INT J COMPUT COMMUN, ISSN 1841-9836 9(3):348-355, June, 2014. Backstepping-based Robust Control for WMR with A Boundary in Prior for the Uncertain Rolling Resistance M. Yue, S. Wang, X.L. Yang Ming Yue*, Shuang Wang, Xiaoli Yang School of Automotive Engineering Dalian University of Technology, Dalian 116024, China *Corresponding author: yueming@dlut.edu.cn Abstract: In this study, we focus on the trajectory tracking control problem of a wheeled mobile robot (WMR) in an uncertain dynamic environment. Concerning the fact that the upper boundary may be usually achieved in prior according to the physical properties of the terrain, this crucial message is utilized to construct the controllers. Firstly, a dynamic model for WMR including the rolling resistance is presented, whose state variables are longitudinal and rotational velocities, as well as the rotational angle of the mobile platform. Secondly, with the aid of backstepping technique, the robust controllers based on the upper boundary are proposed and the globally asymptotic stability of the closed-loop system is proven by the Lyapunov theory in the following. Lastly, a saturation function is applied to replace the signum function, by which the inherent chattering can be suppressed greatly. Numerical simulation results demonstrate that the proposed controllers with upper bound in prior possess robustness characteristics which yields potentially valuable applications for the mobile robot, especially in the unstructured environment. Keywords: Wheeled Mobile Robot (WMR), boundary estimation, uncertain rolling resistance. 1 Introduction In practice, the physical properties of the wheeled mobile robot (WMR) within an unstruc- tured environment are seldom considered in lots of trajectory tracking control issues. Among these physical properties, the rolling resistance acting on the robot wheels is such a crucial physi- cal behavior that can effect the control performances greatly. Particulary, in some rough terrain, such as loose soil and sand conditions, it is difficult to obtain high-performance of trajectory tracking without considering the influences of the rolling resistance. Therefore, when the WMR explores in an outdoor environment and even planetary surface, the rolling resistance may not be neglected for designing the system controllers [1]. Many efforts are devoted to the rolling resistance acting on the WMR’s wheels recently. Most References (see [2,3]) had investigated the interactions of soil and wheels and then some new me- chanical structures had also been reported to enhance the contact between the wheels and terrain. However, there are only a few investigations to deal with rolling resistance problem by control methods. After analysing of the terrain properties, we notice that the boundary (especially for the upper bound) of off-road rolling resistance can be calculated in advance according to the geo- logic parameters when the terrain of the exploration is determined. For practical implementation, the rolling resistance appears drastically changing, which is similar to the disturbances to the mobile benchmark, but with the aid of the boundary message in prior, the robust controller for WMR can be designed to make it possible to against the uncertain dynamics directly. It should be mentioned that, compared with modifying the system mechanical structure, the adopted con- trol approach yields much more flexibility for the mobile robot. Certainly, we are also aware of adaptive scheme to update the upper bound, but the control algorithm with adaptive scheme Copyright © 2006-2014 by CCC Publications Backstepping-based Robust Control for WMR with A Boundary in Prior for the Uncertain Rolling Resistance 349 may make the regulation process more elastic and cause further time-delay usually [4, 5]. To overcome this drawback, a powerful robust control approach is determine to employ to deal with the uncertain problem of the unavoidable rolling resistance. On another hand, if the rolling resistance is concerned, the dynamics of the WMR must be considered. For the past decades, most efforts have been done to the WMR with nonholonomic constraint on kinematic model, all of which the system velocities were assumed to be the control inputs [6, 7]. But unfortunately, there are only a few researches address the dynamic behaviors of nonholonomic system, where control inputs are transformed to be the system actuators, i.e., in most cases the driving torques of motors [8–10]. Since the rolling resistance is described in torque form when it takes place in practice, the dynamic properties must be considered when designing the system controllers to overcome this kind of uncertain dynamics. With a widely survey of the current dynamic models for the WMR, we observe that a dynamic model proposed by Watanabe et.al [11] is so succinct and effective to describe the relationship between the rolling resistance and robot posture on the level of dynamics. Then, we utilize this dynamic model to establish the control system in this investigation. Particularly, it should be noted that the rolling resistance is somewhat difference from the external disturbances and un- modeled dynamics, because the rolling resistance lies in the movement process around a constant value; while the external disturbances and unmodeled dynamics possess white noise property which is around zero. The objective of this study is to address the rolling resistance behaviors of the WMR in detail, and derives the dynamic model to discuss the influence of rolling resistance. Based on addressed dynamic model, a simple and effective approach to stabilize the trajectory tracking system via backstepping techniques is developed. In order to illustrate the efficacy of the control approach, numerical simulations for a practical WMR have been performed. 2 WMR dynamics The typical WMR can be described in Fig.1. It is assumed that the mobile robot is driven by two independent wheels as well as a passive auxiliary wheel is adopted to support the workbench. To begin with, some notations is introduced to help the controlling system design, as follows: l is the distance between the driving wheels and the symmetry axis; d is the distance between point C(x, y) and the mass center of the robot P(xc, yc), which is assumed to be on the symmetry axis; r is the radius of the driving wheels; mp and mw are the masses of vehicle body and wheel; Ic is the inertia moment of vehicle body w.r.t. vertical axis through C; Iw and Im are the inertia moments for the wheel w.r.t. wheel axis and diameter, respectively. According to the parallel axis theorem, the equivalent mass m = mp + 2mw and rotation inertia I = 2Im + 2mwl2 + mcd2 + Ic can be introduced for simply. The robot position can be described by the coordinates (x, y), which is the midpoint C of the axis of two robot wheels, and the orientation angle ϕ, which is heading angle of body coordinate with respect to fixed frame (see Fig.1). With the hypothesis of pure rolling and non slipping condition, the nonholonomic constraint, i.e., ẋ sin ϕ − ẏ cos ϕ = 0, hold throughout the movements. Let v and w be represent the linear and angular velocities of the mobile robot, and τr and τl express the driving torque of the right and left wheels, respectively. Through the analysis of the forces acting on the mobile robot, we can obtain that Iϕ̈ = τr r l − τl r l (1) mv̇ = τr r + τl r (2) 350 M. Yue, S. Wang, X.L. Yang x y O d auxiliary wheel symmetry axis Figure 1: A wheeled mobile robot τi r θ Figure 2: Rolling resistance generation While rolling on the ground, especially operating on the soft soil or sandy terrain, the robot will suffer unavoidable rolling resistance, as shown in Fig.2. Let the notation τ̃d represent this rolling resistance, and the dynamic behaviors of the wheel can be governed by Iwθ̈i + cθ̇i = kui − τi − τ̃di (3) where τi is the driving input of the wheel (i expresses r or l for left or right wheels, the same hereinafter); c is the viscous friction coefficient and θ is the rolling angle of the wheel; k is driving gain between the motor’s voltage and its output torque; ui is the excited voltage signals for the motor installed in the relative wheel. Let vr, vl denote the right and left linear velocity of the wheel center, and we can further have vr = rθ̇r = v + lϕ̇ and vl = rθ̇l = v − lϕ̇. Then, the relationship between linear velocity of the robot’s platform and angular velocity of the robot wheels can be formulated by r(θ̇r + θ̇l) = 2v and r(θ̇r − θ̇l) = 2lϕ̇. With the analysis, we can ultimately obtain the dynamics described by the linear velocity v and orientation angle ϕ as follows: v̇ = − 2cv mr2 + 2Iw + kr mr2 + 2Iw (ur + ul) − r mr2 + 2Iw (τ̃dr + τ̃dl) (4) ϕ̈ = − 2cl2 Ir2 + 2Iwl2 ϕ̇ + krl Ir2 + 2Iwl2 (τ̃dr − τ̃dl) (5) If the system state variables are chosen as x = [v ϕ ϕ̇]T , the driving input is selected as u = [ur ul] T , the disturbance vector is defined as τ̃d = [τ̃dr τ̃dl]T , and the output variables are y = [v ϕ]T , a new dynamic model for control system design can be rewritten by:{ ẋ = Ax + Bu + Dτ̃d y = Cx (6) with A =   a1 0 0 0 0 1 0 0 a2   , B =   b1 b1 0 0 b2 −b2   , D =   d1 d1 0 0 d2 −d2   , C = [ 1 0 0 0 1 0 ] , where a1 = − 2c mr2 + 2Iw , a2 = − 2cl2 Ir2 + 2Iwl2 , b1 = kr mr2 + 2Iw , b2 = krl Ir2 + 2Iwl2 , d1 = − r mr2 + 2Iw , d2 = − rl Ir2 + 2Iwl2 . Backstepping-based Robust Control for WMR with A Boundary in Prior for the Uncertain Rolling Resistance 351 3 Control system design 3.1 System decoupling Noticing that the system (6) is a coupled system, for the convenience of designing the system controller, the proposed dynamic model should be decoupled in the first place. To achieve this objective, a new control input vector should be introduced as follow:[ ur ul ] = [ 1 −1 0 1 ] [ u1 u2 ] (7) With these modified inputs u1 and u2, the system (6) is transformed into two independent subsystems: v̇ = a1v + b1u1 + τ̃dv (8) ẇ = a2w + b2u1 − 2b2u2 + τ̃dw (9) where τ̃dv = d1(τ̃dr + τ̃dl) and τ̃dw = d2(τ̃dr + τ̃dl), which represent disturbances for the two independent subsystems. Supposing the robot moves on a special terrain ground, although the rolling resistance is in a changeable state, the maximum amplitude for τ̃dr and τ̃dl could be estimated in advance from the terrain property. Let the notation τ̄dr and τ̄dl represent the upper bound values of τ̃dr and τ̃dl respectively, and then it can be achieved that τ̄dv = max |d1(τ̃dr + τ̃dl)| (10) τ̄dw = max |d2(τ̃dr + τ̃dl)| (11) 3.2 Control scheme design Considering the decoupled form of the system (8) and (9), a backstepping technique can be applied to derive the robot controller due to the high dimension feature of the system. The process of control design procedures may be divided into two steps which is given as bellows. Step 1. Linear velocity control After the trajectory tracking system has been decoupled, the control algorithms for linear and angular velocities can be derived stage by stage. Here, the control input u1 is used to control the linear velocity, such that the robot can track an desired trajectory from an arbitrary original point with the required performances. Suppose that the desired linear velocity vd, and the tracking error may be expressed by ve = vd − v. Then, a theorem can be given as follow: Theorem 1. For the linear velocity system (4) with the bounded linear velocity, the tracking error ve will globally converge to zero, i.e., lim t→∞ ve(t) = 0, if the control law is given by u1 = 1 b1 [v̇d − a1v + c1ve + sgn(ve)τ̄dv] (12) where c1 is a positive constant and sgn(·) is a signum function. Proof: Consider a candidate Lyapunov function as V1 = 1 2 v2e (13) 352 M. Yue, S. Wang, X.L. Yang Differentiating V1 with respect to time yields V̇1 = vev̇e = ve(v̇d − a1v − b1u1 − τ̃dv) (14) Substituting (12) into (14), it can be ultimately obtained V̇1 = −c1v2e ≤ 0. Obviously, by Lyapunov stability theorem, the propose Theorem 1 is proved. 2 Step 2. Angular velocity control Let ϕd represents the desired orientation angle of the robot to be tracked. Then, the tracking error of the orientation angle can be defined as ϕe = ϕd −ϕ and ϕ̇e = ϕ̇d −ϕ̇. In terms of angular velocity tracking control, it is to design the control input u2 such that the ϕe and ϕ̇e converge to zero as t → ∞. In order to realize this purpose, two virtual control inputs are introduced, which are defined as z1 = ϕe and z2 = ϕ̇e + c2ϕe where c2 is an arbitrary positive constant. Similar to the linear velocity control, based on the above definitions, a Theorem could be given as Theorem 2. For the angular velocity system (5) with the bounded angular velocity, the tracking error ϕe and ϕ̇e will both globally converge to zero simultaneously , i.e., lim t→∞ ||ϕe ϕ̇e|| = 0, if the control law is given by u2 = 1 2b2 [a2w + b2u1 − ϕ̈d − c2z1 + c3z2 + sgn(z2)τ̄dw] (15) where c3 is a positive constant and sgn(·) is a signum function. Proof: Consider a candidate Lyapunov function as V2 = 1 2 z21 (16) Differentiating V2 with respect to time achieves V̇2 = z1ż1 = z1(ϕ̇ − ϕ̇d) = z1z2 − c2z21 (17) Notice that only Lyapunov function V2 can not guarantee the global stability of the system. In order to stabilize the system, another virtual input z2 is designed such that the entire track- ing errors converge to zero. Hence, another candidate Lyapunov function constructed by z2 is established for this purpose. The new candidate Lyapunov function is put forward as V = 1 2 z21 + 1 2 z22 (18) Differentiating V with respect to time obtains V̇ = V̇2 + z2ż2 = z1z2 − c2z21 + z2(a2w + b2u1 − 2b2u2 + c2ż1 − ϕ̈d) (19) Substituting (12) and (15) into (19), one can obtain that V̇ = −c2z21 − c3z 2 2 ≤ 0. Obviously, according to Lyapunov stability theorem, the Theorem is proved. 2 Since the systems (8) and (9) are decoupled, the proposed controller (12) and (15) may guarantee the stabilization of the linear and angular velocity subsystems overall. To sum up, we can summarize the investigation results and ultimately give a theorem as follow: Theorem 3. For the trajectory tracking system (4) and (5), the tracking errors ve, ϕe and ϕ̇e will all globally converge to zero, i.e., lim t→∞ ||ve ϕe ϕ̇e|| = 0, with the controller (12) and (15). Backstepping-based Robust Control for WMR with A Boundary in Prior for the Uncertain Rolling Resistance 353 The proof can be easily achieved according to the above analysis. From the controller designing process, it can be observed that the candidate Lyapunov func- tions are established step by step, which make the control design process be simple and con- venience. This is a backstepping approach which is usually applied for controller design of a sophisticated nonlinear systems. Meanwhile, it should be emphasized that there are inherent chattering caused by signum function in the proposed controllers. To overcome this drawback, one can replace signum function sgn(·) by saturation functions sat(·) to suppress this behaviors, one of which can be given by: sat(χ) = χ |χ| + ϵ (20) where ϵ is a positive constant and χ represents arbitrary variable. 4 Simulation results In this section we perform numerical simulations to verify the effectiveness of the proposed controllers. The simulated parameters of the mechanical structure of the WMR are supposed to be mc = 14kg, mw = 3kg, Ic = 0.9kgm2, Im = 0.05kgm2, Iw = 0.015kPgm2, l = 0.3m, d = 0.05m, r = 0.1m, k = 0.95, and c = 0.01. From the required performance of the trajectory tracking, the control parameters as selected as follows: c1 = c2 = c3 = 10 and ϵ = 0.002. Suppose the mobile robot starts from initial point [x y ϕ]T = [0 0 0]T ; meanwhile, the desired trajectory of linear and angular velocity are hypothesized to be vd = 1m/s, wd = sin t with the initial state vector x0 = [v0 ϕ0 w0]T = [0.5 0.2 0]T . Moreover, the rolling resistance acting on the left and right wheels are supposed to be a worse situation in simulation than that in practice to verify the effectiveness of the control algorithms. So a synthesis signal consisted of two sinusoid signals with different frequency is introduced to imitate the rolling resistance. Here, assume that the rolling resistances are same on the left and right wheels, i.e., τ̃dr = τ̃dl = sin wdt +sin 2wdt where wd denotes the resistance frequency. According to practical experiment, the τ̄dv and τ̄dw can be set as the value of 0.75Nm and 3Nm respectively in this simulation circumstance. In addition, the saturation should be considered for actual driving motor, so a maximum value 10Nm is introduced to restrict the control outputs, i.e., |τr| ≤ 10Nm and |τl| ≤ 10Nm. 0 2 4 6 8 10 12 14 16 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 x /m y / m Figure 3: Actual trajectory on the plane υ e / m /s With resistant consideration without resistant estimation -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 6 8 10 12 14 16 18 20 t /s Figure 4: Tracking error for υe Under the proposed simulation circumstance, the actual tracking process for the desired trajectory is plotted in Fig.3. It indicates the proposed control method can make the mobile robot to track the desired trajectory with satisfactory tracking performances. Besides, we simulate two cases, that is, with and without estimation, to illustrate the signification of control with and 354 M. Yue, S. Wang, X.L. Yang without upper bound information. As shown in Fig.4, the tracking error relative to the linear velocity without boundary is described as the dashed line, as well as the case with boundary is plotted by the dashed line. The results state the tracking errors become much smaller (reduced to 20% of the amplitude) when τ̄dv and τ̄dw are adopted. Particularly, owing to the sat(·), the inherent chattering is also suppressed. Lower chattering may be achieved with smaller ϵ, but it will cause more violent changes for the control inputs. Furthermore, to exhibit the better tracking performance with the upper bound, the tracking errors of ϕ̇e and ϕe are also addressed in Fig.5 and Fig.6, respectively. It is observed that the amplitudes of ϕe and we are reduced to 21% and 18% nearly compared with the case without upper bound estimation, which yields that the derived controller can provide better performances for tracking system and the robustness is enhanced in the following. φ e / r a d 0 2 4 6 8 10 12 14 16 18 20 -0.05 0 0.05 0.1 0.15 0.2 t /s With resistant consideration without resistant estimation Figure 5: Tracking errors for ϕe 0 2 4 6 8 10 12 14 16 18 20 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 t /s ω e / r a d /s With resistant consideration without resistant estimation Figure 6: Tracking errors for ωe 5 Conclusions The trajectory tracking issues were discussed in this study, and the main contribution can be summarized as follows: 1) a decoupled dynamic model was used to describe the complicated nonlinear behaviors of WMR where the rolling resistances are concerned. The decoupled model consisted of two parts: a first-order model for linear velocity and a second-order model for orientation angle of the robot, which not only kinematic postures but also dynamics behaviors were all included; 2) based on the fact that the upper bound of rolling resistance which can be achieved in prior, a robust controller was developed to obtain the better tracking performances. The backstepping methodology was employed to derive the system controller considering the higher dimensional property of the tracking system; and 3) a saturation function was introduced to alleviate the inherent chattering caused by signum function. Such treatment was simply but effective for a practical WMR system. Acknowledgement This research was supported by grants from the National Natural Science Foundation of China (No. 61175101), and the Fundamental Research Funds for the Central Universities (No. DUT12LK45). 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