applied sciences

Article Smooth Fractional Order Sliding Mode Controller for Spherical Robots with Input Saturation

Ting Zhou, Yu-gong Xu and Bin Wu * School of Mechanical, Electronic and , Beijing Jiaotong University, Beijing 100044, China; [email protected] (T.Z.); [email protected] (Y.-g.X.) * Correspondence: [email protected]; Tel.: +86-138-1171-6598

 Received: 15 February 2020; Accepted: 12 March 2020; Published: 20 March 2020 

Abstract: This study considers the control of spherical robot linear motion under input saturation. A fractional sliding mode controller that combines fractional order calculus and the hierarchical sliding mode control method is proposed for the spherical robot. Employing this controller, an auxiliary system in which a filter was used to gain smooth control performance was designed to overcome the input saturation. Based on the theorem, the closed-loop system was globally stable and the desired state was achieved using the fractional sliding mode controller. The advantages of the proposed controller are illustrated by comparing the simulation results from the fractional order sliding mode controllers and the integer order controller.

Keywords: fractional sliding mode control; spherical robot; linear motion; input saturation

1. Introduction Spherical robots are a new type of robot with a ball-shaped exterior shell. Their advantages, such as the lower energy consumption and enhanced locomotion compared to traditional wheeled or legged mobile robots have motivated many researchers to develop them with different structures in recent decades [1–6]. The inner parts are hermetically sealed inside the spherical shell, resulting in increased reliability in hostile environments, making them suitable for security and exploratory tasks [7,8]. Among the current structures of spherical robots, the pendulum-driven type is popular in industry and academia owing to its easy implementation and strong driving torque. Linear motion is the most important motion mode for the execution of a task. However, a pendulum-driven spherical robot in a state of linear movement has underactuated and strong nonlinearity, which is similar to a class of underactuated systems such as the ball-beam system and the pendulum system. In order to overcome these problems in spherical robots or similar underacted systems, many control methods have been implemented such as adaptive control, linearization, back-stepping, sliding mode control, trajectory planning, and the intelligent control method [9–19]. In all these methods, the sliding mode control approach is suitable for motion control systems owing to its unparalleled merits, such as fast response, robustness to parametric uncertainties, and resistance to disturbances [20]. To cope with the control problem of this underactuated system, the hierarchical sliding mode control method was developed [21]. This method designs the first layer sliding surface for each subsystem and then achieves the second surface by a linear or other combination of all first-layer sliding surfaces. This method demonstrates excellent convenience in the design of sliding mode controllers for underactuated systems, and it ensures the stability of the entire system, as well as of each subsystem [22,23]. Yue et al. proposed a hierarchical sliding mode controller for the velocity control of a pendulum-driven spherical robot in [24,25]. However, the traditional integral sliding surface requires

Appl. Sci. 2020, 10, 2117; doi:10.3390/app10062117 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 2117 2 of 17 a long time to track the desired velocity and has a significant overshoot. It is worth noting that none of this research on spherical robot control systems considered input saturation. The driving capacity is relatively limited in actual spherical robots, and input saturation will be an inevitable phenomenon. Without considering the input saturation in the controller design, the control performance of a spherical robot may be weak or instability may occur in the whole system. The above research also neglects the movement of the shell in the opposite direction at the start-up stage due to the reaction force acting on the robot. As can be seen from aforementioned works, this phenomenon is caused by the conservation of momentum. Although this situation may be improved when values of control targets are higher, the problem cannot be wholly solved. In recent years, fractional order calculus has been widely used in science and engineering. Many fractional-order controllers have been developed to enhance control performance [26–30]. For example, Podlubny proposed a fractional PIλDµ controller in [26]. Yin et al. presented a fractional sliding mode controller and showed its benefits over integral controllers [27]. H. Delavari et al. developed a fuzzy fractional sliding surface for a and optimized the controller parameters using the genetic algorithm, achieving better performance than with integral sliding mode controllers [28]. Ebrahimkhani S. et al. put forward a fractional integral sliding mode controller for a wind power generation system to stabilize and enhance the robustness [29]. Zhang et al. studied a fractional sliding mode controller for a permanent magnet synchronous motor [30], while Rahmani M. et al. combined the neural network and fractional order sliding mode control and presented a peristaltic motion controller for a bionic peristaltic robot [31]. Kumar G. et al. studied a fractional sliding control with a PIλ-Dµ sliding surface for a two-tank hybrid system and achieved better performance than that of a controller with a PDµ sliding surface [32]. In [33], a fractional PIλDλ sliding controller is proposed; the PSO algorithm was used to tune the controller parameters. Furthermore, the benefits of a fractional PIλDλ sliding surface are shown in comparison the PDµ sliding surface in [34]. Zhong et al. combined fractional calculus and second-order sliding mode control to give a fractional PDDµ sliding controller for a class of nonlinear systems [35]. Aghababa M.P. et al. proposed a fractional order sliding mode controller for the finite-time stabilization of a nonautonomous fractional order underactuated system with model uncertainties and external noise [36]. Narayan et al. focused on the finite-time convergence of a fractional order nonholonomic chained system and proposed a fractional sliding order controller [37]. Based on the fractional model of a flexible underactuated manipulator, Mujumdar et al. presented a fractional PIλ sliding mode controller [38]. To summarize, most literature has focused on the application of fractional-order sliding mode controllers in fractional-order underactuated systems, and only very few papers have considered applying fractional sliding mode control to integer-order underactuated systems [39]. Existing papers show that the fractional sliding surfaces have better control effects than integer ones. The fractional sliding mode control has faster response and convergence speed in the initial stage due to the fractional operator. This means that the fractional sliding mode controller will yield a high output, which can more easily lead to input saturation. Moreover, in spherical robots, the output of the actuator remains constant, and when the robot system is under input saturation, it can yield a wave motion in the dynamic response process. Though many methods regarding input saturation have been developed [40–43], the wave motion in the dynamic response process cannot be solved in a satisfying way, since the method cannot completely avoid input saturation. For this reason, it is not enough to improve the spherical robot speed control under input saturation by simply designing a sliding surface with faster response and convergence speed. The opposite direction at the start-up stage and wave motion in the dynamic response process also need to be taken into consideration. Focusing on this topic, the novelty of this paper can be summarized as follows:

(1) A fractional PIλDµ sliding control method based hierarchical sliding control and fractional calculus is proposed to improve the control performance; (2) A novel fractional PIλDµ sliding controller with an auxiliary system is proposed to deal with input saturation; Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 17 Appl. Sci. 2020,, 10,, x 2117 FOR PEER REVIEW 3 of 17 Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 17 (3) Smooth dynamic response is achieved by adding a filter, which can decrease the output of the (3)(3) controllerSmooth dynamic in the initial response stage is and achieved make fullby adding use of the a filter, filter, fractional which sliding can decrease surface. the outputoutput ofof thethe controller in the initial stage and make full use of the fractional sliding surface. In this paper, R+ and Z are the set of positive real numbersnumbers and integerintegers,s, respectively,respectively, Rn and Rn×n are the n-dimensional++ Euclidean space, and the matrices of order n. . is the norm. n Forn a realn n×nn are theInIn thisthisn-dimensional paper,paper,R R and Euclidean Z are thethe space, setset ofof positivepositiveand the real realmatricesnumbers numbers of andorder and integers, integers, n. . γ respectively, isrespectively, the norm. R RForandand a R realR× .  areare thethen n-dimensional-dimensional Euclidean Euclidean space, space, and and the the matrices matrices of order of ordern. . γnis. the∞γ norm.xtis1 the Fornorm. a real For number a real number x, x=∈≤min{ n  Z | x  n} is the ceiling ,R and kΓ=k(x ) txt−−1 e d t is the Gamma number x, x minnZxn |  is the ceiling function, andx 1 ()tx ∞te d t is the GammaT x, x = minn Z x n is the ceiling function, and Γ(x) = ∞ t e dt is the0 −− Gamma function. A is =∈≤{ } 0 − Γ=− xt1 numberd e x, T {x∈ min| ≤ nZxn} | is the ceiling function, and ()x te d t is the Gamma function.the transpose AT is of the matrix transposeA, eig( ofA matrix) is the A matrix, eig(A eigenvalues.) is the matrix eigenvalues. 0 function. AT is the transpose of matrix A, eig(A) is the matrix eigenvalues. 2. Motion Motion EquationE Equationquation and ControlC ontrol SystemS ystem 2. Motion Equation and Control System When a spherical robot is moving in linear motionmoti on mode, the dynamical momodel del can be simplified simplified to a a twotwo-dimensional two-dimensionalWhen-dimensional a spherical ring-penduluring ring-pendulumrobot-pendulu is movingm system. in system. linear Figure moti Figureon 1 1 showsmode, shows athe diagram a diagramdynamical of ofthe mo the relateddel related can simplified be simplified simplified 2D model.2Dto a model. two-dimensional ring-pendulum system. Figure 1 shows a diagram of the related simplified 2D model.

Figure 1. Diagram of Spherical RobotRo bot in Linear Motion.Motion . Figure 1. Diagram of Spherical Robot in Linear Motion. wherewhereWhere r is radiusr is radius of sphere, of sphere, M isM weightis weight of sshell, ofhell shell,, m mis isweight weight of of inner inner suspension, suspension, l lisis the the ddistance distanceistance fromwhere weight r is radius centercenter of ofof sphere, massmass toto M spheresphere is weight center, center, of ϕ shell,φis is rolling rolling m is angleweight angle of ofof the theinner spherical spherical suspension, robot, robot,θ l isθis swinging-upisthe swinging-swinging distance- upanglefrom angle weight of the of innerthe center inner suspension of suspensionmass to relative sphere relative to center, the to shell the φ ofis shellshel rolling thel sphericalof theangle spherical robot,of the gspherical robot, is the accelerationg isrobot, the acceleration θ is of swinging- gravity, ofτ gravity, τ is torque of motor, and τ is rolling friction between the shell and ground. By ignoring the gravity,isup torque angle τ of ofis motor, torquethe inner and of motor,suspensionτ f is rolling and relative frictionf is rolling betweento the friction shel thel between of shell the and spherical the ground. shell robot, and By ground. ignoring g is the By theacceleration ignoring damping the ofof the actuator and the relative slidingτ between the spherical robot and the ground, the dynamic equation dampinggravity, τ ofis torquethe actuator of motor, and and the relativef is rolling slidingslidin frictiong between between the thespherical shell and robot ground. and the By ground,ignoring thethe dynamicofdamping the linear equation of motionthe actuator of can the belinear and written themotion relative as [ 44can]: slidinbe writteng between as [[44]:44 ]:the spherical robot and the ground, the dynamic equation of the linear motion can be written as [44]: 2 ϕϕ τ τ (5MMmrmrl+ 3 m ) r / 3 mrl cosθ   00cos−mrlθθ cos     f  2 ++=   (1) + 2 θ ϕϕ  − θθ     τ τ (1) (5Mmrmrlmrl 3cos )θ / 3 ml cos θθθ 000cosmrl   mgl sinf  τ  ++= 00    mgl sin   2     τ (1) mrlcosθ ml θθθ 00 mgl sin   To make the controller method more convenconvenienient,t, the state variables are redefined as ToTo makemake thethe controllercontroller methodmethod moremore convenient,convenient,the thestate statevariables variables are are redefined redefined as as x==[][] x x x x TTϕϕ  θθ  xxxxx[][]11234 2 3 4  (2) ==TTϕϕθθ xxxxx[][]1234  (2) Therefore, EquationEquation (1) trtransformsansforms into the following statestate–space–space equation: Therefore, Equation (1) transformsxx= into the following state–space equation: Therefore, Equation (1) transformsxx12 into the following state–space equation:  xx = xxfxbx212=+ f 1(() x )  b 1 (()sat() x )sat(ττ ) + ggx 1 (() x ) f  .  21 1 1 f  x1 =xfxbx x=+2 () ()sat()ττ + gx () (3)  . xx3421= 1 1 f   34  x2 = f1=(x) + b1(x)sat(τ) + g1(x)τ f (3) . xxx34=+ f( x )  b ( x )sat(ττ ) + g ( x ) (3)  xfxbx442 2() 2 ()sat() gx 2 ()f  x3 = x4  . xfxbx =+() ()sat()ττ + gx ()  x = 42f (x) + b ( 2x)sat(τ) + g 2(x)τf The details of the nonlinear functions4 2 in EqEquation2uation (3) are 2as followsf The details of the nonlinear functions in Equation (3) are as follows The details of the nonlinear functions in Equation (3) are as follows

 2 bcx4 sin x3+bd sin x3 cos x3  f1(x) = 2 2  ac b cos x3  c b cos x−3 b (x) = − (4)  1 ac b2 cos2 x  − 3  g (x) = c 1 ac b2−cos2 x − 3 Appl. Sci. 2020, 10, 2117 4 of 17

2  ad sin x3+bx4 sin x3 cos x3  f2(x) = − 2 2  ac b cos x3  a b cos x−3 b2(x) = − 2 2 (5)  ac b cos x3  −  g (x) = b cos x3 2 ac b2 cos2 x − 3 The input sat(τ) is defined as   τmax, τ > τmax  sat(τ) =  τ, τ τ τ (6)  min max  ≥ ≥ τmin, τ < τmin where a = (5M + 3m) r2/3, b = mrl, l = ml, and d = mgl. The overall system can be regarded as the shell subsystem and pendulum subsystem. Because of the existence of input saturation, there is a difference value, ∆τ, between the design input τ and the actual control input sat(τ): ∆τ = sat(τ) τ (7) − The objective is to design a fractional order hierarchical sliding mode controller to control the state x2 to track the desired state. With the help of this controller, better control performance and increased robustness under input saturation are achieved.

3. Preliminaries This section presents some basic definitions and results of fractional calculus, which will be used later. Among three common ways of defining fractional calculus, i.e., Grunwald-Letnikov, Riemann-Liouville, and Caputo [45], this paper adopts the Riemann-Liouville definition. The Riemann-Liouville fractional integral of a f (t) is defined as:

Z t p 1 f (σ) D− f (t) = dσ, (t > t0, p> 0 ) (8) t0 t ( ) 1 p Γ p t0 (t σ) − − Also, the Riemann-Liouville fractional derivative of the continuous function f (t) is defined as:

q Z t p 1 d f (σ) D f (t) = dσ, q 1 < p < q (9) t0 t ( ) q p q+1 Γ q p dt t0 (t τ) − − − −   where p R+ is the derivative order or integral order, q = p Z; and t is the initial value. ∈ ∈ 0 Property 1 ([45]): For Riemann-Liouville fractional calculus, if function f(t) is continuous and p > q > 0, then

p q p q D [ D− f (t)] = D − f (t), t > t (10) t0 t t0 t t0 t 0

Property 2 ([46]): If 0 < q < 1, the following equality holds for Riemann-Liouville fractional calculus

q q q q D [ D− f (t)] = D− [ D f (t)] = f (t) (11) t0 t t0 t t0 t t0 t

q Lemma 1 ([47]): The fractional integration operator D− with q > 0 is bounded

q D− f (x) K f (x) , 1 γ (12) k k ≤ k kγ ≤ ≤ ∞ where K is a positive constant. Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 17

Dqq[()][()]() D−− ft== D qq Dft ft tttt00 tttt 0 0 (11)

Lemma 1 ([47]): The fractional integration operator D−q with q > 0 is bounded

−q ≤≤≤∞γ Dfx() Kfx ()γ ,1 (12) where K is a positive constant.

LemmaAppl. Sci. 20202 ([48]):, 10, 2117 Consider a fractional linear time-invariant system with a different differential order5 of as 17 follows:

Lemma 2 ([48]): Consider a fractional linear time-invariant system with a different differential order as follows: Dqi xt()= A xt () (13)

qi ∈ n =∈< π κ (14) 21 arg(eig(A)) > π (14) where the parameter κ is the lowest common multiple of qi. 2κ where the parameter κ is the lowest common multiple of qi. Assumption 1. The state xi in Equation (3) is assumed to satisfy the following inequality Assumption 1. The state xi in Equation (3) is assumed to satisfy the following inequality

p pDx( )≤=δ , i 1,2,3,4 D (x ) i δ, i = 1, 2, 3, 4 (15) (15) k i k ≤ where δ is a positive constant. where δ is a positive constant.

4. Fractional Order Hierarchical Sliding Mode Controller 4. Fractional Order Hierarchical Sliding Mode Controller 4.1. Design of the Fractional Hierarchical Sliding Mode Controller 4.1. Design of the Fractional Hierarchical Sliding Mode Controller The hierarchical sliding mode control method is usually used to design the controller for The hierarchical sliding mode control method is usually used to design the controller for underactuated systems. Underactuated systems consist of serval subsystems. A first layer sliding underactuated systems. Underactuated systems consist of serval subsystems. A first layer sliding surface is given for each subsystem, and then a second layer sliding surface is proposed. From surface is given for each subsystem, and then a second layer sliding surface is proposed. From Equation (3), the whole system is divided into an inner suspension system and a spherical shell Equation (3), the whole system is divided into an inner suspension system and a spherical shell system. system. Without considering the input saturation, the desired state of the system can be assumed to Without considering the input saturation, the desired state of the system can be assumed to be: be: = T xxxxxddddd[]1234 (16)

Therefore, the tracking error variables are written as Therefore, the tracking error variables are written as =− =− exxexx111222dd, (17) e =exxexxx=−x , e, = =−x x 1 3334441 − 1d dd2 2 − 2d (17) e3 = x3 x3d, e4 = x4 x4d Without considering input saturation, −the shell and− pendulum subsystem are rewritten in EquationsWithout (18) consideringand (19) respectively. input saturation, the shell and pendulum subsystem are rewritten in Equations (18) and (19) respectively. = xx12  (18) ( . xfxbxgx =+() ()ττ + () x1=21x2 1 1 f . (18) x2 = f1(x) + b1(x)τ + g1(x)τ f = xx34 ( .  (19) x3 =xfxbxgx x=+4 () ()ττ + () .  42 2 2 f (19) x4 = f2(x) + b2(x)τ + g2(x)τ f Based on hierarchical sliding mode control method, the first-layer sliding surface for the shell subsystem is given as: λ 1 µ s1 = k1D − e2 + e2 + k2D e2 (20) The first-layer sliding surface for the inner suspension system is as follows:

λ 1 s2 = k3D − e4 + e4 (21) where λ, µ (0, 1), k , k , and k are positive constants. To accelerate the response velocity, a ∈ 1 2 3 fractional order operator is used in the two first-level sliding surfaces. The first-layer sliding surface in Equation (20) has a fractional PIλDµ structure, while the structure of Equation (21) is PIλ. According Appl. Sci. 2020, 10, 2117 6 of 17 the definition of fractional integral in Equation (9), the fractional integral operator has one more weight function compared to the integer order operator. The weight function has a larger value in the initial stage, but it decreases over time [45]. This means that fractional-order integrals have larger integration coefficients in the initial stage and smaller integration coefficients afterwards. If the fractional-order integral sliding mode surface is simply added to the integral-order integral sliding mode surface to form a fractional-order integral sliding mode surface, it will, on the one hand, have a faster response than the integer-order integral sliding mode surface in the initial stage, but on the other hand, may cause greater overshoot. For this reason, the first-layer sliding surface adds a fractional differential term to reduce the overshoot and further improve the response speed. The integer differential term is sensitive to errors, a feature which makes it possible to reduce the overshoot and accelerate the response speed. This also will reduce the robustness of the system. Following Property 2, the fractional differential term can be treated as a fractional integral of the integer order differential term. This means that the robustness of the control system can be improved by adjusting the value of fractional differential order µ. By differentiating the first layer sliding surfaces (20) and (21) with respect to time, and letting . . s1 = 0 and s2 = 0, then:

. k Dλe + k Dµ+1e + f + b τ + g τ x = 0 (22) 11 2 12 2 1 1 1 f − 2d . k Dλe + f + b τ + g τ x = 0 (23) 12 2 2 2 2 f − 4d Then, the equivalent control law of each subsystem is as follows:

1 λ µ+1 . τ = b − (k D e + λ D e + f + g τ x ) (24) 1 − 1 11 2 12 2 1 1 f − 2d 1 λ . τ = b − (k D e + x f g τ ) (25) 2 − 2 2 4 4d − 2 − 2 f The total control law must include some portions of the equivalent control law of each subsystem to guarantee that the first sliding simultaneously converges to zero. Furthermore, the total control law also needs to include some portions of switch control law. Then, the total control law τ is defined as

τ = τ1 + τ2 + τsw (26) where τsw represents the switch law. To achieve the switch law, the second layer sliding surface is designed as follows:

S = ηs1 + ξs2 (27) where η and ξ are positive constants. From variable structure theory, the exponential reaching rate is selected to obtain better dynamic performance. By calculating the differentiation of the second sliding surface (27), . λ µ+1 . S = η(k11D e2 + k12D e2 + f1 + b1τ + g1τ f x2d)+ . − ξ(k Dλe + f + b τ + g τ x ) (28) 12 2 2 2 2 f − 4d = αsign(S) βS − − where α and β are positive constants. Considering the differentiation of the second layer sliding surface (27), the switch control law is given as αsign(S) βS + ηb2τ1 + ξb1τ2 τsw = − − (29) ηb1 + ξb2 Appl. Sci. 2020, 10, 2117 7 of 17

Placing Equations (24), (25), and (29) into Equation (26), the total control law is given as follows:

αsign(S) βS + ηb1τ1 + ξb2τ2 τ = − − (30) ηb1 + ξb2

4.2. A Novel Fractional PIλDµ Sliding Mode Controller Equation (18) uses the fractional differential term in the sliding surface; this term leads to faster response and lower overshoot. However, it may lead to a high controller output when the desired object varies in impulse form due to the explosion of the derivation pertaining to the desired speed, x2d. The large output of the controller will cause the actuator to saturate rapidly. It also makes the shell subsystem tracking the desired speed become wave-like, with an opposite speed response at the start stage. Therefore, inspired by backstepping method m [49], this study applies a filter to resolve this problem. Below, the aforementioned results are extended to deal with the input saturation problems of the spherical robot system and to achieve smoother dynamic response. Let x2d pass the filter x2d. Then, the new desired state is redefined in Equation (31) . εx2d + x2d = x2d (31) where ε is a designed positive constant, and the initial value of the filter is x2d = 0. Then, the error e2 may be redefined as: e = x x (32) 2 2 − 2d The new equivalent control law of each subsystem is updated as:

. 1 λ µ+1 τˆ = b − (λ D e + λ D e + f + g τ x ) (33) 1 − 1 11 2 12 2 1 1 f − 2d 1 λ . τˆ = b − (λ D e + x f g τ ) (34) 2 − 2 2 4 4d − 2 − 2 f In order to handle the input saturation, an auxiliary system was designed. The total control law needs to include some terms of the auxiliary system. Hence, the new total control law τˆ can be updated as: αsign(S) βS + ηb1τˆ1 + ξb2τˆ2 Sˆ τˆ = − − − (35) ηb1 + ξb2  n o .  1 2 ˆ ˆ ˆ  [S(ηb1 + ξb2)∆τ + 2 ∆τ ] /S χS + ∆τ, S ρ Sˆ =  − − ≥ (36)  0, Sˆ < ρ where χ is a positive constant, and ρ is a positive constant whose value is small.

4.3. Stability Analysis of Each Surface

Theorem 1. For the dynamic system of a spherical robot which satisfies the constraint of Equation (14), and the control parameters α, β, and χ satisfy α > 1, β > 0, χ > 1/2, the first-level sliding surface described in Equations (20) and (21) are asymptotically stable, and the second sliding surface Equation (27) is uniformly bounded by the control laws defined in Equations (35) and (36).

Proof. Choose the Lyapunov function as

1 1 V = S2 + Sˆ2 (37) 1 2 2  Appl. Sci. 2020, 10, 2117 8 of 17

The derivative of Equation (37) as

. . . V1 = SS + SˆSˆ . λ µ+1 = S[η(k1D e2 + k2D e2 + f1 + b1(τ + ∆τ) + g1τ f x2d)+ . − (38) λ . ξ(k3D e4 + f2 + b2(τ + ∆τ) + g2τ f x4d)] + SˆSˆ − . = S[ ηb τ ξb τ + (ηb + ξb )τ + (ηb + ξb )∆τ] + SˆSˆ − 1 1 − 2 2 1 2 1 2 Equation (38) needs to take the following two cases into account.

Case 1. If Sˆ ρ, by putting Equation (36) into Equation (38), one can obtain ≥ . . . V1 = SS + SˆSˆ . λ µ+1 = S[η(k1D e2 + k2D e2 + f1 + b1(τˆ + ∆τ) + g1τ f x2d)+ . . − ξ(k Dλe + f + b (τˆ + ∆τ) + g τ x )] + SˆSˆ 3 4 2 2 2 f − 4d = S[ ηb τˆ ξb τˆ + (ηb + ξb )τˆ + (ηb + ξb )∆τ] − 1 1 − 2 2 1 2 1 2 − (39) S(ηb + ξb )∆τ 1 ∆τ2 k Sˆ2 + Sˆ∆τ 1 2 − 2 − 3 αS2 β S 1 ∆τ2 χ 1 Sˆ2 + 1 Sˆ2 + 1 ∆τ2 ≤ − − | | − 2 − 2 2 2 αS2 (χ 1 )Sˆ2 β S ≤ − − − 2 − | | u V ≤ − 1 1 n o where u = min α, χ 1 . 1 − 2

Case 2. If Sˆ < ρ, one can obtain

. . . V1 = SS + SˆSˆ = S[ ηb τˆ ξb τˆ + (ηb + ξb )τˆ + (ηb + ξb )∆τ + Sˆ] − 1 1 − 2 2 1 2 1 2 = αS2 β S + S(ηb + ξb )∆τ + SSˆ − − | | 1 2 (40) αS2 β S + 1 S2 + 1 [(ηb + ξb )∆τ]2 + 1 S2 + 1 Sˆ2 ≤ − − | | 2 2 1 2 2 2 (α 1)S2 1 Sˆ2 + 1 [(ηb + ξb )∆τ]2 ≤ − − − 2 2 1 2 u V + 1 [(ηb + ξb )∆τ]2 ≤ − 2 1 2 1 2 n o where u = min α 1, 1 . 2 − 2 Thus, one can conclude that the second sliding surface S is uniformly ultimately bounded. When η, ξ satisfy the condition η, ξ > 0, the convergence of the function is independent of the specific value of η, ξ. Considering the following two sliding surfaces:

S1 = η1s1 + ξs2 (41)

S2 = η2s1 + ξs2 (42) where η , η , ξ are positive constant and η , η . Here, one supposes that |S | |S | 1 2 1 2 1 ≥ 2 R t R t (S2 S2)d = [ 2s2 2s2 + s s ( )]d 0 1 2 σ 0 η1 1 η2 1 2ξ 1 2 η1 η2 σ − R t − − (43) = [ (η η )2s2 + 2(η η )s S ]dσ > 0 0 − 1 − 2 1 1 − 2 1 1 From Equation (43) Z t Z t 2 2 (η1 η2) s1d < 2(η1 η2)s1S1dσ (44) 0 − 0 − Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 17 when η, ξ satisfy the conditionηξ,0> , the convergence of the function is independent of the specific valueAppl. Sci. of 2020 η, ξ, .10 Considering, x FOR PEER REVIEW the following two sliding surfaces: 9 of 17 Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 17 ηξ> Sss=+ηξ when η, ξ satisfy the condition ,0, the convergence1112 of the function is independent of the specific(41) ηξ> valuewhen ofη, ξη ,satisfy ξ. Considering the condition the following,0, the two convergence sliding surfaces: of the function is independent of the specific =+ηξ value of η, ξ. Considering the following twoSss sliding2212 surfaces: (42) Sss=+ηξ 1112 (41) where η1, η2, ξ are positive constant and η1 ≠ Sssη2.=+ Here,ηξ one supposes that |S1| ≥ |S2| 1112 (41) =+ηξ ttSss2212 (42) ()SSd2−= 2σηη [ 22 s −+− 22 s 2()] ξηησ ss d 0012Sss 11211212=+ηξ 2212 (42) where η1, η2, ξ are positive constant and η1 t≠ η2. Here, one supposes that |S1| ≥ |S2| (43) =−− [ (ηη )22ssSd + 2( ηη − ) ] σ > 0 1 2 1 2 121 1211 1 2 where η , η , ξ are positive constanttt and η 0≠ η . Here, one supposes that |S | ≥ |S | ()SSd2−= 2σηη [ 22 s −+− 22 s 2()] ξηησ ss d 00tt12 11211212 From Equation (43) ()SSd2−= 2σηη [ 22 s −+− 22 s 2()] ξηησ ss d (43) 0012t 11211212 =−− [ (ηη )22ssSd + 2( ηη − ) ] σ > 0 ttt 121 1211 (43) η −<−ηηησ0 22 22 () 121=−− [s (ηηdsSd ) 2()ssSd 1211 + 2( ηη − ) ] σ > 0 (44) 000 121 1211 From Equation (43) Appl. Sci.Then2020 , 10, 2117 9 of 17 From Equation (43) tt ()η −<−ηηησ22s dsSd 2() tt121 1211 (44) tt00222sS t2 sS 2 σσ()η<<−<−ηηησs11dsSd 2()11∞∞ σ s1 dd121 1211 d (45)(44) Then 0000ηη−−  0 ηη Then Z t Z t ()12Z t () 12 2 2s1S1 2 s1 S1 s dσ < dσ < k k∞k k∞ dσ (45) Then 1 ( ) ( ) The actuator directly drives0 thett inner0 ηsuspension,1 2sSη2 so0 tone2 sS canη1 seeη2 that x3 and x4 are bounded. 2 σσ<<− 11 11∞∞− σ α −1 s1 dd d . (45) λ <∞ 00ttηη2sS−−  0 t2 ηηsS11∞∞ By LemmaThe actuator 1, 24De directly drives can be the 2obtained. innerσσ<< suspension,() 12Therefore,11 so the one ()fact 12 can that see sliding that σx andsurfacex are s2 bounded. is bounded By s1 dd d 3 4 (45) α 1 00()ηη−−  0 () ηη <∞ Lemmacan be confirmed. 1, λ2D − e Then,4 < accordingcan be obtained. to Assumption Therefore,12 1, theonefact can thatobtain 12 sliding s2 surface. Baseds2 ison bounded the conclusion can be The actuator directly∞ drives the inner suspension, so one can. see that x3 and x4 are bounded. confirmed. <∞ Then, accordingα − to Assumption<∞ 1, one can obtain s2 < . Based on the conclusion that that. TheSS, actuator, oneλ directly obtains1 <∞ drivesss11, the. inner suspension, so one can see that x3 and x4 are bounded. By Lemma 1, 24De. can be obtained. Therefore, the fact that∞ sliding surface s2 is bounded S, S 0,> 0,k2>0, k2 >k30,>0, k 3 >,(0,1)0, λ,=µ ,( and0, 1) the, and constraint the constraint= Equation Equation (14) is (14) satisfied, is satisfied, then the then error the of error the with the Barbalat lemma [50], lims1 0∈; similarly, lims2 0 is achieved. oftwo the sliding two sliding surfaces surfaces in Equations in Equations (20)t→∞ and (20)= (21) and decays (21) decays asymptoticallyt→∞ asymptotically= toward toward zero. zero. with the Barbalat lemma [50], lims1 0 ; similarly, lims2 0 is achieved. t→∞ t→∞ λμ∈ Proof.TheoremWhen 2. If thek1>0, first-layer k2>0, k3>0, sliding ,(0,1) surfaces, and remain the constraint on the respective Equation (14) manifold, is satisfied,s = then0 and thes error== 0 of; thethe Proof: When the first-layer sliding surfaces remain on the respective manifold,1 s12 0 and Theorem 2. If k1>0, k2>0, k3>0, λμ,(0,1)∈ , and the constraint Equation (14) is satisfied, then the error of the firsttwo= sliding sliding surfaces mode surfacein Equationss1 is rewritten(20) and (21) into decays the state asymptotically equation form toward as follows:zero. s2 0 ; the first sliding mode surface s1 is rewritten into the state equation form as follows: two sliding surfaces in Equations (20) and (21) decays asymptotically toward zero. 11−−λλ 1 − λ = Proof: When the first-layerD sliding(())De surfaces remain01 on the  De respective () manifold, s1 0 and 22= μ   s = 0 (47) s = Proof0 ; the: firstWhen sliding the modefirst-layer surface sliding s issurfaces rewritten−− remain into the on state the equationrespective form manifold, as follows: 1 and 2 De()221 kk121 k 2 e =   s2 0 ; the first sliding mode surface s1 is rewritten into the state equation form as follows: ==11−−λλTT1−λ 1 − λ By redefining the vectorYyyD [][()](())De22 Dee01, Equation  De ()(47) is rewritten as: −−λλ12= 2 2 − λ  11μ   1 (47) D (())De22−−01  De () De()−22λ = kk121 k 2 e μ 1   (47) D (y1 ) −−01y1 De()22= kk121 k 2  e μ TT1−λ (48) By redefiningredefining thethe vectorvectorYYyy===[][()] [y1 y2]T = Dee[D1−−λ(e2) e2,] TEquation, Equation (47) (47) is rewritten is rewritten as: as: 12Dy()2 −kk12 2 21 k 2 y 2 ==TT1−λ By redefining the vectorYyy[][()]12 Dee 2 2, Equation (47) is rewritten as: 1−λ Equation (48) becomes: D (y1 ) 01y1 −λ = 1μ 01y (48) D (y1 ) −−kk1 k y1 Dy()2 1=−λ12− 2 2 μ s −−1 (48) Dydet()2 kk121 k= 2 0 y 2 − μ (49) Equation (48) becomes: kk12sk+1 Equation (48) becomes: 2 Equation (48) becomes: −λ Then s1 −1 −λ = det1 μ −1 0 (49) −−+skkλμ − λ+ 1112sk1 2 = detks++= ksμ k 0 0 (49) 221−kk + (50) 12sk1 2 Then ππ AssumingThen that sj==ωω(cos + j sin ) is a root of Equation (50), one gets 11−+λμ++= − λ Then 22ks221 ks k 0 (50) 1 λ+11−+µλμ++=1 λ − λ k s ks221+ k kss + k k=00 (50) ππ2 − 2 − 1 sj==ωω(cos + j sin ) Assuming that s = jω = ω (cosπππ + j sin π ) isis a a root root of of Equa Equationtion (50), one gets Assuming that sj==ωω| |(cos222 + j sin2 ) is a root of Equation (50), one gets 22 1+µ λ (1+µ λ)π (1+µ λ)π k ω (cos − + j sin( − ))+ 2 − 2 2 (51) | |1 λ (1 λ)π (1 λ±)π k ω − (cos − + j sin( − )) + k = 0 2| | 2 ± 2 1 Separating the real and imaginary parts of Equation (51)

1+µ λ (1 + µ λ)π 1 α (1 λ)π k ω − cos − + k ω − cos − + k = 0 (52) 2| | 2 2| | 2 1

(1 + µ λ)π (1 λ)π k sin( − ) + k sin( − ) = 0 (53) 2 ± 2 2 ± 2 Appl. Sci. 2020, 10, 2117 10 of 17

Taking the sum of squares of Equations (52) and (53)

2 2(1+µ λ) + 2 2 2λ + 2 + 2 µπ + k2 ω − k2 ω − k1 2k2 cos 2 | | (1+µ λ)π| | (1 λ)π (54) 2k1k2(cos 2− + cos −2 ) = 0

Because of the parameters to hold the conditions that k > 0, k > 0 and λ, µ (0, 1), Equation (54) 1 2 ∈ has no real solutions. This means that Equation (54) has no purely imaginary roots. Combining the constraint of Equation (14), one determines that the errors in the sliding surface can decay asymptotically toward zero. Similarly, the first sliding mode surfaces s2 is rewritten as

1 λ 1 D − e4 = e4 (55) −k3

1 1 λ The eigenvalue of Equation (55) satisfies arg(eig( )) = π > − π. So, one may conclude that − k3 2 state e4 can converge asymptotically to zero.

Remark 1. Chattering phenomena occur when the control law τ is applied to the system. Many approaches have been developed to reduce the chattering caused by the switch function, such as the robust adaptive approach, the higher-order sliding mode method, and the application of continuous or other switch functions [51–53]. This study chooses the hyperbolic tangent function instead of the sign function in the switching law to weaken chattering. Accordingly, the total control law Equation (28) can be written as:

αtanh(S) + βS ηb1τ1 ξb2τ2 Sˆ τ = − − − (56) ηb1 + ξb2

where es e s ( ) = − tanh S s − s (57) e + e−

Remark 2. A selection range of controller parameters is given in Theorems 1 and 2, although a method by which to choose all parameters to gain better control performance still needs to be discussed. Hence, the following steps to adjust the controller parameters are based on simulations.

Step 1: Set λ = µ = 0, k2 = 0, and ε = 0 to make the first layer sliding surface the same as an integer order integral sliding surface. Then, select a large enough value of α and β in the switch law τsw to achieve good robustness and a high convergence rate. Step 2: Select the same value for the parameters in the second layer surface. Because the first layer sliding surface should satisfy the condition that s1 = 0 and s2 = 0 at the same time. It is better to choose the same value for the parameters of the second layer surface. Step 3: Select the initial value of k1 and k3. It is better to make k3 larger than k1. Because the pendulum subsystem is the direct actuator for the shell subsystem. The parameter k3 is the coefficient of the integral term in the sliding surface s2 of the pendulum subsystem, a fast response rate can be achieved by increasing the value of it. Step 4: Increase the value of fractional integral order λ to gain a faster responding rate, then increase the value of µ to shorten the adjustment time and reduce the overshoot. Step 5: Increase the value of ε to smooth the dynamic response of the spherical robot. A higher value of ε will cause a longer adjustment time.

5. Simulation Study This section presents the results of numerical simulations in the MATLAB software, which were performed to demonstrate the effectiveness of the proposed novel fractional PIλDµ sliding mode controller (NFO-PID SMC). Table1 lists the main physical parameters of the spherical robot. To test Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 17

Step 3: Select the initial value of k1 and k3. It is better to make k3 larger than k1. Because the pendulum subsystem is the direct actuator for the shell subsystem. The parameter k3 is the coefficient of the integral term in the sliding surface s2 of the pendulum subsystem, a fast response rate can be achieved by increasing the value of it. Step 4: Increase the value of fractional integral order λ to gain a faster responding rate, then increase the value of μ to shorten the adjustment time and reduce the overshoot. Step 5: Increase the value of ε to smooth the dynamic response of the spherical robot. A higher value of ε will cause a longer adjustment time.

5. Simulation Study

Appl.This Sci. 2020section, 10, 2117presents the results of numerical simulations in the MATLAB software, which were11 of 17 performed to demonstrate the effectiveness of the proposed novel fractional PIλDμ sliding mode controller (NFO-PID SMC). Table 1 lists the main physical parameters of the spherical robot. To test the performance of the controller, the desired angular velocity of the spherical robot was set to x d = 10 the performance of the controller, the desired angular velocity of the spherical robot was set to2 x2d = rad/s. State x is the rolling angle of the spherical robot; thus, the desired state x was set to x d = 10 rad/s. State x11 is the rolling angle of the spherical robot; thus, the desired state x1d1d was set to x1d1 = 1010t rad.t rad. The The shell shell system system was was driven driven by by the the eccentric eccentric moment moment produced produced by by the the rotation rotation between between the the shellshell and and pendulum pendulum subsystems. subsystems. When When the the shell shell keep keepss moving moving at at a a constant velocity, thethe swinging-upswinging- upangular angular velocity velocity of of the the inner inner suspension suspension relative relative to the to shellthe shell should should be zero be [zero20]. This[20]. way,This theway, desired the desiredstate vector state vector of the controlof the control system system of Equation of Equation (3) can (3) be can determined. be determined. xtx=[10 0 0] dd2 (52)(58)

TableTable 1. 1.MainMain physical physical parameters parameters of ofthe the spherical spherical robot. robot.

Parameter M/kg m/kg r/m τl/m · ττmax/N· m τmin/N m Parameter M/kg m/kg r/m l/m max /N m min /N m· · Value 2.5 8 0.15 0.09 2.3 2.3 Value 2.5 8 0.15 0.09 2.3 −2.3 −

YueYue et et al. al. [19] [19 ]proposed proposed an an integer-type integer-type orde orderr sliding sliding mode mode surface surface for for spherical spherical robot robot speed speed control.control. For For comparison, comparison, an an integer-order integer-order sliding sliding controller controller (IO-SMC) (IO-SMC) based based on on the the method method proposed proposed inin this this paper paper was was implemented implemented for for the the spherical spherical ro robotbot system. system. The The integer-type integer-type order order sliding sliding mode mode surfacesurface equation equation can can be be written written as as follows: follows: s =+ke e s11112= k1e1 + e2 (53)(59) =+ ss22334= kke3e3 + ee4 (54)(60) TableTable 2 2summarizes summarizes the the parameters parameters of of the the fraction fractional-orderal-order hierarchical hierarchical sliding sliding mode mode controller controller withwith the the PI PIλDλμD slidingµ sliding surface. surface. In Inorder order to tohighlight highlight th thee differences differences between between the the controllers, controllers, some some parametersparameters were were kept kept the the same same for for both both controllers controllers.. Furthermore, Furthermore, two two additional additional controllers controllers without without thethe filter filter were were used used by by setting setting ε ε= =0.0. To To verify verify the the robustness robustness of of the the four four controllers, controllers, suppose suppose the the frictionfriction force force ττ ==0 0 at at time time tt == 00 s, s, and and then then set set the the friction friction force force τ == 11 NN·mm atatt t= = 1515 s.s. f f f ·

TableTable 2. 2.ParametersParameters of ofthe the controllers. controllers.

Parameter Parameterk1 k2 k1 k3k2 kη3 η ζ ζ α β α λ βμ ελ µ ε Value 1Value 0.31 12 0.3 12 2 2 2 2 5 5 5 0.1 50.1 1 0.1 0.1 1

Figure 2 shows the curves of the velocity response of the spherical shell, while Figure 3 shows the curvesFigure of 2the shows errors. the Figure curves 4 ofshows the velocity the contro responsel output. of Figures the spherical 5–7 show shell, the while time Figure evolutions3 shows of allthe sliding curves mode of the surfaces. errors. Table Figure 34 summarizesshows the control the performance output. Figures of the5– two7 show controllers the time in evolutions terms of overshootof all sliding and modeadjustment surfaces. time. Table 3 summarizes the performance of the two controllers in terms of overshoot and adjustment time.

Table 3. Performances of the controllers.

Controller Overshoot (%) Adjustment Time (s) NFO-PID SMC 0 3.4 IO-PI SMC 35 6.1 Appl. Sci. 2020, 10, 2117 12 of 17 Appl. Sci. 20202020,, 1010,, xx FORFOR PEERPEER REVIEWREVIEW 1212 of 1717

..

((a)) ((b))

FigureFigure 2. 2. ComparisonComparison of of the the curves curves of of the the velocity velocity of of shell shell response response among four controllers: controllers: ( (a).). TimeTime Time intervalinterval 00 0~30 ~ 30 s; s;( (bb).).). Time Time intervalinterval 0~700 ~ 7 s. s..

Figure 3. Curves of error e 2 . Appl. Sci. 2020, 10, x FOR PEER REVIEW FigureFigure 3. 3. CurvesCurves of of error error e2.. 13 of 17

Table 3. Performances of the controllers.

Controller Overshoot (%) Adjustment time (s)(s) NFO--PID SMC 0 3.4 IOIO--PI SMC 35 6.1

5.1. Tracking Performanceerformance Analysis The results in Figure 2a indicate that all four controllers can track the desired angular velocity. The NFO--PID SMC has a shorter adjustment time and does not have overshoot. The adjustment time isis shortenedshortened fromfrom 6.16.1 ss toto 3.43.4 ss.. InIn Figure 2b, itit cancan bebe seenseen thatthat thethe controllercontroller withoutwithout a filterfilter causes wave--likelike behavior and has a larger opposite speed at the start stage. However, the proposed method yields a smoother dynamic response, showing that the filter is effective for the smoothsmooth controlcontrol of sphericalspherical robotrobotss.. ThoughThough thethe filterfilter isis usedused inin thethe IOIO SMC,SMC, thethe wavewave--likelike responseresponse stillstill occursoccurs inin thethe control process, but this phenomenonFigure is 4. hardly The output observed of two incontrollers controllers. the NFO. --PID SMC. Figure 3 proves thatthat x thethe NFONFO--PID SMC and IO SMC can both track the output of the filter x22dd ;; furthermore,furthermore, thethe NFONFO-- Figure 4 shows that the NFO-PID SMC controller can provide the spherical robot with smooth PID SMC controller forces error ee22 toto beginbegin toto convergeconverge earlierearlier thanthan thethe IOIO SMCSMC does.does. This case also input. The output of the two controllers reaches the maximum limit in a very short time, but the showsshows thatthat thethe NFONFO--PID SMC has better tracking performance than IO SMC. output of IO-SMC stays in the maximum value for a longer time than the NFO-PID SMC. This explains why the wave motion still occurs in the IO SMC. Figure 5 highlights the better dynamic response of the second-layer sliding surfaces of the NFO-PID SMC controller compared to the IO- SMC controller. Figures 6 and 7 show that the shell and pendulum subsystems are asymptotically stable, and the fractional sliding surface proposed in Equations (24) and (25) converges to zero faster than the integral sliding surface. This indicates that the fractional sliding mode surface proposed in this paper is effective. In summary, the NFO-PID SMC controller successfully tracks the desired velocity, and the goal of better control performance is achieved.

Figure 5. Curves of second-layer sliding mode surfaces S. Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 17

Figure 4. The output of two controllers.

Figure 4 shows that the NFO-PID SMC controller can provide the spherical robot with smooth input. The output of the two controllers reaches the maximum limit in a very short time, but the output of IO-SMC stays in the maximum value for a longer time than the NFO-PID SMC. This explains why the wave motion still occurs in the IO SMC. Figure 5 highlights the better dynamic response of the second-layer sliding surfaces of the NFO-PID SMC controller compared to the IO- SMC controller. Figures 6 and 7 show that the shell and pendulum subsystems are asymptotically stable, and the fractional sliding surface proposed in Equations (24) and (25) converges to zero faster than the integral sliding surface. This indicates that the fractional sliding mode surface proposed in Appl.this Sci.paper2020 , is10 , effective. 2117 In summary, the NFO-PID SMC controller successfully tracks the desired13 of 17 velocity, and the goal of better control performance is achieved.

Appl. Sci. 2020, 10, x FOR PEERFigureFigure REVIEW 5.5. Curves Curves ofof second-layersecond-layer slidingsliding modemode surfacessurfaces SS.. 14 of 17 Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 17

Figure 6. Curves of first-layer sliding mode surfaces s1 Figure 6. Curves of first-layerfirst-layer sliding mode surfaces ss11 .

Figure 7. Curves of first-layer sliding mode surfaces s2. Figure 7. Curves of first first-layer-layer sliding mode surfaces s2..

5.2.5.1.5.2. Robustness TrackingRobustness Performance A Analysisnalysis Analysis AsTheAs shown shown results i nin in Figure Figure Figure 3 3,2 ,the athe indicate robot robot system system that all actuated actuated four controllers by by the the four four can controllers controllers track the can desiredcan recover recover angular to to the the velocity. desired desired velocityThevelocity NFO-PID almost almost SMC at at the the has same same a shorter time time when when adjustment subjected subjected time to toand friction friction does interference interference not have overshoot. at at the the time time The of of 15 adjustment 15 s s. .This This shows shows time thethe robustness robustness of of the the sliding sliding mode mode control control method. method. The The results results of of Figure Figure 3 3 after after 15 15 s s indicate indicate that that the the NFONFO--PIDPID SMC SMC has has lower lower recover recoveryy speedspeed compared compared to to IO IO SMC. SMC. The The same same situation situation also also can can be be observedobserved in in Figure Figuress 6 6 and and 7 7. .This This situation situation, ,caused caused by by the the weigh weigh function function in in the the fractional fractional integral integral operationoperation, ,decrease decreasess over over time. time. The The ability ability of of the the integra integra term term to to tolerate tolerate extra extra disturbers disturbers is is weakened. weakened. FigureFigure 5 5 shows shows that that the the second second--layerlayer sliding sliding surfaces surfaces of of the the two two controllers controllers reach reach a a new new steady steady--statestate  whichwhich is is not not equal equal to to zero zero when when the the rolling rolling friction friction changes changes from from 0 0 to to 1 1 N Nmm. .The The new new steady steady--statestate of of eacheach second second--layerlayer sliding sliding surface surface is is the the same. same. All All the the results results show show that that NFO NFO--PIDPID SMC SMC has has the the same same robustnessrobustness as as IO IO SMC, SMC, though though a a fractional fractional differential differential term term is is included. included.

6.6. Conclusions Conclusions λ μ InIn this this study study, , aa novel novel fractional fractional PIPIλDDμ slidingsliding controller controller isis proposed proposed toto improveimprove the the control control performanceperformance of of the the spherical spherical robot robot linear linear motion motion with with input input saturation saturation. .The The f fractionalractional sliding sliding surface surface isis applied applied in in the the control control design design to to achieve achieve quick quick tracking tracking of of the the control control target target. .A Ann auxiliary auxiliary system system is is designeddesigned toto handlehandle inputinput saturation.saturation. ByBy addingadding aa filterfilter inin thethe desireddesired velocityvelocity toto reducereduce thethe initialinitial λ μ valuevalue and and gain gain a a smoothsmooth dynamic dynamic response response, ,simulation simulation result resultss show show that that the the novel novel fractional fractional PI PIλDDμ slidingsliding mode mode controller controller hashas smaller smaller overshoot overshoot and and aa shortershorter adjustmentadjustment timetime thanthan thethe integerinteger one. one. TheThe adjustmentadjustment timetime ofof sphericalspherical robotrobot systemsystem decreasedecreasess byby 44%,44%, andand presentspresents nono overshoot.overshoot. TheThe wavewave motion motion in in the the dynamic dynamic response response process process is is efficiently efficiently supp suppressed.ressed. To To conclude, conclude, the the presented presented Appl. Sci. 2020, 10, 2117 14 of 17 is shortened from 6.1 s to 3.4 s. In Figure2b, it can be seen that the controller without a filter causes wave-like behavior and has a larger opposite speed at the start stage. However, the proposed method yields a smoother dynamic response, showing that the filter is effective for the smooth control of spherical robots. Though the filter is used in the IO SMC, the wave-like response still occurs in the control process, but this phenomenon is hardly observed in the NFO-PID SMC. Figure3 proves that the NFO-PID SMC and IO SMC can both track the output of the filter x2d; furthermore, the NFO-PID SMC controller forces error e2 to begin to converge earlier than the IO SMC does. This case also shows that the NFO-PID SMC has better tracking performance than IO SMC. Figure4 shows that the NFO-PID SMC controller can provide the spherical robot with smooth input. The output of the two controllers reaches the maximum limit in a very short time, but the output of IO-SMC stays in the maximum value for a longer time than the NFO-PID SMC. This explains why the wave motion still occurs in the IO SMC. Figure5 highlights the better dynamic response of the second-layer sliding surfaces of the NFO-PID SMC controller compared to the IO-SMC controller. Figures6 and7 show that the shell and pendulum subsystems are asymptotically stable, and the fractional sliding surface proposed in Equations (24) and (25) converges to zero faster than the integral sliding surface. This indicates that the fractional sliding mode surface proposed in this paper is effective. In summary, the NFO-PID SMC controller successfully tracks the desired velocity, and the goal of better control performance is achieved.

5.2. Robustness Analysis As shown in Figure3, the robot system actuated by the four controllers can recover to the desired velocity almost at the same time when subjected to friction interference at the time of 15 s. This shows the robustness of the sliding mode control method. The results of Figure3 after 15 s indicate that the NFO-PID SMC has lower recovery speed compared to IO SMC. The same situation also can be observed in Figures6 and7. This situation, caused by the weigh function in the fractional integral operation, decreases over time. The ability of the integra term to tolerate extra disturbers is weakened. Figure5 shows that the second-layer sliding surfaces of the two controllers reach a new steady-state which is not equal to zero when the rolling friction changes from 0 to 1 N m. The new steady-state of · each second-layer sliding surface is the same. All the results show that NFO-PID SMC has the same robustness as IO SMC, though a fractional differential term is included.

6. Conclusions In this study, a novel fractional PIλDµ sliding controller is proposed to improve the control performance of the spherical robot linear motion with input saturation. The fractional sliding surface is applied in the control design to achieve quick tracking of the control target. An auxiliary system is designed to handle input saturation. By adding a filter in the desired velocity to reduce the initial value and gain a smooth dynamic response, simulation results show that the novel fractional PIλDµ sliding mode controller has smaller overshoot and a shorter adjustment time than the integer one. The adjustment time of spherical robot system decreases by 44%, and presents no overshoot. The wave motion in the dynamic response process is efficiently suppressed. To conclude, the presented control method can be extended to a class of underactuated systems, and further work should focus on the implementation of the proposed method in a real spherical robot to verify its effectiveness.

Author Contributions: Methodology, T.Z. and B.W.; Simulation; T.Z.; Writing, T.Z., Y.-g.X. and B.W. All authors have read and agree to the published version of the manuscript. Funding: This research was funded by The Fundamental Research Funds for the Central Universities under Grant No.2019JBM408. Conflicts of Interest: The authors declare no conflict of interest. Appl. Sci. 2020, 10, 2117 15 of 17

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