Decentralized Adaptive Robust Control of Robot ... - IEEE Xplore
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 20, NO. 5, SEPTEMBER 2012
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Decentralized Adaptive Robust Control of Robot Manipulators Using Disturbance Observers Zi-Jiang Yang, Member, IEEE, Youichirou Fukushima, and Pan Qin
Abstract—In this paper, we propose a decentralized adaptive robust controller for trajectory tracking of robot manipulators. In each local controller, a disturbance observer (DOB) is introduced to compensate for the low-passed coupled uncertainties, and an adaptive sliding mode control term is employed to handle the fastchanging components of the uncertainties beyond the pass-band of the DOB. In contrast to most of the local controllers using DOB for robot manipulators that are based on linear control theory, in this study, by some special nonlinear damping terms, the boundedness of the signals of the overall nonlinear system is first ensured. This paves the way to analyze how the DOB and adaptive sliding mode control play in a cooperative way in each local subsystem to achieve an excellent control performance. Simulation results are provided to support the theoretical results. Index Terms—Adaptive control, decentralized control, disturbance observer, nonlinear control, robot manipulators.
I. INTRODUCTION
R
OBOT manipulators possess many uncertainties with inherent strong nonlinearities. To improve the tracking performance of robot manipulators, significant effort has been made to seek advanced control strategies, such as adaptive control and robust control approaches [1], [2]. However, since most approaches are based on a centralized control structure which requires rather complicated hardware configuration and tedious computation, their practical application may be computationally demanding. A decentralized control system based only on the local information is highly desirable. In fact, the majority of contemporary robots are still controlled by the decentralized (independent joint) proportional-integral-derivative (PID) law in favor of its simple computation and low-cost setup [3], [4]. Due to strong nonlinearities in dynamics, advanced decentralized control techniques are still attracting much attention to achieve a satisfactory trajectory performance of robot manipulators [5]–[11]. The fundamental uncertainties encountered in the decentralized controller design are the strength of the interconnections Manuscript received March 11, 2011; revised July 20, 2011; accepted August 01, 2011. Manuscript received in final form August 01, 2011. Date of publication August 30, 2011; date of current version June 28, 2012. Recommended by Associate Editor F. Caccavale. Z.-J. Yang is with the Department of Intelligent Systems Engineering, College of Engineering, Ibaraki University, Ibaraki 316-8511, Japan (e-mail: yoh@mx. ibaraki.ac.jp). Y. Fukushima is with the Department of Electrical and Electronic Systems Engineering, Graduate School of Information Science and Electrical Engineering, Kyushu University, Nishi-ku 812-8581, Japan (e-mail: [email protected]). P. Qin is with the Faculty of Mathematics, Kyushu University, Nishi-ku 8128581, Japan. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2011.2164076
among the subsystems. Many papers have been devoted to accounting for uncertain interconnections among the subsystems in the last decades. In [12] and [13] the adaptive decentralized control methods based on the high gain stabilization technique were proposed, and in [14], a decentralized adaptive variable structure-based control scheme was proposed, to stabilize interconnected systems where the strength of the interconnections are bounded by polynomials of the state variables. Recently, a decentralized controller has been proposed [15] and applied to multi-machine power systems [16], in which the lumped modeling error, disturbance and interconnections are estimated by the high-gain observer. Additionally, backstepping designbased decentralized adaptive control methods were developed in [17], [18] for systems which do not satisfy the matching condition. For manipulator tracking tasks, decentralized approaches are not straightforward since the overall system cannot be decomposed into subsystems such that the states and control inputs are fully decoupled from one another, because of the inherent coupling such as moment of inertia and Coriolis force. It has been however, found that in the manipulators the strength of the interconnections is bounded by second-order polynomials in states [5]–[10]. Inspired by this, with a third-order feedback control term, adaptive sliding mode controller [6], and adaptive controller [10] with local model estimation have been proposed. The former may require a strong sliding mode control gain to counteract the interconnections. The latter while being computationally demanding, cannot estimate the interconnections from the other subsystems. As an alternative popular approach for compensating for external disturbances and model mismatch, a disturbance observer (DOB) is often included into a motion controller. Early works of decentralized control of robot manipulators using the DOB can be found in [19]–[22]. At each joint of the robot manipulator, the nonlinearities are treated as a lumped disturbance to a nominal local linear plant. A DOB is closed around each joint to estimate and cancel the disturbance term. Each joint is then treated as a linear plant for which a linear controller can be designed. The attractiveness of the DOB based algorithms lies in their simplicity and transparency of design. Unfortunately the stability properties of these algorithms for nonlinear robot manipulators were not extensively studied [23], [24]. Most recently, there have been some works where rigorous stability analysis of nonlinear control systems using the DOB is performed [25]–[32]. In this paper, we propose a decentralized adaptive robust controller for trajectory tracking of robot manipulators. In each local controller, a DOB is introduced to compensate for the lowpassed coupled uncertainties, and an adaptive sliding mode control term is employed to handle the fast-changing components of the uncertainties beyond the pass-band of the DOB. In con-
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trast to the DOB-based linear local controllers for robot manipulators, in this study, by some special nonlinear damping terms, the boundedness of the signals of the overall nonlinear system is first ensured. Differing from some existing works [5]–[11] where only the control performance of the overall system is analyzed, we also analyze how the DOB and adaptive sliding mode control play in a cooperative way to achieve an excellent control performance in each local subsystem, provided the boundedness of the overall system signals. Simulation results are provided to support the theoretical results. II. PROBLEM STATEMENT Consider an -link rigid manipulator governed by the following dynamic model described by the Lagrange-Euler vector equation:
where Substituting
is the position-tracking error vector and is a constant matrix. and into (1), we have (7) (8)
According to (7), (4) and (8), we have [5]–[7], [9], [10] (9) where are some positive constants. For a vector-valued signal , we define a truncated norm as . Then we have [6]. for Lemma 1: Let Assumptions 1 and 2 hold. If there is a constant such that exists, then for all we have
(1) where is the joint angle vector; is the is the input torque vector; inertia matrix; is the centrifugal and Coriolis is the grativity torque; is the torque; friction force torque. The following assumptions are imposed for the system. Assumption 1: All joints of the robot manipulators under consideration are revolute such that Property 1 given later holds. and the time Assumption 2: The reference trajectory and are bounded signals. derivatives The system model (1) has the following properties that will be used in control system analysis and design [2]. is symmetric and posiProperty 1: The inertia matrix tive-definite such that
(10) for some . In some existing works of decentralized control for robot is demanipulators, each local controller signed based on its own local information [5]–[10]. The stability analysis is based on the overall system model (7) by virtue of Property 4 (passive property). This is in contrast to the conventional decentralized linear control techniques using the DOB [19]–[22] which do not discuss the stability of the overall nonlinear system. In this paper, we propose a class of nonlinear local controllers with the DOB. Differing from the existing works, we first ensure the boundedness of the signals of the overall nonlinear system model. And then provided the boundedness of the overall system signals, we analyze the control performance achieved in each local subsystem.
(2) for some constants Property 2: The matrix
III. CONTROLLER DESIGN
. satisfies
A. Introduction of DOB (3)
. for some constant Property 3: The gravity vector satisfy vector
and the friction force
Recall the overall system model (7). We introduce here a constant nominal inertia for each joint and lump all the other components of the dynamics as a disturbance term . Then we have the following vector-valued nominal linear system model with a disturbance term
(4) for some constants Property 4: The matrix metric, i.e.,
. is skew sym-
(11) where and
is the nominal inertia matrix, is the disturbance vector expressed as (12)
(5) Rewriting each subsystem of (11) as is Notice that this property implies that the map passive [1]. Our task is to design a decentralized controller where at each joint a local controller using only the local information is constructed, so that tracks its reference accurately. To this end, we first define an auxiliary error as
Since calculation of by direct differentiation is usually conthrough taminated with high frequency noise, we may pass a low-pass filter to obtain its estimate as
(6)
(14)
(13)
YANG et al.: DECENTRALIZED ADAPTIVE ROBUST CONTROL OF ROBOT MANIPULATORS USING DISTURBANCE OBSERVERS
In this paper, for convenience of expression, denotes not only the Laplace operator, but also a differential operator if the initial condition effects of the DOB are not concerned. This is the so-called DOB studied extensively in the literature [19]–[22]. Although a high-order filter may improve the estimation performance, in this a study, we adopt a simple second-order filter (15) . where at low-frequenHowever, we can only expect cies due to limited pass-band of the DOB. If is fast changing, the estimation error cannot be neglected and hence can even degrade the control performance significantly. Moreover, the may disturb the other joints DOBs’ outputs mutually. To ease the analysis, a straightforward and simple idea is to saturate the output of the DOB as for for for (16) is a selected upper bound of . Usually, it is where commended to choose a sufficiently large so that the saturation action is not active after the transient performance. However, as will be shown by the numerical example, even when is not so large such that is really saturated and hence the estimation error is not sufficiently small, the control performance is still satisfactory, owing to the adaptive sliding mode control term included in the local controller (17) given below. The key point is that the DOB and the adaptive sliding mode control term work in a cooperative manner. That is, if one works more, the other one works less, and vice versa. The advantage of the idea is that we need not a perfect DOB or a perfect sliding mode control. However, owing to their cooperative effects, the problems of high-gain or chattering can be avoided. This will be confirmed later by the numerical examples.
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may local system. Since the DOBs’ outputs disturb mutually through the interconnections of the joints, , we to suppress the interconnections of employ the damping term . Finally, the last term of is a smoothed version of sliding mode control term with an , this term adaptively updated gain . Notice that when approaches a hard switching control action. and play the The nonlinear damping terms key roles to ensure the boundedness of the overall system sigor a larger nals, as will be shown later. Intuitively, a larger causes a stronger control action. Additionally, the sliding mode control term is adopted to further suppress the estimation beyond the pass-band of DOB. error IV. PERFORMANCE ANALYSIS Each local controller (17) uses only the local information of its corresponding local subsystem. However, since all of the local subsystems are strongly interconnected, we cannot easily see if the local subsystems are all bounded. We should first ensure the boundedness of the overall system signals, based on the properties and assumptions given in the previous section. Then provided the boundedness of the overall system signals, we can analyze the control performance of each local subsystem. Therefore, the performance analysis includes two phases, i.e., analysis of the overall system, and then that of each local subsystem. A. Analysis of the Overall System The results of analysis are given in Theorem 1. The proof is along the spirit of [6], but with modifications specified by the newly designed controller in this study. Theorem 1: Let Assumptions 1 and 2 hold. For the robot manipulator (1) controlled by the proposed adaptive decentralized robust controller (17), there exists a constant , such that is bounded as and hence all the internal signals are bounded, provided the following condition:
B. Description of the Controller Motivated by the aforementioned discussions, we design the following local controller using only the local information for each subsystem to ensure the boundedness of the overall system signals and to achieve a satisfactory control performance for each subsystem:
(19) where
, and
. Remark 1: The first inequality of (19) is easily satisfied for a suffiently large . The second inequality of (19) can be rewritten as
(17) are the control gains, and is an where adaptive parameter (adaptive sliding mode control gain) updated as (18) where is the adaptive gain and is the leakage parameter that prevents from growing to be unbounded. is a simple PD control term. is a The term damping term to suppress the effects of neglected uncertainties of the global system model summarized as in (7) and (8) [6], [9], [10]. is a compensation term by DOB for each
(20) is not far away from Typically, the value of need not be very large unity. And for a sufficiently large to satisfy the condition if is not extremely large. Proof: The proof process includes two steps. , The first step is to show that there exists a constant . The conclusion is proved such that is bounded as by contradiction. To this end, according to (19) we first let a positive constant satisfy
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(21) Notice that the above inequality can be satisfied if is sufficiently large. is not bounded. Thus there always Now assume the signal such that . exists a smallest time Consider a Lyapunov function candidate according to Property 2
Clearly, the last two inequalities are in contradiction, according to (21). This implies that the assumption of is false. Thus the error signal vector is bounded and satisfies for all . The next step is to show that all the internal signals are bounded. Rewrite the adaptive law (18) as the output of a stable lowpass filter driven by : (27)
(22) Taking the derivative along the trajectory of the closed-loop system, and using Property 4, the Schwartz inequality and Lemma 1, we have
is bounded since is bounded. FurWe can conclude that thermore, according to Assumption 2, and (6) and (9), we con, and are bounded. Each local controller clude that is bounded. Therefore, all the internal signals are bounded. Remark 2: The condition (19) is always satisfied for a sufficiently large bound . The results of Theorem 1 only tell us that the error signal vector , the adaptive gain vector and hence all the internal signals can be made to be bounded. It should be emphasized here that at the present stage our purpose is only to ensure the boundedness of the overall system signals. And hence a conservative bound of the overall error system is acceptable. Later, we will show that each individual error signal can be made sufficiently small by virtue of the corresponding local controller. Remark 3: Considering the derivation process of the bound, we can see that the damping terms and edness of play the key roles to ensure the boundedness of the overall system signals. Even when the smoothed sliding mode control term is removed, the boundedness of the overall system signals is still ensured. The sliding mode control term is introduced to further reduce the control error of each subsystem in the cases where the signals are fast so that the DOB cannot cope with them sufficiently.
(23) By (19), (21) and the assumption that there exists a smallest time such that , we can say that there exists a time instant such that
B. Analysis of Each Subsystem Provided the boundedness of the overall system signals, we are ready to analyze how the DOBs and adaptive sliding mode control techniques bring improvement in each subsystem. Substituting the local controller (17) into the subsystem (13), we have
(24) However, according to (23), we have . Therefore, for all
(28)
, for all , we have Owing to (16) and Theorem 1, Define
and
are bounded.
(29)
(25) But the definition of
Then we consider the following Lyapunov function:
leads to (26)
(30)
YANG et al.: DECENTRALIZED ADAPTIVE ROBUST CONTROL OF ROBOT MANIPULATORS USING DISTURBANCE OBSERVERS
where of (28), we have
. Taking the derivative along the trajectory
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has decayed out sufficiently such that for a relatively small conwe have stant
(37) and (38) (31) Using the relation
Comparing (29) and (37), it is expected that can be much . Then by rewriting (36), we have, smaller than Corollary 1: Let the assumptions and results of Theorem 1 hold. The auxiliary error of each subsystem satisfies
(32) (39) we have
(33) where
The results of Corollary 1 imply that a sufficiently small ultican be achieved by the cooperamate error tive effects of the DOB and the adaptive sliding mode control. includes also , However, since the Lyapunov function while being i.e., the error term of the adaptive parameter, ensured by (35) may not be very small, and hence the transient performance may be conservative. Moreover, the adaptive parameter usually exhibits a long transient phase. This motivated us to seek a less conservative evaluation of the transient performance, despite the effects of the adaptive laws. We will show here that the nonlinear damping terms play important roles to achieve a reliable transient performance. From (28), we have
(34) and hence we obtain the following transient performance:
(40) and hence (41)
(35) is bounded by Equation (35) implies that as . In addition, since , we have the following. Theorem 2: Let the assumptions and results of Theorem 1 hold. For each subsystem of the decentralized robot manipulator model (13) controlled by the proposed decentralized adaptive robust controller (17), all the internal signals are bounded, and the auxiliary error of each subsystem satisfies
where (42) This leads to Theorem 3: Let the assumptions and results of Theorem 2 hold. The auxiliary error of each subsystem satisfies (43)
(36) Inspection of (34) indicates that can be reduced by choosing sufficiently small values of and , and a large value included in may not of . However, according to (29), be small since the initial value is usually set to be zero. We then discuss the control performance after a short transient be an effective time-constant of the phase of DOB. Let DOB depending on , until which the initial value of
The dominator of (42) clearly illustrates the effects of nonlinear damping terms. The term suppresses the peaking (if it occurs) and the uncertainty . On the other of suppresses the growing of due to the unhand, the term . certainty To summarize, Corollary 1 characterizes a better ultimate achieved mainly by the DOB bound of the auxiliary error and the adaptive sliding mode control term. See the second term on the right-hand side of (39). On the other hand, Theorem 3
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characterizes a reliable transient performance ensured mainly by the nonlinear damping terms, despite the effects of the adaptive laws.
TABLE I PHYSICAL PARAMETERS OF THE ROBOT MANIPULATOR
C. Comments and Guidelines of Parameter Design The proposed controller (17) includes the following design parameters: linear control gains [defined in (6)] and , nonlinear control gains and , smoothing factor of sliding mode control , adaptive gain and leakage parameter , and timeconstant of DOB. We will draw some guidelines of parameter design by commenting on the aforementioned analysis results. , In the controller (17), the amplitude of the linear term is defined in (6), does not tend to be very large in where generic cases compared to the other terms, and hence the gains and can be chosen to be relatively large. This term also contributes to achieve a better transient performance and a smaller ultimate tracking error. can be relatively small if the feedThe nonlinear term back error is small. However, if is large at some time points may bedue to some reasons, its nonlinear feedback gain come very large. Therefore, a modest should be chosen. may be large, Since the amplitude of the DOB’s output it is recommendable to choose a modest nonlinear control gain to avoid a noisy or large control signal. In Remark 1 we have mentioned that the condition of Theorem 1 may still be maintained for a modest . The results of Corollary 1 imply that a small smoothing factor leads to a small ultimate tracking error. However, as well known in the literature, a less smooth switching function may cause the chattering problem. Therefore, should not be too small. It should be commented here that even if is not so small, at least the low-passed components of the uncertainties can be compensated for by the DOB. The results of (34), (38), and Corollary 1 imply that a small leakage parameter and a large adaptive gain may lead to a small ultimate tracking error. However, according to (27), too small an may lead to a large , which may cause a large control effort. Moreover, too large a may make change sensitively and hence make the control signal oscillatory. defined in Usually, a small of DOB leads to a small sensitive to noise. Owing to (37). A very small may make enhancement by the adaptive sliding mode control term, need not be very small. The DOB and adaptive sliding mode control should play in a cooperative way.
A. Description of the Manipulator Model Consider a two-DOF planar robot arm
(44) whose entries are given as follows:
(45)
(46) where the physical parameters are given in Table I, and the 9.807 m/s . Notice that the gravity acceleration is given as frictions and are not neglible, compared to the gravity effects. To carry out more realistic simulations, we consider the following simulation conditions. The controller is implemented at 0.5 ms. To verify the noise effects, a sampling period of a uniformly distributed noise between 0.0001 and 0.0001 rad is added to each position measurement. The measurements of and are obtained by pseudodifferentiations of the position measurements due to . The amplitudes of and are limited within the interval of Nm. And each control signal is quantized by 12 bits, such that the quantization Nm. step is B. Tracking Performance for Slow Reference Trajectories
V. SIMULATION EXAMPLE Extensive numerical studies have been performed to support the theoretical results obtained in this paper. For the sake of comparison, we use the example in [10]. Considering some more realistic situations, the simulation conditions are modified by adding the position measurement noises, pseudodifferentiations of position measurements, control signal quantizations with a finite number of bits, control signal saturations and unmodeled frictions. The numerical results of [10] are not reproduced here due to the limitation of space.
We first investigate the tracking performance for the following slow reference trajectories studied in [10], [6]:
rad
(47)
To show that the controller is robust against nonzero initial tracking errors, the initial conditions are given as rad rad/s
(48)
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Fig. 2. Tracking performance for fast reference trajectories. Fig. 1. Tracking performance for slow reference trajectories.
According to the guidelines of controller parameter design, the design parameters are given as follows:
(49) . The initial adaptive parameters are set as Additionally, the saturation levels of the DOBs are chosen as [see (16)]. The first simulation results are shown in Fig. 1, where from the top to the bottom are respectively the position-tracking er, auxiliary errors , control signals , DOBs’ rors , and adaptive parameters . It can be outputs found in Fig. 1 that the proposed controller delivers a very excellent position-tracking performance. Since the reference trajectories are not so fast, it is considered that the uncertainties and are well compensated. We have verified that the position-tracking error levels are much smaller than those in [10] for the same physical parameters, reference trajectories and initial conditions, even under the more realistic conditions in the presence of measurement noises, input signal quantization errors and unmodeled frictions. Notice that the local adaptive controllers in [10] estimate only the local uncertainties, but cannot estimate the interconnections from the other subsystems. In contrast, in our proposed controller, the DOBs estimate the interconnections from the other subsystems with small errors if the reference trajectories are not so fast. We have also investigated the performance by the method in [6], although the results are not shown here due to the limitation of space. We have verified that the results of Fig. 1 are much better. C. Tracking Performance for Fast Reference Trajectories For fast-changing trajectories, i.e., large amplitudes of and , the amplitudes of the entries of the disturbance vector (12) are likely large so that the decentralized control of robot
Fig. 3. Tracking performance for fast reference trajectories (w = w = 70).
manipulators are more challenging. To confirm this, we here consider the following reference trajectories:
(50) is replaced by a faster Notice that compared to (47), only one. However, due to the interconnections, the disturbance term of joint 2 becomes also more significant. The results of the controller given by (49) are shown in Fig. 2. It can be seen that in this case, the DOBs’ outputs are much more significant and fast-changing than those shown in Fig. 1. The control performance is still satisfactory, owing to the excellent performance of the proposed controller. As suggested in (16), the DOB outputs are saturated to ease is not so large and hence the performance analysis. When is really saturated, the estimation error may not be sufficiently small. In this case, however, owing to the adaptive sliding mode control terms, the control performance may not degrade significantly. To confirm this, we repeat the simulation of Fig. 2 under the same conditions except that the saturation . The results are shown levels are reduced to hits the in Fig. 3. It can be verified in Fig. 3 that although
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REFERENCES
Fig. 4. Tracking performance in the case of step-like changes of some parameters.
saturation level, the control results of Fig. 3 are still similar to those in Fig. 2. Finally, for the same controller used for the results of Fig. 2, we investigate the control performance when the values of in Table I changed to be 150% of their original 6.0 kg, values at the time instant of 40 s, i.e., 3.0 kg, 1.5 kgm , 1.2 kgm at 40 s. The results are shown in Fig. 4. It can be verified that due to the cooperative effects of the nonlinear damping terms, DOB, and the adaptive sliding mode control term, the control performance is not sensitive. However, it can be observed the amplitudes of the DOBs’ outputs after 40 s are quite different from those before 40 s.
VI. CONCLUSION In this study, a decentralized adaptive robust controller for trajectory tracking of robot manipulators has been proposed. For manipulator tracking tasks, design of a decentralized controller is not straightforward since all of the subsystems are strongly interconnected. To tackle these interconnections, at each subsystem, a DOB is introduced to compensate for the low-passed coupled uncertainties, and an adaptive sliding mode control term is employed to handle the fast-changing components of the uncertainties beyond the pass-band of the DOB. By some special nonlinear damping terms, the boundedness of the overall system signals is ensured. Furthermore, we have analyzed how the DOB and adaptive sliding mode control play in a cooperative way in each local subsystem to achieve an excellent control performance, i.e., the control performance of each subsystem has been analyzed rigorously. The theoretical claims have been confirmed by the simulation results. Compared to most centralized adaptive control methods for robot manipulators, the proposed controller while delivering a very excellent control performance, is considered to be very simple and requires moderate computational burden.
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[26] Z. J. Yang, Y. Fukushima, S. Kanae, and K. Wada, “Robust nonlinear output-feedback control of a magnetic levitation system by K-filter approach,” IET Control Theory Appl., vol. 3, no. 7, pp. 852–864, 2009. [27] Z. J. Yang, S. Hara, S. Kanae, K. Wada, and C. Y. Su, “An adaptive robust nonlinear motion controller combined with disturbance observer,” IEEE Trans. Control Syst. Technol., vol. 18, no. 2, pp. 454–462, Mar. 2010. [28] Z. J. Yang, S. Hara, S. Kanae, and K. Wada, “Robust output feeddback control of a class of nonlinear systems using a disturbance observer,” IEEE Trans. Control Syst. Technol., vol. 19, no. 2, pp. 256–268, Mar. 2011. [29] Y. Yu and Y. S. Zhong, “Robust backstepping output tracking control for SISO uncertain nonlinear systems with unknown virtual control coefficients,” Int. J. Control, vol. 83, no. 6, pp. 1182–1192, 2010.
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Decentralized Adaptive Robust Control of Robot Manipulators Using Disturbance Observers Zi-Jiang Yang, Member, IEEE, Youichirou Fukushima, and Pan Qin
Abstract—In this paper, we propose a decentralized adaptive robust controller for trajectory tracking of robot manipulators. In each local controller, a disturbance observer (DOB) is introduced to compensate for the low-passed coupled uncertainties, and an adaptive sliding mode control term is employed to handle the fastchanging components of the uncertainties beyond the pass-band of the DOB. In contrast to most of the local controllers using DOB for robot manipulators that are based on linear control theory, in this study, by some special nonlinear damping terms, the boundedness of the signals of the overall nonlinear system is first ensured. This paves the way to analyze how the DOB and adaptive sliding mode control play in a cooperative way in each local subsystem to achieve an excellent control performance. Simulation results are provided to support the theoretical results. Index Terms—Adaptive control, decentralized control, disturbance observer, nonlinear control, robot manipulators.
I. INTRODUCTION
R
OBOT manipulators possess many uncertainties with inherent strong nonlinearities. To improve the tracking performance of robot manipulators, significant effort has been made to seek advanced control strategies, such as adaptive control and robust control approaches [1], [2]. However, since most approaches are based on a centralized control structure which requires rather complicated hardware configuration and tedious computation, their practical application may be computationally demanding. A decentralized control system based only on the local information is highly desirable. In fact, the majority of contemporary robots are still controlled by the decentralized (independent joint) proportional-integral-derivative (PID) law in favor of its simple computation and low-cost setup [3], [4]. Due to strong nonlinearities in dynamics, advanced decentralized control techniques are still attracting much attention to achieve a satisfactory trajectory performance of robot manipulators [5]–[11]. The fundamental uncertainties encountered in the decentralized controller design are the strength of the interconnections Manuscript received March 11, 2011; revised July 20, 2011; accepted August 01, 2011. Manuscript received in final form August 01, 2011. Date of publication August 30, 2011; date of current version June 28, 2012. Recommended by Associate Editor F. Caccavale. Z.-J. Yang is with the Department of Intelligent Systems Engineering, College of Engineering, Ibaraki University, Ibaraki 316-8511, Japan (e-mail: yoh@mx. ibaraki.ac.jp). Y. Fukushima is with the Department of Electrical and Electronic Systems Engineering, Graduate School of Information Science and Electrical Engineering, Kyushu University, Nishi-ku 812-8581, Japan (e-mail: [email protected]). P. Qin is with the Faculty of Mathematics, Kyushu University, Nishi-ku 8128581, Japan. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2011.2164076
among the subsystems. Many papers have been devoted to accounting for uncertain interconnections among the subsystems in the last decades. In [12] and [13] the adaptive decentralized control methods based on the high gain stabilization technique were proposed, and in [14], a decentralized adaptive variable structure-based control scheme was proposed, to stabilize interconnected systems where the strength of the interconnections are bounded by polynomials of the state variables. Recently, a decentralized controller has been proposed [15] and applied to multi-machine power systems [16], in which the lumped modeling error, disturbance and interconnections are estimated by the high-gain observer. Additionally, backstepping designbased decentralized adaptive control methods were developed in [17], [18] for systems which do not satisfy the matching condition. For manipulator tracking tasks, decentralized approaches are not straightforward since the overall system cannot be decomposed into subsystems such that the states and control inputs are fully decoupled from one another, because of the inherent coupling such as moment of inertia and Coriolis force. It has been however, found that in the manipulators the strength of the interconnections is bounded by second-order polynomials in states [5]–[10]. Inspired by this, with a third-order feedback control term, adaptive sliding mode controller [6], and adaptive controller [10] with local model estimation have been proposed. The former may require a strong sliding mode control gain to counteract the interconnections. The latter while being computationally demanding, cannot estimate the interconnections from the other subsystems. As an alternative popular approach for compensating for external disturbances and model mismatch, a disturbance observer (DOB) is often included into a motion controller. Early works of decentralized control of robot manipulators using the DOB can be found in [19]–[22]. At each joint of the robot manipulator, the nonlinearities are treated as a lumped disturbance to a nominal local linear plant. A DOB is closed around each joint to estimate and cancel the disturbance term. Each joint is then treated as a linear plant for which a linear controller can be designed. The attractiveness of the DOB based algorithms lies in their simplicity and transparency of design. Unfortunately the stability properties of these algorithms for nonlinear robot manipulators were not extensively studied [23], [24]. Most recently, there have been some works where rigorous stability analysis of nonlinear control systems using the DOB is performed [25]–[32]. In this paper, we propose a decentralized adaptive robust controller for trajectory tracking of robot manipulators. In each local controller, a DOB is introduced to compensate for the lowpassed coupled uncertainties, and an adaptive sliding mode control term is employed to handle the fast-changing components of the uncertainties beyond the pass-band of the DOB. In con-
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trast to the DOB-based linear local controllers for robot manipulators, in this study, by some special nonlinear damping terms, the boundedness of the signals of the overall nonlinear system is first ensured. Differing from some existing works [5]–[11] where only the control performance of the overall system is analyzed, we also analyze how the DOB and adaptive sliding mode control play in a cooperative way to achieve an excellent control performance in each local subsystem, provided the boundedness of the overall system signals. Simulation results are provided to support the theoretical results. II. PROBLEM STATEMENT Consider an -link rigid manipulator governed by the following dynamic model described by the Lagrange-Euler vector equation:
where Substituting
is the position-tracking error vector and is a constant matrix. and into (1), we have (7) (8)
According to (7), (4) and (8), we have [5]–[7], [9], [10] (9) where are some positive constants. For a vector-valued signal , we define a truncated norm as . Then we have [6]. for Lemma 1: Let Assumptions 1 and 2 hold. If there is a constant such that exists, then for all we have
(1) where is the joint angle vector; is the is the input torque vector; inertia matrix; is the centrifugal and Coriolis is the grativity torque; is the torque; friction force torque. The following assumptions are imposed for the system. Assumption 1: All joints of the robot manipulators under consideration are revolute such that Property 1 given later holds. and the time Assumption 2: The reference trajectory and are bounded signals. derivatives The system model (1) has the following properties that will be used in control system analysis and design [2]. is symmetric and posiProperty 1: The inertia matrix tive-definite such that
(10) for some . In some existing works of decentralized control for robot is demanipulators, each local controller signed based on its own local information [5]–[10]. The stability analysis is based on the overall system model (7) by virtue of Property 4 (passive property). This is in contrast to the conventional decentralized linear control techniques using the DOB [19]–[22] which do not discuss the stability of the overall nonlinear system. In this paper, we propose a class of nonlinear local controllers with the DOB. Differing from the existing works, we first ensure the boundedness of the signals of the overall nonlinear system model. And then provided the boundedness of the overall system signals, we analyze the control performance achieved in each local subsystem.
(2) for some constants Property 2: The matrix
III. CONTROLLER DESIGN
. satisfies
A. Introduction of DOB (3)
. for some constant Property 3: The gravity vector satisfy vector
and the friction force
Recall the overall system model (7). We introduce here a constant nominal inertia for each joint and lump all the other components of the dynamics as a disturbance term . Then we have the following vector-valued nominal linear system model with a disturbance term
(4) for some constants Property 4: The matrix metric, i.e.,
. is skew sym-
(11) where and
is the nominal inertia matrix, is the disturbance vector expressed as (12)
(5) Rewriting each subsystem of (11) as is Notice that this property implies that the map passive [1]. Our task is to design a decentralized controller where at each joint a local controller using only the local information is constructed, so that tracks its reference accurately. To this end, we first define an auxiliary error as
Since calculation of by direct differentiation is usually conthrough taminated with high frequency noise, we may pass a low-pass filter to obtain its estimate as
(6)
(14)
(13)
YANG et al.: DECENTRALIZED ADAPTIVE ROBUST CONTROL OF ROBOT MANIPULATORS USING DISTURBANCE OBSERVERS
In this paper, for convenience of expression, denotes not only the Laplace operator, but also a differential operator if the initial condition effects of the DOB are not concerned. This is the so-called DOB studied extensively in the literature [19]–[22]. Although a high-order filter may improve the estimation performance, in this a study, we adopt a simple second-order filter (15) . where at low-frequenHowever, we can only expect cies due to limited pass-band of the DOB. If is fast changing, the estimation error cannot be neglected and hence can even degrade the control performance significantly. Moreover, the may disturb the other joints DOBs’ outputs mutually. To ease the analysis, a straightforward and simple idea is to saturate the output of the DOB as for for for (16) is a selected upper bound of . Usually, it is where commended to choose a sufficiently large so that the saturation action is not active after the transient performance. However, as will be shown by the numerical example, even when is not so large such that is really saturated and hence the estimation error is not sufficiently small, the control performance is still satisfactory, owing to the adaptive sliding mode control term included in the local controller (17) given below. The key point is that the DOB and the adaptive sliding mode control term work in a cooperative manner. That is, if one works more, the other one works less, and vice versa. The advantage of the idea is that we need not a perfect DOB or a perfect sliding mode control. However, owing to their cooperative effects, the problems of high-gain or chattering can be avoided. This will be confirmed later by the numerical examples.
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may local system. Since the DOBs’ outputs disturb mutually through the interconnections of the joints, , we to suppress the interconnections of employ the damping term . Finally, the last term of is a smoothed version of sliding mode control term with an , this term adaptively updated gain . Notice that when approaches a hard switching control action. and play the The nonlinear damping terms key roles to ensure the boundedness of the overall system sigor a larger nals, as will be shown later. Intuitively, a larger causes a stronger control action. Additionally, the sliding mode control term is adopted to further suppress the estimation beyond the pass-band of DOB. error IV. PERFORMANCE ANALYSIS Each local controller (17) uses only the local information of its corresponding local subsystem. However, since all of the local subsystems are strongly interconnected, we cannot easily see if the local subsystems are all bounded. We should first ensure the boundedness of the overall system signals, based on the properties and assumptions given in the previous section. Then provided the boundedness of the overall system signals, we can analyze the control performance of each local subsystem. Therefore, the performance analysis includes two phases, i.e., analysis of the overall system, and then that of each local subsystem. A. Analysis of the Overall System The results of analysis are given in Theorem 1. The proof is along the spirit of [6], but with modifications specified by the newly designed controller in this study. Theorem 1: Let Assumptions 1 and 2 hold. For the robot manipulator (1) controlled by the proposed adaptive decentralized robust controller (17), there exists a constant , such that is bounded as and hence all the internal signals are bounded, provided the following condition:
B. Description of the Controller Motivated by the aforementioned discussions, we design the following local controller using only the local information for each subsystem to ensure the boundedness of the overall system signals and to achieve a satisfactory control performance for each subsystem:
(19) where
, and
. Remark 1: The first inequality of (19) is easily satisfied for a suffiently large . The second inequality of (19) can be rewritten as
(17) are the control gains, and is an where adaptive parameter (adaptive sliding mode control gain) updated as (18) where is the adaptive gain and is the leakage parameter that prevents from growing to be unbounded. is a simple PD control term. is a The term damping term to suppress the effects of neglected uncertainties of the global system model summarized as in (7) and (8) [6], [9], [10]. is a compensation term by DOB for each
(20) is not far away from Typically, the value of need not be very large unity. And for a sufficiently large to satisfy the condition if is not extremely large. Proof: The proof process includes two steps. , The first step is to show that there exists a constant . The conclusion is proved such that is bounded as by contradiction. To this end, according to (19) we first let a positive constant satisfy
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(21) Notice that the above inequality can be satisfied if is sufficiently large. is not bounded. Thus there always Now assume the signal such that . exists a smallest time Consider a Lyapunov function candidate according to Property 2
Clearly, the last two inequalities are in contradiction, according to (21). This implies that the assumption of is false. Thus the error signal vector is bounded and satisfies for all . The next step is to show that all the internal signals are bounded. Rewrite the adaptive law (18) as the output of a stable lowpass filter driven by : (27)
(22) Taking the derivative along the trajectory of the closed-loop system, and using Property 4, the Schwartz inequality and Lemma 1, we have
is bounded since is bounded. FurWe can conclude that thermore, according to Assumption 2, and (6) and (9), we con, and are bounded. Each local controller clude that is bounded. Therefore, all the internal signals are bounded. Remark 2: The condition (19) is always satisfied for a sufficiently large bound . The results of Theorem 1 only tell us that the error signal vector , the adaptive gain vector and hence all the internal signals can be made to be bounded. It should be emphasized here that at the present stage our purpose is only to ensure the boundedness of the overall system signals. And hence a conservative bound of the overall error system is acceptable. Later, we will show that each individual error signal can be made sufficiently small by virtue of the corresponding local controller. Remark 3: Considering the derivation process of the bound, we can see that the damping terms and edness of play the key roles to ensure the boundedness of the overall system signals. Even when the smoothed sliding mode control term is removed, the boundedness of the overall system signals is still ensured. The sliding mode control term is introduced to further reduce the control error of each subsystem in the cases where the signals are fast so that the DOB cannot cope with them sufficiently.
(23) By (19), (21) and the assumption that there exists a smallest time such that , we can say that there exists a time instant such that
B. Analysis of Each Subsystem Provided the boundedness of the overall system signals, we are ready to analyze how the DOBs and adaptive sliding mode control techniques bring improvement in each subsystem. Substituting the local controller (17) into the subsystem (13), we have
(24) However, according to (23), we have . Therefore, for all
(28)
, for all , we have Owing to (16) and Theorem 1, Define
and
are bounded.
(29)
(25) But the definition of
Then we consider the following Lyapunov function:
leads to (26)
(30)
YANG et al.: DECENTRALIZED ADAPTIVE ROBUST CONTROL OF ROBOT MANIPULATORS USING DISTURBANCE OBSERVERS
where of (28), we have
. Taking the derivative along the trajectory
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has decayed out sufficiently such that for a relatively small conwe have stant
(37) and (38) (31) Using the relation
Comparing (29) and (37), it is expected that can be much . Then by rewriting (36), we have, smaller than Corollary 1: Let the assumptions and results of Theorem 1 hold. The auxiliary error of each subsystem satisfies
(32) (39) we have
(33) where
The results of Corollary 1 imply that a sufficiently small ultican be achieved by the cooperamate error tive effects of the DOB and the adaptive sliding mode control. includes also , However, since the Lyapunov function while being i.e., the error term of the adaptive parameter, ensured by (35) may not be very small, and hence the transient performance may be conservative. Moreover, the adaptive parameter usually exhibits a long transient phase. This motivated us to seek a less conservative evaluation of the transient performance, despite the effects of the adaptive laws. We will show here that the nonlinear damping terms play important roles to achieve a reliable transient performance. From (28), we have
(34) and hence we obtain the following transient performance:
(40) and hence (41)
(35) is bounded by Equation (35) implies that as . In addition, since , we have the following. Theorem 2: Let the assumptions and results of Theorem 1 hold. For each subsystem of the decentralized robot manipulator model (13) controlled by the proposed decentralized adaptive robust controller (17), all the internal signals are bounded, and the auxiliary error of each subsystem satisfies
where (42) This leads to Theorem 3: Let the assumptions and results of Theorem 2 hold. The auxiliary error of each subsystem satisfies (43)
(36) Inspection of (34) indicates that can be reduced by choosing sufficiently small values of and , and a large value included in may not of . However, according to (29), be small since the initial value is usually set to be zero. We then discuss the control performance after a short transient be an effective time-constant of the phase of DOB. Let DOB depending on , until which the initial value of
The dominator of (42) clearly illustrates the effects of nonlinear damping terms. The term suppresses the peaking (if it occurs) and the uncertainty . On the other of suppresses the growing of due to the unhand, the term . certainty To summarize, Corollary 1 characterizes a better ultimate achieved mainly by the DOB bound of the auxiliary error and the adaptive sliding mode control term. See the second term on the right-hand side of (39). On the other hand, Theorem 3
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characterizes a reliable transient performance ensured mainly by the nonlinear damping terms, despite the effects of the adaptive laws.
TABLE I PHYSICAL PARAMETERS OF THE ROBOT MANIPULATOR
C. Comments and Guidelines of Parameter Design The proposed controller (17) includes the following design parameters: linear control gains [defined in (6)] and , nonlinear control gains and , smoothing factor of sliding mode control , adaptive gain and leakage parameter , and timeconstant of DOB. We will draw some guidelines of parameter design by commenting on the aforementioned analysis results. , In the controller (17), the amplitude of the linear term is defined in (6), does not tend to be very large in where generic cases compared to the other terms, and hence the gains and can be chosen to be relatively large. This term also contributes to achieve a better transient performance and a smaller ultimate tracking error. can be relatively small if the feedThe nonlinear term back error is small. However, if is large at some time points may bedue to some reasons, its nonlinear feedback gain come very large. Therefore, a modest should be chosen. may be large, Since the amplitude of the DOB’s output it is recommendable to choose a modest nonlinear control gain to avoid a noisy or large control signal. In Remark 1 we have mentioned that the condition of Theorem 1 may still be maintained for a modest . The results of Corollary 1 imply that a small smoothing factor leads to a small ultimate tracking error. However, as well known in the literature, a less smooth switching function may cause the chattering problem. Therefore, should not be too small. It should be commented here that even if is not so small, at least the low-passed components of the uncertainties can be compensated for by the DOB. The results of (34), (38), and Corollary 1 imply that a small leakage parameter and a large adaptive gain may lead to a small ultimate tracking error. However, according to (27), too small an may lead to a large , which may cause a large control effort. Moreover, too large a may make change sensitively and hence make the control signal oscillatory. defined in Usually, a small of DOB leads to a small sensitive to noise. Owing to (37). A very small may make enhancement by the adaptive sliding mode control term, need not be very small. The DOB and adaptive sliding mode control should play in a cooperative way.
A. Description of the Manipulator Model Consider a two-DOF planar robot arm
(44) whose entries are given as follows:
(45)
(46) where the physical parameters are given in Table I, and the 9.807 m/s . Notice that the gravity acceleration is given as frictions and are not neglible, compared to the gravity effects. To carry out more realistic simulations, we consider the following simulation conditions. The controller is implemented at 0.5 ms. To verify the noise effects, a sampling period of a uniformly distributed noise between 0.0001 and 0.0001 rad is added to each position measurement. The measurements of and are obtained by pseudodifferentiations of the position measurements due to . The amplitudes of and are limited within the interval of Nm. And each control signal is quantized by 12 bits, such that the quantization Nm. step is B. Tracking Performance for Slow Reference Trajectories
V. SIMULATION EXAMPLE Extensive numerical studies have been performed to support the theoretical results obtained in this paper. For the sake of comparison, we use the example in [10]. Considering some more realistic situations, the simulation conditions are modified by adding the position measurement noises, pseudodifferentiations of position measurements, control signal quantizations with a finite number of bits, control signal saturations and unmodeled frictions. The numerical results of [10] are not reproduced here due to the limitation of space.
We first investigate the tracking performance for the following slow reference trajectories studied in [10], [6]:
rad
(47)
To show that the controller is robust against nonzero initial tracking errors, the initial conditions are given as rad rad/s
(48)
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Fig. 2. Tracking performance for fast reference trajectories. Fig. 1. Tracking performance for slow reference trajectories.
According to the guidelines of controller parameter design, the design parameters are given as follows:
(49) . The initial adaptive parameters are set as Additionally, the saturation levels of the DOBs are chosen as [see (16)]. The first simulation results are shown in Fig. 1, where from the top to the bottom are respectively the position-tracking er, auxiliary errors , control signals , DOBs’ rors , and adaptive parameters . It can be outputs found in Fig. 1 that the proposed controller delivers a very excellent position-tracking performance. Since the reference trajectories are not so fast, it is considered that the uncertainties and are well compensated. We have verified that the position-tracking error levels are much smaller than those in [10] for the same physical parameters, reference trajectories and initial conditions, even under the more realistic conditions in the presence of measurement noises, input signal quantization errors and unmodeled frictions. Notice that the local adaptive controllers in [10] estimate only the local uncertainties, but cannot estimate the interconnections from the other subsystems. In contrast, in our proposed controller, the DOBs estimate the interconnections from the other subsystems with small errors if the reference trajectories are not so fast. We have also investigated the performance by the method in [6], although the results are not shown here due to the limitation of space. We have verified that the results of Fig. 1 are much better. C. Tracking Performance for Fast Reference Trajectories For fast-changing trajectories, i.e., large amplitudes of and , the amplitudes of the entries of the disturbance vector (12) are likely large so that the decentralized control of robot
Fig. 3. Tracking performance for fast reference trajectories (w = w = 70).
manipulators are more challenging. To confirm this, we here consider the following reference trajectories:
(50) is replaced by a faster Notice that compared to (47), only one. However, due to the interconnections, the disturbance term of joint 2 becomes also more significant. The results of the controller given by (49) are shown in Fig. 2. It can be seen that in this case, the DOBs’ outputs are much more significant and fast-changing than those shown in Fig. 1. The control performance is still satisfactory, owing to the excellent performance of the proposed controller. As suggested in (16), the DOB outputs are saturated to ease is not so large and hence the performance analysis. When is really saturated, the estimation error may not be sufficiently small. In this case, however, owing to the adaptive sliding mode control terms, the control performance may not degrade significantly. To confirm this, we repeat the simulation of Fig. 2 under the same conditions except that the saturation . The results are shown levels are reduced to hits the in Fig. 3. It can be verified in Fig. 3 that although
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REFERENCES
Fig. 4. Tracking performance in the case of step-like changes of some parameters.
saturation level, the control results of Fig. 3 are still similar to those in Fig. 2. Finally, for the same controller used for the results of Fig. 2, we investigate the control performance when the values of in Table I changed to be 150% of their original 6.0 kg, values at the time instant of 40 s, i.e., 3.0 kg, 1.5 kgm , 1.2 kgm at 40 s. The results are shown in Fig. 4. It can be verified that due to the cooperative effects of the nonlinear damping terms, DOB, and the adaptive sliding mode control term, the control performance is not sensitive. However, it can be observed the amplitudes of the DOBs’ outputs after 40 s are quite different from those before 40 s.
VI. CONCLUSION In this study, a decentralized adaptive robust controller for trajectory tracking of robot manipulators has been proposed. For manipulator tracking tasks, design of a decentralized controller is not straightforward since all of the subsystems are strongly interconnected. To tackle these interconnections, at each subsystem, a DOB is introduced to compensate for the low-passed coupled uncertainties, and an adaptive sliding mode control term is employed to handle the fast-changing components of the uncertainties beyond the pass-band of the DOB. By some special nonlinear damping terms, the boundedness of the overall system signals is ensured. Furthermore, we have analyzed how the DOB and adaptive sliding mode control play in a cooperative way in each local subsystem to achieve an excellent control performance, i.e., the control performance of each subsystem has been analyzed rigorously. The theoretical claims have been confirmed by the simulation results. Compared to most centralized adaptive control methods for robot manipulators, the proposed controller while delivering a very excellent control performance, is considered to be very simple and requires moderate computational burden.
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