Thiết kế bộ điều khiển thích nghi cho robot hai bánh tự cân bằng sử dụng chiến lược điều khiển trượt tầng và mạng nơron RBT

Nội dung text: Thiết kế bộ điều khiển thích nghi cho robot hai bánh tự cân bằng sử dụng chiến lược điều khiển trượt tầng và mạng nơron RBT

1. P-ISSN 1859-3585 E-ISSN 2615-9619 SCIENCE - TECHNOLOGY DESIGNING AN ADAPTIVE CONTROLLER FOR TWO-WHEELED SELF-BALANCING MOBILE ROBOT USING HIERARCHICAL SLIDING CONTROL STRATEGY AND RADIAL BASIS FUNCTION NEURAL NETWORK THIẾT KẾ BỘ ĐIỀU KHIỂN THÍCH NGHI CHO ROBOT HAI BÁNH TỰ CÂN BẰNG SỬ DỤNG CHIẾN LƯỢC ĐIỀU KHIỂN TRƯỢT TẦNG VÀ MẠNG NƠRON RBF Hoang Thi Tu Uyen1, Kim Dinh Thai2, Pham Viet Anh2, Dinh Xuan Minh3, Le Xuan Hai2,* unstable system that comprises a two-wheel chassis and an ABSTRACT inverted pendulum body. Despite the inverted pendulum This paper presents a novel adaptive controller for two-wheeled self- structure of the TWSBMR causing postural instability, it balancing mobile robots combining sliding mode control and hierarchical sliding provides many benefits for driving efficiency. In contrast to control techniques. In addition, the radial basis function neural networks a three-wheeler or a four-wheeler, it can pass through (RBFNN) are also applied to approximate the uncertain components in the narrow spaces and allows the driver to maintain an upright system. The stability of the closed-loop control system is proven based on the posture on inclined terrains and steer on the spot. As a Lyapunov principle. The simulation results show that the proposed controller's result, TWSBMR is widely used in practical applications, response quality is excellent even if the system is affected by unexpected such as unmanned navigation vehicles [1-3], personal external disturbances. transporters [4, 5], wheeled humanoids [6], and robot Keywords: Two-wheeled self-balancing mobile robot, Sliding mode control, wheelchairs for the disabled [7, 8]. Hierarchical sliding control, Radial basis function neural network. The TWSBMR is characterized by highly nonlinear and TểM TẮT inherently unstable dynamics and is classified as an underactuated system [9]. With just two actuator inputs of Bài bỏo này trỡnh bày về một bộ điều khiển thớch nghi mới cho robot hai both wheels, it implements three movements: pitch, yaw, bỏnh tự cõn bằng bằng việc kết hợp những kỹ thuật điều khiển trượt và điều and forward. Therefore, controlling a robot to move as khiển trượt tầng. Bờn cạnh đú, mạng nơ ron RBF cũng được sử dụng để xấp xỉ cỏc desired while maintaining upright posture is a thành phận phi tuyến trong hệ thống. Tớnh ổn định của hệ thống điều khiển challenging topic that has drawn the attention of vũng kớn được chứng minh dựa theo nguyờn lý Lyapunov. Những kết quả mụ researchers around the world [9]. Similar to other phỏng cho thấy chất lượng đỏp ứng của bộ điều khiển đề xuất là rất tốt ngay cả underactuated systems, control methods for TWSBMR are khi hệ thống chịu ảnh hưởng bởi nhiễu ngoài khụng biết trước. diverse, ranging from simple linear control techniques to Từ khúa: Xe hai bỏnh tự cõn bằng, điều khiển trượt, điều khiển trượt tầng, complex nonlinear control techniques. Several studies mạng nơ ron RBF. based on linear control techniques for TWSBMR have been reported in the literature, such as PID control [10- 1Faculty of Automation, Industrial University of Ho Chi Minh City 12], pole placement [13, 14], and linear quadratic 2HaUI Institute of Technology, Ha Noi University of Industry regulator (LQR) [15, 16]. However, linear control 3Faculty of Electrical Engineering, Hanoi University of Industry approaches cannot maintain postural stability as soon as *Email: hailx@haui.edu.vn the TWSBMR enters a zone of nonlinear behavior with Received: 28/9/2021 large pitch angles due to intentional maneuvers or Revised: 28/10/2021 external disturbances [17]. Many nonlinear control Accepted: 15/11/2021 methods have been developed to address this issue, such as feedback linearization [18, 19], model predictive control (MPC) [20, 21], and sliding mode control (SMC) 1. INTRODUCTION [22-24], fuzzy control [25-29], neuro-adaptive control [30- A two-wheeled self-balancing mobile robot (TWSBMR) 33]. Consequently, nonlinear control approaches remain or two-wheeled inverted pendulum robot is a naturally favored and more efficient for TWSBMR control. Website: Vol. 57 - Special (Nov 2021) ● Journal of SCIENCE & TECHNOLOGY 39
2. SCIENCE - TECHNOLOGY P-ISSN 1859-3585 E-ISSN 2615-9619 Among the nonlinear control methods, the SMC angle in pitch motions. Hence, TWSBMR's motion is scheme is an excellent candidate for controlling characterized by the state vector q  x θ ψT . underactuated, nonlinear systems. SMC is designed in two steps. First, design an appropriate sliding surface that determines the system's behavior during sliding. Subsequently, a control action is designed to lead all state trajectories to the sliding surface in finite-time and then force them to remain there. Once trajectories are established on the sliding surface, the system becomes insensitive to modeling errors and external disturbances. There have been some other variations based on the SMC scheme to design a controller for TWSBMR, such as discrete-time SMC [34, 35], higher-order SMC [36]. For instance, H. Aithal and S. Janardhanan [36] have proposed a second-order SMC method for trajectory tracking of a two- Figure 1. Schematic of the two-wheeled self-balancing mobile robot wheeled mobile robot. Although this approach has the advantage of eliminating chattering, it is computationally The relationship between a wheel's torque and the heavy compared to conventional SMC and requires prior current flowing through it is described as follows knowledge of system parameters. T K i L m L (1) In this paper, we present a novel control method for TR K m i R TWSBMR, named adaptive hierarchical sliding mode control (AHSMC), combining SMC, hierarchical SMC (HSMC) where TL, TR, iL, iR is the torque, the electric current of the and radial basis function neural networks (RBFNN). The left and right wheel, respectively. HSMC technique is used in this study since it is well suited With model parameters as listed in Table 1, TWSBMR's to underactuated systems and is also highly sustainable equation of motion is as follows [37]. First, the controller is designed based on SMC and Mq C D q  G Bτ (2) HSMC techniques, abbreviated as HSMC, to stabilize the system states on the sliding surface. The RBF neural where MCD,, ℝ is the inertia matrix, centrifugal network is then employed to approximate TWSBMR's and Coriolis force matrix, damping matrix, respectively. uncertainty components. The RBF network is used here G ℝ is the gravity matrix, B ℝ is the input because it can approximate any nonlinear function with transformation matrix, and τ  i i T is the input matrix. arbitrary precision when hidden layer nodes are large LR enough [38]. Moreover, it has a simple structure with only m11 m 12 0 one input layer, one hidden layer and one output layer, and M M q m21 m 22 0 , it is very suitable for real-time applications. As a result, the 0 0 m proposed controller is robust against parametric 33 uncertainties and external disturbances, allowing fast 0 c12 c 13 convergence and high tracking accuracy. C C q, q 0 0 c , 23 The remainder of the paper is organized as follows: c c c Section 2 presents the dynamic model of the system, while 31 32 33 the control design steps are introduced in Section 3; Section 4 provides simulation results, and some concluding d11 d 12 0 0 remarks are drawn in Section V. D d21 d 22 0 , G G q mB glsin  , 2. DYNAMIC MODEL 0 0 d33 0 It is necessary to build a dynamic model of the system before designing the controller for TWSBMR. A reliable KKm m dynamic model is a prerequisite for any model-based r r control design. This study uses the TWSBMR mathematical BKK m m model as detailed in [39], which is built from the Euler- K d K d Lagrange equation. m m 2r 2r The TWSBMR comprises three rigid bodies, two wheels on either side and an inverted pendulum, as shown in Fig. The elements of the matrices M, C, D are as follows 1. In the fixed coordinate system {N}, x denotes the robot's 2J m = m +2m + , m m m lcosθ , m I m l2 displacement in straight motions, ψ denotes the robot's 11 B W r2 12 21 B 22 2 B rotation angle in yaw motions, and θ denotes the body's tilt 40 Journal of SCIENCE & TECHNOLOGY ● Vol 57 - Special (Nov 2021) Website:
3. P-ISSN 1859-3585 E-ISSN 2615-9619 SCIENCE - TECHNOLOGY 2 J d 2 2 u1 i L i R m33 I 3 2K m W 2 I 3 I 1 m B l sin  (4) r 2 u2 i R i L   c12 m B l  sin  , c13 m B lψsinθ , Then, Eq. (3) is rewritten as follows 2  c I I m l ψsinθcosθ x1 x 2 23 3 1 B x f g u 2 1 1 1  2  c31 m B lψsinθ , c32 I 3 I 1 m B l ψsinθcosθ , x 3 x 4 (5) 2  x f g u c33 I 3 I 1 m B l θsinθcosθ 4 2 2 1  2 x5 x 6 2cα 2cα d cα d11 , d12 d 21 , d22 2cα , d33 x f g u r2 r 2r2 6 3 3 2 Table 1. Model parameters of TWSBMR Definition of tracking errors e x x x x; e e Symbol Definition 1 1 d d 2 1  d Distance between the two wheels e3 x 4 θ d θ θ d; e 4 e 3 l Length of pendulum e5 x 5 ψ d ψ ψ d; e 6 e 5 r Radius of wheels Where xd, θd, ψd are the reference values of x, θ and ψ, mB Mass of the pendulum body (except wheels) respectively. mW Mass of each wheel Hence, Eq. (5) is rewritten in error form as follows J, K Mass moment of inertia (MOI) of each wheel w.r.t. the e1 e 2 wheel axis and the vertical axis.   e2 f 1 g 1 u 1 x d I1, I2, I3 MOI of the pendulum body w.r.t.  e3 e 4 (6) From (2), it is deduced that   e4 f 2 g 2 u 1 θ d x f1 b 1 b 1 e 5 e 6 iR θ f b b (3)   2 2 2 e6 f 3 g 3 u 2 ψ d iL ψ f c c 3 1 1 3.1. Hierarchical sliding mode controller design m22 m21 This section applies the hierarchical sliding mode m12 K m m11 K m r r control strategy [37] to design a controller that stabilizes where b1 , b2 , forward and pitch movements. m12 m 21 m 11 m 22 m12 m 21 m 11 m 22 Step 1: Considerring the first subsystem in (6) K d c m , 1 e1 e 2 2ra33 (7)   e2 f 1 g 1 u 11 x d   m22 d 11 m 12 d 21 x c 12 d 12 m 22 m 12 d 22 θ where u11 is the virtual control signal to ensure this m c m c ψ m m glsinθ 22 13 12 23 12 B subsystem is stable, i.e. lime1 0 f t 1 m m m m 12 21 11 22 The first-level sliding surface for first subsystem is   m21 d 11 m 21 m 11 x c 12 d 12 m 21 d 22 θ defined as  s c e e (8) m11 c 23 m 21 c 13 ψ m 11 m B glsinθ 1 1 1 2 f2 m12 m 21 m 11 m 22 where c1 is arbitrary positive constant. c x c θ c d ψ The control law for (7) consists of two components, the f 31 32 33 33 equivalent control law and switching control law, which are 3 m 33 designed as follows: 3. CONTROLLER DESIGN u11 u 11eq u 11sw (9) Let us define c1 e 2 f 1 T T u x x x x x x x x x θ θ ψ ψ 11eq  1 2 3 4 5 6  g1 (10) k s  sign() s  x Two virtual control signals are defined as follows u 1 1 1 1 d 11sw g1 Website: Vol. 57 - Special (Nov 2021) ● Journal of SCIENCE & TECHNOLOGY 41
4. SCIENCE - TECHNOLOGY P-ISSN 1859-3585 E-ISSN 2615-9619 where k1, η1 are arbitrary positive constants. Differentiating both sides of (18) with respect to time yields Differentiating s1 with respect to time yields    VSS * (19) s1 c 1 e 2 f 1 g 1 u 11eq u 11sw x d where k , η are arbitrary positive constants.  2 2 c1 e 2 f 1 g 1 u 11eq g 1 u 11sw x d The common control signal (u1) for the first and second k1 s 1 η 1 sign s 1 subsystem in (33) is defined as: Considering the following Lyapunov function as follows: u1 u 11eq u 11sw u 12eq u 12sw (20) 1 2 Substituting (28), (29) and (34) into (33) and simplifying V s 12 1 yields Differentiating V1 with respect to time yields λ g β g u u  1 1 1 2 11sw 12sw   2 S V1 s 1 s 1 k 1 s 1  1 s 1 sign s 1 0   (21) β1 g 2 u 11eq λ 1 g 1 u 12eq λ 1 x d β 1 θ d Thus, s is stable according to the Lyapunov criterion, 1 k2 S η 2 sign S i.e. lims1 0 . According to the definition of the sliding t The control signals u11sw, u 12sw are designed so that surface, the state error lime1 0 . t λ g u β g u 1 1 12eq 1 2 11eq Step 2: Considerring the second subsystem in (6): u12sw u 11sw λ1 g 1 β 1 g 2 e e (22) 3 4   (11) k2 S η 2 sign S λ 1 x d β 1 θ d  e 4 f 2 g 2 u 12  d λ1 g 1 β 1 g 2 The first-level sliding surface for second subsystem is Then, defined as:   2 V S* S k2 S  2 S 0 (23) s2 c 2 e 3 e 4 (12) Substituting (10), (14) and (22) into (20), we derive the where c is arbitrary positive constant. 2 control signal for the first two subsystems in (33) as follows: The control law for (11) consists of two components, the λ1 f 1 β 1 f 2 λ 1 c 1 e 2 β 1 c 2 e 4 equivalent control law and switching control law, which are u1 designed as follows: λ1 g 1 β 1 g 2 (24)   u u u (13) k2 S η 2 sign S λ 1 x d β 1 θ d 12 12eq 12sw λ g β g c e f 1 1 1 2 2 4 2 u12eq (14) 3.2. Sliding mode controller design g2 Considerring the 3rd subsystem in (6): Step 3: Considerring first and second subsystem in (6) e e  5 6 e1 e 2 (25) e f g u   6 3 3 2 e2 f 1 g 1 u 1 x d (15)  The sliding surface for this subsystem is defined as: e3 e 4   s3 e 6 c 3 e 5 (26) e4 f 2 g 2 u 1 θ d where c is a arbitrary positive constant. The second-level sliding surface for two subsystems is 3 defined as: Taking the derivative of s3 with respect to time yields s3 f 3 g 3 u 2  d c 3 e 6 (27) S λs11 βs 12 λce 121 e 2 βce 123 e 4 (16) Using a Lyapunov candidate function as follows: where λ1, β1 are arbitrary positive constants. Differentiating S with respect to time yields 1 2 V3 s 3 (28)  2 S λs11 βs 12  λce 121  e  2 βce 123  e  4 (17) Differentiating both sides of (28) with respect to time   λce122 βce 124 λf 11 gu 11 x d βf 12 gu 21 θ d yields    Let us consider the following Lyapunov function as: V3 s 3 s 3 s 3 f 3 g 3 u 2 ψ d c 3 e 6 1 VS 2 (18) The control signal for (25) is defined as: 2 1  u2 g 3 f 3 ψ d c 3 e 6 k 3 sign s 3 η 3 s 3 (29) 42 Journal of SCIENCE & TECHNOLOGY ● Vol 57 - Special (Nov 2021) Website:
5. P-ISSN 1859-3585 E-ISSN 2615-9619 SCIENCE - TECHNOLOGY where k3, η3 are arbitrary positive constants. Fig. 2 shows the proposed TWSBMR control structure  2 diagram, in which the three networks RBF1, RBF2 and RBF3 Then V3 k 3 s 3  3 s 3 0 . are RBFNNs used to approximate functions f1, f2, f3 Thus, s3 is stable according to the Lyapunov criterion, respectively. These networks have the same structure, as i.e. lims3 0 and lime5 0 depicted in Fig. 3. The RBFNN's inputs include a position t t T T     For reducing chattering at high frequencies, the sign(.) vector q  x θ ψ and a velocity vector q x θ ψ . function in (24), (29) is replaced by a sat(.) function defined The output of the RBF1, RBF2 and RBF3 networks as follows: ˆ ˆ ˆ denoted f1,, f 2 f 3 , are approximations of f1, f2, f3, respectively. sign x , x 1 sat x (30) The RBFNN's hidden layer includes l nodes x, x 1 T h  h1 h 2  h l  , defined as follows: By substituting the virtual control signals u1, u2 provided   in (24), (29) into (4), we determine the corresponding q c1i ,q c 1i q c 2i ,q c 2i exp current signals for the TWSBMR's wheel motors. 2 bi (31) 3.3. Adaptive rule design hi , i 1,2,  ,l. l q c1j ,q c 1j q c 2j ,q  c 2j Considering the dynamic model of TWSBMR in (27): the exp  2 matrices C, D, G contain uncertain components that are j 1 bj difficult to determine in practice. Additionally, the functions f1, f2, f3 in (32) contain the elements of these matrices. where .,. is the scalar product operator defined in Therefore, SMC and HSMC controllers designed in the normed space 〈ℝ, ‖. ‖〉. previous section are unlikely to achieve high accuracy in actual TWSBMR control. As a result, to increase the It is noted that with sufficient nodes in the hidden layer robustness of HSMC and SMC controllers with model (l), an RBF neural network can approximate any nonlinear uncertainty and the effects of unknown external function with arbitrary precision. disturbances, this section proposes an AHSMC controller As a result, the RBF1, BRF2 and RBF3's outputs are using RBF neural network (RBFNN) [38] for adaptively presented as follows: estimation the functions f1, f2, f3. T f1 W 1 h  1 T f2 W 2 h  2 (32) T  f3 W 3 h 3 where ε1, ε2, ε3 are minor errors, T Wi  w i1 w i2 w il  with i = 1, 2, 3 is the ideal weights vector between the hidden and the output layer. ˆ ˆ ˆ Let WWW1,, 2 3 denote estimations of WWW1,, 2 3 respectively. fˆ Wˆ T h 1 1 ˆ ˆ T f2 W 2 h (33) ˆ ˆ T f3 W 3 h Figure 2. Proposed TWSBMR control structure diagram The estimation errors for WWW1,, 2 3 are determined as: WWW ˆ 1 1 1  ˆ WWW2 2 2 (34) WWW ˆ 3 3 3 The control signals in (47) and (50) are rewritten as follows: ˆ ˆ   λ1 f 1 β 1 f 2 λ 1 c 1 e 2 β 1 c 2 e 4 k2 S η 2 sat S λ 1 x d β 1 θ d uˆ1 (35) Figure 3. RBF neural network structure for approximating functions f1, f2, f3 λ1 g 1 β 1 g 2 λ 1 g 1 β 1 g 2 Website: Vol. 57 - Special (Nov 2021) ● Journal of SCIENCE & TECHNOLOGY 43
6. SCIENCE - TECHNOLOGY P-ISSN 1859-3585 E-ISSN 2615-9619 ˆ 1 ˆ  S η sat S k S S λ ε β ε u2 g 3 f 3ψ d c 3 e 6 k 3 sat s 3 η 3 s 3 (36) 2 2 1 1 1 2 V (44) αStrWWTT W  trWW  W  Theorem: An updated law for neural network's weight 1 1 1 2 2 2  matrices are selected as follows: Next, we consider the following Lyapunov function   WWFˆ  λ Sh α S W ˆ 1 1 1 1 1 12 1  T 1  V3 s 3 tr W 3 F 3 W 3 (45) ˆ  ˆ 2 2 WWF2 2 2 β 1 Sh α S W 2 (37) Taking the time derivative of V3 gives  ˆ ˆ T 1  W3 W 3 F 3 sh α s 3 W 3 V s s tr W  F W  (46) 3 3 3 3 3 3 where F1, F2, F3 and α, σ are preselected positive Eq. (27) gives us: constants.  T 1  V33323 s g uˆ f ψ d36 c e tr W 333 F W (47) If the following conditions are satisfied Substituting (36) into (47) results in: 2 2 ε WW1 2 N1 F F T 1  S α V s k sat s η s s f fˆ tr W  F W  3 33 3 33333 333 k2 4k 2 (38) 2 Tˆ T T 1   ε W3 s33 k sat s 33333 ηs sWhε 33 Wh trWFW 333 s N2 σ F 3 (48) η3 4η 3 2 T T 1  s k sat s ηs sε sWh trWFW   33 3 33 333 333 where ε λ ε β ε , ε ε then the closed- N1 1 1 1 2 N2 3  s k sat s ηs2 sε trW T shFW 1 ˆ loop system will be stable according to Lyapunov criterion. 33 3 3333 33 33 Proof: Substituting (37) into (48) gives Considerring the Lyapunov function in quadratic form V s k sat s ηs2 sε σstrW  T W W  (49) as follows: 3 33 33333 3 333 2 1 1 1 T   2 T 1   T 1  Notice that tr W3 W 3 W 3 . V S tr W1 F 1 W 1 tr W 2 F 2 W 2 (39) F 2 2 2 Using the Cauchy-Schwart inequality, we have Here, the matrix trace operator tr(X) is defined as 2 tr WT W W  W W  W  ,,, i 1 2 3 the sum of all the elements along the main diagonal of i i i iF iFF i matrix X. It can be deduced from (44), (49) that Taking the derivative of both sides of (39) with respect 2 SηsatS2 kS 2 Sλε 1 1 βε 1 2  to time yields  V 2 2  (50) T 1 T 1  α S W W W  α S W  W W  V SS tr W  F W  tr W  F W  (40) 1FFFF 1FF 1 2 2 2 1 1 1 2 2 2  2 Using the estimated control signal uˆ from RBFNN, Eq. V s k sat s η s2 s ε σ s W  W W  (51) 1 3 33 33333 3 3FF 3F 3 (21) is rewritten as follows: Hence  ˆ   S λce112 βce 124 λf 11 βf 12 λg 11 βgu 121 λx 1d βθ 1d (41) 2 2  1  1 SηsatSαSW2 1 W 1 αSW 2 W 2 Substituting (32), (33), (34) and (35) into (41), one can FF2FF 2  2 2 find out that: V    WW1 2  TT  S k S ε α FF S η2 sat S k 2 S λ 1 W 1 h β 1 W 2 h λ 1 ε 1 β 1 ε 2 (42) 2 N1 (52) 4 From this it can be deduced that:  2 2  2 1 W3 Sη2 sat S k 2 S S 2 λ 1 ε 1 β 1 ε 2   F V3 s 3 k 3 sat s 3 σ s 3 WW3 3 s 3 η 3 s 3 ε N2 σ F 2F 4 T 1 T 1  V trWFW   trWFW   1 1 1 2 2 2 Thus, if condition (38) is satisfied, one can deduce that: TT  Sλ1 W 1 h Sβ 1 W 2 h 2  2 WW1 2 2 FF Sη sat S k S S λ ε β ε k2 S ε N1 α 0  2 2 1 1 1 2 4 V 0 V (43) T 1ˆ  T 1 ˆ 2  tr W1 Sλ 1 h F 1 W 1 tr W 2 Sβ 1 h F 2 W 2 W V3 0   3 F η3 s 3 ε N2 σ 0 Substituting (37) into (43), we obtain: 4 44 Journal of SCIENCE & TECHNOLOGY ● Vol 57 - Special (Nov 2021) Website:
7. P-ISSN 1859-3585 E-ISSN 2615-9619 SCIENCE - TECHNOLOGY This means that all subsystems in (5) are asymptotically stable at the origin by Lyapunov's criteria. The theorem has been proved. 4. SIMULATION RESULTS To verify the effectiveness of the AFHSMC controller, we conducted some simulations using Matlab software. As a comparison of both controllers, simulations were also performed with the HSMC controller. For the simulations, the TWSBMR system parameters are assumed to be known as 2 mb = 116(kg), J = 16.25(kg.m ), r = 0.1(m), l = 0.23(m), 2 I1 = 0.26(kg.m ), d = 0.19(m), mw = 11.4(kg), I2 = 0.165 2 2 (c) Yaw motion (kg.m ), I3 = 0.2(kg.m ), by = 5(Ns.m), brx = 3.68 (Ns.m), g = 10(m/s2), α = 56o Figure 4. Simulation results without unknown external disturbance In addition, the controller parameters are determined Figure 4 shows the system's responses in the absence of empirically, i.e., error and trial, to achieve the best possible external disturbances. For both AFHSMC and HSMC control quality. controllers, all responses rapidly approached and stabilized at their reference values. These results show that AFHSMC's λ1 = 3, β1 = 0.1, c1 = 5, c2 = 0.01, k2 = 0.01, η2 = 10, λ = 12, control quality is better than HSMC's, especially with regard k3 = 3, η3 = 2, l = 15, F1 = F2 = 5, F3 = 20, α = 0.1, σ = 0.65 to pitch and yaw motions. We verified the system's responses using the following Figure 5 shows that AFHSMC outperforms HSMC under reference values: xd = 1(m), θd = 0(degree), external disturbances. In the case of TWSBMR systems ψd = (180/π)x0.1sin(2πt) (degree) equipped with the HSMC controller, all system outputs are We invested two experiments in a simulated knocked out of steady equilibrium as soon as external noise environment with/without the effect of unknown external appears. Meanwhile, with the AFHSMC controller, all disturbances. system outputs are almost unaffected by external noises. These results are because the AFHSMC controller has strong adaptability to the model uncertainty and the effects of unknown external disturbances with the proposed adaptive law. (a) Straight motion (a) Straight motion (b) Pitch motion (b) Pitch motion Website: Vol. 57 - Special (Nov 2021) ● Journal of SCIENCE & TECHNOLOGY 45
8. SCIENCE - TECHNOLOGY P-ISSN 1859-3585 E-ISSN 2615-9619 [6]. M. Stilman, J. Wang, K. Teeyapan, R. Marceau, 2009. Optimized control strategies for wheeled humanoids and mobile manipulators. 2009 9th IEEE-RAS International Conference on Humanoid Robots, pp. 568-573. [7]. H. Uustal, J. L. Minkel, 2004. Study of the Independence IBOT 3000 Mobility System: an innovative power mobility device, during use in community environments. Archives of physical medicine and rehabilitation, vol. 85 12, pp. 2002-10. [8]. M. Nikpour, L. Huang, A. M. Al-Jumaily, 2020. Stability and Direction Control of a Two-Wheeled Robotic Wheelchair Through a Movable Mechanism. IEEE Access, vol. 8, pp. 45221-45230. [9]. R. P. M. Chan, K. A. Stol, R. Halkyard, 2013. Review of modelling and control of two-wheeled robots. Annu. Rev. Control., vol. 37, pp. 89-103. (c) Yaw motion [10]. J. Meng, A. Liu, Y. Yang, Z. Wu, Q. Xu, 2018. Two-Wheeled Robot Figure 5. Simulation results with unknown external disturbance Platform Based on PID Control. in 2018 5th International Conference on Thus, the simulation results demonstrate that the Information Science and Control Engineering (ICISCE), pp. 1011-1014. proposed controller can accurately control the position and [11]. V. Mudeng, B. Hassanah, Y. T. K. Priyanto, O. Saputra, 2020. Design and orientation of the TWSBMR while maintaining a minimal Simulation of Two-Wheeled Balancing Mobile Robot with PID Controller. pitch angle. Furthermore, the proposed controller is robust International Journal of Sustainable Transportation Technology, Volume 3, Issue against external noise as the TWSBMR moves. 1, pp 12-19 5. CONCLUSION [12]. C. Ben Jabeur, H. Seddik, 2020. Design of a PID optimized neural This paper has proposed an adaptive controller for networks and PD fuzzy logic controllers for a two wheeled mobile robot. Asian TWSBMR combining SMC, hierarchical SMC (HSMC) and RBF Journal of Control. neural network. The controller SMC-HSMC acts as the [13]. A. Fahmi, et al., 2020. Modelling and Control using Pole Placement central controller to ensure the stable system's state on the Method for Self Balancing Robot. Solid State Technology, vol. 63, pp. 1314-1324. sliding surface. An adaptive rule is designed to [14]. Y. Amano, 2014. Stability Control for Two-Wheeled Mobile Robot Using approximate the uncertain components in the system. Robust Pole-Placement Method. Consequently, the proposed controller is robust in actual TWSBMR control under uncertain model parameters or [15]. N. Uddin, T. A. Nugroho, W. A. Pramudito, 2017. Stabilizing Two- unexpected external disturbances. The simulation results wheeled robot using linear quadratic regulator and states estimation. 2017 2nd show that the system responses quickly converge to their International conferences on Information Technology, Information Systems and reference and are little affected by unknown external Electrical Engineering (ICITISEE), pp. 229-234. disturbances. The stability of the proposed control system [16]. P. Oryschuk, A. Salerno, A. M. Al-Husseini, J. Angeles, 2009. is also rigorously proven according to Lyapunov's principle. Experimental Validation of an Underactuated Two-Wheeled Mobile Robot. In the subsequent studies, we will test the proposed IEEE/ASME Transactions on Mechatronics, vol. 14, pp. 252-257. algorithm for actual TWSBMR and verify the effectiveness of [17]. S. Kim, S. Kwon, 2017. Nonlinear Optimal Control Design for the proposed controller with many real-life scenarios. Underactuated Two-Wheeled Inverted Pendulum Mobile Platform. IEEE/ASME Transactions on Mechatronics, vol. 22, pp. 2803-2808. [18]. K. Pathak, J. Franch, S. K. Agrawal, 2005. Velocity and position control of REFERENCES a wheeled inverted pendulum by partial feedback linearization. IEEE Transactions [1]. Y.S. Ha, S. i. Yuta, 1996. Trajectory tracking control for navigation of the on Robotics, vol. 21, pp. 505-513. inverse pendulum type self-contained mobile robot. Robotics Auton. Syst., vol. 17, [19]. M. Muhammad, S. Buyamin, M. N. Ahmad, S. W. Nawawi, Z. Ibrahim, pp. 65-80. 2012. Velocity tracking control of a Two-Wheeled Inverted Pendulum robot: A [2]. F. Grasser, A. D'Arrigo, S. Colombi, A. C. Rufer, 2002. JOE: a mobile, comparative assessment between partial feedback linearization and LQR control inverted pendulum. IEEE Trans. Ind. Electron., vol. 49, pp. 107-114. schemes. International Review on Modelling and Simulations, vol. 5, pp. 1038- 1048. [3]. T. Takei, R. Imamura, S. i. Yuta, 2009. Baggage Transportation and Navigation by a Wheeled Inverted Pendulum Mobile Robot. IEEE Transactions on [20]. M. ệnkol, C. k. Kasnakp lu, 2018. Adaptive Model Predictive Control of a Industrial Electronics, vol. 56, pp. 3985-3994. Two-wheeled Robot Manipulator with Varying Mass. Measurement and Control, vol. 51, pp. 38 - 56. [4]. S. C. Lin, C. C. Tsai, H. C. Huang, 2011. Adaptive Robust Self-Balancing and Steering of a Two-Wheeled Human Transportation Vehicle. Journal of [21]. D. P. Nam, N. H. Quang, D. C. H. Phi, T. N. Anh, D. Bao, 2019. Robust Intelligent & Robotic Systems, vol. 62, pp. 103-123. Model Predictive Control Based Kinematic Controller for Nonholonomic Wheeled Mobile Robotic Systems. Lecture Notes in Networks and Systems Springer [5]. P. Petrov, M. Parent, 2010. Dynamic modeling and adaptive motion International Publishing. control of a two-wheeled self-balancing vehicle for personal transport. 13th International IEEE Conference on Intelligent Transportation Systems, pp. 1013- [22]. A. Sinha, P. Prasoon, P. Bharadwaj, A. Ranasinghe, 2015. Nonlinear 1018. Autonomous Control of a Two-Wheeled Inverted Pendulum Mobile Robot Based on 46 Journal of SCIENCE & TECHNOLOGY ● Vol 57 - Special (Nov 2021) Website:
9. P-ISSN 1859-3585 E-ISSN 2615-9619 SCIENCE - TECHNOLOGY Sliding Mode. 2015 International Conference on Computational Intelligence and [38]. J. Liu, 2017. Intelligent Control Design and MATLAB Simulation. Networks, pp. 52-57. Springer. [23]. N. Esmaeili, A. Alfi, H. Khosravi, 2017. Balancing and Trajectory Tracking [39]. S. Kim, S. Kwon, 2015. Dynamic modeling of a two-wheeled inverted of Two-Wheeled Mobile Robot Using Backstepping Sliding Mode Control: Design pendulum balancing mobile robot. International Journal of Control, Automation and Experiments. Journal of Intelligent & Robotic Systems, vol. 87, pp. 601-613. and Systems, vol. 13, pp. 926-933. [24]. J. Mu, X. g. Yan, S. K. Spurgeon, Z. Mao, 2017. Nonlinear sliding mode control of a two-wheeled mobile robot system. Int. J. Model. Identif. Control., vol. 27, pp. 75-83. THễNG TIN TÁC GIẢ [25]. M. Begnini, D. W. Bertol, N. A. Martins, 2017. A robust adaptive fuzzy Hoàng Thị Tỳ Uyờn1, Kim Đỡnh Thỏi2, Phạm Việt Anh2, variable structure tracking control for the wheeled mobile robot: Simulation and 3 2 experimental results. Control Engineering Practice, vol. 64, pp. 27-43. Đinh Xuõn Minh , Lờ Xuõn Hải 1 [26]. G. R. Yu, Y. K. Leu, H. T. Huang, 2017. PSO-based fuzzy control of a self- Khoa Tự động húa, Trường Đại học Cụng nghiệp Thành phố Hồ Chớ Minh balancing two-wheeled robot. 2017 Joint 17th World Congress of International 2Viện Cụng nghệ HaUI, Trường Đại học Cụng nghiệp Hà Nội Fuzzy Systems Association and 9th International Conference on Soft Computing 3Khoa Điện tử, Trường Đại học Cụng nghiệp Hà Nội and Intelligent Systems (IFSA-SCIS), pp. 1-5. [27]. Á. Odry, I. Kecskộs, E. Burkus, P. Odry, 2017. Protective Fuzzy Control of a Two-Wheeled Mobile Pendulum Robot: Design and Optimization. WSEAS Transactions on Systems and Control archive, vol. 12. [28]. T. Zhao, Q. Yu, S. Dian, R. Guo, S. Li, 2019. Non-singleton General Type-2 Fuzzy Control for a Two-Wheeled Self-Balancing Robot. International Journal of Fuzzy Systems, pp. 1-14. [29]. M. Muhammad, A. A. Bature, U. Zangina, S. Buyamin, A. Ahmad, M. A. Shamsudin, 2019. Velocity control of a two-wheeled inverted pendulum mobile robot: a fuzzy model-based approach. Bulletin of Electrical Engineering and Informatics. [30]. C. C. Tsai, H. C. Huang, S. C. Lin, 2010. Adaptive Neural Network Control of a Self-Balancing Two-Wheeled Scooter. IEEE Transactions on Industrial Electronics, vol. 57, pp. 1420-1428. [31]. A. I. Glushchenko, V. A. Petrov, K. A. Lastochkin, 2018. On Development of Neural Network Controller with Online Training to Control Two-Wheeled Balancing Robot. 2018 International Russian Automation Conference (RusAutoCon), pp. 1-6. [32]. A. Chhotray, D. R. Parhi, 2019. Navigational Control Analysis of Two- Wheeled Self-Balancing Robot in an Unknown Terrain Using Back-Propagation Neural Network Integrated Modified DAYANI Approach. Robotica, vol. 37, pp. 1346 - 1362. [33]. A. Maity, S. Bhargava, 2020. Design and Implementation of a Self- Balancing Two-Wheeled Robot Driven by a Feed-Forward Backpropagation Neural Network. IRJET Journal Volume 7, Issue 9. [34]. A. Filipescu, V. Mợnzu, B. Dumitrascu, A. Filipescu, E. Minca, 2011. Trajectory-tracking and discrete-time sliding-mode control of wheeled mobile robots. 2011 IEEE International Conference on Information and Automation, pp. 27-32. [35]. P. N. Mali, N. G. Kulkarni, 2015. Discrete-Time Sliding Mode Control for Wheeled Mobile Robot Trajectory-Tracking. IOSR Journal of Electrical and Electronics Engineering (IOSR-JEEE) Volume 10, Issue 3 Ver. III. [36]. H. Aithal, S. Janardhanan, 2013. Trajectory tracking of two wheeled mobile robot using higher order sliding mode control. 2013 International Conference on Control, Computing, Communication and Materials (ICCCCM), pp. 1-4. [37]. D. Qian, J. Yi, D. Zhao, 2008. Hierarchical sliding mode control for a class of SIMO under-actuated systems. Control and Cybernetics, vol. 37, pp. 159-175. Website: Vol. 57 - Special (Nov 2021) ● Journal of SCIENCE & TECHNOLOGY 47