On the Boundedness Property of the Inertia Matrix and Skew-Symmetric Property of the Coriolis Matrix for Vehicle-Manipulator Systems

Transcription

1 O the Boudedess Property of the Ierta Matrx ad Skew-Symmetrc Property of the Corols Matrx for Vehcle-Mapulator Systems Pål Joha From Departmet Mathematcal Sceces ad echology, Norwega Uversty of Lfe Sceces, Ås, Norway Emal: Igrd Schjølberg Appled Cyberetcs, SINEF, rodhem, Norway Ja ommy Gravdahl, Krst Ytterstad Petterse, hor I. Fosse Departmet of Egeerg Cyberetcs, Norwega Uversty of Scece ad echology, rodhem, Norway hs paper addresses the boudedess property of the erta matrx ad the skew-symmetrc property of the Corols matrx for vehcle-mapulator systems. hese propertes are wdely used cotrol theory ad Lyapuov-based stablty proofs ad thus mportat to detfy. he skew-symmetrc property does ot deped o the system at had, but o the parametersato of the Corols matrx, whch s ot uque. It s the authors experece that may researchers take ths assumpto for grated wthout takg to accout that several parametersatos exst. I fact, most researchers refer to refereces that do ot show ths property for vehclemapulator systems, but for other systems such as sgle rgd bodes or fxed-base mapulators. As a result, the otherwse rgorous stablty proofs fall apart. I ths paper we lst some relevat refereces ad gve the correct proofs for some commoly used parametersatos for future referece. Depedg o the choce of state varables, the boudedess of the erta matrx wll ot ecessarly hold. We show that dervg the dyamcs terms of quas-veloctes leads to a erta matrx that s bouded ts varables. o the authors best kowledge we derve for the frst tme the dyamc equatos of vehcle-mapulator systems wth o-eucldea jots for whch both propertes are true. 1 Itroducto hs paper s motvated by a geeral cocer that some frequetly used propertes of the erta ad Corols matrces for vehcle-mapulator systems are assumed true based o the proofs for other systems. We show that the proofs of these propertes for fxed-base robot mapulators or sgle rgd bodes caot be geeralsed to vehcle-mapulator systems drectly. I fact, we show that the most commoly used equatos for vehcle-mapulator systems do ot possess both the boudedess ad skew-symmetrc propertes. here s thus a eed to clarfy to what extet these propertes are true ad fd rgorous mathematcal represetatos of these systems for use smulatos ad cotroller desg. o ths ed we preset a reformulato of the dyamc equatos for vehcle-mapulator systems for whch both the boudedess ad the skew-symmetrc propertes are true. Lyapuov based cotrollers are based o several assumptos that make the cotroller desg both more coveet ad physcally meagful. Ufortuately, these propertes are almost uversally take for grated. As these assumptos are ot always true, the stablty proofs fall apart. he frst property s cocered wth the boudedess of the erta matrx M,.e. the exstece of lower ad upper bouds o ts sgular values. For a gve robotc mapulator there may exst oe mathematcal represetato for whch the erta matrx s bouded ad aother for whch t s ot. For the most commo mathematcal represetato of vehcle-mapulator systems ths property s ot true. he secod problem that we are cocered wth s to fd a parametersato of the Corols matrx C so that the matrx Ṁ 2C s skew-symmetrc. Such a parametersato s easy to fd for fxed-base robots or for sgle rgd bodes, but ot always for vehcle-mapulator systems. Partcularly we fd that such a parametersato s rather hard to fd, especally together wth the boudedess property. he skewsymmetrc property of the Corols matrx s most cases assumed true wthout ay further proof. I the authors vew, ths s a strog weakess because ths property depeds o how we choose to represet the Corols matrx. It s thus ot suffcet to refer to a arbtrary proof of skew-symmetry: oe must refer to a proof for the specfc parametersato of the Corols matrx chose. Most papers o the topc of

2 vehcle-mapulator systems refer to Atoell 1, Fosse ad Fjellstad 2, Caudas de Wt et al. 3 or Schjølberg ad Fosse 4 for ths proof. However, oe of these refereces actually show the proof. Also refereces take from the fxed-base robotcs lterature such as Murray et al. 5 ad Scavcco ad Sclao are commoly ad wrogly used for vehcle-mapulator systems. he proof ca be foud Schjølberg 7, but oly for systems where the boudedess property does ot hold. We preset ths proof, ad correct some mstakes made, so that ths proof s correctly preseted for future referece. For the formulato preseted Egelad ad Petterse 8 the dyamcs possesses the skew-symmetrc property ad, based o the proof Schjølberg ad Fosse 4, we show that ths property ca be show also whe the dyamcs are wrtte terms of global state varables. 2 Propertes of the dyamcs I ths secto we lst some mportat propertes of dyamcal systems matrx form that play mportat roles system aalyss as well as cotroller desg. Assume for ow that we ca wrte the dyamc equatos of a mechacal system the form M(q) q+c(q, q) q=τ (1) where q s the state of the system, M(q) s the erta matrx, ad C(q, q) s the Corols matrx. he followg propertes ca be assocated wth the erta ad Corols matrces 9: Property 1. (he boudedess property) he erta matrx M(q) s uformly bouded q,.e. there exst costats d 1 ad d 2, such that < d 1 M(q) d 2 <, q R where s the duced 2-orm for matrces (see 1),.e. a max-boud o the maxmum sgular value ad a mboud o the mmum sgular value of the matrx. Property 2. (he skew-symmetrc property) he matrx (Ṁ(q) 2C(q, q)) s skew-symmetrc. Property 1 s true whe there are o sgulartes preset. hus, f the Euler agles are used to represet the atttude of the vehcle, as Fosse 11, Schjølberg 7 ad Børhaug 9, ths s ot satsfed. he exstece of the boudares d 1 ad d 2 s the bass of ga cotroller desg ad global Lyapuov stablty, ad s used several mapulator cotrol laws such as robust cotrol, 1. Property 2 s true for a certa parametersato of the Corols matrx. Such a represetato s well kow for robotc mapulators o a fxed base 5, ad for vehcles wth o mapulator attached 11. hs property s frequetly used to cacel the o-leartes of the Corols matrx from Lyapuov fuctos. 3.1 he Model of Schjølberg 7 I ths secto we preset the dyamc equatos as they are ormally preseted the uderwater robotcs lterature. he detals ca be foud Schjølberg 7. he dyamcs ca be wrtte as ξ=j(ξ)ζ, (2) M(q) ζ+c(q,ζ)ζ=τ (3) where ξ = η q deotes the posto, ζ = ν q the velocty, M(q) R (+) (+) s the erta matrx, ad C(q,ζ) R (+) (+) s the Corols matrx. he velocty trasformato matrx s gve by J(ξ)= R b(θ) Θ (Θ) R (+) (+), (4) I where R b (Θ) SO(3) s the rotato matrx ad Θ = φ θ ψ the Euler agles. Θ (Θ) s gve by (zyx-sequece) 1 sφtaθ cosφtaθ Θ (Θ)= cosφ sφ. (5) sφ cosθ cosφ cosθ Θ (Θ), ad thus also J(ξ), are ot defed for θ=±π/2. Let ν deote the lear ad agular velocty of body represeted the ertal frame, ad P (q) R (+) be the trasformato matrx of lk, that gves the relato ν = P (q)ζ. he erta matrx of the vehcle-mapulator system ca the be wrtte as 8 M(q)= P (q)i P (q) () where I R deotes the costat postve-defte dagoal erta tesor of lk expressed F ad we thus sum from the base b to the ed of the cha,.e., lk. We ote that the erta matrx M(q) depeds oly o the jot varables q ad s depedet of the posto η of the vehcle. he Corols matrx s gve by 7 C(q,ζ)= Ṗ (q)i P (q) P (q)w (ζ)p (q) (7) where W (ζ) s a skew-symmetrc matrx 7. We wll use the framework of Egelad ad Petterse 8 to fd a expresso for W (ζ). hs s show Secto Vehcle-Mapulator Dyamcs I ths secto we revew some commoly used approaches for modellg vehcle-mapulator systems. 3.2 Multbody Dyamcs erms of Quas-Veloctes I ths secto we derve the dyamcs of a robotc mapulator mouted o a free-floatg base terms of quas-

3 veloctes. he approach s based o Egelad ad Petterse 8, but a few errors from ths paper have bee corrected ad we also provde some more detals the dervato. Frst, wrte the lear ad agular veloctes ν of each lk represeted the ertal framef as ν ν =,v ν,ω ad the dyamcs ca be wrtte as 8 ν d dt K ν = ν ζ. (8) ˆν +,ω K ˆν,v ˆν,ω ν } = τ. (9) We ow derve the dyamcs matrx form followg the approach Egelad ad Petterse 8, but addto we show the explct expressos for the matrces whch were ot show Egelad ad Petterse 8 ad we correct a error s the expresso of the Corols matrx. Frst wrte ad ( ) d K dt ν = d dt (I ν )=I ν = I ν ζ+ ν ζ, (1) ˆν K,ω ν ˆν,v,v ˆν K = K ν ν,ω,v,ω K ν ν,ω ν,v K ν,v ν,ω,ω K ν =,v ν,v ν. (11) ν,v ν,ω,ω Substtutg (1) ad (11) to (9) we get ν d dt ν ν I ζ+ ν ν ν I ζ + ν K ν ˆν +,ω K ˆν,v ˆν,ω I ν ζ ν ν I ν ν,v ν,v ν } = τ ν,v ν,v ν ν,ω = τ,ω ν,v ν ζ=τ ν,ω (12) where we have used the relato (8). he erta matrx s the gve by () wth P (q)= ν gve by (7) wth W (ν )= ν,v ad the Corols matrx s ν,v. (13) ν,ω 3.3 Geeral Multbody Dyamcs I ths secto we exted the formulato from the prevous secto to clude more geeral structures ad also mechasms where the posto of the vehcle eeds to be cluded the dyamcs. he approach s based o Dudam ad Stramgol 12 ad From et al. 13 where the dyamcs of vehcle-mapulator systems are derved ad the boudedess property holds. Usg stadard otato 5, we ca descrbe the pose of each framef relatve tof as a homogeeous trasformato matrx g SE(3). hs pose ca also be descrbed usg the vector of jot coordates q as g = g b g b = g b g b (q). he base pose g b ad the jot postos q thus fully determe the cofgurato state of the robot. I a smlar way, the spatal velocty of each lk ca be expressed usg twsts 5: ν ν ( ) =,v =ν ν b + νb = Ad gb ν b b + J (q) q,ω (14) where ν,v ad ν,ω are the lear ad agular veloctes, respectvely, of lk relatve to the ertal frame, J (q) R s the geometrc Jacoba of lk relatve tof b ad the adjot s defed as Ad g := R ˆpR R R. he velocty state s thus fully determed gve the twst ν b b of the base ad the jot veloctes q. hs llustrates how the kematcs of the system ca be aturally descrbed terms of the (global) state varables Q=g b,q} ad v=ν b b, q}. Gve a mechasm wth coordates formulated ths geeralsed form, we ca wrte ts ketc eergy as K (Q,v) = 2 1v M(Q)v wth M(Q) the erta matrx coordates Q. he dyamcs of ths system the satsfes M(Q) v+c(q,v)v=τ (15) wth τ the vector of gravtatoal forces, frcto, ad other exteral forces (collocated wth v). From expresso (14) for the twst of each lk the mechasm, we ca derve a expresso for the total ketc eergy. he ketc eergyk of lk the follows as K = 1 ( ν b b 2 + J (q) q) Ad I gb Ad gb = 2 1 (ν ) b b q ν b M (q) b q ( ) ν b b + J (q) q (1)

4 wth Ad M (q) := gb I Ad gb Ad g b I Ad gb J J Ad g b I Ad gb J Ad g b I Ad gb J (17) 3.2 ad 3.3. Frst, for the Corols matrx gve (7) we ca wrte (Ṁ 2C)= d dt ( P (q)i P (q) ) where J (q) s the geometrc Jacoba of lk. he total ketc eergy of the mechasm s gve by the sum of the ketc eerges of the mechasm lks ad the o-ertal base, that s, ( K (q,v)= 1 Ib ) 2 v + M (q) v (18) =1 }} erta matrx M(q) =2 2 ( P (q)i Ṗ (q) P (q)w P (q) ) P (q)w P (q) (2) ad (Ṁ 2C) s skew-symmetrc, for skew-symmetrc W. hus, the formulatos gve Sectos 3.1, 3.2 ad 3.3 all satsfy the skew-symmetrc property. wth M(q) the erta matrx of the total system. We see that from (17) we ca reformulate the expresso Egelad ad Petterse 8 for the erta matrx () wth P (q)= Ad gb Ad gb J R (+). (19) Smlarly the Corols matrx ca be foud by (7) where W s gve by (13) ad P by (19). C(q,ζ) s thus also welldefed. 3.4 he Boudedess Property he dyamcs as preseted Schjølberg 7 ad Secto 3.1 do ot satsfy Property 1. Due to the sgularty there exst solated pots the cofgurato space where the erta matrx s sgular. Eve though ths s the most commo formulato of vehcle-mapulator systems the lterature ths fact s ormally ot addressed Lyapuov stablty proofs. he formulato Egelad ad Petterse 8 ad Secto 3.2 s globally vald ad the erta matrx s bouded the whole cofgurato space. For systems where the cofgurato of o-eucldea jots eeds to be cluded the dyamcs, there does ot seem to be a straght forward way to clude the trasformato betwee the local ad global state varables wthout troducg sgulartes to the formulato. hs s, however, possble wth the formulato preseted Secto 3.3 where the erta matrx s bouded for the whole cofgurato space also for o-euclde jots wth a Le group topology, such as SO(3) ad SE(3). hese formulatos allow us to use the matrx represetato g SE(3) of the cofgurato space ad the structure of the cofgurato mafold s thus mataed. 3.5 he Skew-Symmetrc Property Schjølberg 7 shows that for the formulato preseted Secto 3.1 the skew-symmetrc property holds bodyfxed coordates. Based o ths proof we ca show that ths property also holds for the approaches preseted Sectos 4 Coclusos he boudedess property of the erta matrx ad the skew-symmetrc property of the Corols matrx both deped o the choce of mathematcal represetato. he proofs of such propertes thus eed to be based o the partcular represetato chose. I other words, a referece to a proof for a dfferet choce of state varables or a dfferet parametersato of the matrces s ot vald. I ths paper we have show that several wdely used formulatos of vehclemapulator dyamcs do ot possess these propertes. We have also show that some of the most commoly used refereces used for example stablty proofs of Lyapuov-based cotrol laws fact do ot show these propertes. As a result, may of the cotrol laws preseted the lterature are ot vald. For several formulatos of vehcle-mapulator dyamcs commoly foud the lterature we have studed whether the boudedess ad skew-symmetrc propertes hold. Whe we fd the dyamc equatos to satsfy these propertes we have also cluded the proofs for future referece. hese proofs have ot prevously bee preseted correctly for vehcle-mapulator systems. Fally we propose a modfed verso of the dyamc equatos that satsfy both propertes for geeral multbody systems. Refereces 1 Atoell, G., 2. Uderwater robots. Moto ad force cotrol of vehcle-mapulator systems. Sprger- Verlag. 2 Fosse,. I., ad Fjellstad, O. E., Nolear modellg of mare vehcles degrees of freedom. Iteratoal Joural of Mathematcal Modellg Systems, 1, o. 1, pp Caudas de Wt, C., Daz, E., ad Perrer, M., Robust Nolear Cotrol of a Uderwater Vehcle/Mapulator System wth Composte Dyamcs. I IEEE Iteratoal coferece o robotcs ad automato, Isttute of electrcal egeerg INC (IEEE), pp

5 4 Schjolberg, I., ad Fosse,. I., Modellg ad cotrol of uderwater vehcle-mapulator systems. Proceedgs of the 3rd Coferece o Mare Craft Maeuverg ad Cotrol, Southampto, UK, pp Murray, R. M., L, Z., ad Sastry, S. S., A Mathematcal Itroducto to Robotc Mapulato. CRC Press. Scavcco, L., ad Sclao, B., 25. Modellg ad Cotrol of Robot Mapulators. Sprger. 7 Schjolberg, I., 199. Modelg ad cotrol of uderwater robotc systems. PhD thess, Departmet of Egeerg Cyberetcs, Norwega Uversty of Scece ad echology, rodhem, Norway. 8 Egelad, O., ad Petterse, K. Y., Free-floatg robotc systems. Cotrol Problems Robotcs ad Automato, 23, pp Borhaug, E., 28. Nolear cotrol ad sychrozato of mechacal systems. PhD thess, Departmet of Egeerg Cyberetcs, Norwega Uversty of Scece ad echology, rodhem, Norway. 1 Ghorbel, F., Srvasa, B., ad Spog, M. W., O the uform boudedess of the erta matrx of seral robot mapulators. Joural of Robotc Systems, 15, o Fosse,. I., 211. Hadbook of Mare Craft Hydrodyamcs ad Moto Cotrol. Joh Wley & Sos Ltd. 12 Dudam, V., ad Stramgol, S., 28. Sgulartyfree dyamc equatos of ope-cha mechasms wth geeral holoomc ad oholoomc jots. IEEE rasactos o Robotcs, 24(3), Jue, pp From, P. J., Dudam, V., Gravdahl, J.., ad Sastry, S., 29. Modelg ad moto plag for mechasms o a o-ertal base. I the Proceedgs of Iteratoal Coferece of Robotcs ad Automato, Kobe, Japa.