Robust Tracking Under Nonlinear Friction Using Time-Delay Control With Internal Model

Transcription

1 1406 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 17, NO. 6, NOVEMBER 2009 Robust Tracking Under Nonlinear Friction Using Time-Delay Control With Internal Model Gun Rae Cho, Student Member, IEEE, Pyung Hun Chang, Member, IEEE, Sang Hyun Park, and Maolin Jin, Member, IEEE Abstract In this paper, the robustness problem in time-delay control (TDC) is considered in the presence of the nonlinear friction dynamics of robot manipulators. As a remedy for this problem, the TDC is enhanced with a compensator based on internal model control (IMC). The robustness and stability of the proposed method have been analyzed to be effective against friction while preserving the positive attributes of the TDC: It is simple, efficient, and easily applicable because it does not require a complete plant model. Through experiments, the proposed method achieves accuracy levels that are times better than that of the TDC, thereby confirming its effectiveness under friction. Index Terms Friction, internal model control (IMC), robot manipulator, robust tracking control, time-delay control (TDC). I. INTRODUCTION T IME-DELAY control (TDC) [1], [2] is a control technique that compensates system uncertainties for example, unmodeled dynamics, parameter variations, and disturbances by utilizing a time-delayed signal of system variables, and inserts the desired dynamics into the plant. Owing to the effectiveness and efficiency of time-delay estimation (TDE), the TDC is robust and yet is characterized by a compact structure and relatively simple gain selection procedure. The TDC has been applied to many mechanical systems, with consistently good results [3] [7]. It has been observed, however, that the TDC reveals noticeable performance degradation in the presence of nonlinear friction dynamics [8]. For example, Coulomb friction and static friction often cause large tracking errors as a consequence of their rapid dynamics when a plant crosses zero velocity. This friction is not only common in practice but also has significant effects on robot dynamics. Coulomb friction is everywhere and accounts for as much as 30% of the maximum motor torque in some robot drive trains, such as a PUMA arm [9]. Static friction is dominant in plants that incorporate, for example, pneumatic valves [10] and pneumatic cylinder systems [8]; yet, it can occur in any plant, although its level is insignificant. Manuscript received March 13, 2008; revised August 07, Manuscript received in final form October 07, This work was supported in part by the Ministry of Knowledge Economy under the Center for Intelligent Robot (CIR) Support Program, by the Agency for Defence Development, and by the Unmanned Technology Research Center (UTRC), Korea Advanced Institute of Science and Technology. Recommended by Associate Editor K. Kozlowski. G. R. Cho, P. H. Chang, and S. H. Park, are with the Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daejeon , Korea ( grcho@mecha.kaist.ac.kr; hyunmu@kaist.ac.kr; phchang@mecha.kaist.ac.kr). M. Jin is with the Research Institute of Industrial Science and Technology (RIST), Pohang, Korea ( mulim@rist.re.kr). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TCST Recently, several studies have sought to compensate friction using the LuGre model [11] [13]. Since the friction model, however, requires a complex and time-consuming process of parameter identification, its inclusion to TDC for friction compensation significantly undermines the simplicity of TDC and is thus not recommendable. As a nonmodel-based friction compensation method, the TDC has been modified with the switching action of sliding-mode control [8], [14], [15]. This approach, however, is associated with some drawbacks. First, the use of discontinuous input tends to cause chattering in the tracking response. In addition, extra gain tuning is needed for the switching action, which imposes a further burden on the design of the controller. This paper introduces an enhanced controller: a TDC with an internal model (TDCIM), in which an internal-model-control (IMC)-based compensator is combined with the TDC. The key idea of this combination is twofold: first, to improve the TDC using the so-called perfect control property of the IMC, and second, to benefit the IMC by exempting from the necessity of a rather complete plant model. To elaborate, the IMC part, on the one hand, is expected to endow the further robustness against friction without any extra gain to tune by virtue of its perfect control property, which enables one to eliminate the effect of any type of disturbance perfectly [16], [17], thereby enhancing robustness (interested readers are referred to [18] and [19] for demonstrations of the robustness of the IMC). The TDC part, on the other hand, is meant to provide a simple structure that makes it unnecessary to have a rather complete plant model (including friction). In short, it is expected that the TDCIM gains a synergy effect coming from the complementary use of the TDC and the IMC. This paper is organized as follows. In Section II, the TDC is briefly reviewed, and its problems are analyzed. Section III presents the enhanced controller and the analysis of its properties. The robustness of the TDCIM is verified experimentally in Section IV. Finally, the results are summarized, and conclusions are drawn in Section V. II. PROBLEMS OF TDC DUE TO FRICTION We summarize the TDC law for robot manipulators [2] and analyze its problems concerning the TDE error. A. Review of the TDC The dynamics of -DOF robot manipulators is generally described as follows: where denote the joint angle, joint angular velocity, and joint angular acceleration, respectively; is (1) /$ IEEE

2 CHO et al.: ROBUST TRACKING UNDER NONLINEAR FRICTION USING TDC WITH INTERNAL MODEL 1407 the inertia matrix; represents the terms due to Coriolis and centrifugal forces; denotes the terms due to gravity; is the nonlinear friction; indicates the unmodeled dynamics and disturbances; and is the input torque. By introducing a constant diagonal matrix,, which represents the known part of, one can rewrite (1) as follows: (2) where denotes the total sum of the nonlinear dynamics of robot manipulators, friction, and disturbances and is described as follows: (3) The control objective of the TDC, like the computed torque method, is to achieve the following error dynamics: where, denotes the desired trajectory, and signifies the value of at time. To this end, the control torque is designed based on the computed torque method as follows: where denotes the estimated value of, and and represent the diagonal gain matrices of decoupled PD controllers. In essence, therefore, the controller in (5) attempts to cancel in (2) by and inject the desired dynamics in (4). Whereas the computed torque method incorporates real-time computation of based on a robot dynamic model, the TDC uses the TDE described as follows. Under the assumption that is continuous or piecewise continuous and that the time delay is sufficiently small, the following approximation holds: providing an effective estimation of, i.e., which is the essential idea of the TDC. The TDE can be obtained by using (2) as follows: Note that (9) is a causal relationship. Using the TDE for (9) leads (5) to (4) (5) (6) (7) (8) (9) in (10) Combining (10) with (6), we obtain the final form of the TDC (11) Fig. 1. Block diagram of the TDC. (A) Dashed box denotes feedforward function processing. (B) Feedback linearization using the TDE. (C) PD-type feedback of. Fig. 1 shows the block diagram of the closed-loop system with the TDC. Notice that the TDC may be viewed as consisting of three functions: 1) feedforward function processing ; 2) feedback linearization using the TDE; and 3) the PD-type feedback of. This viewpoint, which will be used in Section III-B, is particularly useful to combining the TDC and the IMC. Owing to the TDE, the TDC has a simple structure and is efficient approximately as efficient as a typical PID control. Furthermore, since is selected as a constant diagonal matrix, the TDC can be designed as if it consisted of individual joint controllers by using each diagonal element of,, and, respectively. B. Problems of the TDC Due to the TDE Error If the time delay is set infinitesimally small, a perfect estimation of would be possible by using the TDE. Because of digital implementation, however, the smallest value for is the sampling time, which is finite. Therefore, the estimation error exists owing to a finite. Substituting (5) into (2) and considering (8), one can derive the following relationship: (12) The LHS of the aforementioned equation denotes the estimation error. Now, define the TDE error as follows: (13) Substituting (6) into (13), we obtain the error dynamics of the TDC (14) which clearly shows the influence of the TDE error on the tracking error. The of (3) has nonlinear dynamics, as surveyed in [21], and includes Coulomb friction and static friction, which change very rapidly around. This rapid dynamics could change significantly even within one sampling time; in this case, the approximation in (7) does not hold anymore. Accordingly, the TDE in (9) becomes inaccurate, causing a large by virtue of (13), and thus causing a large tracking error according to (14). III. TDCIM In this section, we propose an enhanced controller, the TDCIM, which improves the robustness of the TDC against the

3 1408 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 17, NO. 6, NOVEMBER 2009 Fig. 2. Block diagram of the IMC, where denotes the IMC controller, signifies the plant, and implies the internal model. Fig. 3. Simplified block diagram of the TDC with the plant linearized by the TDE. rapidly changing nonlinear friction using the IMC. We shall keep this paper self-contained by introducing the IMC briefly in the early stage of this section. A. IMC: A Perfect Controller The IMC, proposed by Garcia and Morari [16], is shown in Fig. 2. It consists of the controller and the internal model. The overall transfer function of the IMC is given as (15) The IMC controller is designed based on the inverse of the model,, including a low-pass filter for robust stability and performance against modeling errors [16], [17], [20]. From (15), one can easily derive the property of the perfect control. Assuming that is realizable and that the IMC system is closed-loop stable, then the perfect reference tracking control can be achieved from (15) for all despite any disturbance [16], [17]. Suppose that we can apply the ideal IMC to the system controlled by the TDC. Then, the perfect control property of the IMC should be able to resolve the problem due to nonlinear friction this is the first key contribution of this paper. The essential idea behind the TDCIM came from [20], which suggested the IMC structure for robot manipulators. It consists of the computed torque method to linearize the robot dynamics and the linear IMC to compensate the modeling error. However, the method in [20] still has a drawback: the need for whole plant models, a substantial barrier against its application to real robots. Therefore, we propose the TDCIM to overcome the drawback of the conventional IMC by using the TDE this is the second key contribution of this paper. B. Derivation of the TDCIM 1) Derivation of the Linear Dynamics From the TDC System: As was pointed out with Fig. 1, the TDC provides three essential functions: the feedback linearization part, the PD-type feedback, and the feedforward part. These functions, when combined together, lead to the linear form with the perturbation error as follows. Feedback linearization is achieved by using the TDE. As was mentioned, the TDC in (5) attempts to cancel in (2) by. Substituting (13) into (12), one can express the closed-loop dynamics as (16) Fig. 4. Simplified block diagram of the TDC with the plant linearized by the TDE and PD feedback. Clearly, the closed-loop dynamics due to the TDC may be regarded as a linear equation,, subject to disturbance coming from the TDE error. Accordingly, the TDC block diagram in Fig. 1 can be simplified with (16), as shown in Fig. 3. Referring to the PD-type feedback and the feedforward function in Fig. 1, one can divide the control input in (6) into a feedforward input and a feedback input (17) where represents the feedforward function and denotes the PD-type feedback action. Substituting (17) into (16) leads to (18) which is shown in Fig. 4. As Fig. 4 plainly shows, the closed-loop dynamics due to the TDC may be regarded as a cascade combination of a feedforward controller and the plant ; as a result, the closed-loop dynamics of the TDC can be converted into a form to which the IMC is applicable. However, the TDE error affects as the input disturbance to without any compensation. Therefore, we have tried to compensate the TDE error using the IMC, as will be explained in the following section. Note, in addition, that (18) consists of decoupled second-order linear dynamics. 2) Compensator Design Based on the IMC: Now, according to the design procedure of the IMC [16], [17], [20], the compensator based on the IMC is selected and combined with the system in Fig. 4; accordingly, the IMC feedback loop is added for the completion of the TDCIM. From Fig. 4, the internal model can be apparently selected as follows: (19) Equation (19) shows obviously that is invertible; therefore, the IMC controller can be determined as (20) which the TDC already possesses, as shown in Fig. 4. Adding the IMC feedback, we can achieve the TDCIM, as shown in Fig. 5. Matching Fig. 5 with Fig. 2, one can easily predict that,

4 CHO et al.: ROBUST TRACKING UNDER NONLINEAR FRICTION USING TDC WITH INTERNAL MODEL 1409 all the terms within the bracket on the RHS represent the additional terms due to the IMC feedback. The effect and contribution of these terms due to the IMC feedback are made clear from Fig. 5 that shows that the internal model dynamics becomes (25) Fig. 5. Simplified block diagram of the TDCIM. whereas the plant dynamics (26) in the TDCIM, the TDE error would be compensated by the following IMC property: the robustness against in Fig. 2. Recovering the original block diagram of Fig. 1 in Fig. 5 leads to Fig. 6, an expanded block diagram of the closed-loop system due to the TDCIM. Note that a time-delay term is included in the IMC feedback denoted by a dashed box in Fig. 6 to represent the causal relationship for digital implementation. A comparison of Fig. 6 with Fig. 1 reveals that the structure of the TDCIM is slightly more complex than that of the TDC. It also shows that, in the TDCIM,, the combined value of the reference and IMC feedback, is used instead of, the reference of the TDC shown in Fig. 1. From Fig. 6, the control law of the TDCIM is obtained as (21) (22) (23) and denotes the output of the internal model and can be obtained from because in Fig. 6. The control law described earlier shows that the TDCIM can be designed by choosing only,, and, not requiring extra gains to tune. Furthermore, because of the use of the TDE, it does not need a complete computation of robot dynamics. Therefore, it is easy to design, fully sharing the positive attributes of the TDC. The stability of the TDCIM is analyzed in the Appendix. C. Function of the IMC Feedback To determine the properties of the TDCIM, it is essential to examine the net contribution of the IMC feedback. Substituting (23) into (22) and subtracting (6) lead to the control input of the TDCIM (see (24), shown at the bottom of the page). Therefore, Subtracting (26) from (25) at time leads to (27) Clearly, the terms due to the IMC feedback amount to the TDE error at one sampling time before. Therefore, substituting (27) into (24) and the result into (21), one can describe the control input and input torque of the TDCIM, respectively, as (28) (29) The TDCIM can thus be thought of as a TDC with an additional compensator using the TDE error at,. Comparing the RHS of (29) with (5) and (8), one may view the TDCIM as the combination of with a new estimator consisting of and. By using (13), the new estimator can be expressed as (30) from which the estimation error of due to the TDCIM can be obtained as. This indicates that the new estimator reduces the influence of the TDE error and improves the estimation performance by using, the TDE error at. By substituting (29) into (2), one can derive the following relationship: (31) The error dynamics due to the TDCIM can be easily obtained by substituting (6) into (31) as follows: (32) (24)

5 1410 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 17, NO. 6, NOVEMBER 2009 Fig. 6. Overall block diagram of the TDCIM. Fig. 7. Performance of the TDC and TDCIM against a discontinuous. (b) and (c) Estimation error of each controller. (d) and (e) Simulation results of the error dynamics in (14) and (32), respectively, where N m,,,, and. (a) H and the TDE. (b) Estimation error in TDC:. (c) Estimation error in TDCIM:. (d) TDC: Simulation of (14). (e) TDCIM: Simulation of (30). Comparison of (32) with (14) shows the impact of the IMC feedback on error dynamics, displaying how the TDCIM diminishes the tracking error by reducing the TDE error. The compensator using the IMC feedback can work effectively even when changes discontinuously. Fig. 7(d) and (e) shows the simulations of the respective error dynamics due to each of (14) and (32). When changes discontinuously with N mat, the TDCIM yields a substantially small tracking error as compared with the TDC (both controllers are designed as,,, and ). The improvement of the TDCIM can be accounted for by the estimation errors shown (conceptually) in Fig. 7(b) and (c). Compared with the estimation error of the TDC in Fig. 7(b), the TDCIM exerts a counteraction with, as seen in Fig. 7(c) (negative direction), eliminating the effect of the TDE error. This explains the better robustness, as shown in Fig. 7(e). Note that rapidly changing uncertainty like includes nonlinear friction such as Coulomb friction and static friction. D. Some Practical Issues 1) Design Procedure of the TDCIM: The TDCIM has quite a few design variables; one can design the controller only by choosing the gains (,, and ) and the sampling time. Thus, the design procedure of the TDCIM is simple and is introduced briefly in the following. Step 1) Determine the desired error dynamics in (4) for each joint. After choosing the desired natural frequency and the desired damping ratio, one can systematically design and whose diagonal elements are and, respectively. Step 2) Select a sampling time interval of the closed-loop system with consideration of the speed of the controller hardware [22]. Commonly, faster sampling provides better performance because a smaller makes the TDE in (9) and the additional compensator using the IMC in (27) more exact.

6 CHO et al.: ROBUST TRACKING UNDER NONLINEAR FRICTION USING TDC WITH INTERNAL MODEL 1411 Step 3) Tune the diagonal matrix. After letting, one can tune each. Set as a small positive value at first, and then increase just before the closed-loop system has noisy response. Certainly, the tuned has to satisfy the stability condition explained in the Appendix. Following the aforementioned design procedure, one can design the TDCIM by tuning only one gain matrix without any information about the plant dynamics. The procedure is relatively easy compared with that of conventional control schemes, e.g., PID control needs to tune three gains for each joint, and the control scheme based on the computed torque control (e.g., [20]) needs a whole model of the plant dynamics. 2) Use of Low-Pass Filter: It is necessary to use a low-pass filter for the TDCIM in Fig. 6. Without a low-pass filter, the IMC used as a compensator for the TDCIM may become unstable when there are modeling errors [16], [20]. In addition, the amplification of noise due to numerical differentiation [23] in the TDCIM also necessiates a low-pass filter. Specifically, the TDCIM in Fig. 6 uses the second-derivative signals of the plant outputs, commonly obtained by numerical differentiation of the position information of the plant, e.g., the Euler numerical differentiator [3]. In this case, one needs to use low-pass filters to attenuate the influence of noise. Fortunately, the of the TDC has an implicit function of the low-pass filter (the same goes for the TDCIM). As described in [23] and [24], one can achieve a first-order low-pass filter effect by simply reducing from to, with, and is given as follows: (33) where denotes the cutoff frequency of the low-pass filter for the th joint. If the aforementioned method is not enough, one can add a low-pass filter explicitly to the IMC controller in Fig. 6 by replacing with, as introduced in [17] and [20]. denotes the low-pass filter matrix whose diagonal elements are given as (34) where denotes the cutoff frequency of the low-pass filter for the th joint and signifies the order of the low-pass filter (commonly, for a second-order system). In this case, however, the filter dynamics remains in the overall dynamics of the closed-loop system see [17] and [20] and may influence the control performance. Therefore, one should design the low-pass filter carefully with consideration of the tradeoff between attenuability of noise effect and performance degradation due to the filter dynamics. IV. EXPERIMENT A. Experimental Setup The improved tracking performance of the TDCIM over the TDC is verified with the experiments of a 2-DOF planar industrial robot shown in Fig. 8. The parameters of the robot are provided in Table I; it shows that each joint of the robot has rela- Fig. 8. Two-DOF planar robot system. TABLE I SPECIFICATIONS OF THE 2-DOF PLANAR ROBOT tively high gear ratio, which causes noticeable friction. The controllers are implemented with a desktop computer equipped with a real-time OS, i.e., QNX. The desired trajectories for both joints are set as follows: (35) where and. The TDCIM is implemented with the implicit digital low-pass filter (cutoff frequency at 20 Hz). The TDC is also implemented in the same way for comparison purposes. For both controllers, the error dynamics in (14) and (32) is designed to have the same natural frequencies and damping ratios as and, respectively, resulting in PD gains and, where denotes the th joint. The time delay is set with the sampling time as. for each controller is tuned, as shown in Fig. 9, which also describes the stable regions of according to the stability analysis described in the Appendix (see [4] for the stability analysis of TDC). As pointed with in Fig. 9(a), the of the TDCIM is tuned to minimize the tracking error. For comparison with the TDC, the of the TDC is selected in two ways: first, the same as that of the TDCIM (point in Fig. 9(b)), and second, the best tuned that minimizes the tracking error (point in Fig. 9(b)). The of the TDCIM would be too conservative for the TDC since the TDCIM has a smaller stable range than the TDC, as shown in Fig. 9(a) and (b). Incidentally, the derivative terms and are calculated by using Euler numerical differentiation, as introduced in [3]. B. Experimental Results The experimental comparison is carried out, and the results are shown in Fig. 10. The maximum tracking errors and

7 1412 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 17, NO. 6, NOVEMBER 2009 Fig. 9. Stable gain regions according to the stability analysis in the Appendix, and the gain values used in the experiment whose results are shown in Fig. 10. (a) TDCIM: Stable region of. (b) TDC: Stable region of. Fig. 10. Experimental comparison of the TDCIM and the TDC. In each figure, TDCIM and TDC denote the results with gain set in Fig. 9, whereas TDC: best tuned is with gain set in Fig. 9(b). (a) Joint 1: Position response. (b) Joint 2: Position response. (c) Joint 1: Position error. (d) Joint 2: Position error. (e) Joint 1: Input torque. (f) Joint 2: Input torque. the mean deviations of tracking errors are listed in Tables II and III, respectively. Fig. 10(c) and (d) clearly shows that the TDCs both the case using the same gain as that of the TDCIM and the best tuned case yield large tracking errors due to Coulomb friction at every instant slightly after each joint crosses zero velocity. Of the two cases, the TDC with the best tuned has the smaller tracking error but still shows noticeable performance degradation due to Coulomb friction. Coulomb friction belongs to a class of nonlinear friction having rapidly changing characteristics. In comparison, however, the TDCIM reduces the tracking error vastly. The maximum error of joint 1 due to the TDCIM is less than 1/16 of that of the TDC, and the maximum error of joint 2 due to the TDCIM is less than 1/21. Fig. 10(e) and (f) shows that the TDCIM has a torque profile that is slightly larger than that of the TDC, without any unwanted effects such as

8 CHO et al.: ROBUST TRACKING UNDER NONLINEAR FRICTION USING TDC WITH INTERNAL MODEL 1413 TABLE II MAXIMUM TRACKING ERRORS: Replacing in (36) with (1) at yields the following: (37) TABLE III MEAN DEVIATIONS OF TRACKING ERRORS: where and. From (31), holds, and substituting these equations into (37) yields the following: (38) (39) (40) chattering. The significant performance improvement has been achieved without the cost of excessively high gains or any other side effect. V. CONCLUSION Although having excellent robustness against various uncertainties, the TDC also revealed the robustness problem under very rapid dynamics of friction. We have shown that the robustness problem comes from the TDE error, and it can be effectively resolved by enhancing the TDC with the IMC concept, i.e., by the TDCIM. We have analyzed its properties and stability, confirming enhanced robustness due to the IMC concept. Through the experiments with a 2-DOF planar robot, the robustness of the TDCIM is verified when the TDE error becomes large due to the rapid dynamics of friction. The TDCIM preserves the positive attributes of the TDC with little additional cost: It has a simple structure similar to the TDC, requiring neither additional hardware nor extra gains to tune; the inclusion of several lines of program code is all that costs. Therefore, it can be applied to real systems as easily as the TDC has been. APPENDIX STABILITY OF THE TDCIM We have analyzed the stability of the TDCIM by inspecting the boundedness of, as used in [4]. The error dynamics of the TDCIM in (32) indicates that has an exponentially stable dynamics subject to with appropriate and. Thus, the boundedness of guarantees the BIBO stability of the TDCIM and can be shown in similar manner with the stability analysis of the TDC in [4] and [25]. At first, we derived the dynamics of. Substituting (30) into (29) and recalling from (2), one can rewrite the input torque as (36) where. Equation (38) can be represented in discrete-time domain as the following state-space form: (41) where is considered as a forcing function. Equation (41) is the second-order discrete equation and asymptotically stable if the eigenvalues of in (41) remain in the unit circle [4], [25]. Hence, a stable can be determined by evaluating according to the unit-circle criterion. In doing so, is not evaluated in time domain,but is evaluated in the whole joint space of a given robot. For is a function of, only and the robot posture at any time is determined within the joint space, i.e.,, where denotes the working range of the robot in any set of time and represents the whole joint space. Therefore, stable gain in is always stable in. The stable region of in can be easily obtained with computer simulations by using a simple programming code. The procedure consists of the following steps. 1) Given an, determine if it is a stable by examining while changing within the joint space with adequately small step size. 2) Increase from while, for each, repeating Step 1) until begins to violate the unit-circle criterion. The stable region of for the robot in Fig. 8 has been obtained by following the aforementioned procedure. In this case, is a function of and is given as follows: kg m (42) where. We examined for each with jointspace range of and step size, and we obtained a stable gain region, as described in Fig. 9(a).

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