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Design and Implementation of Integral Sliding-Mode Control on an Underactuated Two-Wheeled Mobile Robot Jian-Xin Xu, Fellow, IEEE, Zhao-Qin Guo, and Tong Heng Lee
Abstract—This paper presents a novel implementation of an integral sliding-mode controller (ISMC) on a two-wheeled mobile robot (2 WMR). The 2 WMR consists of two wheels in parallel and an inverse pendulum, which is inherently unstable. It is the first time that the sliding-mode control method is employed for real-time control of a 2 WMR platform and several critical issues are addressed. First, the 2 WMR is underactuated, which uses only one actuator to achieve position control of the wheels while balancing the pendulum around the upright position. ISMC is suitable for control of the underactuated 2 WMR, because ISMC has an extra degree of freedom in control when sliding mode is achieved. In this paper, we utilize this extra degree of freedom to implement a linear nominal controller, which is found adequate in stabilizing the sliding manifold in a range around the equilibrium. Second, the 2 WMR system is in presence of both matched and unmatched uncertainties. The implemented ISMC, with an integral sliding surface and a switching term, is able to completely nullify the influence from the matched uncertainties. The implemented linear nominal controller stabilizes the sliding manifold that is subject to unmatched uncertainties. Third, references design are addressed when implementing ISMC on the 2 WMR. The effectiveness of ISMC is verified through intensive simulation and experiment results. Index Terms—Integral sliding-mode controller (ISMC), linear controller, steady-state error, trajectory planning, underactuated system.
I. I NTRODUCTION
T
HE development and control of 2 WMR or wheeled inverted pendulum (WIP) is a popular research topic in recent years [1]–[14]. However, most of the published works are based on theoretical analysis and results are obtained by simulations. Only few researchers have implemented their proposed control algorithms on real-time platforms. Prototypes and products of two-wheeled mobile vehicle or robot have been designed in some universities and research institutes [1]–[9]. The 2 WMR usually consists of two wheels in parallel and an inverse pendulum. The control objective of the 2 WMR is to perform motion control of the wheels while stabilizing the Manuscript received September 27, 2012; revised March 15, 2013 and June 20, 2013; accepted August 13, 2013. Date of publication September 18, 2013; date of current version January 31, 2014. The authors are with the Graduate School for Integrative Sciences and Engineering, National University of Singapore, Singapore 117456, and also with the Department of Electrical and Computer Engineering, National University of Singapore (NUS), Singapore 117583 (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2013.2282594
Fig. 1. Prototype of the two-wheeled mobile robot.
pendulum around the upright position that is an unstable equilibrium point. This type of systems that have fewer numbers of actuators than the degrees of freedom (DOF) to be controlled are called underactuated systems. Due to the difference in mechanical configuration, underactuated 2 WMRs can be classified into the class without input coupling and the class with input coupling [10]. Since the existing works mostly focus on studying control of underactuated systems without input coupling, this work is devoted to the development and control of an underactuated 2 WMR with input coupling. A prototype of 2 WMR is built in our lab as shown in Fig. 1. The motor shaft coupler is fixed at the center of the wheel and the motor housing is rigidly connected to the pendulum, thus the torque generated by the motor directly acts on both the wheels and the pendulum with the same size but opposite directions, which results in the input coupling of the 2 WMR system. Stabilizing algorithms based on Lyapunov theory, passivity, feedback linearization, etc., are developed for underactuated systems in absence of uncertainties [15]–[18]. The controller design and stability prove are based on the accurate mathematical models without considering any uncertainties. However, uncertainties and model mismatch between the nominal mathematical models and the real-life plants are inevitable. Furthermore, some of the control algorithms are too complicated to be implemented. Since uncertainties could affect system performance or even devastate system stability, researchers are motivated to explore robust control designs for underactuated systems with uncertainties [19]–[24]. Real-time control of 2 WMRs and similar underactuated systems are presented in [1]–[9], [19], [20]. In [1]–[4], fullstate feedback linear controller is employed. However, the robustness of the linear controller is limited. In [5], [6], a
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novel adaptive output recurrent cerebellar model articulation controller is proposed, which is a model free design. Functions are used to approximate the system model, thus, the designed control algorithm is complex in mathematics and not evident in physics idea, furthermore, there are plenty of controller parameters to be determined. In [7], a fuzzy traveling and position control algorithm is proposed, however it is limited applicable to the 2 WMR without input coupling. In [8], [9], two-wheeled self-balancing vehicles are developed. The basic principle for riding the two-wheeled vehicle is that the traveler’s body leads forward to make the wheels accelerate and leads backward to make the wheels slow down. Essentially, the mobility of the scooter is not autonomous because the traveler is involved in control. Sliding-mode control (SMC) is a well-known robust control approach for systems with model uncertainties and external disturbances [19]–[31]. For control of underactuated systems, SMC with a linear sliding surface has been proposed in [12], [19], [20], [22]. However, the implementation of the SMC with the linear sliding surface could be problematic. First, the sliding surface parameters affect the system performance in a complicated manner, thus, it is hard to predict the system responses based on the information of the chosen parameters. Second, the determination and tuning of the SMC parameters could be challenging considering the non-affine structure of the sliding manifold in the controller parameters. Other types of SMC for controlling underactuated systems have been discussed in [21], [23]–[25]. In [21], an SMC design based on the cascade normal form is proposed, and the validity holds under certain assumptions. However, the 2 WMR studied in our work does not meet these assumptions. Second-order SMC designs for underactuated systems are discussed in [23]–[25]. The design of second-order SMC requires that the derivative of the defined sliding variable is known. In [23]–[25], the SMC design requires that the derivatives of all system states are known. However, in this work, the derivatives of the velocity states are not available because the 2 WMR system is in presence of both parametric and external uncertainties. Integral-type sliding-mode designs are proposed in [33] for controlling systems with both matched and unmatched uncertainties. The sliding mode exists from the very beginning, therefore the system is more robust against perturbations than the other SMC systems with reaching phase [33]. The ISMC is constructed by a nominal control part and a switching term. With the switching term, the matched uncertainties can be perfectly rejected. With the freedom to design a nominal control for the sliding manifold, ISMC can be easily incorporated with other robust control methods, such as linear matrix inequality (LMI), H∞ , and linear quadratic regulator (LQR) to deal with the unmatched uncertainties. Furthermore, ISMC provides one more degree of freedom in choosing an appropriate projection matrix to reduce the effect of the unmatched uncertainties. In this paper, an ISMC is proposed for control of the 2 WMR. First, an integral-type sliding surface is defined and the control law is derived by using Lyapunov theory. The resulting sliding manifold is still underactuated with a nominal controller to be further designed. To make the control algorithm simple and implementable, a linear controller is adopted as the nominal
Fig. 2.
Model of the two-wheeled mobile robot.
controller. It is found that the linear controller is adequate to stabilize the sliding manifold around the equilibrium. In implementations, regulation and setpoint control of the 2 WMR are considered. In the existing works [2]–[9], the control tasks are achieved only when the 2 WMRs are placed on a flat surface. In this paper, the control tasks are achieved not only when the 2 WMR is placed on a flat surface but also on an inclined surface. The particular characteristics of the underactuated 2 WMR system are investigated, according to which, references for both the wheel and the pendulum are designed. The paper is organized as follows. In Section II, the 2 WMR dynamic model is introduced. In Section III, the ISMC design is detailed. In Section IV, intensive simulation investigations are conducted to verify the effectiveness of the proposed controller. In Section V, the implementation of ISMC on the real platform is given. Conclusions are drawn in Section VI. Throughout this paper, a function F (λ1 , λ2 , . . . , λn ) will be written as F , where λ1 , λ2 , . . . , λn can be either parameters or variables. II. P ROBLEM F ORMULATION A. System Model Fig. 2 shows the model of the 2 WMR. The wheel motion is defined along the surface. The wheels displacement and velocity are denoted by x and x, ˙ respectively, with rightward as positive direction. θ is the tilting angle of the pendulum with the upright position as zero point and clockwise rotation as positive direction. θ˙ is angular velocity of the pendulum. ϕ is the slope angle of the inclined road, for traveling on flat surface, ϕ = 0. fr is the friction between the wheels and the ground. u is the control input to the system and physically represents the torque generated by the motor which acts on the wheels with clockwise rotation as positive direction. Note that the motor driving the wheel is directly mounted on the pendulum, there is a reaction torque −u applied to the pendulum. τf is the joint friction, which also acts on both the wheel and the pendulum as τf and −τf , respectively. Other system parameters are as: the mass of the wheels mw = 1.551 kg, the mass of the pendulum mp = 1.6 kg, the rotation inertia of the wheels Iw = 0.005 kg · m2 , the rotation inertia of the pendulum Ip = 0.027 kg · m2 , the
XU et al.: DESIGN AND IMPLEMENTATION OF INTEGRAL SLIDING-MODE CONTROL
radius of the wheel r = 0.08 m, the distance between Center of Gravity (COG) of the pendulum and the center of the wheel l = 0.13 m, the acceleration of gravity g = 9.81 m/s2 . Lagrangian mechanics method is used to derive the mathematical model of the 2 WMR system, which leads to a secondorder nonlinear model given by a¨ x + bθ¨ − mp l sin(θ + ϕ)θ˙2 + sin ϕ(mp + mw )g 1 = (u + τf − rfr ) r b¨ x + cθ¨ − mp lg sin θ = −u − τf
(1)
(2)
where a = mw + mp + (Iw /r2 ), b = mp l cos(θ + ϕ) and c = I p + mp l 2 . B. Control Objective The control objective for the 2 WMR is to achieve setpoint control of the wheel, while balance the pendulum at an equi˙ librium (θ = θe , θ˙ = 0). Define x = [x1 , x2 , x3 , x4 ]T = [x, x, ˙ T and the reference signal for x is chosen as r = θ, θ] [xr , vr , θr , 0]T with x˙ r = vr . We obtain the error states as e = [e1 , e2 , e3 , e4 ]T = x − r = [x1 − xr , x2 − vr , x3 − θr , x4 ]T . Now the control objective is to ensure the convergence of e. The error dynamic model of the 2 WMR is obtained as e˙ = η(e) + g(e) [u + dm (e, t)] + du (e, t)
(3)
where η u is the system nonlinear term, dm is the lumped matched uncertainties, du is the lumped unmatched uncertainties. We have η(e) = [e2 g(e) = [0
η1 (e) g1 (e)
η2 (e)]T
e4
0 g2 (e)]T
d m = τf du (e, t) = [0
du1 (e, t)
0
du2 (e, t)]T
where mp l 2 ce4 sin(e3 + θr + ϕ) − bg sin(e3 + θr ) ac − b2 c(mp + mw )g sin ϕ − ac − b2 mp l −be4 2 sin(e3 + θr + ϕ) + ag sin(e3 + θr ) η2 = 2 ac − b b(mp + mw )g sin ϕ + ac − b2 c 1 b g1 = + r ac − b2 ac − b2 1 −b −a g2 = + 2 r ac − b ac − b2 −c b du1 = fr , du2 = fr ac − b2 ac − b2 η1 =
and b = mp l cos(e3 + θr + ϕ).
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C. Trajectory Planning Without loss of generality, we consider a setpoint control task for the 2 WMR, i.e., the 2 WMR is supposed to reach a desired position xd and stop there. We simply use a linear segment and two parabolic blends to construct a smooth trajectory for the 2 WMR, which also yields a smooth reference signal for the wheel velocity [10]. The reference inputs are computed by the following equations: ⎧ vm t, 0 < t < t1 ⎪ ⎨ t1 vm , t1 ≤ t ≤ t2 (4) vr (t) = vm ⎪ ⎩ vm − t3 −t2 (t − t2 ), t2 ≤ t ≤ t3 0, t3 ≤ t ≤ ts xr (t) + vr Ts , if xr (t) < xd xr (t + Ts ) = (5) xd , if xr (t) ≥ xd where xd is the desired setpoint, Ts is the sampling time. For both simulations and experimental testings in the later work, the parameters are specified as t1 = 1 s, t2 = 15 s, t3 = 16 s, ts = 20, vm = 0.1 m/s, xd = 1.5 m. D. Analysis of the Pendulum Equilibrium Point At the equilibrium point, the wheel acceleration is zero (¨ x = 0), the pendulum angular velocity and acceleration are zero (θ˙ = 0, θ¨ = 0), meanwhile the joint friction does not exist (τf = 0), the dynamic (1) and (2) become 1 sin ϕ(mp + mw )g = (τ − rfr ) r −mp lg sin θ = − τ. From the above equations, the pendulum equilibrium point is obtained as θe = arcsin
r sin ϕ(mp + mw )g + rfr . mp lg
(6)
Remark 1: The varying θe is an inherent characteristic of the 2 WMR system with input coupling. The equilibrium θe depends on the size of friction fr and slope ϕ. When the 2 WMR travels under the same circumstance, θe is fixed and irrelevant to controller parameters or control tasks. Considering that our control objective is setpoint control, the 2 WMR finally stops at the desired setpoint, thus we have fr = 0, and θe = arcsin
r sin ϕ(mp + mw ) . mp l
(7)
It is reasonable to choose the reference position for the pendulum as θr = θe . For 2 WMR traveling on a flat surface, ϕ = 0, we have θr = θe = 0. For 2 WMR traveling on an inclined surface, the value of θe can be calculated according to (7) only when the system parameters involved are known. In this paper, if part of the system parameters involved are unknown, estimated values of the unknown parameters are used to obtain the estimated equilibrium point, denoted as θˆe . In such a situation, the reference position for the pendulum is designed as θr = θˆe .
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Considering that the 2 WMR experiences modeling uncertainties due to the unmodeled frictions and variation of system parameters, robustness should be an important concern in the controller design. The following nonlinear integral-type sliding surface is proposed in [33] to handle systems with matched and unmatched uncertainties: t σ(e, t) = se(t) − se(t0 ) −
sdu −ρ ≤ |σsg| |dm | + sg
III. I NTEGRAL S LIDING -M ODE C ONTROL D ESIGN
[sη(e) + sg(e)κ(e, t)] dτ = 0
≤ − ρ0 |σsg| < 0. Since σ(e(t0 ), t0 ) = 0, we can conclude that the controller (9) using the gain function (10) guarantees that the sliding mode σ = 0 can be maintained ∀t ∈ [t0 , ∞). In the sliding mode, σ(t) = 0, σ(t) ˙ = 0, and define ed as the state vector in the sliding mode. The equivalent control is derived from σ˙ = 0, which is
t0
(8) where κ(e, t) is a nominal control, s is a 1 × 4 projection vector with freedom to design, and sg(e) = 0. Here, we define s = [s1 , s2 , s3 , s4 ], to satisfy sg(e) = 0, we have cs2 − bs4 = 0.
First, we investigate the effect of frictions to the 2 WMR system. From (3), we can see that the joint friction τf is a matched uncertainty, while the ground friction fr is an unmatched uncertainty. The control law is designed as (9)
where the switching gain function is ρ = ρm + ρu + ρ0
(10)
where ρm is the upper bound of the matched uncertainty dm , ρu is the upper bound of {sg}−1 sdu , and ρ0 is a positive constant. Theorem 1: With the nonlinear integral-type sliding surface (8) and the controller (9), the global attractiveness of the sliding manifold is achieved. In the sliding mode, the matched uncertainties will be completely nullified. Further, the influence of unmatched uncertainties can be reduced with the freedom in choosing the projection vector s. Proof: Differentiating the sliding surface (8) with respect to time t using (3) one obtains ˙ − sη(e) − sg(e)κ(e) σ(t) ˙ = se(t)
sdu +u−κ . = sg dm + sg
(11)
We choose a non-negative quadratic function V = σ 2 /2. Differentiating V with respect to time t yields
Substituting the ISMC law (9) into the above we obtain
sdu ˙ − ρsgn(sgσ) V = σsg dm + sg
Substituting the above ueq (t) into (3), one obtains the sliding manifold as
(12)
(13)
where δ is the resulting unmatched uncertainty and ⎡ ⎤ ⎡ ⎤ 0 0
g2 du1 − g1 du2 ⎢ s4 ⎥ gs ⎢δ ⎥ δ =⎣ 1⎦= I − du = ⎣ ⎦. 0 0 sg s2 g1 + s4 g2 δ2 −s2 (14) From (3) and (13), it can be seen that the matched uncertainty dm is completely nullified. We can choose s2 and s4 to minimize the effect of the unmatched uncertainty δ in the sliding manifold. Referring to (14), when s2 = 0 and s4 = 0, the unmatched uncertainties in the sliding manifold only exist in the wheel subsystem, when s4 = 0 and s2 = 0, the unmatched uncertainties only exist in the pendulum subsystem. Since the pendulum subsystem is much more sensitive to uncertainties than the wheel subsystem, it is preferred to choose s = [0, 0, 0, s4 ] and s4 = 0. B. ISMC for System With Parameter Uncertainties From the practical point of view, the load of the pendulum mp , COG l and slope angle of the traveling surface ϕ are most ˆ as the estimation of likely to vary. Define p = [mp , l, ϕ] and p p = [m ˆ p , ˆl, ϕ], ˆ for the 2 WMR system with parameter uncertainties, the nonlinear system term η(e) and the input vector g(e) in (3) are expressed as η(e, p) and g(e, p). The dynamic model of the 2 WMR system with parameter uncertainties is expressed as e˙ = η(e, p) + g(e, p)(u + dm ) + du .
V˙ = σ σ. ˙ Substituting σ˙ in (11) into the above we have
sdu ˙ +u−κ . V = σsg dm + sg
sdu . sg
e˙ d (t) = η(ed ) + g(ed )κ(ed ) + δ
A. ISMC for System With Unmodeled Frictions
u(t) = κ(e, t) − ρ(e, t)sgn(sgσ)
ueq (t) = κ − dm −
(15)
ˆ (e, p ˆ ) and g ˆ (e, p ˆ ), the estimation of η(e, p) and Define η g(e, p), respectively, we have the estimation errors caused by ˆ , p) = η(e, p) − η(e, p ˆ) the parameter uncertainties as Δη(e, p ˆ , p) = g(e, p) − g(e, p ˆ ), and the system dynamic and Δg(e, p model (15) becomes ˆ , p) ˆ (e, p ˆ ) + Δη(e, p e˙ = η ˆ ) + Δg(e, p ˆ , p)] (u + dm ) + du . + [ˆ g(e, p
(16)
XU et al.: DESIGN AND IMPLEMENTATION OF INTEGRAL SLIDING-MODE CONTROL
The integral sliding surface is designed as t σ(e, t) = se(t) − se(t0 ) −
ˆ ) + sˆ ˆ )κ(e, t)] dτ [sˆ η (e, p g(e, p
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Since σ(x(t0 ), t0 ) = 0, we can conclude that the sliding mode σ = 0 can be maintained ∀t ∈ [t0 , ∞). In the sliding mode, the equivalent control is derived from σ˙ = 0, which is
t0
= 0.
(17)
The control law is u(t) = κ(e, t) − ρ(e, t)sgn(sgσ)
(18)
ueq (t) =
Substituting the above ueq (t) into (16), one obtains the sliding manifold as
where the switching gain function is as (10) with ρm ≥ |dm | sdu sΔη sΔgκ + + . ρu ≥ sg sg sg
(19)
sgn [sg(e, p)] = sgn [s4 · g2 (e, p)] . Refer to the expression of g2 in Section II-B, we have
1 b g2 = − a + 0. The optimal solution is k = R−1 P g0
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Fig. 3. Time responses of x, θ and u under linear control. In simulations, the 2 WMR system is considered in absence of frictions and system uncertainties.
Fig. 5. Time responses of x, θ, u and σ under the ISMC. In simulations, the ˙ 2 WMR system is considered with the joint friction τf = 0.2θ˙ + 0.3sgn(θ), which is a matched uncertainty.
Fig. 4. Time responses of x, θ and u under linear control. In simulations, the ˙ 2 WMR system is considered in presence of frictions, τf = 0.2θ˙ + 0.3sgn(θ) ˙ and fr = 0.5x˙ + sgn(x).
where P is the solution of the Riccati equation AT0 P + P A0 + Q − P g0 R−1 g0T P = 0. Choosing {q1 , q2 , q3 , q4 } = {50, 0.1, 500, 1}, R = 1, we obtain the feedback gains as k = [−7.0711, −9.6708, −27.0228, −2.8418]T . The initial states of the 2 WMR are as x = [0, 0, 0.1, 0]T . The simulation results are shown in Fig. 3. The wheel reaches the desired setpoint smoothly with a small overshoot, the pendulum angular stays around zero. Next, the linear controller is applied to the 2 WMR system ˙ and fr = in presence of the frictions, τf = 0.2θ˙ + 0.3sgn(θ) 0.5x˙ + sgn(x). ˙ The simulation results are shown in Fig. 4. It is found that the pendulum and the wheel keep vibrating around the desired positions, which are not satisfactory responses and indicates the limited robustness of the linear controller. B. ISMC for System With Matched Uncertainties We consider the joint friction exists in the system and τf = ˙ which is a matched uncertainty. ISMC is 0.2θ˙ + 0.3sgn(θ), applied with s = [0, 0, 0, 1], ρ = 0.1 + 0.2|x4 | + 0.3, and the nominal linear controller κ uses the same feedback gains as in the pervious subsection. We set θr = 0 and γc = 0 since fr = 0 and ϕ = 0. The simulation results are shown in Fig. 5. The 2 WMR reaches the desired setpoint smoothly and the pendulum is balanced at θe = 0. The 2 WMR responses are almost the same as in Fig. 4 despite the system is in presence of the joint friction τf , which demonstrates the effectiveness of ISMC in rejecting matched uncertainties. It is noted that control signal shows switching behavior, which can be explained as the following. In the ideal sliding mode, we have σ = 0. To make the system states stay on the switching surface, an infinite
switching frequency is needed, which is impossible to achieve in any digital implementation. Due to the finite sampling frequency in implementations, the “chattering” phenomenon occurs. Remark 3: In this paper, the DC motor is controlled by a discontinuous pulse width modulation (PWM) signal. The characteristic of the PWM control is its switching (on–off) operation mode, which is achieved by electronic power switchers. Therefore, the implementation of the switching type control signal is not a problem. Furthermore, it may even be more advantageous to employ the ISMC than other continuous controllers because the ISMC naturally generates a discontinues control signal while other continuous controllers are designed to generate continuous signals which however are forced to become discontinuous in real implementation [31], [32]. C. ISMC for System With Both Matched Uncertainties and Unmatched Uncertainties Two type of unmatched uncertainties exist in the system, one is due to the external disturbance and the other is due to the uncertain system parameters. First, we consider the 2 WMR system with the ground ˙ and the joint friction τf = 0.2θ˙ + friction fr = 0.5x˙ + sgn(x) ˙ ISMC is applied with ρ = 0.1 + 0.2|x4 | + 0.3 + 0.3sgn(θ). br/(b + ar)(0.5|x2 | + 1) and the nominal controller in (33). Other control parameters are chosen the same as in the preceding simulation. The simulation results are shown in Fig. 6. The 2 WMR reaches the desired setpoint at t = 20 s and the pendulum is finally balanced at the upright position, i.e., θ = 0, which indicates that the proposed ISMC is also robust to unmatched unknown friction. At the time interval 3 ∼ 15 s, the 2 WMR reaches a steady state that the 2 WMR travels with the constant speed 0.1 m/s, the pendulum is balanced at θ = 0.041 rad and the tracking error of the wheel position exists. The results are consistent with the analysis in Section II-D. When fr = 0, the equilibrium of the pendulum is not the upright position, but related with the size of the ground friction. Since the ground friction is
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Fig. 6. Time responses of x, θ and u under the ISMC with and without the compensation term γc . In simulations, the 2 WMR system is considered in ˙ and the ground friction presence of the joint friction τf = 0.2θ˙ + 0.3sgn(θ), fr = 0.5x˙ + sgn(x) ˙ which is an unmatched uncertainty.
unknown to the designer, θr = 0 is used in the controller design, which yields e3 = θ − θr = 0 and e1 = 0. Although the ground friction brings a tracking error of the wheel position during the traveling, the system performance is still satisfactory. When the 2 WMR stops at the desired setpoint at t = 20 s, the ground friction disappears, so does the tracking error of the wheel position. Next, the 2 WMR system with parameter uncertainties is considered. The actual values of the uncertain parameters are as [mp , l, ϕ] = [2.0 kg, 0.18 m, π/15 rad], which are assumed to be unknown to the designer. Estimated values of the uncertain ˆ = [1.6 kg, 0.13 m, 0 rad], are used in parameters, [m ˆ p , ˆl, ϕ] sliding surface and controller designs. The frictions are also considered to exist in the system. ISMC is applied with θr = 0 and the nominal controller is designed as in (33). First, γc = 0 is applied. ISMC shows the robustness to the parameter uncertainties. The pendulum balances at a new equilibrium position θ = 0.26 rad. However, the tracking performance of the 2 WMR is not satisfactory. The tracking error of the wheel position in steady state is e1,s |γc =0 = −0.9259 m. Next, γc = −6.5471 is computed according to (35) and used in (33). The simulation results for the two cases, with and without the compensation term γc , are shown in Fig. 7. By adding the compensation term to the control input, the 2 WMR tracks the planned trajectory better and reaches the desired setpoint without steady-state error. The simulation results are consistent with the theoretical analysis in Sections II-D and III-D. V. I MPLEMENTATION AND E XPERIMENT R ESULTS In simulations, an ideal model of the 2 WMR is used. To stabilize the 2 WMR system in absence of uncertainties, the feedback gains for the nominal linear controller can be chosen in a wide range as long as A0 − g0 kT is Hurwitz. However, considering the existence of mismatch between the real-time system model and the mathematical model (1) and (2), the feedback gains obtained from simulations may not function well on the real-time platform, thus need to be adjusted through experimental testings on the 2 WMR prototype.
Fig. 7. Time responses of x, θ and u under ISMC with and without the compensation term γc . In simulations, the 2 WMR system is considered with uncertain parameters mp , l and ϕ.
Fig. 8. Experimental testing results for regulation task: time responses of x, θ and u under ISMC and linear controller. The mobile robot is placed on flat surface.
A. Regulation Task For implementation, first, we consider a simple regulation task that is to balance the robot at the original position on a flat surface, i.e., xr = 0, vr = 0, and ϕ = 0. Since there exists backlash in the driving mechanism of the 2 WMR [10], strong vibrations are observed by applying the linear controller with the feedback gains obtained from simulations. To reduce the vibrations, the feedback gains are adjusted to k = [−10, −0.5, −35, −3]T . ISMC is applied with the projection vector as s = [0, 0, 0, 0.05] and the feedback gains for the nominal linear controller as k = [−10, −0.5, −35, −3]T . For comparison, the linear controller alone is also applied to the 2 WMR. Fig. 8 shows the experimental results for the 2 WMR under the ISMC and
XU et al.: DESIGN AND IMPLEMENTATION OF INTEGRAL SLIDING-MODE CONTROL
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Fig. 9. Experimental testing results for regulation task: time responses of x and θ under the ISMC proposed in this work. The 2 WMR is placed on a flat surface. A disturbance is added to the system at t = 18 s.
the linear controller. When the linear controller is applied, the 2 WMR is stabilized at the first few seconds, however, becomes unstable in 10 seconds. By applying the ISMC, the 2 WMR is consistently stabilized. The wheels stay around the original place and the pendulum is balanced around θ = 0, which verifies the effectiveness of the ISMC in handling system uncertainties. A testing is conducted to check the robustness of the ISMC with respect to an exceptional disturbance. The experiment results are shown in Fig. 9. At t = 18 s, we push the 2 WMR to the right about 0.15 m. The 2 WMR is finally stabilized around the original position and the transit responses show small oscillations. For comparison, several other existing methods, including the fuzzy traveling and position controller (FTPC) proposed in [7], and the sliding-mode controller proposed in [19], [20], are used to control the 2 WMRs. The experimental results for the 2 WMR system under the FTPC [7] and SMC [19], [20] can be found in [10] (Figs. 11 and 12). By comparing the results, it is evident that the ISMC proposed in this work provides a better performance than the existing methods [7], [19], [20] when controlling the 2 WMR. Next, the robot is placed on an inclined surface and the slope angle ϕ is unknown. ISMC is applied with θr = 0 and the nominal linear controller is designed as in (33). For the first trial, we set γc = 0. The pendulum is balanced around θe = 0.1 rad, however, steady-state error of the wheel position exists, and e1,s |γc =0 = −0.35 m. For the second trial, we use γc = −3.5, which is computed according to (35). Experiment results for the two cases, with and without the compensation term, are shown in Fig. 10. The steady-state error for the wheel position is eliminated under ISMC with the compensation term, which is consistent with the theoretical analysis and simulation results. B. Reaching a Setpoint First, we consider the mobile robot travels on a flat surface, i.e., ϕ = 0. The planned trajectory for the wheeled mobile robot
Fig. 10. Experimental testing results for regulation task: time responses of x, θ and u under ISMC with and without the compensation term γc . The 2 WMR is placed on an inclined surface.
Fig. 11. Experimental testing results for setpoint task: time responses of x, θ and u under ISMC. The mobile robot travels on a flat surface. The pre-planned reference trajectory (5) is applied.
is the same as we used for simulation. ISMC is applied with θr = 0, γc = 0. All other controller parameters are chosen the same as for the regulation task. Experiment results are shown in Fig. 11. The 2 WMR reached the desired setpoint and stays there afterward. ISMC shows the effectiveness for setpoint control of the 2 WMR system. However, it is observed that the trajectory of the wheels x1 is not smooth enough. When the real position of the 2 WMR x1 surpasses the given reference xr , the 2 WMR would stop for a while or travel backwards, which are not the desired motions. Considering our objective for the 2 WMR is traveling forward to arrive the desired position, a
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Fig. 12. Experimental testing results for setpoint task: time responses of x, θ and u under ISMC. The mobile robot travels on a flat surface. The modified reference trajectory (36) is applied.
Fig. 13. Experimental testing results for setpoint task: time responses of x, θ and u under ISMC with γc = 0. The mobile robot travels on an inclined surface with ϕ = 2.5◦ . The pre-planned reference trajectory (5) is applied.
modified reference trajectory xr,n for the wheel position is applied [10], as the following: xr,n (t + Ts ) ⎧ ⎨ xr,n (t) + vr Ts , = x1 (t) + vr Ts , ⎩ xd ,
if x1 (t) ≤ xr,n (t) < xd if xr,n (t) < x1 (t) < xd if xr,n (t) ≥ xd or x1 (t) ≥ xd .
(36)
Another test is conducted with applying the modified reference trajectory, and the design of ISMC is the same as in the preceding test. Experiment results are shown in Fig. 12. We can see that the response is much smoother and the 2 WMR arrives the desired position in a shorter time. Next, we consider the mobile robot travels on an inclined surface and the slope angle ϕ is unknown. ISMC is applied with θr = 0 and the nominal linear controller is designed as in (33). For the first trial, we set γc = 0, the experiment results are shown in Fig. 13. The pendulum is balanced around 0.05 rad. However, steady-state error exists for the wheel position, and e1,s |γc =0 = −0.17 m. For the second trial, γc = −1.7 is computed according to (35) and applied in (33). To have a smooth response, similarly, the modified trajectory in (36) is applied. The experiment results are shown in Fig. 14. We can see the robot reaches the desired position smoothly without steady-state error. VI. C ONCLUSION In this paper, an ISMC is proposed for regulation and setpoint control of an underactuated 2 WMR system with both matched and unmatched uncertainties. The ISMC is constructed by a nominal control part and a switching term. With the switching term, the matched uncertainties are perfectly rejected. With the freedom to design a nominal control for the sliding manifold,
Fig. 14. Experimental testing results for setpoint task: time responses of x, θ and u under ISMC with γc = −1.7. The mobile robot travels on an inclined surface. The modified reference trajectory (36) is applied.
ISMC is incorporated with a linear controller. The main contribution of this paper is that for the first time ISMC is applied to a real time platform of 2 WMR. Regulation and setpoint control tasks are achieved not only when the 2 WMR is placed on a flat surface but also on an inclined surface. Strategies have been proposed to handle many practical problems regarding the implementation such as trajectory planning, eliminating the steady-state error. Simulation and experiment results are provided to validate the effectiveness and robustness of the ISMC.
XU et al.: DESIGN AND IMPLEMENTATION OF INTEGRAL SLIDING-MODE CONTROL
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Zhao-Qin Guo received the B.S. degree from Huazhong University of Science and Technology, Wuhan, China, in 2008 and the Ph.D. degree from the National University of Singapore, Singapore, in 2013. She is currently a Research Fellow in the Department of Electrical and Computer Engineering, National University of Singapore. Her research interests include control theory and applications, particularly fuzzy logic control and sliding-mode control with application to underactuated systems and power systems. Dr. Guo was a recipient of the NUS Graduate School for Integrative Sciences and Engineering Research Scholarship in 2008–2012. Tong Heng Lee received the B.A. degree (with first class honors) in engineering tripos from the University of Cambridge, Cambridge, U.K., in 1980 and the Ph.D. degree from Yale University, New Haven, CT, USA, in 1987. He is the Past Vice President for Research of the National University of Singapore, Singapore, where he is currently a Professor with the Department of Electrical and Computer Engineering and also a Professor with the Graduate School for Integrative Sciences and Engineering. He is currently an Associate Editor of Control Engineering Practice. He is the Deputy Editor-in-Chief of the IFAC journal Mechatronics. His research interests are in the areas of adaptive systems, knowledge-based control, intelligent mechatronics, and computational intelligence. Dr. Lee is currently an Associate Editor of the IEEE T RANSACTIONS ON S YSTEMS , M AN , AND C YBERNETICS and the IEEE T RANSACTIONS ON I NDUSTRIAL E LECTRONICS.
