Sensory Redundant Parallel Moble Mechansm Shraga Shoval and Moshe Shoham* Department of Industral Engneerng & Management, Academc College of Judea and Samara, Arel., *Faculty of Mechancal Engneerng, Technon, Hafa, Israel Abstract Ths paper presents a novel desgn for a moble robot based on the knematcs of parallel mechansms. The robot conssts of 3 legs, each equpped wth an asynchronous drvng unt. The legs are connected to the drvng unts wth sphercal jonts and to the upper plate wth a revolute jont. Three addtonal encoders, attached to the upper revolute jonts provde redundant data. Ths data s used by a knematc model for accurate estmaton of the robot s confguraton and poston n space, even n rough terrans, where conventonal odometry fals. Smulaton results show the advantages of the desgn, and suggest a method for detecton of rregulartes of surfaces n unknown envronments. 1. Introducton Parallel manpulators consst of several actuators connected n parallel to a base and a movng platform. The structure of parallel manpulators makes them partcularly sutable for applcatons where accuracy, rgdty and hgh pay load to - weght rato are mportant, snce ther stffness and dynamc performance are largely superor to those acheved wth conventonal seral archtectures. The maneuverablty as well as rgdty and accuracy are functons of the number of legs, dmensons of the mechansm, and the type of jonts between the plates and the legs. The basc conceptual mechancs s known as the Stewart Platform [Stewart, 196] (even though an earler desgn was suggested by Gough). Snce then, many manpulators were developed based on ths mechansm [Ftcher and MacDowel 198, Hunt 1983, Yang and Lee 1984, Fchter 1986, Kohl et. al. 1988, Hudgens and Tesat 1988, Tsa and Tahmaseb 1983]. Fg. 1 shows some typcal parallel mechansms named accordng to the number of jonts at each platform. Fgure 1: Three types of parallel mechansms A major drawback of parallel robots s ther lmted workng envelope that makes them unsutable for a wde varety of applcatons. To overcome ths shortcomng, several researchers have suggested usng moble jonts between the legs and the statonary platform, turnng the mechansm nto a sem-moble robot. Such a mechansm developed by Ben Horn and Shoham [1994], s shown n Fg.. It conssts of the followng components: three lnks of fxed length havng a sphercal jont on one end and a revolute jont on the other end, three actuators whch move planarly on a statonary platform and an output platform havng sx degrees-of-freedom (DOF). Fgure : Sem-moble parallel mechansm [Ben Horn and Shoham, 1996] To further ncrease moblty, Ben Horn and Shoham [1999, ] suggest a more flexble desgn. Ths mechansm, shown n fgure 3, s based on 3 nflatable legs, an upper platform and 3 asynchronous drvng unts for the three legs. Ths mechansm turns the robot to an autonomous agent, wth theoretcally unlmted workspace. The upper jont of each leg s a 1
revolute jont, whle the lower jont, whch connects the leg to the drvng unt, s a sphercal jont. Ths confguraton offers sx DOF for the upper plate where the control parameters are the postons (X,Y) of the three drvng unts. The mechansm s desgned for applcatons that requre a lght weght and easy deployable robot. Gven the requred trajectory for the TCP (6 parameters) the nverse knematcs model can generate the requred path of each drvng unt, subject to ts non-holonomc constrants. constrants and Secton gves concludng remarks.. Dead reckonng for parallel moble mechansm Fg. 4 s a schematc descrpton of the parallel moble mechansm. The upper plate s connected to each of the three legs wth revolute jonts. These three legs are drven by three asynchronous unts that are connected to the legs wth sphercal jonts. Controlled moton of the three drvng unts determnes the pose (poston and orentaton) of the upper plate. Z T Upper plate Revolute jonts Y p r Sphercal jonts (X3,Y3) (X,Y) (X1,Y Fgure 4: Schematc descrpton of the moble parallel mechansm X Fgure 3: The nflatable moble parallel robot [Ben Horn and Shoham, ] The robot s poston can be determned ether by absolute methods (trangulaton, map matchng GPS etc.) or by a relatve method (.e. odometry). In order to acheve maxmum autonomy, the use of absolute postonng should be restrcted to the begnnng of moton and to the nstances where relatve postonng s not suffcently accurate. In ths paper we propose a modfed knematc desgn of the parallel moble robot whch allows for mproved poston estmate based on odometry, and also mproves both the accuracy of the robot and ts robustness to external dsturbances due to slppage of the drvng wheels, uneven surfaces, bumps holes etc. In Secton we descrbe the problem of dead reckonng for the parallel moble robot. Secton 3 detals the mathematcal and geometrcal model of the robot. Secton 4 shows smulaton results of the robot operatng under varous moton Gven the coordnates of pont T on the upper plate, the poston of the three drvng unts (X 1,Y 1, X,Y and X 3,Y 3 ) s determned by the nverse knematcs model (see Ben-Horn et. al. [] for a detaled soluton of the path when non-holonomc constrants are consdered). The lateral poston of the drvng unts can then be determned ether by odometry or trangulaton. Usng trangulaton for each drvng unt requres preparaton of the surroundngs that also restrcts the moton space. Also, trangulaton can become complex, due to possble nterference of the legs wth the trangulaton procedure by obstructng the beacons. Usng odometry, on the other hand, provdes full autonomy for the robot, wth no preparaton or pre-knowledge of the envronment. However, odometry generates unbounded errors that reduce ts effectveness over relatvely long travel ranges. The presence of bumps, holes, or slppage n one or more of the drvng wheels can create unpredctable errors that ncrease wth travel dstance. For example, f one of the drvng unts s subjected to angular error of 1 o (due to slppage of one drvng wheel), a travel of 9ft generates a lateral error of 1.ft n the poston of the upper plate. Ths error also affects the orentaton of the upper plate as shown n
Fg.. In ths example the lmbs length s ft, and the upper plate s an equlateral trangle of ft. At the end of the 9ft trajectory the upper plate has an angular error of.6 o. Ths error ncreases wth travel untl the robot stops, reaches a sngularty or other undesrable confguraton. 3. The mechancal and geometrc model of the modfed desgn Fg. 6 shows schematc descrpton of the moble parallel robot already shown n Fg. 4. However, ths desgn ncludes 3 addtonal encoders attached to the upper revolute jonts, to measure the rotaton angle between the upper plate and the legs. These angles are - η 1, η, η 3 correspondng to the three legs 1, and 3. Z v w u E3 E1 η3 E η1 η Y D1 1 o 9 ft D3 D X Fgure 6: The modfed desgn of the robot Fgure : The effect of 1 o angular offset on the robot confguraton To determne the accurate confguraton of the upper plate, the absolute poston of each drvng unt, as well as the drect knematc model are requred. Tahmaseb and Tsa [1994] show that the above parallel mechansm has 16 possble drect knematc solutons, whch requre extensve computatonal effort. Furthermore, determnng the absolute poston of each drvng unt s subject to odometrc errors and cannot provde a relable poston estmate. In the followng secton we propose a new approach to the desgn and moton control of the parallel moble robot that reles on odometrc data and on addtonal measurements taken from on board encoders. It requres mnor mechancal modfcatons to the orgnal desgn, and suggests a geometrc model for the new desgn. Ths approach does not requre prelmnary setup of the envronment, allowng a full autonomous moton. Furthermore, our approach detects odometrc errors n real tme, and suggests a fast procedure to correct these errors before affectng the overall robot confguraton. The poston of the three drve unts (U 1,V 1,W 1, U,V,W and U 3,V 3,W 3 ) can be derved n the U-V-W coordnate system (attached to the center of the upper plate) accordng to Eq (1-3) [Tahmaseb and Tsa, 1994]: p U = r cos α cosη + cosα Eq ( 3 p V = r sn α cosη + snα Eq () 3 W = r snη Eq (3) where π π α = + ( + 3 p the length of the edge of the equlateral trangle constructng the upper plate. r the length of the legs ndex of the legs n a cyclc order Based on the poston of the drve unts as determned n the upper plate coordnate system, the Eucldean dstances between the drve unts l 1, l and l 3 s gven by Eq (4): l Eq (4) = ( U 1 U + + ( V 1 V + + ( W 1 W + ( n cyclc order) Returnng to the world coordnate system (X,Y,Z), the poston of the drve unts s determned by the odometrc system where 3
D 1 =(X 1,Y 1,Z 1 ), D =(X,Y,ZDDDand D 3 =(X 3,Y 3,Z 3 ). The dstances between the drve unts can also be derved accordng to Eq (): l Eq () = ( X 1 X + + ( Y 1 Y+ + ( Z 1 Z + If the odometrc system s accurate, the dstances derved n the upper plate coordnate system n Eq (4) s equal to the dstances derved n the world coordnate system accordng to Eq (). If, however, these dstances are dfferent and assumng that the encoder readngs of the revolute angles are accurate, then the odometrc calculaton s faulty. Before proceedng wth the rest of the calculatons, let us assume that odometrc errors occur wth a sngle drvng unt at a tme. For example, f the robot s travelng over slppery terran, the momentary effect s on a sngle drve unt (the unt that drove over the slppery spot). Ths assumpton s realstc when the samplng rate s fast enough as shown by Borensten [199]. In hs work, Borensten shows that the effect of odometrc errors on a 4 DOF planner moble robot can be detected separately for each drvng unt, then corrected before the overall robot confguraton s affected. Accordng to ths assumpton, errors n the odometrc procedure of each drvng unt can be detected and corrected n real tme by comparng the dstances calculated by the X-Y-Z and U-V-W coordnate systems. An odometrc error n one drve unt affects two dstances accordng to Eq (). Fg. 7 shows a case where the poston of two drvng unts D 1 and D are accurately determned, but the poston of D 3 as determned by odomety has an offset l. Ths offset affects the values of two dstances: l 1 (dstance between D 1 and D ) and l (dstance between D 1 and D 3 ). The length of l 3 s not effected by the faulty poston estmate of D 3 (see Eq (). The real locaton of drve unt 3 s gven by the ntersecton of the two crcles - C 1 and C where the radus for C 1 s l 1 / and for C s l /. Gven that the centers of these crcles - D 1 and D are known, the locaton of D 3 can be calculated. If the above procedure s performed at relatvely short ntervals (.e. every mllseconds) the postons of the three drvng unts reman accurate. As mentoned, ths procedure assumes that there s a sngle odometrc error at a tme. In the unlkely event that two (or three) drvng unts are subjected to odometrc errors smultaneously, the proposed procedure cannot be mplemented and addtonal measures must be taken (.e. re-calbraton of the robot s poston). C D l o3 =l t3 l t1 l o1 l o D 1 l t C 1 D 3 -actual D 3 -odometry Fgure 7: Determnng the poston of D 3 by the ntersecton of C and C 1 4. Smulaton results To nvestgate the valdty of our approach, a smulaton was developed. The smulaton determnes the angles between the three legs and the upper platform (η 1, η, η 3 ) based on the actual poston of the three drvng unts (n a real system these angles are measured by the encoders attached to the revolute jonts). The smulaton then calculates these angles based on the odometrc nformaton gven by each drvng unt. Based on the dfferences of these calculatons the smulaton then corrects the estmated poston of each drvng unt and updates the robot s overall model to determne the poston and orentaton of the upper platform. A random nose s added to the calculaton of the angles η 1, η and η 3, smlar to nose expected from the encoders. The nose n the real system s due to encoder resoluton and backlash. In the smulaton shown n ths secton, random nose s n the order of ±1 o. The addton of the ±1 o ensures that the smulaton closely matches the performance expected of a real system. In the frst experment, an ntal odometrc error of o s ntroduced to one of the drvng unts. Ths error adds a constant value to the ntal orentaton of that drvng unt, resultng n a constant growng lateral error, smlar to the example shown n Fg.. The robot s travelng for m at a speed of.66 m/sec wth a samplng rate of Hz. As shown n Fg. 8, the odometrc path does not detect the ntal orentaton error 4
whle the path calculated by the algorthm s close to the actual path of the drvng unt. Y [m] 1 Path of the drvng unt 1 X [m] Odometry Calculated Fgure 8: Path of the drvng unt subjected to ntal orentaton error. In the second smulaton, an odometrc error s generated after 1 seconds. Ths causes a lateral error n one of the drvng unts of.cm nto the center of the robot. Such an error can be generated by slppage of the wheels on wet a surface, movng over bumps or by an external dsturbance applyng an unexpected force on the drvng unt. As shown n Fg. 9a, the odometrc error s mmedately detected by changes n the angle η 1 between leg #1 and the upper platform, and the robot s confguraton s updated accordng to that change. Fgure 9b shows an estmate of the path of drve unt #1 based on odometry alone, and accordng to the suggested sensory-redundant knematc model correcton. As shown, odometry does not detect the error n the poston of leg #1, whle the suggested model detects the odometrc error (based on the unexpected changes n η 1 ) and updates the poston of leg #1 to ts correct value. Ths smulaton shows an mportant feature of the knematc model. In addton to the detecton of the odometrc error, the type and source of the error can be predcted. For example, a sudden change n η 1, (as s the case of ths experment) ndcates that the odometrc error pushed leg #1 nto the center of the robot s structure. Ths mght be the case when leg #1 travels over a bump, s pushed nwards by an external force. A sudden ncrease n η 1 ndcates that leg #1 s pushed away from the center of the robot, ether by external dsturbance, slppage, or by travelng over a hole. Angle [deg] Revolute angle -η 1 [deg] 9 8 7 6 4 1 3 Tme [sev] Y [m] a Path of the drvng unt 1 1 X [m] Odometry Calculated b Fgure 9: Change n η 1 due to odometrc error and ts effect of the path of the drvng unt. Fnally, a smulaton where all drvng unts are subjected to random odometrc errors s shown n Fg.. Ths example s typcal of moton over uneven terrans (.e. paved roads, grass, sand etc), where all drvng unts are contnuously subjected to random dsturbances. Dsturbances cause odometrc errors of up to 1 cm/sec for each drvng unt. All drvng unts are subjected to the dsturbances smultaneously, causng a maxmum lateral error of 1.19 m on the XY plane, and.19 m along the YZ plane. The system detects these odometrc errors as soon as they occur and updates the robot confguraton. If, for example, the robot s msson s to travel along a straght lne, these updates can be ntegrated nto the path plannng and control system. The control system detects the odometrc errors and compensates for them before a sgnfcant lateral error s developed. Y [m] 1 Actual and desgned moton of the three drvng untes 1 X [m] Fgure : Smulaton of random odometrc error for three drvng unts smultaneously
. Conclusons A novel desgn for a moble robot s presented. The knematc desgn combnes technques of parallel mechansms wth conventonal wheeled unts. The robot conssts of three legs, each drven by an asynchronous mechansm connected to the legs wth a sphercal jont. Each leg s also connected to an upper platform wth a revolute jont, resultng n a moble, sx DOF, parallel mechansm. Addtonal encoders measurng the revolute angle of the upper jonts, provde data used by a knematc model to detect and correct postonng errors generated by odometry. Early detecton and correcton of odometrc errors n each leg prevent sgnfcant errors of the upper plate, allowng an autonomous, more accurate and relable moton even on uneven surfaces, where odometry alone generates unbounded poston error. The features of the knematc model can be used for dentfcaton of rregulartes n the envronment. For example, when an odometrc error n detected n a specfc leg (by unexpected changes n the readng of the encoder attached to the upper revolute jont), the source of the odometrc error can be determned. Furthermore, the cause of the error, n terms of bumps or holes, nward or outward slppage and the drecton of external dsturbance, can also be dentfed. 6. References 1. Ben Horn (Dombak) P., 1999, Analyss and Synthess of an Inflatable Parallel Robot, M.Sc. Thess, Technon, Hafa.. Ben Horn (Dombak) P., Shoham, M., and Grossman, G., A Parallel Sx Degrees of-freedom Inflatable Robot, ASME Mechansm and Robotcs Conference, Washngton,. 3. Ben Horn R., Shoham M., Djerass S., 1998, Knematcs, Dynamcs and Constructon of a Planary Actuated Robot Robotcs and Computer-Integrated Manufacturng, Vol, 14, No., pp-163-17. 4. Borensten J., 199, Internal Correcton of Dead-Reckonng Errors wth a Dual Drve Complant Lnkage Moble Robot, Journal of Robotcs Research, Vol. 1, No. 4, pp. 7-73.. Ftcher E.F., 1986, A Stewart Platform-Based Manpulator: General Theory and Practcal Constructon, Internatonal Journal of Robotcs Research, Vol., pp. 17-18. 6. Ftcher E.F., and MacDowel E.D., 198, A Novel Desgn for a Robot Arm, Proceedngs Internatonal Computer Technology, ASME, New York, pp -6. 7. Hudgens J.C., Tesar D., 1988, A Fully-Parallel Sx Degrees-of-Freedom Manpulator: Knematc and Dynamc Model, Trends and Developments n Mechansms, Machnes and Robotcs, Proceedngs of the th Bennal Mechansm Conference, ASME, New York, Vol. 1, No. 3, pp. 9-37. 8. Hunt K.H., 1983, Structural Knematcs of In-Parallel-Actuated Robot Arm, ASME Journal of Mechansms, Transmssons and Automaton n Desgn, Vol., pp. 7-71. 9. Kohl D., Lee S. H., Tsa K. Y., Sandor G. N., 1988, Manpulator Confguraton Based on Rotary-Lnear (R-L) Actuators and Ther Drect and Inverse Knematcs, ASME Journal of Mechansms, Transmssons and Automaton n Desgn, Vol. 1, pp. 397-44.. Stewart D., 196, A Platform wth Sx Degrees of Freedom, Proceedngs of Insttute of Mechancal Engneerng, London England, Vol. 18, pp. 371-386. 11. Tahmseb F., Tsa L.W., 1994, Closed-Form Drect Knematcs Soluton of a New Parallel Mnmanpulator, Transactons of the ASME, Vol. 116, pp. 1141-1147. 1. Tsa L.W., Tahmaseb F., 1983, Synthess and Analyss of a New Class of Sx Degree-of-Freedom Parallel Mnmanpulators, Journal of Robotc Research, Vol., pp. 61-8. 13. Yang D.C., and Lee T.W., 1984, Feasblty Study of a Platform Type of Robotc Manpulator from a Knematc Vewpont, ASME Journal of Mechansms, Transmssons and Automaton n Desgn, Vol. 6, pp. 191-198. 6