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Design of an Error-Based Adaptive Controller for a Flexible Robot Arm Using Dynamic Pole Motion Approach

Design of an Error-Based Adaptive Controller for a Flexible Robot Arm Using Dynamic Pole Motion... Hindawi Publishing Corporation Journal of Robotics Volume 2011, Article ID 726807, 9 pages doi:10.1155/2011/726807 Research Article Design of an Error-Based Adaptive Controller for a Flexible Robot Arm Using Dynamic Pole Motion Approach 1 1 2 Ki-Young Song, Madan M. Gupta, and Noriyasu Homma Intelligent Systems Research Laboratory, College of Engineering, University of Saskatchewan, Saskatoon, Sk, Canada S7N5A9 Research Division on Advanced Information Technology, Cyberscience Center, Tohoku University, Sendai 980-8579, Japan Correspondence should be addressed to Madan M. Gupta, madan.gupta@usask.ca Received 16 July 2011; Accepted 12 October 2011 Academic Editor: Ivo Bukovsky Copyright © 2011 Ki-Young Song et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Design of an adaptive controller for complex dynamic systems is a big challenge faced by the researchers. In this paper, we introduce a novel concept of dynamic pole motion (DPM) for the design of an error-based adaptive controller (E-BAC). The purpose of this novel design approach is to make the system response reasonably fast with no overshoot, where the system may be time varying and nonlinear with only partially known dynamics. The E-BAC is implanted in a system as a nonlinear controller with two dominant dynamic parameters: the dynamic position feedback and the dynamic velocity feedback. For illustrating the strength of this new approach, in this paper we give an example of a flexible robot with nonlinear dynamics. In the design of this feedback adaptive controller, parameters of the controller are designed as a function of the system error. The position feedback K (e,t) and the velocity feedback K (e,t) are continuously varying and formulated as a function of the system error e(t). This approach for formulating the adaptive controller yields a very fast response with no overshoot. 1. Introduction the Lyapunov function has been considered as one of the most effective way for designing controllers for nonlinear Recently, there has been an increasing interest in the design systems. of feedback controllers: from the design of conventional In this paper, we introduce a new notion of con- approaches to the design of intelligent-based approaches. troller called error-based adaptive controller (E-BAC) with a One such approach is on the design of adaptive controller novel conception based upon dynamic pole motion (DPM) for controlling a complex dynamic system containing non- approach. In general, for the design of E-BAC, we consider linearity like flexible joints. During the past, there has been two dominant parameters, the position feedback K (e, t)and a common practice to approximate a nonlinear system by a velocity feedback K (e, t), and a proper design of these two linear system in limited operating ranges and then make use feedback parameters will yield a faster and stable response of of the conventional controller design approaches. However, the system with no overshoot. The feedback parameters are the nonlinearity of a system is inevitable since many adapted by the system error e(t) and its states x(t). systems in practice involve nonlinear relationships among The rest of the paper is organized as follows. In Section 2, the variables such as electromechanical systems, hydraulic we introduce some important observations of a step response systems, and pneumatic systems [1]. For decades, various for a typical linear second-order system. In Section 3,we schemes of adaptive control have been proposed, and adap- describe the notion of dynamic pole motion (DPM) and the tive control for nonlinear systems with complex dynamics design of error-based adaptive controller (E-BAC) in detail. has received great attention. However, not many of these A flexible robotic joint control is presented in Section 4 with approaches are suitable for complex nonlinear systems [2– E-BAC and DPM as a case study. Section 5 concludes this 4]. Up to the present, inverse optimal controller [5–7] using paper with a discussion and future works. 2 Journal of Robotics 2. Some Important Observations in the Step −Cω + jω n d Response for a Second-Order Linear System jω In our study, we consider a typical open-loop second-order plant G (s)definedas jω p d G (s) = , (1a) ω (1 − C ) p n s + as + b r 0 + u ∑ 1 x y −Cω + as + b − u p − (1b) K s Figure 1: Definition of the various parameters in the complex σ-jω K plane: natural frequency: ω , damping ratio: ζ, damped natural −1 frequency: ω = ω (1 − ζ ), and θ = cos ζ. d n ⎡ ⎤ ⎡ ⎤ transient parameters in the design of a controller are usually ⎣ ⎦ ⎣ ⎦ state vector: x = = C , considered [1, 8, 9]: x˙ (1c) π − θ π − θ −1 output: y = Cx = (b + K )x . 1 1 1 rise time: T = = , θ = cos (ζ), ω 1 − ζ Equations (1a), (1b), and (1c) represent a typical second- 4 settling time: T = (2% criterion), order system with position (K ) and velocity (K ) feedbacks. 1 2 ζω As shown in (1b), with position and velocity feedback −ζπ/ 1−ζ controller, the transfer function of the closed-loop system is maximum overshoot: M = e × 100 (%), given by 2 4 2 ( ) bandwidth: ω = ω 1 − 2ζ + 4ζ − 4ζ +2. BW n Y (s) C (5) = ,(2) R(s) s + (a + K )s + (b + K ) 2 1 It is important to note that in the step response of the second- order system, the dominant transient parameters T , T ,and where C = b + K . This transfer function can be compared r s M are dependent upon the natural frequency (ω ) and the with a general linear second-order system model as p n damping ratio (ζ) of the system. Thus, the positions of the poles of the system are determined by the values of ω and ζ 2 n (b + K ) ω = . (3) as shown in Figure 1. 2 2 2 s + (a + K )s + (b + K ) s +2ζω s + ω 2 1 n As shown in Figure 2, it is also to be noted that, in typical transient responses an underdamped system (ζ< 1) yields a Thus, we see that faster rise time (T ) at the expense of a large overshoot (M ) r p and a large settling time (T ), whereas an overdamped system (ζ> 1) yields no overshoot, that is, M = 0, but it yields large ω = (b + K )  K , ω : system natural frequency, 1 p n p T and T . r s ( ) 2ζω = a + K  K , ζ: system damping ratio, n 2 v (4) 3. Development of an Error-Based Adaptive Controller (E-BAC): Some Design Criteria where the parameters K and K are defined as position feed- p v back and velocity feedback, respectively. For the design of an appropriate feedback controller, let us Generally the dynamic behavior of a second-order system consider the system error e(t) as an important signal in our can be described in terms of two dominant parameters, feedback design. In our design methodology developed in the natural frequency (ω ) and the damping ratio (ζ). The this paper, we will make the parameters of the feedback transient response of a typical control system often exhibits controller as functions of the error. From the transient damped oscillations before reaching the steady state. In responses shown in Figure 2(b), we can emphasize that for specifying the transient response characteristics of a second- large errors a small ζ and a large ω , (i.e., an underdamped order control system to a unit-step input, the following dynamics with large bandwidth) will yield a very fast Journal of Robotics 3 1.4 r (t) y(t) + v(t) Underdamped plant 1.2 system u(e, t) x (t) x (t) 2 1 u (e, t) 0.8 v K (e, t) Σ v x x 0.6 u (e, t) K (e, t) Overdamped system 0.4 e(t) Desired system response 0.2 + − Error-based adaptive controller 0123456789 10 Figure 3: The proposed error-based adaptive controller (E-BAC): (a) The system response curves x (t) = x˙ (t). K (e, t)and K (e, t) are defined in (8) and (12). 2 1 p v Overdamped system 0.8 result into a small bandwidth of the system. For small xx 0.6 errors, a large damping ratio in the system will avoid Underdamped system any overshoot in the system response. 0.4 0.2 Design of Parameters for the E-BAC (i) Position feedback K controls the natural frequency −0.2 Desired error response 2 ω ,(K = ω ), and, therefore, the bandwidth of the n p −0.4 system. 0123456789 10 (ii) Velocity feedback K controls the damping ratio ζ, t v (K = 2ζω ). v n (b) The error response curves of the systems Thus, we design the adaptive controller parameters which, Figure 2: System responses to a unit-step input with two different in this case, are the position feedback K (e, t)and velocity locations of poles (i) underdamped case (ζ< 1) and (ii) over- feedback K (e, t) as functions of the error, e(t). This pro- damped case (ζ> 1). The desired system response curve initially follows the underdamped curve for large errors and then settles cedure for designing the adaptive controller will introduce down to a steady-state value (following the overdamped curve) for a dynamic motion in the poles of the system keeping decreasing errors. the system response at an acceptable level. Here thus, we introduce a new notion of the movable poles and give it the name Dynamic Pole Motion (DPM). The proposed novel E- BAC is illustrated in Figure 3. response with a very small rise time T . On the other hand, This novel design philosophy for adaptive controller is for small errors a large ζ and a small ω (i.e., an overdamped translated into the following linguistic algorithm: system with a small bandwidth) will inhibit any overshoot. Since ζ and ω are dependent upon the parameters of As error decreases from a large value to a position feedback (K ) and velocity feedback (K ), if we p v small value, K (e, t)(=ω (t)) is continuously define K (e, t)and K (e, t) as functions of the system error, p v decreased from a very large value to a small value, e(t) = r(t) − y(t), then we can achieve a very fast dynamic and simultaneously, K (e, t)(=2ζ(t)ω (t)) is in- v n response with no overshoot. creased from a small value to a large value. From these qualitative observations on the transient response of the step response, we derive the following design This linguistic control algorithm causes a larger bandwidth criteria for the E-BAC [10]. with a smaller damping ratio for large errors and smaller bandwidth with larger damping ratio for small errors. Hence, Design Criteria for the Error-Based Adaptive Controller as discussed above and shown in Figures 2 and 3, during the operation of the system a desired transient response from the (E-BAC) systems can be achieved by varying ω and ζ as functions (i) If the system error is large, then keep the damping of error. As given in (4), ω and ζ are dependent upon the ratio ζ very small and natural frequency ω very n position feedback K and velocity feedback K ,respectively. p v large. A large ω and small ζ will result into a large n Some typical response curves for a second-order closed-loop bandwidth of the system, thereby a shorter rise time system with varying K and K are shown in Figure 4. p v and fast response. (ii) If the system error is small, then keep the damping 3.1. Design of E-BAC Parameters K (e, t) and K (e, t). Using p v ratio ζ large and natural frequency ω small. This will the design criteria for the adaptive controller stated above, e(t) y(t) 4 Journal of Robotics 1.2 Table 1: Various possible functions and their graphic experessions for feedback gains K (e, t)and K (e, t). p v K (e, t) K (e, t) K = 0.5, ζ = 1, ω = 1.84 p n 0.8 K = 1, ζ = 0.92, ω = 1.73 p n 0.6 K = 1.5, ζ = 0.86, ω = 1.87 p n K (1 + α|e|) pf K = 2, ζ = 0.8, ω = 2 0.4 p n pf K = 2.5, ζ = 0.75, ω = 2.12 p n 0.2 e(t) K = 3, ζ = 0.72, ω = 2.24 p n 0123456789 10 K (e, t) t (s) (a) A family of system response curves with various values of K and a constant K = 3 v 2 K (1 + αe ) pf pf 1.2 e(t) 0.8 K = 0.5, ζ = 0.43, ω = 1.73 K (e, t) v n v K (e, t) K = 1, ζ = 0.58, ω = 1.73 v n 0.6 K = 1.5, ζ = 0.72, ω = 1.73 v n K = 2, ζ = 0.86, ω = 1.73 v n 0.4 K vf K = 2.5, ζ = 1, ω = 2 v n vf 1+ β|e| K = 3, ζ = 1, ω = 3 0.2 v n 0123456789 10 e(t) t (s) K (e, t) (b) A family of system response curves with various values of K and a vf constant K = 1 Figure 4: System response curves of a second-order system varying 1 vf K (position feedback) and K (velocity feedback). 1+ βe p v one can develop many types of functions for K (e, t)and e(t) K (e, t), which satisfy the design criteria with respect to the K (e, t) system error and time. Here, we give one such function for vf K (e, t)and K (e, t) by defining the system error as p v e(t) = r(t) − y(t),(6) K exp(−βe ) vf where the system output y(t)isgiven by y(t) = K (e, t)x (t). (7) p 1 e(t) Many other functions can be derived for K (e, t)and K (e, t), for example, p v Thus, we define the position feedback K (e, t) and the veloc- using the hyperbolic tangent and cosine functions. ity feedback K (e, t) gains as functions of e(t)as K (e, t) = K 1+ αe (t) , p pf K (e, t)and K (e, t), and exp(·) is the exponential function. p v (8) The other possible functions for K (e, t)and K (e, t)are p v ( ) ( ) K e, t = K exp −βe t , v vf given in Table 1. where α and β are some gain constants which decide the slope of the functions and affect the system response (see 3.2. Design of the Error-Based Adaptive Controller (E-BAC). Figure 5), K and K are the final steady-state values of The error-based adaptive control signal u(e, t)isderived as a pf vf Response Response Journal of Robotics 5 4. A Case Study: Control of K (e, t) a Flexible Robot Arm Using E-BAC In this section, we present the design and simulation studies of the proposed error-based adaptive controller (E-BAC) for K a flexible joint of a robot arm. pf 4.1. Modeling of a Single Link Flexible Robot. As shown in Figure 6, a single link manipulator with flexible joint consists e(t) of an actuator connected through a gear train (harmonic (a) Changes in slope for K (e, t) = K (1 + p pf 2 drive) with the ratio n to a rigid link with length l,mass m, αe )for variousvaluesof α: the direction and moment of inertia ml /3. of arrows indicates the increasing value of α from negative to positive values Let us symbolize the rotor inertia of the actuator J , the viscous damping of the actuator B , the relative angular K (e, t) displacement of the joint actuator θ , a torque to the motor vf shaft τ , and the relative displacement of the end effector (load) θ . The joint flexibility is modeled by a linear torsional spring with stiffness k. The dynamics of the manipulator with a flexible joint can be represented by Euler-Lagrange equation defining τ = r as [11, 12] ml mgl θ e(t) ¨ ˙ θ + B θ + sin θ + k θ + = 0, L L L L L 3 2 n (b) Changes in slope for K (e, t) = (12) K exp[−βe (t)] for various values of β:the vf k θ ¨ ˙ J θ + B θ + θ + = r. direction of arrows indicates the increasing M M M M L n n value of β Equation (12) can be rewritten using the state variables x Figure 5: The change of the slopes of K (e, t)and K (e, t)curvesfor p v (i = 1, 2, 3, 4) defining as various values of α and β. ˙ ˙ x (t) = θ , x (t) = θ , x (t) = θ , x (t) = θ . 1 M 2 M 3 L 4 L (13) 1:n Thus, we have M k x˙ (t)= x (t), 1 2 Actuator M τ J M M ˙ ( ) ( ) ( ) ( ) ( ) x t =−a x t − a x t − a x t + br t , 2 1 1 2 2 3 3 x˙ (t) = x , 3 4 Figure 6: A schematic diagram of a single link manipulator with a flexible joint. x˙ (t) =−a x (t) − a x (t) − a sin(x (t)) − a x (t), 4 4 1 5 3 6 3 7 4 (14) where function of the error e(t)and time t using the following two steps: 1 k B k b = , a = , a = , a = , 1 2 3 J J n J J n M M M M (15) position feedback control: u (e, t) = K (e, t)x (t), p p 1 3g 3k 3k 3B a = , a = , a = , a = . 4 5 6 7 (9) 2 2 2 mnl ml 2l ml velocity feedback control: u (e, t) = K (e, t)x (t), v v 2 The block diagram of the system is shown in Figure 7. This system is a nonlinear and time varying system since where K (e, t)and K (e, t)are defined in (8) and (12), res- p v the sine function in the feedback loop of the system causes pectively. Thus, the total feedback signal u(e, t)isgiven by nonlinearity in the system. The output of the system is dependent on the amplitude of the control signal, and if we u(e, t) = u (e, t) + u (e, t), (10) p v use the conventional design tools, this nonlinearity causes some problems in designing an effective controller. In this paper, we present a novel approach to the design of a and the control signal v(t) (see Figure 3)isdefinedas controller for this nonlinear timevarying system by using the error-based adaptive controller (E-BAC) and dynamic pole ( ) ( ) ( ) v t = r t − u e, t . (11) motion (DPM). 6 Journal of Robotics x ˙ = x ∑ x ˙ 1 2 x ∑ x ˙ x ˙ = x x 2 1 − 4 3 4 3 r + 1 1 1 1 s s s s − y − − 1 Sine function Figure 7: Block diagram of the single link manipulator with a flexible joint. The system has both linear and nonlinear feedbacks. g -plane [g (t) = σ (t)+ jω() t ] 1.5 g (t) g (t) 1 3 at t = T 0.8 :1 ω :12.364 BW 0.5 0.6 at t =0 ζ:0.22 0.4 ω :69.42 BW −0.5 0.2 g (t) g (t) −1 T s −1.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 t (s) −2 −1 −0.8 −0.6 −0.4 −0.20 0 .2 Input σ (t) Output Error Figure 8: The sketch of dynamic pole motion (DPM) of the single link manipulator with a flexible joint system, (18), without Figure 10: The system response with the E-BAC to a step reference a controller. Four poles are moving in the g(t) = σ(t)+ jω(t) input. Rise time T = 0.14 seconds; settling time T = 0.36 seconds. r s plane with varying x (t). Note that for certain values of x (t)two 3 3 dynamic poles move towards the right-hand side of the g-plane causing instability in the system. first formulate the dynamic characteristic equation of the system. In this study for the nonlinear and timevarying robot arm, a new notion of timevarying complex variable g(t) = + 1 r v Compensator − (g + g )(g + g )(g + g )(g + g ) σ(t)+ jω(t)(g-plane) is applied instead of the time invariant (G ) 1 2 3 4 complex variable s = σ + jω (s-plane). The g-plane has u = (u + u ) p v E-BAC the same properties of s-plane with an additional property of timevarying. The dynamic characteristic equation of Figure 9: Schematic diagram of the flexible joint of a robot arm the single-link manipulator with a flexible joint shown in with a compensator (G )and E-BAC. Figure 7 and described in (16)isgiven by 4 3 2 g (t)+2g (t)+ 3+ ψ(t) g (t)+ 2+ ψ(t) g(t)+2+ψ(t) = 0, 4.2. Design of Adaptive Controller for the Systems. In this case (17) study, for simplicity we set the value of the parameters a (i ∈ where ψ(t) = sin(x (t))/x (t). [1, 7]) and b equal to 1. Thus, (14)can be rewrittenas 3 3 The dynamic roots of this characteristic equation of the x˙ (t) = x (t), 1 2 transfer function can be calculated as x˙ (t) =−x (t) − x (t) − x (t) + r(t), 2 1 2 3 g (t) = −1 − −2ψ(t) − 3 ± 2 ψ (t) − 4 , 1,2 (16) 2 x˙ (t) = x (t), 3 4 (18) sin(x (t)) g (t) = −1+ −2ψ(t) − 3 ± 2 ψ (t) − 4 . 3,4 x (t) =−x (t) − 1 − x (t) − x (t). 4 1 3 4 x (t) Now we design an error-based adaptive controller for the The nonlinear function, ψ(t) = sin(x (t))/x (t), covers the 3 3 single link robotic manipulator with a flexible joint. We range −0.22 <ψ(t) < 1 for all values of x (t)over [−∞,∞]. jω(t) Amplitude Journal of Robotics 7 g -plane [g (t) = σ (t)+ jω(t)] g (t) −10 g (t) −50 −20 0 0 −30 −50 −1 0.8 0.6 −2 σ (t) −100 −40 g (t) 2 g (t) 0.4 e(t) −3 0.2 −103−102−101−100 −99 −98 −150 −110 −100 −90 −80 −70 −60 −50 −40 −30 −20 σ (t) (a) The dominant poles are moving from initial position (b) 3D sketch of the dynamic poles and zeros motion of the controlled g (0) =−10.24 ± j45.14 to final position g (0.5) =−37.9557 3,4 3 system. The location of the dynamic poles and zeros is a function of the and g (0.5) =−19.2109, respectively. This dynamic pole motion system error causes the system response to become from underdamped response to overdamped response Figure 11: The sketch of motions of dynamic poles and zeros of the system with a compensator and an E-BAC (×: poles, :zeros). Thus, the roots of the dynamic characteristic equation are K (e, t) are designed such that they yield a small moving in the g-plane (g(t) = σ(t)+ jω(t)). damping ratio with large bandwidth for large The moving roots of the dynamic characteristic equation errors and a large damping ratio with small are named as dynamic poles. (Note that in linear time- bandwidth for small errors. invariant dynamic systems since the parameters of the system For achieving a good controller for this fourth-order flexible remain constant, the poles and zeros of the system are robot arm, we must first add a compensator to relocate two time invariant.) The plot of the four dynamic poles of this dynamic poles far away from the origin in the left-half of the flexible robot arm without a controller is shown in Figure 8. g-plane. In this case study, we relocate g (t)and g (t)far away 1 2 From this figure, it is clear that for some values of x two from the jω(t)-axis. These two relocated poles far away in the dynamic poles move towards the right-hand side (RHS) of left-side of the g-plane will induce very small time constants, the g(t) = σ(t)+ jω(t) plane causing instability in the system. thereby, will have negligible effect in the system dynamic The design criteria of our proposed error-based adaptive response. The other two poles that are closer to the imaginary controller (E-BAC) for a system are as below. axis are dominant poles and will cause an influence in the system dynamic response. Then, an E-BAC is added in the Design Criteria of E-BAC for the Flexible Joint of a Robot Arm. feedback loop with a position feedback K (e, t) and a velocity For designing an E-BAC, we should consider the following feedback K (e, t) defined in the previous sections. In this important points. study, the compensator (G ) used introduces two zeros in the forward loop with the position of zeros being g(t) =−102 ± (1) For introducing the stability in the robot arm system, j1.18. This compensator provides a control over the plant we should move the dynamic poles on the left-hand poles [g (t)and g (t)] keeping them far away from thejω(t)- 1 2 side (LHS) on g(t) = σ(t)+ jω(t)plane forall values axis. The feedback controller, E-BAC, provides a control over of x (t). the two plant poles [g (t)and g (t)]. The diagram of the 3 4 (2) Realization of DPM using E-BAC. system with a compensator and an E-BAC is illustrated in Figure 9. (a) For achieving the fast response time, the system The control input signal v(t) is derived using (11)as must have a large bandwidth for large errors v(t) = r(t) − u(t), (19) and small bandwidth for small errors. Thus, the position feedback K , the bandwidth parame- where ter, must be a function of the system error e(t). u(t) = u (e, t) + u (e, t) , p v (b) For no overshoot in the system response, damping should be adjusted continuously as (20) u (e, t) = K 1+ αe (t) x (t), p pf 3 a function of the system error. The position ( ) ( ) ( ) u e, t = exp −βe t x t . v 4 feedback K (e, t) and the velocity feedback jω(t) jω(t) 8 Journal of Robotics Bandwidth at t = 0 (s) −30 −35 −35 −40 −40 −45 −45 0.05 −50 0.04 −50 0.03 −55 0.02 −60 −55 0.01 Frequency (Hz) −60 −1 0 1 2 10 10 10 10 Frequency (Hz) (a) Bandwidth, ω (t) BW (b) ω (0) ≈ 70 Hz BW Bandwidth at t = 0.05 (s) Bandwidth at t = 0.5 (s) −30 −30 −35 −35 −40 −40 −45 −45 −50 −50 −55 −55 −60 −60 −1 0 1 2 −1 0 1 2 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) (c) ω (0.05) ≈ 32 Hz (d) ω (0.5) ≈ 18 Hz BW BW Figure 12: The variation of the bandwidth of the controlled system. (a) 3D sketch of the variation of the bandwidth with respect to the frequency at each time interval, (b) the bandwidth at t = 0 (sec), (c) the bandwidth at t = 0.05 (sec), and (d) the bandwidth at t = 0.5(sec). x (t) = x and x (t) = x˙ are the states of the system, K In the design of the E-BAC, we have arbitrarily chosen 3 4 pf and K are the steady-state values of feedbacks K (e, t)and the gains α = 2, β = 1, K = 70, and K = 5.5. With these vf p pf vf K (e, t), respectively, α and β are some gain constants for values, the controlled system responded as an underdamped K (e, t)and K (e, t), respectively, r(t) is the reference input of system for large error at t = 0, which continuously moved p v the system, and e(t) = [y(t)−r(t)] = (r(t)−K (e, t)x (t)) is towards an overdamped system with decreasing error. p 3 the system error. As described in the design criteria, the objective of the 4.3. Simulation Results. Using the gains α = 2, β = 1, K = pf embedded E-BAC is to design the control u(t) to make the 70, and K = 5.5, the initial positions of the dynamic vf system output y(t) follow the reference input signal r(t)as poles of the system are placed at g (0) =−100.87 ± j1.18 1,2 closely as possible with fast rise timeT and small settling (relocated to far from jω(t)-axis by the compensator), and timeT with no overshootM . Thus, we continuously change g (0) =−10.24 ± j45.1. During the operation of the s p 3,4 the dynamics of the close-loop system: initially for large system, as error is decreased to zero, the final positions errors, we make large bandwidth and very small damping of the dynamic poles are moved to −100.5639, −99.2695, ratio ζ(t), and as error decreases, the damping ratio ζ(t) −37.9557, and −19.2109 on the g-plane. The zeros are is continuously increased and the system bandwidth is located at around −100 near the relocated poles by the decreased. compensator, and the zeros attract the relocated poles not Magnitude (dB) Magnitude (dB) t(s) Magnitude (dB) Magnitude (dB) Journal of Robotics 9 controller, the system is unstable due to the nonlinearity in the feedback loop of the system. However, as shown in Figure 10, with E-BAC the trajectory response of the 0.8 system is very fast, T = 0.36 seconds, without any overshoot (M = 0%). Also, as shown in Figures 12 and 13, in this step 0.6 response, the initial bandwidth of the system is very high 0.4 (≈70 Hz) which settles down to about 18 Hz in the steady- state situation. The bandwidth of the system changes from 0.2 a large value to a small value. Similarly but contrarily, the 0.5 damping ratio ζ(t) varies from 0.221 (t = 0) to 1 (t = 0.5). 0.5 0.4 0.3 e(t) 0.2 From the simulation studies, it is shown that the proposed 0.1 1 t(s) E-BAC is able to control nonlinear time varying systems. Conventionally, a proper design of the controller guarantees Figure 13: The variation of the damping ratio ζ(t)withrespect to that the changing pole position is always positioned in the system error as a function of time. The value of ζ(t) changes the left-hand side (LHS) on g-plane. In this novel design from a low value, ζ(t) = 0.221, to a high value, ζ(t) = 1. approach, the dynamic poles are always located in LHS on g-plane, thus the stability of the controlled system is assured. Further work is under way to extend this E-BAC design to affect the dominant poles. The results of the simulation philosophy for higher-order partially known and unknown study of this case study are shown in Figure 10. Further, the complex dynamic systems. maps of the dynamic poles motion (DPM) are illustrated in Figure 11. The output response initially follows the trajectory References with a large bandwidth and a small damping ratio, which settles down with a large damping and a smaller bandwidth. [1] K. Ogata, Modern Control Engineering, Prentice Hall, Upper It is clear from the figure that the dynamic motion of Saddle River, NJ, USA, 4th edition, 2002. poles of the system is decided by the value of the system [2] Y. C. Chang, “An adaptive H∞ tracking control for a class of error. The initial positions of the dominant dynamic poles nonlinear multiple-input-multiple-output (MIMO) systems,” IEEE Transactions on Automatic Control,vol. 46, no.9,pp. are placed to generate a low damping ratio ζ(t) and large 1432–1437, 2001. bandwidth of the system. Thus, initially the system is under- [3] D. G. Taylor, P. V. Kokotovic, R. Marino, and I. Kanellakopou- damped. Thereafter, the dynamic poles are optimized and los, “Adaptive regulation of nonlinear systems with unmod- shifted as the system error decreases reducing the bandwidth eled dynamics,” IEEE Transactions on Automatic Control, vol. and increasing ζ(t). The final positions of the dominant 34, no. 4, pp. 405–412, 1989. dynamic poles make the system an overdamped system. [4] Y. Liu and X. Y. Li, “Robust adaptive control of nonlinear sys- Thus, the bandwidth becomes small, but ζ(t) becomes high. tems represented by input-output models,” IEEE Transactions The variation of the bandwidth at each time interval is shown on Automatic Control, vol. 48, no. 6, pp. 1041–1045, 2003. in Figure 12, and the variation of the damping ratio ζ(t)is [5] M. Jankovic, R. Sepulchre, and P. V. Kokotovic, “Global adap- shown with respect to the system error at each time interval tive stabilization of cascade nonlinear systems,” Automatica, vol. 33, no. 2, pp. 263–268, 1997. in Figure 13. [6] R.Sepulchre,M.Jankovic, andP.V.Kokotovic,“Integrator forwarding: a new recursive nonlinear robust design,” Auto- 5. Discussion and Conclusions matica, vol. 33, no. 5, pp. 979–984, 1997. [7] E. D. Sontag, “A ’universal’ construction of Artstein’s theorem In this paper, we have proposed the design of an error-based on nonlinear stabilization,” Systems and Control Letters, vol. adaptive controller (E-BAC) for controlling the dynamic 13, no. 2, pp. 117–123, 1989. response of a nonlinear system. The proposed E-BAC is the [8] N. S. Nise, Control Systems Engineering,JohnWiley&Sons, controller with continuously changing feedback parameters New York, NY, USA, 3rd edition, 2000. [9] W. J. Palm III, System Dynamics, McGraw-Hill, New York, NY, as functions of the system error: initially for large errors an USA, 2005. underdamped system with large bandwidth which is forced [10] K. Y. Song, M. M. Gupta, D. Jena, and B. Subudhi, “Design to approach to become an overdamped system with small of a robust neuro-controller for complex dynamic systems,” in bandwidth for small errors. In the beginning, for large errors Proceedings of the Annual Meeting of the North American Fuzzy the system is underdamped, thus, it makes the system faster Information Processing Society (NAFIPS ’09), Cincinnati, Ohio, with a wider bandwidth. As the error decreases, the value of USA, June 2009. the feedback gains K decreases and K increases. The design p v [11] M. W. Spong, K. Khorasani, and P. V. Kokotovic, “An integral of this adaptive controller is conceptually error-based and manifold approach to the feedback-control of flexible joint can be used to handle the complexity of systems. In order robots,” IEEE Journal of Robotics and Automation, vol. 3, no. to support the novel controller, we introduce the notion of 4, pp. 291–300, 1987. [12] L. Jin, M. M. Gupta, and P. N. Nikiforuk, “Dynamic recurrent dynamic pole motion (DPM). neural networks for modeling flexible robot dynamics,” in As a case study, we present a flexible joint of robot Proceedings of the 10th IEEE International Symposium on arm, which is a nonlinear dynamic system, and this system Intelligent Control, pp. 105–110, August 1995. is controlled by the proposed E-BAC. 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Design of an Error-Based Adaptive Controller for a Flexible Robot Arm Using Dynamic Pole Motion Approach

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Copyright © 2011 Ki-Young Song et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Publishing Corporation Journal of Robotics Volume 2011, Article ID 726807, 9 pages doi:10.1155/2011/726807 Research Article Design of an Error-Based Adaptive Controller for a Flexible Robot Arm Using Dynamic Pole Motion Approach 1 1 2 Ki-Young Song, Madan M. Gupta, and Noriyasu Homma Intelligent Systems Research Laboratory, College of Engineering, University of Saskatchewan, Saskatoon, Sk, Canada S7N5A9 Research Division on Advanced Information Technology, Cyberscience Center, Tohoku University, Sendai 980-8579, Japan Correspondence should be addressed to Madan M. Gupta, madan.gupta@usask.ca Received 16 July 2011; Accepted 12 October 2011 Academic Editor: Ivo Bukovsky Copyright © 2011 Ki-Young Song et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Design of an adaptive controller for complex dynamic systems is a big challenge faced by the researchers. In this paper, we introduce a novel concept of dynamic pole motion (DPM) for the design of an error-based adaptive controller (E-BAC). The purpose of this novel design approach is to make the system response reasonably fast with no overshoot, where the system may be time varying and nonlinear with only partially known dynamics. The E-BAC is implanted in a system as a nonlinear controller with two dominant dynamic parameters: the dynamic position feedback and the dynamic velocity feedback. For illustrating the strength of this new approach, in this paper we give an example of a flexible robot with nonlinear dynamics. In the design of this feedback adaptive controller, parameters of the controller are designed as a function of the system error. The position feedback K (e,t) and the velocity feedback K (e,t) are continuously varying and formulated as a function of the system error e(t). This approach for formulating the adaptive controller yields a very fast response with no overshoot. 1. Introduction the Lyapunov function has been considered as one of the most effective way for designing controllers for nonlinear Recently, there has been an increasing interest in the design systems. of feedback controllers: from the design of conventional In this paper, we introduce a new notion of con- approaches to the design of intelligent-based approaches. troller called error-based adaptive controller (E-BAC) with a One such approach is on the design of adaptive controller novel conception based upon dynamic pole motion (DPM) for controlling a complex dynamic system containing non- approach. In general, for the design of E-BAC, we consider linearity like flexible joints. During the past, there has been two dominant parameters, the position feedback K (e, t)and a common practice to approximate a nonlinear system by a velocity feedback K (e, t), and a proper design of these two linear system in limited operating ranges and then make use feedback parameters will yield a faster and stable response of of the conventional controller design approaches. However, the system with no overshoot. The feedback parameters are the nonlinearity of a system is inevitable since many adapted by the system error e(t) and its states x(t). systems in practice involve nonlinear relationships among The rest of the paper is organized as follows. In Section 2, the variables such as electromechanical systems, hydraulic we introduce some important observations of a step response systems, and pneumatic systems [1]. For decades, various for a typical linear second-order system. In Section 3,we schemes of adaptive control have been proposed, and adap- describe the notion of dynamic pole motion (DPM) and the tive control for nonlinear systems with complex dynamics design of error-based adaptive controller (E-BAC) in detail. has received great attention. However, not many of these A flexible robotic joint control is presented in Section 4 with approaches are suitable for complex nonlinear systems [2– E-BAC and DPM as a case study. Section 5 concludes this 4]. Up to the present, inverse optimal controller [5–7] using paper with a discussion and future works. 2 Journal of Robotics 2. Some Important Observations in the Step −Cω + jω n d Response for a Second-Order Linear System jω In our study, we consider a typical open-loop second-order plant G (s)definedas jω p d G (s) = , (1a) ω (1 − C ) p n s + as + b r 0 + u ∑ 1 x y −Cω + as + b − u p − (1b) K s Figure 1: Definition of the various parameters in the complex σ-jω K plane: natural frequency: ω , damping ratio: ζ, damped natural −1 frequency: ω = ω (1 − ζ ), and θ = cos ζ. d n ⎡ ⎤ ⎡ ⎤ transient parameters in the design of a controller are usually ⎣ ⎦ ⎣ ⎦ state vector: x = = C , considered [1, 8, 9]: x˙ (1c) π − θ π − θ −1 output: y = Cx = (b + K )x . 1 1 1 rise time: T = = , θ = cos (ζ), ω 1 − ζ Equations (1a), (1b), and (1c) represent a typical second- 4 settling time: T = (2% criterion), order system with position (K ) and velocity (K ) feedbacks. 1 2 ζω As shown in (1b), with position and velocity feedback −ζπ/ 1−ζ controller, the transfer function of the closed-loop system is maximum overshoot: M = e × 100 (%), given by 2 4 2 ( ) bandwidth: ω = ω 1 − 2ζ + 4ζ − 4ζ +2. BW n Y (s) C (5) = ,(2) R(s) s + (a + K )s + (b + K ) 2 1 It is important to note that in the step response of the second- order system, the dominant transient parameters T , T ,and where C = b + K . This transfer function can be compared r s M are dependent upon the natural frequency (ω ) and the with a general linear second-order system model as p n damping ratio (ζ) of the system. Thus, the positions of the poles of the system are determined by the values of ω and ζ 2 n (b + K ) ω = . (3) as shown in Figure 1. 2 2 2 s + (a + K )s + (b + K ) s +2ζω s + ω 2 1 n As shown in Figure 2, it is also to be noted that, in typical transient responses an underdamped system (ζ< 1) yields a Thus, we see that faster rise time (T ) at the expense of a large overshoot (M ) r p and a large settling time (T ), whereas an overdamped system (ζ> 1) yields no overshoot, that is, M = 0, but it yields large ω = (b + K )  K , ω : system natural frequency, 1 p n p T and T . r s ( ) 2ζω = a + K  K , ζ: system damping ratio, n 2 v (4) 3. Development of an Error-Based Adaptive Controller (E-BAC): Some Design Criteria where the parameters K and K are defined as position feed- p v back and velocity feedback, respectively. For the design of an appropriate feedback controller, let us Generally the dynamic behavior of a second-order system consider the system error e(t) as an important signal in our can be described in terms of two dominant parameters, feedback design. In our design methodology developed in the natural frequency (ω ) and the damping ratio (ζ). The this paper, we will make the parameters of the feedback transient response of a typical control system often exhibits controller as functions of the error. From the transient damped oscillations before reaching the steady state. In responses shown in Figure 2(b), we can emphasize that for specifying the transient response characteristics of a second- large errors a small ζ and a large ω , (i.e., an underdamped order control system to a unit-step input, the following dynamics with large bandwidth) will yield a very fast Journal of Robotics 3 1.4 r (t) y(t) + v(t) Underdamped plant 1.2 system u(e, t) x (t) x (t) 2 1 u (e, t) 0.8 v K (e, t) Σ v x x 0.6 u (e, t) K (e, t) Overdamped system 0.4 e(t) Desired system response 0.2 + − Error-based adaptive controller 0123456789 10 Figure 3: The proposed error-based adaptive controller (E-BAC): (a) The system response curves x (t) = x˙ (t). K (e, t)and K (e, t) are defined in (8) and (12). 2 1 p v Overdamped system 0.8 result into a small bandwidth of the system. For small xx 0.6 errors, a large damping ratio in the system will avoid Underdamped system any overshoot in the system response. 0.4 0.2 Design of Parameters for the E-BAC (i) Position feedback K controls the natural frequency −0.2 Desired error response 2 ω ,(K = ω ), and, therefore, the bandwidth of the n p −0.4 system. 0123456789 10 (ii) Velocity feedback K controls the damping ratio ζ, t v (K = 2ζω ). v n (b) The error response curves of the systems Thus, we design the adaptive controller parameters which, Figure 2: System responses to a unit-step input with two different in this case, are the position feedback K (e, t)and velocity locations of poles (i) underdamped case (ζ< 1) and (ii) over- feedback K (e, t) as functions of the error, e(t). This pro- damped case (ζ> 1). The desired system response curve initially follows the underdamped curve for large errors and then settles cedure for designing the adaptive controller will introduce down to a steady-state value (following the overdamped curve) for a dynamic motion in the poles of the system keeping decreasing errors. the system response at an acceptable level. Here thus, we introduce a new notion of the movable poles and give it the name Dynamic Pole Motion (DPM). The proposed novel E- BAC is illustrated in Figure 3. response with a very small rise time T . On the other hand, This novel design philosophy for adaptive controller is for small errors a large ζ and a small ω (i.e., an overdamped translated into the following linguistic algorithm: system with a small bandwidth) will inhibit any overshoot. Since ζ and ω are dependent upon the parameters of As error decreases from a large value to a position feedback (K ) and velocity feedback (K ), if we p v small value, K (e, t)(=ω (t)) is continuously define K (e, t)and K (e, t) as functions of the system error, p v decreased from a very large value to a small value, e(t) = r(t) − y(t), then we can achieve a very fast dynamic and simultaneously, K (e, t)(=2ζ(t)ω (t)) is in- v n response with no overshoot. creased from a small value to a large value. From these qualitative observations on the transient response of the step response, we derive the following design This linguistic control algorithm causes a larger bandwidth criteria for the E-BAC [10]. with a smaller damping ratio for large errors and smaller bandwidth with larger damping ratio for small errors. Hence, Design Criteria for the Error-Based Adaptive Controller as discussed above and shown in Figures 2 and 3, during the operation of the system a desired transient response from the (E-BAC) systems can be achieved by varying ω and ζ as functions (i) If the system error is large, then keep the damping of error. As given in (4), ω and ζ are dependent upon the ratio ζ very small and natural frequency ω very n position feedback K and velocity feedback K ,respectively. p v large. A large ω and small ζ will result into a large n Some typical response curves for a second-order closed-loop bandwidth of the system, thereby a shorter rise time system with varying K and K are shown in Figure 4. p v and fast response. (ii) If the system error is small, then keep the damping 3.1. Design of E-BAC Parameters K (e, t) and K (e, t). Using p v ratio ζ large and natural frequency ω small. This will the design criteria for the adaptive controller stated above, e(t) y(t) 4 Journal of Robotics 1.2 Table 1: Various possible functions and their graphic experessions for feedback gains K (e, t)and K (e, t). p v K (e, t) K (e, t) K = 0.5, ζ = 1, ω = 1.84 p n 0.8 K = 1, ζ = 0.92, ω = 1.73 p n 0.6 K = 1.5, ζ = 0.86, ω = 1.87 p n K (1 + α|e|) pf K = 2, ζ = 0.8, ω = 2 0.4 p n pf K = 2.5, ζ = 0.75, ω = 2.12 p n 0.2 e(t) K = 3, ζ = 0.72, ω = 2.24 p n 0123456789 10 K (e, t) t (s) (a) A family of system response curves with various values of K and a constant K = 3 v 2 K (1 + αe ) pf pf 1.2 e(t) 0.8 K = 0.5, ζ = 0.43, ω = 1.73 K (e, t) v n v K (e, t) K = 1, ζ = 0.58, ω = 1.73 v n 0.6 K = 1.5, ζ = 0.72, ω = 1.73 v n K = 2, ζ = 0.86, ω = 1.73 v n 0.4 K vf K = 2.5, ζ = 1, ω = 2 v n vf 1+ β|e| K = 3, ζ = 1, ω = 3 0.2 v n 0123456789 10 e(t) t (s) K (e, t) (b) A family of system response curves with various values of K and a vf constant K = 1 Figure 4: System response curves of a second-order system varying 1 vf K (position feedback) and K (velocity feedback). 1+ βe p v one can develop many types of functions for K (e, t)and e(t) K (e, t), which satisfy the design criteria with respect to the K (e, t) system error and time. Here, we give one such function for vf K (e, t)and K (e, t) by defining the system error as p v e(t) = r(t) − y(t),(6) K exp(−βe ) vf where the system output y(t)isgiven by y(t) = K (e, t)x (t). (7) p 1 e(t) Many other functions can be derived for K (e, t)and K (e, t), for example, p v Thus, we define the position feedback K (e, t) and the veloc- using the hyperbolic tangent and cosine functions. ity feedback K (e, t) gains as functions of e(t)as K (e, t) = K 1+ αe (t) , p pf K (e, t)and K (e, t), and exp(·) is the exponential function. p v (8) The other possible functions for K (e, t)and K (e, t)are p v ( ) ( ) K e, t = K exp −βe t , v vf given in Table 1. where α and β are some gain constants which decide the slope of the functions and affect the system response (see 3.2. Design of the Error-Based Adaptive Controller (E-BAC). Figure 5), K and K are the final steady-state values of The error-based adaptive control signal u(e, t)isderived as a pf vf Response Response Journal of Robotics 5 4. A Case Study: Control of K (e, t) a Flexible Robot Arm Using E-BAC In this section, we present the design and simulation studies of the proposed error-based adaptive controller (E-BAC) for K a flexible joint of a robot arm. pf 4.1. Modeling of a Single Link Flexible Robot. As shown in Figure 6, a single link manipulator with flexible joint consists e(t) of an actuator connected through a gear train (harmonic (a) Changes in slope for K (e, t) = K (1 + p pf 2 drive) with the ratio n to a rigid link with length l,mass m, αe )for variousvaluesof α: the direction and moment of inertia ml /3. of arrows indicates the increasing value of α from negative to positive values Let us symbolize the rotor inertia of the actuator J , the viscous damping of the actuator B , the relative angular K (e, t) displacement of the joint actuator θ , a torque to the motor vf shaft τ , and the relative displacement of the end effector (load) θ . The joint flexibility is modeled by a linear torsional spring with stiffness k. The dynamics of the manipulator with a flexible joint can be represented by Euler-Lagrange equation defining τ = r as [11, 12] ml mgl θ e(t) ¨ ˙ θ + B θ + sin θ + k θ + = 0, L L L L L 3 2 n (b) Changes in slope for K (e, t) = (12) K exp[−βe (t)] for various values of β:the vf k θ ¨ ˙ J θ + B θ + θ + = r. direction of arrows indicates the increasing M M M M L n n value of β Equation (12) can be rewritten using the state variables x Figure 5: The change of the slopes of K (e, t)and K (e, t)curvesfor p v (i = 1, 2, 3, 4) defining as various values of α and β. ˙ ˙ x (t) = θ , x (t) = θ , x (t) = θ , x (t) = θ . 1 M 2 M 3 L 4 L (13) 1:n Thus, we have M k x˙ (t)= x (t), 1 2 Actuator M τ J M M ˙ ( ) ( ) ( ) ( ) ( ) x t =−a x t − a x t − a x t + br t , 2 1 1 2 2 3 3 x˙ (t) = x , 3 4 Figure 6: A schematic diagram of a single link manipulator with a flexible joint. x˙ (t) =−a x (t) − a x (t) − a sin(x (t)) − a x (t), 4 4 1 5 3 6 3 7 4 (14) where function of the error e(t)and time t using the following two steps: 1 k B k b = , a = , a = , a = , 1 2 3 J J n J J n M M M M (15) position feedback control: u (e, t) = K (e, t)x (t), p p 1 3g 3k 3k 3B a = , a = , a = , a = . 4 5 6 7 (9) 2 2 2 mnl ml 2l ml velocity feedback control: u (e, t) = K (e, t)x (t), v v 2 The block diagram of the system is shown in Figure 7. This system is a nonlinear and time varying system since where K (e, t)and K (e, t)are defined in (8) and (12), res- p v the sine function in the feedback loop of the system causes pectively. Thus, the total feedback signal u(e, t)isgiven by nonlinearity in the system. The output of the system is dependent on the amplitude of the control signal, and if we u(e, t) = u (e, t) + u (e, t), (10) p v use the conventional design tools, this nonlinearity causes some problems in designing an effective controller. In this paper, we present a novel approach to the design of a and the control signal v(t) (see Figure 3)isdefinedas controller for this nonlinear timevarying system by using the error-based adaptive controller (E-BAC) and dynamic pole ( ) ( ) ( ) v t = r t − u e, t . (11) motion (DPM). 6 Journal of Robotics x ˙ = x ∑ x ˙ 1 2 x ∑ x ˙ x ˙ = x x 2 1 − 4 3 4 3 r + 1 1 1 1 s s s s − y − − 1 Sine function Figure 7: Block diagram of the single link manipulator with a flexible joint. The system has both linear and nonlinear feedbacks. g -plane [g (t) = σ (t)+ jω() t ] 1.5 g (t) g (t) 1 3 at t = T 0.8 :1 ω :12.364 BW 0.5 0.6 at t =0 ζ:0.22 0.4 ω :69.42 BW −0.5 0.2 g (t) g (t) −1 T s −1.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 t (s) −2 −1 −0.8 −0.6 −0.4 −0.20 0 .2 Input σ (t) Output Error Figure 8: The sketch of dynamic pole motion (DPM) of the single link manipulator with a flexible joint system, (18), without Figure 10: The system response with the E-BAC to a step reference a controller. Four poles are moving in the g(t) = σ(t)+ jω(t) input. Rise time T = 0.14 seconds; settling time T = 0.36 seconds. r s plane with varying x (t). Note that for certain values of x (t)two 3 3 dynamic poles move towards the right-hand side of the g-plane causing instability in the system. first formulate the dynamic characteristic equation of the system. In this study for the nonlinear and timevarying robot arm, a new notion of timevarying complex variable g(t) = + 1 r v Compensator − (g + g )(g + g )(g + g )(g + g ) σ(t)+ jω(t)(g-plane) is applied instead of the time invariant (G ) 1 2 3 4 complex variable s = σ + jω (s-plane). The g-plane has u = (u + u ) p v E-BAC the same properties of s-plane with an additional property of timevarying. The dynamic characteristic equation of Figure 9: Schematic diagram of the flexible joint of a robot arm the single-link manipulator with a flexible joint shown in with a compensator (G )and E-BAC. Figure 7 and described in (16)isgiven by 4 3 2 g (t)+2g (t)+ 3+ ψ(t) g (t)+ 2+ ψ(t) g(t)+2+ψ(t) = 0, 4.2. Design of Adaptive Controller for the Systems. In this case (17) study, for simplicity we set the value of the parameters a (i ∈ where ψ(t) = sin(x (t))/x (t). [1, 7]) and b equal to 1. Thus, (14)can be rewrittenas 3 3 The dynamic roots of this characteristic equation of the x˙ (t) = x (t), 1 2 transfer function can be calculated as x˙ (t) =−x (t) − x (t) − x (t) + r(t), 2 1 2 3 g (t) = −1 − −2ψ(t) − 3 ± 2 ψ (t) − 4 , 1,2 (16) 2 x˙ (t) = x (t), 3 4 (18) sin(x (t)) g (t) = −1+ −2ψ(t) − 3 ± 2 ψ (t) − 4 . 3,4 x (t) =−x (t) − 1 − x (t) − x (t). 4 1 3 4 x (t) Now we design an error-based adaptive controller for the The nonlinear function, ψ(t) = sin(x (t))/x (t), covers the 3 3 single link robotic manipulator with a flexible joint. We range −0.22 <ψ(t) < 1 for all values of x (t)over [−∞,∞]. jω(t) Amplitude Journal of Robotics 7 g -plane [g (t) = σ (t)+ jω(t)] g (t) −10 g (t) −50 −20 0 0 −30 −50 −1 0.8 0.6 −2 σ (t) −100 −40 g (t) 2 g (t) 0.4 e(t) −3 0.2 −103−102−101−100 −99 −98 −150 −110 −100 −90 −80 −70 −60 −50 −40 −30 −20 σ (t) (a) The dominant poles are moving from initial position (b) 3D sketch of the dynamic poles and zeros motion of the controlled g (0) =−10.24 ± j45.14 to final position g (0.5) =−37.9557 3,4 3 system. The location of the dynamic poles and zeros is a function of the and g (0.5) =−19.2109, respectively. This dynamic pole motion system error causes the system response to become from underdamped response to overdamped response Figure 11: The sketch of motions of dynamic poles and zeros of the system with a compensator and an E-BAC (×: poles, :zeros). Thus, the roots of the dynamic characteristic equation are K (e, t) are designed such that they yield a small moving in the g-plane (g(t) = σ(t)+ jω(t)). damping ratio with large bandwidth for large The moving roots of the dynamic characteristic equation errors and a large damping ratio with small are named as dynamic poles. (Note that in linear time- bandwidth for small errors. invariant dynamic systems since the parameters of the system For achieving a good controller for this fourth-order flexible remain constant, the poles and zeros of the system are robot arm, we must first add a compensator to relocate two time invariant.) The plot of the four dynamic poles of this dynamic poles far away from the origin in the left-half of the flexible robot arm without a controller is shown in Figure 8. g-plane. In this case study, we relocate g (t)and g (t)far away 1 2 From this figure, it is clear that for some values of x two from the jω(t)-axis. These two relocated poles far away in the dynamic poles move towards the right-hand side (RHS) of left-side of the g-plane will induce very small time constants, the g(t) = σ(t)+ jω(t) plane causing instability in the system. thereby, will have negligible effect in the system dynamic The design criteria of our proposed error-based adaptive response. The other two poles that are closer to the imaginary controller (E-BAC) for a system are as below. axis are dominant poles and will cause an influence in the system dynamic response. Then, an E-BAC is added in the Design Criteria of E-BAC for the Flexible Joint of a Robot Arm. feedback loop with a position feedback K (e, t) and a velocity For designing an E-BAC, we should consider the following feedback K (e, t) defined in the previous sections. In this important points. study, the compensator (G ) used introduces two zeros in the forward loop with the position of zeros being g(t) =−102 ± (1) For introducing the stability in the robot arm system, j1.18. This compensator provides a control over the plant we should move the dynamic poles on the left-hand poles [g (t)and g (t)] keeping them far away from thejω(t)- 1 2 side (LHS) on g(t) = σ(t)+ jω(t)plane forall values axis. The feedback controller, E-BAC, provides a control over of x (t). the two plant poles [g (t)and g (t)]. The diagram of the 3 4 (2) Realization of DPM using E-BAC. system with a compensator and an E-BAC is illustrated in Figure 9. (a) For achieving the fast response time, the system The control input signal v(t) is derived using (11)as must have a large bandwidth for large errors v(t) = r(t) − u(t), (19) and small bandwidth for small errors. Thus, the position feedback K , the bandwidth parame- where ter, must be a function of the system error e(t). u(t) = u (e, t) + u (e, t) , p v (b) For no overshoot in the system response, damping should be adjusted continuously as (20) u (e, t) = K 1+ αe (t) x (t), p pf 3 a function of the system error. The position ( ) ( ) ( ) u e, t = exp −βe t x t . v 4 feedback K (e, t) and the velocity feedback jω(t) jω(t) 8 Journal of Robotics Bandwidth at t = 0 (s) −30 −35 −35 −40 −40 −45 −45 0.05 −50 0.04 −50 0.03 −55 0.02 −60 −55 0.01 Frequency (Hz) −60 −1 0 1 2 10 10 10 10 Frequency (Hz) (a) Bandwidth, ω (t) BW (b) ω (0) ≈ 70 Hz BW Bandwidth at t = 0.05 (s) Bandwidth at t = 0.5 (s) −30 −30 −35 −35 −40 −40 −45 −45 −50 −50 −55 −55 −60 −60 −1 0 1 2 −1 0 1 2 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) (c) ω (0.05) ≈ 32 Hz (d) ω (0.5) ≈ 18 Hz BW BW Figure 12: The variation of the bandwidth of the controlled system. (a) 3D sketch of the variation of the bandwidth with respect to the frequency at each time interval, (b) the bandwidth at t = 0 (sec), (c) the bandwidth at t = 0.05 (sec), and (d) the bandwidth at t = 0.5(sec). x (t) = x and x (t) = x˙ are the states of the system, K In the design of the E-BAC, we have arbitrarily chosen 3 4 pf and K are the steady-state values of feedbacks K (e, t)and the gains α = 2, β = 1, K = 70, and K = 5.5. With these vf p pf vf K (e, t), respectively, α and β are some gain constants for values, the controlled system responded as an underdamped K (e, t)and K (e, t), respectively, r(t) is the reference input of system for large error at t = 0, which continuously moved p v the system, and e(t) = [y(t)−r(t)] = (r(t)−K (e, t)x (t)) is towards an overdamped system with decreasing error. p 3 the system error. As described in the design criteria, the objective of the 4.3. Simulation Results. Using the gains α = 2, β = 1, K = pf embedded E-BAC is to design the control u(t) to make the 70, and K = 5.5, the initial positions of the dynamic vf system output y(t) follow the reference input signal r(t)as poles of the system are placed at g (0) =−100.87 ± j1.18 1,2 closely as possible with fast rise timeT and small settling (relocated to far from jω(t)-axis by the compensator), and timeT with no overshootM . Thus, we continuously change g (0) =−10.24 ± j45.1. During the operation of the s p 3,4 the dynamics of the close-loop system: initially for large system, as error is decreased to zero, the final positions errors, we make large bandwidth and very small damping of the dynamic poles are moved to −100.5639, −99.2695, ratio ζ(t), and as error decreases, the damping ratio ζ(t) −37.9557, and −19.2109 on the g-plane. The zeros are is continuously increased and the system bandwidth is located at around −100 near the relocated poles by the decreased. compensator, and the zeros attract the relocated poles not Magnitude (dB) Magnitude (dB) t(s) Magnitude (dB) Magnitude (dB) Journal of Robotics 9 controller, the system is unstable due to the nonlinearity in the feedback loop of the system. However, as shown in Figure 10, with E-BAC the trajectory response of the 0.8 system is very fast, T = 0.36 seconds, without any overshoot (M = 0%). Also, as shown in Figures 12 and 13, in this step 0.6 response, the initial bandwidth of the system is very high 0.4 (≈70 Hz) which settles down to about 18 Hz in the steady- state situation. The bandwidth of the system changes from 0.2 a large value to a small value. Similarly but contrarily, the 0.5 damping ratio ζ(t) varies from 0.221 (t = 0) to 1 (t = 0.5). 0.5 0.4 0.3 e(t) 0.2 From the simulation studies, it is shown that the proposed 0.1 1 t(s) E-BAC is able to control nonlinear time varying systems. Conventionally, a proper design of the controller guarantees Figure 13: The variation of the damping ratio ζ(t)withrespect to that the changing pole position is always positioned in the system error as a function of time. The value of ζ(t) changes the left-hand side (LHS) on g-plane. In this novel design from a low value, ζ(t) = 0.221, to a high value, ζ(t) = 1. approach, the dynamic poles are always located in LHS on g-plane, thus the stability of the controlled system is assured. Further work is under way to extend this E-BAC design to affect the dominant poles. The results of the simulation philosophy for higher-order partially known and unknown study of this case study are shown in Figure 10. Further, the complex dynamic systems. maps of the dynamic poles motion (DPM) are illustrated in Figure 11. The output response initially follows the trajectory References with a large bandwidth and a small damping ratio, which settles down with a large damping and a smaller bandwidth. [1] K. Ogata, Modern Control Engineering, Prentice Hall, Upper It is clear from the figure that the dynamic motion of Saddle River, NJ, USA, 4th edition, 2002. poles of the system is decided by the value of the system [2] Y. C. Chang, “An adaptive H∞ tracking control for a class of error. The initial positions of the dominant dynamic poles nonlinear multiple-input-multiple-output (MIMO) systems,” IEEE Transactions on Automatic Control,vol. 46, no.9,pp. are placed to generate a low damping ratio ζ(t) and large 1432–1437, 2001. bandwidth of the system. Thus, initially the system is under- [3] D. G. Taylor, P. V. Kokotovic, R. Marino, and I. Kanellakopou- damped. Thereafter, the dynamic poles are optimized and los, “Adaptive regulation of nonlinear systems with unmod- shifted as the system error decreases reducing the bandwidth eled dynamics,” IEEE Transactions on Automatic Control, vol. and increasing ζ(t). The final positions of the dominant 34, no. 4, pp. 405–412, 1989. dynamic poles make the system an overdamped system. [4] Y. Liu and X. Y. Li, “Robust adaptive control of nonlinear sys- Thus, the bandwidth becomes small, but ζ(t) becomes high. tems represented by input-output models,” IEEE Transactions The variation of the bandwidth at each time interval is shown on Automatic Control, vol. 48, no. 6, pp. 1041–1045, 2003. in Figure 12, and the variation of the damping ratio ζ(t)is [5] M. Jankovic, R. Sepulchre, and P. V. Kokotovic, “Global adap- shown with respect to the system error at each time interval tive stabilization of cascade nonlinear systems,” Automatica, vol. 33, no. 2, pp. 263–268, 1997. in Figure 13. [6] R.Sepulchre,M.Jankovic, andP.V.Kokotovic,“Integrator forwarding: a new recursive nonlinear robust design,” Auto- 5. Discussion and Conclusions matica, vol. 33, no. 5, pp. 979–984, 1997. [7] E. D. Sontag, “A ’universal’ construction of Artstein’s theorem In this paper, we have proposed the design of an error-based on nonlinear stabilization,” Systems and Control Letters, vol. adaptive controller (E-BAC) for controlling the dynamic 13, no. 2, pp. 117–123, 1989. response of a nonlinear system. The proposed E-BAC is the [8] N. S. Nise, Control Systems Engineering,JohnWiley&Sons, controller with continuously changing feedback parameters New York, NY, USA, 3rd edition, 2000. [9] W. J. Palm III, System Dynamics, McGraw-Hill, New York, NY, as functions of the system error: initially for large errors an USA, 2005. underdamped system with large bandwidth which is forced [10] K. Y. Song, M. M. Gupta, D. Jena, and B. Subudhi, “Design to approach to become an overdamped system with small of a robust neuro-controller for complex dynamic systems,” in bandwidth for small errors. In the beginning, for large errors Proceedings of the Annual Meeting of the North American Fuzzy the system is underdamped, thus, it makes the system faster Information Processing Society (NAFIPS ’09), Cincinnati, Ohio, with a wider bandwidth. As the error decreases, the value of USA, June 2009. the feedback gains K decreases and K increases. The design p v [11] M. W. Spong, K. Khorasani, and P. V. Kokotovic, “An integral of this adaptive controller is conceptually error-based and manifold approach to the feedback-control of flexible joint can be used to handle the complexity of systems. In order robots,” IEEE Journal of Robotics and Automation, vol. 3, no. to support the novel controller, we introduce the notion of 4, pp. 291–300, 1987. [12] L. Jin, M. M. Gupta, and P. N. Nikiforuk, “Dynamic recurrent dynamic pole motion (DPM). neural networks for modeling flexible robot dynamics,” in As a case study, we present a flexible joint of robot Proceedings of the 10th IEEE International Symposium on arm, which is a nonlinear dynamic system, and this system Intelligent Control, pp. 105–110, August 1995. is controlled by the proposed E-BAC. 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Published: Feb 29, 2012

References

  • An adaptive H∞ tracking control for a class of nonlinear multiple-input-multiple-output (MIMO) systems,
    Y. C. Chang
  • Adaptive regulation of nonlinear systems with unmodeled dynamics,
    D. G. Taylor, P. V. Kokotovic, R. Marino, and I. Kanellakopoulos
  • Robust adaptive control of nonlinear systems represented by input-output models,
    Y. Liu and X. Y. Li
  • Global adaptive stabilization of cascade nonlinear systems,
    M. Jankovic, R. Sepulchre, and P. V. Kokotovic
  • Integrator forwarding: a new recursive nonlinear robust design,
    R. Sepulchre, M. Jankovic, and P. V. Kokotovic
  • A 'universal' construction of Artstein's theorem on nonlinear stabilization,
    E. D. Sontag
  • Design of a robust neuro-controller for complex dynamic systems,
    K. Y. Song, M. M. Gupta, D. Jena, and B. Subudhi
  • An integral manifold approach to the feedback-control of flexible joint robots,
    M. W. Spong, K. Khorasani, and P. V. Kokotovic
  • Dynamic recurrent neural networks for modeling flexible robot dynamics,
    L. Jin, M. M. Gupta, and P. N. Nikiforuk