SM PID controller using fuzzy tuning approach for manipulator

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Sliding Mode PID-Controller Design for Robot Manipulators by Using Fuzzy Tuning Approach Mohammad Ataei1, S Ehsan Shafiei2 Electronic Department- Engineering Faculty, University of Isfahan, Isfahan, Iran E-mail: Electrical and Robotic Engineering Faculty, Shahrood University of Technology, Shahrood, Iran E-mail: Abstract: In this paper, a chattering free sliding mode control (SMC) for a robot manipulator including PID part with a fuzzy tunable gain is designed The main idea is that the robustness property of SMC and good response characteristics of PID are combined with fuzzy tuning gain approach to achieve more acceptable performance For this purpose, in the first stage, a PID sliding surface is considered such that the robot dynamic equations can be rewritten in terms of sliding surface and its derivative and the related control law of the SMC design will contain a PID part The stability guarantee of this sliding mode PID-controller is proved by a lemma using direct Lyapunov method Then, in the second stage, in order to decrease the reaching time to the sliding surface and deleting the oscillations of the response, a fuzzy tuning system is used for adjusting both controller gains including sliding controller gain parameter and PID coefficient Finally, the proposed methodology is applied to a two-link robot manipulator including model uncertainty and external disturbances as a case study The simulation results show the improvements of the results in the case of using the proposed method in comparison with the conventional SMC Key Words: Sliding Mode Control, Robot manipulator, PID control, Fuzzy control, Lyapunov theory INTRODUCTION A robot manipulator is a nonlinear system with high coupling term whose dynamics consists of uncertainty and encountered with payload changes, friction and disturbance [1] On the other hand, sliding mode control (SMC) as a nonlinear technique with the capabilities of robustness against the model uncertainties and ability of the disturbance rejection has been considered in many researches [2-4] Although the robustness of the SMC is one of its main characteristics, this is achieved only in the sliding phase and the system is sensitive to the structured uncertainties and external disturbances in the reaching phase to the sliding surface Therefore, different approaches for improving the performance of the SMC has been proposed which one of them is intelligent control method such as fuzzy control system [5-7] Because of the relations between SMC and fuzzy control, [8], the combination of these two approaches has been considered as a research topic in last years [9-13] such that the advantages of both approaches can be used One simple way to decrease the sensitivity of sliding mode controller to the parametric uncertainties and external disturbances is using of high control gain which decrease also the reaching time and tracking error However, high control gain increases the oscillations in the control signal that may lead to the excitation of high frequency unmodeled dynamics which is an undesired phenomenon To overcome this drawback, the fuzzy logic can be used for tuning of this gain In this regard, in [14], a nonlinear sliding surface and fuzzy logic have been used in the design of a fuzzy terminal SMC for a robot manipulator Also, a fuzzy variable sliding surface based method has been proposed in [15] in order to improve the tracking performance In [16], in addition to using variable sliding surface, the idea of fuzzy gain tuning and boundary layer has been presented to achieve more improvements In this paper, in addition to using the integral term in the sliding surface, [17, 18], at first, the SMC witch is including PID part is designed and its stability guarantee is proved in a lemma Then, in order to improve the controller performance, a fuzzy system is used to tune the gain of reaching phase and also PID part gain Thus, a chattering free SMC is achieved in which the tracking error and reaching time to sliding surface has been reduced without need to variable sliding surface The reminder of the paper is organized as follows In the section 2, the mathematical model of the robot manipulator is given The SMC including the PID loop to which is denoted as SMC-PID is presented in section The design of fuzzy SMC-PID is described in section In section 5, the simulation results are provided and finally, summary and some conclusions are presented THE SYSTEM MATHEMATICAL MODEL The dynamical equation of an n-link robot manipulator in the standard form is as follows [1]: M ( q ) q + C ( q, q ) q + G ( q ) + τ d = τ (1) where M (q) ∈ R n×n is a symmetry and bounded positive definite matrix which is called inertial matrix Moreover, q, q, q ∈ Rn are the position, velocity, and angular acceleration of the robot joint, respectively The matrix C (q, q) ∈ R n×n is the matrix of Coriolis and centrifugal forces such that the matrix H ( q ) − 2C ( q, q ) is asymmetry, i.e., for a nonzero n × vector x we will have: x T [ H (q) − 2C ( q, q )]x = Also, G (q ) ∈ R n is the gravity vector, τ d ∈ R n is the bounded disturbance vector such that τ d ≤ TD and τ ∈ R n is the control input vector In the following, H (q ) , C (q, q ) and G (q ) are shown by H, C, and G, respectively SLIDING MODE CONTROL WITH PID The objective of tracking control is design a control law for obtaining the suitable input torque τ such the position vector q can track the desired trajectory q d In this regard, the tracking error vector is defined as follows: e = qd − q (2) In order to apply the SMC, the sliding surface is considered as the relation (3) which contains the integral part in addition to the derivative term: t s = e + λ1e + λ edt (3) where λi is diagonal positive definite matrix Therefore, s = is a stable sliding surface and e → as t → ∞ The robot dynamic equations can be rewritten based on the sliding surface (in term of filtered error) as follows: Ms = −Cs + f + τ d − τ (4) law should be designed such that the following sliding condition is satisfied [2]: [ t (5) Now, the control input can be considered as follows: τ = fˆ + K v s + K sgn( s ) (6) t fˆ = Mˆ (q d + λ1e + λ e) + Cˆ (q d + λ1e + λ edt ) + Gˆ K ii = [F + K v s + TD + η ]i estimation of (7) and f (10) , i = 1,2, , n (11) Then, the sliding condition (10) is satisfied by equation (4) Proof: Consider the following Lyapunov function candidate: T s Ms (12) Since M is positive definite, for s ≠ we have V > and V= by differentiating of the relation (12) and regarding the symmetric property of M, it can be written: V= T s Ms + s T Ms (13) By substituting (4) in (13) and considering that s T ( M − 2C ) s = , we have: V= T s Ms − s T Cs + s T ( f + τ d − τ ) (14) T = s ( f +τ d −τ ) V = s T ( f + τ d − fˆ − K v s − K sgn( s )) ~ = s T ( f + τ d − K v s) − t K v s = K v e + K v λe + K v λ edt is the outer PID tracking loop, and K v , K are diagonal positive definite matrices and are defined such that the stability conditions are guaranteed The sgn(s) is also the sign function We have also: t ~ ~ ~ ~ f = M (q d + λ1e + λ e) + C (q d + λ1e + λ edt ) + G ≤ F (8) ~ ~ ~ f = f − fˆ , M = M − Mˆ , C = C − Cˆ ,and ~ G = G − Gˆ F can also be selected as the following where relation: t ~ ~ ~ F = M (q d + λ1e + λ e) + C (q d + λ1e + λ edt + G (9) In order to reach the system states (e, e) to the sliding surface s = in a limited time and remain there, the control n (15) K ii s i i =1 an s≠0 By replacing the relation (6) in (14), V can be rewritten as: Where is for This subject is proved in the following lemma Lemma- In the SMC design of a system with dynamic equation (1) and sliding surface (3), if the control input τ is selected as (6), by considering F as (9) and K = diag ( K11 , K 22 , , K nn ) with the following components: Where f = M (q d + λ1e + λ e) + C (q d + λ1e + λ edt ) + G ] d T s Ms < −η ( s T s )1 / 2 dt Since the following inequality (16) is valid and regarding the relation (11), we have: ~ F + K v s + TD ≥ f + τ d − K v s ~ K ii ≥ [ f + τ d − K v s ] i + η i (16) (17) Finally, it can be concluded that: n η i si V ≤− (18) i =1 This indicates that V is a Lyapunov function and the sliding condition (10) has been satisfied The use of sign function in the control law leads to high oscillations in control torque which is undesired phenomenon and is called chattering To overcome this drawback, there are some solutions which one of them is using the following saturation function instead of sign function in the discontinuous part of the control law: = ϕ s ϕ s ≥ϕ −ϕ < s < ϕ −1 s ≤ −ϕ 0.6 0.4 0.2 -1 As it was mentioned before, by using a high gain in SMC (K), the sensitivity of the controller to the model uncertainties and external disturbances can be reduced Moreover, a high gain in PID part of the control system ( K v ) can reduce the reaching time to sliding surface and tracking error However, increasing the gain causes the increment of the oscillations in the input torque around the sliding surface Therefore, if this gain can be tuned based on the distance of the states to the sliding surface, a more acceptable performance can be achieved In other words, the value of gain should be selected high when the state trajectory is far from the sliding surface and when the distance is decreasing, its value should be decreased This idea can be accomplished by using fuzzy logic in combination with SMC to tune the gain adaptively For this purpose, two-input one-output fuzzy system is designed whose inputs are s and s which are the distances of the state trajectories to the sliding surface and its derivative, respectively The membership functions of these two inputs are shown in figure (1) The output of the fuzzy system is denoted by K fuzz and has been shown in figure K = N ⋅ K fuzz (20) K v = N v ⋅ K fuzz (21) These factors can be selected by trial and error such that the stability condition (17) is satisfied Degree of membership NSZEPS PB 0.8 0.6 0.4 -0.5 input variable "s" 0.5 (a) Fig 1: The membership functions for a) input 0.5 S Degree of membership M s B 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 output variable K 0.8 Fig 2: The membership functions of the output Tab 1: The fuzzy rule base for tuning s s N Z P K fuzz K fuzz NB NS Z PS PB B B B B M S M S M S M B B B B The maximum values of K and Kv are limited according to the system actuators power, and the minimum value of K should not be less than the provided amount in relation (17) The fuzzy base rule has been given in table (1) in which the following abbreviations have been used: NB: Negative Big; NS: Negative Small; Z: Zero; PS: Positive Small; PB: Positive Big; M: Medium For example, when s is negative small (NS) and s is positive (P), then K fuzz is small (S) THE CASE STUDY AND SIMULATION RESULTS In order to show the effectiveness of the proposed control law, it is applied to two-links robot with the following parameters: -1 inpu variable sd (b) 0.2 -0.5 Continue Fig 1: The membership functions for b) input (2) For applying these gains to the control input, the normalization factors N and N v as the following relations are used: NB P 0.8 THE DESIGN OF FUZZY SMC-PID Z (19) By this, there is a boundary layer ϕ around the sliding surface such that when the state trajectory reach to this layer will be remaining there N Degree of membership sat s s M (q) = C ( q, q ) = α + β + 2γ cos q β + γ cos q β + γ cos q β − γq sin q − γ (q1 + q ) sin q γq1 sin q (22) (23) αδ cos q1 + γδ cos(q1 + q 2) γδ cos(q1 + q ) (24) 150 input1(N.m) G (q) = where α = (m1 + m2 )a12 , β = m a 22 , γ = m a1 a , δ = g a , and m1 , m , a1 = , a = are the masses and lengths of the first and second links, respectively The masses are assumed to be in the end of the arms and the gravity acceleration is considered g = 9.8 Moreover, the masses are considered with 10% uncertainty as follow: , ∆m1 ≤ m = m 20 + ∆m , ∆m ≤ 50 -50 10 10 time(sec) 100 input2(N.m) m1 = m10 + ∆m1 100 (25) 50 -50 where m10 = and m 20 = , and Mˆ , Cˆ , and Gˆ are time(sec) estimated The desired state trajectory is: Fig 4: The control inputs in the case of using conventional SMC − cos π t (26) cos π t 0.15 and the disturbance torque is considered as follows: τd = (27) 0.5 sin 2πt 0.5 0.5 The values 15 0 , λ2 = 15 40 the ϕ and η of 0 0 (28) 40 are selected as 0 10 (29) Error1(rad) 200 100 -100 10 10 time(sec) 100 50 -50 -100 0.1 0.05 time(sec) 0 10 time(sec) Error2(rad) 10 time(sec) input1 (N.m) Nv = 0.15 1.5 0.5 -0.5 Fig 5: The tracking errors in the case of using Fuzzy SMC-PID In order to show the improvement due to the proposed method of this paper (Fuzzy SMC-PID), the simulation results of applying this method are compared with the related results of the conventional SMC The tracking error and control law in the case of conventional SMC have been shown in figures (3) and (4), respectively The corresponding graphs for the case of applying fuzzy SMC-PID are also provided in figures (5), and (6) -0.05 0.5 -0.5 input2 (N.m) , 10 and N v are selected as follow: 50 0 1.5 ϕ = 0.167 and η = [0.1 0.1]T Moreover, the factors N N= time(sec) The design parameters are determined as follow: λ1 = 0.1 0.05 -0.05 Error2 (rad) which leads to TD = 0.5 sin 2πt Error1(rad) qd = 10 time(sec) Fig 3: The tracking errors in the case of using conventional SMC Fig 6: The control inputs in the case of using Fuzzy SMC-PID As it is seen in these figures, the proposed fuzzy SMC-PID has faster response and less tracking error in comparison with conventional SMC In order to show more clearly the difference between the tracking errors in two cases, the enlarged graphs have been provided in figures (7) and (8) Error1(rad) 0.01 0.005 -0.005 -0.01 10 10 time(sec) -3 x 10 Error2(rad) -5 time(sec) Fig 7: The enlargement of the tracking errors in the case of using conventional SMC -4 Error1 (rad) x 10 -5 10 10 time(sec) -3 Error2 (rad) x 10 0.5 -0.5 -1 time(sec) Fig 8: The enlargement of the tracking errors in the case of using Fuzzy SMC-PID CONCLUSION In this paper, design of a sliding mode control with a PID loop for robot manipulator was presented in which the gain of both SMC and PID was tuned on-line by using fuzzy approach Then the stability guarantee of the system was proved by direct Lyapunov method The proposed methodology in fact tries to use the advantages of the SMC, PID and Fuzzy controllers simultaneously, i e., the robustness against the model uncertainty and external disturbances, quick response, and on-line automatic gain tuning, respectively Finally, the simulation results of applying the proposed methodology to a two-link robot were provided and compared with corresponding results of the conventional SMC which show the improvements of results in the case of using the proposed method REFERENCES [1] M W Spong, and M Vidiasagar, Robot Dynamics and Control, Wiley, New York, 1989 [2] J J Slotine, and W Li, Applied Nonlinear Control, Englewood Cliffs, NJ: Prentice Hall, 1991 [3] W Perruquetti, and J P Barbot, Sliding Mode Control in Engineering Marcel Dekker, Inc New York, 2002 [4] J Y Hung, W Gao, and J C Huang, Variable Structure Control: A Survey, IEEE Trans Ind Elec., Vol 40, No 1, 2-22, 1993 [5] L X Wang, A Course in Fuzzy Systems and Control, Prentice Hall, NJ, 1997 [6] C C Lee, Fuzzy Logic in Control Systems: Fuzzy Logic Controller-Part I, IEEE Trans Sys Man and Cyb Vol 20, No 2, 404-418, 1990 [7] C C Lee, Fuzzy Logic in Control Systems: Fuzzy Logic Controller-Part II, IEEE Trans Sys Man and Cyb Vol 20, No 2, 419-435, 1990 [8] R Palm, D Driankov, and H Hellendoorn, Model Based Fuzzy Control: Fuzzy Gain Schedulers and Sliding mode Fuzzy Controllers Springer-Verlag Berlin Heidelberg, 1997 [9] J C Lo, and Ya H Kuo, Decoupled Fuzzy Sliding Mode Control, IEEE Trans on Fuzzy systems, Vol 6, No 3, 426-435, 1998 [10] L K Wang, H F Leung, and K S Tam, A Fuzzy Sliding Controllers for Nonlinear Systems, IEEE Trans Ind Elec., Vol 48, No 1, 32-37, 2001 [11] Q P Ha, Q H Nguyen, D C 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