XV International PhD Workshop OWD 2013, October Limitations Caused by Sampling and Quantization in Position Control of a Single Axis Robot

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1 XV International PhD Workshop OWD 2013, Octoer 2013 Liitations Caused y Sapling and Quantization in Position Control of a Single Axis Root Csaa Budai, Budapest University of Technology and Econoics ( , László L. Kovács Dr., HAS-BUTE Research Group on Dynaics of Machines and Vehicles) Astract This paper presents the theoretical and experiental investigation of the dynaics of a single degree-of-freedo rootic ar sujected to digital position control. The experiental setup consists of an industrial root axis, a icrocontroller ased low-level control unit, and a highlevel Matla/Siulink position controller. The stale doain of operation is calculated in the paraeter space of the sapling tie and the gains of the applied proportional-derivative (PD) controller. The calculated staility charts are verified y experients, and the paper reports the liitations on syste staility caused y the digital effects and the applied control setup. icrocontrollers. In this paper we investigate the PD digital position control of a rootic ar controlled y a custo, icrocontroller ased hardware. A single axis of an industrial rootic ar is controlled y a icrocontroller, and Siulink Real-Tie Workshop [3] is used to provide a flexile user interface for setting the desired positions of the root. This architecture has the advantage of developing an easy-to-use experiental setup, ut it ay cause further processing delay and synchronization proles that can lead to instailities. This is confired in the second part of the paper, where an explanation is also provided for the differences oserved etween the theoretical and practical staility liits. 1. Introduction Fast and accurate otion/position control is an iportant ojective in various rootic applications. This is typically achieved y using high gain, decentralized controllers that operate at high sapling rates. Often the digital effects, like the teporal and spatial discretization (sapling and quantization) are neglected in the control design, and these systes are treated as continuous-tie systes. However, in soe situations these effects can have a strong influence on the dynaics of the controlled syste. For exaple, sophisticated control algoriths ay require considerale coputational tie, which result in a slower sapling rate. In this case, digital effects can cause an intricate dynaic ehavior. Moreover, the non-sooth effects, like quantization, dry friction, acklash ay lead to chaotic otions even at high sapling rates [1,2]. The use of high sapling frequencies ay e liited y the control hardware in low cost applications, where the sensor readings and the control calculations are ipleented on 2. Experiental Setup The investigated experiental setup consist of a single axis rootic ar (HIRATA MB-H230) controlled y a custo-, self-developed control unit. This low-level controller is ased on a PIC icrocontroller (24FJ128GA010) that counicates with a Siulink Real-Tie Workshop (RTW) ased high-level controller y using RS-232 protocol. The icrocontroller ased platfor processes the easured data fro the otor encoders, and transits the calculated angle to the PC. The RTW ased control software deterines the necessary control force, and transits it via the icrocontroller ased platfor to the H-ridge in the for of a pulse-width-odulation (PWM) signal. The scheatic of the experiental setup can e seen in Fig.1. Here, the single axis industrial rootic ar is shown as a all screw which is connected to the shaft of the DC otor y a rigid clutch. 466

2 The lead of the all screw is l = 20 which gives the travelled distance along the root axis per otor revolutions. By using the previous derivations, this control force can e deterined with respect to the duty cycle of the pulse width odulation signal as rk t U0 F = pc = pc. (1) Rp Fig.1. Scheatic of the Experiental Setup Assuing an ideal (lossless) force-torque transission of the all screw, the end-effector force is f = rm t, where M is the applied otor torque, and r = 2 π / l = 100π is the transission ratio. The t otor torque can e approxiated y considering the steady state otor characteristics. Then applying the Biot-Savart s Law [4] the otor torque is M = ki, where k = N/ A is the otor constant and i = U / R is the current that flows through the otors windings. Here, R = 1.1 Ω is the terinal resistance and the otor. U is the supply voltage of Based on Lenz s Law [4] the otor supply voltage is proportional to its angular velocity. Thus a siple way to control the speed of a DC otor is using a PWM signal to set the average supply voltage. It can e written as U = U 0 pc / p, where U 0 = 24 V is the supply voltage of the H-ridge, p c is the calculated duty cycle, and p = 730 is the axiu of the PWM duty cycle. Note, that the value of p is liited y the settings (e.g., selected PWM frequency) of the icrocontroller, and the quantization of the control force is inversely proportional to this value. In voltage control ode, the otor torque calculated so far is reduced y the ack electrootive force. According to Faraday s Law of Induction [4], this force (torque) is proportional to the angular velocity of the otor shaft. Therefore, it ay e considered as a viscous daping force in odeling the actuated root axis, while the force coponent proportional to the odulated supply voltage ay e seen as the exogenous control force. 3. The Equivalent Mechanical Model By using voltage control, and odeling the ack EMF effect as a viscous daping eleent, the siplified echanical odel of the experiental setup is shown in Fig.2. Fig.2. Equivalent Mechanical Model The equation of otion of this odel is (& ) x&& + x& + C sgn x = F, (2) where is the effective ass that represents the root s inertia, and is the daping induced y the ack EMF effect. Paraeter C 20 N is the easured Coulo friction force, while the control force F is defined y eq. (1). The effective paraeters of this odel were identified in two siple experients: (a) accelerating with constant force, and () easuring how the syste coes to rest due to the viscous daping and dry friction only. By considering a constant positive control force, sgn( x & ) = 1 in eq. (2), then the general solution of the equation of otion ecoes t F C x ( t ) = c1 + c2 e + t, (3) where c1 andc 2 are the constant that depend on the initial conditions. By appropriately selecting the initial position, c 1 can e canceled, while the exponential ter vanishes y tie. For a constant force input, the steady state solution is ( ) x t F C = t. (4) Therefore the effective viscous daping can e calculated as = ( F C )/ s 1443 Ns/, where s v /s is the slope of the linear segent (the steady state velocity) in Fig

3 x ξ acceleration P 1 s v 0 τ s τ F C = deceleration P 2 Fig.3. Typical root otion in paraeter identification experients The steady state velocity can e seen as the initial velocity of our second experient with zero control force. The corresponding decelerating otion starts at P 1 in Fig. 3., where also a new coordinate frae is introduced to descrie this otion. Point P 2 is the point where the root stops due to dry friction and viscous daping. The corresponding tie τ 93 s was deterined as the ean of soe s uncertain values in the region where the tangent of the curve in Fig 3. ecoes zero. Based on eq. (3), in the new ( τ, ξ ) frae the displaceent of the decelerating root is given y the general solution τ C ξ( τ ) = c1 + c2 e τ. (5) Then, with the initial conditions ξ ( τ = 0) = 0 and d ξ / dτ ( τ = 0) = v0 the displaceent function ecoes τ v 0 e C C ξ( τ ) = + 1 τ. t (6) By differentiating this expression with respect to τ and considering the stop condition ( ) d ξ / dτ τ = τs = 0, the effective ass of the odel in eq. (2) can e estiated as τs = 37 kg. v0 ln 1+ C (7) 4. Controlled Dynaics For the analysis of the dynaics of the digitally controlled syste we neglect, the otherwise stailizing, dry friction in eq. (2) and we introduce the control force where xj 1 xj 2 F = kpxj 1 kd, t [ t j, t j + 1), t (8) k p and k d are the control gains, t is the sapling tie and t j denotes the j-th sapling instant. In addition, x j 1 = x( t j 1) x j 2 j 2 and = x( t ) represent the sapled and delayed position feedack data [5,6]. The considered single sapling period delay in eq. (8) corresponds to the fact that the Siulink RTW odel always transits the output force first, and processes the received input later in the sae control cycle [3]. With these assuptions, the piecewise linear syste of equations (2) and (8) can e solved for the consecutive sapling instant, and a linear apping can e created in the for z Az (9) j + 1 = j, where z j = x j x & j t xj 1 xj 2 is the discrete state vector [7,8], and the leading atrix ε 1 1 ε 1 ε ( d + p) d β β A =0 1 εβ ε( d + p) εd (10) is called the (state) transition atrix. Here, β ε = (1 e )/ β, and p = k t 2 /, d = k t / and β = t / are the diensionless proportional and differential gains and the diensionless viscous daping, respectively. To otain this copact for we defined the eleents of the discrete state vector with hoogeneous units, i.e., the velocity is scaled y the sapling tie. By using eq. (9) with eq. (10) the tie series of discrete state values can e deterined for any initial condition z 0, and via that the discrete tie dynaics of the syste can e investigated. p T d 468

4 For the proper selection of the control gains the stale doain of operation has to e deterined. Since eq. (9) define a ulti-diensional geoetric series, y using the trial solution z ( t ) = Ke λt, it can e shown that the syste is asyptotically stale if, and only if, the agnitude of largest eigenvalue λ t µ { λ} = e, Re < 0 of the transition atrix A is less than one. The corresponding staility conditions can analytically further e analyzed y using the Moeius transforation as shown in [9,10]. For revity, here we restrict the investigation to the nuerical coputation of the stale doains. The syste has three control paraeters: p, d and t. With these two practically useful 2D staility charts can e generated. By fixing the sapling tie (to its sallest possile value) the stale doain is represented as a D-shaped area in the p d paraeter plane. In voltage control ode, when there is large internal daping due to the ack EMF effect, a siple proportional controller is also a valid choice. In this case, the staility chart can e plotted in the t k p plane (with diensional paraeters) instead of using the t p plane. These staility charts are illustrated in Fig. 5. and Fig. 6. d STABLE t UNSTABLE s p 1 Fig.5. Diensionless staility chart in p d plane Fig.6. Diensionless staility chart in t k p plane 5. Measureent Results The dynaic analysis presented in the previous section does not consider the effect of the input/output quantization. This effect, however, can have serious effect on the dynaics. Velocity estiation fro sapled and quantized position data introduces an error which results in a noisy signal. To eliinate this noise, filtering is necessary which would introduce tie delay (phase lag) in the syste. In the recent paper, we would like to focus on the possile effects of position quantization only, and we only consider a proportional controller for the experients. The use of advanced velocity estiators and the effect of quantization noise will e the suject of future investigations. The theoretical and experientally identified staility charts are copared in Fig. 7., where t is the sapling tie, and P = k p is the scaled proportional gain that was used during our experients. This scaling includes the position quantization and P has the unit of [N/encoder count]. Note, that the resolution of the otor encoder is 1000 cpr which together with the lead l = 0.02 / revolution gives /count position resolution for the linear otion of the root axis. 469

5 easured signal this results in different effective sapling ties. The average dynaic ehavior ay e approxiated y a odel with two (or ore) sapling tie delays. This was oserved in our experients, where we used the siplest quadrature (detecting the rising edges only) to process the encoder signals. To increase staility for saller sapling ties the encoder resolution can e iproved y quadrature 4X encoding. This is planned as future work. Fig.7. Measured staility chart The results of our initial experients are shown in Fig. 7. Here, the theoretical staility oundary is represented y the solid line and the dots show the liits found experientally at different sapling ties. It can e seen that the easured data fit very well to the theoretical oundary in the range of larger ( s) sapling ties. For saller sapling ties the syste unexpectedly ecae unstale at lower gains. In addition, it was oserved that at t = 1 s and t = 2 s the instaility occurs exactly at the sae gains. These proles could partially e explained y the default settings of the serial counication locks used in our Siulink RTW odel. According to [3], the serial counication lock y default has the axiu sapling frequency of 500 Hz. This explains the sae dynaic ehavior for the experients with 1 s and 2 s sapling ties. For saller sapling ties, the RTW docuentation [3] suggests the use of the option "direct port access". This option is not docuented in detail and the possile iproveent is said to e hardware dependent. Using the direct port access option the experiental results are shown in Fig. 8. Fro this figure it is apparent that staility was iproved in the range 4 s > t > 2 s. The points elow t = 2 s see to fit to a different staility oundary corresponding to the sae PD controller with two sapling tie delays. This additional delay ay e explained y the effect of quantization of the otor encoder. At high sapling rates, the controller ay read the sae position data repeatedly while the output force reains unchanged. Depending on the 470 Fig.8. Measured staility chart (with direct port access) 5. Conclusions In this paper the effect of sapling and quantization was investigated on the syste staility in case of a single axis root sujected to PD digital position control. The theoretical results were copared to experients conducted on a device with a icrocontroller ased control unit that interfaces the root to a high-level Siulink RTW controller. In case of the investigated experiental setup, the effect of output (force) quantization and dry friction did not have iportant influence on the syste staility. This ay e explained y the large viscous daping. On the other hand, it was oserved that sapling and input (position) quantization can cause unexpected instaility proles even in case of high sapling frequencies. It is also noted that using a coercial software, like Siulink Real-Tie Workshop, the default options, and the applied real-tie update echanis have iportant effects on syste staility which requires careful testing and analysis.

6 Biliography [1] Csernák, G., Stépán, G.: Sapling and Round-off, as Sources of Chaos in PD-controlled Systes, Proceedings of The 19th Mediterranean Conference on Control and Autoation. Corfu, Greek, , pp , 2011 [2] Csernák, G., Stépán, G.: Digital control as source of chaotic ehavior, International Journal of Bifurcation and Chaos 20:(5) pp , 2010 [3] MathWorks: Siulink Coder (forerly Real-Tie Workshop), URL: products/siulink-coder/, [4] Petrik O., Hua A., Szász G.: Rendszertechnika Budapest, Tankönyvkiadó, 1986 (in Hungarian) [5] Stépán, G.: Retarded Dynaical Syste, Longan, Harlow, 1989 [6] Insperger T., Stépán G.: Sei-Discretization for Tie-Delay Systes, Budapest, Springer, 2011 [7] Kovács L.L., Kövecses J., Stépán, G.: Analysis of effects of differential gain on dynaic staility of digital force control, International Journal of Non-Linear Mechanics, Elsevier, vol. 43, , 2008 [8] Insperger, T., Kovács, L.L., Galaos, P., Stépán, G.: Increasing the accuracy of digital force control process using the act-and-wait concept, IEEE/ASME Transactions on Mechatronics, 15(2), pp , 2010 [9] Magyar, B., Hős Cs., Stépán G.: Influence of Control Valve Delay and Dead Zone on the Staility of a Siple Hydraulic Positioning Syste, Matheatical Proles in Engineering 11: 15 p. Paper , 2010 [10] Stépán, G., Steven, A., Maunder, L.: Design principles of digitally controlled roots, Mech. Mach. Theory 25, , 1990 Authors MSc Eng Csaa Budai Dept. of Mechatronics, Optics and Mechanical Engineering Inforatics, Budapest University of Technology and Econoics, H-1521 Budapest, Hungary tel.: (+36-1) e-ail: udaicsaa@ogi.e.hu László L. Kovács Dr HAS-BUTE Research Group on Dynaics of Machines and Vehicles, H-1521 Budapest, Hungary tel.: (+36-1) e-ail: kovacs@.e.hu 471