This paper describes and complements a poster presentation where different algorithms are compared from experimental results. Interest is focused on the robust behavior of simple algorithms used for the motion control of joints in robots.Peer ReviewedPostprint (published version

Bertran Albertí, Eduardo

Munilla, I.

UPCommons. Portal del coneixement obert de la UPC

) .. ·"" . . . . . . ' . . ' . . ·:· .. ·' · ~ . .... . - . . • l . ' • 1,". ~JJll • ~ • • , I , D~ ' ~ L a !· v . . . •no-sity 'lo.J, for York, 'ering >nic 3, COMPARATIVE STUDY OF ALGORITHMS FOR THE MOTION CONTROL OF A ROBOT ARM Munilla, I., Bertran, E. E.T . S . E . TELECOMUNICACID UN!VERSITAT POLITECNICA DE CATALUNYA APT . 30.002 08080 - Barcelona <Spain> Abstract: This paper describes and complements a poster where different algorithms are compared from results. Interest is focused on the robust simple algorithms used for the motion control robots. presentation experimental behaviour of of joints in 1. INTRODUCTION 'Within the functional structure of smart robots there are several elements whose good autonomous performance should be warranted. This is the case of robot arm joints. Decisions about the needed behaviour of the robot arm are taken depending on environment information processed from the outputs of a set of sensors. For instance, vision involves image processing in order to identify objects as well as obstacles within the work area . Scene generation is a slow process. Therefore, the strongly depends trajectory planning on the _ hipothesis that joint movements closely follow the elaborated set points <Lee, 1984). Transient inaCcuracies, while the arm is following any desired trajectory could alter the configuration of the previously seen scene and should not be allowed. Variable load conditions at each joint and ~p constraints in velocity (sampling rate> should also be consl dPred . 1 t h"s bP.en proposed the use of adaptive control algorithms or, alternatively, the use of robust control algorithms, the latter being a compromise between performance and complexity <Luh, 1983; Vukobratovic, 1984) . Some different strategies for adaptive another Congress control are exposed paper presented in <Bertran, 1986> in this In this paper the characteristics of different ~p based control algorithms are compared. At first, robustness is defined in the context of manipulator movements. Then, the performances of PID-like, variable structure and LQ optimal controllers are analized . Sensor needs and behaviour degradation as a sampling time !onction are specially studied . Finally, from a comparative study, a robust controller proposed. structure is These algorithms have also been tested in de servomotors used to drive an experimental robot arm. The importance of degraded conditions has been studied . Experimental results re-enfo1-ce theoretical previsions. In special, it is concluded that the use of partial coupling between control e.lgorithms of the: joiuts is advisory in order to obtain some interesting fault tolerance properties. 663 664 2. ROBUST CONTROL ~NO SENSOR F~lLURES In this section conditions to have a robust control algorithm, capabl~ to support same sensor failures, ar~ studied. Robustness is d~fined as the capability to hold selected properties of the system in spite of the pres.:nce of a cla=>s of perturbat lons <Ack~rmannI 1980>. The perturbations of interest in the case of a robot joint control are mainly inertia variations and sensor failures. The property to be held is the dynamical behaviour of the arm and, specially, its stability margin. DC JDIHT WDR n a ull\ Figure 1 The de servomotor for robot joint movement has been modelled. The model is linear and includes the mechanical da10inant time constant, wich varies according to the load and the arm position, and the electrical non-dominant one. lt has been considered the presence of po~itian and speed sensors, and the model is formulated in sensor coordinates. So it results a third order discrete model with partial state measurements <Fig ll. For sake of simplicity ·af the algorithms, only partial state constant feedback performances are analized. Parameter design equations corresponding to the controlled system characteristic polynomial: z '' + paz'"' + P• z + po = 0 ar-e: p' p • : Kv K [ e l< p {- T / T ... ) -~xp<JTLT .. l 1 /( 1',..-T-1 + K,. K ( T,.,-T •. + T" e>epC-T/T,. l ~TI.I a:<p{-T/T,.,)1/(T,.,-T, ... 1 t exp (-T/T, .. ) +e:<p <-T/T ., l + exp <- T/T , .. l .;,xp < -T/T ., l p .. c -Kv K (e>ep(-T/T,.,l-exp<JTLT~lNLETIJT_N + K"" K EI.~ ex:p<-T/T,.,l 1',,. e:<p(-T/T .. l+ (T,,.-T <.,) e:<pC-T/T,,.l e:<p<JTLT~FF / /( T .,., - T .,1 exp<-T/T.,) exp<-T/T-J Therefore p~ is ind~pendent of the values of K~ and K.,. Loci of stable performance <mapping the 1z1=1 contour of the z-plane> have been plotted in the <pu,po> coefficient plane <Fig. 2l far two different situations : al Sampling time: T 0. 2 T,., bl Sampling time: T « T,., Figure 2 Due to the variations of T,.., a) and bl should be considered as the limit cases far actual situation joint. [n bath cases considered that T w ,._. it in the is also Stability loci can be transferred to I I l I l I I I I I the 3>. <K, us r-e tat ia1 K, . with From Fit! be sele.: compromi charact.: <shaded choose stabillt sen-5ar f sensor syst~m. A way tc position ..-ecanstr still rue a< ThL3 rec of an ap values the ~yst fai lur~. 1nc ve rne n t . . . z, ·.: ... _.·,.·.--,:.: . .-:.; ... ~. : .... ·, .. : · .. ·.·:._, · .. ·.. . . . :: ,·.. .. . ..:· ... . , .... ·. ; .. .. ;. ~·"~:~l: .t + the <K~.I K~> 3>, using relations: parameter plane the simplified <Fig. design K~ K ~ - p.;:o T,.,/[ T,.,-TJ K, ... K '>' (T.,,/T> [p ... +p,+T/T.,,-1) with P .o· ~ -2 + T/T,., Figure 3 From Fig. 3 values for Kv and K~ be selected that represent a can eo; cod compromise between dynamic characteristics and T- variations <shaded area>. Also it is posible to choose values in order to hold stability in front of the speed sensor failure . However , the position sensor system. failure unstabilizes the A way to stabilize the system if the position sensor f6ils is to reconstruct the position <9> from the still measured speed values <n>: B<kT> = T n<kT> + 91[k-llT > This reconstruction and the selection of an appropiate pair of emergency values for Kv and ~J eau stabilize the syste1u ~ uJuo:diately after such "' failure, avoidiug und<=sired joint movement::. . In Fig. 3, the intersection of K~=O and the stability zone decreases from a> to b>. Therefore, using incomplete state feedback, when a lower sampling time is chosen a poor robustness of the system can result. This seems a paradoxical conclusion, and means a compromise between system stability in front of structural variations <T of moderate to high value> and the classical system stability in normal conditions operating with its nominal parameters <T of low value>. 3. COMPARATIVE STUDY OF ALGORITHMS 3. 1. Hardware implementation As is shown in the top of the poster, the equipment used to experiment the different control algorithms is composed by: - A )JC AD'[ 65. Interface hardware: 8-bit A/D & D/A converters, multiplexers <Bertran, 1983>. - Drivers for the joint motors. 3.2. PID regulator Tba ro;,gulator experimented firstly, in order to have a reference for the co1nparison of results, has been the well known PID regulator, using the Ziegler-N!chols rules to tune the reeulator parameters. The use of e~npi ~ ical rules, longer experience and noinimum sensor needs are the main advantages of this regulator. The presence of a pole in the of the s-plane in each joint makes unnecesary the effect regulator. On the other band, origin model, of the the D .:fiect is normally no required due to the dominance of a single pole < 1 r.troduced by the mechanical This regulator is poor in order to compensate non-linear effects in the joint. Dead-bands due to the de motor ar.d mechanical linkages are traduced to sleadyJs~ate positional errors. 665 666 The relnforcem~nt of th~ I ~ffect ln order to try to reduce these errors produces a limitJcycl~I with mean value ~qual to the desired fin~l position <see poster>. In spite that it has been used the first-order d it" ferenc~ method <Houpis, 1985> to obtain a discrete PlO regulator Cf.LC control>, and that this method ls theoretically restricted to systems with a sampling period that locates the poles and zeros in a zone around ~PM° from the positive real a~is in the z-plane, this regulator has demonstrated a good robustness when sampling period is increased over T , ., <see poster>. 3 . 3. Optimal Linear Regulator This kind of regulator has higher sensor needs, due to it's based on a variable-state feedback. As it is shown in the poster, measured variables have been position and speed. As much with this choice as if the speed sensor is substituted by a de motor current one, this regulator has a certain degree of fault-tolerance: If some positional loop element is damaged, the joint do not becomes an integrator <the behaviour of the open loop motor> as in the PID regulator case, because it is very easy to detect the presence of speed or current variations without positional ones. In this case, a fast solution is to nullify the motor set-points in order to avoid ballistic problems in the arm. On the other hand, the choice of a performance inde~ that wheigts speed and position deviations hides the pole placement. Contradictory results good as between optimization and dynamics have been obtained reflected in the poster. The the typical pole-assignement use is of becomes more attractive designer point of view. from 3.4. Variable structure control method the This kind of regulator has the same sen:,;or needs than the Optimal Linear R~gul.:>tor. Th,.. c ontrol dl.3cantt.nuous, and depends chosen :Rtructur~ . chasdn dua. l regul,.tor . In our case control The main a~vantI.ge of this t.;ow l ·;; on the we have action control L~ its insensitivity with re,;pect to lnaccuracles in the system model <Young, 1978; Bengia.min, L9d4l. Considering only the mechanical time-constant, the d~cislon about the control signal is taken in function of the measured values of both position and speed. Swltching line is ,;;hewn in Flg. 4.. Figure 4 Experimental results <see poster) have demonstrated a good robustness in front of system parameter and sampling time variations. But if sampl i ng time is increased the control 3Ct1on is taken with a d~lay versus the desired switching l i ne. So an unestable motion could be followed . Another restriction to the use of this regulator is that none of the sensors can fail, because then it is impossible to decide the appropiate control action. 3.4. Sensor failures In front of an speed sensor failure variable structure control should be avoided as emergency algorithm. Then robustness can be warranted at a J a ln t . ..ip[Jrop K,. (Se• The U:.5• sensor c.J.n be s~nsor = miniml r~dund. used order sisnal contra REFERE Ao:!:<:erm o~sign IEEE T Decemb Beng-ia "Varia Contra Ma.gaz i Bertra J .; :1 Contra 'o'irwle 54.-59 . . . . : .~ .. _~ ·.,:· :~:..;; _t; .;. .. ;·~;~J~:D::~~.:~ .. \J.J.::J~:i.;;.:.; · ~; ,.:; \ :~:: .: .. :~~:I~ ._: : . . ~~J ·, DJJ~: :_:, ; ·.·, . ': ; : .. ·: .· . - .. :.· . . : .. ~ . ~· -.. > .-.. D~ _. : -:.' -;. .. ' . ,,; ;:..:~i~!i J~ " • ~ ~ ;, -· ~ • • • • • • ' • : ' • • • ' - ... • • ~ • ' 0 -: - • - • • .• ,, ' ' -. • ()I •• ~ . t I · . . is the ~ve ion 1th tem 4) . cal the ion oth is / / / ' ::ss che ! ay Sa be che of it :~.e •l!1'1 joint controller level by appropiate sampling time T K~ <see section 2>. selecting and gain The use of some degree of redundancy is necessary only in case of position sensor failure. Of course, robustness can be re-enforced if two position sensors are located at each joint . To minimize sensor needs, this redundancy used at can be implemented and a coordination level, in position order to generate a backup signal submitted to the controllers <see poster) . joint REFERENCES Ackermann, J . <1980> "Parameter Space Design of Robust Control Systems" . IEEE Trans. on AC, vel AC-25, n°6, December 1980, pp. 1053-1072. Bengiamin, "Variable Control". Magazine, N.; Kauffmann, Structure IEEE Control August 1984, pp. B. <1984> Position Systems 3-8. Mart! ne:z, L. ; Bertran, E.; J.; Munilla, Control by lolirwless IJorld , Miguel, "Process 54-59. 1. <1983> Personal September Computer" . 1983. PP · Bertran, E.; San-Martin, "Comparative £itudy of Adaptive Strategies for a Control", this volume. M. < 1986> Different Robot Joint Houpis, C.; Lament, G. "Digital Control Systems" . -Hill. 1985 . <1985> McGraw Luh, J.Y.S . 11983> Controller Design Robots- A Tutorial". SMC, vel SMC-13, n°3, "Conventional for Industrial IEEE Trans. on May/ June 1983 pp . 289-316 . Vukobratovic, M.; Stokic.; Kircanski, N. <1984> "Towards Nonadaptive and Adaptive Robots". AC-29, 841-844. Control of IEEE Trans . n°9, September Man ipulat im1 on AC, vel 1984 . pp . Youn8, K. D. <1978> "Controller Design for a Manipulator Using Theory of Variable Structure Systems" . IEEE Trans. on SMC, vel SMC-8, n°2, February 1978 , pp. 101-109. 667 

Comparative study of algorithms for the motion control of a robot arm

Abstract

Similar works

Full text

Available Versions

UPCommons. Portal del coneixement obert de la UPC