A family of saturated controllers for UWMRs

Journal Pre-proof A family of saturated controllers for UWMRs Javier Moreno-Valenzuela, Luis Montoya-Villegas, Ricardo Pérez-Alcocer, Jesús Sandoval

PII: DOI: Reference:

S0019-0578(20)30007-0 https://doi.org/10.1016/j.isatra.2020.01.007 ISATRA 3448

To appear in:

ISA Transactions

Received date : 6 May 2019 Revised date : 9 December 2019 Accepted date : 3 January 2020 Please cite this article as: J. Moreno-Valenzuela, L. Montoya-Villegas, R. Pérez-Alcocer et al., A family of saturated controllers for UWMRs. ISA Transactions (2020), doi: https://doi.org/10.1016/j.isatra.2020.01.007. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2020 Published by Elsevier Ltd on behalf of ISA.

Journal Pre-proof

A family of saturated controllers for UWMRs∗

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Javier Moreno-Valenzuela†a , Luis Montoya-Villegasa , Ricardo P´erez-Alcocerb , and Jes´ us Sandovalc a

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Instituto Polit´ecnico Nacional-CITEDI, Av. Instituto Polit´ecnico Nacional No. 1310, Colonia Nueva Tijuana, Tijuana, Baja California 22435, M´exico, email: [email protected], [email protected] b CONACYT-Instituto Polit´ecnico Nacional-CITEDI, Av. Instituto Polit´ecnico Nacional No. 1310, Colonia Nueva Tijuana, Tijuana, Baja California 22435, M´exico, email: [email protected] c Tecnol´ogico Nacional de M´exico/Instituto Tecnol´ogico de la Paz, Boulevard Forjadores de Baja California Sur No. 4720, La Paz, Baja California Sur, M´exico, 23080, email: [email protected] January 9, 2020

Abstract

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Input saturation appears in a physical system when a large power dissipation is requested. In this situation, and specifically for unicycle-type wheeled mobile robots, actuators only can deliver a finite amount of power. Thus, in practice the linear and angular velocity input of this class of mobile robots is limited and this should be considered in the control design. In this paper, a family of controllers that produce saturated velocity input for unicycle-type wheeled mobile robots is presented. The proposed family of controllers is designed to satisfy the trajectory tracking control goal. Sufficient conditions to prove the closed-loop system global asymptotic stability are established by using Lyapunov’s theory. Already reported schemes and original designs are shown to satisfy the properties of the given family of controllers. By using two different motion tasks, experimental tests in real-time with five saturated control schemes are presented in order to validate the proposed theory. In order to show the ability of the family of controllers to produce limited control action, experiments have also been carried out with an unsaturated algorithm. Better tracking accuracy is obtained with the original design derived from the proposed class of algorithms. Keywords: Unicycle wheeled mobile robots, velocity input saturation, nonlinear control, trajectory tracking control, real-time experiments.

Introduction

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1

The control of mobile vehicles with nonholonomic constraints has grown in the last decades. This class of wheeled mobile robots is implemented to achieve several tasks as: transport, inspection, paint, solder, draw, and others where the tracking and regulation precision are crucial. A literature review is presented in the next. Visual guidance of mobile robots was carried out in [1]. The authors of [2] and [3] presented some experiments with a trajectory tracking controller for unicycle-type wheeled mobile robots (UWMRs) focused on actuator and power stage dynamics. Mart´ınez-Clark et al. [4] developed a posture regulation ∗ This

work was supported by CONACYT Projects A1-S-24762, 166636, and C´ atedras 1537, TecNM Project, and SIP–IPN. author. E-mail: [email protected]

† Corresponding

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controller for the emergence of a self-organized mobile robot swarm. A new motion controller based on the feedback linearization was presented in [5]. In [6], a sliding mode neuro-adaptive controller was designed. Falsafi et al. [7] designed a controller for UWMRs by using fuzzy logic methods. Similarly, a navigation controller for UWMRs was proposed in [8] by using fuzzy logic. A stabilizing control law was proposed by Hashimoto et al. [9]. Gu et al. [10] proposed a nonlinear trajectory tracking controller in order to make the mobile robot pass through the required waypoints for reconstructing the orbital movement. In [11], a lookahead kinematic adaptive robust controller and a robust adaptive computed torque dynamic controller were designed for the trajectory tracking problem of UWMRs in presence of uncertainties. A tracking controller for the tractor-trailer wheeled robot was proposed in [12], which is based on a PID controller. A problem frequently found in the control of robotic and mechatronic systems is the control input saturation, which appears when the actuators reach their maximum capabilities. A way to avoid this problem is to design controllers that request limited control input. The saturation restrictions are addressed by several researches. For example, Wu and Lian [13] designed a saturated controller applied to switched systems. In [14], a feedback control law adopting the saturated super-twisting algorithm was designed. Furthermore, a H∞ fuzzy controller for electric power steering with actuator saturation and a saturated fault-tolerant attitude tracking controller for a disturbed rigid spacecraft were proposed by the authors Nasri et al. [16] and Sun [15], respectively. The results of [17] showed a saturated prescribed performance controller for EulerLagrange dynamic systems. Finally, the authors of [18] reported an adaptive nested saturation feedback controller for multiple integrator systems.

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It is worthwhile to notice that UWMRs are subjected to nonlinearities due to the saturation restrictions in the actuators. Actually, the consideration of the velocity saturation of the actuators is necessary to design controllers for UWMRs. However, stabilization and trajectory tracking problems for nonholonomic mobile systems with saturated inputs have been studied rarely in the literature and only a few works have addressed these problems. For example, in [19] and [20], solutions based on the backstepping and finite-time control techniques were proposed to limit the velocities and torques, respectively. Jiang et al. [21] designed two controllers using a model-based control strategy. A path following methodology based on a fuzzy-logic set of rules which imitates the human driving behavior was proposed by Antonelli et al. [22]. The authors of [23] presented a saturated fuzzy controller for trajectory tracking. Moreover, a passivity-based output feedback tracking controller providing amplitude-limited control signals to prevent the actuators from saturation was introduced by Shojaei and Chatraei [24].

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Additionally, the mobile robot control community has faced the saturation problem by providing solutions based on adaptive control. The authors of [25] and [26] used saturated adaptive controllers to guide an UWMR during trajectory tracking. Huang et al. [27] proposed global adaptive stabilization and tracking control scheme for a UWMR with input saturation and external disturbances. A saturated output feedback controller for uncertain nonholonomic UWMRs was presented by Shojaei [28]. In [29], a neural adaptive robust output feedback controller was created to solve the trajectory tracking problem of electrically driven UWMRs under model uncertainties and actuator saturation. A robust neural network-based control scheme was presented by the authors of [30] to address the problem of tracking and stabilization simultaneously for a UWMR which presents parametric uncertainties, external disturbances, and input saturation. The results of [31] showed a neuro adaptive robust controller with a real-time self-tuning to deal with unmodeled dynamics and external disturbances.

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On the other hand, control techniques have been considered to deal with the input saturation. Evers and Nijmeijer [32] designed a new practical stabilizing hybrid controller for UWMRs. A saturated controller based on integral sliding mode was proposed in [33] to solve the tracking problem in the presence of input saturation and unknown disturbances. Semiglobal practical stabilizing control schemes for a class of UWMR with input constraints was proposed by Wang [34]. In [35], based on visual servoing a new saturated switching controller was given. Similarly, in [36], a saturated switching controller was reported. Chen et al. [37] introduced a vision-based controller by using calibrated camera parameters. The authors proposed a continuous and saturated controller applying the finite-time control theory and a switching technique. The results of [38] showed a double loop control structure for UWMRs with velocity saturation by using the hyperbolic tangent function and an integral sliding mode controller. A simple tracking controller was proposed by Chen and Jia [39], which incorporates amplitude and rate-bounded feedback signals produced from two first-order filters.

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Chen [40] presented a continuous time-varying saturated controller with a systematic strategy that combines the theory of finite-time stability with the virtual-controller-tracked strategy. The authors of [41] designed a saturated state feedback controller for UWMRs. A saturated synchronous controller was introduced for multiple UWMRs to perform a time-varying formation task in [42]. Liu and Gao [43] proposed a Lyapunovbased predictive tracking controller for UWMRs subjected to control input constraints. In [44], a vision-based strategy was proposed for tracking control and regulation of a UWMR using an uncalibrated monocular camera. Finally, a synthetic-analytic behavior-based control for tracking of input-constrained UWMRs was designed by Meza-S´anchez et al. [45].

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As pointed out in the literature review, just a few strategies have been given for the saturated control of UWMRs. In other words, this problem has not been studied enough whereby original solutions are necessary, including generalizations and experimental assessments. The contributions of this document are:

• The introduction of a novel family of saturated controllers, which includes already reported approaches as well as original designs. • A rigorous stability analysis that guarantees global asymptotic stability for the overall closed-loop system. Derived from this analysis, tuning guidelines ensuring bounding of the control action and the closed-loop stability are also provided.

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• The experimental validation of five saturated control schemes belonging to the given family of controllers. Two of these schemes are reported in [21] and [46], while the remaining schemes correspond to original designs presented for the first time. In addition, an unsaturated controller reported in [49] is also experimentally tested in order to perform a comparison with respect to the control schemes belonging the family of saturated controllers. Among them, three are original designs and two are already known control schemes. Two of the original schemes provided better tracking results than the controllers taken from the literature.

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The remaining of this document is organized as follows. Section II describes the kinematic model for UWMRs, the control problem and the open-loop system. In Section III, the family of saturated controllers together with its respective stability analysis are presented. Section IV presents some designs examples based on the family of saturated controllers. In Section V, experimental results in a mobile robot are given. Besides, comparisons between saturated controllers and an unsaturated scheme are also discussed. Finally, Section VI establishes the conclusions of this work.

Kinematic model

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The kinematic model of the UWMR has been used in several control approaches [21]. Based on this model, the location of the vehicle can be known by the position of the middle point of the virtual axle and its orientation referred to a fixed frame of reference. In Figure 1 the UWMR pose is depicted. Thus, the kinematic model for the UWMR is given as     x˙ cos(θ) 0    y˙  =  sin(θ) 0 V , (1) W 0 1 θ˙

where x(t) and y(t) are the position of the point of interest of the mobile robot in the horizontal plane, θ(t) is the orientation, and the control inputs V (t) and W (t) are the linear and angular velocities, respectively.

2.1

Control problem

The family of saturated controllers presented in this document has been designed to guarantee the position and orientation trajectory tracking control goal formulated by Jiang et al. in the pioneering work [21]. 3

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Figure 1: UWMR. Robot model referred to the coordinate (x, y) of po . The reference trajectory is given by

    x˙ r Vr cos(θr )  y˙ r  =  Vr sin(θr )  , Wr θ˙r

(2)

where xr , yr , θr are the reference pose, and Vr and Wr are linear and angular reference velocities, respectively, bounded by

for all t ≥ 0. Besides, the following limit

≤

|Vr |max ,

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|Vr (t)|

|Wr (t)|

≤

|Wr |max ,

lim Vr (t) 6= 0

(3)

t→∞

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is assumed to be satisfied.

The control objective is to design time-varying state-feedback control inputs of the form V (t, θ, x, y),

such that

W (t, θ, x, y),

(4)

     x ˜(t) xr (t) − x(t) 0 lim  y˜(t)  = lim  yr (t) − y(t)  = 0 t→∞ t→∞ ˜ θr (t) − θ(t) 0 θ(t)

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

(5)

is ensured, while guaranteeing bounded control action in the sense −Vmax ≤ V (t) ≤ Vmax ,

−Wmax ≤ W (t) ≤ Wmax ,

∀ t ≥ 0,

(6)

where Vmax > supt≥0 |Vr (t)| and Wmax > supt≥0 |Wr (t)| are the maximum linear and angular velocities, respectively. Notice from (6) that the saturation limits of V (t) and W (t) are symmetric.

2.2

Open-loop system

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The pose error of the UWMR represented in the body reference frame is given by the following transformation [21]:      ˜ e1 cos(θ) sin(θ) 0 x e2  = − sin(θ) cos(θ) 0 y˜ . (7) e3 0 0 1 θ˜ Then, differentiating the equation (7) with respect to time and substituting (1) and (2) in the resulting expression, the open-loop dynamics of the tracking error for the UWMR vehicle is defined as     e˙ 1 Vr cos e3 − V + W e2 e˙ 2  =  Vr sin e3 − W e1  . (8) e˙ 3 Wr − W 4

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3

Family of saturated controllers

It is noteworthy that the saturated control structures reported in [21] and [46] share a common structure. Inspired in this form, the proposed family of saturated controllers is given by the expression     Vr cos(e3 ) + k1 tanh(e1 ) V = , (9) 1 ,e2 ) sin(e3 ) W Wr + λVr ∂f (e + k2 tanh(e3 ) ∂e2 e3

P1 The partial derivatives

∂f (e1 ,e2 ) ∂e1

constant kf such that kf ≥

∂f (e1 ,e2 ) exist ∂e2 1 ,e2 ) sup∀e1 ,e2 | ∂f (e |. ∂e2

and

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where λ, k1 , k2 are strictly positive control gains and f (e1 , e2 ) is a positive definite and continuously differentiable function. Under a proper selection, the function f (e1 , e2 ) allows ensuring the control objective (5) and as will be seen later the bounds in (6). The function f (e1 , e2 ) must satisfy the following properties: and are continuous. In addition, there is a positive

P2 f (0, 0) = 0.

P3 f (e1 , e2 ) > 0, ∀ e1 6= 0, e2 6= 0, and is radially unbounded. P4

∂f (e1 ,e2 ) e1 ∂e2

P5

∂f (e1 ,e2 ) ∂e1 ∂f (e1 ,e2 ) ∂e1

∂f (e1 ,e2 ) e2 . ∂e1

re-

tanh(e1 ) > 0, ∀ e1 6= 0 and e2 ∈ IR and tanh(e1 ) = 0, ∀ e1 = 0 and e2 ∈ IR.

∂f (e1 ,e2 ) ∂e1

= ke1 h(e1 , e2 ), where k is a positive constant and h(e1 , e2 ) is a positive function bounded satisfying 0 ≤ h(e1 , e2 ) ≤ 1, ∀ e1 , e2 ∈ IR.

P7 The partial derivatives P8 lime2 →0

∂f (e1 ,e2 ) ∂e2

∂ 2 f (e1 ,e2 ) ∂e22

= 0.

and

∂ 2 f (e1 ,e2 ) ∂e1 ∂e2

exist and are continuous.

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P6

=

Moreover, the family of saturated controllers (9) satisfies the constraints in the inequalities stated in (6) as follows = =

|Vr |max |Wr |max

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|V (t)| ≤ Vmax |W (t)| ≤ Wmax

+ k1 , + λ|Vr |max kf

+

k2 .

(10)

where kf is defined in property P1 and the properties | tanh(x)| ≤ 1, | sin(x)| ≤ 1 for all x ∈ IR, are used to obtain (10). Substituting the family of saturated controllers (9) into the tracking error dynamics (8) the overall closed-loop system is given by e˙ 1

= Vr sin(e3 ) − Wr e1 − λVr

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e˙ 2

= −k1 tanh(e1 ) + Wr e2 + λVr

e˙ 3

= −λVr

∂f (e1 , e2 ) sin(e3 ) e2 + k2 tanh(e3 )e2 , ∂e2 e3

∂f (e1 , e2 ) sin(e3 ) e1 − k2 tanh(e3 )e1 , ∂e2 e3

∂f (e1 , e2 ) sin(e3 ) − k2 tanh(e3 ). ∂e2 e3

(11) (12) (13)

Proposition 1. Assume that the properties P1-P8 are fulfilled. Then, the state-space origin [e1 e2 e3 ]T = [0 0 0]T of the closed-loop system (11)–(13) is globally asymptotically stable. Besides, in accordance with (7), is achieved the limit ˜ = 0. lim x ˜(t), y˜(t), θ(t) (14) t→∞

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Proof. The property P1 is required to guarantee the bounding (10) of the control action. Considering the Lyapunov function candidate 1 λf (e1 , e2 ) + e23 , 2

U (e1 , e2 , e3 ) =

(15)

which is positive definite under the properties P2-P3 and the constant λ strictly positive. Obtaining the time derivative, we have λ

∂f (e1 , e2 ) ∂f (e1 , e2 ) e˙ 1 + λ e˙ 2 + e3 e˙ 3 . ∂e1 ∂e2

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U˙ (e1 , e2 , e3 ) =

(16)

Notice that P1 is required to compute U˙ (e1 , e2 , e3 ) in (16). Substituting the closed-loop system (11)-(13) into the expression (16), we get   ∂f (e1 , e2 ) ∂f (e1 , e2 ) sin(e3 ) −k1 tanh(e1 ) + Wr e2 + λVr e2 + k2 tanh(e3 )e2 U˙ (e1 , e2 , e3 ) = λ ∂e1 ∂e2 e3   ∂f (e1 , e2 ) ∂f (e1 , e2 ) sin(e3 ) + λ Vr sin(e3 ) − Wr e1 − λVr e1 − k2 tanh(e3 )e1 ∂e2 ∂e2 e3   ∂f (e1 , e2 ) sin(e3 ) + e3 −λVr − k2 tanh(e3 ) . (17) ∂e2 e3

re-

Expanding equation (17), grouping common terms, and using P4 to cancel similar terms, U˙ (e1 , e2 , e3 ) can be expressed as ✿ ✘✘ ✘✘ ✘ ✿ ✘ ✘ ✘ ∂f (e1 , e2 ) ∂f (e1 , e2 )✘ , e ) sin(e3 ) ∂f (e1 , e2 ) ∂f (e ✘✘1✘ 2 λ k1 tanh(e1 ) + λ ✘✘✘✘ Wr e2 + λ2 Vr e2 ✘✘ ∂e1 ✘ ✘ ∂e1 ∂e1 ∂e e3 2 ✘ ✘ ✘✘ 4 2 3 ✿ ✘ ✘ ✿ ✘ ✿ ✘ ∂f (e1 , e✘ ∂f (e1 , e2 )✘✘✘✘ ∂f (e1 , e2 )✘✘✘✘ 2 )✘✘ ✘ + λ ✘✘✘ Vr sin(e3 ) − λ ✘✘✘ Wr e1 λk2 ✘✘ tanh(e3 )e2 ✘ ∂e2 ✘✘ ∂e2 ✘✘✘ ∂e1

+

− −

Finally, we have

5 4 ✿ ✘✘ ✘ ✿ ✘ ✘ ✘✘ ✘ ✘ ✘ ∂f (e , e ) ∂f (e , e ) ∂f (e , e ) sin(e ) ✘ ✘ 1 2 1 2 3 1 2 ✘✘ λ2 Vr e1 − λk2 ✘✘✘✘ tanh(e3 )e1 ✘2✘✘ ∂e2 e3 ✘∂e ✘✘✘ ∂e2 ✘ ✘✘ 2 ✿ ✘ 1✘ ✘ ∂f (e1 , e2 ) sin(e ) ✘ 3 ✘✘ ✚ λVr e✚ − k2 e3 tanh(e3 ). ❃ 3 ✘✘ ∂e✘ ✘ 2 1 ✘ ✘✘ e✚ ❃ 3 ✚

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U˙ (e1 , e2 , e3 ) = −

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3

∂f (e1 , e2 ) U˙ (e1 , e2 , e3 ) = −λ k1 tanh(e1 ) − k2 e3 tanh(e3 ). ∂e1

(18)

(19)

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Note that from P5 we can conclude that

U˙ (e1 , e2 , e3 ) ≤ 0,

∀ t ≥ 0.

(20)

Therefore, knowing that U (e1 , e2 , e3 ) is a positive definite function and U˙ (e1 , e2 , e3 ) is negative or zero, then U is decreasing or constant for all t ≥ 0. From this fact, we conclude that e1 (t), e2 (t), e3 (t) ∈ L∞ .

To prove that the control inputs V (t), W (t) ∈ L∞ , the assumption that Vr (t), Wr (t) ∈ L∞ , and the fact that sin(e3 ) lim =1 e3 →0 e3 6

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are used in (9). Now, the open-loop equations in (8) are utilized to prove that e˙ 1 (t), e˙ 2 (t), e˙ 3 (t) ∈ L∞ . Therefore, e1 (t), e2 (t), e3 (t) are uniformly continuous functions.

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The next step consists in proving the asymptotic convergence of e1 (t) and e3 (t). Integrating both sides of the equation (19): Z ∞ − U˙ (t)dt = −U (∞) + U (0) 0 Z ∞ Z ∞ ∂f (e1 (t), e2 (t)) = λk1 tanh(e1 (t))dt + k2 e3 (t) tanh(e3 (t))dt, (21) ∂e1 (t) 0 0 and using the fact U (0) ≥ U (∞) ≥ 0, it is possible to prove from the inequality (21) that Z ∞ ∂f (e1 (t), e2 (t)) U (0) tanh(e1 (t))dt ≤ < ∞, ∂e (t) λk1 1 0 and

Z

∞

0

e3 (t) tanh(e3 (t))dt ≤

U (0) < ∞. k2

(22)

(23)

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By using the facts that e1 (t), e3 (t) are uniformly continuous, that the integrals expressed in equations (22) and (23) exist and using property P6, Barbalat’s lemma is invoked (see Lemma 8.2 in ref. [47], p. 323,) to prove that lim e1 (t) = 0, and lim e3 (t) = 0. t→∞

t→∞

Furthermore, to prove that limt→∞ e2 (t) = 0, the second time derivative of e3 (t) in (13) is obtained:

=

−V˙ r λ

∂f sin(e3 ) ∂ 2 f sin(e3 ) ∂2f sin(e3 ) ∂f e3 cos(e3 ) − sin(e3 ) − λVr 2 e˙ 2 − λVr e˙ 1 − λVr e˙ 3 ∂e2 e3 ∂e2 e3 ∂e1 ∂e2 e3 ∂e2 e23

−k2 sech2 (e3 )e˙ 3 .

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e¨3

(24)

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Because e1 (t),e˙ 1 (t),e2 (t),e˙ 2 (t),e3 (t),e˙ 3 (t) ∈ L∞ , the assumption that Vr (t), V˙ r (t) ∈ L∞ , P7, and the fact that   e3 cos(e3 ) − sin(e3 ) lim = 0, e3 →0 e23

it is possible to conclude from the right-hand side of (24) that e¨3 (t) ∈ L∞ . Therefore, e˙ 3 (t) is uniformly continuous. Since limt→∞ e3 (t) = 0 has been proven, Barbalat’s lemma in the version expressed in Lemma 4.2 in ref. [48], p. 123, is employed to prove that lim e˙ 3 (t) = 0.

t→∞

Equation (13) is rewritten for the sake of better appreciation in the coming discussion:

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e˙ 3 = −λVr

∂f (e1 , e2 ) sin(e3 ) − k2 tanh(e3 ). ∂e2 e3

Therefore, the limit stated in (25) implies lim −λVr

t→∞

∂f (e1 (t), e2 (t)) sin(e3 ) = 0, ∂e2 (t) e3

and by using the assumption (3) and lime3 →0

sin(e3 ) = 1 we can conclude that e3

∂f (e1 (t), e2 (t)) = 0, t→∞ ∂e2 (t) lim

7

(25)

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which at the same time implies together with P8 that lim e2 (t) = 0.

t→∞

Finally, since the error transformation matrix in (7) is invertible, the limit in (14) is satisfied. This completes the proof of Proposition 1.

4

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Next, we present five designs belonging to the family of controllers in (9). Two of these designs were taken from the literature [21], [46], identifying the function f (e1 , e2 ). The remaining three schemes are original controllers obtained by proposing functions f (e1 , e2 ) satisfying the properties P1-P8.

Design Examples

4.1 4.1.1

Designs reported in [21] and [46] Design 1 taken from [21]

This design was introduced in [21] and belongs to the family of controllers analyzed in this document. The controller in [21] is a particular case of schemes expressed in (9) with 1 log(1 + e21 + e22 ), 2

re-

f (e1 , e2 ) =

where log(x), x ∈ IR, means the natural logarithm of x.

(26)

The function f (e1 , e2 ) in (26) satisfies the properties P1-P8, which are verified explicitly as follows:

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• P1 The two partial derivatives required

∂f (e1 , e2 ) e1 = ∂e1 1 + e21 + e22

and

urn a

e2 ∂f (e1 , e2 ) = ∂e2 1 + e21 + e22

are continuous functions. In addition, for this case kf = 0.5. • P2 The function (26) evaluated at e1 = e2 = 0 is equal to zero, that is, f (0, 0) =

1 2

log(1 + 0 + 0) = 0.

• P3 The function (26) satisfies

f (e1 , e2 ) =

1 2

log(1 + e21 + e22 ) > 0,

∀

e1 6= 0, e2 6= 0,

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and is radially unbounded clearly.

• P4 By computing the corresponding expressions involved in this property, we have

Therefore,

∂f (e1 ,e2 ) e1 ∂e2

=

∂f (e1 , e2 ) e2 e1 e1 = , ∂e2 1 + e21 + e22 ∂f (e1 , e2 ) e1 e2 e2 = . ∂e1 1 + e21 + e22 ∂f (e1 ,e2 ) e2 ∂e1

as desired. 8

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• P5 For this property, it may be verified that ∂f (e1 , e2 ) e1 tanh(e1 ) = tanh(e1 ) > 0, ∀ e1 6= 0 and e2 ∈ IR, ∂e1 1 + e21 + e22 and

∂f (e1 , e2 ) e1 tanh(e1 ) = tanh(e1 ) = 0, ∀ e1 = 0 and e2 ∈ IR. ∂e1 1 + e21 + e22

pro of

• P6 This is satisfied as follows:

(27)

∂f (e1 , e2 ) e1 = = ke1 h(e1 , e2 ), ∂e1 1 + e21 + e22 where k = 1, h(e1 , e2 ) =

1 , 1+e21 +e22

which at the same time achieves 0 ≤ h(e1 , e2 ) ≤ 1, ∀ e1 , e2 ∈ IR.

• P7 In this condition we have that:

∂ 2 f (e1 , e2 ) 1 2e22 = − 2 2 2 ∂e2 1 + e1 + e2 (1 + e21 + e22 )2 and

re-

∂ 2 f (e1 , e2 ) 2e1 e2 =− ∂e1 e2 (1 + e21 + e22 )2

are continuous functions.

• P8 Finally, the last condition is fulfilled as

lim

e2 = 0. 1 + e21 + e22

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e2 →0

Hence, the close-loop system stability is achieved with the following controller: #   " Vr cos(e3 ) + k1 tanh(e1 ) V , = λVr e2 sin(e3 ) + k2 tanh(e3 ) Wr + 1+e W 2 +e2 e3 1

(28)

2

4.1.2

urn a

which is actually the scheme proposed in [21]. Besides, the upper bounds of the control input (28) are given in (10). Design 2 taken from [46]

The second controller presented in the document was proposed in [46]. This scheme is structured as the family of controllers analyzed. The positive function f (e1 , e2 ) in this case is explicitly given by q f (e1 , e2 ) = 1 + e21 + e22 − 1, (29)

Jo

and satisfies the properties P1-P8 as follows: • P1 The partial derivatives

and

∂f (e1 , e2 ) e1 =p ∂e1 1 + e21 + e22 ∂f (e1 , e2 ) e2 =p ∂e2 1 + e21 + e22

are continuous functions. Consequently, kf = 1.0. 9

Journal Pre-proof

• P2 The function (29) satisfies the equation f (0, 0) =

f (e1 , e2 ) = and is radially unbounded.

p 1 + e21 + e22 − 1 > 0,

• P4 In this property we have

∀

e1 6= 0, e2 6= 0,

pro of

• P3 The function f (e1 , e2 ) satisfies

√ 1 + 0 + 0 − 1 = 0.

∂f (e1 , e2 ) e2 e1 e1 = p , ∂e2 1 + e21 + e22

and therefore P4 is accomplished.

∂f (e1 , e2 ) e1 e2 e2 = p , ∂e1 1 + e21 + e22

• P5 The product analyzed in this property is positive definite, i.e.,

re-

and

∂f (e1 , e2 ) e1 tanh(e1 ) > 0, ∀ e1 6= 0 and e2 ∈ IR, tanh(e1 ) = p ∂e1 1 + e21 + e22 ∂f (e1 , e2 ) e1 tanh(e1 ) = 0, ∀ e1 = 0 and e2 ∈ IR. tanh(e1 ) = p ∂e1 1 + e21 + e22

lP

• P6 The structure and conditions required in this property are fulfilled as ∂f (e1 , e2 ) e1 = ke1 h(e1 , e2 ), = p ∂e1 1 + e21 + e22 1 , 1+e21 +e22

where k = 1 and h(e1 , e2 ) = √

which clearly satisfies 0 ≤ h(e1 , e2 ) ≤ 1, ∀ e1 , e2 ∈ IR.

• P7 The partial derivatives for this condition are continuous as shown

urn a

∂ 2 f (e1 , e2 ) ∂e22

∂ 2 f (e1 , e2 ) ∂e1 e2

=

1 e22 p −p , 2 2 1 + e1 + e2 (1 + e21 + e22 )3

e1 e2 = −p . (1 + e21 + e22 )3

• P8 This last property is satisfied because

e2 lim p = 0. 1 + e21 + e22

e2 →0

Jo

Therefore, the control law reported in [46] belongs to the family of controllers studied in this work: #   " Vr cos(e3 ) + k1 tanh(e1 ) V = W + √ λVr e2 sin(e3 ) + k tanh(e ) . (30) r 2 3 W 2 2 e3 1+e1 +e2

Similarly to the Design 1, the upper bounds of V (t) and W (t) are established using the equation (10).

10

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4.2

Original designs

We have proven that some already reported control schemes match the conditions established for the introduced family of controller (9). Now, three original designs are presented which satisfy the properties given in Proposition 1. 4.2.1

Design 3

pro of

An original saturated controller is proposed by using the following positive definite function  q  1 f (e1 , e2 ) = log cosh βe21 + βe22 + 1 − 1 , α

(31)

where α and β are strictly positive constants. Let us discuss how the function (31) satisfies the properties P1-P8. • P1 The partial derivatives

p  βe21 + βe22 + 1 − 1 p α βe21 + βe22 + 1

∂f (e1 , e2 ) = ∂e1

βe1 tanh

∂f (e1 , e2 ) = ∂e2

βe2 tanh

p  βe21 + βe22 + 1 − 1 p α βe21 + βe22 + 1

re-

and

are continuous functions. Besides, the constant kf =

√ β α .

lP

• P2 The function f (e1 , e2 ) is null at e1 = 0 and e2 = 0, that is,

√ f (0, 0) = α1 log cosh α 0 + 0 + 1 − 1 = 0. • P3 The function f (e1 , e2 ) in (31) is positive definite   p f (e1 , e2 ) = α1 log cosh βe21 + βe22 + 1 − 1 > 0,

e1 6= 0, e2 6= 0,

urn a

and is radially unbounded.

∀

• P4 This property is satisfied as:

 p 2 + βe2 + 1 − 1 βe e tanh βe 1 2 1 2 ∂f (e1 , e2 ) p e1 = , 2 2 ∂e2 α βe1 + βe2 + 1 p  2 + βe2 + 1 − 1 βe e tanh βe 1 2 1 2 ∂f (e1 , e2 ) p e2 = . 2 2 ∂e1 α βe1 + βe2 + 1

Therefore, the partial derivatives are equal as required.

and

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1 ,e2 ) • P5 For both conditions required by this property, the positive definiteness of the product ∂f (e tanh(e1 ) ∂e1 is achieved, such that, p  2 + βe2 + 1 − 1 βe tanh(e ) tanh βe 1 1 1 2 ∂f (e1 , e2 ) p tanh(e1 ) = > 0, ∀ e1 6= 0 and e2 ∈ IR, ∂e1 α βe21 + βe22 + 1

p  2 + βe2 + 1 − 1 βe tanh(e ) tanh βe 1 1 1 2 ∂f (e1 , e2 ) p tanh(e1 ) = = 0, ∀ e1 = 0 and e2 ∈ IR. 2 2 ∂e1 α βe1 + βe2 + 1 11

Journal Pre-proof

• P6 The function h(e1 , e2 ) is computed by

where k =

β α

p  βe21 + βe22 + 1 − 1 βe1 tanh ∂f (e1 , e2 ) p = = ke1 h(e1 , e2 ), ∂e1 α βe21 + βe22 + 1 √  tanh βe21 +βe22 +1−1 √ 2 and h(e1 , e2 ) = and 0 ≤ h(e1 , e2 ) ≤ 1, ∀ e1 , e2 ∈ IR. 2 βe1 +βe2 +1

pro of

• P7 None of the second partial derivatives are discontinuous: p  p  2 2 2 2 + βe2 + 1 − 1 2 + βe2 + 1 − 1 2 βe βe β e sech β tanh 2 1 2 1 2 ∂ f (e1 , e2 ) p = + 2 2 ∂e22 α (βe21 + βe22 + 1) α βe1 + βe2 + 1 p  2 2 2 2 β e2 tanh βe1 + βe2 + 1 − 1 − , α(βe21 + βe22 + 1)3/2 p p   2 2 2 2 + βe2 + 1 − 1 2 + βe2 + 1 − 1 2 β e e sech βe β e e tanh βe 1 2 1 2 1 2 1 2 ∂ f (e1 , e2 ) = − . ∂e1 e2 α (βe21 + βe22 + 1) α(βe21 + βe22 + 1)3/2 • P8 This is directly achieved by noting that

re-

lim

p  βe21 + βe22 + 1 − 1 p = 0. α βe21 + βe22 + 1

βe2 tanh

e2 →0

lP

Finally, the control input has the following mathematical structure in accordance with (9):     Vrcos(e 3 ) + k1 tanh(e 1)  √ V . βe2 tanh βe2 +βe2 +1−1 sin(e ) = 3 W √ 21 22 Wr + λVr + k2 tanh(e3 ) e3 α

(32)

βe1 +βe2 +1

As pointed out in the above schemes, the upper bounds for the controller (32) are given in (10). 4.2.2

Design 4

urn a

Similarly to Design 3 in equation (32), the second original saturated controller is endowed with a novel function constituted by p  (p if e21 + e22 + 1 − 1 > π, e21 + e22 + 1 − 1 − π + 1, p  p (33) f (e1 , e2 ) = 1 − cos e21 + e22 + 1 − 1 , if e21 + e22 + 1 − 1 ≤ π, which satisfies the properties P1-P8, as follows: • P1 The partial derivatives

Jo

and

  √ e1 2 , ∂f (e1 , e2 )  e21 +e 2 +1 = e1 sin √e21 +e22 +1−1  ∂e1 √2 2  , e1 +e2 +1

  √ e2 2 , ∂f (e1 , e2 )  e21 +e 2 +1 = e2 sin √e21 +e22 +1−1  ∂e2 √2 2  , e1 +e2 +1

p  e21 + e22 + 1 − 1 > π p  if e21 + e22 + 1 − 1 ≤ π if

p  e21 + e22 + 1 − 1 > π p  if e21 + e22 + 1 − 1 ≤ π if

are continuous functions. Besides, the constant kf = 1.0. 12

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• P2 We can show this property by observing that
√ f (0, 0) = 1 − cos 0 + 0 + 1 − 1 = 0,


√ 0 + 0 + 1 − 1 ≤ π.

if

and is radially unbounded.

pro of

• P3 The function (33) is always positive for any value e1 6= 0 and e2 6= 0. Thus, p  p   e21 + e22 + 1 − 1 − π + 1, if e21 + e22 + 1 − 1 > π, p  p  f (e1 , e2 ) = 1 − cos e21 + e22 + 1 − 1 , if e21 + e22 + 1 − 1 ≤ π, • P4 Developing each one of the following products:  e2 e1   √e2 +e2 +1 , ∂f (e1 , e2 ) 1 2 e1 = e2 e1 sin √e21 +e22 +1−1  ∂e2 √2 2  , e1 +e2 +1

 e1 e2   √e2 +e2 +1 , ∂f (e1 , e2 ) 1 2 e2 = e1 e2 sin √e21 +e22 +1−1  ∂e1 √2 2  ,

p

 e21 + e22 + 1 − 1 > π,  p e21 + e22 + 1 − 1 ≤ π, if if

p

re-

e1 +e2 +1

 e21 + e22 + 1 − 1 > π, p  if e21 + e22 + 1 − 1 ≤ π, if

we show that the property P3 is achieved.

lP

1 ,e2 ) • P5 For this property, the product ∂f (e tanh(e1 ) should be a positive definite function, which is ∂e1 proven below  e tanh(e ) p  1 1   √e2 +e2 +1 , if e21 + e22 + 1 − 1 > π ∂f (e1 , e2 ) 1 2  √ p  tanh(e1 ) = e2 +e2 +1−1 e1 tanh(e1 ) sin  ∂e1 e21 + e22 + 1 − 1 ≤ π √2 21 2  , if

e1 +e2 +1

> 0, ∀ e1 6= 0 and e2 ∈ IR,

=

1

2

urn a

∂f (e1 , e2 ) tanh(e1 ) ∂e1

 e tanh(e ) 1 1   √e2 +e2 +1 ,

e1 tanh(e1 ) sin  √2 

√  e21 +e22 +1−1

e1 +e22 +1

p  e21 + e22 + 1 − 1 > π p  if e21 + e22 + 1 − 1 ≤ π if

,

= 0, ∀ e1 = 0 and e2 ∈ IR.

• P6 In particular for this property,

 e1   √e2 +e2 +1 , ∂f (e1 , e2 ) 1 √ 2  = ke1 h(e1 , e2 ) = e1 sin e21 +e22 +1−1  ∂e1 √2 2  , e1 +e2 +1

Jo

where k = 1 and

 1   √e2 +e2 +1 , 1√ 2   h(e1 , e2 ) = sin e21 +e22 +1−1  √2 2  , e1 +e2 +1

p  e21 + e22 + 1 − 1 > π, p  if e21 + e22 + 1 − 1 ≤ π, if

p  e21 + e22 + 1 − 1 > π, p  if e21 + e22 + 1 − 1 ≤ π,

if

p  = h(e1 , e2 ) ≤ 1, if e21 + e22 + 1 − 1 > π, and 0 ≤ p  h(e1 , e2 ) ≤ 1, if e21 + e22 + 1 − 1 ≤ π. clearly 0 ≤ √

1 e21 +e22 +1

13

sin

 √ e21 +e22 +1−1

√

e21 +e22 +1

=

Journal Pre-proof

• P7 The function (33) have continuous second partial derivatives  2  √ 1 2 − 2 e22 3/2 +e2 +1 (e1+e2 +1) √ ∂ 2 f (e1 , e2 )  e21√ √   = sin e2 +e2 +1−1 e22 cos e21 +e22 +1−1 e22 sin e21 +e22 +1−1  ∂e22  √ 12 22 − + 3/2 e21 +e22 +1 e1 +e2 +1 (e21 +e22 +1)

1

2

√  e1 e2 sin e21 +e22 +1−1 e21 +e22 +13/2

is continuous as well.

• P8 This property is fulfilled by noting that  e2   √e2 +e2 +1 = 0, 1 √ 2  lim e2 sin e21 +e22 +1−1 e2 →0  √2 2  = 0, e1 +e2 +1

p  e21 + e22 + 1 − 1 > π, p  if e21 + e22 + 1 − 1 ≤ π,

if

pro of

which is continuous, and  −e1 e2  2 ∂ f (e1 , e2 )  (e21 +e22 +1 )3/2 = e1 e2 cos√e2 +e2 +1−1 1 2  ∂e1 e2  − e2 +e2 +1

 p e21 + e22 + 1 − 1 > π, p  if e21 + e22 + 1 − 1 ≤ π,

if

p  e21 + e22 + 1 − 1 > π, p  if e21 + e22 + 1 − 1 ≤ π. if

re-

Hence, the control law derived from the family (9) is then expressed by     Vr cos(e3 ) + k1 tanh(e1 ) V = , 3) W + k2 tanh(e3 ) Wr + λVr η sin(e e3

lP

which satisfies the Proposition 1, where   √ e2 2 , ∂f (e1 , e2 )  e21 +e  2 +1  η= = e2 sin √e21 +e22 +1−1  ∂e2 √2 2  , e1 +e2 +1

(34)

p  e21 + e22 + 1 − 1 > π, p  if e21 + e22 + 1 − 1 ≤ π. if

It is possible to verify that the controller (34) has the upper bound for V (t) and W (t) given in (10). Design 5

urn a

4.2.3

A third original saturated controller is proposed by using the following positive definite function f (e1 , e2 ) =

−1 + 1. e21 + e22 + 1

(35)

The function (35) satisfies the properties P1-P8 as will be explained below.

and

Jo

• P1 The partial derivatives

2e1 ∂f (e1 , e2 ) = 2 2 ∂e1 (e1 + e22 + 1) ∂f (e1 , e2 ) 2e2 = 2 2 ∂e2 (e1 + e22 + 1)

are continuous functions. In addition, kf = 0.6495. • P2 The function (35) satisfies the first condition as f (0, 0) =

−1 + 1 = 0. 0+0+1 14

(36)

Journal Pre-proof

• P3 This property establishes that the function (35) is positive as −1 e21 +e22 +1

f (e1 , e2 ) =

∀

+ 1 > 0,

e1 6= 0, e2 6= 0,

and is radially unbounded. • P4 We prove this property by observing that

pro of

∂f (e1 , e2 ) 2e1 e2 e1 = 2, 2 ∂e2 (e1 + e22 + 1) ∂f (e1 , e2 ) 2e2 e1 e2 = 2. 2 ∂e1 (e1 + e22 + 1) • P5 This property is satisfied as follows

∂f (e1 , e2 ) 2e1 tanh(e1 ) tanh(e1 ) = 2 > 0, ∀ e1 6= 0 and e2 ∈ IR, ∂e1 (e21 + e22 + 1) 2e1 tanh(e1 ) ∂f (e1 , e2 ) tanh(e1 ) = 2 = 0, ∀ e1 = 0 and e2 ∈ IR. ∂e1 (e21 + e22 + 1)

re-

and

• P6 This condition is fulfilled as

∂f (e1 , e2 ) 2e1 = 2 = ke1 h(e1 , e2 ), ∂e1 (e21 + e22 + 1)

• P7 For this condition,

1

2

(e21 +e22 +1)

. It is clear that 0 ≤ h(e1 , e2 ) ≤ 1 ∀ e1 , e2 ∈ IR.

lP

where k = 2 and h(e1 , e2 ) =

∂ 2 f (e1 , e2 ) −8e22 2 = + 3 2 2 2 2 2 ∂e2 (e1 + e2 + 1) (e1 + e22 + 1)

and

urn a

−8e2 e1 ∂ 2 f (e1 , e2 ) = 3 ∂e1 e2 (e21 + e22 + 1)

are continuous functions.

• P8 This property is proven as

2e2

lim

e2 →0 (e2 1

2

+ e22 + 1)

= 0.

Jo

Consequently, the saturated controller (9) with the function introduced in (35) is expressed as #   " Vr cos(e3 ) + k1 tanh(e1 ) V sin(e3 ) = W + λV 2e2 + k2 tanh(e3 ) . 2 r r 2 2 W (e +e +1) e3 1

(37)

2

Similarly to all the previous designs, the upper bound of V (t) and W (t) generated by the controller (37) are given in (10).

15

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Reference trajectory Position Orientation

yr(t)


r(t)

~ x(t) ~ y(t) ~
(t)

Error transformation (7)

e1(t) e2(t) e3(t)


(t) Saturated/Unsaturated controller (9)/(38)

V(t) W(t)

UWMR Pioneer P3-DX

y(t)

x(t)

Vr(t) Wr(t)

pro of

Velocities

xr(t)

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re-

Figure 2: Control diagram. Green lines indicate the measured signals.

Figure 3: Experimental benchmark for saturated/unsaturated controllers tests.

Real-time experimental results

urn a

5

In this section, the experimental benchmark to evaluate the performance of each one of the controllers (28), (30), (32), (34) and (37) given in Section 4 is described. More specifically, the five controllers discussed in Section 4 and the unsaturated controller expressed in [49] are tested by specifying two different desired trajectories. Besides, the accuracy evaluation of the tested controllers is computed using some performance index.

Jo

All the controllers are implemented in the mobile robot Pioneer P3-DX manufactured by MobileRobots Inc. The block control diagram given in Figure 2 provides a description on how our implementations were carried out. The UWMR Pioneer P3-DX is controlled by linear and angular velocity inputs and has an internal PC (operating under Ubuntu GNU/Linux 12.04 LTS operative system). The ROS Indigo Igloo (Robot Operating System) is used to receive the control velocities and convert them into voltages requested by each motor located at each one of the two wheels. The control voltages for both electrical motors are unavailable for measurement. The robot motors are driven by an internal controller, which converts linear and angular velocities into voltages. The data acquisition of the vehicle pose is performed at a frequency of 100 [Hz] by using the Vision System OptiTrack. The control action is computed and commanded to the UWMR through a remote PC running Matlab-Simulink (R2015a) which includes a toolbox to enable the interface with ROS. This toolbox allows sending the control action to the vehicle computer. Figure 3 shows the implemented platform configuration.

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5.1

Description of the unsaturated controller [49]

In the document [49] a trajectory tracking controller for UWMRs described by (1) was introduced. Specifically, the controller in [49] is expressed by 

   Vr cos(e3 ) + k1 e1 V = . 3) W Wr + λVr e2 sin(e + k2 e3 e3

(38)

5.2

pro of

As can be seen, unlike the family of saturated controllers (9), the unsaturated controller (38) does not limit the error e1 in the linear velocity calculus. Furthermore, in the angular velocity calculus the error 1 ,e2 ) is e2 , which is unbounded as a result of the e3 is not limited as well. Finally, the equivalent term ∂(e∂e 2 nonexistence value for kf .

Results for circle path

re-

The real-time performance of the controller (9) is evaluated with each one of five functions f (e1 , e2 ) analyzed in Section 4 and with respect to the unsaturated controller given in (38), two sets of trajectory tracking experiments were carried out. The first set of experiments was intended to track a circular path with a diameter of 1.0 [m] for 80 [s], which is generated using the kinematic reference model given in (2), and the reference velocities given in Table 1. In addition, the maximum levels of control input, the parameters kf associated to property P1 for each design in Section 4 and the gains defined in order to satisfy the bounds given in the expression (10) for each design of the family of saturated controllers are described in Table 1 as well.

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urn a

Unsaturated controller Design 1 Design 2 Design 3 Design 4 Design 5

Table 1: Controller parameters. Parameter Value for the circular path Vmax ± 0.75 [m/s] Wmax ± 2.0 [rad/s] Vrmax 0.5 [m/s] Wrmax 1.0 [rad/s] k1 , k2 0.25, 0.7 λ 1.0 kf , λ 0.5, 1.2 kf , λ 1.0, 0.6 kf , λ, α, β 2.0, 0.3, 1.0, 4.0 kf , λ 1.0, 0.6 kf , λ 0.6495, 0.9238

Value for the lemniscate path ±0.75 [m/s] ±2.0 [rad/s] 0.6322 [m/s] ± 1.8320 [rad/s] 0.1178, 0.15 1.0 0.5, 0.0569 1.0, 0.0285 2.0, 0.0142, 1.0, 4.0 1.0, 0.0285 0.6495, 0.0438

The UWMR Pioneer P3-DX initial conditions for the first set of experiments were: x(0) = 2.00 [m], y(0) = 2.00 [m],

Jo

θ(0)

= −2π [rad].

Observe that the same values for k1 and k2 are set in all the controllers, which is useful to appreciate the performance of system when different functions f (e1 , e2 ) are used from a controller with respect to others. The path drawn by the UWMR in the horizontal plane using the unsaturated controller and the five controllers is illustrated in Figure 4, where a circular path was defined as a reference. The position and orientation errors are shown in Figure 5. The control inputs depicted in Figure 6 represents the velocities applied to the UWMR. Note in Figure 6 that the unsaturated controller exceeds the admissible limits of maximum velocity.

17

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2.5

2

re-

1

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y [m]

1.5

0.5

-0.5 -1

urn a

0

-0.5

Unsaturated controller

0

Design 1

0.5

1

1.5

Design 3

Design 4

2

2.5

x [m]

Design 2

Design 5

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Figure 4: Experimental results: path depicted by the robot.

18

Reference

pro of

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1

x ˜ [m]

0 -1 -2 -3 0

10

20

0

10

20

30

y˜ [m]

0

-2

6

2 0 0

30

urn a

4

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-1

θ˜ [rad]

50

60

70

80

re-

1

40 Time [s]

10

20

Unsaturated controller

30

40 Time [s]

50

60

70

80

40 Time [m]

50

60

70

80

Design 1

Design 2

Design 3

Design 4

Design 5

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Figure 5: Experimental results: position and orientation errors when the desired pose trajectory draws a circle in the the horizontal plane.

19

1

pro of

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Exceeded limit

Vmax

V [m/s]

0.5

0

−Vmax

-1 0

10

20

30

40 Time [s]

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6 5 4 3

50

60

70

80

50

60

70

80

Exceeded limit

2 1 0 Reference

urn a

W [rad/s]

re-

-0.5

10

20

Unsaturated controller

30

Wmax

40 Time [s] Design 1

Design 2

Design 3

Design 4

Design 5

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Figure 6: Experimental results: velocity control inputs when the desired pose trajectory draws a circle in the the horizontal plane. Notice that for the experimental implementation of the controller (38) the generated control inputs exceed the admissible limits.

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pro of

Table 2: Circular path: RMS values for the position and orientation errors and control inputs for the experiments when the desired pose trajectory draws a circle in the the horizontal plane. 60 [s] ≤ t ≤ 80 [s] RMS x ˜ [m] y˜ [m] θ˜ [rad] V [m/s] W [rad/s] Design 1 0.0073 0.0067 0.0292 0.4986 0.9860 Design 2 0.0106 0.0101 0.0409 0.4971 0.9770 Design 3 0.0072 0.0063 0.0207 0.5001 0.9884 Design 4 0.0135 0.0105 0.0244 0.4965 0.9864 Design 5 0.0081 0.0069 0.0375 0.4994 0.9837 The performance of the five designed controllers is evaluated using the root mean square (RMS) performance index for the position and orientation errors as well as for the control inputs in the time lapse 60 [s] ≤ t ≤ 80 [s], when the transients are considered settled. The results are shown in Table 2. The best results are obtained with the Design 3 in equation (32) because it reacts to low level errors in e1 (t) and e2 (t) through the gain β.

5.3

Results for a lemniscate path

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re-

The second set of experiments was implemented for 200 [s] by using as reference pose which is defined by the next equations:   2πt xr (t) = 0.9 sin [m], (39) 20   4πt [m], (40) yr (t) = 0.9 sin 20 θr (t) = atan2 (y˙ r , x˙ r ) [rad], (41) where the reference velocities were settled by:

Vr =

p x˙ 2r + y˙ r2 ,

urn a

x˙ r y¨r − x¨r y˙ r Wr = θ˙r = . x˙ 2r + y˙ r2

(42) (43)

Notice that the reference positions (39) and (40) draws a lemniscate path in the horizontal plane. The initial conditions for the UWMR were given as x(0) = y(0) = θ(0) =

−1.00 [m], −1.80 [m], −π [rad].

Jo

Besides, the bounds of the linear and angular reference velocities, the maximum limit values of control input, the parameters kf described in property P1 and the gains selected in order to satisfy the equation (10) for each design of the family of saturated controllers are shown in Table 1. Notice from Table 1 that the gains k1 and k2 have the same numerical values for all experiments. The performance obtained is depicted in Figure 7, where the vehicle path is drawn for the five designs belonging to the family of saturated controllers and the unsaturated controller. Figure 8 and 9 show the results associated with the position and orientation errors and the control inputs, respectively. As in the case of the circular path, note that in Figure 9, the unsaturated controller exceeds the admissible limits of maximum velocity again. ˜ and the control inputs V (t) and W (t) Table 3 displays the RMS values of the error signals x ˜(t), y˜(t), θ(t) in the time lapse 160 [s] ≤ t ≤ 200 [s], when the transients are considered settled. The best results are 21

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1.5

1

0.5

re-

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Figure 8: Experimental results: position and orientation errors when the desired pose trajectory draws a lemniscate in the the horizontal plane.

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Figure 9: Experimental results: velocity control inputs when the desired pose trajectory draws a lemniscate in the the horizontal plane. Notice that for the experimental implementation of the controller (38) the generated control inputs exceed the admissible limits.

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Table 3: Lemniscate path: RMS values for the position and orientation errors and control inputs for the experiments when the desired pose trajectory draws a lemniscate in the the horizontal plane. 160 [s] ≤ t ≤ 200 [s] RMS x ˜ [m] y˜ [m] θ˜ [rad] V [m/s] W [rad/s] Design 1 0.1007 0.1475 0.1518 0.4546 0.7742 Design 2 0.1128 0.1363 0.1793 0.4556 0.7772 Design 3 0.0890 0.1489 0.1489 0.4534 0.7728 Design 4 0.1288 0.1575 0.1545 0.4524 0.7720 Design 5 0.0749 0.1299 0.1467 0.4545 0.7740 obtained with Design 5 in equation (37). Notice that this scheme behaves like a high gain controller when the errors e1 , e2 and e3 are small, which increases the rejection to disturbances such as delays and friction.

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Conclusions

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In this paper, a family of saturated controllers for UWMR vehicles was proposed. The controllers satisfying the properties of the family of algorithms were able to guarantee asymptotic trajectory tracking while actuators saturation were avoided, that is, the control input requested to actuators did not exceed the maximum linear and angular velocity permitted for the UWMR. Five designs that satisfy the properties imposed in the close-loop system stability analysis were presented. Two of these schemes were taken from the literature while the other three controllers were original designs.

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Real-time experiments were carried out to validate the proposed theory and to show the functionality of the family of saturated controllers. Performance comparisons were achieved with respect to the unsaturated controller [49].

References

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The obtained results for the family of controllers showed a good trajectory tracking performance while control inputs were bounded for all time. The controller in [49] failed in keeping the control inputs V and W within the admissible limits. Finally, some of the original schemes provided better tracking results than the two controllers taken from the literature.

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[46] Y. Su and C. Zheng, “Global asymptotic stabilization and tracking of wheeled mobile robots with actuator saturation”. In proceedings of 2010 IEEE International Conference on Robotics and Biomimetics, Tianjin, China, December, 2010, pp. 345-350. [47] H. K. Khalil, Nonlinear Systems, 3rd ed. (Prentice–Hall, Upper Saddle River, 2002). [48] J. J. E. Slotine and W. Li, Applied nonlinear control, 1st ed. (Prentice–Hall, Englewood-Cliffs, 1991).

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[49] W. Dixon, D. Dawson, E. Zergeroglu and A.Behal, Nonlinear Control of Wheeled Mobile Robots, 1st ed. (Springer-Verlag, London, 2001).

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Input saturation in the control of mobile robots is a problem found in practice.

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Only a few solutions for this problem have been studied.

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A novel family of saturated controllers, which includes already reported approaches as well as original designs.

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A rigorous stability analysis that guarantees global asymptotic stability is given.

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An experimental validation of five control schemes belonging to the given family of controllers is provided.

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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: