Robust approximate fixed-time tracking control for uncertain robot manipulators

Mechanical Systems and Signal Processing 135 (2020) 106379

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Robust approximate fixed-time tracking control for uncertain robot manipulators Yuxin Su a,⇑, Chunhong Zheng b, Paolo Mercorelli c a

School of Electro-Mechanical Engineering, Xidian University, Xi’an 710071, China School of Electronic Engineering, Xidian University, Xi’an 710071, China c Institute of Product and Process Innovation, Leuphana University of Lüneburg, D-21339 Lüneburg, Germany b

a r t i c l e

i n f o

Article history: Received 29 May 2019 Received in revised form 22 August 2019 Accepted 15 September 2019

Keywords: Fixed-time stability Robot control Robust control Terminal sliding mode control Trajectory tracking

a b s t r a c t This paper addresses the problem of robust trajectory tracking for uncertain robot manipulators within a priori fixed-time. A new sliding surface is first proposed and a robust control is developed for ensuring global approximate fixed-time convergence. Global approximate fixed-time convergence of tracking errors is proven that the position tracking errors globally converge to an arbitrary small set centered on zero within a uniformly bounded time and then go to zero exponentially. It is also proved that there exists a uniformly bounded a priori convergence time and such a bound is independent of the initial states. Advantages of the proposed approach include approximate fixed-time stability featuring faster transient and higher steady-state tracking precision and singularity-free. Numerical simulations and experimental results validate the effectiveness and improved performance of the proposed approach. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Trajectory tracking of robot manipulators may be recognized as one of its basic aim in robot control and at the same time finds its main application in robotic and automation field. High-precision trajectory tracking of robot manipulators is a topic that continues to challenge control theoreticians and engineers [1–4]. Finite-time stable control of robot manipulators offers an effective solution to this problem due to it may give rise to fast transient and high steady-state precision performances besides finite-time convergence [5–8]. This observation is supported by several works for finite-time tracking of robot manipulators. They can be divided into two categories, i.e. geometric homogeneity technique (GHT) and terminal sliding mode (TSM) control. GHT for regulation of robots is first reported in Ref. [9]. GHT-based controls developed in Refs. [10–12] produce global finite-time tracking for robot manipulators. These controls suffer from the drawback that they require the knowledge of robot dynamics is exactly known. TSM control (TSMC) presents a model-free design for finite-time tracking of uncertain robot manipulators. TSMC is first introduced in Ref. [13], where the control is designed to ensure that the closed-loop system trajectories not only reach the sliding surface in finite time but also, during the sliding phase, converge to the origin in finite time. Following this idea, various TSM schemes have been proposed. More specifically, in Ref. [14], a robust TSMC for global finite-time tracking of robot manipulators is proposed. A simple version of this TSMC is reported in Ref. [15]. The main disadvantage of these TSM approaches is the singularity problem [16,17]. The singularity limits the applications of the TSMC and seriously deteriorates ⇑ Corresponding author. E-mail address: [email protected] (Y. Su). https://doi.org/10.1016/j.ymssp.2019.106379 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

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Y. Su et al. / Mechanical Systems and Signal Processing 135 (2020) 106379

the performance of the control. This observation is supported by a review of literature which yields several different approaches to address singularity problem. A new sliding surface is proposed in Ref. [17] and adopted in Refs. [18,19], for nonsingular TSM finite-time tracking of robot manipulators. Along this line, a nonlinear term is added to produce a nonsingular fast TSMC finite-time tracking scheme [20]. This fast TSMC is later extended in Ref. [21] for finite time fault tolerant control of robot manipulators. The methods to avoid singularity by introducing some rigorous constraints on the sliding surface near the equilibrium are presented in Ref. [22,23]. But unfortunately it is recently shown that this approach is untenable [24]. A recursive derivative and integral TSM design is developed in Ref. [25]. An integral sliding surface is proposed and a continuous TSMC is presented in Ref. [26] for global finite-time tracking of robots. This approach is later extended for task space finite-time tracking [27]. More recently, a new integral nonsingular fast TSM surface is formulated and robust fault tolerant control for robot manipulators is proposed [28]. The minor weakness remains for these finite-time stable controls for robot manipulators is that the convergence time of the tracking errors relies heavily on the initial condition. This implies that the convergence time may grow unboundedly while the initial condition tends to infinity, which contradicts with the essence of the finite-time control implying that the controlled trajectories of the system become and remain zero within a finite time. Recently, fixed-time stable control of dynamical systems has been proposed to overcome this weakness. Compared with the finite-time control, the most appealing feature of the fixed-time control is that the convergence time is uniformly bounded a priori and independent of initial condition [29–31]. Moreover, besides the fixed-time convergence, fixed-time control can give rise to faster transient and higher precision. Recognizing these advantages, several fixed-time controls for second-order systems can be found in the literature. In particular, in Ref. [32], a nonsingular TSMC is proposed for fixed-time stabilization of double integrator systems and is applied to consensus tracking of second-order multi-agent networks. This scheme is later extended in Ref. [33] for fixed-time stabilization of nonlinear second-order systems. In Ref. [34], a nonsingular fast fixed-time TSMC is constructed and is applied to chaos suppression in power system. In Ref. [35], the bi-limit homogeneity technique of Ref. [29] is used for controller and observer-controller designs for fixed-time stabilization of double integrator systems. In Ref. [36], a fixed-time stabilization strategy for uncertain nonholonomic systems is proposed by adding one power integrator and switching techniques. In Ref. [37], a model-based nonlinear inverse dynamics control is proposed by using bi-limit homogeneity technique for global fixed-time tracking of robot manipulators. The main drawback of Ref. [37] is that it requires the dynamics of robot manipulators is exactly known. In a realistic scenario, exact knowledge of the dynamic model of robot manipulators can never be assumed. This implies that a robust control that does not refer to exact modeling information is more desirable. This paper presents a robust nonsingular TSMC for global approximate fixed-time tracking of robot manipulators in the presence of parametric uncertainties and external disturbances. The contribution of this paper is twofold. A new sliding surface is proposed to eliminate the singularity completely. A simple easy-going robust TSMC is proposed. Global approximate fixed-time tracking stability is proven that the position tracking errors globally converge to an arbitrary small set centered on zero within a uniformly bounded time and then go to zero exponentially. The appealing advantages of the proposed control are that it offers an easy-going robust control for global approximate fixed-time tracking of uncertain robot manipulators featuring faster transient and higher steady-state precision tracking. The remainder of this paper is organized as follows. In Section 2, some preliminaries including the model and properties of robot manipulators and fixed-time stability of dynamical systems are introduced. The control design and stability analysis are presented in Section 3. In Section 4, we illustrate our design via simulation performed on a two-DOF robot. The proposed approach is further experimental validated with a two-DOF robotic system in Section 5. Finally, a conclusion is included in Section 6.

2. Problem statement and preliminaries 2.1. Robot model and properties The dynamics of an n-DOF rigid robot manipulator with revolute joints can adequately be described as [38]

€ þ C ðq; q_ Þh_ þ g ðqÞ þ dðt Þ ¼ s M ðqÞq

ð1Þ

_ q € 2 R denotes the position, velocity, and acceleration, respectively, M ðqÞ 2 R where q; q; is the symmetric and positive definite inertia matrix, C ðq; q_ Þ 2 Rnn denotes the centrifugal-Coriolis matrix, g ðqÞ 2 Rn is the gravitational term, dðtÞ 2 Rn is the bounded external disturbance satisfying k dðt Þ k  dM with dM is a known positive constant, and s 2 Rn denotes the torque input vector. Recalling that robot manipulators are being considered, the following property can be established. n

nn

Property 1 [38]. The matrices M ðqÞ and C ðq; q_ Þ and the gravitational term g ðqÞ are bounded by

Mm  k M ðqÞ k  MM ;

8 q 2 Rn

ð2Þ

k C ðq; q_ Þ k  C M k q_ k ;

8q; q_ 2 Rn

ð3Þ

Y. Su et al. / Mechanical Systems and Signal Processing 135 (2020) 106379

8 q 2 Rn

k g ðqÞ k  GM ;

3

ð4Þ

where M m , M M , C M , and GM are some known positive constants, and k  k denotes the standard Euclidean norm. The subsequent development is based on the assumption that position q and velocity q_ are available and the desired trajectory qd 2 Rn be any C 2 for the system (1). Additionally, the following assumption will be exploited [17,18,20]. Assumption 1. The model parameters can be described as

M ðqÞ ¼ M 0 ðqÞ þ DM ðqÞ

ð5Þ

C ðq; q_ Þ ¼ C 0 ðq; q_ Þ þ DC ðq; q_ Þ

ð6Þ

g ðqÞ ¼ g 0 ðqÞ þ Dg ðqÞ

ð7Þ

where M 0 ðqÞ, C 0 ðq; q_ Þ, and g 0 ðqÞ denote the nominal parts, and DM ðqÞ, DC ðq; q_ Þ, and Dg ðqÞ the model uncertainties. Our objective is to design a singularity-free TSMC for the uncertain robot manipulators subject to bounded disturbances such that the trajectory tracking is approximate fixed-time ensured within a predefined time that independents of initial condition. To quantify this objective, a position tracking error eðtÞ 2 Rn is defined as

e ¼ q  qd

ð8Þ

2.2. Preliminaries The concept and one lemma on fixed-time stability of dynamical systems are first introduced. A lemma that will play a key role in the subsequent stability analysis is also revisited. Definition 1 (Fixed-time stability [30]). Consider the following system

x_ ¼ f ðxÞ;

x 2 Rn

xð0Þ ¼ x0 ;

ð9Þ

n

with f : U 0 ! R is a nonlinear function and can be discontinuous on an open neighborhood U 0 of the origin. The origin of (9) is said to be fixed-time stable in a neighborhood U  U 0 , if it is finite-time stable and the convergence time T ðx0 Þ is uniformly upper bounded, i.e., for any x0 2 Rn , there exists a known positive constant T max such that T ðx0 Þ  T max . When U ¼ Rn , global fixed-time stability is obtained. Lemma 1. [30, Lemma 1]. Consider the system (9). Suppose there exist a positive definite continuous function V ðxÞ : U ! R, real numbers a; b > 0, 0 < p < 1, and q > 1 and an open neighborhood U  U 0 of the origin such that V_ ðxÞ þ aV p ðxÞ þ bV q ðxÞ  0; x 2 U 0 ff0g. Then, the origin of system (9) is fixed-time stable. Moreover, if U ¼ Rn , V is proper

and V_ takes negative value on Rn ff0g, then the origin is globally fixed-time stable. The convergence time is uniformly upper bounded by

T  T max ¼

1

að1  pÞ

þ

1 bðq  1Þ

ð10Þ

Lemma 2 [32]. For any n1 ; n2 ; . . . ; nn  0, the following inequality holds n X

nri

n X



i¼1 n X i¼1

!r ni

;

for 0 < r  1

ð11Þ

i¼1

nri

n

1r

n X

!r ni

;

for r  1

i¼1

3. Control development 3.1. Control formulation Before constructing the controller, a nonlinear function sp ðxÞ is first introduced as follows [39,40]

ð12Þ

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Y. Su et al. / Mechanical Systems and Signal Processing 135 (2020) 106379

( s p ð xÞ ¼

jxjp sgnðxÞ; jxj  d dp1 x; j xj < d

ð13Þ

where 0 < p < 1 and 0 < d  1 are two positive constant parameters to be designed, and sgnðÞ is the standard signum function. The first derivative of sp ðxÞ with respect to x is given by

(

p

f ð xÞ ¼

pjxjp1 ; jxj  d dp1 ; j xj < d

ð14Þ q

Another nonlinear function sig ðxÞ is introduced as follows q

sig ðxÞ ¼ jxjq sgnðxÞ

ð15Þ

with q > 1. Comparing (13) with (15), it is clear that the function sp ðxÞ introduces a linear term for the case of jxj < d. Otherwise, the p control design with sp ðxÞ may suffer from the singularity problem due to the first derivative of sp ðxÞ (i.e. f ðxÞ given by (14)) contains a negative power ð p  1Þ on x (see (14) for details), which may go to infinity while x goes to zero. This may lead to the requested control effort goes to infinity. q An example of the nonlinear function sp ðxÞ and sig ðxÞ is illustrated in Fig. 1 with d ¼ 0:01, and p ¼ 0:5 for sp ðxÞ, q ¼ 1:5 for q sig ðxÞ, respectively. From Fig. 1, it is clear that the function sp ðxÞ has the characteristics of enlargement of small argument q and reduction of large argument. While for the function sig ðxÞ, it has a reverse characteristics, that is, the enlargement of large argument and reduction of small argument. q To aid the subsequent control design and analysis, we define the vectors S p ðeÞ; Sig ðeÞ 2 Rn and diagonal matrices F p ðeÞ; H q ðeÞ 2 Rnn as follows: T

S p ðeÞ ¼ ½sp ðe1 Þ;    ; sp ðen Þ

ð16Þ

 q T q q Sig ðeÞ ¼ sig ðe1 Þ;    ; sig ðen Þ

ð17Þ

 p  p F p ðeÞ ¼ diag f ðe1 Þ;    ; f ðen Þ

ð18Þ

  H q ðeÞ ¼ diag je1 jq1 ;    ; jen jq1

ð19Þ

where 0 < p < 1 and q > 1, diagð  Þ denotes a diagonal matrix, and ei ; i ¼ 1; 2; . . . ; n denotes the ith component of the vector e. q On the basis of the properties of the nonlinear functions sp ðÞ and sig ðÞ, a novel sliding surface is designed as q s ¼ e_ þ K 1 S p ðeÞ þ K 2 Sig ðeÞ

ð20Þ

nn

where K 1 ; K 2 2 R are constant positive definite diagonal gain matrices. Based on the subsequent analysis, we propose the following TSMC to solve the above stated problem:

s ¼ s0 þ s1 þ s2

ð21Þ

q

Fig. 1. Illustration of the nonlinear functions sp ðxÞ and sig ðxÞ.

Y. Su et al. / Mechanical Systems and Signal Processing 135 (2020) 106379

5

with

s0 ¼ M 0 ðqÞq€ d þ C 0 ðq; q_ Þq_ þ g 0 ðqÞ 

ð22Þ 

s1 ¼ K 0 Sig r ðsÞ  M0 ðqÞ K 1 F p ðeÞ þ qK 2 H q ðeÞ e_ s2 ¼ 

s u ksk

ð23Þ ð24Þ

and

u¼

 1  2 k þ b0 þ b2 k q_ k þ ck s0 þ s1 k 1c

ð25Þ

where K 0 2 Rnn is a constant positive definite diagonal control gain matrix, r > 1 and k are positive gains, and b0 and b2 , and c are known positive constants, respectively, defined by subsequent (34) and (33). 3.2. Error system development Let us first develop the closed-loop dynamics of the robot manipulators. Based on Assumption 1, the dynamic equation of the robot manipulator can be rewritten as

€ þ C 0 ðq; q_ Þq_ þ g 0 ðqÞ ¼ s þ q M 0 ðqÞq

ð26Þ

with the lumped uncertainty qðtÞ 2 R is defined as n

q ¼ DM ðqÞq€  DC ðq; q_ Þq_  Dg ðqÞ  dðtÞ

ð27Þ

In light of (1) and (5), it follows that

€ ¼ Eðs  C ðq; q_ Þ  g ðqÞ  dðtÞÞ DM ðqÞq

ð28Þ

where E 2 Rnn is defined by Refs. [38,41]

E ¼ I n  M 0 ðqÞM 1 ðqÞ

ð29Þ

and I n denotes the n  n identity matrix. Similar to Refs. [38,41], once M 0 ðqÞ is chosen as

M0 ¼

2 In m1 þ m2

ð30Þ

where m1 and m2 are two positive constants defined by

m1  k M 1 ðqÞ k  m2

ð31Þ

then E is upper bounded by Refs. [38,41]

kEkc

ð32Þ

with c is a known positive constant given by

c ¼ ðm2  m1 Þ=ðm1 þ m2 Þ < 1

ð33Þ

By Property 1, Assumption 1, (27), (28), and (32), the uncertainty qðtÞ can be upper bounded as follows:

k q k  b0 þ b2 k q_ k þ ck s k 2

ð34Þ

where b0 and b2 are two known positive constants. Remark 1. Compared with the work of Refs. [17,18,20], a term related to control input s is added in the upper bound of qðtÞ. Hence, the assumption on the boundedness of the control input s of Refs. [17,18,20] is no longer needed. In fact, the control proposed in Refs. [17,18,20] does not meet that required boundedness. Now let us formulate the closed-loop dynamics for s. For this purpose, after taking the time derivative of (20) and multiplying both sides of the equation by M 0 ðqÞ, we have

  M 0 ðqÞs_ ¼ M 0 ðqÞ€e þ M 0 ðqÞ K 1 F p ðeÞ þ qK 2 H q ðeÞ e_

ð35Þ

€ from (26) into the resulting equation, yields Upon applying the definition of e given by (8) to (35) and substituting M 0 ðqÞq

  € d  C 0 ðq; q_ Þq_  g 0 ðqÞ þ M 0 ðqÞ K 1 F p ðeÞ þ qK 2 H q ðeÞ e_ M 0 ðqÞs_ ¼ s þ q  M 0 ðqÞq

Substituting the proposed control law (21) into (36) and using (22)–(25), the closed-loop dynamics for s take

ð36Þ

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Y. Su et al. / Mechanical Systems and Signal Processing 135 (2020) 106379

M 0 ðqÞs_ ¼ 

 1 s  2 r k þ b0 þ b2 k q_ k þ ck s0 þ s1 k þ q  K 0 Sig ðsÞ 1c ksk

ð37Þ

Note that the system (37) is discontinuous, its solutions are understood in the Filippov sense [42]. 3.3. Stability analysis Now, we are in a position to state the following result. Theorem 1. Given the uncertain robotic system described by (1), the robust controller defined by (21)–(25) ensures that the position tracking errors globally converge to an arbitrarily small set d centered on zero within a bounded convergence time

T  Ts þ Te

ð38Þ

and then go to zero exponentially, in which T s and T e are two known positive constants, respectively, defined by

! 2 1 nðr1Þ=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir þ T s  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m1 þ m2 k ðr  1Þkmin ðK 0 Þ ðm1 þ m2 Þ 2ð1pÞ=2 ð2nÞ1q=2 þ ð1  pÞkmin ðK 1 Þ ðq  1Þkmin ðK 2 Þ

Te 

ð39Þ

! ð40Þ

with kmin ðAÞ is the minimum eigenvalue of the positive definite diagonal matrix A. Proof. Define a proper positive definite Lyapunov function candidate as follows:

V¼

1 T s M 0 ðqÞs 2

ð41Þ

Note from (30) that M 0 ðqÞ is a constant matrix. Differentiating V with respect to time along (37), we have

V_ ¼ sT M 0 ðqÞs_

ð42Þ

After substituting M 0 ðqÞs_ from (37) into (42) leads to

 1 sT s  2 r k þ b0 þ b2 k q_ k þ ck s0 þ s1 k þ sT q  sT K 0 Sig ðsÞ 1c ksk   1 2 r ¼ k s k k þ b0 þ b2 k q_ k þ ck s0 þ s1 k þ sT q  sT K 0 Sig ðsÞ 1c

V_ ¼ 

Upon applying (25) to (43), we have

 2 V_ ¼ c k s ku  k s k k þ b0 þ b2 k q_ k þ ck s0 þ s1  2  c k s ku  k s k k þ b0 þ b2 k q_ k þ ck s0 þ s1

 r k þ sT q  sT K 0 Sig ðsÞ  r k þ k s kk q k  sT K 0 Sig ðsÞ

ð43Þ

ð44Þ

Substituting (34) into (44), it follows that

    2 2 r V_  ck s k u  k s k k þ b0 þ b2 k q_ k þ ck s0 þ s1 k þ k s k b0 þ b2 k q_ k þ ck s k  sT K 0 Sig ðsÞ  sT K 0 Sig ðsÞ  kk s k  c k s ku þ ck s kk s2 k r

ð45Þ

Note that in the derivation of (45) we have used the fact from (21) that k s k  k s0 þ s1 k þ k s2 k.

Applying (24) and (25) to (43) and noting from (25) that u > 0, we obtain the following upper bound on V_ r V_  sT K 0 Sig ðsÞ  kk s k ¼ 

n X

k0i jsi jrþ1  kk s k  kmin ðK 0 Þ

i¼1

n X

jsi jrþ1  kk s k

ð46Þ

i¼1

where k0i and kmin ðK 0 Þ are the ith element and minimum eigenvalue of the diagonal matrix K 0 , respectively. Now invoking (12) of Lemma 2, the final upper bound on V_ is

V_  nð1rÞ=2 kmin ðK 0 Þk s krþ1  kk s k

ð47Þ

By virtue of (30) and (41), we have

V¼

1 k s k2 m1 þ m2

ð48Þ

Y. Su et al. / Mechanical Systems and Signal Processing 135 (2020) 106379

7

Applying (48) to (47), it follows that

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffi V_ þ k m1 þ m2 V þ nð1rÞ=2 kmin ðK 0 Þðm1 þ m2 Þðrþ1Þ=2 V ðrþ1Þ=2  0

ð49Þ

By Lemma 1, global fixed-time convergence of s to zero directly follows. Moreover, the uniformly upper bounded convergence time is given by Ref. (39) in Theorem 1. Once the sliding surface s ¼ 0 is achieved within T s , the close-loop system dynamics are govern by q e_ ¼ K 1 S p ðeÞ  K 2 Sig ðeÞ

ð50Þ

The convergence of position tracking errors can be analyzed by the following two cases. Case 1: (k e k  d). For this case, in light of (13), (16), and (17), the dynamics (50) can be explicitly rewritten as p q e_ ¼ K 1 Sig ðeÞ  K 2 Sig ðeÞ

ð51Þ

For system (51), the proper positive definite Lyapunov function candidate is proposed as

V1 ¼

1 T e e 2

ð52Þ

The time derivative of V 1 along (51) is

V_ 1 ¼ eT e_

ð53Þ

After substituting (51) into (52) yields p q V_ 1 ¼ eT K 1 Sig ðeÞ  eT K 2 Sig ðeÞ ¼ 

n X

k1i jei jpþ1 

n X

i¼1

k2i jei jqþ1

ð54Þ

i¼1

where k1i and k2i denote the ith elements of the diagonal matrices K 1 and K 2 , respectively. By virtue of Lemma 2, the upper bound of V_ 1 can be written as

V_ 1  kmin ðK 1 Þk e kpþ1  nð1qÞ=2 kmin ðK 2 Þk e kqþ1

ð55Þ

Upon applying (52)–(55), it follows that

V_ 1  2ðpþ1Þ=2 kmin ðK 1 ÞV ðpþ1Þ=2  2ðqþ1Þ=2 nð1qÞ=2 kmin ðK 2 ÞV ðqþ1Þ=2

ð56Þ

Invoking Lemma 1 again, the convergence of the position tracking error to the arbitrary small d centered on zero directly follows and the convergence time is explicitly given by (40) in Theorem 1. Case 2: (k e k < d). In light of (13) and (16), the dynamics (50) can be explicitly rewritten as q e_ ¼ dp1 K 1 e  K 2 Sig ðeÞ

ð57Þ

With the proper positive definite Lyapunov function candidate V 1 defined by (52), we have q V_ 1 ¼ dp1 eT K 1 e  eT K 2 Sig ðeÞ  dp1 kmin ðK 1 Þk e k2 

n X

k2i jei jqþ1  2dp1 kmin ðK 1 ÞV 1

ð58Þ

i¼1

By Lyapunov stability theory [43], it is clear that the position tracking errors converge to zero exponentially. This completes the proof. h Remark 2. The proposed TSMC does not refer to modeling parameters in the control law formulation, which would give rise to global approximate fixed-time tracking of robot manipulators in the presence of parametric uncertainties and bounded disturbances. The improved performances such as faster transient and higher steady-state tracking precision are expected. Remark 3. The proposed fixed TSMC can be easily extended to a class of uncertain second-order nonlinear dynamical systems for global approximate fixed-time tracking, which represents a broader class of problems:

x_ 1 ¼ f 1 ðx1 ; x2 Þ; x_ 2 ¼ f 2 ðx1 ; x2 Þ þ g ðx1 ; x2 Þ þ Dðt; x1 ; x2 Þ þ Bðx1 ; x2 Þs

ð59Þ

where x1 ; x2 2 Rn , f 1 ; f 2 2 Rn are smooth vector functions and g 2 Rn represents time-invariant uncertainties and disturbances satisfying k g k 2 ‘1 , Dðt; x1 ; x2 Þ 2 Rn is time-variant uncertainty term, and B 2 Rnn is a non-singular input matrix.

8

Y. Su et al. / Mechanical Systems and Signal Processing 135 (2020) 106379

Remark 4. It is clear that the proposed control is discontinuous, which may induce potential chattering. To eliminate this chattering, the commonly-used boundary layer technique [43] is adopted. That is, the discontinuous control input (24) is modified as

s2 ¼ 

s u k s k þ s0

ð60Þ

where s0 is a positive constant. Similar to the commonly-used boundary modification for discontinuous control [43], this modified continuous control only assures bounded tracking result. Remark 5. The proposed control can be tuned by the following guidance: Based on the estimate of the inertia matrix M ðqÞ, we first choose two positive constants m1 and m2 such that the inequality (31) is satisfied. With this, we can choose M 0 ðqÞ and c in accordance with (30) and (33), respectively. Then according to the amplitude of the parametric uncertainty and the bounded external disturbances, we can select the two positive constants b0 and b2 such that the uncertainty qðtÞ defined by (27) to satisfy the upper bound (34). After that, we can select the other control gains K 0 , K 1 , K 2 , r > 1, k, 0 < p < 1, 0 < d  1, and q > 1 by trial-and-error for a good tracking performance. Normally, d can be chosen as d ¼ 0:01 for most robotic systems and smaller p is helpful for fast transient but too small p may induce chattering, due to the fact that the nonlinear function sp ðxÞ defined by (13) is continuous but non-smooth (see (14) for details). Hence, there exists a trade-off for p between better q performance and lower chattering. q may choose as small as possible in the range of q > 1 due to the function sig ðxÞ has the characteristics of enlargement for large argument and reduction for small argument (see Fig. 1). Large q may result in excessive requested control inputs and this may lead to the harmful actuator saturation. r > 1 can be chosen in a similar way to q. The remaining gains K 0 , K 1 , K 2 , and k can be selected as large as possible for faster transient and higher steady-state tracking precision. But note that too large K 0 , K 1 , K 2 , and k also may induce the actuator saturation. 4. Numerical simulations Comparisons on the two-DOF robot used in Refs. [17,20] are conducted to demonstrate the improved performances of the proposed controller. The dynamics are given in the form (1) with the following entries



M¼ C¼

p1 þ 2p2 cosðq2 Þ p3 þ p2 cosðq2 Þ p3 þ p2 cosðq2 Þ

p2 sinðq2 Þq_ 1 0



ð61Þ

p4 2p2 sinðq2 Þq_ 1 p2 sinðq2 Þq_ 2

g ¼ ½p5 cosðq1 Þ þ p6 cosðq1 þ q2 Þ;

ð62Þ

p6 cosðq1 þ q2 ÞT

ð63Þ

with

p1 ¼ ðm1 þ m2 Þr21 þ m2 r 22 þ J 1 ; p4 ¼ p3 þ J 2 ;

p2 ¼ m2 r 1 r 2 ;

p5 ¼ ðm1 þ m2 Þr1 g 1 ;

p3 ¼ mr 22

p6 ¼ m2 r 2 g 1

ð64Þ

The parameters are given in SI units and summarized as [17]: m1 ¼ 0:5, m2 ¼ 1:5, r1 ¼ 1:0, r2 ¼ 0:8, J 1 ¼ J 2 ¼ 5:0, and g 1 ¼ 9:8. The nominal value of m1 and m2 are m10 ¼ 0:4 and m20 ¼ 1:2. The parametric uncertainties of this robotic system can be easily calculated with Assumption 1 by inserting the real systematic parameters and the nominal ones into (61)–(64). The desired trajectories qd ¼ ½qd1 ; qd2 T ðradÞ are the same as that in Refs. [17,20] with

qd1 ¼ 1:25  ð7=5Þexpðt Þ þ ð7=20Þexpð4t Þ qd2 ¼ 1:25 þ expðt Þ  ð1=4Þexpð4t Þ

ð65Þ

h iT The sampling period is T ¼ 1 ms. The initial conditions are qð0ÞT ; q_ ð0ÞT ¼ ½1:0; 1:5; 0; 0T . Note that the simulation conditions are the same as [17] for a fair comparison. Taking the practical implementation into account, the maximum torques of actuators are limited as of s max ¼ ½100; 50T Nm. Comparison with the nonsingular TSMC proposed by Feng et al. in Ref. [17] is first conducted and performed with the same conditions as Ref. [17]. According to Theorem 2 of Ref. [17], the TSMC is given by

ðe_ Þ

ð66Þ

s ¼ s0 þ u0 þ u1

ð67Þ

s ¼ e þ Sig

m=n

where m > n are all positive odd integers and, s0 is the same as in (22), and u0 and u1 are defined as

Y. Su et al. / Mechanical Systems and Signal Processing 135 (2020) 106379

u0 ¼ 

u1 ¼ 

n 2m=n M 0 ðqÞSig ðe_ Þ m

ð68Þ

  _ 2 þ k1 k M 1 0 ðqÞ k b0 þ b1 k q k þ b2 k q k ksk

9

s

ð69Þ

where b0 , b1 , and b2 are known positive upper bounds, and k1 is a positive design constant. For the robot, the upper bounds are calculated as b0 ¼ 12, b1 ¼ 2:2, and b2 ¼ 2:8. The lower and upper bounds of the inverse inertial matrix defined by (31) are given by m1 ¼ 0:09 and m2 ¼ 0:2 and hence d ¼ 0:3793. The other control gains of the proposed fixed-time TSMC (Fixed TSMC) are chosen by trial-and-error until a good tracking is obtained. They are d ¼ 0:01, p ¼ 0:5, q ¼ 1:2, r ¼ 1:2, k ¼ 1, K 0 ¼ diagð2; 2Þ, K 1 ¼ diagð2; 2Þ, and K 2 ¼ diagð2; 2Þ. The gains of the TSMC presented in Ref. [17] are m ¼ 5, n ¼ 3, and k1 ¼ 4. The position tracking error comparisons are illustrated in Fig. 2. For a clear comparison, the requested control inputs of these two controls are shown in Figs. 3 and 4, respectively. As we see, both controllers ensure that the robot complete the desired position tracking correctly, and after a transient due to errors in initial conditions, the positioning tracking errors tend to zero. Obviously, the proposed Fixed TSMC obtains a much faster transient in comparison with the TSMC. To eliminate the chattering, the numerical simulation with the modified continuous control (60) is performed with s0 ¼ 0:005. The simulation results are illustrated in Figs. 5 and 6. Clearly, this modified continuous Fixed TSMC also achieves a quite favorable faster transient and the requested control inputs are continuous.

Fig. 2. Position tracking errors of Fixed TSMC and TSMC.

Fig. 3. Requested inputs of TSMC.

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Y. Su et al. / Mechanical Systems and Signal Processing 135 (2020) 106379

Fig. 4. Requested inputs of Fixed TSMC.

Fig. 5. Position tracking errors of the modified continuous Fixed TSMC.

Fig. 6. Requested inputs of the modified continuous Fixed TSMC.

Y. Su et al. / Mechanical Systems and Signal Processing 135 (2020) 106379

11

After that, the nonsingular fast TSMC (FTSMC) proposed by Ref. [20] is also selected for comparison. The disturbances are the same as Ref. [20] and given as dðt Þ ¼ ½2sinðt Þ þ 0:5sinð200pt Þ; cosð2tÞ þ 0:5sinð200ptÞ . The control input of the FTSMC s and s0 is the same as that in (67) and (22), respectively, and u0 is T

   h u0 ¼ M 0 ðqÞ k2 s þ C1 I 2 þ C1 diag jejC1 I2 2

2I2 C2

Sign

ðe_ Þ

i

ð70Þ

with k2 is a positive gain to be selected. u1 is the same as (69) with the following defined s C

C

s ¼ e þ Sign 1 ðeÞ þ Sign 2 ðe_ Þ

ð71Þ

    where diag jejC1 I2 is defined as diag je1 jC11 1 ; je2 jC12 1 and C1i ; ði ¼ 1; 2Þ denotes the ith diagonal elements of the positive h iT C C C definite constant diagonal gain matrix C1 , and the vector Sign 1 ðeÞ ¼ sig 11 ðe1 Þ; sig 12 ðe2 Þ . Here they are chosen the same as that in Ref. [20]: k1 ¼ 1, k2 ¼ 2, C1 ¼ diagð2; 2Þ, C2 ¼ diagð5=3; 5=3Þ, b0 ¼ 12, b1 ¼ 2:2, and b2 ¼ 2:8. We compare the modified continuous Fixed TSMC and FTSMC as the same conditions in Ref. [20]. The initial conditions of h iT Ref. [20] are changed as qð0ÞT ; q_ ð0ÞT ¼ ½3:0; 2:5; 0; 0T . First, the control gains of the proposed Fixed TSMC are kept unchanged. The obtained position tracking errors and requested inputs are shown in Figs. 7 and 8, respectively. From the comparisons, it can be clearly seen that the proposed Fixed TSMC obtains a faster transient over the FTSMC for this large initial conditions and bounded disturbances. After that, due to the control inputs of the proposed Fixed TSMC are much smaller than those of the FTSMC and the control capability of the proposed Fixed TSMC does not take full apply, we increase the control gain K 0 of the Fixed TSMC to K 0 ¼ diagð5; 4Þ and the others are kept unchanged. The simulation results are illus-

Fig. 7. Position tracking errors of modified continuous Fixed TSMC and FTSMC.

Fig. 8. Requested inputs of modified continuous Fixed TSMC and FTSMC.

12

Y. Su et al. / Mechanical Systems and Signal Processing 135 (2020) 106379

trated in Figs. 9 and 10. Obviously, with this large control gain, the transient of the proposed Fixed TSMC is further reduced. Note that this much improved performance of the proposed Fixed TSMC is obtained without an excessive control effort (see Fig. 10). One may argue that the given desired trajectories for the above simulations may become constant after a short time interval and the time-varying property of which is not so obvious. To clarify this argument, the numerical simulation comparison on desired sinusoidal trajectories is further carried out. The strong changed desired sinusoidal trajectories qd ¼ ½qd1 ; qd2 T ðradÞ are given as

qd1 ¼ 1:25 þ ðp=3Þsinð2t þ p=4Þ qd2 ¼ 0:5 sinð2t þ p=4Þ

ð72Þ

This comparison is accomplished with the modified continuous Fixed TSMC and the continuous fast TSMC (FTSMC) of Ref. h iT [20]. The initial conditions are qð0ÞT ; q_ ð0ÞT ¼ ½1:0; 1:5; 0; 0T . The same control gains as the results presented in Figs. 7 and 8 are selected. To clearly illustrate this desired sinusoidal trajectories, the position tracking with the modified continuous Fixed TSMC is shown in Fig. 11. The position tracking errors and requested control efforts are illustrated in Figs. 12 and 13, respectively. Obviously, for this strong changed desired sinusoidal trajectories, both the modified continuous Fixed TSMC and FTSMC of Ref. [20] also assure a satisfactory tracking and the modified continuous Fixed TSMC achieves a much faster transient over the FTSMC of Ref. [20].

Fig. 9. Position tracking error comparison with the large gains of continuous Fixed TSMC.

Fig. 10. Requested inputs with the large gains of continuous Fixed TSMC.

Y. Su et al. / Mechanical Systems and Signal Processing 135 (2020) 106379

Fig. 11. Position tracking of sinusoidal trajectories with continuous Fixed TSMC.

Fig. 12. Position tracking error comparison of sinusoidal trajectories.

Fig. 13. Requested inputs of sinusoidal trajectories.

13

14

Y. Su et al. / Mechanical Systems and Signal Processing 135 (2020) 106379

The numerical simulations clearly demonstrate that the proposed robust control offers a much improved control design for faster transient and higher steady-state tracking of robot manipulators in the presences of parametric uncertainties and bounded disturbances.

Fig. 14. Picture of the robotic system.

Fig. 15. Schematic representation of the robotic system.

Fig. 16. Desired positions.

Y. Su et al. / Mechanical Systems and Signal Processing 135 (2020) 106379

15

5. Experimental results The proposed Fixed TSMC is further validated experimental on an industrial robot manipulator. The diagram of the robotic system is shown in Fig. 14 and is drawn schematically in Fig. 15. It consists of a motor, a dSPACE (DS1103) control board, and a multilink robot. The servo is operated in torque mode, so the motor acts as torque source and it accepts an analog voltage as a reference of torque signal. The feedback position is measured by an incremental encoder. The controller executes programs at a sampling frequency 1 kHz. According to the used motors, the supply torques are limited within     T s max ¼ ½30; 30T Nm. The desired positions are set as qd ¼ p=4sinð0:5tÞ 1  exp 0:05t3 ; 1  expð0:1tÞ ðradÞ and illustrated in Fig. 16. All the initial conditions are zero. Similar to the numerical simulations, we conduct the comparisons with the modified continuous versions of the proposed Fixed TSMC, TSMC of Ref. [17], and FTSMC of Ref. [20]. The layer width is s0 ¼ 0:005. The upper bounds for this robotic system are b0 ¼ 15, b1 ¼ 3, and b2 ¼ 4. The other gains of all controls are chosen by trial-and-error until a good tracking performance was obtained and they are summarized in Table 1.

Table 1 Control gains for experiment. Controller

Gains

Fixed TSMC TSMC FTSMC

d ¼ 0:01, p ¼ 0:5, q ¼ 1:2, c ¼ 0:5, r ¼ 1:5, k ¼ 1, K 0 ¼ diagð4; 3Þ, K 1 ¼ diagð2; 2Þ, K 2 ¼ diagð2; 2Þ m ¼ 5, n ¼ 3, k1 ¼ 4 k1 ¼ 1, k2 ¼ 3, C1 ¼ diagð2; 2Þ, C2 ¼ diagð5=3; 5=3Þ

Fig. 17. Position tracking errors of the continuous Fixed TSMC and FTSMC.

Fig. 18. Position tracking errors of the continuous Fixed TSMC and TSMC.

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Y. Su et al. / Mechanical Systems and Signal Processing 135 (2020) 106379

Fig. 19. Requested inputs of the first joint.

Fig. 20. Requested inputs of the second joint.

The experimental position tracking error comparisons are shown in Figs. 17 and 18. The requested control inputs are illustrated in Figs. 19 and 20. As we see, the three controls ensure a favorable position tracking result. Similar to the numerical simulations, the proposed Fixed TSMC obtains an improved performance over the FTSMC and TSMC. Note that the favorable result of the proposed Fixed TSMC is obtained without an excessive control effort.

6. Conclusions A new sliding surface and terminal sliding mode control have been proposed for global approximate fixed-time tracking of uncertain robot manipulators with bounded disturbances. Advantages of the proposed approach include the elimination of the singularity completely and an ability to ensure approximate fixed-time tracking stability featuring the position tracking errors globally faster converge to an arbitrary small set centered on zero within a uniformly predefined time and then go to zero exponentially. Numerical simulations and experimental results demonstrate the improved performance of the proposed approach. The developed approach offers an improved tracking control design for uncertain robot manipulators, and also solves the global approximate fixed-time tracking problem for a large class of uncertain second-order nonlinear systems. Future efforts will focus on finding a robust control for fixed-time stable trajectory tracking of uncertain robot manipulators subject to bounded external disturbances and actuator constraints. References [1] A. Ferrara, G.P. Incremona, Design of an integral suboptimal second-order sliding mode controller for the robust motion control of robot manipulators, IEEE Trans. Control Syst. Technol. 23 (6) (2015) 2316–2325. [2] G.P. Incremona, A. Ferrara, L. Magni, MPC for robot manipulators with integral sliding modes generation, IEEE Trans. Mechatronics 22 (3) (2017) 1299– 1307.

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