Researching and developing some control laws for a wheeled mobile robot in the presence of slippage

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: 1. Library of Graduate University of Science and Technology 2. 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 1. 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 L R L 2 2 r r b               (1.1) Where R L,  are angular coordinates of the right and left wheel respectively. Thereby, showing the kinematic model as follows [4]: M M cos sin x y            (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 M M0 cos sinr x y b        (1.3) L M M0 cos sinr x y b        (1.4) M M0 sin cosx y    (1.5) By stark contrast, in the presence of slippage, the linear velocity along the longitudinal axis is computed as follows: R L 2     (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: R L 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]: F 2 F 1 F 3  Left wheel Right wheel Caster wheel a 2b F 4   L  R Wheel Shaft G M Platform θ Figure 1.1. One wheeled mobile robot and slippage. MM cos sin sin cos x y                 (1.8) Due to slippage, the nonholonomic constraint is represented as follows [32]: R R M Mcos sinr x y b         (1.9) L L M Mcos sinr x y b         (1.10) M Msin cosx y     (1.11) 1.5. Dynamic model This dynamic model subjected to slippage, model uncertainties, and external disturbances is expressed as follows: d       Mv Bv Bv Qγ C G τ τ (1.23) Property 1: M has the invertible feature, is positive definition, and satisfies the following inequality 2 2T 1 2M M x x Mx x with M1 and M2 being upper and lower bound of M and satisfying 2 1M M 0  . Property 2: matrix  2  M B v is skew-symmetric, that is to say  T 2 0   x M B v x with 2 1 x R . 1.6. Conclusion for Chapter 1 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 3 published material. CHAPTER 2. DESIGNING AN ADAPTIVE CONTROL LAW BASED ON A THREE- LAYER NEURAL NETWORK 2.1. Problem statement Due to the fact that the control law in Chapter 3 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 3. 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: D 0 D 0 . cos( . ) . sin( . ) D D x T t R t x y T t R t y           (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   3 T 1 2, , ,..., Ny y y   y W V  T T W σ V x where 1 T 1 21, , ,..., Nx x x   x is the input vector, and 1 2 xD yD axis OY C xM D (target) P M axis OX yM x P O axis MY axis MX Figure 2.1. Illustrating the target in the body coordinate system M-XY. ijw   W and ijv    V 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) = 𝐖𝐓(𝐕𝐓𝐱) + , (2.3) where  is the vector of optimal errors. Assumption 2.1:  is bounded. Especially, ‖‖ ≤ 𝒃 where 𝒃 expresses an upper bound of . Let 𝐟(𝐱, �̂�, �̂�) = �̂�(𝐱, �̂�, �̂�) = �̂�(�̂�𝐓𝐱) denote an estimation of f(x), where �̂�, �̂� are estimation 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: 1 D M 2 D M cos sin sin cos x y y y                       ζ (2.6) Taking the first order derivative with respect to time of (2.6) yields D D cos sin sin cos x y                ζ hv χ (2.7) where R L          v , 2 2 1 1 1 1 1 1 2 2 2 2 r r b b r r b b                           h , 2 1 2 2 R L R L b                       χ . Taking the second order derivative with respect to time of (2.6) yields 1 2   ζ hv Ψ Ψ (2.8) where, D D D D 1 D D D D cos sin sin cos sin cos cos sin x y x y x y x y                         Ψ hv D D 2 D D sin cos cos sin x y x y                     Ψ . Remark 2.2: If 1 ≠ 0, then h is an invertible matrix. Let us define the position tracking error vector as   T 1 2 de e e = ζ - ζ (2.9) where dζ 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   T d ,0Cζ . 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 2Ψ in (2.2) depends on the velocities and accelerates of slippage directly, so it is uncertain. Thus, on auxiliary variable is proposed as follows:  1 d 1   κ = h ζ Λe Ψ (2.14) On the other hand, (2.23) can be rewritten as follows: d   Mv τ Bv d τ (2.15) where     d Qγ C G Bv . Next, one control law is chosen via the computing-torque method as follows:  1 ˆˆ ˆ ˆ, ,     τ Mh Kφ f x W V (2.19) where K is a 2 × 2 diagonal, constant, positive definite matrix and is chosen arbitrarily. 𝐟(𝐱, �̂�, �̂�) is 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 T T1ˆ ˆ ˆˆ   W H σφ σV xφ φ W (2.24)  T T2ˆ ˆ ˆ V H xφ W σ φ V (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 1. 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. Figure 2.2. structure of the three-layer neural network. ∑ ∑ Input layer Hidden layer Output layer x1 x2 y1 y2  1  yN3 wN2N3  v1  v2  vN2  w1  wN3 x N1 v 11 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:     D D 2 3cos 0,2 0,5 3sin 0,2 x t y t       (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. Figure 2.4. the timelines of slip velocities. 2.7. Conclusion for chapter 2 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. 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 time (s) v e lo c it ie s o f w h e e l s lip ( m /s ) velocities of wheel slip longitudinal slip of the right wheel longitudinal slip of the left wheel lateral slip controller WMR subject to slippag e Target (xD, yD) e + - Eq. (3.2) Three-layer neural network v  Eq. (4.6) Bảng 2.1. Các tham số của rô bốt di động [21]. Variable Meaning value r Radius of each wheel 0,065 (m) b Haft of the wheel shaft 0,375 (m) IG The inertia of moment of the platform about the vertical axis through G. 15,625 (kg.m2) IW The inertia of moment of the wheel about the rotational axis . 0,0025 (kg.m2) ID The inertia of moment of the wheel about the diameter axis . 0,005 (kg.m2) mG The mass of the platform 30 kg mW The mass of each wheel 1 kg C The distance between M and P 0,5 m a The distance between M and G 0,3 m Figure 2.5. control performance comparison between two control methods in example 2.1. Figure 2.6. Comparison of tracking errors between two control methods in example 2.1. Figure 2.7. the torques of the proposed method in example 2.1. CHAPTER 3. DESIGNING A ROBUST ADAPTIVE BACKSTEPPING CONTROL LAW BASED A GAUSSIAN WAVELET NETWORK 3.1. Problem statement Even though the control method in chapter 2 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 kine