3 Þ  p11 ð2 sin ð2ðq2 q3 Þ þ sin ðq2  2q3 ÞÞÞq_ 2 c32 ¼ p9 sin ðq2 Þ  p10 sin ðq3 Þ  p11 ð2 sin ð2ðq2  q3 Þ þ sin ðq2  2q3 ÞÞÞq_ 1 þ ðp7 sin ð2ðq2 þ q3 Þ þ 2p9 sin ðq2 Þ  p10 ð sin ðq3 Þ  sin ð2q2 þ q3 ÞÞ  2p11 ð sin ð2ðq2  q3 Þ þ sin ðq2  2q3 ÞÞÞq_ 2 þ p7 sin ð2ðq2 þ q3 ÞÞq_ 3

Rererences

ð54Þ

[1] Bascetta L, Rocco P. Revising the robust-control design for rigid robot manipulators. IEEE Transactions on Robotics 2010;26(1):180–6. [2] Liuzzo S, Tomei P. A global adaptive learning control for robotic manipulators. Automatica 2008;44(5):1379–84. [3] Sage HG, De Mathelin MF, Ostertag E. Robust control of robot manipulators: a survey. International Journal of Control 1999;72(6):1498–522. [4] Lewis FL, Dawson DM, Abdallah CT. Robot Manipulator Control: Theory and Practice. New York: Marcel Dekker; 2004. [5] Bernstein DS, Michel AN. A chronological bibliography on saturating actuators. International Journal of Robust and Nonlinear Control 1995;5(5):375–80. [6] Dixon WE. Adaptive regulation of amplitude limited robot manipulators with uncertain kinematics and dynamics. IEEE Transactions on Automatic Control 2007;52(3):488–93. [7] Su YX, Müller PC, Zheng CH. Global asymptotic saturated PID control for robot manipulators. IEEE Transactions on Control Systems Technology 2010;18 (6):1280–8. [8] Loria A, Nijmeijer H. Bounded output feedback tracking control of fully actuated Euler–Lagrange systems. Systems and Control Letters 1998;33(3): 151–161. [9] Aguinga-Ruiz E, Zavala-Rio A, Santibanez V, Reyes F. Global trajectory tracking through static feedback for robot manipulators with bounded inputs. IEEE Transactions on Control Systems Technology 2009;17(4):934–44. [10] Lefeber AAJ, Nijmeijer, H. Globally bounded tracking controllers for rigid robot systems. Proceedings of the 4th European Control Conference; 1997, Brussels, Belgium. [11] Dixon WE, de Queiroz MS, Zhang F, Dawson DW. Tracking control of robot manipulators with bounded torque inputs. Robotica 1999;17(2):121–9. [12] Moreno-Valenzuela J, Santibanez V, Campa R. A class of OFT controllers for torque-saturated robot manipulators: Lyapunov stability and experimental evaluation. Journal of Intelligent Robotic Systems 2008;51(1):65–88. [13] Su Y.X., Zheng C.H.. A simple repetitive learning control for asymptotic tracking of robot manipulators with actuator saturation. Proceedings of the 18th IFAC World Congress, Milano, Italy; 2011: 6886–6891. [14] Bhat SP, Bernstein DS. Continuous finite-time stabilization of the translational and rotational double integrators. IEEE Transactions on Automatic Control 1998;43(5):678–82. [15] Hong Y, Xu Y, Huang J. Finite-time control for robot manipulators. Systems and Control Letters 2002;46(4):243–53. [16] Qian CJ, Lin W. Recursive observer design, homogeneous approximation, and nonsmooth output feedback stabilization of nonlinear systems. IEEE Transactions on Automatic Control 2006;51(9):1457–71. [17] Perruquetti W, Floquet T, Moulay E. Finite-time observers: application to secure communication. IEEE Transactions on Automatic Control 2008;53(1): 356–360. [18] Orlov Y, Aoustin Y, Chevallereau C. Finite time stabilization of a perturbed double integrator—part I continuous sliding mode-based output feedback synthesis. IEEE Transactions on Automatic Control 2011;56(3):614–8. [19] Man Z, Paplinski AP, Wu H. A robust MIMO terminal sliding mode control scheme for rigid robotic manipulators. IEEE Transactions on Automatic Control 1994;39(12):2464–9. [20] Tang Y. Terminal sliding mode control for rigid robots. Automatica 1998;34(1): 51–56. [21] Feng Y, Yu X, Man Z. Non-singular terminal sliding mode control of rigid manipulators. Automatica 2002;38(12):2159–67. [22] Yu SH, Yu XH, Shirinzadeh B, Man ZH. Continuous finite-time control for robotic manipulators with terminal sliding mode. Automatica 2005;41(11): 1957–1964. [23] Jin M, Lee J, Chang PH, Choi C. Practical nonsingular terminal sliding-mode control of robot manipulators for high-accuracy tracking control. IEEE Transactions on Industrial Electronics 2009;56(9):3593–601. [24] Zhao DY, Li SY, Gao F. A new terminal sliding mode control for robotic manipulators. International Journal of Control 2009;82(10):1804–13. [25] Barambones O, Etxebarria V. Energy-based approach to sliding composite adaptive control for rigid robots with finite error convergence time. International Journal of Control 2002;75(5):352–9.

98

Y. Su, J. Swevers / Robotics and Computer-Integrated Manufacturing 30 (2014) 91–98

[26] Parra-Vega V, Hirzinger G. Chattering-free sliding mode control for a class of nonlinear mechanical systems. International Journal of Robust and Nonlinear Control 2001;11(12):1161–78. [27] Liu HT, Zhang T. Neural network-based robust finite-time control for robotic manipulators considering actuator dynamics. Robotics and ComputerIntegrated Manufacturing 2013;29(2):301–8.

[28] Su YX. Global continuous finite-time tracking of robot manipulators. International Journal of Robust and Nonlinear Control 2009;19(17):1871–85. [29] Su YX, Zheng CH. Global finite-time inverse tracking control of robot manipulators. Robotics and Computer-Integrated Manufacturing 2011;27(3):550–7. [30] Slotine JJE, Li W. Applied nonlinear control. NJ: Englewood Cliffs: Prentice Hall; 1991.