Towards a dynamic actuator model for a hexapod robot

Transcription

1 Towrds dynmic ctutor model for hexpod robot Dve McMordie, Chris Prhcs nd Mrtin Buehler mcmordie cprhcs Ambultory obotics Lbortory, Centre for Intelligent Mchines, McGill University, Montrel, CANADA ABSTACT We describe model predicting the output torque of the bttery-mplifier-ctutor-ger combintion used on the hexpod robot Hex, bsed on requested PWM (Pulse- Width-Modultion) duty cycle to the mplifier, bttery voltge, nd speed. The model is broken into independent components, ech experimentlly vlidted: power source (bttery), mplifier,, nd (plnetry) ger. The resulting ggregte model shows <6 % Full Scle MS error in predicting output torque in the first qudrnt of opertion (positive torques). Understnding the key ingredients nd the ttinble ccurcies of torque production models in our commonly used bttery-mplifier-ctutor-ger combintions is criticl for mobile robots, in order to minimize sensing, nd thus spce, size, weight, power consumption, filure rte, nd cost of mobile robots. Keywords DC permnent mgnet brush, PWM, plnetry ger, current, voltge, model, hexpod, robot I. Introduction Hex is simple nd highly mobile hexpod robot []driven by six Mxon E brush-type W DC s [3], combined with Mxon :1 plnetry gers, ech plced t the hip of complint fibreglss leg. Hex s suite of dynmic behviours includes running speeds up to. m/s [], climbing up to 3 degree inclines, bounding [6], pronking [7][8][9] nd stir climbing [1][11][1]. Currently, ech behviour exploits proportionl-derivtive control of the legs to trck position nd velocity trjectories generted by clock-driven stte mchine. While this prdigm hs resulted in impressive open-loop behviours, precise control of leg torques my enble enhnced stbility nd efficiency in certin behviours vi online feedbck control. Torque control for ech leg ctutor my be chieved in one of three wys: (1) vi feedbck from torque cell plced between the ger output shft nd the leg ttchment; () vi current sensing in the mplifier nd (3) vi estimtion bsed on speed nd mplifier duty cycle. Since solutions (1) nd () require dditionl design complexity nd expense due to sensor integrtion, it is nturl to ttempt (3) first. To this end we present simple model for the reltionship between speed, mplifier duty fctor nd output torque, bsed on collection of models of ech component of the ctutor system. II. Modelling Approch Physics-bsed nlytic models hve the dvntge tht they provide modulr, provble hypotheses bout the opertion of the ctutor, which cn led both to improved model ccurcy, nd insights into how to improve the ctutor. Wht s more, nlytic models re typiclly much lower-dimension thn utomticlly generted purely numericl or computtionl models, providing simpler implementtion in softwre. In this section, we propose simple physics-bsed models for ech system component, which will then be evluted experimentlly in the subsequent section. Vb Bttery Model Amplifier Model Motor Model b mp L Ib Vs Vmp d V I ξ Ger Model efficiency: η η(, ω) 33.6:1 plnetry ger Figure 1: Compound ctutor model A single bttery-mplifier- combintion my be represented by bove electricl circuit. To obtin the complete model for torque s function of speed nd duty cycle, the bove circuit ws broken into the three simpler models shown bove, ech of which ws fit nd vlidted with experimentl dt. Bttery Model The bttery model shown bove is representtive of the ctutor system found in the robot. This simple internl resistnce model hs been used during testing nd shows tht the bttery hs the behviour of Thévenin-equivlent voltge source for short time intervls during dischrge. However, this model cnnot be used to relibly predict bttery current bsed on voltge (or vice-vers) since the internl resistnce of the bttery ppers to chnge over the course of dischrge. For the purpose of the ggregte model, bttery internl resistnce nd internl voltge re ignored becuse circuitry onbord Hex mesures V s nd I b, removing the need to estimte V s. The foregoing rgument suggests tht models generted with fixed supply voltge will generlize to vrible bttery voltges in the robot, so long s the bttery terminl voltge cn be mesured nd used in the model. While n estimte of I b is currently vilble on Hex, it is not necessrily desirble to depend on this

2 eport Documenttion Pge Form Approved OMB No Public reporting burden for the collection of informtion is estimted to verge 1 hour per response, including the time for reviewing instructions, serching existing dt sources, gthering nd mintining the dt needed, nd completing nd reviewing the collection of informtion. Send comments regrding this burden estimte or ny other spect of this collection of informtion, including suggestions for reducing this burden, to Wshington Hedqurters Services, Directorte for Informtion Opertions nd eports, 11 Jefferson Dvis Highwy, Suite 1, Arlington VA -3. espondents should be wre tht notwithstnding ny other provision of lw, no person shll be subject to penlty for filing to comply with collection of informtion if it does not disply currently vlid OMB control number. 1. EPOT DATE. EPOT TYPE 3. DATES COVEED -. TITLE AND SUBTITLE Towrds Dynmic Actutor Model for Hexpod obot. CONTACT NUMBE b. GANT NUMBE c. POGAM ELEMENT NUMBE 6. AUTHO(S) d. POJECT NUMBE e. TASK NUMBE f. WOK UNIT NUMBE 7. PEFOMING OGANIZATION NAME(S) AND ADDESS(ES) Defense Advnced eserch Projects Agency,371 North Firfx Drive,Arlington,VA, PEFOMING OGANIZATION EPOT NUMBE 9. SPONSOING/MONITOING AGENCY NAME(S) AND ADDESS(ES) 1. SPONSO/MONITO S ACONYM(S) 1. DISTIBUTION/AVAILABILITY STATEMENT Approved for public relese; distribution unlimited 13. SUPPLEMENTAY NOTES The originl document contins color imges. 1. ABSTACT see report 1. SUBJECT TEMS 11. SPONSO/MONITO S EPOT NUMBE(S) 16. SECUITY CLASSIFICATION OF: 17. LIMITATION OF ABSTACT. EPOT unclssified b. ABSTACT unclssified c. THIS PAGE unclssified 18. NUMBE OF PAGES 19. NAME OF ESPONSIBLE PESON Stndrd Form 98 (ev. 8-98) Prescribed by ANSI Std Z39-18

3 mesurement, nd hence it is substituted for in the derivtion tht follows. Amplifier Model Gte drive nd Control Logic commnd voltge Amplifier Model 16 uh 16 uh uf uf V Motor Model Figure : Detil of mplifier nd Figure shows the electricl connection of the PWM mplifier (Apex Microtechnology SA6 [1]) nd. Normlly the PWM mplifier would be modelled together with the, s the winding inductnce of the is necessry prt of the step down converter circuit, filtering out the PWM crrier frequency to produce smooth voltge output. However, in this ppliction second-order LC lowpss filter with roll-off frequency 8. khz strongly ttenutes the 1 khz PWM crrier component, leving only smooth voltge wveform proportionl to the modulting signl, d, (eq 1) nd supply voltge t the terminls. As result of this filter, the PWM switching wveform is completely removed from the output, nd the system my be modelled s n idel trnsformer whose output voltge is governed by the buck converter eqution []: V d Vmp, d [-1,1] (1) Assuming no power loss cross the idel trnsformer, we cn write I b V mp V I or V d V, mp where d is the duty fctor. Using this equlity, n expression my be derived for the terminl voltge in terms of the source voltge, terminl current nd duty cycle: V d Vmp d (Vs I b mp ) () V d (V d I ) (b) s Motor Model Perhps the simplest model for brush-type DC permnent mgnet is the liner circuit model, derived by mens of Kirchoff s voltge lw: di (3) V ξ L I dt In ddition, the bck emf, ξ, nd output torque,, s function of rmture current re given by: ξ K () K TI (b) Eq. 3 is further simplified by neglecting di nd substituting for bck emf, simplifiction tht relies on the smll winding inductnce (1 µh), nd on the low frequency commnd signl: V () I mp I L L dt ξ By combining the bove equtions, n expression for the terminl current in terms of source voltge, speed nd duty cycle is found: (6) I d mp Output torque, bsed on rmture current, is simply modeled s: (7) K ti K t d Ger Model Though more sophisticted models exist for plnetry gers, we begin with the simplest model likely to produce ccurte prediction of torque: N η (8) lod ω ωlod N where N is the ger rtio nd η is the ger efficiency. Bsed on eq s. 1-6, the complete model predicting output torque s function of commnd signl nd speed is given by: (9) N η K t d mp Though this model undoubtedly neglects certin effects t ech stge, it is simple, requires little sensing, nd my be esily implemented in softwre. Before proceeding with more detiled modeling it is instructive to exmine the performnce of this simple model. III. Dynmometer Overview Figure 3: Dynmometer test setup To fcilitte testing of s under dynmiclly relistic conditions, the uthors constructed dynmometer consisting of the ctutor system under test, torque nd speed mesurement pprtus, nd lod cpble of torques nd speeds well beyond the rnge of the device under test (DUT). The ctutor system ws fully instrumented for mesurement of supply voltge nd current, terminl voltge, current nd cse temperture using Ntionl Instruments PCI-636E 16-bit dt cquisition crd. mp

4 Tc o r e Tc s e _ lo d Using Ntionl Instruments LbVIEW progrm ws developed to perform mesurements, proportionl-integrl (PI) control of lod speed nd control of the device-under-test commnd signl t rte of Hz. Accurcies for ech mesurement device used in this set-up re given in Tble 1. IV. Model Vlidtion Figure 6 shows bttery voltge during the pronking git with the results of Thévenin model overlid. The source voltge nd internl resistnce were fit using the MATLAB function polyfit. Bttery voltge during pronking -- ctul nd predicted bsed on bttery current DUT Power Supply Lod Power Supply 3 Actul Liner model Hex Motor Driver Bord v 1.3 v i DUT Tc s e Lod cell conditioning Brekout Box Lod Motor ω E-Series DAQ bord Lod Motor Drive Bttery Voltge Figure : Dynmometer electricl test set-up Device under test (DUT) Adjustble torque rm Figure : Dynmometer mechnicl test setup The physicl design of the dynmometer provides rigid connection between the s nd lod cell. The lod cell is ttched to the DUT by mens of vrible length torque rm. The torque rm is ttched directly to the gerhed, providing direct nd rigid link with which to mesure the rection force t the lod cell. The DUT nd torque rm re mounted on bering, to isolte the lod cell s the sole rection point for output torque. The torque rm length my be vried to suit number of different torque rnges. Tble 1: Instruments used / prmeters mesured Vrible Type Mfr. P/N Accurcy (% FS) ω Encoder Agilent HEDS-.3 % Lod Cell Sensotec 31/13-.% V s V. Divider 1. % I b Current Shunt % I Hll effect C Mgnetics Lod Motor Lod Cell nge of possible lod cell positions C1-3 1 % V V. Divider % t(s) Figure 6: Bttery voltge -- ctul nd predicted bsed on bttery current The models described bove were fit to dynmometer dt from single experimentl run. During this tril, lod speed ws held constnt t rd/s increments between zero nd. rd/s by mens of proportionl-integrl controller. This PI speed controller ws not expected to hold the speed exctly constnt when DUT torque ws high. Insted, speed control ws used merely to ensure complete coverge of the first qudrnt of the torque speed curve. For ech speed, commnded duty cycle ws driven by sinusoid commnd from zero to 1 % nd bck to zero. Model fits were performed on only the first qudrnt of dt (positive torques). As the simple models exhibited slightly lrger errors in the second qudrnt, modeling for this qudrnt will be the subject of future work. Figure 7 shows the region of the torque-speed curve swept during the test. Speed (rd/s) Speed vs. Torque Torque (Nm) Figure 7: Speed versus torque for the DUT We begin by exmining the input-output reltionship of the PWM mplifier described in eqution 1. The SA6 mplifier genertes lockednti-phse PWM signl in response to commnd signl in the pproximte rnge [., 8.] V with the centre roughly t 6. V. To determine the exct offset 3

5 voltge nd scle fctor, s well s to vlidte the buck converter eqution for this circuit, input voltge ws swept cross its rnge while mesuring output voltge. Amplifier Output Voltge (V) Amplifier open-circuit output voltge vs. commnd signl Actul output voltge Estimted ouput voltge Amplifier Commnd Voltge (V) Figure 8: Output of mplifier vs. commnd voltge ctul nd estimted By fitting line to the liner region of the output voltge versus commnd signl dt using the MATLAB function polyfit, the gin nd offset of the mplifier were found to be.776 V nd 9.6 V/V respectively. Figure 8 shows the qulity of the fit; the mplifier ppers to be liner throughout the middle region while diverging slightly from the model close to ± 1 % duty fctor. Terminl voltge during experiment Using the bove mpping, the remining prmeter of the mplifier model (eq ), internl resistnce, ws determined by fitting estimted terminl voltge during n experiment to ctul, using the MATLAB Nelder-Mede minimiztion progrm lsqcurvefit. The minimiztion resulted in MOSFET internl resistnce of. Ohms, close to the Apex SA6 dtsheet vlue of. Ohms [1]. Motor terminl voltge (V) Motor terminl voltge erorr (V) 1 1 Actul nd Estimted Motor Terminl Voltge Actul Estimted Figure 9: Estimted nd ctul terminl voltge Motor current estimtion To clculte the estimted current, the output of eq 3 ws fit to the rel current using ctul terminl voltge nd speed of the, nd by vrying the rmture resistnce,. A vlue of 1.6 Ω ws found, s compred to the dtsheet vlue of 1.33 Ω. This model, together with prmeters listed below, resulted in the following plot of estimted vs. ctul current using eq. to determine the estimted current. Motor Current (A) Est. Motor Current Error (A) 1 1 Actul nd Estimted Motor Current Est current Actul Current Figure 1: Estimted nd ctul current during experiment Output Torque Output torque ws mesured with lod cell positioned on torque rm 87. mm from the xis of the device under test. Using the sme process s for the other models, predicted torque ws fit to ctul torque by djusting the torque constnt, nd ssuming gerhed efficiency of 8 %. The resulting torque constnt,.16, is close to the dtsheet vlue of.161. Mesured torque is compred to tht predicted from the mesured nd estimted currents in Figure 11 using the torque constnt found. Torque (Nm) Est. Torque Error (Nm) 3 1 Torques -- rel nd predicted Actul Bsed on ctul current Bsed on est current Figure 11: Motor torque-- ctul nd predicted bsed on mesured nd estimted current

6 Torque (Nm) Summry of model fit To ssess the performnce of ech model, MS errors between the predicted nd ctul results for ech model were clculted for the first qudrnt nd for the first nd second qudrnts together. The results re displyed in Tble. In prticulr, in the first qudrnt, MS error for the torque estimte ws.%. Tble : MS estimtion errors (FS full-scle) QI error QI, QII error Model V/A/Nm % FS V/A/Nm % FS V d (V d i V s ω mp ) s i K t d mp Unfortuntely, the surprising ccurcy of current nd torque estimtion did not extend to the second qudrnt. MS prediction error in the first nd second qudrnts together ws 1.1 % full-scle, nerly double tht of the first qudrnt. Although the exct cuse of the discrepncy is not yet obvious, Figure 1 shows tht the mesured torque is consistently lrger in mgnitude thn the estimted torque in the second qudrnt. Since second qudrnt opertion involves substntil current flow through the MOSFET body diodes, it is expected tht the circuit model will be slightly different for second qudrnt opertion. Est. Torque Error (Nm) 1 - Torques -- rel nd predicted Actul Bsed on ctul current Bsed on est current Figure 1: Mesured nd estimted torques for the first nd second qudrnts. V. Conclusions & Future Work While severl issues still remin to be solved, this work hs resulted in model cpble of predicting output torque within. % FS during firstqudrnt opertion, while requiring no sensing t ll, besides velocity. Future work will focus both on improving the ccurcy of the models presented in this work nd on other res of ctutor modeling. For exmple, model predicting core temperture s function of cse temperture nd input current will be sought to help void filures. Also, precise mesurements of the efficiency of the over the operting rnge my help improve the efficiency of the gits used on Hex. Acknowledgements This work is supported by DAPA/SPAWA contrct number N661--C-86. The uthors would like to thnk members of the Hex tem nd in prticulr Uluç Srnli for their contributions. eferences [1] Apex Microtechology, SA6 Dtsheet ( SA6U ev. F [] Ksskin, J.G., Schlecht, M.F., Verghese, G.C., Principles of Power Electronics, eding, MA: Addison-Wesley, p 111. [3] Mxon Precision Motors, High Precision Drives nd Systems (1 Product Ctlog), p 7. [] U. Srnli, M. Buehler, nd D. E. Koditschek, "Hex: A Simple nd Highly Mobile Hexpod obot," Int. J. obotics eserch, (7): , July 1. [] Joel D. Weingrten, Mrtin Buehler, ichrd Groff, Dniel E. Koditschek, Automted git genertion nd optimiztion for legged robots IEEE 3 Int. Conf. obotics nd Automtion, Tipei, Tiwn (submitted) [6] D. Cmpbell nd M. Buehler, "Preliminry Bounding Experiments in Dynmic Hexpod," In Bruno Sicilino nd Polo Drio, editors, Experimentl obotics VIII, Springer-Verlg,. (in press) [7] D. McMordie nd M. Buehler, "Towrds Pronking with Hexpod obot", th Int. Conf. on Climbing nd Wlking obots, Krlsruhe, Germny, September - 6, 1. [8] H. Komsuoglu, D. McMordie, U. Srnli, N. Moore, M. Buehler, nd D. E. Koditschek, "Proprioception Bsed Behviorl Advnces in Hexpod obot", 1 IEEE Int. Conf. on obotics nd Automtion (ICA), pp 36-36, Seoul, Kore, My 1-6, 1. [9] D. McMordie, Towrds pronking with hexpod robot, M. Eng. Thesis, McGill University,. [1] E.Z. Moore nd M. Buehler, "Stble Stir Climbing in Simple Hexpod", th Int. Conf. on Climbing nd Wlking obots, Krlsruhe, Germny, September - 6, 1. [11] E. Z. Moore, D. Cmpbell, F. Grimminger, nd M. Buehler, " elible Stir Climbing in the Simple Hexpod 'Hex'," IEEE Int. Conf. on obotics nd Automtion (ICA), Vol 3, pp -7, Wshington, D.C., U.S.A., My 11-1, [1] D. Cmpbell, M. Buehler, Stir Descent in the Simple Hexpod Hex, IEEE 3 Int. Conf. obotics nd Automtion, Tipei, Tiwn (Submitted) [13] A. E. Fitzgerld, C. Kingsley, Jr., S. D. Umns, Electric Mchinery, fifth edition. New York: McGrw-Hill, 199, p 39.