Summary of Engineering Doctoral Dissertation: Researching and developing some control laws for a wheeled mobile robot in the presence of slippage

The objective of the thesis is to propose some new control methods to compensate for the negative effects of pattern uncertainty, external noise and wheel slip. Analyze and build dynamic and kinetic models of mobile robots when model uncertainties exist. Demonstrating the correctness and effectiveness of new control methods by the Lyapunov stability standard and Barbalat lemma. Advanced domestic and foreign control methods for mobile robots in the presence of uncertainty models, external noise, and wheel slip. Then propose new control methods.

MINISTRY OF EDUCATION AND TRAINING VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY - NGUYEN VAN TINH RESEARCHING AND DEVELOPING SOME CONTROL LAWS FOR A WHEELED MOBILE ROBOT IN THE PRESENCE OF SLIPPAGE ENGINEERING DOCTORAL DISSERTATION Major: Control and Automation Technology Code: 9.52.02.16 SUMMARY OF ENGINEERING DOCTORAL DISSERTATION Ha Noi, 2018 This work is completed at: Graduate University of Science and Technology Vietnam Academy of Science and Technology Supervisor 1: Dr Pham Minh Tuan Reviewer 1: …………………………………………………………………… ………………………………………………………………………………… Reviewer 2: …………………………………………………………………… ………………………………………………………………………………… Reviewer 3: …………………………………………………………………… ………………………………………………………………………………… This Dissertation will be officially presented in front of the Doctoral Dissertation Grading Committee, meeting at: Graduate University of Science and Technology Vietnam Academy of Science and Technology At ………… hrs …… day …… month…… year …… This Dissertation is available at: Library of Graduate University of Science and Technology National Library of Vietnam ABSTRACT The necessary of the thesis In these years, there is a growing recognition that mobile robots have the capability to operate in a wide area and further the ability to manipulate in an automatic and smart way without any actions taken by human Hence, this project concentrated on researching and developing some control laws for wheeled mobile robots The researching problems of this thesis The author concentrated on radical control methods in order to deal with wheel slipping whenever there exist slippage, model uncertainties, and external disturbances Object of study So as to easily demonstrate the validity and performance of the proposed control methods, the object of study was selected to be one three-wheel mobile robot To be specific, this robot consists of two differential driving wheels and one caster wheel used to make gravity balance The purpose of researching Proposing a number of radical control approaches so as to cope with the negative effects of model uncertainties, external disturbances, and above all slippage Approaches of study The approach of study is illustrated as the following order:  Analyzing and building the kinematic and dynamic model of the mobile robot with the occurrence of model uncertainties, external disturbances, and above all slippage  Researching, analyzing state-of-the-art control methods which were designed both domestic and foreign for this topic After that, some radical control methods were proposed  Proving the correctness and efficiency of the proposed control approaches via Lyapunov standard and Barbalat lemma  Demonstrating the above-mentioned control methods through Matlab/Simulink tool Scientific and practical benefits of this project Scientific benefits: Building novel control approaches for a wheeled mobile robot with the purpose of compensating for the negative effects of model uncertainties, external disturbances, and above all slippage Practical benefits: the proposed control methods in this project could be applied for wheeled mobile robots operating in warehouses with the slippery floor and/or in orchards with wet land Structure of the thesis Chapter 1: Overviewing domestic and foreign studies in recent years, and then showing a process by which the kinematic and dynamic model of a wheeled mobile robot are established in the presence of model uncertainties, external disturbances, and slippage Chapter 2: Designing an adaptive control law based a three-layer neural network Chapter 3: Designing a robust adaptive backstepping control law based a Gaussian wavelet network Chapter 4: Designing a backtepping control law ensuring finite-time convergence at dynamic level CHAPTER OVERVIEWING AND MATHEMATIC MODELS 1.1 Problem statement Motion control problem is fairly important in the field of mobile robot because the performance of control laws affects the efficiency of the application of mobile robots in production and life Thus, this problem is chosen as the goal of this project These days, motion control problem for wheeled mobile robots has attracted the consideration of researchers all over the world Needless to say, a wheeled mobile robot is one of the system subjected to nonholonomic constraint [1] Furthermore, it is one multi input – multi output nonlinear system [2] It is thanks to the recent advances in control theory as well as engineering, there were a large number of different control methods applied such as sliding mode control [3-4], robust control [5], … These control laws were under the assumption of pure rolling and no slippage Notwithstanding, in application practice, the violation of the above assumption can still happen That is to say, there exists slippage [12-13] Slippage is one of the key factors making the visible degradation of control performance Therefore, in such circumstances, so as to heighten control performance, a controller must be capable of compensating slippage 1.2 Domestic study In Vietnam, until now, there have been reports researching autonomous vehicles such as the group of the authors from University of Transport and Communications in Hanoi studying swarm robots [14-15] One group of authors at Hanoi University of Science and Technology researched on building a mathematical model for one four-wheel electrical car considering the interaction between wheels and road [18] Nevertheless, there have been still not many the studying results of addressing slippage for wheeled mobile robots to be published 1.3 Foreign study There have been reports researching on control problems compensating slippage for wheeled mobile robots It is due to slippage that the performance of closed system deteriorates and even the state of the system is unstable Frequently, so as to cope with slippage, the values of friction parameter and sideslip angle must be always measured in real-time accurately Specifically, the authors in [12] addressed slippage through compensating slip-ratios of wheels Gyros and accelerometers were utilized in [13] so as to compensate slippage in real-time The study in [19] reported a robust controller by which both slip-kinematic and slip-dynamic models were taken care thanks to the framework of differential flatness 1.4 Kinematic model Let us consider one wheeled mobile robot under a nonholonomic constraint as Fig 1.1 Without slippage, the linear and angular velocities are calculated as follows [21]:  r R  L     r R  L    2b      (1.1) Where R ,L are angular coordinates of the right and left wheel respectively L F3 Left wheel Caster wheel F4 F2 Wheel Shaft  M G θ a R 2b Platform  F1 Right wheel Figure 1.1 One wheeled mobile robot and slippage Thereby, showing the kinematic model as follows [4]:  xM   cos   yM   sin      (1.2) The nonholonomic constraint always assures the two following factors:  The direction of the linear motion is always perpendicular to the wheel shaft  Both the linear and angular motion of this robot fully depend on the pure rolling of the differential driving wheels Specifically, the mathematical model of this constraint is shown as follows [32]:  rR  xM cos  yM sin   b (1.3)  rL  xM cos  yM sin   b (1.4)   xM sin   yM cos (1.5) By stark contrast, in the presence of slippage, the linear velocity along the longitudinal axis is computed as follows:  R L (1.6) with  R ,  L being the longitudinal slip coordinates of the right and left wheels respectively Next, the actual yaw rate is computed as follows:  L  R 2b (1.7) Let us define  as the lateral slip coordinate along the wheel shaft (see Fig 2.1) In this circumstance, the kinematic model of this object is illustrated as follows [30]:  xM   cos   sin    yM   sin    cos     (1.8) Due to slippage, the nonholonomic constraint is represented as follows [32]: 1.5  R  rR  xM cos  yM sin  b (1.9)  L  rL  xM cos  yM sin  b (1.10)    xM sin   yM cos (1.11) Dynamic model This dynamic model subjected to slippage, model uncertainties, and external disturbances is expressed as follows: Mv  Bv  Bv  Qγ  C  G  τd  τ (1.23) Property 1: M has the invertible feature, is positive definition, and satisfies the following inequality 2 M1 x  xT Mx  M2 x with M1 and M2 being upper and lower bound of M and satisfying M2  M1  Property 2: matrix M  2B  v  is skew-symmetric, that is to say xT M  2B  v  x  with x  R 21 1.6 Conclusion for Chapter The attention and attempt of researchers all over the world for compensating slippage has increasingly become more prevalent than ever before However, most the studies were conducted under the assumption that the sideslip angle and friction parameter always are measure exactly in real time It goes without saying that accelerates and velocities are always directly measured via affordable and feasible sensors Yet, it is difficult and expensive to measure the sideslip angle and friction parameter [40] Taking into account all the factors mentioned above, this project is going to offer radical control approaches so as to compensate slippage for a wheeled mobile robot without measuring the sideslip angle and friction parameter In stark contrast, the negative effects of slippage are going to deal with in an indirect way via the proposed controllers here Moreover, the kinematic and dynamic model of the wheeled mobile robot subjected to slippage, model uncertainties, and external disturbances were established successfully These model are going to be used for designing control laws in next chapters This researching result was published in the number published material axis OY axis MX axis MY yD 2 D (target) P yM M 1 C xM xP O xD axis OX Figure 2.1 Illustrating the target in the body coordinate system M-XY CHAPTER DESIGNING AN ADAPTIVE CONTROL LAW BASED ON A THREELAYER NEURAL NETWORK 2.1 Problem statement Due to the fact that the control law in Chapter was designed under such an ideal, the applicability of that control method is very limited Therefore, in this chapter 4, one radical control method is proposed under a more practical condition in order to heighten the applicability in comparison to the method in Chapter To be specific, such a more practical condition involve the following factors:  There exist model uncertainties and external disturbances  The velocities and accelerates of slippage are not measured Let D(xD,yD) be a target which is moving in a known desired trajectory (see Figure 3.1) Without loss generality, the motion equation of D can be supposed as follows:  xD  TD t  R cos(.t )  x0   yD   TD t  R sin(.t )  y0 (2.1) , TD, R, , x0, y0 are constant parameters, and time t varies from zero to infinity We assume that the tool location is at point P So, the requirement of the position tracking control problem is to control the WMR so that P has to track D with the position tracking errors being uniformly ultimately bounded Remark 2.1: In Figure 2.1, we denote (xP, yP) as the position of P Let (xP, yP, ) be the actual posture of the WMR, and (xPd, yPd, d) be the desired one of the WMR The presence of both the longitudinal and lateral slips makes it impossible to control the WMR in the way that the actual posture (xP, yP, ) tracks the desired one (xPd, yPd, d) with an arbitrarily good tracking performance [32] Instead of this, it is fully possible to control the WMR with the purpose of making the actual position (xP, yP) track the desired one (xPd, yPd) with an arbitrarily good tracking performance [32] 2.2 Structure of the three-layer neural network (NN) Admittedly, artificial neural networks have the ability of approximating nonlinear and sufficiently smooth functions with arbitrary accuracy In this subsection, a three-layer NN is introduced briefly [8] As illustrated in Figure 2.2, the output of the NN can be computed as y  W, V    y1 , y2 , , yN3  T    WT σ V T x where x  1, x1 , x2 , , xN1  T is the input vector, and W   wij  and V  vij  are the NN weight matrices (𝐳) = [𝟏, (𝒛𝟏 ), (𝒛𝟐 ), … ]𝐓 with 𝐳 = [𝒛𝟏 , 𝒛𝟐 , … ]𝐓 Next, () is the activation function of the NN In this paper, the activation function is chosen to be the sigmoid kind as (𝒛) = 𝟏/(𝟏 + 𝐞𝐱𝐩(−𝒛)) Let 𝐟(𝐱): 𝐑𝐍𝟏 → 𝐑𝐍𝟑 be a smooth function There exist optimal weight matrices W and V so that: 𝐟(𝐱) = 𝐖 𝐓 (𝐕 𝐓 𝐱) + , (2.3) where  is the vector of optimal errors Assumption 2.1:  is bounded Especially, ‖‖ ≤ 𝒃 where 𝒃 expresses an upper bound of  ̂,𝐕 ̂) = 𝐲̂(𝐱, 𝐖 ̂,𝐕 ̂) = 𝐖 ̂ (𝐕 ̂ 𝐓 𝐱) denote an estimation of f(x), where 𝐖 ̂,𝐕 ̂ are estimation Let 𝐟̂(𝐱, 𝐖 matrices of 𝐖 and 𝐕, respectively, and they are provided by an online weight tuning algorithm to be revealed subsequently 2.3 Expressing the vector filtered tracking errors (FTE) Let O-XY be the global coordinate system, M-XY be the body coordinate system which is attached to the platform of the WMR (see Figure 2) The coordinate of the target is represented in M-XY as follows:     cos ζ   1       sin  sin    xD  yM  cos   yD  yM  Taking the first order derivative with respect to time of (2.6) yields  cos sin    xD  ζ  hv     χ   sin  cos   yD  (2.6) (2.7)   r   r   R   L     1  R     1    L where v    , h   b   b   , χ        R     2b r    r  L  1    2b 2b   Taking the second order derivative with respect to time of (2.6) yields ζ  hv  Ψ1  Ψ2 (2.8) where, Ψ1  hv   xD cos  yD sin   xD  sin   yD  cos    xD sin   yD cos  xD  cos  yD  sin       xD sin   yD cos  Ψ2      xD cos  yD sin   Remark 2.2: If 1 ≠ 0, then h is an invertible matrix Let us define the position tracking error vector as e = e1 e2   ζ - ζd T (2.9) where ζ d is the desired coordinate vector of the target in M-XY According to the requirement of the position tracking control problem mentioned above and Fig 3.2, one can easily set ζd  C ,0 T In order to tackle this problem via the novel proposed control method, first of all, the scheme of entire closed loop system is proposed as Figure 2.3 The vector FTE is defined as follows: φ = e + Λe (2.10) where Λ is one diagonal, positive-definition, and constant matrix It can be chosen arbitrarily 2.4 Structure of the controller Ψ in (2.2) depends on the velocities and accelerates of slippage directly, so it is uncertain Thus, on auxiliary variable is proposed as follows: (2.14) κ = h1  ζ  Λe  Ψ  d On the other hand, (2.23) can be rewritten as follows: (2.15) Mv  τ  Bv  d  τd where d  Qγ  C  G  Bv Next, one control law is chosen via the computing-torque method as follows:   ˆ 1 Kφ  fˆ x, W ˆ ,V ˆ  τ  Mh   (2.19) ̂,𝐕 ̂) is where K is a × diagonal, constant, positive definite matrix and is chosen arbitrarily 𝐟̂(𝐱, 𝐖 the output of the NN in order to approximate 𝐟(𝐱) In this work, let us propose the online weight tuning algorithm for the NN weights as follows:  T ˆ  H σφ ˆ T xφT   φ W ˆ W  σV ˆ  ˆ  H xφT W ˆ Tσ   φ V ˆ V  (2.24)  (2.25) where 𝐇𝟏 is an (𝑵𝟐 + 𝟏) × 𝑵𝟑 positive definition constant matrix 𝐇𝟐 is an (𝑵𝟏 + 𝟏) × 𝑵𝟐 positive definition constant matrix  is positive constants 2.5 Stability Theorem For the WMR subject to wheel slip as in Eq (1.8), let the control input be given by Eq (2.19) and the online weight tuning algorithm be provided by Eqs (2.24) and (2.25) Then, according to Lyapunov theory and LaSalle extension, the stability of the closed-loop system is assured to achieve the desired tracking performance where  as well as the vector of the weight errors are uniformly ultimately bounded [8] and can be kept arbitrarily small v1 v11 x1 w1  vN2 Input layer y1 ∑ y2 v2 x2 xN1 ∑ wN3 wN2N3 Hidden layer  yN3 Output layer Figure 2.2 structure of the three-layer neural network Target (xD, yD) Eq (4.6) Eq (3.2) v Three-layer neural network  WMR subject to slippag e e controller + Figure 2.3 Scheme of the whole closed loop control system 2.6 Simulation results Example 2.1: target D moves in a straight line as follows:  xD   3cos  0, 2t    yD  0,5  3sin  0, 2t  (2.36) Obviously, in Figures 2.5, 2.6, and 2.7, we can easily see that when the accelerations and velocities of the unknown wheel slips were not measured and model uncertainties and unknown bounded disturbances existed, the control approach in [8] could not compensate the undesired effects while the proposed control method effectively dealt with the undesired effects velocities of wheel slip (m/s) velocities of wheel slip 0.4 0.3 0.2 longitudinal slip of the right wheel longitudinal slip of the left wheel lateral slip 0.1 0 10 time (s) Figure 2.4 the timelines of slip velocities 2.7 Conclusion for chapter All in all, in this chapter, an adaptive tracking controller based on a three-layer NN with the online weight updating algorithm was developed to let the WMR track a desired trajectory with one desired tracking performance It has been clear that the convergence of both the position tracking errors and the NN weight errors to an arbitrarily small neighborhood of the origin was ensured by the standard Lyapunov criteria and LaSalle extension The results of the Matlab simulations illustrated the validity and efficiency of the proposed control method Figure 2.7 the torques of the proposed method in example 2.1 CHAPTER DESIGNING A ROBUST ADAPTIVE BACKSTEPPING CONTROL LAW BASED A GAUSSIAN WAVELET NETWORK 3.1 Problem statement Even though the control method in chapter illustrated the efficiency to cope with model uncertainties and external disturbances, the control accuracy, namely the tracking error vector e, still not small enough in compared to the expectation of tasks demanding high-accuracy The reason may be:  There was the classification in a clear way for particular tasks Especially, what control terms are used to deal with the negative effects of slippage at the kinematic level and/or model uncertainties as well as external disturbances at the dynamic level?  There was not robust control term, so the stable criterion is only UUB Specifically, the tracking errors were only ensured to converge to a near-zero compact set rather than asymptotic convergence to zero Therefore, in this chapter, one novel robust adaptive tracking control method based on the backstepping technique [8] is proposed for a wheeled mobile robot so as to compensate slippage, model uncertainties and external disturbances The scheme of this control system is shown in Fig 3.1 In particular, this system consists of two closed control loops The outer loop comprises the kinematic controller In this kinematic controller, the kinematic robust term is utilized for compensating the harmful influence of slippage the output of kinematic is the same as the input of the inner dynamic control loop The inner loop is composed of the dynamic controller Here, the Gaussian wavelet network is employed in order to approximate unknown nonlinear functions due to no prior acknowledgement of the dynamic model of this WMR The dynamic robust term is useful to cope with the negative effects of model uncertainties and external disturbances 3.2 Structure of the Gaussian wavelet network Let us consider a Gaussian wavelet network as Fig 5.2 The outputs of this network with p wavelet basis functions are shown as follows: fk  x   p  w j j  x  with (3.1) j = 1,…, p j where x   x1 , , xn  is the input vector; w j shows a weight with j = 1,…, p;  j  x  denotes T a multidimensional wavelet function as follows  j  x    j  x1  j  x2   j  xn   where  j  xi   i  xi  exp   ij2 xi  cij    2 (3.2)  with i  xi    ij  xi  cij   ij and cij are the dilation and translation parameters, respectively Thanks to the strong approximation ability of the Gaussian wavelet network [41], given any smooth function f  x  , there exists an optimal weight matrix W , an optimal vector ξ  , and an optimal vector c such that f  x   W*T ψ  x, ξ* , c*   ε (3.5) where ε describes one vector of optimal approximation errors 3.3 Designing kinematic control law First of all, the first derivative of (2.6) is computed as follows  cos sin    xD  ζ  hv     χ   sin  cos   yD  (3.7)   L  where χ    R   , h is shown in (2.6)   Assumption 3.3: xD , yD are twice differential T Assumption 3.4: slip velocities are bounded Thus, there exists a known positive  so that χ  Since the velocities of slippage are not measured, χ in (3.7) is uncertain Thus, the kinematic in this method is proposed as follows: t   cos v c  h 1   Λe  Λ I ed  ζ d      sin     sin    xD  r     cos   yD   (3.9) where v c is the desired vector of the vector of angular velocities of the differential driving wheels v ; r is the kinematic robust term proposed as follows: r e e (3.10) where  is the gain of the kinematic robust 3.4 Designing the dynamic control law The dynamic control law is proposed as follows ˆ , ξˆ , cˆ  dˆ τ  Ks  fˆ x, W   (3.14) where K is a diagonal positive-definition constant matrix which can be selected arbitrarily ˆ , ξˆ , cˆ is the output of GWN in (3.6); dˆ shows the dynamic robust term used to eliminate fˆ x, W   total uncertainty due to the inevitable GWN approximation errors , model uncertainties, external disturbances The dynamic robust term dˆ in (3.14) is proposed in the following form: s dˆ   s (3.25) where  is a positive constant 3.5 Stability Theorem 3.1: Let consider the WMR in the presence of the unknown wheel slips, model uncertainties, bounded external disturbances with the kinematics (1.8) and the dynamics (1.23) under a condition that Assumptions 3.1-3.5 hold If the proposed control method as shown Fig 3.2 with the kinematic control law (3.9), dynamic control law (3.14) is utilized, then the position and velocity tracking error vectors, consisting of e and s, asymptotically converge to zero as t   and the control gains are ensured to be bounded for all t  3.6 Simulation results In this subsection, one simulation is performed in order to validate the correctness and efficiency of this proposed control method Furthermore, one comparison between the tracking performance of Chapter and is also carried out Dynamic robust term kinematic controller + Dynamic controller + Gaussian wavelet network kinematic robust term Eq (2.6) Figure 3.1 Scheme of the control system in chapter WMR dilation x1 translation  dilation  xn Figure 3.2 Structure of Gaussian wavelet network – GWN Figure 3.3 Comparison trajectory between chapter and in example 3.1 a) b) Figure 3.4 Comparison between the tracking errors of chapter and a) comparison of e1 in the whole evolution; b) comparison of e2 in the whole evolution; Example 3.1: target D moved on circular path as follows:  xD   3cos  0, 2t    yD  0,5  3sin  0, 2t  (3.51) The simulation results of the both control methods were in Figures 3.3, 3.4, 3.5 Should Figure 3.3 be seen, there will be misleading information that the tracking performances of the these both methods are same as each other Alternatively, when observing Fig 3.4, one can easily see that the tracking errors e1,2 in this chapter are smaller than those in chapter Particularly, the biggest values of e1,2 in this novel method in the steady state are 1.10-3 and 0,6.10-3(m) respectively in comparison with those in chapter being 1,5.10-2 and -1,2.10-2 (m) respectively Therefore, the approximation ability of GWN and the robust capability of the robust term gave this radical control method in this chapter the advantage over the method in chapter However, the drawback of this control method is that the torques in the beginning time are very big in comparison to chapter The Fig 2.5 showed clearly To be specific, this control method at the beginning time requested the torques 100 N.m at the right wheel and 80 N.m at the left wheel respectively On the other hand, the method in chapter only requested the corresponding values being 63 (N.m) and 38 (N.m) respectively It is interesting when looking at Fig 3.6, the efficiency of dealing with chattering at the both wheels This leads to that the cost of energy will be reduced and further the operation time of actuators will be heightened 3.7 Conclusion chapter In this chapter, the robust adaptive tracking control method based on the Gaussian wavelet network with the online update laws has been proposed to let the WMR track a predefined trajectory with the desired tracking performance in the presence of the unknown wheel slips, model uncertainties, and unknown bounded external disturbances, etc It is unnecessary to request prior knowledge of the dynamic parameters of the WMR Besides, it is no need to preliminary offline train for the gains of the Gaussian wavelet network because they can be initialized easily The two robust terms were employed so as to form the robustness of the whole closed-loop control system Especially, the one was used in the kinematic controller, the other was adopted in the dynamic controller It has been shown that both the asymptotic convergence of the position tracking errors and the velocity tracking errors to zero is assured The robust terms and the gains of the Gaussian wavelet network are bounded via standard Lyapunov criteria and LaSalle extension as well as Barbalat’s lemma The results of the computer simulations validated the correctness as well as effectiveness of the radical proposed control approach However, the disadvantage of this control method is that the torques at the beginning time are very big The content of this chapter is cited from the 7th published paper a) b) Figure 3.5 Comparison between the torques of the two control methods a) the right torque; b) the left torque assessment of efficiency of dealing with chattering at the RIGHT wheel 80 Not dealing with chattering dealt with chattering left wheel torque (N.m) right torque (N.m) 100 50 -50 time (s) assessment of efficiency of dealing with chattering at the LEFT wheel Not dealing with chattering dealt with chattering 60 40 20 -20 time (s) Figure 3.6 the assessment of the efficiency of dealing with chattering at the both wheels CHAPTER DESIGNING A BACKTEPPING CONTROL LAW ENSURING FINITETIME CONVERGENCE AT DYNAMIC LEVEL 4.1 Problem statement As mentioned above, the disadvantage of the control method in chapter is that the torques at the beginning time are very big This disadvantage maybe cause the occurrence of torque saturation [43] (see Figure 4.1) Such torque saturation is a phenomenon which happens whenever the maximum ability of a motor for supplying torque is smaller than the requirement of the output of a controller Should this phenomenon occur, then the control performance of the whole system will deteriorate and even the system will be unstable [44] The disadvantage of chapter has motivated the suggestion of the novel control method in this chapter Like to chapter 3, the proposed control method here is based on the backstepping technique The scheme of the entire control system is described in Figure 4.2 4.2 Structure of RBFNN As can be seen in Fig 4.3, the structure of RBFNN is composed of three layers: the input, hidden, and output layer For any one smooth and bounded function vector f  x  : R N1  R N3 , there exists one optimal weight matrix W such that (4.4) f  x   WTσ  ε where ε is one optimal approximation error vector 4.3 Designing the kinematic control law According to (3.9) in chapter 3, the kinematic control law in this control method is proposed as follows: t   cos v c  h 1   Λe  Λ I ed  ζ d      sin    sin    xD   χˆ      cos   yD    (4.6) where v c is the desired vector of the angular velocity vector v and is also the output of the kinematic controller, Λ is a diagonal positive-definition constant matrix; χˆ is the kinematic robust term proposed as follows: χˆ  ˆ e e (4.7) where ˆ is the gain of the kinematic robust term updated online as follows: (4.8) ˆ  H e where H is a positive constant scalar and can be chosen arbitrarily 4.4 Designing the dynamic control law Since there is no prior acknowledge of the dynamic model of this mobile robot, one cannot know f  x  precisely Therefore, one can propose the dynamic control law as follows:    (4.12) ˆ  dˆ τ  Ksig  s   fˆ x, W   ˆ W ˆ T σ is where K is a positive-definition constant matrix and can be selected arbitrarily fˆ x, W the output of the RBFNN (4.3) and is used to approximate f  x     sig  s    s1 sign  s1  , s2 sign  s2    T (4.13) with  is a positive constant    ; dˆ describes the dynamic robust term utilized to compensate model uncertainties, external disturbances, and the unavoidable errors of RBFNN approximation due to the finite number of the hidden neurons Now, the dynamic robust term dˆ is defined as follows:   s dˆ   ˆ   WM    s (4.16) where  is a positive constant and can be chosen arbitrarily; ˆ is the gain of the dynamic robust term and is updated as follows: ˆ  s  (4.17) where  is a positive constant and can be chosen arbitrarily 4.5 Stability Theorem 4.1: Let us one WMR in the presence of slippage, model uncertainties, and external disturbances with kinematic model (1.8) and the dynamic model (1.23) Let Assumptions 3.3-3.5 and 4.1-4.3 hold Should the new control method be illustrated by Fig 4.2 with the kinematic control law (4.6) and the dynamic control law (4.12), then the tracking angular velocity error vector s will converge to zero within a finite time , the tracking position error vector e will asymptotic converge to zero when t   , and further the control signals are ensured to be bounded 4.6 Computer simulation In this subsection, one simulation is carried out to validate the correctness and performance of this new control method Moreover, one comparison between chapter and is implemented in order to show the advantage of this new control method With the purpose of comparison, the both control methods were designed with the same control gains and under the same condition where there exist both model uncertainties and external disturbances and the velocities of slippage were not measured Controller C Motor A A Saturation point C Abbreviation: C – Command at the controller’s output A – Response of the motor Saturation point Figure 4.1 Motor’s Saturation point Dynamic robust term e - Kinematic controller + - + Dynamic controller WMR subject to slippage RBFNN Kinematic robust term Eq (2.6) Figure 4.2 Scheme of the whole control system in chapter x1  Output layer Input layer Hidden layer Figure 4.3 Structure of RBFNN Example 4.1: target D moved in the circular path as follows:  xD   3cos  0, 2t      yD   3sin  0, 2t  (4.49) The simulation results are shown in Figures 4.4, 4.5, 4.6, and 4.7 Especially, Fig 4.4 expressed that the tracking performances of these both are the same as each other in the O-XY plane Fig 4.5 shows the comparison of the tracking position errors between the two control methods The tracking position errors e1,2 of the new control method are slightly smaller than those of the old control method in chapter When it comes to the tracking angular velocity error vector s defined in (3.13) and (4.11) in the dynamic control loop as Fig 4.6, the tracking velocity errors, in the stead state, of the new control method are significantly smaller than those of the old one in chapter Particularly, the biggest values in the steady state of s1,2 in the new one are nearly 5.10-3 (rad/s), but the corresponding values in the old one are roughly 2.5× 10-2 (rad/s) The reason is that the new control method here ensured the finite-time convergence of s to zero at the dynamic level instead of the asymptotic convergence of s to zero as in the old control method in chapter Figure 4.4 Comparison of the tracking performance between the control methods in chapter and chapter Figure 4.5 Comparison the tracking position errors between the two control methods a) b) c) d) Figure 4.6 Comparison between the angular tracking errors of the two control methods a) Comparison in the whole evolution of the angular tracking error s1 ; b) Comparison in the whole evolution of the angular tracking error s2 ; c) Comparison in the steady state of the angular tracking error s1 ; d) Comparison in the steady state of the angular tracking error s2 ; Final and most important, with regard to the torques as Fig 4.7, it is obvious that at the beginning time, the torques of the new control method are significantly smaller than those of old one in chapter To be specific, the former values are 30 (N.m) and 25 (N.m) as against the latter values being 100 (N.m) and 80 (N.m) Thus, the new control method stops the torque saturation occur and further reduced the price of investment in actuators 4.7 Conclusion for Chapter Summary, this novel control method has overcome some disadvantages of the old control one in chapter Specifically:  Avoid the torque saturation at the beginning time  Heighten the accuracy at the dynamic level  No need to prior acknowledge of the upper bound of the model uncertainties, external disturbances, and slippage So sanh mo men quay o banh trai comparison between the right torque 80 chapter chapter left torque (N.m) right torque (N.m) 100 50 chapter chapter 60 40 20 -50 time (s) -20 time (s) Figure 4.7 Comparison between the torques of the control methods in chapters and CONCLUSION AND SUGGESTION Taking into account all the factors mentioned above, the conclusion can be drawn as follows: The main researching contents of this project  Analyzing overviewing, actual situation, and the direction of the development of state-ofthe-art control methods for a wheeled mobile robot so as to track a predefined trajectory  Analyzing the causes and negative effects of model uncertainties, external disturbances, and slippage  Establishing the kinematic and dynamic models of a wheeled mobile robot in the presence of slippage  Designing 03 various control methods so as to overcome the influence of slippage  Carrying out computer simulations via Matlab/Simulink tool in order to validate the correctness of these above-mentioned control methods The contributions of this thesis The project has 03 contributions as follows:  Designing an adaptive control law based a three-layer neural network (chapter2)  Designing a robust adaptive backstepping control law based a GWN (Chapter 3)  Designing a backtepping control law ensuring finite-time convergence at dynamic level (Chapter 4) Future work  Continuing researching and developing radical control methods for a wheeled mobile robot to track a desired trajectory  Establishing experimental systems so as to confirm the proposed control methods LIST OF THE RELEASED PAPERS Nguyễn Văn Tính, Phạm Thượng Cát, Phạm Minh Tuấn, “Mơ hình hóa điều khiển rơ bốt di động non-holonomic có trượt ngang”, Kỷ yếu hội nghị toàn quốc lần thư Điều khiển Tự động hóa – VCCA, 2015, Thái Nguyên, 103-108 N V Tinh, N T Linh, P T Cat, P M Tuan, M N Anh, N P Anh, Modeling and Feedback Linearization Control of a Nonholonomic Wheeled Mobile Robot with Longitudinal, Lateral Slips, In: Proc 2016 IEEE International Conference on Automation Science and Enginerring, TX, USA, 996-1001 Tinh Nguyen, Hung Linh Le, Neural network-based adaptive tracking control for a nonholonomic wheeled mobile robot subject to unknown slips, Journal of Computer Science and Cybernetics, Vietnam Academy of Science and Technology, 2017, 33(1), 1-17 Tinh Nguyen, Linh Le, “Neural network-based adaptive tracking control for a nonholonomic wheeled mobile robot with unknown wheel slips, model uncertainties, and unknown bounded disturbances”, Turkish Journal of Electrical Engineering & Computer Sciences, 2018, 26, 378392 Tinh Nguyen, Kiem Nguyentien, Tuan Do, Tuan Pham, Neural Network-based Adaptive Sliding Mode Control Method for Tracking of a Nonholonomic Wheeled Mobile Robot with Unknown Wheel Slips, Model Uncertainties, and Unknown Bounded External Disturbances, Acta Polytechnica Hungarica, 2018, 15(2), 103-123 Kiem Nguyentien, Linh Le, Tuan Do, Tinh Nguyen, Minhtuan Pham, Robust control for a wheeled mobile robot to track a predefined trajectory in the presence of unknown wheel slips, Vietnam Journal of Mechanics, Vietnam Academy of Science and Technology, 2018, 40(2), 141 –154 Nguyen Tinh, Thuong Hoang, Minhtuan Pham & Namphuong Dao, A Gaussian wavelet network-based robust adaptive tracking controller for a wheeled mobile robot with unknown wheel slips, International Journal of Control, 2018, DOI: 10.1080/00207179.2018.1458156 REFERENCES L Xin, Q Wang, J She, Y 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Performance, IEEE Transactions on Neural Networks, 1996, 7(2), 388-399 46 L Wang, T Chai, L Zhai, Neural network-based terminal sliding mode control of robotic manipulators including actuator dynamics, IEEE Transaction on Industrial Electronics, 2009, 56(9), 3296-3304 47 A K Pamosoaji, P T Cat, K Hong, Sliding-mode and proportional-derivative-type motion control with radial basis function neural network based estimators for wheeled vehicles, International Journal of Systems Sciences, 2014, 45(12) ... thesis In these years, there is a growing recognition that mobile robots have the capability to operate in a wide area and further the ability to manipulate in an automatic and smart way without any... disturbances, and above all slippage Approaches of study The approach of study is illustrated as the following order:  Analyzing and building the kinematic and dynamic model of the mobile robot. .. in recent years, and then showing a process by which the kinematic and dynamic model of a wheeled mobile robot are established in the presence of model uncertainties, external disturbances, and