A neural network based sliding mode controller of folding-boom aerial work platform

Transcription

1 Research Article A neural network based sliding mode controller of folding-boom aerial work platform Advances in Mechanical Engineering 7, Vol. 9() 9 Ó The Author(s) 7 DOI:.77/ journals.sagepub.com/home/ade Haidong Hu, Ning Cai, Lizhen Cui, Yan Ren and Wensheng Yu Abstract Aerial work platform is a special vehicle for carrying personnel to the appointed site in the air for operations. Therefore, the work platform requires high stability. This article proposes a sliding mode controller based on neural network for tracking control of folding-boom aerial work platform. Since the chattering caused by sliding mode controller with high-speed switching control may lead to system performance degradation, continuous control obtained from neural network system replaces discontinuous switching control to eliminate chattering. Furthermore, the whole system is proved to be stable by Lyapunov stability theorem. Finally, numerical results show that the designed controller can eliminate the chattering resulting from switching control in sliding mode controller and inhibit the vibration of work platform when there exists system uncertainty. Moreover, the controller is effective for the reduction of tracking error. Keywords Aerial work platform, flexible multi-body dynamics, trajectory tracking control, neural network based sliding mode control Date received: 9 May 6; accepted: June 7 Handling Editor: Francesco Massi Introduction The folding-boom aerial work platform is a type of engineering vehicle which is used for enhancing the personnel to the designated place for installation and maintenance, as shown in Figure. Therefore, it requires high stability for the security of people working on the platform. As the extensive use of light-long beam in the structure of arm system of aerial work platform, elastic deformation of beam cannot be ignored to guarantee work platform s stable motion. For realizing trajectory tracking of aerial work platform, adaptive neural network controller is adoptedinjiaetal. and self-tuning fuzzy proportional integral derivative (PID) control scheme is proposed in Miao et al. However, the deformation of beam is not considered in the established model. Based on the theory of flexible multi-body dynamics and Lagrange s equation, the model of folding-boom aerial work platform with flexible beam driven by hydraulic cylinder is established, and the vibration existed in flexible beam is shown in Hu et al. Besides, the similar model is obtained, and fuzzy PID is used for the trajectory tracking of work platform in Meng, but this study only gives simulation results, and the stability of system is not proved. In addition, backstepping control method of work platform is proposed in Hu et al. 5 basedonthemodelinhuetal. However, this method can only be used for the accurate models. In fact, School of Information Engineering, Inner Mongolia University of Science & Technology, Baotou, China College of Electrical Engineering, Northwest University for Nationalities, Lanzhou, China School of Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing, China Corresponding author: Ning Cai, College of Electrical Engineering, Northwest University for Nationalities, Lanzhou 7, China. caining9@tsinghua.org.cn Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution. License ( which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages ( open-access-at-sage).

2 Advances in Mechanical Engineering This article is organized as follows. In section Flexible multi-body dynamic equations of work platform, the dynamic equations of aerial work platform is introduced and analyzed. Then, a NNSMC is designed to attain the control objective in section Design of NNSMC. In section Simulation results, the simulation study for tracking control of aerial work platform is carried out, and finally concluding comment is provided in section Conclusion. Figure. Scheme of folding-boom aerial work platform. there exists parameter uncertainty in the model of foldingboom aerial work platform. Sliding mode control (SMC) has robustness to model uncertainty, 6 which has been widely used for the control of nonlinear system. This article proposes an SMC method used for tracking control of aerial work platform. As a robust control scheme, SMC can make the state of the system move along the designed sliding mode surface using the switching control strategy. However, as the discontinuous switching property of SMC, it is difficult to make system state converge to equilibrium point along the sliding mode surface strictly. As a result, the chattering will be occurred. To reduce the chattering, several approaches have been proposed, 7 5 among which, neural network system is an alternative method. 5 Artificial neural network is derived from biological networks, 6 which is a kind of dynamical complex network 7 8 being closely relevant to the graph theoretic methods. As neural network system has capability of approximating any nonlinear function with arbitrary precision on a compact set based on the universal approximation theorem, it can solve the chattering problem existed in SMC. In Sun et al., 9 a neural network based sliding mode adaptive controller is proposed for robot manipulators; however, the link is considered as rigid. In addition, the similar control method is used for flexible links in Tang et al., but the links are driven by DC motors instead of hydraulic cylinder usedinaerialworkplatform. In this article, a neural network based sliding mode control (NNSMC) for tracking control of aerial work platform with flexible beam driven by hydraulic cylinder is presented. The proposed controller combines universal approximation capability of neural network system with the robustness of sliding mode controller. Based on the Lyapunov stability theorem, NNSMC can guarantee the stability of the whole system. In addition, the simulation results show that the aerial work platform s tracking error can be reduced, and the vibration can be inhibited effectively when there exist model parameter uncertainties. Flexible multi-body dynamic equations of work platform The flexible multi-body dynamic equations of aerial work platform can be written as ( G u + U _u + H q + R = Q u M q + Nq + H T u + V T _u ðþ = Q q In equation (), Q q = ½ Q Q Q 5 Q 6 Š T, Q u = ½ Q Q Š T, u = ½ u u Š T, _u = ½ _u _u ŠT, q = ½ q q q q Š T, and q = ½ q q q q Š T, in which u and u are the angles of beams and with respect to their horizontal planes; q, q and q, q are the deflection variables associated with the two former model functions of beams and, respectively. G, M, and H are the mass matrices and can be expressed as follows m + m + m l m + m l l cos (u u ) G = 6 7 m + m l l cos (u u ) m 5 + m l m M = m m 6 7 m 5 m l m l m l H = p p p cos (u u ) 6 7 m l 5 m l p p N is the coefficient matrix corresponding to generalized coordinate q and is given by EI p l 8EI p N = l EI p 6 l 8EI p 7 5 l

3 Hu et al. In addition, U and V are the coefficient matrices corresponding to _u and can be written as " U = m # + m l l sin (u u ) m + m l l sin (u u ) V = m l sin (u u ) p 5 in which Q u = ½ Q Q Š T and Q q = ½ Q Q Q 5 Q 6 Š T are the generalized forces corresponding to u = ½ u u Š T and q = ½ q q q q Š T, respectively. The model of folding-boom aerial work platform has following properties: Property. The matrix D(z) is symmetric positive definite and satisfies \d m kd(z) k d M, 8z R n, R is a column vector and is given by R = p m l sin (u u )(_u + _u )_q + m + m + m gl cos u m + m gl cos u T Design of NNSMC Design of SMC Define generalized coordinates as z = ½ u q Š T, where u = ½ u u Š T and q = ½ q q q q Š T, then _z = ½ _u _q Š T and z = ½ u q Š T.Therefore, the model of aerial work platform can be described as D(z) z + C(z, _z)_z + B(z)=u where D(z)= G H H T, C(z, _z)= U Y _ X M V T Y _ in which " u _Y = _ #, _u " # X = p m l sin (u u )(_u + _u ) In addition, B(z) is given by B(z)= m + m + m gl cos u m + m gl cos u EI p l q 8EI p l q EI p l q 8EI p l q T ðþ where m, m, and m are the mass of beams,, and load, respectively; l and l are the lengths of beams and ; g is the acceleration of gravity; E is the modulus of elasticity of the beam material, and I and I are the moment of inertia of the cross section of beams and. u is the control input and can be written as u = ½u u u u u 5 u 6 = Q T u Q T T q = ½Q Q Q Q Q 5 Q 6 Š T Š T where d m and d M denote the minimum and maximum eigenvalues of D(z), respectively. Property. _D(z) C(z, _z) is a skew symmetric matrix, that is, x T ½ _D(z) C(z, _z)šx =, 8x R n. For convenience, D(z), C(z, _z), and B(z) are written as D, C, and B, respectively. In addition, define tracking error as e = z z d ðþ where z d is reference trajectory; then, the sliding surface can be chosen as s = _e + le ðþ where l = diag½ l l l l l 5 l 6 Š, in which l i is positive constant. Therefore, the tracking control is expressed as follows: design a control law u to make sliding mode occur on sliding surface (equation ()), then tracking error (equation ()) approaches zero asymptotically with a prescribed transient response. 9 Define the reference state _z r = _z s = _z d le, z r = z _s = z d l_e Then, the control law u designed is shown below where u = ^u As K sgn s ^u = ^D z r + ^C_z r + ^B ð5þ wherein ^D, ^C, and ^B are estimates of D, C, and B, respectively, which reflect the model uncertainties. A = diag½ a a a a a 5 a 6 Š is diagonal matrix, in which a i is positive constant. Ksgns is switching control, in which K = diag½ K K K K K 55 K 66 Š is gain matrix with constant K ii..besides,sgns i is the sign function and is given by

4 Advances in Mechanical Engineering 8 < s i. sgn s i = s i = : s i \ Substituting equation (5) into equation () leads to D_s = (C + A)s + Df K sgn s ð6þ where Df is defined as the system uncertainties and is given by Df = DD z r + DC_z r + DB, in which DD, DC, and DB can be expressed as DD = ^D D, DC = ^C C, and DB = ^B B, respectively. For the dynamic model (equation ()) with sliding surface (equation ()), the stability of overall system with sliding mode controller can be ensured by control law (equation (5)). That is to say, the actual trajectory is able to follow the desired trajectory. Although the SMC can guarantee stability of the overall system, there is chattering phenomenon on sliding surface since switching control K sgn s is discontinuous function. Therefore, the control law (equation (5)) cannot guarantee that lim s =. In this article, a t! NNSMC is presented to eliminate the chattering. Design of NNSMC The radial basis function neural network (RBFNN), which has input layer, hidden layer, and output layer, is adopted in NNSMC. Figure shows the structure of RBFNN. Let w w w m f w w w m f W = , f = , w h w h w hm f h y y Y = y m then the output of RBFNN can be expressed as Y = W T f, where W is the weight matrix of the RBFNN and f is radial basis function (RBF) vector. As the universal approximation ability of RBFNN, the RBFNN with enough number of hidden layer neurons can be used to identify the Df defined in equation (6) such that Df = W T f + e ð7þ where W is the optimization weights matrix, and e is the 6 approximation error vector. To eliminate chattering in SMC, neural network output ^k is used for approximating the system uncertainties Df and is given by ^k = ^W T f ð8þ where the RBFNN weight matrix ^W can be adapted during learning process, and s is selected as neural network input. The control law can be chosen as follows u = ^D z r + ^C_z r + ^B As ^k Substituting equation (9) into equation () leads to D_s = (C + A)s + Df ^k ð9þ ðþ Theorem. For the dynamic model (equation ()) with the tracking error (equation ()), the sliding surface is defined as equation (), and the neural network based sliding mode controller is designed as equations (9) and (8), then the actual trajectory can converge to the desired trajectory. Proof. Define ^L =, L = ^W W with klk F L M and L = L ^L and then the selected Lyapunov candidate function is V = st Ds + tr( L T S L) ðþ where S is a matrix and can be written as S =,inwhichg is a 6 6 diagonal positive definite matrix. Since D is also a positive symmetric matrix, V G is positive definite. Define z = ½ s T Š T, the adaptive law of neural network weights can be designed as _^L = SFz T nskk^l z ðþ in which F = ½ f T Š T is the column vector, and n is the positive real number. The time derivative of equation () can be obtained as Figure. Radial basis function neural networks. _V = _st Ds + s T _Ds + s T T D_s + tr( L S L) _ ðþ

5 Hu et al. 5 As _D C is a skew symmetric matrix, namely s T _Ds = s T (C)s ðþ Substituting equation () into equation () leads to _V = s T ½D_s + CsŠ + tr( L T S _ L) ð5þ Substituting equations (7), (8), and () into equation (5) gives h _V = s T (C + A)s + Df ^k i + Cs + tr( L T S _ L) then = s T As + s T (Df ^k)+tr( L T S _ L) = s T As + s T ((W T ^W T )f + e )+tr( L T S _ L) Define the error of neural network weights as W T = W T ^W T In order to ensure _V\, the inequality below can be derived k min kk z e N + n L L L M. F F Therefore, the above inequality is tenable when (e N +(n=)l M =k min)\ kk. z Thus, by adjusting the values of n and k min, _V\ can be ensured. Since _V = only when s =, the overall system is asymptotically stable based on Lyapunov stability theorem. This means that lim s = lim (_e + le)=, which implies t! t! lim z = z d and lim _z = _z d. Thus, work platform s actual t! t! trajectory can converge to the desired trajectory. That completes the proof. Simulation results The choices of simulation parameters and initial conditions are listed in Table. Reference trajectory is given by _V = s T As + s T ( W T f + e )+tr( L T S _ L) Define K Z = as matrix and A e = ½ e T ŠT as column vector, and assume that kk\e e N, where e N is a constant, then then r(t)= ½p= pt=8 Š T _r(t)= ½ p=8 Š T r(t)= ½ Š T _V = z T K Z z + z T e + z T L T F + tr( L T S _ L) = z T K Z z + z T e + tr( L T S _ T L + L Fz T ) Since _ L = _^L, the time derivative of Lyapunov function can be expressed as _V = z T K Z z + z T e + nkktr( z L T (L L)) In addition, according to Schwarz inequality tr( L T (L L)) L klk F L and the following inequality F z T K Z z k min kk z where k min. is the minimum eigenvalue of K Z _V k min kk z + e N kk+ z nkk z L klk F L F F = kkk z min kk e z N + n L L klk F F F kkk z min kk z e N + n L L L M F F F In equation (5), the control parameters are set as A = 5diag½ Š l = diag½ Š K = diag½5 5 Š And ^D, ^C, and ^B are chosen as.9d,.8c, and.7b, respectively, for demonstrating the robustness of SMC to parameter uncertainties. In equation (9), the same control parameters are chosen for the convenience of comparing with SMC. Simultaneously, the parameters in equation () can be chosen as n = :, G = diag½ Š Table. Simulation parameters and initial conditions. Parameter Beam Beam Work platform Length (m) Mass (kg) EI (N m ) Initial angle (rad).9.5 Initial angular velocity (rad/s)

6 6 Advances in Mechanical Engineering Figure. The tracking of u in the NNSMC. Figure 5. The trajectory tracking of the work platform in the NNSMC. Figure. The tracking error of u in the NNSMC. Gaussian basis functions are selected as! f i = exp kk s :, i =,,..., where the center of Gaussian function c i is set at, the width s i at :. In addition, initial weights are set to zero. Figures are simulation results of NNSMC. Figures are simulation results of SMC. From Figures, it can be seen that the proposed NNSMC for the uncertain model can realize the tracking control of work platform and suppress the vibration simultaneously. Figures and 5 reflect that the actual trajectories of u and work platform are able to track the desired steadily, and simultaneously, the vibrations are eliminated effectively. Figures and 6 are the tracking errors of work platform on u and trajectory, respectively. From Figures and 6, we can see that the tracking errors are very small, and the controller has good tracking performance. Figure 7 demonstrates that vibration is effectively restrained. Deformation variable converges to small Figure 6. The trajectory tracking error of the work platform in the horizontal X direction in the NNSMC. Figure 7. Deflection variable q varying with time in the NNSMC.

7 Hu et al. 7 Figure 8. The adjustment of the control input u in the NNSMC. Figure. Sliding surface of u in the NNSMC. Figure 9. The output ^k of neural network in the NNSMC. Figure. The trajectory tracking of the work platform in the SMC. values, which means that the vibration existed in flexible beam is weakened effectively. Figure 8 shows the adjustment of control input u, which realizes tracking control of work platform. From Figure 8, we can see that the adjustment process of u is smooth. The regulation process of the output ^k of neural network, which reflects the approximate process to the system uncertainties Df, as shown in Figure 9. As shown in Figure, the NNSMC in this article can make sliding surface s converge to zero gradually. This reflects that tracking error of u can converge to zero asymptotically. Figure demonstrates that work platform can be driven to desired trajectory by SMC. Although tracking error in Figure is small, there is high-frequency chattering in the control input u during the period of 5 s, as shown in Figure. As the fast control is unrealistic for valve-controlling cylinder used for driving the beam, the implementation of SMC is very difficult. Figure. The trajectory tracking error of the work platform in the horizontal X direction in the SMC.

8 8 Advances in Mechanical Engineering Figure. The adjustment of the control input u in the SMC. gain is needed for compensation of model uncertainties, the chattering phenomenon, which degenerates the system performance, is occurred. In order to eliminate the chattering, we propose NNSMC using continuous control acquired from neural network to replace discontinuous switching control. In the NNSMC, neural network output k can compensate the model uncertainties and the chattering is eliminated effectively. As the weight update law of neural network is obtained from Lyapunov method, the stability and convergence of the whole system can be assured. Simultaneously, the results of simulation demonstrate that the designed NNSMC is effective for eliminating chattering caused by switching control existed in SMC. Moreover, we can see that the good performance of NNSMC is achieved for work platform s vibration restraining and tracking error reducing. The main contribution of this research is the proposition of NNSMC for the model of aerial work platform with flexible beam. Both the theoretical proof and simulation results show that the controller is effective for trajectory tracking and vibration suppressing when there exist system uncertainties. However, the current results are limited in two aspects: () in order to apply to practice, the NNSMC should be converted into digital form further. () There exist difficulties in the construction of experimental platform to verify the effectiveness of NNSMC at present. These limitations will be improved in the future work. Figure. The switching control K sgn(s ) in the SMC. Figure is the adjusted process of switching control K sgn(s ), which reflects that the robustness to parameter disturbance is obtained by the high-speed switching control in SMC. As the unmodeled characteristics of system can be excited easily by the high-frequency input, the performance of control system will be degenerated. Therefore, it is difficult for the application of SMC in practical system. With the comparison between Figures and Figures, we conclude that the presented NNSMC can achieve good tracking performance and eliminate the chattering caused by switching control existed in SMC simultaneously. Conclusion In this article, SMC method is used for the model considering the flexible deformation of beam to reduce tracking error and suppress vibration of work platform when there exist system uncertainties. As the switching Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by National Natural Science Foundation (NNSF) of China (Grants No. 675 and 6576), by Fundamental Research Funds for the Central Universities (Grants No. 96 and 97), and by Natural Science Foundation of Inner Mongolia (Grant No. 5MS6). References. Hu HD, Li E, Zhao XG, et al. Modeling and simulation of folding-boom aerial platform vehicle based on the flexible multi-body dynamics. In: Proceedings of IEEE international conference on intelligence control and information processing, Dalian, China, 5 August, pp New York: IEEE.. Jia PX, Li E, Liang ZZ, et al. Adaptive neural network control of an aerial work platform s arm. In: Proceedings of the th world congress on intelligent control and

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