Recursive dynamics simulator (ReDySim): A multibody dynamics solver

Transcription

1 THEORETICAL & APPLIED MECHANICS LETTERS 2, 6311 (212) Recursve dynamcs smulator (ReDySm): A multbody dynamcs solver Surl V. Shah, 1, a) Paramanand V. Nandhal, 2, b) 2, c) and Subr K. Saha 1) Department of Mechancal Engneerng McGll Unversty, QC H3AC3, Canada 2) Department of Mechancal Engneerng Indan Insttute of Technology, New Delh 1116, Inda (Receved 1 September 212; accepted 25 September 212; publshed onlne 1 November 212) Abstract Recursve formulatons have sgnfcantly helped n achevng realtme computatons and modelbased control laws. The recursve dynamcs smulator (ReDySm) s a MATLABbased recursve solver for dynamc analyss of multbody systems. ReDySm delves upon the decoupled natural orthogonal complement approach orgnally developed for seralchan manpulators. In comparson to the commercally avalable software, dynamc analyses n ReDySm can be performed wthout creatng sold model. The nput parameters are specfed n MATLAB envronment. ReDySm has capablty to ncorporate any control algorthm wth utmost ease. In ths work, the capabltes of ReDySm for solvng openloop and closedloop systems are shown by examples of robotc grpper, KUKA KR5 ndustral manpulator and fourbar mechansm. ReDySm can be downloaded for free from and can be used almost nstantly. c 212 The Chnese Socety of Theoretcal and Appled Mechancs. [do:1.163/ ] Keywords ReDySm, multbody systems, dynamc modelng, recursve dynamcs, DeNOC Multbody dynamcs fnd applcatons n robotcs, automoble, aerospace and many other streams for analyss, smulaton and control. It has evolved a lot n the last two decades and there s a huge scope of research for scentsts and engneers. Computeraded dynamc analyss of multbody systems has been the prme motve of engneers snce evoluton of hgh speed facltes usng computers. Dynamc analyss nvolves force and moton analyses. Force or nverse dynamcs analyss attempts to fnd the drvng and reactve forces for the gven nput moton, whereas the moton analyss or forward dynamcs, obtans system s confguraton under the nput forces. Force analyss helps n desgn and control of multbody systems, whereas moton analyss allows one to study and test a desgn vrtually wthout really buldng a real prototype. An effcent framework s essental for the dynamc analyss of complex multbody systems. There are several software and toolboxes avalable for smulatng multbody systems. They are n the form of toolkt for MATLAB 1,2 and LabVIEW. 3 The commercally avalable tools such as ADAMS 4 and RecurDyn 5 are the general purpose software used for dynamc analyss. These software are well suted for some standard ndustral applcatons, however, they fal to attract research communty manly due to ther nflexblty at user s end for solvng complex problems. The generc nature of ths software many tmes compromse wth accuracy of the results even for smaller system for longer smulaton tme. In ths context, recursve dynamcs algorthms play an mportant role. They are attractve due to smplcty and computatonal unformty regardless of ever growng complex multbody systems. Recursve formulatons a) Correspondng author. Emal: surlvshah@gmal.com. b) Emal: paramanandn@gmal.com. c) Emal: saha@mech.td.ac.n. have sgnfcantly helped n achevng realtme computatons and modelbased control laws. ReDySm 6 8 s a MATLABbased solver, whch s a general purpose platform, essentally consstng of very effcent recursve order (n) nverse and forward dynamcs algorthms for smulaton and control of a treetype system. ReDySm delves upon the decoupled natural orthogonal complement (DeNOC) approach 9 orgnally developed for seralchan manpulators. The specalty of ths solver exsts n ts recursve nature and flexblty n solvng complex problems. The gan n computatonal tme s more as the number of lnks and jonts n the system ncreases. Ths paper manly addresses analyss of fxedbase open and closedloop systems usng ReDySm. The floatngbase module of RedySm s also avalable on and can be used for analyzng legged robots 7 and space robots wth moble or free base. Detals of floatngbase module are not gven due to space lmtaton. Even though, the algorthms n ReDySm are meant for treetype systems as shown n Fg. 1, t can effectvely be used to solve closedloop systems by smply provdng constraned Jacoban matrx resultng out of loopclosure equatons and lettng ReDySm know how a closed loop system s cutopen. The flow chart showng nverse and forward dynamcs algorthms 6 of the treetype systems are gven n Fg. 2. The physcal confguraton of the system s manly defned by the Denavt Hartenberg (DH) 1 parameters, as proposed by Khall and Klenfnger. 11 The user nputs for both the nverse and forward dynamcs are dscussed next. In order to perform nverse dynamcs, the followng nput parameters are requred: Model parameters: (1) number of lnks (n l ); (2) type of system (type),.e., openloop or closedloop; (3) degreesoffreedom (dof) of the system;

2 63112 S. V. Shah, P. V. Nandhal, and S. K. Saha Theor. Appl. Mech. Lett. 2, 6311 (212) Fg. 1. Alnk or body Base Treetype system. (4) actuated jont n the system (a j ); (5) vector contanng number of jont varables assocated wth each jont (n j ); (6) constant DenavtHartenberg (DH) parameters for revolute jonts (a, α and b); (7) parent of each lnk (β); (8) vector d k measured from orgn O k to the centerofmass (COM) C k of the kth lnk; (9) mass of each lnk (m k ); (1) vector of gravtatonal acceleraton (g) n the nertal frame; (11) nerta tensor of each lnk about COM and represented n bodyfxed frame (Ik C); (12) tme perod (T p ) and step sze (dt). The above nput parameters are entered n the functon fle named nputs.m. The jont torques are then obtaned by callng protected functon fle runnv.p. User may call functon fle plot tor.m to see the output torques. The functon fle runnv.p also calculates Lagrange multplers at the cut jonts for closedloop systems. In order to generate nverse dynamcs results, cyclodal jont trajectory s chosen as default. However a user can defne any trajectory through the functon fle trajectory.m. Usng the defned trajectory, the calculated jont torque values are stored n data fle tor.dat. The numercal results of jont trajectores are stored n traj.dat. One can also vsualze nput jont trajectory by usng fle anmate.m. The fle anmate.m s not generc and need to be modfed for anmatng dfferent systems. In order to solve the nverse dynamcs of a closedloop system, frst the trajectores of ndependent jonts are entered n the functon trajectory.m whereas the relatonshp between ndependent and dependent jont trajectores and Jacoban matrx are entered n the functon fle nv kne.m. The nv kne.m fle s specfc to a gven system and the user s requred to modfy t dependng on the type of system to be analyzed. In order to perform forward dynamcs, nput parameters are provded n the fle nputs.m. These nput parameters are nothng but the frst eleven enttes of model parameters, as dscussed earler. In addton to these nput parameters, the followng parameters are requred for the purpose of ntegraton: (1) ntal condtons y = [q T q T E act ] T where q, q and E act are ntal jont postons, jont rates, and actuator energy, respectvely; (2) ntal tme (t ) and fnal tme (t f ) of smulaton, and step sze (dt); (3) relatve tolerance (r tol ) and absolute tolerance (a tol ), and the type of ntegrator, note that one may use ether adaptve solver ode45 (for nonstff problem) or ode15s (for stff problem) or fxed step solver ode5 by specfyng the ndex, 1 or 2, respectvely. These parameters are entered n the functon fle named ntals.m. The jont torque on each jont, whch are nput for forced smulaton, can be entered n the functon fle torque.m. The default value of torque at each jont s kept zero whch lead to the free smulaton. However, n the case of force smulaton, the user can defne proportonal (P), proportonal and dervatve (PD) or modelbased control law for torque nput as the current tme (t), number of lnks (n l ), and vectors of jont postons (θ) and jont veloctes ( θ) are passed to the functon torque.m. One can also ntegrate any user defned control algorthm n ths functon fle. Dynamc smulaton of closedloop system also requres enterng Jacoban and ts tme dervatve n the functon fle jacoban.m. It s worth notng that the vectors of current jont angels (θ) and jont veloctes ( θ) are passed as the nputs to ths functon and outputs are Jacoban matrx and ts tme dervatve. Fnally, smulaton s performed by runnng protected functon fle runfor.p. The output jont motons are stored n data fle statevar.dat whereas the tme hstory s stored n tmevar.dat. The jont motons can be plotted by usng functon fle plot moton.m. The functon fle for kne.m can be used to calculate the poston of the lnk orgn and COM, velocty, angular velocty and the tp poston. Moreover, the total energy can be calculated by runnng the fle energy.p and plotted usng plot en.m. Ths can be used for the purpose of valdatng the smulaton results. The system can also be anmated usng the fle anmate.m. Dynamc analyses of seral, treetype and closedloop systems are presented next usng ReDySm. The planar 4dof robotc grpper and a spatal 6dof ndustral manpulator KUKA KR5 12 are selected as openloop systems, whereas a 1dof planar fourbar mechansm s selected as an example of closedloop system. A treetype robotc grpper, as shown n Fg. 3, can hold objects to be manpulated by a robotc manpulator. Numercal results for nverse dynamcs,.e., to fnd the jont torques for a gven set of nput motons, were obtaned usng the nverse dynamcs module of the ReDySm. The detaled steps are shown n Fg 2(a) where the defntons of all varables are avalable n Ref. 6. The moton for each jont for ths purpose was computed usng θ (T ) θ () θ (t) = θ () + [ T t T ( )] 2π 2π sn T t, (1)

3 63113 Recursve dynamcs smulator Theor. Appl. Mech. Lett. 2, 6311 (212) Jont torques (τ τ ), nerta parameters (M ), twst and moton propagatons (A j and. N ), and ntal condtons q and q Jont motons (q q. q.. ), nerta parameters (M ), twst and moton propagatons (q β and N ) Forward recurson /1.s t /A β t β +N... θ.. t /A. β t β +A β t β +N. θ w~* /M t +ΩΩ M E t Forward recurson Backward recurson /1.s t /A β t β +N... θ.. t /A... β t β +A β t β +N +N. θ θ w~ /M t +ΩΩ M E t /s.1 τ /N w~ T w~ β( /w ~ β( +A β() w ~ Backward recurson Forward recurson /1.s ^ ^ ^ ^ ^^ Ψ /M N, I /N Ψ, Ψ /ΨΨ I ^ T ϕ ~ /ϕϕ N η ~ ^ 1 ϕ ^ /I ϕ ^ ^ ^ T M /M ΨΨ Ψ, η /ΨΨ ^ ϕ η ~ η ^ ^ T ^ M β() /M β() A β() M A β() ~ ~ T η β() /ηη β() A β( η T 1 /1.s.. ~ µ ~.. /A β() (N β() q β() µ µ β() ) ~ T q /ϕϕ ΨΨ ~ µ Jont torques (ττ ) Independent acceleratons q.. (a) Inverse dynamcs (b) Forward dynamcs Fg. 2. Recursve dynamcs algorthms. 6 X 2 θ 3 O 3 ϑ3 3 ϑ2 θ 2 X 3 X 1 5 ϑ3 O 3 ϑ2 5 X 4 2 O 2 O 2 ϑ4 ϑ1 5 ϑ1 Y 1 Y 4 θ 4 O 4 ϑ4 O 4 1 θ 1 X O O 1 1 O O 1 ϑ 5 Fg. 3. Robotc grpper. where θ() and θ(t ) denote the ntal and fnal jont angles, respectvely, as gven n Table 1. The jont trajectory n Eq. (1) s so chosen that the ntal and fnal jont rates and acceleratons for all the jonts are zero. These ensure smooth jont motons. The lengths and masses of all lnks were taken as l 1 =.1 m, l 2 = l 3 = l 4 =.5 m, m 1 =.4 kg, and m 2 = m 3 = m 4 =.2 kg. The sample jont torques for the robotc grpper are then plotted n Fg. 4. In order to valdate the results, a CAD model of the grpper was developed n ADAMS 4 software and used for the computaton of the jont torques. The results were supermposed n Fg. 4, whch show close match wth the values obtaned usng the proposed nverse dynamcs algorthm of the ReDySm. ReDySm took only.25 s on Intel T23@1.66 GHz computng system. The ADAMS software, however, took 1.95 s, whch s longer and s expected as t s general purpose software.

4 63114 S. V. Shah, P. V. Nandhal, and S. K. Saha Theor. Appl. Mech. Lett. 2, 6311 (212) 1..8 ReDySm ADAMS.3.2 τ1/nm.6.4 τ2/nm Fg. 4. Jont torques for robotc grpper. 6 8 ReDySm ADAMS 2 1 θ1 /(Ο) 1 θ2 /(Ο) Fg. 5. Smulated Jont angles for robotc grpper (Jonts 1 and 2). Fg. 6. Jont torques for KUKA KR5. Fg. 7. Smulated jont angles of the KUKA KR5 (Jonts 1, 2 and 3).

5 Energy/J Recursve dynamcs smulator Theor. Appl. Mech. Lett. 2, 6311 (212) Table Total Potental Actuator Knetc Fg. 8. Energy plot of KUKA KR5. Intal and fnal jont angles for robotc grpper. Jonts θ() 9 θ(t ) The forward dynamcs smulaton of the robotc grpper was then carred out usng forward dynamcs module of ReDySm. Numercal results for the acceleraton were obtaned for the freefall of the grpper,.e., t was left to move under gravty wthout any external torques appled at the jonts. The acceleratons were then numercally ntegrated twce usng the ordnary dfferental equaton (ODE) solver ode45 (default n ReDySm) of MATLAB. The ode45 solver s based on the explct RungeKutta formula gven n Dormand and Prnce. 13 The ntal jont angles and rates were taken as θ 1 = 6, θ 2 = θ 3 = θ 4 =, and θ 1 = θ 2 = θ 3 = θ 4 = rad/s respectvely. Fgure 5 shows the comparson of the smulated jont angles and the same obtaned n ADAMS (by usng RKF45 solver) over the tme duraton of 1 s wth the step sze of.1 s. The results obtaned usng ReDySm s n close match wth those obtaned usng ADAMS. The ReDySm took.29 s n contrast to 39 s requred by ADAMS. Dynamc analyss of a spatal 6dof ndustral manpulator KUKA KR5 was performed next. To study the dynamc behavor of the KUKA KR5, ts knematc archtecture 12 was used wth userdefned mass and nerta propertes. The DH parameters consdered for KUKA KR5 and ts assumed mass and nerta propertes are gven n the Table 2. Frst, nverse dynamcs of the KUKA KR5 robot was performed for ntal and fnal jont angles of all the sx jonts as and 6, respectvely. Once agan cyclodal trajectory of Eq. (1) was used as the nput jont motons. The jont torques at frst three jonts are shown n Fg. 6. Note that the torque requred at jont 1 s zero at the begnnng and at the end, as jont 1 s not affected by the gravtatonal acceleraton. Moreover, the torque requrement at jont 2 s maxmum, whch s very obvous as ths jont s requred to overcome the effect of the gravty on lnks 2, 3, 4, 5 and 6. Next, forward dynamcs of the KUKA KR5 robot was performed for the freefall under gravty. For ths, Drvng torque/nm Fg ϑ θ 3 θ2 Fg ϑ λ 1 λ 2 A fourbar mechansm. Ο 1 ϑ1 ReDySm θ 1 SmMechancs Inverse dynamcs results for fourbar mechansm. both ntal jont angles and veloctes are assumed to be zero. Fgure 7 shows the plot of jont angles for the tme perod of 1 s. The law of conservaton of energy was used to valdate the smulated results. Snce there s no dsspaton n the system, the total energy s plotted n Fg. 8, whch remaned unchanged throughout the smulaton tme. Hence, the smulaton results are valdated. Dynamc analyss of a closedloop planar fourbar mechansm was carred out next. Frst, nverse dynamcs was carred out usng ReDySm s nverse dynamcs module. The lnk length of crank, output lnk, coupler and fxed base were taken as.38 m, m, m and.89 5 m respectvely. Also, the mass of the crank, output lnk and coupler were taken as 1.5 kg, 3 kg and 5 kg respectvely. Inerta tensor of each lnk about ts COM was calculated by assumng each lnk as a slender rod. In order to solve the fourbar mechansm, t was cut at an approprate jont, as shown n Fg. 9, to form a treetype system. 14 The opened jont was substtuted wth sutable constrant forces (λ) known as Lagrange multplers. For ths purpose, the Jocoban matrx was provded n nv kne.m functon fle. The constant angular velocty of rad/s was provded as the nput to the jont 1 of the fourbar mechansm. The constant angular velocty value was provded n

6 63116 S. V. Shah, P. V. Nandhal, and S. K. Saha Theor. Appl. Mech. Lett. 2, 6311 (212) Table 2. DH parameters and nerta propertes of KUKA KR5. a /m α /( ) b /m θ /( ) m /kg I,xx /(kg m 2 ) I,yy /(kg m 2 ) I,zz /(kg m 2 ) 1.4 θ θ θ θ θ θ Jont angle/(ο) Smulated jont motons for the fourbar mecha Fg. 11. nsm Ο ReDySm SmMechancs the functon fle trajectory.m. The nverse dynamcs was carred out for the tme perod of 1.33 s. The jont torque at jont 1 was calculated by executng functon fle runnv.p. The jont torque at jont 1 s plotted n Fg. 1. Next, smulaton of the fourbar mechansm was carred out usng ReDySm s forward dynamcs module. The smulaton was done for freefall under gravty for tme perod of.68 s wthout any external torque appled at the jont 1. The Jacoban matrx was provded n jocoban.m functon fle. By executng functon fle runfor.p, the jont angles were calculated, whch are shown n Fg. 11. The results of the ReDySm were valdated wth MATLAB s SmMechancs as shown by crcular markers n Fgs. 1 and 11. An effcent Recursve Dynamcs Smulator (ReDySm) has been developed n MATLAB for θ 1 θ 2 θ 3 analyses of multbody systems. Capablty of the ReDySm s shown for seral, treetype and closedloop systems whose results are valdated wth the other commercal software. The ablty of ReDySm n ncludng desred trajectory and control law, provde flexblty to researcher n ncorporatng ther customzed algorthms. The control aspect s not reported here due to space lmtaton and wll be communcated n future. ReDySm does not requre buldng model n the software envronment before smulaton; users can smply provde the nput parameters n the MATLAB envronment. ReDySm showed consderable mprovement over commercal software such as ADAMS and algorthms avalable n lterature n terms of the computatonal tme. ReDySm can be downloaded free from The users are encouraged to send ther comments and suggestons to redysm@gmal.com or the authors. 1. P. I. Corke, Robot. Automat. Mag. IEEE 3, 24 (1996). 2. M. Toz, and S. Kucuk, Comput. Appl. Eng. Edu. 18, 319 (21). 3. T. J. Mateo Sanguno, and J. M. A. Márquez, Smulaton Tool for Teachng and Learnng 3D Knematcs Workspaces of Seral Robotc Arms wth up to 5dof. Computer Applcatons n Engneerng Educaton, Automated Dynamc Analyss of Mechancal System (ADAMS), Verson 25.., MSC Software (24). 5. RecurDyn. FunctonBay Inc S. V. Shah, Modular framework for dynamc modelng and analyss of treetype robotc systems [PhD Thess]. (Indan Insttute of Technology Delh, Delh, 211). 7. S. V. Shah, S. K. Saha, and J. K. Dutt, ASME J. Nonlnear Computat. Dyn. 7, (212). 8. S. V.Shah, S. K. Saha, and J. K. Dutt, Mech. Mach. Theory 49, 234 (212). 9. S. K. Saha, ASME J. Appl. Mech. 66, 986 (1999). 1. J. Denavt, and R. S. Hartenberg, ASME J. Appl. Mech. 215 (1955). 11. W. Khall, and J. Klenfnger, IEEE Int. Conf. on Robotcs and Automaton, KUKA KR5, Techncal specfcaton, KUKA Robotcs webste, last accessed on 2 May J. R. Dormand, and P. J. Prnce, J. Comput. Appl. Math. 6, 19 (198). 14. H. Chaudhary, and S. K. Saha, ASME J. Mech. Desgn 129, 1234 (27).