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Precise slow motion control of adirect-drive robot arm with velocityestimation and friction compensationG. Liua,*, A.A. Goldenbergb, Y. ZhangbaDepartment of Aerospace Engineering, Ryerson University, 350 Victoria Street,Toronto, Ont., Canada M5B 2K3bRobotics and Automation Laboratory, University of Toronto, 5 Kings College Road,Toronto, Ont., Canada M5S 3G8AbstractPrecise low speed motion control of a robot manipulator calls for precise position andvelocity measurement and joint friction compensation, as well as robustness and adaptabilityof the control scheme. However, precise velocity measurement and friction compensationremain challenging research tasks, especially for very slow motions. In the present work, asimple and efficient method is proposed to estimate velocity from a sampled incremental en-coder pulse train, which is then utilized in the experimental investigation on a proposed robustdecomposition-based friction compensation method. The experimental results on a directdrive robot arm have demonstrated precise motion control at very slow speeds and in thepresence of significant joint friction.? 2004 Elsevier Ltd. All rights reserved.Keywords: Precise motion control; Velocity estimation; Friction compensation; Robust control;Decomposition-based control1. IntroductionPrecise low speed motion control of a direct-drive robot arm relies on preciseposition and velocity measurement and joint friction compensation, as well asrobustness and/or adaptability of the control scheme 6,1214,17,19. However,both precise velocity measurement/estimation and friction compensation remain*Corresponding author. Tel.: +1-416-979-5000; fax: +1-416-979-5056.E-mail address: gjliuryerson.ca (G. Liu).0957-4158/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechatronics.2004.03.002Mechatronics 14 (2004) 821834challenging research topics, especially for very slow motions. Velocity measurementis a crucial issue in the implementation of friction compensation, which is not onlyrequired in model-based friction feedforward compensation, but also in most feed-back compensators 1,11,15,19. The velocity measurement noise, when amplified byfeedback gains, can stimulate unmodeled higher order dynamics.For velocity measurement, generator type tachometers and encoder-basedvelocity measurement electronics are commercially available, but they often provideunsatisfactory outputs at slow velocities due to noise and low resolution. In theliterature, various methods have been proposed to estimate velocity from positionmeasurements, typically the outputs of an optical encoder. The simplest velocityestimation method is the Euler approximation, which takes the difference betweenthe last two sampled positions and divides it by the sampling period. When thepositions are precisely sampled, such as in the case of a laser dynamic calibrator usedin our experiments, the Euler approximation gives the simplest and most efficientvelocity estimation. However, with an incremental encoder pulse train, the sampledpositions contain stochastic errors due to sampling and encoder fabrication toler-ance, which result in large deviation in the velocity estimation when the Eulerapproximation is used, especially when the sampling period is small and the jointspeed is low. In order to reduce the velocity estimation error, one effective way is toincrease the position increment by tracing backwards a few steps before applying theEuler approximation. While this remedy can smooth the velocity estimation, it alsocauses time delay in the estimated velocity. The number of backward steps has to becarefully chosen in order to balance noise level and time delay in the velocity esti-mates. In 6, it is found that three steps is the best for a sampling rate of 2500 Hz intheir experiments with an encoder of 655,360 pulses per revolution. In 5, it isproposed that the number of backward steps should be adjustable in order toaccommodate various speeds, and a discrete time adaptive windowing method ispresented for velocity estimation. Velocity estimation using Kalman filtering ofposition measurements is proposed in 2, with the assumption of Gauss distributionof the position sampling error. Some non-linear observers are also derived forvelocity estimation using full dynamic equations of the plant 7,8. Recently, anadaptive fuzzy logic-based velocity observer is reported in 9, with experimentalverification results.Another critical issue in slow motion control is friction compensation, which hasbeen an important research topic in the motion control of mechanical systemsincluding robot manipulators for decades 1. Most research work on friction com-pensation is concentrated on two major issues: friction modeling and control syn-thesis. A dynamic state variable friction model is presented in 3, and severaladaptive friction compensation methods have been developed based on this statevariable model 4,16. In 10, a decomposition-based friction compensation methodis proposed by applying a decomposition-based control approach to a linearizedparametric friction model derived from the well-known friction model presented in1. In this friction compensation strategy, the friction modeling uncertainties havebeen divided into parametric uncertainty and non-parametric uncertainty. Anadaptive controller is designed to compensate for the parametric friction model822G. Liu et al. / Mechatronics 14 (2004) 821834uncertainty while a robust compensator deals with the non-parametric modelinguncertainty. The overall controller guarantees the uniform ultimate boundedness ofthe system error. While computer simulations are reported in 10, experimentalresults were not available. In the computer simulations, it is assumed that the jointvelocity can be precisely measured.In the present work, a simple and efficient method is proposed to estimate velocityfrom a sampled incremental encoder pulse train. Based on the physical character-istics of the sampled data from incremental encoders, the proposed velocity esti-mation method allows the user to tune the estimation precision and time delay. Inorder to evaluate the proposed method experimentally, a laser dynamic calibrator isused to obtain precise velocity measurement of a direct drive robot joint, which isthen utilized as the reference to evaluate the proposed method. Furthermore, theproposed velocity estimation method is utilized in the experiments on the proposeddecomposition-based friction compensation method. The experimental results on adirect drive robot arm have demonstrated precise motion control at low velocityusing an optical incremental encoder and in the presence of significant joint friction.The rest of the paper is organized as follows: Section 2 presents the proposedvelocity estimation method. In Section 3, the proposed decomposition-based frictioncompensation method is outlined. The experimental setup is introduced and theexperimental comparison results are presented in Section 4. Concluding remarks aregiven in Section 5.2. Velocity estimationThe manufacturers of encoders and motors normally provide the total number ofpulses per revolution, N, as one of the specifications. The resolution of the encoder isthen R 2p=N, correspondingly. The accuracy of velocity estimation for a givenencoder is constrained by the resolution of the encoder. The best velocity estimationalgorithm shall take full advantage of the encoder outputs. Also, a good velocityestimation method should provide a means to allow the user to tune the tradeoffbetween precision and time delay.For an incremental encoder with a resolution of R, the position qt is sampledwith a sampling period of T, and the discrete sampled position at time kT is denotedas hk;k 1;2;. If the encoder output pulse train is evenly distributed atany constant rotation speed, the position error associated with hk can beexpressed ashkj? qkTj R1for a period of time jT, j 1;2;. and j k, and a corresponding position segmentqkT ? qk ? jT), we havehkj? hk ? j ? qkT ? qk ? jTj 2R2G. Liu et al. / Mechatronics 14 (2004) 821834823Hencehk ? hk ? jjT?qkT ? qk ? jTjT?2RjT3SinceqkT ? qk ? jTjT vj4is the average speed over the time segment k ? jT;kT, andhk ? hk ? jjT vj5is the estimation of v using the sampled data hk and hk ? j, we have from Eq. (5)j vj? vjj 2RjT6If we define the velocity estimation resolution asvRRT7which is a constant once the encoder resolution R and sampling period T are fixed.Eq. (6) can be rewritten asj vj? vjj 2jvR8It can be seen from Eq. (8) that the absolute accuracy of the average velocityestimation can be reduced by increasing the number of backward steps j. However,most likely the relative error is more concerned than the absolute accuracy inpractice. We evaluate the relative accuracy defined byrj vj? vj vj?hk ? hk ? j ? qkT qk ? jThk ? hk ? j?2Rhk ? hk ? j?1hk ? hk ? jjj? 2R9Since the difference between the sampled positions can be expressed ashkj? hk ? jj sjR10where sjis an integer, and its value is the number of encoder pulses between hk andhk ? j. From Eq. (9), we haverj vj? vj vj?2rj12For example, to limit the relative accuracy within rj 2%, sj 100, i.e. a minimumof 101 pulses have to be traced backwards.For estimating velocity based on the above analysis, the last step is to determinethe number of steps j that corresponds to the number of encoder pulses sj. Since thenumber of backward steps j depends on the motion speed, more encoder pulses willbe generated at higher speeds. At slow speeds, more steps have to be traced back forthe same amount of pulses. A simple search algorithm as illustrated in Fig. 1 canidentify the number j. However, it should be noted that when the joint is not movingor moving at extremely slow speed, the number of back steps j may not be identifiedor extremely large. To address this problem, we suggest a maximum limit, m, for j.That is, j6m. Naturally, this constraint also sets a limit to the relative accuracy rj,i.e. rjcannot be arbitrarily small once m is chosen.Based on the above analysis, the proposed velocity estimation method is sum-marized as follows:Step 1: Determine the resolution of the encoder R.Step 2: Specify the maximum relative accuracy rj. Set j = 0 j = j + 1 v (k)=(k) - (k-j)/(jT) Yes No |(k) - (k-j)| sjR No Yes j = m Fig. 1. Velocity estimation procedure.G. Liu et al. / Mechatronics 14 (2004) 821834825Step 3: Calculate the corresponding number of pulses sjthat satisfies Eq. (12).Step 4: Select the maximum number of backward steps m.Step 5: Following the procedure as illustrated in Fig. 1, determine in real time therequired backward steps j that satisfies (10) or is limited by the maximumm, and calculate the velocity estimation.The following remarks clarify the applicability of the proposed velocity estimationmethod:Remark 1. The proposed velocity estimation method allows specification of therelative estimation accuracy rj, and the corresponding number of backward steps isdetermined in real time, adapting to the motion speed automatically.Remark 2. In practice the maximum relative accuracy rjand the maximum numberof backward steps m may be tuned by trial and error to achieve the best possibleperformance.Remark 3. In the above analysis, it has been assumed that the encoder pulse train isevenly distributed. In practice, due to fabrication tolerance, the actual encoder res-olution RW R 2p=N, it is equivalent to a lower encoder resolution.3. Decomposition-based friction compensationIn theory, since the unmodeled low velocity friction is obviously bounded, a ro-bust controller can compensate for low velocity friction 6,1214. However, highfeedback gain is required in order to achieve high accuracy, which is always limitedby hardware issues including unmodeled high order plant dynamics and sensormeasurement noise. The key in practical robust control design is to achieve desiredperformance with minimum feedback gains 12. In 10, a friction compensationscheme is synthesized by applying the decomposition-based control design approachdeveloped in 1214. The fundamental strategy of the decomposition-based systemmodeling and control approach is to distinguish between uncertain parameters andvariables of different physical types, and to design a separate compensator for eachof them, while taking into account each specific physical feature. This approachadvocates treating each type of model uncertainty with the most suitable and efficientmeans, including PID, robust, adaptive, and sensor-based control methods. Theoverall controller is generated by synergetic integration of these compensators. In theproposed friction compensation scheme, adaptive control and robust control tech-niques are both applied, and they complement each other in dealing with modeluncertainties.Since the focus of the present work is friction compensation, we consider a singlejoint robot arm with the mathematical model:M q B_ q Fc Fsexp?Fs_ q2sgn_ q Fqq; _ q s13826G. Liu et al. / Mechatronics 14 (2004) 821834where M denotes a constant inertia, qt, _ qt, qt represent the acceleration, velocityand position, respectively. B denotes the viscous friction coefficient. Fcdenotes theCoulomb friction related parameter. Fsdenotes the static friction related parameter.Fsis a positive parameter corresponding to the Stribeck effect. Fqq; _ q reflects theposition dependency of friction and other friction modeling errors, and st is theactuation input. The sign function is defined assgn_ q 1for _ q 00for _ q 0?1for _ q 08:14The dynamic model is formulated to include Coulomb friction, static friction,Stribeck effect, position dependency and other bounded disturbances. In this model,frictional memory and rising static friction as discussed in 1 are assumed negligible.To focus on the friction compensation, the inertia M is assumed accurately known.The friction model parameters, B;Fc, Fs, and Fs, are not accurately known, and theyare not necessarily constant. However, their nominal values are determined off-lineas constants. The nominal values of B, Fc, Fs, and Fsare denoted asB,Fc,Fs, andFs,respectively.The non-parametric friction term Fqq; _ q is bounded asjFqq; _ qj q15where q is a known constant bound for any position q and velocity _ q.Assuming the nominal values of Fsand Fsare close to their actual values, welinearize the Stribeck effect Fsexp?Fs_ q2 at the nominal parameter valuesFsandFs.By ignoring higher order terms, a linearized model is obtained asM q B_ q FcFsexp?Fs_ q2 exp?Fs_ q2Fs?Fsexp?Fs_ q2_ q2Fssgn_ q Fqq; _ q s16In a more compact form, Eq. (16) can be rewritten asM q B_ q FcFsexp?Fs_ q2sgn_ q ? Y_ qF Fqq; _ q s17whereY_ q _ qsgn_ qexp?Fs_ q2sgn_ q ?Fs_ q2exp?Fs_ q2sgn_ q?18and the parametric model uncertaintyP is defined asP B ? BFc? FcFs? FsFs? Fs?T19To incorporate variable parametric model uncertainty compensation,Pisdecomposed asP PcPv20wherePcis a constant unknown vector, andPvcontains variable elements that arebounded as follows:jPvij qii 1;2;3;421G. Liu et al. / Mechatronics 14 (2004) 821834827The control objective is to synthesize a control scheme that provides precisetracking of a given smooth desired trajectory qdt. The position and velocitytracking errors are defined ase q ? qd_ e _ q ? _ qd22a;bFurthermore, a mixed tracking error is defined asr _ e ke23where k is a positive constant.For the control task of tracking the commanded trajectory qdt, a nominalcontrol s0is defined ass0 Ma B_ q FcFsexp?Fs_ q2sgn_ q24where the quantity a is defined asa qd? 2k_ e ? k2e25Applying the approach of decomposition-based control design, an adaptive com-pensator is designed with respect to the constant parametric uncertaintyPc, and arobust compensator forPv. The overall control law is synthesized ass s0 Y_ qupc upv uu26where the compensators upc, upvand uuare defined as follows.upc ?kZt0Y_ qTrds27where k is a constant control gain.upvi?qifijfijif jfijPepi?qifiepiif jfij :i 1;2;3;428where f Y_ qTr, and epiis a positive control parameter for i 1;2;3;4.uu?qrjrjif jrjPe?qreif jrj :29where e is another positive control parameter that can be tuned for achieving bettertracking results.It has been proved in 10 that the tracking error is uniformly ultimately boundedunder the control law defined by (26)(29). Also, the ultimate bound of the trackingerror is determined by the variable parametric uncertainty, the non-parametricuncertainty and control parameters only, and it is not affected by the constant ele-ment of the parametric uncertainty.828G. Liu et al. / Mechatronics 14 (2004) 8218344. ExperimentsThe proposed friction compensation and velocity estimation methods are imple-mented experimentally using a direct drive robot arm, controlled by a PC computerwith a sampling period of 1 ms. A picture of the experiment setup is shown in Fig. 2.The robot arm is actuated by a direct drive DC motor at each joint. For the purposeof the experiments, only one of the two joints is controlled to move at various speedswhile the other joint is fixed. The incremental optical encoder of the motor generates1,024,000 pulses per revolution.4.1. Velocity measurement and estimationFor the experimental evaluation, it is important to determine what accuracy theproposed velocity estimation method actually offers in order to evaluate its effec-tiveness, and a convincible way to obtain an accurate reference for comparison is tointroduce an external velocity measurement device with high accuracy. For thispurpose, a laser dynamic calibrator is integrated into the experimental system. Witha customized interface module and real-time software, the laser calibrator providesprecise measurements of position increments, which are also stored in the data file ofthe experiments. With the high precision position measurements of the laser cali-brator, the angular velocity of the robot joint is calculated to provide the referencefor evaluating the velocity estimation from encoder outputs.In the experiments, the robot joint is controlled to move at various velocities. Theposition readings from the incremental encoder and the laser calibrator, as well asthe velocity signal generated by the electronic differentiator in the motor drive, arestored in a data file for analysis with MATLAB.Fig. 2. An direct drive robot arm and a laser dynamic calibrator.G. Liu et al. / Mechatronics 14 (2004) 821834829The experimental results have demonstrated the effectiveness of the proposedvelocity estimation method. The velocity estimation is close to the average velocitycalculated from the laser dynamic calibrator. As expected, the estimated velocitieshave a time delay from the actual velocities measured by the laser calibrator. Thetime delay is longer at lower speeds and shorter at higher speeds. The velocitymeasurements provided by the motor drive is not as good as velocity estimatesobtained using the proposed estimation method at low speeds. Even at high speedsthe velocity estimation results are comparable to that from the electronic differen-tiator at the 1 kHz sampling rate. Fig. 3 presents the low velocity estimation resultsof the proposed method using experimental data, in comparison with the velocitiesmeasured by the laser dynamic calibrator and electronic differentiator in the motordrive. Fig. 4 shows how the specified relative accuracy affects the velocity estimationresults, which confirms that selecting higher relative accuracy results in a smoothervelocity estimation, but with a longer time delay.4.2. Friction compensationThe nominal values of the parameters for the system model (17) and uncertaintybounds were estimated based on the experimental data obtained using the lasercalibrator as reported in 18, using a similar procedure as in 14. The estimatedparameters are as given in Table 1.The control objective is to achieve high accuracy tracking of positional referencetrajectories in the presence of joint friction. The desired reference trajectories havebeen chosen asqd A1 ? cos0:5pt_ qd 0:5pAsin0:5ptFig. 3. Velocity estimation resultsslow motion.830G. Liu et al. / Mechatronics 14 (2004) 821834where A is a constant that is used to change the reference trajectory. For A 0:003rad, the reference position and velocity are shown in Fig. 5.The uncertainty bounds were estimated based on experimental results. Thedecomposition-based friction compensation method can be intuitively tuned toachieve the best tracking results. The control parameters, including the relativevelocity estimation accuracy rj, are tuned by trial and error to achieve the besttracking results while not exciting vibration of the arm. The control parameters aresummarized in Table 2. Fig. 6 shows the positional tracking result of the decom-position-based friction compensation method, and Fig. 7 shows the estimatedvelocity. As the robot joint reverses its moving direction, the actual velocity is closeto zero, and the velocity estimation contains larger errors as shown in Fig. 7, due tothe limited maximum backward step. The positional tracking error also increases asa result. The relatively large velocity estimation noise at peak velocities in Fig. 7 is adirect result of the relatively large noise in the position measurement as shown inFig. 6.Precise motion control has been achieved using the combination of the proposedvelocity estimation method and friction compensation scheme. During the experi-ments, it was observed that the velocity estimation had substantial effects on thetracking performance. High feedback gains amplify velocity measurement noiseFig. 4. Velocity estimation results for various relative accuracy specifications.Table 1Nominal model parametersM (kgm2)Fs(Nm)Fc(Nm)B (Nms/rad)Fs(s2/rad2)0.765.272.643.6740G. Liu et al. / Mechatronics 14 (2004) 821834831which then stimulates unmodeled higher order dynamics. The proposed velocityestimation method allowed higher feedback gains, which in turn result in moreprecise tracking. For the same set of control parameters as given in Table 2, therobot arm vibrate severely if the electronic differentiator output or simple Eulerapproximation is used for velocity measurement. In order not to excite higher orderdynamics, the control gains have to be reduced and the tracking errors would in-crease as a result.Fig. 5. The commanded position (a) and velocity (b).Table 2control parameterskkeep1ep2ep3ep42001600.10.050.050.050.05832G. Liu et al. / Mechatronics 14 (2004) 8218345. Concluding remarksVelocity estimation is a key factor in precise low speed motion control of thedirect drive robot arm. The smoothness and accuracy of velocity estimation havesubstantial effect on the joint friction compensation and effectiveness of the controlscheme. In this paper, a simple and efficient method is presented to estimate velocityfrom a sampled incremental encoder pulse train, which is then utilized in theexperimental investigation on a proposed robust decomposition-based frictioncompensation method. The experimental results on the direct drive robot arm havedemonstrated precise motion control at low speed and in the presence of significantjoint friction.Fig. 7. Estimated velocity.Fig. 6. Position tracking error.G. Liu et al. / Mechatronics 14 (2004) 821834833AcknowledgementsThis research is supported in part by the National Science and Engineering Re-search Council of Canada.References1 Armstrong-Helouvry B, Dupont P, Canudas de Wit C. A survey of models, analysis tools andcompensation methods for the control of machines with friction. Automatica 1994;30(7):1083138.2 Belanger PR. Estimation of angular velocity and acceleration from shaft encode measurements. In:Proceedings of the 1992 IEEE International Conference on Robotics and
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