IJOER-APR-2016-35

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slide 1: International Journal of Engineering Research ScienceIJOER ISSN: 2395-6992 Vol-2 Issue-4 April- 2016 Page|104 Adaptive Backstepping Tuning Functions Control Design for Industrial Robot Manipulators Tran Xuan Kien 1 Do Duc Hanh 2 1 RD Department Military Institute of Science and Technology Hanoi Vietnam Email: txkien2003gmail.com 2 Control Engineering Department Military Technical Academy Hanoi Vietnam Email: hanhaitcv.ac.vn Abstract— In this paper an adaptive backstepping control design with tuning functions and K-filters for robot manipulators is developed. A stronger stability and convergence performance of the designed control in comparison with backstepping observer is achieved despite the presence of disturbances parameter uncertainties system nonlinearities for a real-time system of a single-link flexible-joint manipulator. Keywords— Tuning functions K-filters Adaptive Observer Backstepping Robot Manipulator Control. I. INTRODUCTION The adaptive backstepping solution to the problem of nonlinear stabilization and tracking in the presence of unknown parameters is a starting point for more elaborate adaptive control designs for feedback systems including robot manipulators 2-4. One of the improvements to be achieved with the tuning functions design 1 is to reduce the dynamic order of the adaptive controller to its minimum the number of parameter estimates is equal to the number of unknown parameters. This minimum-order design is advantageous not only for implementation but also because it guarantees the strongest achievable stability and convergence properties. In the tuning functions procedure the parameter update law is designed recursively. At each consecutive step we design a tuning function as a potential update law. In contrast to adaptive backstepping in 2 these intermediate update laws are not implemented. Instead the controller uses them to compensate for the effect of parameter estimation transients. Only the final tuning function is used as the parameter update law. In this paper we presented an adaptive backstepping tuning functions control design for systems in the output feedback form. In the design different filter structures K-filter and identifiers are applied. The rest of the paper is structured as follows. A design with tuning functions for a single-link flexible-joint robot manipulator is presented in section II. Experiment design and the performance of the designed real-time control for the flexible-joint robot arm are presented in section III. We conclude in section IV. II. TUNING FUNCTIONS FOR A SINGLE-LINK FLEXIBLE-JOINT ROBOT MANIPULATOR 2.1 System Modeling We consider a single-link flexible-joint robot manipulator actuated by a DC motor as shown in Fig. 1 This is depicted in 2 but repeated here for convenience. The dynamic equations of the system are as follows: 2 1 1 1 1 1 1 2 2 1 2 2 1 2 cos 0 t b q J q F q K q mgd q N q K J q F q q K i N N LDi Ri K q u 1 slide 2: International Journal of Engineering Research ScienceIJOER ISSN: 2395-6992 Vol-2 Issue-4 April- 2016 Page|105 K N X Z Y d J1 J2 O q2 q2 N q1 mg FIG. 1. A SINGLE-LINK FLEXIBLE ROBOT MANIPULATOR Where q 1 q 2 are the angular positions of the link and the motor shaft i is the armature current and u is the armature voltage. The inertials J 1 J 2 the viscous friction constants F 1 F 2 the spring constant K the torque constant K the torque constant K t the back-emf constant K b the armature resistance R and inductance L the link mass M the position of link’s centre of gravity d the gear ratio N and the acceleration of gravity g can all be unknown. We assume that only the link position q 1 is measured. The choice of state variables: 5 1 1 2 1 3 2 4 2 q q q q i      The dynamic equations of the system become: 1 2 1 2 2 2 1 1 1 1 3 4 2 2 4 1 4 5 2 2 2 5 5 4 1 cos 1 t b mgd F K y J J J N K F K J N N J J R K u L L L y 2 Clearly 2 is not in the output-feedback. Differentiating y twice we obtain 2 Dy  d D dt is the differentiation operator and 2 1 3 1 1 1 cos mgd F K D y y Dy y J J J N It implies that: 2 1 1 3 1 1 1 cos J N mgd F K D y y Dy y K J J J 3 2 1 1 4 3 1 1 1 cos J N mgd F K D D y D y D y Dy K J J J Differentiating and substituting 3 4 we obtain slide 3: International Journal of Engineering Research ScienceIJOER ISSN: 2395-6992 Vol-2 Issue-4 April- 2016 Page|106 4 3 2 1 2 1 2 1 2 5 2 1 1 2 1 2 1 2 2 1 2 2 2 1 1 2 1 2 1 2 1 2 cos cos cos J J N F F K K F F D y D y D y K K J J J J N J J mgd F K F K mgd mgdK D y Dy D y y J J J N J J J J J J N Finally differentiating and substituting 5 4 we arrive at the input-output description 5 4 3 3 1 2 1 2 1 2 2 1 2 1 2 1 1 2 2 1 2 1 2 2 2 1 2 1 2 1 2 2 2 1 2 1 2 1 2 1 2 1 2 2 1 cos cos t t b t b K K R F F mgd R F F K K K K FF D y u D y D y D y J J NL L J J J L J J J L J J N J J R K K FF FK F K K K F R F mgd D y D y L J J N J J J J N J J J J L L J J 1 2 2 1 2 1 2 1 2 2 2 2 1 1 2 cos cos t b t b R FK F K K K Dy L J J N J J J J L K RF K K mgd R mgd D y y N L L J L J J N 3 It is tedious but straightforward using 3 to find a choice of state variables. 1 2 1 2 3 2. 3. 3 4 4. 5. 4 5 6. 7. 5 0 8. 1 . .cos .cos .cos . .cos x x a y x x a y a y x x a y a y x x a y a y x b u a y y x 4 Where the unknown parameters 1 2 3 8 ..... a a a a are defined as: 1 2 1 2 1 1 2 1 3 0 2 2 1 2 1 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 4 5 2 2 1 2 1 2 1 2 1 2 1 2 2 1 6 t t b b t R F F mgd K R F F K K K K FF a a b a L J J J J J NL L J J J L J J N J J R K K FF FK F K K K F R F mgd a a L J J N J J J J N J J J J L L J J K a 1 2 2 7 8 2 2 2 1 2 1 2 1 2 1 2 1 2 . t b t b R F K F K K K K RF K K mgd R mgd a a L J J N J J J J L N L L J J L J J N 5 Hence the design procedure of theorem is applicable to 4 and an adaptive controller that achieves bounded asymptotic position tracking from all initial conditions and for all position values of the constant J 1 J 2 F 1 F 2 K K t K b R L. We consider systems in the output-feedback form: 1 2 0 1 1 1 2 3 0 2 2 1 1 0 1 1 1 1 0 1 p 1 0 0 1 1 q j j j q j j j q p p p j j p j q p p p j j m j q n n j j n j x x y a y x x y a y x x y a x x y a y b y u x y a y b y u y x  6 slide 4: International Journal of Engineering Research ScienceIJOER ISSN: 2395-6992 Vol-2 Issue-4 April- 2016 Page|107 01 02 03 04 05 0 0 0 0 0 y y y y y 8 5 0 q n p m 1 2 3 4 5 y 0 0 0 0 0 0 0 0 y cos 0 0 0 0 0 0 0 0 y cos 0 0 0 0 0 0 0 0 y cos 0 0 0 0 0 0 0 0 cos i i i i i y y y y y y y y y Where x R is the state u R is the input y R is the output φ σ are smooth nonlinear functions and 1 2 7 8 0 ...... a a a a a b b7 are vectors of unknown constant parameters. Only the output y is available for measurement. We rewrite as 0 1 T x Ax y a b u y e x 8 11 81 44 15 85 0 ... ... ... ... 1. 0 0 ... x y y A I y c y y    9 Filters and observer We start by rewriting 8 as 0 1 T x Ax y a b u y e x 10 Where the p q+m+1 9 dimensional parameter vector is defined by 0 8 1 x b a 11 And 1 2 T row row F y u u y 12 If  were known we would design an observer 0 ˆ ˆ T x A x ky F y u 13 With the vector 1 2 .... T n k k k k chosen so that the matrix 0 1 T A A ke By Hurwitz that is 0 0 0 T T PA A P I P P Then the observer error ˆ x x x would be governed by the exponentially stable system 0 x A x Since  is not known the observer is not implementable but it provides motivation for the subsequent development. We define the state estimate ˆ T x  14 Which employs the filters 0 0 T T T A ky A F y u  15 The state estimate error ˆ x x 16 is readily shown to satisfy . A   slide 5: International Journal of Engineering Research ScienceIJOER ISSN: 2395-6992 Vol-2 Issue-4 April- 2016 Page|108 The nonminimal observer1415 is still nonimplementable because it depends on  . However it has a key property not present in13: the state x satisfies a static relationship with  that is T x    This is easily verified by substituting14 into16. Remark 1. The certainty equivalence counterpart of the estimate14 is ˆ T x  17 it can alternatively be generated via 0 ˆ ˆ T T x A x ky F y u  18 This observer is not a certainty equivalence version of 13 because of the term ˆ T  . To reduce the dynamic order of the filter15 we exploit the structure of Fyu. Denote the first m+1 columns of Ω T by υ 0 satisfy the equations 0 0 0 5 A e u  19 It is easy to show that 0 5 5 A e e Therefore the vectors are generated by only one input filter 0 5 A e u  20 with the algebraic expressions 0 0...5 j j A j   While we always implement the filter20 for analysis we use the equations 19 considering12 15 20. Ω is obtained as 5 1 0 .... T   21 where the matrix is generated by 0 nxq A y22 TABLE 1 K-filter 0 A ky   0 nxq A y 0 5 A e u  23 0 0.... j j A j m   0 T j 24 slide 6: International Journal of Engineering Research ScienceIJOER ISSN: 2395-6992 Vol-2 Issue-4 April- 2016 Page|109 The implemented filters are summarized in Table 1. The total dynamic order of the K-filters is nxq+2. As explained in 1 a further reduction is possible by using the reduced-order observer technique so that the total filter dynamic order becomes n-1q+2. To prepare for the backstepping procedure in the next subsection we consider the equation for the output rewritten from 6 21 y x a25 we need to replace the unavailable state x 2 by available filter signals. We have 2 2 2 2 022 2 0 02 22 2 2 . 0 T x b           26 we obtain the following two important expressions for y 0 2 0 2 T T y        27 where the regressor and truncated regressor are defined as 0222 0   28 and 0 2 2    2.2 Adaptive controller design with tuning functions The tuning functions design with K-filters has many similarities with the state-feedback design presented in section 4.2 in 1. It also uses the same technique for dealing with unknown high frequency gain b m as in section 4.5 in 1. The first obstacle for applying backstepping with output feedback is that the state x 2 is not measured. For this reason 27 is written in a form which suggests that the filter signal be used for backstepping. Indeed by comparing 6 with 0 5 m m A e u   29 we see that both x 2 and 2 m  are separated from the control u by 4 integrators. Therefore the system to which we apply backstepping is 0 2 0 1 01 05 0 05 01 2... 4. T i m i i y k i b u k           30 Our analysis will show that once we stabilize the system 30 all the close-loop signals remain bounded. In the backstepping procedure while multiplying by nonlinear terms we will include nonlinear damping terms in our stabilizing functions. Next we need to choose the stabilizing functions and the tuning functions to achieve a design skew-symmetric form. For system 30 we change coordinates 1 1 0 1 ˆ 2...4 r i i i r i z y y z y i   31 where ˆ  is an estimate of 0 1/ b  The complete tuning function design with K-filters is summarized in Table 2 where the filters given in Table 1 are employed. slide 7: International Journal of Engineering Research ScienceIJOER ISSN: 2395-6992 Vol-2 Issue-4 April- 2016 Page|110 TABLE 2 1 1 1 ˆ 2...4 r i i m i r i z y y z y i   32 1 1    1 1 1 1 1 0 T c z d z     2 2 1 1 1 1 2 0 1 2 2 2 1 1 2 2 2 ˆ ˆ ˆ i r b z c d k z e z y y y y           2 2 1 1 1 1 1 1 1 1 ˆ i i i i i i i i i i i i i j i i j i z c z d z k z z z y y y y             1 1 1 1 1 0 0 0 1 1 1 1 1 1 ˆ i T j i i i i i r j j r m i i j j j j j A ky A y y y y k              33 1 1 1 1 ˆ T r y e z     1 1 i i i i z y    34 Adaptive control law: 5 4 05 1 r u y y    35 Parameter update laws: W z t z   0 1 1 ˆ sgn T r b y e z   36 III. EXPERIMENT DESIGN 3.1 Experiment and simulation setup A 2-DOF flexible-joint robot arm control is implemented based on algorithms shown in section II. The control algorithms are modelled in Simulink run in real-time under Real Time Window Target utility of Real Time Workshop and interface through a PCI1711 card from Advantech with the real world robot arm see Fig. 2. The model constructed the same as in 2 and we repeat here for convenience consisted of MATLAB/Simulink algorithm PCI1711 card DC motor with PWM driver and IMU. Torque transmission between the actuator DC motor and the plant industrial robot arm is of spring drive as a flexible-joint. slide 8: International Journal of Engineering Research ScienceIJOER ISSN: 2395-6992 Vol-2 Issue-4 April- 2016 Page|111 1 q K FIG. 2. MODEL OF CONTROL ALGORITHMS 3.2 Simulation Environment and Experiment Results The proposed algorithms are simulated in MATLAB/Simulink implemented in a real-time hardware using Real-time Workshop with DAQ cards. Model parameters can be viewed modified in run-time allowing to find an optimal control parameter set and controller types as well. FIG. 3. EXPERIMENT RESULT The experiment result for control algorithms is shown in Fig. 3 where the setpoint for robot arm angle changes from -10 0 to +10 0 lines 1 and 2 show changes of the robot arm angle output signals for backstepping tuning functions with K-filters and backstepping observer controllers respectively. The tuning functions controller has shorter response time and better control performance in terms of stability and convergence. IV. CONCLUSION In this paper an adaptive backstepping control with tuning functions and K-filters for robot manipulators is developed and implemented. Applying tuning functions with K – filters the strongest stability and convergence properties can be achieved. The result is shown for a real-time system of a single-link flexible-joint manipulator but the proposed algorithms can be extended and applied to more complex industrial robot manipulators. REFERENCES 1 J M. Krstic I. Kanellakopoulos P. Kokotovic “Nonlinear and Adaptive Control Design” Wiley 1995. 2 Tran Xuan Kien Do Duc Hanh “Adaptive Observer Backstepping Control for Industrial Robot Manipulators Using IMU”. International Journal of Emerging Technology and Advanced Engineering Volume 6 Issue 1 January 2016 102-111. 3 Bo Zhou “Backstepping Based Global Exponential Stabilizaton of a Tracked Mobile Robot with Slipping Pertubation” Journal of Bionic Engineering 2011 8:69-76. 4 Yongming Li “Adaptive Fuzzy Output Feedback Control for a Single-Link Flexible Robot Manipulator Driven DC Motor via Backstepping” Nonlinear Analysis:Real World Applications 2013 14: 483-494.

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