On-Line Trajectory Generation: Nonconstant Motion Constraints

Transcription

1 2012 IEEE Internatona Conference on Robotcs and Automaton RverCentre, Sant Pau, Mnnesota, USA May 14-18, 2012 On-Lne Trajectory Generaton: Nonconstant Moton Constrants Torsten Kröger Abstract A concept of on-ne trajectory generaton for robot moton contro systems enabng nstantaneous reactons to unforeseen sensor events was ntroduced n a former pubcaton. Ths prevousy proposed cass of agorthms requres constant knematc moton constrants, and ths paper extends the approach by the usage of tme-varant moton constrants, such that ow-eve trajectory parameters can now abrupty be changed, and the system can react nstantaneousy wthn the same contro cyce typcay one msecond or ess). Ths feature s mportant for nstantaneous swtchngs between state spaces and reference frames at sensor-dependent nstants of tme, and for the usage of the agorthm as a contro submodue n a hybrd swtched robot moton contro system. Rea-word expermenta resuts of two sampe use-cases hghght the practca reevance of ths extenson. I. INTRODUCTION Sensor ntegraton n the feedback oops of ow-eve moton controers beongs to key technooges for the future advancement of robot arm controers. It s mportant to enabe robot moton controers to nstantaneousy swtch from sensor-guded moton contro e.g., force/torque contro [1] or vsua servo contro [2]) to trajectory-foowng moton contro and vce versa) at unforeseen nstants. Ths way, new event-based robot programmng methodooges can be reazed as robots become enabed to react nstantaneousy n the moment the event s detected. In recent works [3], [4], a concept of on-ne trajectory generaton OTG) was proposed. The resutng agorthms run n parae to oweve moton controers and are abe to compute a trajectory from arbtrary states of moton wthn the same contro cyce that the unforeseen swtchng occurs. The major mtaton of the agorthms descrbed n [3] s that ony constant knematc moton constrants can be apped to them, that s, B = V, A, J, D ) = const Z, 1) where the coumns of the matrx B contan the mum veocty vector V at a dscrete nstant T, the mum acceeraton vector A, the mum jerk vector J, and/or even a vector for the mum dervatves of jerk D. Ths paper extends the agorthms of [3], such that tme-varant vaues for a eements of B can be apped. Ths way, the agorthm can generate a trajectory even f one or more eements of the current state of moton M = P, V, A, J ) = 1M ) T,..., km,..., KM 2) T. Kröger s wth the Artfca Integence Laboratory at Stanford Unversty, Stanford, CA , USA, [email protected]. exceed the vaues of B. In eqn. 2), K represents the number of degrees of freedom DOF) of the robotc system; P contans the poston, V the veocty, A the acceeraton, and J the jerk at the dscrete nstant T. The major benefts of ths extenson w be: The vaues of the knematc moton constrants can be abrupty ncreased or decreased, such that moton trajectory parameters can be adapted on-ne and the system reacts to the change mmedatey wthn one contro cyce commony, one msecond or ess). The agorthm can be used as a contro submodue n a hybrd swtched robot moton contro system, whch s avaabe even f sensors fa. Instantaneous swtchngs between state spaces and reference frames at unforeseen nstants become possbe. One of the prerequstes for the embeddng of robot dynamcs to the OTG concept s set up future work). The next secton ntroduces reated works, Sec. III descrbes the extenson of the OTG agorthm, and Sec. IV dscusses rea-word expermenta resuts of ths extended cass of OTG agorthms. II. RELATED WORK The works most reated to ths paper are [5] [11], a of whch beong to the feds of robot moton contro [12] and trajectory generaton [13], [14] n robotc systems. Macfarane et a. [5] present a jerk-bounded, near-tme-optma trajectory panner that uses quntc spnes, whch are aso computed on-ne but ony for one-dof systems. In [6], Cao et a. use rectanguar jerk puses to compute trajectores, but nta acceeratons dfferent from zero cannot be apped. Compared to the mut-dof approach presented here, the atter method has been deveoped for one-dmensona probems ony. Broquère et a. [7] pubshed a work that uses an on-ne trajectory generator for an arbtrary number of ndependenty actng DOFs. The approach s very smar to the one of Lu [8] and s based on the cassc sevensegment acceeraton profe [15]. Wth regard to [4], t s a Type V on-ne trajectory generaton approach desgned for handng severa DOFs ndvduay. Another very recent concept was proposed by Haddadn et a. [11]; Instead of generatng moton trajectores, vrtua sprngs and dampng eements are setup up used as nput vaues for a Cartesan mpedance controer of the robot. A dsadvantage of [5], [6], [8] s that they cannot cope wth nta acceeraton vaues unequa to zero. A further, recent work of Haschke et a. [9] presents an on-ne trajectory panner n the very same sense as [3] does. The /12/$ IEEE 2048

2 proposed agorthm generates jerk-mted trajectores from arbtrary states of moton, but t suffers from numerca stabty probems, that s, t may happen, that no jerk-mted trajectory can be cacuated. In such a case, a second-order trajectory wth nfnte jerks s cacuated. Furthermore, the agorthm ony aows target veoctes of zero. Ahn et a. [10] proposed a work for the on-ne cacuaton of onedmensona moton trajectores for any gven state of moton and wth arbtrary target states of moton, that s, wth target veoctes and target acceeratons unequa to zero. Sxthorder poynomas are used to represent the trajectory, whch s caed arbtrary states poynoma-ke trajectory ASPOT). The major drawback of ths work s that no knematc moton constrants, such as mum veocty, acceeraton, and jerk vaues, can be specfed. On-ne trajectory generaton agorthm Set-ponts for owereve contro III. THE EXTENDED ALGORITHM Let us frst descrbe the extenson of the OTG agorthm n a generc manner and afterwards concretey by means of the extenson of the OTG Types III V. 1 Fg. 1. Input and output vaues of the Type IX OTG agorthm cf. [3]). A. Forma Descrpton Ths nomencature used here s nherted from [3], [4]. Let us defne a trajectory M t), whch s cacuated at a dscrete tme nstant T, as { 1 ) M t) = m t), 1 V,..., ) m t), V,..., L ) } 3) m t), L V, where the eements m t) are matrces of moton poynomas ) m t) = p t), v t), a t), j t) = 1 m t),..., k m t),..., K m t) ) T. 4) Tme-dscrete vaues are represented by capta etters, tmecontnuous vaues by ower case etters. A trajectory segment of a snge DOF k s descrbed by the moton poynomas k m t) = k p t), kv t), ka t), kj t) ), 5) where k p t) represents the poston progresson, k v t) the veocty progresson, k a t) the acceeraton progresson, and k j t) the jerk progresson. Accordng to eqn. 3), a compete trajectory s descrbed by L segments, and each segment s accompaned by a set of tme ntervas V = { 1 ϑ,..., k ϑ,..., K ϑ }, where k ϑ = [ 1 k t, k t ], 6) such that a snge set of moton poynomas k m t) s ony vad wthn the nterva k ϑ. Fg. 1 shows the nput and output vaues of the OTG agorthm n a generc manner cf. [3]). It s the task of 1 For the OTG Types I II, J and D are rreevant; for the Types III V, the vaues of D are not reevant, because they are not consdered by these types of OTG agorthms cf. [3]) the agorthm to tme-optmay transfer an arbtrary current state of moton M nto the desred target state of moton M trgt = P trgt, V trgt, A trgt, J ) trgt 7) under consderaton of the knematc moton constrants B. The agorthm works memoryess and cacuates ony the state of moton of the next contro cyce, M +1 because an unforeseen sensor event may occur unt T +1 ). The cyce tme commony s n the range of one msecond or ess to be abe to react nstantaneousy, that s, n the same contro cyce an unforeseen swtchng event occurs. The agorthms descrbed n [3] requre constant vaues of B cf. eqn. 1)) and are extended now, such that the eements of B can be tme-varant. The extenson works n the same way as the basc agorthms of [3] do, but here an addtona decson tree s connected upstream of each basc decson tree. The case n whch one or more eements of the nta moton state vaues km exceed ther correspondng constrants of kb has to be consdered n ths extenson, and t may happen at any dscrete tme nstant T wth Z and k {1,..., K}, n whch k V > k V and/or k A > k A and/or k J > k J and/or k D > k D 8) s true. Furthermore, moton states km may occur, whch are wthn ther respectve bounds kb at nstant T, but whch w ead to an unavodabe future exceedng of kb at a tme nstant T +u : k V +u > k V and/or k A +u > k A k J +u > k J and/or k D +u > k D and/or wth u N\{0}. 9)

3 Fg. 2. Ths decson tree s apped pror to the decson trees of the Type III V OTG agorthms descrbed n [3]. Its task s to brng a moton state vaues M nto ther mts B and to guarantee that the eements of M can reman wthn n these bounds. For ths purpose, one or more of the ntermedate acceeraton profes, whch are shown on the eft, are connected upstream of the acceeraton profes seected by the man agorthm cf. [3], [4]). If one or more of the cases n eqns. 8) and 9) are true, the extended agorthm has to seect and parameterze Λ ntermedate trajectory segments k m wth {1,..., Λ} wth correspondng tme ntervas k ϑ wth {1,..., Λ} cf. eqns. 5) and 6)) n order to gude these vaues back nto ther bounds, and furthermore to brng the DOF k of the system nto a state of moton after whch the moton varabes can be kept wthn the constrant vaues of k B. Therefore, ths set of ntermedate trajectory segments s determned and executed pror to the segments that are generated by the basc agorthm: k m wth {Λ + 1,..., L} and k ϑ wth {Λ + 1,..., L}. Correspondng to the type of OTG, veocty profes Type I, II), acceeraton profes Type III V), or jerk profes Type VI IX) are apped, respectvey. As for the basc agorthms [3], [4], t s absoutey essenta that the addtona decson tree and the respectve ntermedate moton profes cover the compete nput space of the agorthm; otherwse there woud be nput vaues W cf. Fg. 1) to whch no trajectory, that s, no output sgna M +1, can be generated. B. Extenson by Means of Type III V OTG Agorthms Based on the OTG Types III V, we w derve the requred extenson n ths subsecton; these types generate jerk-mted trajectores cf. [3]). Fg. 2 shows the respectve decson tree n a mnmzed verson. The executon of the ntermedate trajectory segments shown on the eft of Fg. 2 s fnshed at Λ+1) k t cf. eqns. 6) and 3)), and we have to assure, that ) k V Λ+1) k v k t + k V ) k A Λ+1) k a k t + k A 10) hod for a DOFs k {1,..., K}. Furthermore, we have ) Λ+1) to ensure for each DOF k that f we brng k a k t to zero by appyng the mum possbe jerk ± k J, the mum veocty vaue of k V s not exceeded agan nether postvey nor negatvey). Therefore, the pan condton )) 2 ) Λ+1) Λ+1) ka kv k t k t ± kj k V 11) has to be fufed for a DOFs k {1,..., K}. In the foowng, the extenson s derved n order to et eqns. 10) and 11) be true and to subsequenty appy the basc part of a Type III, IV, or V OTG agorthm cf. [3]). The mnmzed decson tree of Fg. 2 takes advantage of sgn swtchngs. Snce ths tree s executed pror to a basc decson trees of OTG Types III V and aso pror to the further decson tree for the synchronzaton of mutpe DOFs, the etter X has been chosen to repace the actua tree dentfer e.g., 1A, 1B, or 2). Decson X.001 eads to a swtchng of sgns for the nta and for the target state of moton f the current acceeraton vaue k A s negatve. Hence, k A s postve for decson X.002. Ths decson checks whether k A s currenty exceeded. If t s exceeded, we set up a frst ntermedate acceeraton profe segment NegLn), whch brngs A down to k A 2050

4 by appyng k J. The decsons X.003 and X.004 check whether k V s postvey or negatvey exceeded. Snce our current acceeraton vaue s postve, decson X.003 cacuates the veocty vaue that we woud obtan f we were to brng the acceeraton vaue to zero whch ncreases the veocty vaue). If the resutng veocty s then greater than + k V, we decrease the acceeraton to zero by appyng k J agan NegLn), perform a swtchng of sgns, and et the decsons X.005 to X.008 brng the veocty vaue nto ts bounds. Decson X.004 ony checks whether k V s exceeded. If ths s the case, we contnue at decson X.005. For ths decson, we know that the veocty s ess than k V, and the acceeraton s postve no matter f the branch of decson X.003 or X.004 has been taken). If we woud now ncrease the acceeraton to + k A, decson X.005 checks whether the resutng veocty vaue s greater or ess than k V. If t s ess, we know that a smpe acceeraton ncrease brngs the veocty vaue back nto ts mts, but we have to make sure, that t can reman wthn these. For ths purpose, decson X.006 checks whether + k V woud be exceeded f we subsequenty decreased the acceeraton vaue to zero. If ths s not the case eft branch), a smpe PosLn profe segment, whch appes + k J, compes wth the requrements of eqns. 10) and 11). Otherwse rght branch), we woud ncrease the acceeraton vaue to a certan peak vaue, and subsequenty decrease t agan, such that we woud reach + k V exacty after the fu decrease to zero profe segment PosLn- NegLn). The decsons X.007 and X.008 work anaogousy. In the ast step, we have to re-swtch the sgns agan f they have been swtched before decson X.009). Fnay, we can assure that the condtons of eqns. 10) and 11) are fufed and w not be breached agan, and we can contnue wth the basc decson trees of the Type III, IV, or V OTG agorthm as presented n [3]. Dependng on the nput vaues W cf. Fg. 1) and n correspondence to the ntermedate acceeraton profes of the decson tree shown n Fg. 2, Λ {0,..., 5} 12) hods for the Types III V. Once the Λ ntermedate trajectory segments are determned, they have to be parameterzed. Ths s aso done n the same way as for the acceeraton profes as descrbed n [3], [4]. The resutng systems of equatons are of trva nature and can be soved n a straghtforward way wthout any numerca probems. IV. RESULTS Ths secton dscusses expermenta resuts and the benefts acheved wth the extended varant of the Type IV OTG agorthm. For comprehensveness, we start wth a smpe one-dof exampe, and subsequenty we appy the proposed concept to a sx-dof robot manpuator. A. Basc Exampe wth a One-DOF System For a better understandng, we ustrate the functonaty of the extended Type IV OTG agorthm by means of a Fg. 3. Resutng Type IV trajectory at T 0 for one DOF k generated wth the extended OTG agorthm presented n Sec. III for the gven nput vaues of eqn. 13). The vertca dashed nes ndcate the bounds of the snge trajectory segments, and the horzonta dotted nes ndcate the knematc moton constrants kb. concrete exampe wth one DOF k ony. Let us assume some gven nput vaues k W0 at nstant T 0 = 0 ms: kp 0 = 100 mm kp trgt 0 = 300 mm kv 0 = 270 mm/s kv trgt 0 = 100 mm/s ka 0 = 450 mm/s 2 kv0 = 300 mm/s ka0 = 300 mm/s 2 kj0 = 900 mm/s 3. 13) After the cacuaton at the contro cyce of T 0, the trajectory of Fg. 3 resuts from the nput vaues kw0 of eqn. 13). In the frst step, we seect ntermedate trajectory segments by appyng the decson tree of Fg. 2. Here, we woud take the foowng path: X.001 Change of sgns X.002 N egln X.003 N egln Change of sgns X.005 X.007 P oslnhd X.009 Basc decson tree of Type IV. Ths exampe resuts n Λ = 4 ntermedate trajectory 2051

5 segments cf. Fg. 2): N egln = P osln N egln = P osln P oslnhd One segment One segment Two segments 1k m 0 t), 1 k V 0) 2k m 0 t), 2 k V 0) 3k m 0 t), 3 k V ) 0, 4k m 0 t), 4 k V ) 0. At T 0, both condtons, eqns. 10) and 11), are not fufed. These Λ = 4 trajectory segments ead to a new state of moton Λ k m 0 Λ+1) k t 0), whch satsfes eqns. 10) and 11), and we can execute the basc Type IV decson tree of [3]. The resut of ths tree s that P ost rapzeronegt rap acceeraton profe s requred for the tme optma souton; the correspondng nonnear system of equatons can be set up and soved to get a trajectory parameters k M t). Fnay, we obtan L = = 11 trajectory segments, whereas the ffth segment s actuay not exstent, because 5 k a 0 5k t 0 ) = k A, and, thus, 5 k t 0 6 k t 0 hods cf. Fg. 3). B. Rea-Word Expermenta Resuts To hghght the practca reevance of the method proposed n ths paper, we now dscuss rea-word expermenta resuts and show, how the extended OTG agorthm can be apped n a hybrd swtched-system for robot moton contro. For the experments, the same hardware setup as descrbed n [3] has been used: The orgna controer of a sxjont Stäub RX60 ndustra manpuator [16] was repaced, and the frequency nverters were drecty nterfaced. Three PCs runnng wth QNX [17] as rea-tme operatng system perform a contro rate of 10 KHz for the jont controers; a hybrd swtched-system controer s used for Cartesan space contro and runs at a frequency of 1 KHz. If we consder the actuator space contro scheme as Lyapunov stabe [18], we can focus on the task space contro scheme, whch contans the hybrd swtched-system, and the on-ne trajectory generaton submodue. Especay, the works of Brancky [19], [20] and Lberzon [21], [22] provde eementary concepts to deveop and anayze hybrd swtchedsystem contro technques. In partcuar, the stabty anayss s of fundamenta nterest here, because the stabty of a swtched-system cannot be assured by the stabty of each snge sub-controer. Provng the stabty of hybrd swtchng systems can be extremey dffcut and many researchers are workng on anayzng such stabty questons. Brockett [23] expans ths subject for moton contro systems. Žefran and Burdck [24], [25] suggest an approach, n whch a system wth changng dynamcs s consdered, and a hybrd controer s desgned for handng the system n dfferent regmes of dynamcs. One essenta beneft of the OTG submodue s that t can take over contro at any tme and n any state of moton to stabze the system, f the underyng trajectory-trackng controer s stabe. Here, we consder a hybrd swtched robot moton contro system wth sx DOFs { x, y, z, x, y, z } [26]; the resuts are shown n Fgs. 4 and 5. Startng wth a sensorguded moton of a smpe PID zero-force controer [1], the Fg. 4. Poston, veocty, and acceeraton progressons. The swtchng from sensor-guded robot moton contro to trajectory-foowng contro happened at t = 662 ms as the some acceeraton vaues exceeded ther mum vaues cf. eqn. 14)). A sx trajectores constantaneousy reach ther desred target state of moton at T N = 2816 ms. Fg. 5 depcts these trajectores n the veocty-acceeraton-pane of the state space cf. [26]). system detects that the acceeraton amptudes exceed ka = 500 mm/s 2 k, ) { x, y, z} Z ka = 500 /s 2 k, ) { x, y, z } Z { } 14) for z, x, z at t = 662 ms. It s essenta that approprate nput parameters M trgt and M trgt are setup n the moment of swtchng from sensor-guded moton contro to trajectory-foowng contro. A the oca forward dynamcs [27], V may be cacuated from commony s gven may be through mechanca system propertes, and J 2052

6 Fg. 5. Correspondng to Fg. 4, ths dagram ustrates the sx trajectores from the moment of swtchng on.e., n the nterva 662 ms t 2816 ms). A trajectores termnate n an equbrum pont of the underyng contro oops cf. [26]). setup wth regard to the current task agan. The smpest way to setup M trgt woud be to choose V trgt = 0 and A trgt = 0 Z 15) agan. The desred pose P trgt shoud be set to a safe pose n workspace, such that no cosons and no snguartes occur durng the moton. Dependng on the task, t can aso be reasonabe to specfy a desred target veocty vector V trgt 0 n space e.g., n order to synchronze the system wth cooperatng one or to acheve a defned state of moton from whch a safe moton can be contnued). Fg. 4 depcts the poston, veocty, and acceeraton progressons for a sx DOFs { x, y, z, x, y, z }, and Fg. 5 dspays the correspondng trajectores the veoctyacceeraton-pane of the state space from the moment of swtchng on. As one can ceary see n Fg. 4, the trajectores of a sx DOFs are contnuous, and they reach ther desred target state of moton M trgt constantaneousy at T N = 2816 ms. Furthermore, Fg. 5 shows that a sx trajectores termnate n an equbrum pont of the nner contro oops as eqn. 15) was apped n ths experment.e., t s a Type III trajectory, cf. [3], [4]). Furthermore, t woud aso be possbe to swtch ony a seecton of DOFs nstead of a DOFs. The reason, why ony resuts wth decreasng moton constrants are shown, s that ths case s more demandng, and ncreasng vaues can aready be handed by the former agorthm of [3], [4]. For the reason of carty, ony these smpe exampes have been chosen for the smuaton and rea-word expermenta resuts. Due to the concept proposed n ths paper, we are now abe to nstantaneousy swtch to state feedback contro, that s, contro performed by the extended OTG agorthm, n order to stabze the system and contnuousy gude t to a safe pose. As proposed n [26], ths concept may be used to stabze hybrd swtched-systems n a very genera way, as the OTG agorthms can take over contro from arbtrary states of moton at unforeseen nstants. V. CONCLUSION The cass of on-ne trajectory generaton agorthms descrbed n [3] was extended, such that tme-varant knematc moton constrants can be apped to the agorthms. An addtona decson tree s requred to be apped upstream of the basc decson trees; f necessary, ths tree seects ntermedate moton profes to be executed pror to the ones of the basc agorthm. As an exampe, ths tree was derved for agorthm Types III V, whch generate jerkmted moton trajectores. Such an extended OTG agorthm enabes users to onne ncrease or decrease the vaues of knematc moton constrants; moton trajectory parameters can be adapted onne, and the system reacts to the change wthn one contro cyce commony a msecond or ess). If used as a contro submodue n a hybrd swtched-system, the extended OTG agorthm can be used as a state feedback controer, whch s avaabe even f sensors fa. Rea-word expermenta resuts have shown how the extended agorthm can be used n a hybrd swtched-system. Furthermore, nstantaneous swtchngs between state spaces and reference frames at unforeseen nstants become possbe, and t s aso a prerequste for the embeddng of robot dynamcs, whch s part of the future work to be done n ths research drecton. APPENDIX Ths extenson of the OTG Framework has become part of the Refexxes Moton Lbrares [28], whch can be downoaded from [29]. ACKNOWLEDGMENT The research descrbed n ths paper was conducted at the Insttut für Robotk und Prozessnformatk at the Technsche Unverstaet Caroo-Whemna zu Braunschweg, Braunschweg, Germany, headed by Professor Fredrch M. Wah, to whom I woud ke to express my sncere grattude. Furthermore, I woud ke to express my apprecaton to Professor Oussama Khatb, who s currenty hostng me at the Stanford Artfca Integence Laboratory at Stanford Unversty, Stanford, USA. The works of my former dpoma students, Mchaea Hansch, Chrstan Hurnaus, and Adam Tomczek, who worked hard on the frst deas and mpementatons of ths concept, are hghy apprecated. I am ndebted to the Deutsche Forschungsgemenschaft DFG, German Research Foundaton). REFERENCES [1] L. Van,, and J. De Schutter. Force contro. In B. Scano and O. Khatb, edtors, Sprnger Handbook of Robotcs, chapter 7, pages Sprnger, Bern, Hedeberg, Germany, frst edton, [2] F. Chaumette and S. A. Hutchnson. Vsua servong and vsua trackng. In B. Scano and O. Khatb, edtors, Sprnger Handbook of Robotcs, chapter 24, pages Sprnger, Bern, Hedeberg, Germany, frst edton, [3] T. Kröger and F. M. Wah. On-ne trajectory generaton: Basc concepts for nstantaneous reactons to unforeseen events. IEEE Trans. on Robotcs, 261):94 111, February [4] T. Kröger. On-Lne Trajectory Generaton n Robotc Systems, voume 58 of Sprnger Tracts n Advanced Robotcs. Sprnger, Bern, Hedeberg, Germany, frst edton, January

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