Omnidirectional mobile robot robust tracking: Sliding-mode output-based control approaches

Control Engineering Practice 85 (2019) 50–58

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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

Omnidirectional mobile robot robust tracking: Sliding-mode output-based control approaches L. Ovalle a , H. Ríos b ,∗, M. Llama a , V. Santibáñez a , A. Dzul a a

Tecnológico Nacional de México/I.T. La Laguna, División de Estudios de Posgrado e Investigación, Blvd. Revolución y Cuahutémoc S/N C.P. 27000, Torreón, Coahulia, Mexico b CONACYT-Tecnológico Nacional de México/I.T. La Laguna, División de Estudios de Posgrado e Investigación, Blvd. Revolución y Cuahutémoc S/N C.P. 27000, Torreón, Coahulia, Mexico

ARTICLE

INFO

Keywords: Robust control Output-feedback control Omnidirectional mobile robot Position tracking Continuous sliding-mode control

ABSTRACT This work deals with the robust position tracking control problem for an omnidirectional mobile robot. To this aim, four continuous Sliding-Mode Control strategies are presented. The position and orientation of the platform are assumed to be the only available information about the system. To implement the controllers as output-feedback controllers, a High-Order Sliding-Mode Observer is implemented for each output signal. The proposed robust control strategies are able to deal with some classes of external disturbances. The closed-loop stability of each controller is proved by means of Lyapunov functions and homogeneity concepts. Simulations and experiments validate the applicability of the proposed controllers.

1. Introduction The velocity tracking control problem for mobile robots is usually done by means of kinematic controllers, assuming that the driver of the motors can achieve perfect velocity tracking (Dong & Xu, 2001). However, as mobile robots perform tasks where a heavy workload is employed, or a high velocity is needed, this assumption might not be fulfilled (Li & Ye, 2014) and therefore the dynamics of the mechanism should be considered. For mobile robots under highly changing loads, or vehicles where more than one surface is traveled, and therefore the friction of the ground might change, it is important to consider the presence of disturbances, something that is not possible with kinematic controllers. When a feedback control loop for robotic mechanisms is designed, it is often assumed that the drivers provide a perfect response to a generalized force, meaning that the actuator dynamics is often neglected (Kelly, Santibáñez, & Loría, 2006). In this sense, a better accuracy might be achieved if the effect of this dynamics is considered. Both position and velocity tracking control of omnidirectional mobile robots have already been reported in the literature. For instance in Li, Chen, Hung, and Yeh (2008), a kinematic controller based on fuzzy logic is presented to solve the velocity tracking problem for a three wheeled robot. In Lin and Shih (2013), an adaptive controller is shown; in this paper the generalized forces are assumed to be the control inputs to solve the velocity tracking problem in a four wheeled mobile robot. In Treesatayapun (2011) a fuzzy neural network-based controller is proposed for a class of discrete-time nonlinear systems;

the controller ensures an ultimate bound of the position tracking error without knowledge of the mathematical model and experiments are carried out in a three wheeled platform. In Bigelow and Kalhor (2017) an adaptive extended Sliding-Mode controller (SMC) is proposed. The scheme makes use of an evolving linear model; asymptotic stability of the position tracking errors are achieved for four wheeled robot. In Wang, Liu, Yang, Hu, Jiang, and Yang (2018), a model predictive scheme is presented for a three wheeled robot to track a position trajectory. In Li and Zell (2009), a kinematic controller, considering the actuators dynamics and actuator saturation is presented to deal with the position tracking problem for a three wheeled robot. In Peñaloza-Mejía, Márquez-Martínez, Alvarez, Villarreal-Cervantes, and García-Hernández (2015) a position tracking controller that ensures boundedness of the velocities is presented for a three wheeled robot. In Alakshendra and Chiddarwar (2017a), an adaptive sliding-mode controller is proposed for a four wheeled platform; the sliding-mode controller is based on a first-order algorithm and the position tracking error converges to zero asymptotically. In Alakshendra and Chiddarwar (2017b), the position tracking problem for a four wheeled robot is tackled; in this work, a cylinder is mounted atop the robot and the objective is to follow a path while maintaining the cylinder on the surface of the robot, this objective is achieved by a sliding-mode controller and a special type of switching sliding surface. In Ren, Sun, and Ma (2016) a passivity based approach is utilized for the position tracking control of a three wheeled omnidirectional mobile robot considering the generalized forces as control inputs. In Huang and Tsai (2008) a robust controller, based on an integral

∗ Corresponding author. E-mail address: [email protected] (H. Ríos).

https://doi.org/10.1016/j.conengprac.2019.01.002 Received 27 July 2018; Received in revised form 16 November 2018; Accepted 1 January 2019 Available online xxxx 0967-0661/© 2019 Elsevier Ltd. All rights reserved.

L. Ovalle, H. Ríos, M. Llama et al.

Control Engineering Practice 85 (2019) 50–58

backstepping approach for position tracking, and a robust adaptive path following controller has been presented for the three wheeled mechanism considering the dynamics of the actuators. In Viet, Doan, Hung, Kim, and Kim (2012), a robust controller for a mobile manipulator based on an omnidirectional mobile robot is presented; the control scheme is based on a kinematic controller for the manipulator and a first-order sliding-mode approach for the robust control for the position tracking of the mobile base, which is a three wheeled robot. In SiraRamírez, López-Uribe, and Velasco-Villa (2013), a linear observer-based output feedback controller with active disturbance rejection is presented taking into account that the generalized forces are the control inputs and an ultimate bound on the position tracking error is achieved. However, the application of output-feedback controllers based on SMCs for this system seems to be absent in the literature. In this sense, this work presents the application of four different control schemes to solve the position tracking control problem for this system, considering external disturbances and/or parameter uncertainties. All of the controllers make use of a High-Order Sliding-Mode Observer (HOSMO) to deal with incomplete information about the state vector. Then, the design of four controllers to solve the position tracking problem are presented. The stability of every controller is studied by means of Lyapunov functions and homogeneity concepts. All of the results are validated by real-time experiments. This work is structured as follows. In Section 2 the problem statement is shown, while Section 3 introduces some mathematical preliminaries; then, on Section 4, the HOSMO is designed. In Section 5 the design procedure for the controllers is presented. Simulation results are shown in Section 6, while experimental results are shown in Section 7; then, some conclusions are given in Section 8. Lastly, some preliminary results and the proof of the main results are given in the Appendix.

Fig. 1. Mobile robot diagram.

After a series of algebraic manipulations, and considering the relation ̇ 𝝋̇ = 𝐄𝑇 𝐑(𝜃)𝝃, it is possible to show that the dynamics (1) considering the actuator dynamics (2) is given by: ̇ 𝝃̇ + 𝑫 𝝃̇ = 𝝉 + 𝜹, 𝐌𝝃̈ + 𝑪(𝝃) (3) [ ] [ ] where 𝝉 𝑇 = 𝜏1 𝜏2 𝜏3 is a vector of generalized forces, 𝜹𝑇 = 𝛿1 𝛿2 𝛿3 is a vector containing some Lipchitz continuous disturbance forces for each ̇ 𝑫 ∈ R3×3 are given as configuration variable and the matrices 𝑴, 𝑪(𝜉), follows:

2. Problem statement

𝑴 = 𝑴 𝑹 + (𝐼2 + 𝐽𝑚 𝑟2𝑒 )𝑬𝑬 𝑇 ,

Consider the posture dynamic model of an omnidirectional mobile robot in the inertial frame (Campion, Bastin, & Dandrea-Novel, 1996) (see Fig. 1):

[𝑘 𝑘 ] 4 ̇ [𝐼2 + 𝐽𝑚 𝑟2𝑒 ]𝜃𝑩, 𝑫 = 𝑟2𝑒 𝑎 𝑏 + 𝑘𝑣 𝑬𝑬 𝑇 , 2 𝑅 𝑟 𝑎 1 0⎤ ⎡0 𝑩 = ⎢−1 0 0⎥ . ⎢ ⎥ 0 0⎦ ⎣0

̇ = 𝑪(𝝃)

̈ = 𝐄𝝉 𝝓 , 𝐑(𝜃)𝐌𝐑 𝝃̈ + 𝐄𝐼2 𝝋 (1) [ ] 𝑇 where 𝝃 = 𝑥 𝑦 𝜃 represents the configuration variables in the task [ ] space, 𝝋𝑇 = 𝜑1 𝜑2 𝜑3 𝜑4 represents the angular position of each wheel, [ ] 𝝉 𝝋 = 𝜏1 𝜏2 𝜏3 𝜏4 is a vector containing the torques of each wheel, 𝐄 represents the transpose of the jacobian matrix for the system, given by: ⎡1 ⎢ 1 ⎢1 𝐄= 𝑟 ⎢1 ⎢1 ⎣

1 −1 1 −1

Notice that 𝑴 is a constant diagonal matrix whose entries are denoted as 𝑀1 , 𝑀2 and 𝑀3 . Consider the following function to transform armature voltages into generalized forces:

𝑇

𝐿⎤ ⎥ −𝐿⎥ , −𝐿⎥ ⎥ 𝐿⎦

𝝉=

sin 𝜃 cos 𝜃 0

[ ] Thus, consider the following change of variables 𝒙𝑇 = 𝑥𝑝 𝑥𝑣 , [ ] [ ] [ ] [ ] 𝑇 ̇ 𝒙𝒑 = 𝑥1 𝑥2 𝑥3 = 𝑥 𝑦 𝜃 , and 𝒙𝒗 = 𝑥4 𝑥5 𝑥6 = 𝑥̇ 𝑦̇ 𝜃 . Therefore, the model (3) has a state-space representation of the form: [ ] [ ] 𝒙𝒗 𝑑 𝒙𝒑 = , 𝒇 (𝑥𝑣 ) + 𝑮[𝜏 + 𝛿] 𝑑𝑡 𝒙𝒗 where [ ]𝑇 𝒇 (𝒙𝒗 ) = 𝑓1 (𝒙𝒗 ) 𝑓2 (𝒙𝒗 ) 𝑓3 (𝒙𝒗 ) = −𝑴 −1 [𝑪(𝒙𝒗 ) + 𝑫]𝒙𝒗 ,

0⎤ 0⎥ . ⎥ 1⎦

and 𝑮 = 𝑴 −1 = diag{𝑀1−1 𝑀2−1 𝑀3−1 } = diag{𝑔1 𝑔2 𝑔3 }.

According to Kelly et al. (2006), if all motors are considered identical, it is possible to consider the following actuator dynamics ̈ + 𝑘𝑣 𝝋̇ + 𝐽𝑚 𝝋

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𝑇

𝐌𝐑 = diag{𝑚1 + 4𝑚2 , 𝑚1 + 4𝑚2 , 4𝑚2 (𝑙12 + 𝑙22 ) + 𝐽1 + 4𝐽3 }, 𝑚1 is the mass of the body, 𝑚2 is the mass of each wheel, 𝐽1 is the inertia of the body, 𝐽2 is the inertia of the wheels over the shaft of the motor, 𝐽3 is the inertia of the wheels perpendicular to the shaft of the motor, 𝑙1 is the length over 𝑅1 , 𝑙2 is the length over 𝑅2 , 𝑟 is the radius of the wheels, and 𝐑(𝜃) is a rotation matrix for a planar motion ⎡ cos 𝜃 𝐑(𝜃) = ⎢− sin 𝜃 ⎢ ⎣ 0

𝑘𝑎 𝑟𝑒 𝑇 𝑹 (𝜃)𝑬𝒖. 𝑅𝑎

𝑘 𝑘𝑎 𝑘𝑏 1 𝝋̇ + 𝝉 𝝓 = 𝑎 𝐮, 𝑅𝑎 𝑟 𝑒 𝑅𝑎 𝑟2𝑒

The position tracking error is defined as 𝝐 𝒑 = 𝒙𝒅 − 𝒙𝒑 , where 𝒙𝒅 𝑇 = [𝑥𝑑1 𝑥𝑑2 𝑥𝑑3 ] represents a desired trajectory, which is assumed to be twice differentiable with a bounded second derivative. The time derivative of the position tracking error, 𝝐 𝒗 = 𝝐̇ 𝒑 = 𝒙̇ 𝒅 − 𝒙𝒗 , represents a velocity tracking error. Therefore, the tracking error dynamics is given by [ ] [ ] 𝝐𝒗 𝑑 𝝐𝒑 = . (5) 𝒙̈ 𝒅 − 𝒇 (𝒙𝒗 ) − 𝑮[𝝉 + 𝜹] 𝑑𝑡 𝝐 𝒗

(2)

where 𝐽𝑚 is the inertia of the shaft of the motors, 𝑘𝑏 is the back EMF constant, 𝑘𝑎 is the torque constant, 𝑅𝑎 is the armature resistance, 𝑘𝑣 is the viscous friction of the motor and 𝑟𝑒 is the gear ratio and 𝐮 = [ ] 𝑢1 𝑢2 𝑢3 𝑢4 represents the armature voltages. 51

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Remark 1. Assumption 1 implies that either functions 𝑓̃𝑖 are Lipschitz or the velocity estimation errors and control inputs are bounded, and disturbances 𝛿𝑖 are Lipschitz. From the practical point of view, velocities, accelerations and control efforts are always physically bounded; then, Assumption 1 is fulfilled.

The objective of this work is to design a series of robust continuous control laws assuming that velocity information, i.e the variable 𝒙𝒗 , is not available for measurement and considering the generalized forces 𝝉 as the control inputs. 3. Preliminaries

Define the velocity observation error 𝑒𝑣𝑖 = 𝑥𝑣𝑖 − 𝑥̂ 𝑣𝑖 , an auxiliary [ ] error 𝑒𝜁𝑖 = 𝛿̃𝑖 − 𝜁𝑖 and the observation error vector 𝒆𝒊 = 𝑒𝑝𝑖 𝑒𝑣𝑖 𝑒𝜁𝑖 . Then, the following theorem deals with the convergence properties of the observers:

Consider a time-dependent differential equation: 𝑑𝒙(𝑡) = 𝒇 (𝑡, 𝒙(𝑡), 𝒘(𝑡)) , 𝑡 ≥ 𝑡0 , 𝑡0 ∈ R, (6) 𝑑𝑡 where 𝒙(𝑡) ∈ R𝑛 is the state vector and 𝒘(𝑡) ∈ R𝑞 is the external disturbance; 𝒇 ∶ R × R𝑛 → R𝑛 is a continuous function with respect to 𝒙 and piece-wise continuous with respect to 𝑡, 𝑓 (𝑡, 𝟎, 𝟎) = 0, ∀ 𝑡 ∈ R. The solution of the system (6), for an initial condition 𝒙𝟎 ∈ R𝑛 , at time instant 𝑡0 ∈ R𝑛 , is denoted as 𝒙(𝑡, 𝑡0 , 𝒙𝟎 ) and defined on some finite time interval [𝑡0 , 𝑡0 + 𝑇 ), such that 0 < 𝑇 < ∞. Let Ω be an open neighborhood of the origin in R𝑛 , 𝟎 ∈ Ω and 𝑤(𝑡) = 0.

Theorem 1. Let the observers (7) be applied to the system (3) and Assumption 1 be satisfied. If the gains 𝑘𝑖 , 𝑘𝑖+3 and 𝑘𝜁𝑖 are chosen as: 1

The proof of all the theorems introduced in this paper are given in the Appendix. Remark 2. Since the origin of (24) is GUFTS, 𝑒𝜁𝑖 (𝑡) = 0, for all 𝑡 ≥ 𝑇0 , implying that 𝛿̃𝑖 (𝑡) = 𝑔𝑖 𝛿𝑖 (𝑡), for all 𝑡 ≥ 𝑇0 .

1. Uniformly Stable (US) if for any 𝜀 > 0 there is 𝛿(𝜀) such that for any 𝒙𝟎 ∈ Ω, if ‖𝒙𝟎 ‖ ≤ 𝛿(𝜀) then ‖𝒙(𝑡, 𝑡0 , 𝒙𝟎 )‖ ≤ 𝜀 for all 𝑡 ≥ 𝑡0 for any 𝑡0 ∈ R; 2. Uniformly Asymptotically Stable (UAS) if it is US and attractive in Ω, i.e. for any 𝒙𝟎 ∈ Ω, there exists 𝛿 > 0 such that lim𝑡→∞ 𝒙(𝑡) = 0, for all ‖𝒙𝟎 ‖ ≤ 𝛿 and any 𝑡0 ∈ R; 3. Uniformly Exponentially Stable (UES) if it is US and exponentially converging from 𝛺, i.e. for any 𝑥0 ∈ 𝛺 there exist 𝑘, 𝜎 > 0 such that |𝑥(𝑡, 𝑡0 , 𝑥0 )| ≤ 𝑘|𝑥0 |𝑒−𝜎 (𝑡−𝑡0 ) for all 𝑡 ≥ 𝑡0 for any 𝑡0 ∈ R; 4. Uniformly Finite-Time Stable (UFTS) if it is US and finite-time converging from Ω, i.e. for any 𝒙𝟎 ∈ Ω there exist 0 ≤ 𝑇𝑥0 < +∞ such that 𝒙(𝑡, 𝑡0 , 𝒙𝟎 ) = 0 for all 𝑡 ≥ 𝑡0 + 𝑇𝑥0 , for any 𝑡0 ∈ R. The function 𝑇0 (𝒙𝟎 ) = inf {𝑇𝒙𝟎 ≥ 0 ∶ 𝒙(𝑡, 𝑡0 , 𝒙𝟎 ) = 0 ∀ 𝑡0 ≥ 𝑡0 + 𝑇𝑥0 } is called the setting-time of the system (6).

5. Robust controller design This section presents four different sliding-mode based control schemes to deal with the position tracking control problem. Every controller assumes that the velocity tracking error, 𝜖𝑣 is not available for measurement. Therefore, the estimate 𝜖̂𝑣 = 𝑥̇ 𝑑 − 𝑥̂ 𝑣 is employed instead. The control law is given by: ⎤ ⎡ −1 ⎢ 𝑔 [𝑓1 (𝑥̂ 𝑣 ) − 𝑥̈ 𝑑1 + 𝜁1 + 𝑣1 ]⎥ 1 ⎡𝜏1 ⎤ ⎢ ⎥ ⎢𝜏 ⎥ = ⎢ −1 [𝑓 (𝑥̂ ) − 𝑥̈ + 𝜁 + 𝑣 ]⎥ , 2 2 𝑣 𝑑 2 2 2 ⎥ ⎢ ⎥ ⎢ 𝑔2 ⎣𝜏3 ⎦ ⎢ ⎥ ⎢ −1 [𝑓 (𝑥̂ ) − 𝑥̈ + 𝜁 + 𝑣 ]⎥ 3 𝑣 𝑑 3 3 ⎦ ⎣𝑔 3 3 𝑅𝑎 + −1 𝑬 𝑹 (𝜃)𝝉, 𝒖= 𝑘𝑎 𝑟𝑒

If Ω = R𝑛 , then 𝒙 = 0 is said to be globally US (GUS), UAS (GUAS) or UFTS (GUFTS), respectively.

This section presents the design of an HOSMO to deal with incomplete information of the state-vector; in particular, the signal 𝒙𝒗 is assumed as unavailable for measurement. Let us introduce the following notation ⌈𝑠⌋𝛾 ∶= |𝑠|𝛾 sign(𝑠) for any 𝑠 ∈ R and any 𝛾 ∈ R≥0 . Then, consider the following HOSMO (CruzZavala & Moreno, 2016): 2

(7)

for 𝑖 = 1, 3; where 𝑥̂ 𝑝𝑖 represents the position estimations, 𝑥̂ 𝑣𝑖 are the velocity estimations, 𝜁𝑖 are some auxiliary variables, 𝑒𝑝𝑖 = 𝑥𝑝𝑖 − 𝑥̂ 𝑝𝑖 represents the position estimation error while 𝑘𝑖 , 𝑘𝑖+3 and 𝑘𝜁𝑖 are the observer gains to be designed for 𝑖 = 1, 3. Thus, consider the following assumption over the class of perturbations to be considered:

(11)

5.1. Sliding surface-based super-twisting controller Notice that the generalized forces act on the accelerations of the system and therefore each position variable 𝜖𝑝𝑖 has a relative degree equal to two with respect to the inputs 𝜏𝑖 . In this sense, it is necessary to consider the following sliding surface 𝑠𝑖 = 𝑐𝑖 𝜖𝑝𝑖 + 𝜖𝑣𝑖 ,

Assumption 1. There exist some known and positive constants 𝜂1 , 𝜂2 , 𝜂3 > 0 such that the following conditions are satisfied: |𝑑 | | ̃| | 𝛿3 | ≤ 𝜂3 , | 𝑑𝑡 | | |

(10)

with 𝑬 + as the right pseudoinverse of the matrix 𝑬 and 𝑣1 , 𝑣2 , 𝑣3 as auxiliary control variables. Thus, the closed-loop dynamics is given by: [ ] [ ] [ ] 𝝐𝒗 𝝐𝒗 𝑑 𝝐𝒑 = = , (12) ̃ 𝒗 − 𝒆𝜻 𝒗−𝜹+𝜻 𝑑𝑡 𝝐 𝒗 [ ] [ ] [ ] 𝑇 with 𝜹̃ = 𝛿̃1 𝛿̃2 𝛿̃3 , 𝜻 𝑇 = 𝜁1 𝜁2 𝜁3 , 𝒆𝑇𝜻 = 𝑒𝜁1 𝑒𝜁2 𝑒𝜁3 and 𝒗𝑇 = [𝑣1 𝑣2 𝑣3 ]. Notice that the proposed controllers are given with generalized forces as control inputs; nevertheless, it is possible to use (11) to consider the armature voltages as the control inputs. The rest of this section is dedicated to design the functions 𝑣𝑖 by means of sliding-mode algorithms to ensure the stability of the closed-loop dynamics (12).

4. Observer design

|𝑑 | |𝑑 | | ̃| | | | 𝛿1 | ≤ 𝜂1 , | 𝛿̃2 | ≤ 𝜂2 , | 𝑑𝑡 | | 𝑑𝑡 | | | | | ̃ ̃ where 𝛿𝑖 ∶= 𝑓𝑖 + 𝑔𝑖 𝛿𝑖 , and 𝑓̃𝑖

(9)

the observation error 𝒆𝒊 = 𝟎 is GUFTS for all 𝑖 = 1, 3.

Definition 1 (Khalil, 2002; Polyakov, 2012). The origin of the system (6), 𝒙 = 𝟎, is said to be:

⎤ 𝑥̂ 𝑣𝑖 + 𝑘𝑖 ⌈𝑒𝑝𝑖 ⌋ 3 ⎡𝑥̂̇ 𝑝𝑖 ⎤ ⎡ ⎥ 1 ⎢𝑥̂̇ ⎥ = ⎢⎢ , ⎢ ̇𝑣𝑖 ⎥ ⎢𝜁𝑖 + 𝑘𝑖+3 ⌈𝑒𝑝𝑖 ⌋ 3 + 𝑓𝑖 (𝑥̂ 𝑣 ) + 𝑔𝑖 𝜏𝑖 ⎥⎥ ⎣ 𝜁𝑖 ⎦ ⎣ 𝑘𝜁𝑖 ⌈𝑒𝑝𝑖 ⌋0 ⎦

1

𝑘𝑖 = 2𝜂𝑖3 , 𝑘𝑖+3 = 1.5𝜂𝑖 2 , 𝑘𝜁𝑖 = 1.1𝜂𝑖 , ∀𝑖 = 1, 3,

with 𝑐𝑖 > 0, in order to design a Super-Twisting algorithm (Shtessel, Edwards, Fridman, & Levant, 2014) to solve the position tracking problem. It is worth noting that the achievement of a sliding-mode on 𝑠𝑖 = 0 implies that the system dynamics collapses to 𝜖𝑣𝑖 = −𝑐𝑖 𝜖𝑝𝑖 . However, the variable 𝜖𝑣𝑖 is not available for measurement and the

(8)

∶= 𝑓𝑖 (𝑥𝑣 ) − 𝑓𝑖 (𝑥̂ 𝑣 ). 52

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Control Engineering Practice 85 (2019) 50–58 Table 1 Selection of gains for the Nonsingular Terminal SMC algorithm. Set 𝑐𝑖 𝑘̄ 1𝑖

surface 𝑠̂𝑖 = 𝑐𝑖 𝜖𝑝𝑖 + 𝜖̂𝑣𝑖 is employed instead. Notice that 𝑠̂𝑖 can be written as 𝑠̂𝑖 = 𝑠𝑖 + 𝑒𝑣𝑖 . Then, the following design of 𝑣𝑖 is proposed 1

1

𝑣𝑖 = 𝑘𝑖+3 ⌈𝑒̂𝑝𝑖 ⌋ 3 − 𝑐𝑖 𝜖̂𝑣𝑖 − 𝑘̄ 1𝑖 ⌈𝑠̂𝑖 ⌋ 2 + 𝜈𝑖 , 0

𝜈̇ 𝑖 = −𝑘̄ 2𝑖 ⌈𝑠̂𝑖 ⌋ .

(13) (14)

𝑘̄ 1𝑖 = 1.5𝜂̄𝑖 ,

2

28.7𝜂̄𝑖

3

−1∕2 7.7𝜂̄𝑖 −1∕2 𝜂̄𝑖

−1∕2

2∕3 25𝜂̄𝑖 2∕3 19𝜂̄𝑖 2∕3 13𝜂̄𝑖 2∕3 7𝜂̄𝑖

1 2

𝑘̄ 2𝑖 = 1.1𝜂̄𝑖 ,

(15)

3 4

with 𝜂̄𝑖 = 2.1𝜂𝑖 ; then, the position tracking error 𝜖𝑝𝑖 = 0 is GUES.

𝑘̄ 2𝑖

2∕3

2.5𝜂̄𝑖

4.5𝜂̄𝑖

2∕3

2𝜂̄𝑖

2∕3 7.5𝜂̄𝑖 2∕3 16𝜂̄𝑖

2𝜂̄𝑖

4.4𝜂̄𝑖

7𝜂̄𝑖

Table 2 Selection of gains for the Continuous Twisting SMC algorithm. Set 𝑘̄ 1𝑖 𝑘̄ 𝑖2 𝑘̄ 3𝑖

Theorem 2. Let Assumption 1 hold, and the observer (7) and controller (10), with 𝑣𝑖 given by (13)–(14), be applied to system (3). If the control gains 𝑘̄ 1𝑖 and 𝑘̄ 2𝑖 are selected according to: 𝑐𝑖 > 0,

20𝜂̄𝑖

4

Thus, the following theorem deals with the stability of the position tracking error dynamics (12).

1 2

−1∕2

1

1∕2 15𝜂̄𝑖 1∕2 10𝜂̄𝑖 1∕2 7.5𝜂̄𝑖 1∕2 5𝜂̄𝑖

𝑘̄ 4𝑖

2.3𝜂̄𝑖

1.1𝜂̄𝑖

2.3𝜂̄𝑖

1.1𝜂̄𝑖

2.3𝜂̄𝑖

1.1𝜂̄𝑖

2.3𝜂̄𝑖

1.1𝜂̄𝑖

5.2. Continuous singular terminal SMC Theorem 5. Let Assumption 1 hold, and the observer (7) and controller (10), with 𝑣𝑖 given by (21)–(22), be applied to system (3). If the control gains 𝑘̄ 1𝑖 , 𝑘̄ 2𝑖 , 𝑘̄ 3𝑖 and 𝑘̄ 4𝑖 are selected according to one of the possible selections given in Table 2, with 𝜂̄𝑖 = 2.1𝜂𝑖 ; then, the position tracking error 𝜖𝑝𝑖 = 0 is GUFTS.

Let us introduce the following sliding surface: 3

𝑠̂𝑖 = 𝑐𝑖 ⌈𝜖𝑝𝑖 ⌋ 2 + 𝜖̂𝑣𝑖 , and consider the following design for 𝑣𝑖 : 1

1

𝑣𝑖 = 𝑘𝑖+3 ⌈𝑒𝑝𝑖 ⌋ 3 − 𝑘̄ 1𝑖 ⌈𝑠̂𝑖 ⌋ 2 + 𝜈𝑖 ,

(16)

0 𝜈̇ 𝑖 = −𝑘̄ 2𝑖 ⌈𝑠̂𝑖 ⌋ .

(17)

It is important to highlight the following properties of the proposed controllers:

Then, the following theorem deals with the stability of the tracking errors 𝜖𝑝𝑖 and 𝜖𝑣𝑖 .

1. All of the proposed control strategies are robust against Lipchitz continuous disturbances and provide continuous control efforts. 2. The Sliding Surface-based Super-Twisting controller provides uniform exponential stability of the position errors. 3. In contrast, the Continuous Singular Terminal, Continuous Nonsingular Terminal and Continuous Twisting SMCs provide uniform finite-time stability of the position errors. Evidently, this is a stronger property than GUAS or GUES and represents the main theoretical difference between these controllers and the Sliding Surface-based Super-Twisting controller. 4. The Sliding Surface-based Super-Twisting controller, the Continuous Singular Terminal and the Continuous Nonsingular Terminal SMCs have three design gains. However, the Continuous Singular and Nonsingular Terminal SMCs are more complex to implement due to the nonlinearities in the surfaces. 5. The Continuous Twisting SMC has four control gains while the rest of them possess three parameters. This means that such a controller is the most complex to implement and design. 6. A clear trade-off between implementation/design complexity and stability performance exists among the proposed controllers.

Theorem 3. Let Assumption 1 hold, and the observer (7) and controller (10), with 𝑣𝑖 given by (16)–(17), be applied to system (3). If the control gains 𝑘̄ 1𝑖 and 𝑘̄ 2𝑖 are selected according to 1

𝑐𝑖 > 0,

𝑘̄ 1𝑖 = 1.5𝜂̄𝑖2 ,

𝑘̄ 2𝑖 = 1.1𝜂̄𝑖 ,

(18)

with 𝜂̄𝑖 = 2.1𝜂𝑖 ; then, the position tracking error 𝜖𝑝𝑖 = 0 is GUFTS. 5.3. Continuous nonsingular terminal SMC Let us introduce the following sliding surface: 2

𝑠̂𝑖 = 𝑐𝑖 ⌈𝜖̂𝑣𝑖 ⌋ 3 + 𝜖𝑝𝑖 , and consider the following design for 𝑣𝑖 : 1

1

𝑣𝑖 = 𝑘𝑖+3 ⌈𝑒𝑝𝑖 ⌋ 3 − 𝑘̄ 1𝑖 ⌈𝑠̂𝑖 ⌋ 3 + 𝜈𝑖 ,

(19)

0

𝜈̇ 𝑖 = −𝑘̄ 2𝑖 ⌈𝑠̂𝑖 ⌋ .

(20)

Then, the following theorem deals with the stability of the tracking errors 𝜖𝑝𝑖 and 𝜖𝑣𝑖 . Theorem 4. Let Assumption 1 hold, and the observer (7) and controller (10), with 𝑣𝑖 given by (19)–(20), be applied to system (3). If the control gains 𝑐𝑖 , 𝑘̄ 1𝑖 and 𝑘̄ 2𝑖 are selected according to one of the possible selections given in Table 1, with 𝜂̄𝑖 = 2.1𝜂𝑖 ; then, the position tracking error 𝜖𝑝𝑖 = 0 is GUFTS.

6. Simulation results

In order to show the robustness properties of the proposed algorithms, some simulations are carried out considering model (3) with the parameters given by Table 3.

5.4. Continuous twisting SMC

For comparison purposes, a PID controller given as

Consider the following design of 𝑣𝑖 : 1

1

1

𝑣𝑖 = 𝑘𝑖+3 ⌈𝑒𝑝𝑖 ⌋ 3 − 𝑘̄ 1𝑖 ⌈𝜖𝑝𝑖 ⌋ 2 + 𝑘̄ 2𝑖 ⌈𝜖̂𝑣𝑖 ⌋ 2 + 𝜈𝑖 , 0

0

𝜈̇ 𝑖 = −𝑘̄ 3𝑖 ⌈𝜖𝑝𝑖 ⌋ −𝑘̄ 4𝑖 ⌈𝜖̂𝑣𝑖 ⌋ .

𝑡

1

𝑣𝑖 = 𝑘𝑖+3 ⌈𝑒𝑝𝑖 ⌋ 3 − 𝑘𝑝𝑖 𝜖𝑝𝑖 − 𝑘𝑣𝑖 𝜖̂𝑣𝑖 − 𝑘𝐼𝑖 (21)

∫0

𝜖𝑝𝑖 (𝜏)𝑑𝜏,

is considered. Notice that this controller represents a regular PID 1 controller, with the addition of the term 𝑘𝑖+3 ⌈𝑒𝑝𝑖 ⌋ 3 . Such a term is necessary to compensate the observer dynamics. The controller gains are given as 𝑘𝑝1 = 𝑘𝑝2 = 13, 𝑘𝑝3 = 50, 𝑘𝑣1 = 𝑘𝑣2 = 1, 𝑘𝑣3 = 5, 𝑘𝐼1 = 𝑘𝐼2 = 1.75 and 𝑘𝐼3 = 0.75.

(22)

Notice that the right hand of (22) represents a twisting controller that is integrated through 𝜈𝑖 . Then, the following theorem deals with the stability of the tracking errors 𝜖𝑝𝑖 and 𝜖𝑣𝑖 . 53

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Control Engineering Practice 85 (2019) 50–58

Table 3 Parameters of the dynamic model. Symbol

Value

Units

𝑚1 𝑚2 𝐽1 𝐽2 𝐽3 𝑙1 𝑙2 𝑟 𝐽𝑚 𝑘𝑏 𝑘𝑎 𝑅𝑎 𝑘𝑣 𝑟𝑒

2.8 0.38 0.0608 3.24 × 10−4 4.69 × 10−4 0.1524 0.1505 0.042 5.7 × 10−7 0.0133 0.013 1.9 0.001 64

kg kg kg m2 kg m2 kg m2 m m m kg m2 Vs∕rad Nm∕A Ω Nm s∕rad –

Fig. 3. Orientation response of the system.

Fig. 4. Temporal evolution of the control inputs.

Fig. 2. Position response of the system.

For simulation purposes, the disturbance terms take the following form: ⎡ 5 sin( 𝜋 𝑡) + 0.1 ⎤ ⎥ 10 ⎡𝛿1 ⎤ ⎢ ⎥ ⎢ 𝜹 = ⎢𝛿2 ⎥ = ⎢−5 cos( 𝜋 𝑡) + 0.1⎥ . ⎢ ⎥ ⎢ 10 ⎥ ⎣𝛿3 ⎦ ⎢ 5 cos( 𝜋 𝑡) + 0.1 ⎥ ⎦ ⎣ 10 Only simulations for the PID controller, Sliding Surface-based SuperTwisting controller and the Singular Terminal SMC are presented. Notice that the robustness of the rest of the controllers is similar. The legend ss-ST is used to represent the Sliding Surface-based SuperTwisting controller while STSMC is used for the Continuous Singular Terminal SMC. Observer (7) is used in all of the simulations with gains: 𝑘1 = 𝑘2 = 2.42, 𝑘3 = 7.19, 𝑘4 = 𝑘5 = 2.08, 𝑘6 = 10.22, 𝑘7 = 𝑘8 = 1.97 and 𝑘9 = 51.11. The parameters for the ss-ST and STSMC are taken as 𝑘11 = 𝑘12 = 1.7, 𝑘13 = 10.22, 𝑘21 = 𝑘22 = 1.42 and 𝑘23 = 51.11. The simulations consider the following reference signals

Fig. 5. RMS value for the state vector.

considered. This RMS value is computed according to the expression: √ 𝑡 1 𝜖𝑅𝑀𝑆 = 𝝐 (𝜏)𝑇 𝝐 𝒑 (𝜏)𝑑𝜏, 𝛥𝑇 ∫𝑡−𝛥𝑇 𝒑 which represents a measure of performance over the last 𝛥𝑇 seconds. Fig. 5 shows the RMS values for the position signal with 𝛥𝑇 = 5[seg]. Notice that the performance of both robust controllers is very similar, compensating the effect of the disturbances.

𝑥𝑑1 (𝑡) = 0.5 sin(𝜔𝑡) [m], 𝑥𝑑2 (𝑡) = 0.5 cos(𝜔𝑡) [m], 𝑥𝑑3 (𝑡) = −(𝜔𝑡 + 𝜋∕2) [rad], 𝜔 = 𝜋∕10 [rad∕seg],

7. Experimental results

as desired positions for 𝑥, 𝑦, and 𝜃 respectively. Fig. 2 shows the response of the system in the x–y plane, while the response for the orientation is shown in Fig. 3. Notice that the PID controller achieves only stability of the error signal in the presence of disturbances. The control signals are shown in Fig. 4, all of these are continuous as expected. In order to provide some quantitative measure for the performance of the controllers, the RMS value for the position tracking error is

To further prove the applicability of the proposed controllers, realtime experiments were carried out on a Nexus four wheeled robot, shown in Fig. 6, whose dynamics are given by (3), and whose parameters are given in Table 3. The robot position and orientation are tracked and accurately measured using an OptiTrack camera system with six synchronized infrared 54

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Control Engineering Practice 85 (2019) 50–58

Fig. 7. Position response of the system. Fig. 6. Nexus robot.

Table 4 Parameters of the controllers. Gain

ss-ST

STSMC

NTSMC

TSMC

𝑘̄ 11 𝑘̄ 21 𝑘̄ 31 𝑘̄ 41

1.53 1.15 – – 1 1.53 1.15 – – 1 6.3 19.4 – – 1

1.53 1.15 – – 0.3 1.53 1.15 – – 0.3 6.3 19.4 – – 0.3

6.02 1.67 – – 2.08 6.02 1.67 – – 2.08 31.2 19.4 – – 0.605

3.92 3.24 0.96 0.42 – 3.92 3.24 0.96 0.42 – 14.36 8.57 6.76 3.23 –

𝑐1 𝑘̄ 12 𝑘̄ 22 𝑘̄ 32 𝑘̄ 42 𝑐2 𝑘̄ 13 𝑘̄ 23 𝑘̄ 33 𝑘̄ 43 𝑐3

Fig. 8. Orientation response of the system.

the PID controller. The ss-ST and STSMC controllers have the longestlasting transient responses, nonetheless as shall be latter demonstrated, after a finite time these controllers have an adequate response. Notice that, according to Remark 2, all of the proposed strategies, even the PID controller, should show robustness to uncertainties and disturbances such as irregular surfaces due to the robustness that the observer and the controller itself provide. Fig. 8 shows the response for the orientation variable. It is noteworthy that the PID controller provides a good performance in this task; however, all of the proposed controllers are capable of driving the orientation error close to zero. Due to practical issues, such as discretization effects and measurement noise, it is not possible to achieve a perfect tracking of the desired trajectories; however, all of the controllers provide an acceptable performance. The main purpose of Fig. 9, where the norm of the position estimation error is presented, is to validate the results from Theorem 2. Notice that this norm does not vanish. This is due to the effects of the friction forces, irregularities in the terrain and noise from the vision system that can be seen as additional disturbances. Nonetheless, this error is, in fact, small enough to get a good performance from the proposed schemes. The input signals are shown in Fig. 10. For each of these signals, the results are continuous, which is one of the objectives of this work. The control efforts are given as armature voltages; these were computed using the control law (10) and the relation from generalized forces to armature voltages (11). The RMS value, 𝜖𝑅𝑀𝑆 , is shown in Fig. 11. Notice that the PID controller seems to have the fastest transient response. Nonetheless, the state signals for the position variables does not approach the reference trajectory nearly as well as the rest of the controllers do. The transient response of the ss-ST and STSMC controllers, which are the slowest of

cameras connected to a ground station. Note that the proposed controllers only require position measurements; then, it is also possible to obtain such data by means of GPS, odometry, IMUs and/or any other group of sensors capable of measuring the position and orientation of the system. Note that disturbances, such as parameter uncertainties, unmodeled dynamics, irregular surfaces or measurable noise; are inherent to the experimental setup. In this sense, the controllers are experimentally tested under disturbances. All of the experiments were carried out in an output-feedback scheme by means of the observer (7), where the observer gains were designed as: 𝑘1 = 𝑘2 = 3.42, 𝑘3 = 6.21, 𝑘4 = 𝑘5 = 3.35, 𝑘6 = 8.2158, 𝑘7 = 𝑘8 = 5.5 and 𝑘9 = 33. The following legend is used: ss-ST is the Sliding Surface-based Super-Twisting Controller, STSMC is the Continuous Singular Terminal Controller, NTSMC is the Continuous Nonsingular Terminal SMC and TSMC is the Continuous Twisting SMC. The gains of the controllers are given in Table 4. The experiments consider the following reference signals 𝑥𝑑1 (𝑡) = 0.5 sin(𝜔𝑡) [m], 𝑥𝑑2 (𝑡) = 0.5 cos(𝜔𝑡) [m], 𝑥𝑑3 (𝑡) = −(𝜔𝑡 + 𝜋∕2) [rad], 𝜔 = 𝜋∕10 [rad∕seg], ired positions for 𝑥, 𝑦, and 𝜃 respectively. For comparison purposes, a PID controller is implemented as in Section 6. The controller gains are given as 𝑘𝑝1 = 𝑘𝑝2 = 13, 𝑘𝑝3 = 50, 𝑘𝑣1 = 𝑘𝑣2 = 1, 𝑘𝑣3 = 5, 𝑘𝐼1 = 𝑘𝐼2 = 1.75 and 𝑘𝐼3 = 0.75. Fig. 7 shows the position response of the system. Notice that every controller proposed is capable of, after a transient response, drive the positions to a region close to the desired trajectory; this is not the case with 55

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Control Engineering Practice 85 (2019) 50–58

8. Conclusions This work contributes with four strategies to robustly solve the position tracking control problem for an omnidirectional robot by means of continuous controllers. The proposed strategies are given by means of four different continuous controllers: (1) Sliding Surface-based Super Twisting, (2) Singular Terminal SMC (3) Nonsingular Terminal SMC and (4) Continuous Twisting SMC. All of the controllers used an HOSMO to deal with incomplete state information. The stability of the proposed controllers was studied by means of Lyapunov functions and homogeneity properties. Some simulations and experimental results validate the feasibility and workability of the proposed continuous control strategies in the presence of some disturbances. Conflict of interest

Fig. 9. Position estimation error.

None declared. Acknowledgments The authors gratefully acknowledge the financial support from TecNM research projects. L. Ovalle and H. Ríos also acknowledge the financial support from CONACYT, Mexico, 591548 and 270504, respectively. Appendix Definition 2 (Bernuau, Efimov, Perruquetti, & Polyakov, 2014). Let 𝒓 = (𝑟1 , … , 𝑟𝑛 ) with 𝑟𝑖 > 0 be a vector of weights and 𝜆 > 0. Define the dilation matrix Λ𝒓 (𝜆) = 𝑑𝑖𝑎𝑔(𝜆𝑟𝑖 ), for 𝑖 = 1, 𝑛. The function 𝑔 ∶ R𝑛 → R is called r-homogeneous, if for any 𝒙 ∈ R𝑛 the relation 𝑔(Λ𝒓 (𝜆)𝒙) = 𝜆𝑞 𝑔(𝑥), holds for some 𝑞 ∈ R and all 𝜆 > 0. The function 𝒇 ∶ R𝑛 → R𝑛 is called r-homogeneous, if for any 𝒙 ∈ R𝑛 the relation 𝒇 (Λ𝒓 (𝜆)𝒙) = 𝜆𝑞 Λ𝒓 (𝜆)𝒇 (𝒙), holds for some 𝑞 ≥ − min1≤𝑖≤𝑛 𝑟𝑖 and all 𝜆 > 0. In both cases, the constant 𝑞 is called the degree of homogeneity.

Fig. 10. Temporal evolution of the control inputs.

Homogeneous systems possess, to some degree, robustness with respect to external disturbances and are therefore very important when dealing with some Sliding-Mode algorithms (Bernuau et al., 2014). In this sense, let us consider the following differential inclusion: (23)

𝒙̇ ∈ 𝒇 (𝒙, 𝒘), R𝑛

where 𝒙 ∈ is the state vector and 𝒘 ∈ represents a vector of external perturbations. 𝒇 is a set valued map defined as 𝒇 ∶ R𝑛+𝑚 → R𝑛 . Let R+ denote the set of all non-negative real numbers. For a (Lebesgue) measurable function, 𝑑 ∶ R+ → R𝑚 , define the norm ‖𝒅‖[𝑡0 ,𝑡1 ) =ess sup𝑡∈[𝑡0 ,𝑡1 ) ‖𝒅‖, where ess sup represents the essential supremum, then ‖𝒅‖∞ = ‖𝒅‖[0,∞) . The set with the property ‖𝒅‖∞ < ∞ is denoted as ∞ ; and 𝐷 = {𝒅 ∈ ∞ ∶ ‖𝒅‖∞ ≤ 𝐷} for any 𝐷 > 0. A continuous function 𝛼 ∶ R+ → R+ belongs to the class  if it is strictly increasing and 𝛼(0) = 0; it belongs to class ∞ if it is also unbounded. A continuous function 𝛽 ∶ R+ × R+ → R+ belongs to the class  if for each fixed 𝑠, 𝛽(⋅, 𝑠) belongs to the class , and 𝛽(𝑟, ⋅) is strictly decreasing to zero for any fixed 𝑟 ∈ R+ ; while 𝛽 ∈ 𝑇 if for each fixed 𝑠, 𝛽(⋅, 𝑠) ∈ , and for each fixed 𝑟 there exists 0 < 𝑇 (𝑟) < ∞ such that 𝛽(𝑟, 𝑠) is decreasing to zero with respect to 𝑠 < 𝑇 (𝑟), and 𝛽(𝑟, 𝑠) = 0 for all 𝑠 ≥ 𝑇 (𝑟).

Fig. 11. RMS value for the state vector.

Table 5 Average of RMS values. ss-ST

STSMC

NTSMC

TSMC

PID

0.58

0.65

0.25

0.27

0.29

R𝑚

the proposed controllers, is dependent on the parameter 𝑐𝑖 , which is a free parameter that might be tuned freely to shape this behavior. In order to provide a quantitative analysis of the controllers, the average of the RMS values are shown in Table 5. Notice that the PID controller has a lower RMS value than that of the ss-ST and STSMC controllers, this is to be expected due to the poor transient response that these controllers have. However, as it is shown in Fig. 11, these controllers eventually have a better response than that of the PID controller. On the other hand, the other two algorithms, the TSMC and NTSMC controllers, have a better performance, with the TSMC controller slightly outperforming the NTSMC controller.

Definition 3 (Khalil, 2002). The system (23) is said to be finite-time input-to-state stable (FT-ISS) if for any 𝒘 ∈ ∞ and any 𝒙𝟎 ∈ R𝑛 there exist some functions 𝛽 ∈ 𝑇 and 𝛾 ∈  such that ‖ ‖ ‖ ‖ ‖𝒙(𝑡, 𝒙𝟎 , 𝒘)‖ ≤ 𝛽(‖𝒙𝟎 ‖, 𝑡) + 𝛾(‖𝒘‖∞ ), ∀𝑡 ≥ 0, ‖ ‖ ‖ ‖ holds. Let us denote an extended discontinuous function 𝒇̃ (𝒙, 𝒘) (𝒇 𝑇 (𝒙, 𝒘), 𝟎𝒎 )𝑇 . 56

=

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Control Engineering Practice 85 (2019) 50–58

Theorem 6 (Bernuau et al., 2014). Let 𝒇̃ be homogeneous with weights 𝒓 = (𝑟1 , … , 𝑟𝑛 ) > 0 and 𝒓̃ = (̃𝑟1 , … , 𝑟̃𝑚 ) ≥ 0 with a degree 𝑞 ≥ − min1≤𝑖≤𝑛 𝑟𝑖 , i.e. 𝒇 (Λ𝒓𝒙 , 𝝀𝒓̃ 𝒘) = 𝜆𝑞 Λ𝒓 𝒇 (𝒙, 𝒘). Assume that the system (23) is GUAS for 𝒘 = 𝟎. Let

Proof of Theorem 3. The closed-loop dynamics is given by: ⎡𝜖 ⎤ ⎡ 𝜖̂𝑣𝑖 − 𝑒𝑣𝑖 ⎤ ⎥ 1 𝑑 ⎢ 𝑝𝑖 ⎥ ⎢ 𝜖̂ = −𝑘̄ ⌈𝑠̂ ⌋ 2 + 𝜈̄𝑖 ⎥ . 𝑑𝑡 ⎢ 𝑣𝑖 ⎥ ⎢⎢ 1𝑖 𝑖 0 ⎣ 𝜈̄𝑖 ⎦ ⎣−𝑘̄ 2𝑖 ⌈𝑠̂𝑖 ⌋ + 𝛿̄𝑖 ,⎥⎦

‖𝒇 (𝒛, 𝒘) − 𝒇 (𝒛, 𝟎)‖ ≤ 𝜎(‖𝒘‖) ∀𝒛 ∈ S𝑟 , with 𝜎(𝑎) =

{

𝑐𝑎𝜌min , 𝑐𝑎𝜌max ,

with the definition of 𝜈̄𝑖 and 𝛿̄𝑖 given in Theorem 2. In Fridman, Moreno, Bandyopadhyay, Kamal, and Chalanga (2015), the following Lyapunov function candidate has been proposed

𝑖𝑓 𝑎 ≤ 1, 𝑖𝑓 𝑎 > 1,

𝑉 (𝜖𝑝𝑖 , 𝜖̂𝑣𝑖 , 𝜈̄𝑖 ) = 𝝔𝑇 𝑷 𝝔,

for some 𝑐 > 0 and 𝜌max ≥ 𝜌min > 0. Then, the system (23) is ISS if 𝑟̃min > 0, where 𝑟̃min = min1≤𝑗≤𝑚 𝑟̃𝑗 . If the homogeneity degree 𝑞 ≤ 0, then the system (23) is FT-ISS.

⎛ 𝜖̂𝑣𝑖 − 𝑒𝑣𝑖 ⎞ 1 ⎜ ̄ ⎟ 2 𝜈̄𝑖 ⎟ , 𝒇̃ = (𝜖𝑝𝑖 , 𝜖̂𝑣𝑖 , 𝜈̄𝑖 , 𝑒𝑣𝑖 ) = ⎜−𝑘1𝑖 ⌈𝑠̂𝑖 ⌋ + 0 ⎜ −𝑘̄ 2𝑖 ⌈𝑠̂𝑖 ⌋ ⎟ ⎜ ⎟ 0 ⎝ ⎠

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It is possible to show, based on (Cruz-Zavala & Moreno, 2016), that the observation error dynamics admits the following proper, smooth and homogeneous of degree 5 Lyapunov function: [ ] 5 5 3 2 𝑉𝑖 (𝑧1𝑖 , 𝑧2𝑖 , 𝑧3𝑖 ) = 𝛽1𝑖 |𝑧1𝑖 | 3 − 𝑧1𝑖 𝑧2𝑖 + |𝑧2𝑖 | 2 5 5 [ ] 5 5 3 2 1 3 +𝛽2𝑖 |𝑧2𝑖 | 2 − 𝑧2𝑖 𝑧3𝑖 + |𝑧3𝑖 | 3 + 𝛽3𝑖 |𝑧3𝑖 |5 , 5 5 5

is homogeneous of degree 𝑞 = −1 with weights 𝒓 = (3, 2, 1) and 𝑟̃ = 2 and the condition ‖𝒇 (𝜖𝑝𝑖 , 𝜖̂𝑣𝑖 , 𝜈̄𝑖 , 𝑒𝑣𝑖 ) − 𝒇 (𝜖𝑝𝑖 , 𝜖̂𝑣𝑖 , 𝜈̄𝑖 , 0)‖ = ‖𝑒𝑣𝑖 ‖ ≤ 𝜎(‖𝑒𝑣𝑖 )‖, holds. Thus all the conditions of Theorem 6 are satisfied with 𝑐 ≥ 1, 𝜌max = 𝜌min = 1. Therefore, one may conclude that the origin of (27) is FT-ISS with respect to 𝑒𝑣𝑖 . Then, since 𝑒𝑣𝑖 = 0 is GUFTS, it follows that the position tracking error 𝜖𝑝𝑖 = 0 is GUFTS.

𝑒𝜁

𝑒

where 𝑧1𝑖 = 𝑒𝑝𝑖 , 𝑧2𝑖 = 𝑘 𝑣𝑖 , 𝑧𝑖+6 = 𝑘 𝑖 , and 𝛽1𝑖 , 𝛽2𝑖 and 𝛽1𝑖 are positive 𝑖+3 𝜁𝑖 constants. The positivity of the Lyapunov function follows from the Young inequality. After a series of algebraic manipulations, the derivative of the Lyapunov function can be bounded as: ][ ] [ 4] 2 [ 𝜉 1+ 𝑖 𝑉̇ 𝑖 (𝑧1𝑖 , 𝑧2𝑖 , 𝑧3𝑖 ) ≤ −𝑘̃ 1𝑖 𝛽1𝑖 [⌈𝑧1𝑖 ⌋ 3 − 𝑧2𝑖 ]2 + 𝑘̃ 3𝑖 𝜃3𝑖 − 𝛽3𝑖 |𝑧1𝑖 | 3 𝑘𝜁𝑖 ] [ ][ 1 1 3 ] [ ] [ +𝑘̃ 2𝑖 ⌈𝑧1𝑖 ⌋ 3 − 𝑧3𝑖 𝛽1𝑖 ⌈𝑧1𝑖 ⌋ 3 − 𝑧3𝑖 − 𝛽2𝑖 ⌈𝑧2𝑖 ⌋ 2 − 𝑧33𝑖 ,

Proof of Theorem 4. The closed-loop dynamics is given by: ⎡𝜖 ⎤ ⎡ 𝜖̂𝑣𝑖 − 𝑒𝑣𝑖 ⎤ ⎥ 1 𝑑 ⎢ 𝑝𝑖 ⎥ ⎢ 𝜖̂ = −𝑘̄ ⌈𝑠̂ ⌋ 3 + 𝜈̄𝑖 ⎥ . 𝑑𝑡 ⎢ 𝑣𝑖 ⎥ ⎢⎢ 1𝑖 𝑖 0 ⎣ 𝜈̄𝑖 ⎦ ⎣−𝑘̄ 2𝑖 ⌈𝑠̂𝑖 ⌋ + 𝛿̄𝑖 ,⎥⎦

5

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𝜈̄̇ 𝑖 = −𝑘̄ 2𝑖 ⌈𝑠̂𝑖 ⌋0 − 𝛿̄𝑖 ,

(26)

5

𝑉 (𝜖𝑝𝑖 , 𝜖̂𝑣𝑖 , 𝜈̄𝑖 ) = 𝛽1 |𝜖𝑝𝑖 | 3 + 𝛽2 𝜖𝑝𝑖 𝜖̂𝑣𝑖 + 𝛽3 |𝜖̂𝑣𝑖 | 2 +𝛽4 𝜖𝑝𝑖 ⌈𝜖̂𝑣𝑖 ⌋2 − 𝛽5 𝜖̂𝑣𝑖 𝜈̄𝑖3 + 𝛽6 |𝜈̄𝑖 |5 ,

(30)

with 𝛽, 𝛾 > 0. In Kamal et al. (2016), it has been proven that, for 𝑒𝑣𝑖 = 0, the function 𝑉 is a Lyapunov function for system (24) such that, for some selection of 𝑐𝑖 , 𝑘̄ 1𝑖 and 𝑘̄ 2𝑖 , 𝑖 = 1, 3 from Table 1, 𝑉̇ ≤ −𝜅𝑉 4∕5 , with some positive 𝜅. Therefore, (𝜖𝑝𝑖 , 𝜖̂𝑣𝑖 , 𝜈̄𝑖 ) = 𝟎 is UFTS. Notice that the function

Proof of Theorem 2. Define 𝜈̄𝑖 = 𝜈𝑖 − 𝑒𝜁𝑖 ; then, using (24), the time derivative of 𝑠̂𝑖 along the trajectories of (12) is given by: 1

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with the definition of 𝜈̄𝑖 and 𝛿̄𝑖 given in Theorem 2. In Kamal, Moreno, Chalanga, Bandyopadhyay, and Fridman (2016), the following Lyapunov function candidate has been proposed

whose negativity can be determined based on (Cruz-Zavala & Moreno, 2016), for some constants 𝑘̃ 1𝑖 , 𝑘̃ 2𝑖 and 𝑘̃ 3𝑖 . This implies that (𝑒𝑝𝑖 , 𝑒𝑣𝑖 , 𝑒𝜁𝑖 ) = 0 is GUAS. However, since (24) is homogeneous of degree −1, GUFTS of (𝑒𝑝𝑖 , 𝑒𝑣𝑖 , 𝑒𝜁𝑖 ) = 0 is implied.

𝑠̂̇ 𝑖 = 𝑐𝑖 𝜖𝑣𝑖 + 𝑥̈ 𝑑 − 𝑥̂̇ 𝑣𝑖 = −𝑘̄ 1𝑖 ⌈𝑠̂𝑖 ⌋ 2 + 𝜈̄𝑖 ,

⌋2 ]𝑇

with 𝝔 = [⌈𝜖𝑝𝑖 ⌋ 𝑠̂𝑖 ⌈𝜈̄𝑖 and some 𝑷 = 𝑷 𝑇 > 0 . In Fridman et al. (2015), it has been proven that, for 𝑒𝑣𝑖 = 0, the function 𝑉 is a Lyapunov function for system (24) such that, for some 𝑐𝑖 , 𝑘̄ 1𝑖 and 𝑘̄ 2𝑖 , 𝑖 = 1, 3, 𝑉̇ ≤ −𝜅𝑉 3∕4 , with some positive 𝜅. Therefore, (𝜖𝑝𝑖 , 𝜖̂𝑣𝑖 , 𝜈̄𝑖 ) = 𝟎 is UFTS. Notice that the function

2

⎡𝑒̇ 𝑝𝑖 ⎤ ⎡ 𝑒𝑣𝑖 − 𝑘𝑖 ⌈𝑒𝑝𝑖 ⌋ 3 ⎤ 1⎥ ⎢𝑒̇ 𝑣𝑖 ⎥ = ⎢⎢ 3 . ⎢ ⎥ ⎢𝑒𝜁𝑖 − 𝑘𝑖+3 ⌈𝑒𝑝𝑖 ⌋ ⎥⎥ ⎣𝑒̇ 𝜁𝑖 ⎦ ⎣ −𝑘 ⌈𝑒 ⌋0 + 𝛿̃̇ ⎦ 𝜁𝑖 𝑝𝑖 𝑖

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2 3

Proof of Theorem 1. The observation error dynamics is given by:

⎛ 𝜖̂𝑣𝑖 − 𝑒𝑣𝑖 ⎞ 1 ⎜ ̄ ⎟ 3 𝜈̄𝑖 ⎟ , 𝒇̃ = (𝜖𝑝𝑖 , 𝜖̂𝑣𝑖 , 𝜈̄𝑖 , 𝑒𝑣𝑖 ) = ⎜−𝑘1𝑖 ⌈𝑠̂𝑖 ⌋ + ⎜ −𝑘̄ 2𝑖 ⌈𝑠̂𝑖 ⌋0 ⎟ ⎟ ⎜ 0 ⎝ ⎠

with 𝛿̄𝑖 = 𝑘𝜁𝑖 ⌈𝑒𝑝𝑖 ⌋0 − 𝛿̃̇ 𝑖 . Notice that |𝛿̄𝑖 | ≤ 𝑘𝜁𝑖 + 𝜂𝑖 = 2.1𝜂𝑖 , which rises from Assumption 1 and (9). For the dynamics (25)–(26), it has been proven that the function

is homogeneous of degree 𝑞 = −1 with weights 𝒓 = (3, 2, 1) and 𝑟̃ = 2 and the condition

𝑉 (𝜖̂𝑣𝑖 , 𝜈̄𝑖 ) = 𝝇(𝜖̂𝑣𝑖 , 𝜈̄𝑖 )𝑇 𝑷 𝝇(𝜖̂𝑣𝑖 , 𝜈̄𝑖 ),

‖𝒇 (𝜖𝑝𝑖 , 𝜖̂𝑣𝑖 , 𝜈̄𝑖 , 𝑒𝑣𝑖 ) − 𝒇 (𝜖𝑝𝑖 , 𝜖̂𝑣𝑖 , 𝜈̄𝑖 , 0)‖ = ‖𝑒𝑣𝑖 ‖ ≤ 𝜎(‖𝑒𝑣𝑖 )‖,

1

with 𝝇 = [⌈𝑠̂𝑖 ⌋ 2 , 𝜈̄𝑖 ]𝑇 and some 0 < 𝑷 = 𝑷 𝑇 ∈ R2×2 , is a Lyapunov function. Then, according to Levant (1998), if the gains are selected according to (15), the time derivative of 𝑉 , along the trajectories of (25)–(26), is bounded by 1

(27)

holds. Thus all the conditions of Theorem 6 are satisfied with 𝑐 ≥ 1, 𝜌max = 𝜌min = 1. Therefore, one may conclude that the origin of (27) is FT-ISS with respect to 𝑒𝑣𝑖 . Then, since 𝑒𝑣𝑖 = 0 is GUFTS, it follows that the position tracking error 𝜖𝑝𝑖 = 0 is GUFTS.

1

2 𝑉̇ (𝜖̂𝑣𝑖 , 𝜈̄𝑖 ) ≤ −𝜅𝜆min (𝑷 )𝑉 2 (𝜖̂𝑣𝑖 , 𝜈̄𝑖 ),

Proof of Theorem 5. The closed-loop dynamics is given by:

for some 𝜅 > 0. Therefore, (𝑠̂𝑖 , 𝜈̄𝑖 ) = 0 is GUFTS. Moreover, if the conditions for Theorem 2 hold then, 𝑠̂𝑖 (𝑡) = 𝑠𝑖 (𝑡) for all 𝑡 ≥ 𝑇 > 0. Therefore, the system dynamics collapse to the linear differential equation 𝜖𝑣𝑖 = −𝑐𝑖 𝜖𝑝𝑖 whose origin, 𝜖𝑝𝑖 = 0, is GUES for every 𝑐𝑖 > 0.

⎤ 𝜖̂𝑣𝑖 − 𝑒𝑣𝑖 ⎡𝜖 ⎤ ⎡ ⎥ 1 1 𝑑 ⎢ 𝑝𝑖 ⎥ ⎢ 𝜖̂𝑣𝑖 = ⎢−𝑘̄ 1𝑖 ⌈𝜖𝑝𝑖 ⌋ 2 − 𝑘̄ 2𝑖 ⌈𝜖̂𝑣𝑖 ⌋ 2 + 𝜈̄𝑖 ⎥ . 𝑑𝑡 ⎢ ⎥ ⎢ 0 0 ⎣ 𝜈̄𝑖 ⎦ ⎣ −𝑘̄ 3𝑖 ⌈𝜖𝑝𝑖 ⌋ − 𝑘̄ 4𝑖 ⌈𝜖̂𝑣𝑖 ⌋ + 𝛿̄𝑖 , ⎥⎦ 57

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L. Ovalle, H. Ríos, M. Llama et al.

Control Engineering Practice 85 (2019) 50–58

with the definition of 𝜈̄𝑖 and 𝛿̄𝑖 given in Theorem 2. In Torres-González, Sanchez, Fridman, and Moreno (2017), the following Lyapunov function candidate has been proposed 5

𝑉 (𝜖𝑝𝑖 , 𝜖̂𝑣𝑖 , 𝜈̄𝑖 ) = 𝛽|𝜖𝑝𝑖 | 3 + 𝜖𝑝𝑖 𝜖𝑣𝑖 +

2 2 1 𝑐 |𝜖 | 5 − 𝜖𝑣𝑖 𝜈̄𝑖3 + 𝛾|𝜈̄𝑖 |5 , 5 𝑖 𝑣𝑖 𝑘̄ 1𝑖

Dong, W., & Xu, W. L. (2001). Adaptive tracking control of uncertain nonholonomic dynamic system. IEEE Transactions on Automatic Control, 46(3), 450–454. Fridman, L., Moreno, J. A., Bandyopadhyay, B., Kamal, S., & Chalanga, A. (2015). Continuous nested algorithms: The fifth generation of sliding mode controllers. In Recent advances in sliding modes: From control to intelligent mechatronics (pp. 5–35). Springer. Huang, H. -C., & Tsai, C. -C. (2008). Adaptive robust control of an omnidirectional mobile platform for autonomous service robots in polar coordinates. Journal of Intelligent and Robotic Systems, 51(4), 439–460. Kamal, S., Moreno, J. A., Chalanga, A., Bandyopadhyay, B., & Fridman, L. M. (2016). Continuous terminal sliding-mode controller. Automatica, 69, 308–314. Kelly, R., Santibáñez, V., & Loría, J. (2006). Control of robot manipulators in joint space. Springer Science & Business Media. Khalil, H. (2002). Nonlinear systems. New Jersey, U.S.A.: Prentice Hall. Levant, A. (1998). Robust and exact differentiation via sliding mode technique. Automatica, 34(3), 1379–1384. Li, T. H. S., Chen, C. -Y., Hung, H. -L., & Yeh, Y. -C. (2008). A fully fuzzy trajectory tracking control design for surveillance and security robots. In 2008 IEEE international conference on systems, man and cybernetics (pp. 1995–2000). Singapore, Singapore. Li, D., & Ye, J. (2014). Adaptive robust control of wheeled mobile robot with uncertainties. In 2014 IEEE 13th international workshop on advanced motion control (pp. 518–523). Li, X., & Zell, A. (2009). Motion control of an omnidirectional mobile robot. In J. Filipe, J. Cetto, & J. Ferrier (Eds.), Informatics in control, automation and robotics, Vol. 24 (pp. 181–193). Springer. Lin, L. -C., & Shih, H. -Y. (2013). Modeling and adaptive control of an omni-mecanumwheeled robot. Intelligent Control and Automation, 4(2), 166–179. Peñaloza-Mejía, O., Márquez-Martínez, L. A., Alvarez, J., Villarreal-Cervantes, M. G., & García-Hernández, R. (2015). Motion control design for an omnidirectional mobile robot subject to velocity constraints. Mathematical Problems in Engineering, 2015. Polyakov, A. (2012). Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Transactions on Automatic Control, 57(8), 2106–2110. Ren, C., Sun, Y., & Ma, S. (2016). Passivity-based control of an omnidirectional mobile robot. Robotics and Biomimetics, 3(1), 10. Shtessel, Y., Edwards, C., Fridman, L., & Levant, A. (2014). Sliding mode control and observation, Vol. 10. Springer. Sira-Ramírez, H., López-Uribe, C., & Velasco-Villa, M. (2013). Linear observer-based active disturbance rejection control of the omnidirectional mobile robot. Asian Journal of Control, 15(1), 51–63. Torres-González, V., Sanchez, T., Fridman, L. M., & Moreno, J. A. (2017). Design of continuous twisting algorithm. Automatica, 80, 119–126. Treesatayapun, C. (2011). A discrete-time stable controller for an omni-directional mobile robot based on an approximated model. Control Engineering Practice, 19(2), 194–203. Viet, T. D., Doan, P. T., Hung, N., Kim, H. K., & Kim, S. B. (2012). Tracking control of a three-wheeled omnidirectional mobile manipulator system with disturbance and friction. Journal of Mechanical Science and Technology, 26(7), 2197–2211. Wang, C., Liu, X., Yang, X., Hu, F., Jiang, A., & Yang, C. (2018). Trajectory tracking of an omni-directional wheeled mobile robot using a model predictive control strategy. Applied Sciences, 8(2).

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with 𝛽, 𝛾 > 0. In Torres-González et al. (2017), it has been proven that, for 𝑒𝑣𝑖 = 0, the function 𝑉 is a Lyapunov function for system (24) such that, for some selection of 𝑘̄ 1𝑖 , 𝑘̄ 2𝑖 , 𝑘̄ 3𝑖 and 𝑘̄ 4𝑖 , 𝑖 = 1, 3 from Table 2, 𝑉̇ ≤ −𝜅𝑉 4∕5 , with some positive 𝜅. Therefore, (𝜖𝑝𝑖 , 𝜖̂𝑣𝑖 , 𝜈̄𝑖 ) = 0 is UFTS. Notice that the function 𝜖̂𝑣𝑖 − 𝑒𝑣𝑖 ⎛ ⎞ 1 1 ⎜ ̄ ̄ 2𝑖 ⌈𝜖̂𝑣𝑖 ⌋ 2 + 𝜈̄𝑖 ⎟⎟ 2 −𝑘 − 𝑘 ⌈𝜖 ⌋ ⎜ 1𝑖 𝑝𝑖 ̃ 𝒇 = (𝜖𝑝𝑖 , 𝜖̂𝑣𝑖 , 𝜈̄𝑖 , 𝑒𝑣𝑖 ) = , ⎜ −𝑘̄ 3𝑖 ⌈𝜖𝑝𝑖 ⌋0 − 𝑘̄ 4𝑖 ⌈𝜖̂𝑣𝑖 ⌋0 ⎟ ⎜ ⎟ 0 ⎝ ⎠ is homogeneous of degree 𝑞 = −1 with weights 𝒓 = (3, 2, 1) and 𝑟̃ = 2 and the condition ‖𝒇 (𝜖𝑝𝑖 , 𝜖̂𝑣𝑖 , 𝜈̄𝑖 , 𝑒𝑣𝑖 ) − 𝒇 (𝜖𝑝𝑖 , 𝜖̂𝑣𝑖 , 𝜈̄𝑖 , 0)‖ = ‖𝑒𝑣𝑖 ‖ ≤ 𝜎(‖𝑒𝑣𝑖 )‖, holds. Thus all the conditions of Theorem 6 are satisfied with 𝑐 ≥ 1, 𝜌max = 𝜌min = 1. Therefore, one may conclude that the origin of (27) is FT-ISS with respect to 𝑒𝑣𝑖 . Then, since 𝑒𝑣𝑖 = 0 is GUFTS, it follows that the position tracking error 𝜖𝑝𝑖 = 0 is GUFTS. References Alakshendra, V., & Chiddarwar, S. S. (2017a). Adaptive robust control of Mecanumwheeled mobile robot with uncertainties. Nonlinear Dynamics, 87(4), 2147–2169. Alakshendra, V., & Chiddarwar, S. S. (2017b). Simultaneous balancing and trajectory tracking control for an omnidirectional mobile robot with a cylinder using switching between two robust controllers. International Journal of Advanced Robotic Systems, 14(6), 1729881417738728. Bernuau, E., Efimov, D., Perruquetti, W., & Polyakov, A. (2014). On homogeneity and its application in sliding mode control. Journal of The Franklin Institute, 351(4), 1866– 1901. Bigelow, F. F., & Kalhor, A. (2017). Robust adaptive controller based on evolving linear model applied to a Ball-Handling mechanism. Control Engineering Practice, 69, 85–98. Campion, G., Bastin, G., & Dandrea-Novel, B. (1996). Structural properties and classification of kinematic and dynamic models of wheeled mobile robots. IEEE Transactions on Robotics and Automation, 12(1), 47–62. Cruz-Zavala, E., & Moreno, J. (2016). Lyapunov functions of continuous and discontinuous differentiators. In Proceedings of the 10th IFAC symposium on nonlinear control systems (pp. 660–665). Monterey, CA, USA.

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