Robust control of a wheeled mobile robot by voltage control strategy

Transcription

1 Nonlinear Dyn (215) 79: DOI 1.17/s ORIGINAL PAPER Robust control of a wheeled mobile robot by voltage control strategy Mohammad Mehdi Fateh Aliasghar Arab Received: 29 June 213 / Accepted: 15 August 214 / Published online: 6 September 214 Springer Science+Business Media Dordrecht 214 Abstract A nonholonomic wheeled mobile robot is an uncertain nonlinear system. To guarantee stability and overcome uncertainties, the torue-based control approaches become computationally extensive. To simplify the control problem, this paper develops a novel robust control approach based on the voltage control strategy. To improve the precision, the dynamics of motors are taken into account. In addition, a novel statespace model for the electrically driven nonholonomic wheeled mobile robot in the workspace is presented. The most important advantage of the proposed control law is that it is free from the robot dynamics, thereby the controller is simple, fast response, and robust with ignorable tracking error. The control approach is verified by stability analysis. Simulations and experimental results show the effectiveness of the proposed control applied to a nonholonomic wheeled mobile robot driven by permanent magnet DC motors. A comparison with an adaptive feedback linearizing control approach and the control confirms the superiority of the proposed approach in terms of precision, simplicity of design, and computations. Keywords Wheeled mobile robot Robust control Voltage control strategy Uncertainty M. M. Fateh (B) A. Arab Department of Electrical and Robotic Engineering, Shahrood University, Shahrood, Iran mmfateh@shahroodut.ac.ir 1 Introduction There is a great attention to employ mobile robots in several applications. Mobile robots play an interesting role in hazardous environments would be unsafe for human. The wheeled nonholonomic robots are into a class of mobile robots that have operated as automated guided vehicles, rescue robots, and electrically wheelchairs. There has been an increasing interest to control the nonholonomic wheeled robots in the presence of uncertainties. The uncertainties may include model uncertainty containing the kinematics and dynamics, external disturbances raised from unknown environment, disturbed reference trajectories, and noises. To overcome uncertainties, valuable classic control approaches have been presented. For example, one can address H control [1], sliding mode control [2], adaptive control [3] adaptive sliding mode control [4], adaptive feedback linearization control [5], robust control [6], and robust adaptive control [7]. Alternatively, intelligent control has become an interesting topic for controlling complex systems by function approximation or obtaining rules from the experts knowledge using some powerful tools such as fuzzy logic, neural networks, and intelligent algorithms. There are valuable research works in this field such as adaptive neural control [8], adaptive fuzzy control [9], fuzzy neural control [1], fuzzy control design using particle swarm optimization [11], and fuzzy control design using genetic algorithm [12] to name a few. Downloaded from

2 336 M. M. Fateh, A. Arab The tracking control is the goal of the presented control approaches for mobile robots. An approach to couple path planning and control for mobile robot navigation in a hybrid control framework was presented [13]. An experimental study of dynamic-based trajectory tracking of a four-wheeled autonomous ground vehicle was reported [14]. A main trend in the control of wheeled robots is the use of two main control loops namely the kinematic control and the torue control [8]. Many control approaches have used the mentioned structure to control a wheeled robot. The purpose of kinematic controller was to produce velocity output for the robot to make the tracking error converge to zero. A torue controller is designed based on the system dynamics such that the velocity of the mobile robot converges to the generated desired velocity by the kinematic controller. The early approach proposed for the kinematic controller is the backstepping control [15]. To improve its performance, adaptive kinematic controller [16], fuzzy control [17], and neural network [18] were developed. The dynamic controller includes many control approaches particularly robust control to overcome uncertainties. Considering the actuator dynamics and the constraints for voltage and current of motors as practical issues is reuired in the control of a nonholonomic mobile robot. A trajectory tracking control for a nonholonomic mobile robot by the integration of a kinematic controller and neural dynamic controller was investigated, the wheel actuator dynamics is integrated with mobile robot dynamics and kinematics so that the actuator input voltages are the control inputs [19]. An adaptive nonlinear model-based prediction control was reported for trajectory tracking of wheeled mobilerobots in which robot dynamics including the actuator dynamics are subject to various uncertainties [2]. Near minimum-time direct voltage control algorithms were proposed for wheeled mobile robots with current and voltage constraints so that voltages to the motors are controlled directly without velocity/torueservo modules, while satisfying the current constraints [21]. In the mentioned control methods that are based on the torue control strategy, the voltage or current of motors must be controlled so as to provide a desired torue as a reference torue for the torue control loop that drives the robot. Actually, two control loops are used, one for producing the reference torue and another for tracking the reference signal. In the proposed control loop in this paper, the reference torue is not provided, thus one control loop is omitted. Actually, the motor voltages as the inputs of the integrated system are directly controlled as the control laws. So far, control of wheeled robots has used the torue control; however, the torue control approaches would have some shortcomings. This type of control is nonlinear, coupled, and computationally extensive due to the characteristics of the robot dynamics. In addition, the dynamics of the robot s motors are excluded from the control problem, as they become dominant in high-velocity and high-precise applications. It has been shown that the dynamics of actuators should be taken into consideration to ensure a reliable and meaningful control performance [22]. Furthermore, the torue commands are assumed as the inputs of the robotic system; however, the robot is driven by the motors with voltage inputs in practice. Therefore, giving the voltage commands as the control inputs is realized to be more practical than the torue commands. The main contribution of this paper is to employ a voltage control loop in replace of the torue control loop. This idea simplifies the control problem and improves the precision by taking the dynamics of motors into account. A novel robust control law is then proposed, which can efficiently overcome uncertainties since it is free from the robot dynamics. This advantage makes it simple, fast response, and robust with ignorable tracking error. In addition, a novel state-space model of nonholonomic robot system in the workspace is presented. The proposed control is compared with an adaptive feedback linearizing control approach [5] and traditional control through simulations and experimental results. Voltage control strategy has shown superiority to torue control strategy in the control of robot manipulators [23]. The torue control strategy considers the joint torues as the control inputs, as the voltage control strategy focuses on the motor voltages as the control inputs of the robotic system. As a result, the voltage control strategy pays attention to the model of motors which is much simpler than the model of robot manipulator used for the torue control strategy. Recently, various control approaches in voltage control strategy such as robust fuzzy control [24], robust control of flexible-joint robots [25], and robust control by estimation of uncertainty [26] were proposed for robot manipulators. In this paper, the voltage control is developed for the nonholonomic wheeled mobile robot in the

3 Robust control of a wheeled mobile robot 337 presence of uncertainties. The novel control design is called as the robust voltage control (RVC). The rest of this paper is organized as follows. Section 2 presents the dynamics of an electrically driven nonholonomic wheeled mobile robot. Section 3 develops the voltage control and formulates the robust control approach. Section 4 presents the stability analysis and performance evaluation. Section 5 illustrates the simulation results, and finally, the Sect. 6 concludes the paper. 2 Modeling An electrically driven nonholonomic mobile robot can be modeled via kinematics, dynamics, and actuator modeling. Kinematics deals with the geometric relationships without considering the role of forces. Dynamics studies the motion caused by forces. Actuator modeling relates the control input to the torue input of the robot. Consider a nonholonomic mobile robot consists of two active wheels or joints. Each wheel is driven by a permanent magnet DC motor through a coupling as shown in Fig. 1. OXY is the reference coordinate system, and O r X r Y r is a coordinate system attached to the robot its origin is located in the middle between the right and left driving wheels. The position C is the center of mass of the mobile robot. The length d is the distance from O r to C. The length 2b is the distance between the two driving wheels, and r w is the radius of the wheel. The position of the robot in the reference frame is represented by = [ xyθ ] T in which [xy] is the position of Or in the reference frame, and θ is the heading direction taken counterclockwise from the axis X r to the axis of X. The forward kinematics f k can be represented by = f k (ϕ) (1) ϕ =[ϕ r ϕ l ] T represents the angular position of right and left wheels, ϕ r and ϕ l, respectively. By taking the time derivative of (1), the velocity in the reference frame is obtained as = J() ϕ (2) J () R 2 3 is the Jacobian matrix expressed [3] as J() = r w 2 cosθ cosθ sinθ sinθ (3) 1/b /b The dynamics of robot in the workspace is described as [3] M () + C (, ) + F ( ) + G () +τ d + A T () λ = B () τ (4) M() R 3 3 denotes the inertia matrix which is a symmetric and positive definitive matrix. C (, ) R 3 is the vector of centrifugal and Coriolis torues, G() R 3 is the vector of gravitational torues, F( ) R 3 is the vector of frictions, and τ d R 3 is the disturbance. B () R 3 2 is a matrix which transforms the input torues from the joint space to workspace. A() R 3 is the vector associated with constraint, and λ R is the constraint force. Consider the velocity vector denoted as P = [ ] T υω υ is the linear velocity and ω is the angular velocity in the reference frame. Thus, P = T ϕ ϕ (5) T ϕ = r w 2 [ b 1 b ] (6) Fig. 1 Schematic top view of the nonholonomic two-wheeled mobile robot Property 1 The matrix M () is symmetric and positive definite.

4 338 M. M. Fateh, A. Arab Property 2 The matrix Ṁ () 2C (, ) is skew symmetric matrix. Assumption 1 It is assumed that the wheels of the robot do not slide. This is expressed by the nonholonomic constraint [8] ẋsinθ ẏcosθ = (7) Assumption 2 All kinematic euality constraints are assumed to be independent of time such that A() = (8) From (2), ϕ can be calculated as ϕ = J() (9) J () is the pseudoinverse of Jacobian matrix defined as [ ] J () = J () T J () J () T = 1 [ ] cosθ sinθ b (1) r w cosθ sinθ b By taking the time derivative of (9), one can calculate ϕ = J () +J() (11) Using (9), J() is computed as J() = θ r w [ ] sinθ cosθ sinθ cosθ (12) The electric motors provide the joint torues via the dynamics J m ϕ m + B m ϕ m + r g τ R = τ m (13) τ m R 2 is the motor torues and τ R R 2 is the robot torue; J m, B m, and r g are the 2 2 diagonal matrices for the motor coefficients, namely the inertia, damping, and reduction gear, respectively. The motor velocities ϕ m R 2 and wheel velocities ϕ are related through the gears as ϕ m = r g ϕ (14) Substituting (9) and (11) into(13), and using (14), yields to J m rg J () +[J m rg J () + B m rg J () ] +r g τ R = τ m (15) In order to obtain the motor voltages as the inputs of system, we consider the electrical euation of geared permanent magnet DC motors in the matrix form L a İ a + R a I a + K b ϕ m + ζ = U (16) U R 2 is the motor voltages and I a R 2 is the vector of motor currents. R a, L a, K b R 2 2 represent the diagonal matrices for the coefficients of armature resistance, inductance, and back-emf constant, respectively, and ζ is external disturbance. L a İ a + R a I a + K b r g J () + ζ = U (17) The motor torue vector τ m is produced by the motor current vector, τ m = K m I a. (18) K m R 2 2 is a diagonal matrix of the torue constants. Using (5) (18), the state-space model of the electrically driven mobile robot is then derived as ż = f s (z) + bu (19) b = and [ 6 2 I 2 2 ], z = and f s (z) = I a f s1 f s2 f s3 f s1 = z 2 ( f s2 = J m rg J (z 1) + r g B M (z 1 )). ( ( K m z 3 J m rg J(z 1 ) + B m rg J (z 1) ) ( )) +r g B C (z 1, z 2 ) z 2 r g B F (z 2 )+G (z 1 )+τ d +A T (z 1 ) λ ( ) f s3 = La K b rg J (z 1) z 2 + R a z 3 + ζ The state-space euation (19) shows a highly coupled nonlinear large multivariable system. Complexity of the model has been a serious challenge in the literature of mobile robot modeling and control.

5 Robust control of a wheeled mobile robot Proposed control laws The proposed control scheme includes two interior control loops. The inner loop controls the voltages of motors while the outer loop controls the motion of the mobile robot. Motion control is not straightforward because the two-wheeled mobile robot is nonholonomic and MIMO systems [27]. The structure of the proposed voltage control is depicted in Fig Motion control The backstepping controller is a commonly used kinematic level controller for the motion control loop in the nonholonomic wheeled mobile robot. The objective of the motion controller was to follow the desired trajectory d d = [x d y d θ d ] T (2) The position error in the robot coordinate frame as shown in Fig. 2 can be expressed as e p = [ e x e y e θ ] T (21) According to Fig. 2, one can represent e p = T ( d ) (22) cos θ sin θ T = sin θ cos θ (23) 1 From (9) and (6), it can be calculated that = T p P (24) cos θ T p = sin θ 1 (25) Let us define a desired velocity vector in the reference coordinate frame P d = [ υ d ω d ] T (26) Based on the backstepping method, the motion controller [18]isgivenby P c = υc ω c υ = d cos(e θ ) + k x e x ω d + k y υ d e y + k θ υ d sin(e θ ) (27) and k x, k y, and k θ are positive and selected as control design parameters. As a result of applying control law (27) [16], if P P c, then e p. Using(24), one can compute c as y d Y y e x e θ θ θ d c = T p P c (28) Considering (27), if e p, then from (22) d. Furthermore, from this reasoning, it can be concluded that a bounded c implies a bounded d.this leads us to design the inner control loop for tracking c. 3.2 Robust voltage control O x Fig. 2 Viewpoint of error coordinate x d X The proposed control approach is based on the voltage control strategy [18] using the model of motor which is much simpler than the model of mobile robot. Therefore, the control problem is taken from the robot control to the motor control. As a result, the control law will be free from the robots dynamics.

6 34 M. M. Fateh, A. Arab Fig. 3 Structure of proposed controller K b rg J() can be represented as K b r g J() = J e K (29) k b r g r w k K = b r g r w bk b r g r w, J e = [ ] cos θ sin θ 1 cos θ sin θ (3) Using (29), the motor euation (17) can be rewritten as ˆR a I a + J e + μ = U (31) ˆR a and are the nominal values of R a and K, respectively, and μ is called the lumped uncertainty expressed by μ = L a İ a + (R a ˆR a )I a + J e (K ) + ζ (32) In fact, the lumped uncertainty includes the parametric uncertainty (R a ˆR a )I a + J e (K ), unmodeled dynamics L a İ a, and the external disturbance ζ. A robust control law is proposed as ˆR a I a + J e ( c + K p ( c ) ) + U r = U (33) K p is the controller gain matrix and U r is a robust controller to overcome the lumped uncertainty, and let us define e c = c (34) Applying (33) to the system (31), one can obtain the closed-loop system ė c + K p e c = J e [μ U r] (35) Assume that the lumped uncertainty is bounded as μ ρ(t) (36) Using the Cauchy Schwarz ineuality, μ R a ˆR a. Ia + J e. K. + L a İa + ζ (37) Assume that R a ˆR a γ1, K γ2, L a γ 3, ζ γ 4 (38) γ 1,γ 2,γ 3, and γ 4 are positive known constants. Using (38) and J e = 2, the upper bound of uncertainty, ρ(t), can be calculated as ρ(t) = γ 1 I a + γ 2 + γ 3 İa + γ4 (39) In order to design a robust controller, we suggest a positive definite function of the form V (e c ) =.5e T c e c (4) V (e c )>fore c = and V () =. Wetake the time derivative of V (e) to get V (e c ) = e T c ėc (41)

7 Robust control of a wheeled mobile robot 341 Substituting (35) forė c into (41) yields V (e c ) = e T c ( ) k p e c + J e (μ U r) (42) Since e T c k pe c < ife c =, in order to satisfy V (e c )<, it is sufficient that e T c J e (μ U r) (43) Hence, e T c Since e T c J e μ et c J e μ e T c J e U r (44) J e. μ <ρ(t) e T c (45) By using (45) to satisfy ineuality (44), one can propose e T c J e U r = ρ(t) Thus, ( U r = ρ(t).sgn ec T e T c ) J e J e Since J e =, wehaveec T and ec T J e (46) (47) J e = if e T c =, J e = if ec T =. On the other hand, ( ) sgn ec T J e is not defined for ec T J e =. Therefore, U r is suggested to be U r = { ( ) ρ(t).sgn ec T J e if e c = if e c = (48) In the case of e c = as stated by (4), V (e c ) =. Therefore, the closed-loop system is stable regardless of the control law U r. Thus, one can suggest U r = if e c =. Considering (33) and (48), the final control law is given by ˆR a I a +J e ( c +K p ( c ) ) +ρ(t)(ec T U= J e)/ ec T J e e c = ˆR a I a +J e ( c +K p ( c ) ) e c = (49) It can be concluded that the error vector e c in the closedloop system (35) approaches zero using the control law (49) since V (e c ) is negative. The chattering problem will be reduced if control law (49) is modified as ˆR a I a +J e ( c +K p ( c ) ) +ρ(t)(e T c J e )/ ec T U= ˆR a I a +J e ( c +K p ( c ) ) J e e c >ε e c ε (5) The control law depends only on the nominal parameters of motors and the kinematic parameters of robot. It is free from the dynamics of robot. 4 Stability analysis In order to analyze the stability, the boundedness of all states denoted by z in system (19) under control law (5) must be verified. Consider the positive definite function (4), V (e c ) =.5ec T e c. As proven above, V (e c ) < ife c =. As a result, e c converges to zero. We have V (e c ) = ec T ėc. Conseuently, V (e c ) converges to zero if ė c be bounded. The following proof is given to verify the boundedness of ė c. Proof Considering (35), let us define f(t) = J e [μ U r] for t T (51) T is the operating time of the desired trajectory. Then, system (35) is represented as ė c + K p e c = f(t) (52) Since K p is diagonal, for the ith element of vector e c, one can write ė ci + K pi e ci = f i (t) (53) For a given period T, the generalized Fourier series for f i (t) can be expressed as f i (t)= 1 ( nπt 2 a i+ a ni cos T a i = 2 T a ni = 2 T b ni = 2 T T T T n=1 ) + ( ) nπt b ni sin T n=1 (54) f i (t)dt (55) ( ) nπt f i (t)cos dt (56) T ( ) nπt f i (t)sin dt (57) T

8 342 M. M. Fateh, A. Arab The response of system to the sinusoid part of f i (t) is a sinusoid function, and the response to the constant part of f i (t) is an exponential function. Thus, if each of a i, a ni or b ni be unbounded, the response will be unbounded. However, the response of linear system (53) to the input f i (t) is unbounded if f i (t) is unbounded. However, it is proven that e c as the response of system is bounded. Thus, f i (t) must be bounded. From (52), we have ė c = K p e c + f(t).the boundedness of e c and f(t) implies the boundedness of ė c. As a result of boundedness of e c, d is bounded. Since d is bounded, then will be bounded. Since ė c is bounded, c is bounded. The boundedness of ė and c yields to the boundedness of. Recalling (51), f(t) = J e [μ U r ], f(t), J e and μ are bounded. Thus, U r is bounded. Note that μ is bounded as stated by (37), is constant, and J e is bounded since J e = [ J e () T J e () ] Je () T and ( J e () ) = 2. As given by (48), U r = ρ(t).sgn ec T J e if e c =. Thus, U r = ρ. The boundedness of U r implies the boundedness of ρ. Then, considering ρ in (39) verifies the boundedness of I a and İ a. In summary, it can be concluded that system is stable and the tracking error asymptotically converges to zero. 5 Simulation and experimental results The proposed control law (RVC) given by (5) is applied to the electrically driven two-wheeled mobile robot with a symbolic representation in Fig. 1. The value of ε in the control law is set to.1. The dynamics of robot in (4) has the details [16]: m c m c dsinθ M() = m c m c dsinθ m c dsinθ m c dsinθ I B() T = 1 [ ] cosθ sinθ b, r w cosθ sinθ b m c d θcosθ C(, ) = m c d θcosθ G() = A() = [ sin(θ) cos(θ) ] λ = m c (ẋ cos θ +ẏ sin θ) θ (58) Table 1 Dynamical parameters of the robot b (m) d (m) I ( kgm 2) m c (kg) r w (m) Table 2 Parameters of the motors r g R a ( ) L a (mh) K m (Nm/A) K b K m The dynamical parameters are given in Table 1.The frictional torue vector is given by F( ) = 5 +.5sign( ) (59) The motor parameters are given in Table 2. The maximum voltage of motors is set to U max =24 V. The nominal parameters are assumed to be 8 % of the real values. The external disturbance ζ = [ ] T ζ 1 ζ 2 is given as an example by { ζ1 = 2 if 1 < t < 4 otherwise (6) ζ 2 = 2 if 1 < t < 4 otherwise The values of γ 1 =.4,γ 2 =.5,γ 3 =, and γ 4 = 2 are selected to cover uncertainties and disturbance. In order to show the advantages of proposed method, a comparison with an adaptive feedback linearizing control approach [5] is presented, as well. 5.1 Simulation 1 The proposed control (RVC) is compared with a traditional control and an adaptive feedback linearizing control (AFL) [5] in tracking a circular path. The control laws are simulated to evaluate the control performances. The circular path is shown in Fig. 4 and expressed by d = [ 2sin(ω d t) 2 2 cos(ω d t) t ω ddt + θ() ] T t 6s (61) ω d =.1 rad/s and the operating time for the desired path is 6 s. The desired trajectory is sufficiently smooth such that all its derivatives up to the reuired order are bounded. The design parameters of the RVC are set to K p = 1, the kinematic control gains,

9 Robust control of a wheeled mobile robot 343 Y (m) RVC AFL Desired path X (m) Fig. 4 Circular path in the x y plane d RVC AFL 2 (a) Control Effort of the Proposed (b) Control Effort of AFL Fig. 6 Voltages of motors for tracking the circular path Y (m) RVC AFL Desired path Fig. 5 Performances of controllers for tracking the circular path K x = 15, K y = 2, K θ = 15. The control design parameters of the control are chosen by trial-anderror method to achieve good performance. The AFC parameters are given by [5]. The tracking performances are satisfactory as shown in Fig. 4. The tracking error is highlighted as shown in Fig. 5. The initial tracking error is given as d () () = [.5m.5m rad ]. The norm of tracking error in the RVC after 1 s converges asymptotically to an ignorable value of about.5. The control efforts promptly reply to disturbances as shown in Fig. 6a. When starting, the voltages of motors are limited to 24 V. Then, it goes rapidly down. The voltages of motors relatively remain constant due to the desired linear, and angular velocities are constant. The sudden changes detected on the Fig. 6 are because of responding to the disturbances, and Fig. 6b represents the control effort of AFL. The proposed con X (m) Fig. 7 Mobile robot tracks the rectangular path trol shows the best performance as shown in Fig. 5. The norm of velocity error is reduced smoothly to an ignorable value, as well. However, the AFL control has shown much more errors with oscillations and ripples. The RVC is free from the robot dynamics, as the AFL is dependent on the robot dynamics. The performance of control is better than the AFL but is not as good as the RVC in terms of tracking error. 5.2 Simulation 2 A comparison between the RVC, traditional control, and the AFL control [5] is presented for tracking a rectangular path with round corners in the x y plane as shown in Fig. 7. The path is expressed as

10 344 M. M. Fateh, A. Arab p d = { [.3 (m/s).1 (rad/s) ] T on the corners [.1 (m/s) (rad/s) ] T on the sides (62).8.6 RVC AFL The AFL controller for the wheeled mobile robot is designed as follows: V = ˆD [σ r l f ˆl ] (63) σ r = ÿ d + β 1 (ẏ d ẏ) + β 2 (y d y) + (64) V is the voltage command of the motors, β 1 and β 2 are a positive diagonal gains, ˆD is the estimation of system matrix D, ˆl is the estimation of l, and y is defined as output vector y = [ ] T x +.1 cos θ y +.1sinθ. The system matrix D, l f, l and robust term are presented in [5]asfollows: cos θ sin θ α5 α D =. 6 (65) sin θ cos θ α 5 α 6 l f = ẏ (66) α 1 υ d cos θ α 2 ωd 2 cos θ l =.1α 3 υ d ω d sin θ +.1α 4 ω d sin θ α 1 υ d sin θ α 2 ωd 2 sin θ (67) +.1α 3 υ d ω d cos θ.1α 4 ω d cos θ 2(y d y) ˆϖ 2 = (68) 2 ˆϖ y d y + ϒ(t) ϒ(t) is a strictly positive time function as ϒ(t) = k ϒ ϒ(t), ϒ() > and ˆϖ = Y ϖ ˆα ϖ, Y ϖ = [ P 1 ] and ˆα ϖ = k αϖ Y T y d y, k αϖ > [. The estimation of ] parameter vector α is ˆα = ˆα1 ˆα 2 ˆα 3 ˆα 4 ˆα 5 ˆα 6, and the adaptive law is defined as ˆα = t Ɣ α W T α (y d y) dt + ˆα() (69) Ɣ α is a diagonal positive matrix and W α is the regression matrix of the form [ ] w11 w W α = 12 w 13 w 14 w 15 w 16 w 21 w 22 w 23 w 24 w 25 w 26 w 11 = υ d cos θ, w 12 = ωd 2 cos θ, w 13 =.1υ d ω d sin θ, w 14 =.1ω d sin θ, (7) d Fig. 8 Performances of the controllers for tracking the rectangular path w 15 =ˆα 5 (o 1 cos 2 θ + o 2 cos θ sin θ), w 16 =ˆα 6 (o 1 sin 2 θ o 2 cos θ sin θ), w 21 = υ d sin θ, w 22 = ωd 2 sin θ, w 23 =.1υ d ω d cos θ, w 24 =.1ω d cos θ, w 25 =ˆα 5 (o 1 cos θ sin θ + o 2 sin 2 θ), w 26 =ˆα 6 ( o 1 cos θ sin θ + o 2 cos 2 θ) o 1 = σ r1 ẏ 1 w 11 ˆα 1 w 12 ˆα 2 w 13 ˆα 3 w 14 ˆα 4, o 2 = σ r2 ẏ 2 w 21 ˆα 1 w 22 ˆα 2 w 23 ˆα 3 w 24 ˆα 4 The control design parameters are set to K p = 5, β 1 = 8, β 2 = 16, K x = 15, K y = 2, and K = 15. These parameters are chosen by trial-and-error method to achieve a satisfactory performance. The initial tracking error is given as d () () = [.5m.5m rad ]. The tracking performance of RVC is better than the AFL control and the control as shown in Fig. 8. The control performance is not good. The norm of tracking error in the RVC is reduced to 8 1 4, as the norm of tracking error in the AFL is reduced to that is 25 times larger than its value in the RVC. It is worthy to note that the RVC is less computational than AFL, as well. The control efforts for the RVC and the AFL are presented in the Fig. 9 (a) and Fig. 9 (b), respectively.

11 Robust control of a wheeled mobile robot (a) Control Effort of the Proposed (b) Control Effort of AFL Fig. 9 Voltages of motors for tracking the rectangular path d norm of error RVC AFL Fig. 11 Performances of the controllers for tracking 8-shaped path Y (m) 5 RVC AFL Desired path X (m) Fig. 1 Mobile robot tracks the 8-shaped path 2 (a) Control Effort of the Proposed (b) Control Effort of AFL Fig. 12 Voltages of motors for tracking 8-shaped path 5.3 Simulation 3 A comparison between the RVC, control, and the AFL control is presented for tracking an 8-shaped path as shown in Fig. 1. The initial tracking error is given as d () () = [ m 1 m π 4 rad ]. The tracking performance of RVC is better than the AFL and as shown in Fig. 11. The norm of tracking error in the RVC is not reduced to a constant value but mean suare of error for the RVC, the AFL, and the is.96, 1.21, and 1.91, respectively. It is worthy to note that the RVC is less computational than AFL, as well. The control efforts for the RVC and the AFL are presented in the Fig. 12a, b, respectively. 5.4 Experimental results The RVC is tested in tracking a circular path. In addition, a comparison between the RVC and control for tracking an 8-shaped path is presented. The setup is shown in Fig. 13. The specification of the host computer adopted here is the Intel Core-i5 CPU 2.6 GHz with 4G RAM. The euipment for the experiments is listed as follows: a camera with resolution and 3 fps, a 2.4 GHz freuency wireless Zig-Bee radio module that connected to the host computer and the mobile robot. The mobile robot consists of the PIC-based driver board, permanent magnet DC motors, gearboxes, receiver, and structural mechanism.

12 346 M. M. Fateh, A. Arab.5.4 d Fig. 13 Experimental setup Fig. 15 Norm of error in tracking the circular path by the RVC Fig. 14 Mobile robot tracks a circular path in practice A software was developed using Microsoft Visual Studio 212 C# for image processing, communicating, and controlling. The mobile robot tracks well the circular path in Fig. 14 with a tracking error shown in Fig. 15. The norm of tracking error comes down from about.5 to about.4 at the end. The voltage of motors is under the permitted values as shown in Fig. 16. In the next experiment, the mobile robot tracks the 8- shaped path in Fig. 17. A comparison between the RVC and control for the 8-shaped path is presented in Fig. 18. The tracking performance of RVC is better than the performance of control as shown in Fig. 19. The voltage of motors is under the permitted values as shown in Fig. 2. The satisfied control performance in experimental results verifies that the proposed method can bear the measurement noise Fig. 16 Voltage of motors for tracking circular path by the RVC Fig. 17 Mobile robot tracks the 8-shaped path in practice

13 Robust control of a wheeled mobile robot 347 Y (m) Desired path Proposed X (m) Fig. 18 A comparison between the RVC and control for tracking 8-shaped path 2 (a) Control effort of proposed (b) Control effort of Fig. 2 Control efforts in tracking the 8-shaped path d Proposed Fig. 19 A comparison on norm of error for tracking 8-shaped path path. Compared with the AFL control, the RVC has an advantage of being free from the robot dynamics. The RVC has shown a better tracking performance than the control, as well. The RVC reuires only some parameters of motors and kinematic parameters, as the AFL control is highly dependent on the dynamics of the robot. The RVC is simpler and less computational with using only one control loop while both the AFL control and control have two control loops. The stability analysis has proven that the RVC can guarantee stability, overcome uncertainties, and obtain good tracking performance in the sense that the tracking error converges asymptotically to an ignorable value as illustrated by simulations. The proposed control law is simple, fast response, and robust with ignorable tracking error. 6 Conclusion This paper has studied the use of voltage control in replace of the torue control for the nonholonomic wheeled mobile robot driven by permanent magnet DC motors. The whole robotic system including the robot and electric motors has been considered in the control problem. Compared with torue-based control approaches, the proposed voltage control law (RVC) is computationally simpler and has a simpler design. Through simulations and experimental results, the RVC is compared with the AFL control and traditional control. The control performances were tested in three paths of circular path, rectangle path, and an 8-shaped References 1. Chen, H., Ma, M.M., Wang, H., Liu, Z.Y., Cai, Z.X.: Moving horizon H tracking control of wheeled mobile robots with actuator saturation. IEEE Trans. Control Syst. Technol. 17, (29) 2. Yang, J.-M., Kim, J.-H.: Sliding mode control for trajectory tracking of nonholonomic wheeled mobile robots. IEEE Trans. Robot. 15(3), (1999) 3. Fukao, J.T., Nakagawa, H., Adachi, N.: Adaptive tracking control of a nonholonomic mobile robot. IEEE Trans. Neural Netw. 16(5), (2) 4. Chen, C., Li, T.S., Yeh, Y., Chang, C.C.: Design and implementation of an adaptive sliding-mode dynamic controller for wheeled mobile robots. Mechatronics 19, (29)

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