Discrete-time direct adaptive control for robotic systems based on model-free and if then rules operation

Transcription

1 Int J Adv Manuf Technol (213) 68: DOI 1.17/s ORIGINAL ARTICLE Discrete-time direct adaptive control for robotic systems based on model-free and if then rules operation Chidentree Treesatayapun Received: 8 June 212 / Accepted: 15 January 213 / Published online: 8 February 213 Springer-Verlag London 213 Abstract An adaptive controller for a class of nonlinear discrete-time systems is proposed for robotic systems under the assumption that the parameters and structure of system dynamics are all unknown. This controller is designed with the concept of model-free adaptive control requiring only the input output of the unknown plant. The robotic system has been generalized to be a nonaffine discrete-time system under reasonable assumptions. The adaptive scheme called fuzzy rules emulated network (FREN) is implemented as a direct controller. The IF THEN rules for FREN have been defined by the knowledge according to the relation between input and output of the robotic system without any compensator for the unknown mathematical model or nonlinearities. The underlying physical specifications of robotic system such as the operating range, maximum joint velocity, and so on have been considered to initialize the membership functions and adjustable parameters of FREN. The adaptation scheme is developed according to convergence analysis established for both adjustable parameters and the tracking error. The performance of the proposed controller is validated by the experimental system with a 7-degrees-of-freedom robotic arm operated in velocitymode control. Keywords Adaptive control Robotic systems Discrete-time Fuzzy logic C. Treesatayapun ( ) Department of Robotic and Advanced Manufacturing, CINVESTAV-Saltillo, Ramos Arizpe 259, Mexico treesatayapun@gmail.com; chidentree@cinvestav.edu.mx 1 Introduction In several industrial applications, motion control of robotic manipulators is a difficult one because of the highly nonlinear dynamic systems which generally effected by unknown system dynamics and uncertainties [1]. The main challenge is to overcome the nonlinearities and the typical coupling effects of robotic systems. Recently, various controllers have been proposed and developed in order to acquire the best performance. The feedback linearization has been implemented in the robotics context with a torquemode controller [2]. However, this controller requests the well-defined dynamics of the robotic systems. Different adaptive control schemes have been developed to deal with the unknown dynamics [3] and references therein. Unfortunately, the linearization of unknown parameters must be held in the assumptions that those unknown parameters are constant or slowly time varying and linear in structure. The robust controller such as the sliding-mode control (SMC) [4] has been magnified especially for manipulator driving systems [5]. The general drawback of SMC is the control chattering which can be occurred by mechanism coupling and undesirable high-frequency dynamics. To reduce the chattering, the integration between SMC and fuzzy logics had been introduced in [6, 7] and references therein. Generally, those control laws have been designed in continuous time domain and then implemented on discrete-time domain by the digital computer under the assumption of very short sampling time. Furthermore, the discretization of the control law designed within continuous time domain may cause stability and performance degradation. It is reasonable to design the control law in discrete-time domain which can be directly implemented in a digital computer [8, 9]. Recently, the implementation on the direct discretetime controller [1] has been introduced validated with

2 576 Int J Adv Manuf Technol (213) 68: a 5-degrees-of-freedom (DOF) (operated only two joints) robotic arm. However, the part of discrete-time dynamic model is still necessary in order to complete the control law. Model-free adaptive control (MFAC) has been continuously developed for several systems with unknown or illdefined model especially for a class of discrete-time domain [11, 12]. Without any mathematic model of the plant, a linearization concept based on pseudo-partial derivative (PPD) is composed as the equivalent system. Due to the comparison with other adaptive controllers, only realtime measured data of the controlled plant are necessary to establish MFAC. Unlike model reference adaptive control based on neural networks, the off-line tuning phase can be excluded because the time-varying PPD can be tuned by real-time measurement data only [13]. For a class of nonlinear systems with unknown parameters, an artificial neural network compensation has been implemented for an adaptive decoupling control but the convergence analysis has not been intensively considered [14]. Furthermore, the data-based control scheme is almost reasonable to directly design the controller without any mathematical model of underlying physical plant. Recently, the convergence analysis of data-based controller [15] has been introduced by an artificial neural network based on a fuzzy inference system. In this article, the alternative method to design a controller for robotic systems is introduced under the assumption that the parameters and structure of system dynamics are all unknown. The robotic systems are generalized in the nonaffine discrete-time system which related only input (control effort) and measured output. The simple design procedure is introduced with the relation between input output of the robotic systems according to the human knowledge in the format of IF THEN rules. Those IF THEN rules can be directly included to an adaptive network called fuzzy rules emulated network (FREN). The proposed controller can be designed without any mathematical model of the robotic systems. The convergence analysis is established for both adjustable parameters inside FREN and the tracking error. The experimental system with a 7-DOF robotic arm (Mitsubishi PA1) is carried out to verify the performance of the proposed algorithm. 2 Model-free robotic systems Most robotic systems especially in the industrial sector are velocity-mode controllers. To simplify, in this work, we consider that the control effort which will be fed to the robot is the velocity command denoted by u and the output is the joint position denoted by x. This concept can be illustrated in Fig. 1 as a single-joint robotic system. Fig. 1 Single-joint robotic system concept For the system based on a digital computer controller, generally, we presume that the relation between joint position and velocity command can be defined by the formulation in a class of nonaffine discrete-time systems as x(k + 1) = f N (μ, u), (1) where f N ( ) is an unknown nonlinear function, μ = [xx(k 1) x(k n) u(k 1) u(k 2) u(k m)] T, and k stands for the time index. Parameters n and m denote as the order of system s output and input, respectively. In this work, those parameters are assumed to be unknown. To continue the controller design, these following assumptions are necessary: Assumption 2a System (Eq. 1) is invertible and stabilized, i.e., bounded output implies bounded input. Assumption 2b Let g u = f N (μ k,u k ) u k, thus < g u G u, k whenμ μ and u u two compact sets. By taking the Taylor expansion of function f N ( ) with respect to the control effort u around (μ, u(k 1)), Eq. 1 can be obtained as x(k + 1) = f N (μ, u(k 1)) + f N(μ, u k ) uk u k =u(k 1) [u u(k 1)]+ε u, = f N (μ, u(k 1)) f N(μ, u k ) uk u k =u(k 1) u(k 1) + ε u + f N(μ, u k ) u k uk u, =u(k 1) (2)

3 Int J Adv Manuf Technol (213) 68: where ε u stands for other high-order terms. Let us define F N (μ) = f N (μ, u(k 1)) f N(μ, u k ) uk u k =u(k 1) u(k 1) + ε u, (3) and G N (μ) = f N(μ, u k ) uk u ; (4) k =u(k 1) thus, Eq. 2 can be rewritten as x(k + 1) = F N (μ) + G N (μ)u. (5) Remark Both assumptions 2a and 2b are reasonable for robotic systems. In this work, let us assume that these nonlinear functions F N and G N are all unknown. The control effort u is directly determined by an adaptive network called FREN [16, 17] which will be discussed next. 3 FREN and controller design 3.1 Network architecture Let us consider a general IF THEN rule defined by the following syntax: RULE i: IF e IS A i THEN u i = β i φ i where e denotes a variable which can be clarified as the input and u stands as the output. This rule represents that if a crisp value e belongs to the fuzzy set A i with the membership function μ Ai, then the output of this rule can be calculated by u i = β i μ Ai (e) where β i is a designed parameter for the ith rule. Furthermore, this parameter can be initially defined by using the human knowledge related to the system. The example on how to set this parameter will be represented at the end of this section. According to that IF THEN rule and a SISO system where e and u are input and output, respectively, the network construction has been introduced to emulate the fuzzy inference calculation. This network is called FREN. FREN can be decomposed into four layers as shown in Fig. 2. The calculation inside this network can be explained as follows: Layer 1: Layer 2: The input e of this layer is sent to each node in the next layer directly. Thus, there is no computation in this layer. Each node in this layer contains a membership function corresponding to one linguistic level Fig. 2 Structure of FREN Layer 3: Layer 4: (e.g., negative, nearly zero, etc.). The output at the ith node is calculated by φ i = μ Ai (e), (6) where μ Ai ( ) denotes a membership function of a fuzzy set A at the ith rule (i = 1, 2,...,N), where N denotes the number of rules. This layer has N nodes calculation and theoutput u i can be determined by u i = φ i β i, (7) where β i is a parameter of the ith node. The output u is obtained in this layer as N u = u i. (8) i=1 Let us define β = [β 1 β 2 β N ] T and φ as the FREN basis function vector φ = [φ 1,φ 2,,φ N ] T ; thus, the output u can be alternatively given by u = β T φ. (9) In the case of adaptation, the design parameter will be tuned for every time index k as β i ; thus, we have u = β T φ. (1) This network decomposition allows the designer to intuitively set the initial value of FREN s parameters. 3.2 IF THEN rules for velocity-mode controller The closed-loop system is designed by FREN as the direct adaptive controller which determines the control effort u from the error e when e = x d x, (11) where x d denotes the desired position. Figure 3 illustrates the system configuration. It is clear that the knowledge about the controlled plant is directly integrated to the

4 578 Int J Adv Manuf Technol (213) 68: Fig. 3 System configuration adaptive controller FREN by IF THEN rules format. In this work, only on-line leaning mechanism is applied with the associate of some designed parameters which will be discussed next. FREN is an adaptive network which can be designed by the IF THEN rules related on the human knowledge according to the system. In general, let us consider the single-joint robotic system based on velocity-mode controllers depicted in Fig. 4. The controller should determine the velocity command to force the joint reaching the desired position. With the error defined by Eq. 11 and the velocity concept illustrated in Fig. 4, those IF THEN rules can be formulated as follows: Here, PL, PS, Z, NS, and NL denote positive large, positive small, zero, negative small, and negative large, respectively, and v i is the velocity obtained at the ith rule for i = 1, 2,, 5. By referring to Eq. 1, the control effort can be determined by u = 5 u i. (12) i=1 The initial setting of parameters β i (Eq. 1) can be given according to the robot system s specification related to the range of velocity. The example will be discussed about this initial setting on the experimental setup section. If e is PL (Far on clockwise position) If e is PS (Close on clockwise position) If e is Z (Reach the desired position) If e is NS (Close on counterclockwise position) If e is NL (Far on counterclockwise position) Then u 1 = β PL φ 1, (Higher velocity in counterclockwise direction) Then u 2 = β PS φ 2, (Low velocity in counterclockwise direction) Then u 3 = β Z φ 3, (Stop the movement) Then u 4 = β NS φ 4, (Low velocity in clockwise direction) Then u 5 = β NL φ 5. (Higher velocity in clockwise direction) 3.3 Parameters adaptation The gradient descent method with the proposed timevarying step size is introduced to adjust these parameter β i for i = 1, 2,, 5. The cost function E, whichis needed to be minimized, can be defined as E = 1 2 e2. (13) At time index k + 1, all adjustable parameters β i can be determined by E(k + 1) β i (k + 1) = β i η, (14) β i where η is a time-varying learning rate. According to the gradient descent technique, the learning rate plays the Fig. 4 IF THEN rules concept based on FREN with velocity mode

5 Int J Adv Manuf Technol (213) 68: importance role about system stability and convergence. Small learning rate can provide the satisfied system stability but the convergence rate can be slow. On the other hand, the faster convergence rate can be obtained by the higher learning rate but the system stability cannot be assured. In this work, we introduce the determination method to obtain the possible biggest learning rate when the system stability can be guaranteed. By applying the chain rule through Eqs. 13, 11,and5,we obtain E(k + 1) β i E(k + 1) x(k + 1) u = x(k + 1) u β i, = [x d (k + 1) x(k + 1)]y p φ i. (15) Thus, the tuning law can be rewritten as β i (k + 1) = β i + η i e(k + 1)y p φ i, (16) where y p denotes x(k+1) u. Let us consider the system formulation in Eq. 5 again; clearly, we have y p = G N (μ). (17) Term G N (μ) is assumed to be unknown and follows the assumption 2b, but the tuning law needs this information to adjust the parameter. To overcome this problem, we need to design the new adaptation law which will be represented in the next subsection with the convergence analysis. 3.4 Convergence and stability analysis The key of this work is to determine the learning rate η for every time index k. By substituting the control effort u givenbyeq.1 into the system formulation (Eq. 5), we have For the convenient presentation, let us select learning to be η = γ G 2 u φt φ, (21) where G u is the positive upper bound of f N (μ k,u k ) u k as given by assumption2b and <γ <2isthe designed parameter. AccordingtoEq.21, only the upper bound G u is needed to be estimated; thus, the convergence of tunable parameters can be demonstrated with the following lemma: Lemma 2 The convergence of tunable parameters With the system given by Eq. 1, let the control effort u be generated by Eq. 1 and the adjustable parameters be tuned by Eq. 16. If the learning rate η is given by Eq. 21 where <γ <2andG u is the upper bound given by assumption 2b, then the convergence of adjustable parameters β i is guaranteed. Proof Substitute Eq. 21 into Eq. 16 and rearrange with Eq. 5 for the next time index error Eq. 19; thus, we have [ ] γ β(k + 1) = I G 2 u φt φ G2 N φt φ β, +η[x d (k + 1) F N ]y p φ, [ = I γ G2 N ] G 2 β + η[x d (k + 1) u F N ]y p φ, = ξ 1 β + ξ 2, (22) x(k + 1) = F N + G N β T φ. (18) Thus, the next time step error can be rewritten as e(k + 1) = x d (k + 1) F N G N β T φ. (19) By substituting Eq. 19 into Eq. 16, the adaptation law can be obtained as [ β(k + 1) = β + η x d (k + 1) F N ] G N β T φ y p φ], = [ ] I ηg 2 N φt φ β +η[x d (k + 1), F N ]y p φ. (2) Fig. 5 Experimental setup with PA-1

6 58 Int J Adv Manuf Technol (213) 68: Fig. 6 PA-1 q i for i = 1, 2,, 7 where ξ 1 = I γ G2 N and ξ G 2 2 = η[x d (k + 1) u F N ]y p φ. By setting the designed parameter γ as the above and G N G u, thus ξ 1 < 1, k that proof is complete. According to the result obtained by the previous lemma, only the convergence of the adjustable parameters is ensured with the setting parameter < γ < 2 but the closedloop system stability is still not discussed yet. Next, we will introduce the proof of system performance based on the proposed controller with the following theorem: Theorem 1 System stability(closed-loop system convergence) Let the desired trajectory x d be bounded and the upper bound of G N be known as G u. Determine the control effort u by Eq. 1 and tune parameters by Eq. 16 with the varying learning rate given by Eq. 21 when <γ <2. Then, the tracking error e defined by Eq. 11 is bounded for the nonlinear system given by Eq. 1. Proof Let the Lyapunov function candidate be defined as V= 1 2 e2 ; (23) thus, the change of Lyapunov function can be given as V = V(k+ 1) V, = 1 2 e2 (k + 1) 1 2 e2, = 1 2 [e e]2 1 2 e2, = e[e + e ]. (24) 2 Let us consider the next time index error be written by e(k + 1) = e + e, (25) Fig. 7 Block diagram of PA-1 with FREN controllers

7 Int J Adv Manuf Technol (213) 68: Fig. 8 PA-1 Tracking performance for joint q 1 : x 1 and e x 1 e 1.4 Angular Position [rad] Time index Fig. 9 Control effort for joint q 1 : u Angular Velocity [rad/sec] Time index

8 582 Int J Adv Manuf Technol (213) 68: Fig. 1 PA-1 tracking performance for joint q 2 : x 2 and e x 2 e Angular Position [rad] Time index Fig. 11 Control effort for joint q 2 : u Angular Velocity [rad/sec] Time index

9 Int J Adv Manuf Technol (213) 68: Fig. 12 PA-1 tracking performance for joint q 3 : x 3 and e x 3 e Angular Position [rad] Time index Fig. 13 Control effort for joint q 3 : u Angular Velocity [rad/sec] Time index

10 584 Int J Adv Manuf Technol (213) 68: Fig. 14 PA-1 tracking performance for joint q 4 : x 4 and e x 4 e Angular Position [rad] Time index Fig. 15 Control effort for joint q 4 : u Angular Velocity [rad/sec] Time index

11 Int J Adv Manuf Technol (213) 68: Fig. 16 PA-1 tracking performance for joint q 5 : x 5 and e x 5 e Angular Position [rad] Time index Fig. 17 Control effort for joint q 5 : u Angular Velocity [rad/sec] Time index

12 586 Int J Adv Manuf Technol (213) 68: Fig. 18 PA-1 tracking performance for joint q 6 : x 6 and e x 6 e Angular Position [rad] Time index Fig. 19 Control effort for joint q 6 : u Angular Velocity [rad/sec] Time index

13 Int J Adv Manuf Technol (213) 68: Fig. 2 PA-1 tracking performance for joint q 7 : x 7 and e x 7 e Angular Position [rad] Time index Fig. 21 Control effort for joint q 7 : u Angular Velocity [rad/sec] Time index

14 588 Int J Adv Manuf Technol (213) 68: where e can be estimated by e T e,β β, (26) when e,β =[ e(k+1) β 1 the chain rule, we have e(k + 1) β i = e(k+1) β 2 e(k + 1) x(k + 1) u x(k + 1) u β i, e(k+1) β l ]T.Byusing = G N φ i. (27) Thus, Eq. 26 can be rearranged as e G N φ T β. (28) From the tuning law obtained by Eq. 2, the change of adjustable parameters can be rewritten as β = β(k + 1) β, = ηg 2 N φ 2 β + η[x d (k + 1) F N ]G N φ. (29) By substituting Eq. 29 into Eq. 28,wehave e ηg 3 N φ 2 φ T β η[x d (k + 1) F N ]G 2 N φt φ, [ = ηg 2 N φ 2 x d (k + 1) + F N ] +G N φ T β, = ηg 2 N φ 2 [e(k + 1)]. (3) By using Eq. 25, we can rearrange Eq. 3 as e = ηg2 N φ 2 e 1 + ηg 2. (31) N φ 2 By substituting Eq. 31 into Eq. 24, we obtain V = ηg2 N φ 2 e ηg 2 N φ 2 [ ] 1 ηg2 N φ ηG 2. (32) N φ 2 With the learning rate in Eq. 21, the change of Lyapunov function candidate can be rearranged by V = γ 1 + γ [ ] 2 GN G u [ GN G u ] 2 1 γ ( γ [ ] 2 GN G u [ GN ] ) 2 e2 G u. (33) Remark According to Eq. 33, the sign or direction of G N is not needed. 4 Experimental results In this experimental setup, the proposed control algorithm is implemented to control a 7-DOF Mitsubishi PA-1 robotic arm system depicted in Fig. 5. With seven moving joints, Fig. 6 represents the positions for every joint variables q 1,q 2,,q 7 used in this work. The overall system configuration is illustrated in Fig. 7. SevenFRENs are designed to control each of the joints independently. This robotic system is operated in the velocity-mode control which means that FRENs must generate the velocity commands to move every joint to follow the desired trajectory. To begin the controller design, the suitable IF THEN rules are all needed to be specified first. Based on the knowledge of PA-1, those IF THEN rules can be given as follows: If e is PL If e is PS If e is Z If e is NS If e is NL Then u 1 = β PL φ 1, Then u 2 = β PS φ 2, Then u 3 = β Z φ 3, Then u 4 = β NS φ 4, Then u 5 = β NL φ 5, where u denotes the velocity command determined by FREN and e is the position error. In this design, the physical limit of the robotic arm such as the allowable maximum velocity can be included for consideration as follows: Joint # [ ] u rad M s Joint # [ ] u rad M s q 1 1. q 4 2., q 2 1. q , q 3 2. q , q

15 Int J Adv Manuf Technol (213) 68: Thus, those linear parameters β can be initialized for every joint as follows: Parameter Value [rad/s] Parameter Value [rad/s] Parameter Value [rad/s] β q1 PL(1) 1 β q2 PL(1) 1 β q3 PL(1) 2, β q1 PS(1).5 β q2 PS(1).5 β q3 PS(1).75, β q1 Z(1) β q2 Z(1) β q3 Z(1), β q1 NS(1).5 β q2 NS(1).5 β q3 NS(1).75, β q1 NL(1) 1 β q2 NL(1) 1 β q3 NL(1) 2, β q4 PL(1) 2 β q5 PL(1) 6.2 β q6 PL(1) 6.2, β q4 PS(1) 1.25 β q5 PS(1) 3.5 β q6 PS(1) 3.75, β q4 Z(1) β q5 Z(1) β q6 Z(1), β q4 NS(1) 1.25 β q5 NS(1) 3.5 β q6 NS(1) 3.75, β q4 NL(1) 2 β q5 NL(1) 6.2 β q6 NL(1) 6.2, β q7 PL(1) 6.2, β q7 PS(1) 4, β q7 Z(1), β q7 NS(1) 4, β q7 NL(1) 6.2. The tracking performance for every joints q 1,q 2,,q 7 can be illustrated in Figs. 8, 9, 1, 11, 12, 13, 14, 15, 16, 17, 18, 19, and 2, respectively, with their error signals. According to those tracking performance illustrations and the decreasing of error signals during the on-line adaptation, the validation of tuning algorithm is demonstrated and the satisfied results can be obtained. Regarding the error signals displayed in Figs. 8, 1 2, those error signals are asymptotically decreased in relation to increasing time index. This behavior demonstrates the efficiency of adaptive algorithm when all adjustable parameters β have been tuned to handle the controlled robotic arm. This controller is working as the velocity mode which means that the control signal should be normally determined in the first derivative of tracking signal. Unfortunately, this robotic arm is one of the nonlinear systems; thus, the controller must compensate those unknown nonlinearities. Considering velocity commands u illustrated in Figs. 9, 11 21, those control efforts are not completely cosine signals because the controller successfully compensates nonlinearities to gain the superior performance. 5Conclusion In this article, an adaptive controller has been introduced for robotic systems which have been generalized in the nonaffine discrete-time system. This design has been directly conducted without any requirement of a mathematical model. Only the relation between input and output of robotic systems has been used to define the IF THEN rules for an adaptive network FREN. The membership functions and adjustable parameters have been initialized according to robotic physical systems and based on knowledge. Theoretically, those adjustable parameters and tracking error have been guaranteed by the convergence analysis. Furthermore, the experimental setup with the 7-DOF robotic arm has validated the proposed controller s performance. In this work, the decoupling system for each joint has not been considered but the experimental results demonstrate that the proposed controller can handle the system performance. Normally, the control effort (velocity) should be the cosine function of the joint position but the time varying of the control effort is not completely cosine because of the nonlinearities of robotics systems. This controller has automatically tuned the adjustable parameters and compensated the unknown nonlinearities to perform the superior tracking performance. Acknowledgment This work has been supported by CONACyT Mexico, grant SEP-CONACyT # References 1. Adam MJK (1999) Basics of robotics: theory and components of manipulators and robots. Springer, Wien, NJ

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