Geometric Properties of Singular Extremals for a Submerged Rigid Body

Transcription

1 Geometrc Propertes o Sngular Extremals or a Submerged Rgd Body M. Chyba, T. Haberorn, R.N. Smth, G.R. Wlens Department o Mathematcs, Department o Ocean & Resources Engneerng Unversty o Hawa, Honolulu, HI Emal: mchyba@math.hawa.edu, haberor@math.hawa.edu, ryan@ore.hawa.edu, grw@math.hawa.edu September 5, 2007 Abstract In ths paper we examne the submerged rgd body as motvaton or the extenson o the noton o decouplng vector elds to nclude moton n a vscous lud. We gve a relatonshp between decouplng vector elds and the sngular extremals or the system whch lns the optmzaton o the moton plannng problem to the nherent geometrc ramewor. We end by showng how ths connecton s appled to the creaton o tme ecent traectores whch are mplementable onto a test-bed autonomous underwater vehcle (AUV. Introducton Practcal applcatons o autonomous robots such as the Mars Rover Sprt or the AUV ABE motvate research on the control o mechancal systems. From a mathematcal pont o vew such robots all nto the class o smple mechancal systems; ther Lagrangan s o the orm netc mnus potental energy. These systems can be characterzed by derental geometrc propertes through the study o geometrc control. Recent treatments o ths topc ound n [3] dsplay that ths geometrc ramewor s the correct archtecture to study the practcal applcatons o smple mechancal systems. The otherwse complex nonlnearty o such systems can be exploted usng derental AMS Classcatons: 49-XX,70EXX,93CXX Research supported n part by NSF grant DMS Research supported by NSF grant DMS Research supported by NSF grant DMS-03064

2 geometrc control propertes. Ths allows us to consder the moton plannng problem and nvestgate optmzaton o such systems. Due to smplcatons or alternate control ormulatons, optmzaton has been prevously unaddressable or systems such as underwater vehcles. Under ths geometrc archtecture, the tme mnmum problem or a submerged rgd body has been examned n [7, 8, 9, 0, ] whch manly ocus on the condtons or an extremal to be sngular. Here we revst these results and generalze them to nclude the rgd body submerged n a vscous lud. We show a relatonshp between these sngular extremals and the geometrc noton o decouplng vector elds whch gves nsght nto the moton plannng problem or the submerged rgd body. Decouplng vector elds are computed or the ully-actuated and under-actuated sceneros o a sx degree-o-reedom (DOF underwater vehcle submerged n an deal lud. It s an open queston and area o current research to characterze and classy decouplng vector elds or such a vehcle submerged n a real lud. It s the geometrc propertes o the sngular extremals and ther relatonshp to decouplng vector elds whch we examne to help answer ths open queston. 2 Equatons o Moton We derve the equatons o moton or a controlled rgd body mmersed n an deal lud (ar and n a real lud (water. By real lud, we mean a lud whch s vscous and ncompressble wth rotatonal low. Here, we consder water to be vscous lud (real lud n order to emphasze the ncluson o the dsspatve terms n the equatons o moton. Ths motvaton comes rom our desre to apply our results to the desgn o traectores or test-bed underwater vehcles. In the sequel, we denty the poston and the orentaton o a rgd body wth an element o SE(3: (b,r. Here b = (b,b 2,b 3 t R 3 denotes the poston vector o the body, and R SO(3 s a rotaton matrx descrbng the orentaton o the body. The translatonal and angular veloctes n the body-xed rame are denoted by ν = (ν,ν 2,ν 3 t and Ω = (Ω,Ω 2,Ω 3 t respectvely. Notce that our notaton ders rom the conventonal notaton used or marne vehcles. Usually the veloctes n the body-xed rame are denoted by (u,v,w or translatonal moton and by (p,q,r or rotatonal moton, and the spatal poston s usually taen as (x, y, z. However, snce ths paper ocuses on the theory, the chosen notaton wll prove more ecent especally or the use o summaton notaton n our results. It ollows that the nematc equatons or a rgd body are gven by: ḃ = Rν ( Ṙ = R ˆΩ (2 where the operator ˆ : R 3 so(3 s dened by ŷz = y z; so(3 beng the space o sew-symmetrc 3 3 matrces. To derve the dynamc equatons o moton or a rgd body, we let p be the total translatonal momentum and π be the total angular momentum, n the nertal rame. Let P and Π be the respectve quanttes n the body-xed rame. It ollows that ṗ = 2

3 =, π = = ( ˆx + l = τ where (τ are the external orces (torques, gven n the nertal rame, and x s the vector rom the orgn o the nertal rame to the lne o acton o the orce. To represent the equatons o moton n the body-xed rame, we derentate the relatons p = RP, π = RΠ+ ˆb p to obtan Ṗ = ˆPΩ+E F (3 Π = ˆΠΩ+ ˆPν + = (R t (x b R t + E T (4 where E F = R t ( = and E T = R t ( l = τ represent the external orces and torques n the body-xed rame respectvely. To obtan the equatons o moton o a rgd body n terms o the lnear and angular veloctes, we need to compute the total netc energy o the system. The netc energy o the rgd body, T body, s gven by: T body = 2 ( v Ω t ( mi3 mˆr CG mˆr CG J b ( v Ω where m s the mass o the rgd body, I 3 s the 3 3-dentty matrx and r CG s a vector whch denotes the locaton o the body s center o gravty wth respect to the orgn o the body-xed rame. J b s the body nerta matrx. Based on Krchho s equatons [7] we have that the netc energy o the lud, T lud, s gven by: T lud = 2 ( v Ω t ( M C t C J ( v Ω where M,J and C are respectvely reerred to as the added mass, the added mass moments o nerta and the added cross-terms. These coecents depend on the densty o the lud as well as the body geometry. Summarzng, we have obtaned that the total netc energy o a rgd body submerged n an unbounded deal or real lud s gven by: t ( I I 2, (7 ( v Ω T = 2 ( I I 2 I t 2 I 22 I t 2 I 22 ( v Ω ( mi3 + M mˆr CG +C t = mˆr CG +C J b + J Ths can also be wrtten as T = 2 (νt I ν + 2ν t I 2 Ω + Ω t I 22 Ω. Usng P = T ν and Π = T Ω, we have: ( P Π ( mi3 + M mˆr CG +C t = mˆr CG +C J b + J ( ν Ω (5 (6 (8. (9 The netc energy o a rgd body n an nterconnected-mechancal system s represented by a postve-semdente (0, 2-tensor eld on the conguraton space Q. The sum over all the tensor elds o all bodes ncluded n the system s reerred to as the 3

4 netc energy metrc or the system. In ths paper, the mechancal system s composed o only one rgd body, and the netc energy metrc s actually a Remannan metrc gven by on Q = SE(3 R 3 : ( M 0 G = (0 0 J For the rest o ths paper, we tae the orgn o the body-xed rame to be C G, n other words, ˆr CG = 0. Moreover, we assume the body to have three planes o symmetry wth body axes that concde wth the prncpal axes o nerta. Ths mples that J b, M and J are dagonal, whle C s zero. We have the equatons P = (mi 3 + M ν = Mν and Π = (J b + J Ω = JΩ where M = mi 3 + M and J = J b + J. It ollows rom equatons (3 and (4 that M ν = Mν Ω+E F ( J Ω = JΩ Ω+Mν ν + = (R t (x b R t + E T (2 The terms Mν Ω, JΩ Ω and Mν ν account or the Corols and centrpetal eects. These eects can also be expressed n the language o derental geometry va a connecton, see [3] or a treatse on ane derental geometrc control. A Remannan metrc determnes a unque ane connecton whch s both symmetrc and metrc compatble. Ths Lev-Cvta connecton provdes the approprate noton o acceleraton or a curve n the conguraton space by guarenteeng that the acceleraton s n act a tangent vector eld along γ. Ths settng or acceleraton s handled by et bundles whch can be studed n depth n [24]. Explctly, γ(t = (b(t,r(t s a curve n SE(3, and γ (t = (ν(t,ω(t s ts pseudo-velocty, the acceleraton s gven by ( γ γ ν + M = ( Ω Mν, Ω+ J ( Ω JΩ+ν Mν (3 where denotes the Lev-Cvta connecton and γ γ s the covarant dervatve o γ wth respect to tsel. The ane connecton ormulaton o our system wll be used later n our paper to establsh a connecton between sngular extremals and decouplng vector elds. Gravty, buoyancy and dsspatve orces can be modeled by addng external orces and torques and τ. We assume the vehcle to be neutrally buoyant, whch means that the buoyancy orce and the gravtatonal orce are equal. Snce the orgn o the body-xed rame s C G, the only moment due to the restorng orces s the rghtng moment r CB R t ρgv, where r CB s the vector rom C G to the center o buoyancy C B, ρ s the lud densty, g the acceleraton o gravty, V the volume o dsplaced lud and the unt vector pontng n the drecton o gravty. Addtonal hydrodynamc orces experenced by a rgd body submerged n a real lud are due to drag eects. We assume here that orm drag s domnant or our applcaton (specc AUV test-bed and our estmatons o ths nclude any other drag terms (such as lud sheer stresses due to rotatonal vscous low. In ths paper, we mae the assumpton that we have a drag orce D ν (ν and a drag momentum D Ω (Ω, we 4

5 neglect the o-dagonal terms. The contrbuton o these orces s quadratc n the veloctes, more precsely we have Drag = C D ρa ν ν where C D s the drag coecent, ρ s the densty o the lud and A s the proected surace area o the obect. The drag orce and momentum are then non derentable unctons whch causes dcultes n theoretcal analyss. To overcome ths, some assume the vehcle to move n a sngle drecton, hence ν ν = ν 2. We do not want to mae ths assumpton, because at least rotatons are needed n both drectons. Expermental results or our test-bed vehcle suggest that the total drag orce versus velocty can be approxmated by a cubc uncton wth no quadratc or constant term. Ths s what we assume here. To summarze, the translatonal drag s gven by D ν (ν = dag(d ν ν 3 + D 2 by D Ω (Ω = dag(d Ω Ω3 + D2 ν Ω where D ν, D Ω ν ν and the rotatonal drag are the constant drag coecents. DEFINITION 2.. Under our assumptons, the equatons o moton n the body-xed rame or a rgd body submerged n a real lud are gven by: M ν = Mν Ω+D ν (νν + ϕ ν J Ω = JΩ Ω+Mν ν + D Ω (ΩΩ r CB R t ρgv +τ Ω (4 where M accounts or the mass and added mass, J accounts or the body moments o nerta and the added moments o nerta. The matrces D ν (ν,d Ω (Ω represent the drag orce and drag momentum, respectvely. The term r CB R t ρgv s the rghtng moment nduced by the buoyancy orce. Fnally, ϕ ν = (ϕ ν,ϕ ν2,ϕ ν3 t and τ Ω = (τ Ω,τ Ω2,τ Ω3 t account or the control. For a rgd body movng n deal lud (ar, we neglect the drag eects: D ν (ν = D Ω (Ω = 0. REMARK 2.2. In (4 we assume that we have three orces actng at the center o gravty along the body-xed axes and that we have three pure torques about these three axes. We wll reer to these controls as the sx DOF controls. Ths s not realstc rom a practcal pont o vew snce underwater vehcle controls may represent the acton o the vehcle s thrusters or actuators. The orces rom these actuators generally do not act at the center o gravty and the torques are obtaned rom the momenta created by the orces. As a consequence, to set up experments wth a real vehcle, we must compute the transormaton between the sx DOF controls and the controls correspondng to the thrusters. We address such a transormaton or our actual test-bed vehcle n [4]. Together, equatons (, (2 and (4 orm a rst-order ane control system on the tangent bundle T SE(3 whch represents the second-order orced ane-connecton control system on SE(3 ( γ γ M = ( D ν (νν + ϕ ν J ( D Ω (ΩΩ r CB R t. (5 ρgv +τ Ω Introducng σ = (ϕ ν,τ Ω, equaton (5 taes the orm: γ γ = Y(γ(t+ 6 = I (γ(tσ (t (6 ( M 0 wth I beng column o the matrx I = and Y(γ(t accounts or the external orces (a restorng orce r CB R t ρg, a drag momentum D Ω (ΩΩ, and a 0 J 5

6 drag orce D ν (νν. In the absence o these external orces the equatons o moton n (5 represent a let-nvarant ane-connecton control system on the Le group SE(3, ( γ γ M = ϕ ν J. (7 τ Ω More generally, ust as equaton (5 on SE(3 s equvalent to equatons (, (2 and (4 on T SE(3, a orced ane-connecton control system on a manold Q s equvalent to an ane control system on T Q wth a drt. Ths equvalence s realzed va the geodesc spray o an ane-connecton and the vertcal lt o tangent vectors to Q. DEFINITION 2.3. Let v T q Q T Q, then the vertcal lt at v s a map vlt v : T q Q T v T Q. For w T q Q, we dene vlt v (w = d dt (v+ tw t=0. In components, vlt v (w = ( 0 w T v T Q. DEFINITION 2.4. The geodesc spray o s the vector eld S, on T Q, that generates geodesc low. Speccally, or v T q Q, S(v = d dt γ v(t t=0 where γ v s the unque -geodesc such that γ v (0 = q and γ v (0 = v. From Equaton (3, n the specal case o our Lev-Cvta connecton, the geodesc spray s gven by: ν S(b,R,ν,Ω = Ω M ( Ω Mν J ( Ω JΩ+ν Mν. For ths presentaton o S(b, R, ν, Ω, the components are expressed relatve to the standard let-nvarant bass o vector elds on T SE(3 rather than coordnate vector elds. Equatons ( and (2 can be used to recover expressons or ḃ and Ṙ. Now, the ane control system on T SE(3 wth ts assocated drt s as ollows. We denote by η = (b,b 2,b 3,φ,θ,ψ t the poston and orentaton o the vehcle wth respect to the earth-xed reerence rame. The coordnates φ,θ,ψ are the Euler angles or the body rame. We ntroduce χ = (η,ν,ω, and let χ 0 = χ(0 and χ T = χ(t be the ntal and nal states or our submerged rgd body. Then our equatons o moton can be wrtten as: where the drt Y 0 s gven by Y 0 = χ(t = Y 0 (χ(t+ 6 = Y (tσ (t (8 Rν ΘΩ M [Mν Ω+D ν (νν] JΩ Ω+Mν ν + D Ω (ΩΩ r CB R t ρgv (9 where Θ s the transormaton matrx between the body-xed angular velocty vector (Ω,Ω 2,Ω 3 t and the Euler rate vector ( φ, θ, ψ t, see [3]. 6

7 The nput vector elds are gven by Y = (0,0 t, or n other words Y = vlt(i. In [3, p224] the authors show that traectores or the ane-connecton control system on Q map bectvely to traectores or the ane control system on TQ whose ntal ponts le on the zero-secton. The becton maps the traectory γ : [0,T] Q to the traectory ϒ = γ : [0,T] T Q. In local coordnates, the equatons o moton or a submerged rgd body are derved as ollows. The coordnates correspondng to translatonal and rotatonal veloctes n the body rame are ν = (ν,ν 2,ν 3 t and ( Ω = (Ω(,Ω 2,Ω 3 t. Equatons ( and (2 can R 0 ν be wrtten n local coordnates as η = where 0 Θ Ω and R(η = cosψ cosθ R2 R 3 snψ cosθ R 22 R 23 snθ cosθ snφ cosθ cosφ (20 snφ tanθ cosφ tanθ Θ(η = 0 cosφ snφ (2 0 snφ cosθ cosφ cosθ where R 2 = snψ cosφ +cosψ sn θ snφ, R 3 = snψ sn φ +cosψ cosφ snθ, R 22 = cosψ cosφ +snφ snθ snψ and R 23 = cosψ snφ +snψ cosφ snθ. Notce that the transormaton depends on the conventon used or the Euler angles. Our choce relects the act that the rgd body goes through a sngularty or an nclnaton o ± π 2. In the sequel we denote the dagonal elements o the added mass matrx, the nerta matrx, and the added nerta matrx respectvely by {M ν,m ν 2,M ν 3 }, {J b,j b2,j b3 } and {J Ω,J Ω 2,J Ω 3 }, respectvely. The restorng orces n local coordnates are: r CB R t ρgv = ρgv where r CB = (x B,y B,z B. y B cosθ cosφ z B cosθ snφ z B snθ x B cosθ cosφ x B cosθ sn φ + y B sn θ (22 LEMMA 2.5. The equatons o moton or a submerged rgd body n a real lud wth external orces expressed n coordnates are gven by the ollowng ane control sys- 7

8 tem: ν = ν 2 = m+m ν m+m ν 2 ḃ = ν cosψ cosθ + ν 2 R 2 + ν 3 R 3 (23 ḃ 2 = ν snψ cosθ + ν 2 R 22 + ν 3 R 23 (24 ḃ 3 = ν snθ + ν 2 cosθ snφ + ν 3 cosθ cosφ (25 φ = Ω + Ω 2 snφ tanθ + Ω 3 cosφ tanθ (26 θ = Ω 2 cosφ Ω 3 snφ (27 ψ = snφ cosθ Ω 2 + cosφ cosθ Ω 3 (28 [ (m+m ν 3 ν 3 Ω 2 +(m+m ν 2 ν 2 Ω 3 + D ν (ν +ϕ ν ] (29 [(m+m ν 3 ν 3 Ω (m+m ν ν Ω 3 + D ν (ν 2 +ϕ ν2 ] (30 ν 3 = m+m ν 3 Ω = Ω 2 = Ω 3 = [ (m+m ν 2 ν 2 Ω +(m+m ν ν Ω 2 + D ν (ν 3 +ϕ ν3 ] (3 I b + J Ω [(I b2 I b3 + J Ω 2 J Ω 3 Ω 2 Ω 3 +(M ν 2 M ν 3 ν 2 ν 3 +D Ω (Ω +ρgv ( y B cosθ cosφ + z B cosθ sn φ+τ Ω ] (32 I b2 + J Ω 2 I b3 + J Ω 3 [(I b3 I b + J Ω 3 J Ω Ω Ω 3 +(M ν 3 M ν ν ν 3 +D Ω (Ω 2 +ρgv (z B sn θ + x B cosθ cosφ+τ Ω2 ] (33 [(I b I b2 + J Ω J Ω 2 Ω Ω 2 +(M ν M ν 2 ν ν 2 +D Ω (Ω 3 +ρgv ( x B cosθ snφ y B snθ+τ Ω3 ] (34 where D ν (ν = D ν ν 3 +D 2 ν ν and D Ω (Ω = D Ω Ω3 +D2 Ω Ω. ϕ ν = (ϕ ν,ϕ ν2,ϕ ν3 and τ Ω = (τ Ω,τ Ω2,τ Ω3 represent the control. As mentoned prevously, the control represents the actuaton o thursters. A consequence s that the components o the control are bounded. We here put a bound on the 6 DOF control, assumng each component s ndependently bounded rom the others. See [4] or a dscusson about translatng these bounds to the actual control or our test-bed vehcle. DEFINITION 2.6. An admssble control s a measurable bounded uncton (ϕ ν,τ Ω : [0,T] F T where: F = {ϕ ν R 3 α mn ν T = {τ Ω R 3 α mn Ω ϕ ν α max ν τ Ω α max Ω, α mn ν, α mn Ω < 0 < α max ν, =,2,3} < 0 < αω max, =,2,3} (35 8

9 3 Sngular Extremals In ths secton we study the sngular arcs as dened by the Maxmum Prncple or the tme mnmal problem. 3. Maxmum Prncple Assume that there exsts an admssble tme-optmal control σ = (ϕ ν,τ Ω : [0,T] F T, such that the correspondng traectory χ = (η,ν,ω s a soluton o equatons (23-(34 and steers the body rom χ 0 to χ T. For the mnmum tme problem the Maxmum Prncple, see [23], mples that there exsts an absolutely contnuous vector λ = (λ η,λ ν,λ Ω : [0,T] R 2, λ(t 0 or all t, such that the ollowng condtons hold almost everywhere: η = H λ η, ν = H λ ν, Ω = H λ Ω, λη = H η, λ ν = H ν, λ Ω = H Ω (36 where the Hamltonan uncton H s gven by: H(χ,λ,σ = λ t η(rν,θω t + λ t νm [Mν Ω+D ν (νν + ϕ ν ] +λ t Ω J [JΩ Ω+Mν ν + D Ω (ΩΩ r B R t ρgv +τ Ω ] (37 Furthermore, the maxmum condton holds: H(χ(t,λ(t,σ(t = max σ F T H(χ(t,λ(t,σ (38 The maxmum o the Hamltonan s constant along the solutons o (36 and must satsy H(χ(t,λ(t,σ(t = λ 0, λ 0 0. A trple (χ,λ,σ that satses the Maxmum Prncple s called an extremal, and the vector uncton λ( s called the adont vector. The maxmum condton (38, along wth the control doman F T, s equvalent almost everywhere to (M, J dagonal and postve, =, 2, 3: ϕ ν (t = α mn ν τ Ω (t = α mn Ω λ ν (t < 0 and ϕ ν (t = α max ν λ ν (t > 0 (39 λ Ω (t < 0 and τ Ω (t = α max Ω λ Ω (t > 0 (40 Clearly, the zeros o the unctons λ ν determne the structure o the solutons to the Maxmum Prncple, and hence o the tme-optmal control. DEFINITION 3.. We denote the th swtchng uncton by: =,...,6. δ (t = λ t (ty, (4 DEFINITION 3.2. We say that a component σ o the control s bang-bang on a gven nterval [t,t 2 ] ts correspondng swtchng uncton δ s nonzero or almost all t [t,t 2 ]. A bang-bang component o the control only taes values n {α mn ν,α max ν } 9

10 σ = ϕ ν and n {α mn Ω,αΩ max } σ = ϕ Ω or almost every t [t,t 2 ], =,,6. On the other hand, there s a nontrval nterval [t,t 2 ] such that a swtchng uncton s dentcally zero, the correspondng component o the control s sad to be sngular on [t,t 2 ]. A sngular component control s sad to be strct the other controls are bang. Assume a gven component o the control to be pecewse constant; or example, when the component s bang-bang. Then, we say that t s [t,t 2 ] s a swtchng tme or ths component, or each nterval o the orm ]t s ε,t s + ε[ [t,t 2 ], ε > 0, the component s not constant. 3.2 Swtchng Functons LEMMA 3.3. The rst dervatve o the swtchng uncton δ, =,...,6 s an absolutely contnuous uncton. Usng Y 0,...,Y 6 and σ,...,σ 6 rom equaton (8, the rst and second dervatves o δ are gven by: δ (t = λ t (t[y 0,Y ](χ(t (42 δ (t = λ t (tad 2 Y 0 Y (χ(t+ 6 = where [, ] denotes the Le bracet o vector elds. λ t (t[y,[y 0,Y ]](χ(tσ (t (43 Proo. It s a standard act that the dervatve o δ along an extremal s gven by δ (t = λ t (t[y 0,Y ](χ(t+ 6 = λ t (t[y,y ](tσ (t. The vector elds Y are vertcal lts; t ollows that ther Le bracets are zero. Derentatng once more, we obtan (43. REMARK 3.4. Instead o the Le bracets, we can use the Posson bracets. Indeed, we wrte the Hamltonan uncton as H = H = H σ where H 0 = λ t Y 0,H = λ t Y, equatons (42, (43 become: δ (t={h 0,H }(χ(t and δ (t={h 0,{H 0,H }}(χ(t+ 6 = {H,{H 0,H }}(χ(tσ (t. Another drect consequence o the orm o the nput vector elds Y s the symmetrc property descrbed n Lemma 3.5. It wll play a maor role when computng the second dervatve o the swtchng unctons. Notce that ths lemma holds wth or wthout external orces. LEMMA 3.5. For, =,,6, we have [Y,[Y 0,Y ]] = [Y,[Y 0,Y ]] (44 Proo. It comes rom the act that [Y,[Y 0,Y ]] s a multple o partal dervatves commute. 2 Y 0 χ 6+ χ +6 and that the To derve conclusons about the sngular arcs or our system, such as ther order, we need to explctly descrbe the Le bracets nvolved n (42 and (43. Let S = ( R 0 0 Θ be the transormaton matrx between the coordnates expressed n the nertal rame and the coordnates expressed n the body-xed rame and let S be the -th column. We begn by dervng the results or the smpled case o a rgd body movng n the ar. 0

11 For our computatons, we ntroduce U = {,2,3} and V = {4,5,6}. The next three propostons are a result o straghtorward but heavy computatons. We decded to omt these computatons snce only the results are mportant or the rest o the paper. The vectors e, U represent the standard bass or R 3. PROPOSITION 3.6. For a rgd body movng n the ar, we have that: ( m+m ν S Ω ε [Y 0,Y ] ar =, U \{, } m+m ν e ν ε, U \{, } I b + J Ω ( m+mν m+m ν e or U where ε = sgn( and [Y 0,Y ] ar = ε, U \{ 3, 3}, U \{ 3, 3} or V where ε = sgn( +3. ( I b 3 +J Ω 3 ε S (m+m ν ν (m+m ν (I b 3 + J Ω 3 e Ω I b + J Ω ( I b + J Ω I b 3 + J Ω 3 e (45 (46 To study the Le bracets [Y,[Y 0,Y ]] ar, let us ntroduce a new pece o notaton. Wthout loss o generalty we may assume rom Lemma 3.5. We dene: B, U [Y,[Y 0,Y ]] ar = B 3 U, V (47 B 3, 3, V Then, we get the ollowng Proposton. PROPOSITION 3.7. For a rgd body movng n the ar, we have that ( B = B 3, 3 0 = I b + J Ω ( m+m ν m+m ν e B 3 = (m+m ν (I b 3 + J Ω 3 ( e 0 (48. (49 where, or B, 3, 3 or B 3, 3, and, 3 or B, 3. We now extend the computatons to the model n real lud. Remember here that we consder dsspatve orces actng on the vehcle. However, notce that the restorng orces do not play any role n the expresson o the Le bracets, yet the drag orces have a sgncant mpact.

12 PROPOSITION 3.8. For a rgd body movng n a real lud, we have that: [Y 0,Y ] real = [Y 0,Y ] ar D ν ν 2+D2 ν (m+m ν e (50 or U, and [Y 0,Y ] real = [Y 0,Y ] ar D ( 3 Ω Ω 2 +D( 32 Ω e (I b 3 +J Ω (5 or V. Moreover: [Y,[Y 0,Y ]] real = [Y,[Y 0,Y ]] ar + ( ( 6D ν ν (m+m ν 3 6D ( 3 Ω Ω 3 (I b 3 +J Ω 3 3 Y, U Y, V 0 U, V REMARK 3.9. More explctly, or the Le bracets o order 2 the above proposton says that: [Y,[Y 0,Y ]] real = 0, = 3; U, V (52 [Y,[Y 0,Y ]] real = ( 6D ν ν (m+m ν 3 Y, = ;, U (53 ( ( 3 6D Ω Ω 3 [Y,[Y 0,Y ]] real = (I b 3 + J Ω Y, = ;, V ( [Y,[Y 0,Y ]] real = (m+mν (m+m ν (m+m ν (m+m ν Y (55 or ;, U ; U \{, } [Y,[Y 0,Y ]] real = (I b 3 + J Ω 3 (I b 3 + J Ω 3 (I b 3 + J Ω 3 (I b 3 + J Ω 3 or ;, V ; V \{, } [Y,[Y 0,Y ]] real = (m+m ν (I b 3 + J Ω 3 or U ; V ; U \{,( 3} Y (56 Y (57 An mportant consequence o the prevous computatons that we wll explot n ths paper s stated n Proposton

13 PROPOSITION 3.0. For a rgd body movng n the ar, we have that: [Y,[Y 0,Y ]] ar (χ = 0, =,...,6. (58 In a real lud, the prevous Le bracet s not zero but satses: [Y,[Y 0,Y ]] real (χ Span{Y }, =,...,6. (59 Proo. Ths result s a drect consequence o our computatons on Le bracets. Indeed, equaton (58 comes rom the act that (48 mples that B and B 3, 3 equal zero. The actors multplyng Y n (59 are gven by (53 and ( Order o the Sngular Arcs We now demonstrate that Proposton 3.0 can be stated n terms o the order o sngular extremals. DEFINITION 3.. Along a strct σ -sngular arc, let q be such that d2q dt 2q δ s the lowest order dervatve n whch σ appears explctly wth a nonzero coecent. We dene q as the order o the sngular control σ. Ths denton assumes the well nown result that a sngular control σ rst appears explctly n an even order dervatve o δ, see [22]. PROPOSITION 3.2. Let χ be an extremal that s strctly sngular or the component σ o the control. Then, or a submerged rgd body the order o the sngular control s at least 2. Proo. Let χ be a strct σ -sngular extremal. By denton, the uncton δ s dentcally zero along the extremal. The sngular control σ s obtaned rom equaton (43 provdng that the term λ t [Y,[Y 0,Y ]](χ s non zero. However, rom Proposton 3.0, ths s zero or movement n ar and s a multple o λ t Y or moton n a real lud. But snce along a σ -sngular extremal we have δ = λ t Y = 0, then λ t [Y,[Y 0,Y ]](χ s zero n a real lud as well. Ths means that we must compute at least the ourth dervatve o the swtchng uncton to obtan the sngular control as a eedbac. REMARK 3.3. For a rgd body movng n ar, the term λ t [Y,[Y 0,Y ]](χ s dentcally zero everywhere. We then say that the order s ntrnsc. For a real lud, λ t [Y,[Y 0,Y ]](χ s zero only along the sngular arc. To determne the exact order o strct sngular controls, we need to compute the ourth dervatve o the swtchng unctons. The coecent o the sngular control σ n δ (4 s represented by the ollowng Le bracets: λ t [Y,[Y 0,[Y 0,[Y 0,Y ]]]]. The computatons n 3-dmensons are very complcated due to the complexty o the equatons. Based on prevous results n [7] on a smpled 2-dmensonal model, we state the ollowng conecture. CONJECTURE 3.4. For a 3-dmensonal rgd body movng n a real lud, the sngular arcs are o the ollowng orders: 3

14 . m+m ν = m+m ν. The ϕ ν -sngular arcs are o nnte order. The τ Ω -sngular arcs are o ntrnsc order m+m ν m+m ν. The ϕ ν -sngular and τ Ω -sngular arcs are o order 2. REMARK 3.5. The order o the sngular arcs n the translatonal veloctes s related to the symmetry o the rgd body. 3.4 Chatterng Arcs It has been establshed n [26] that there s a close relatonshp between the exstence o chatterng arcs and sngular extremals o order 2. Such arcs are very nterestng rom a theoretcal pont o vew, however these arcs are mpossble to mplement n practce. Let us consder a smpled stuaton to carry out the computatons such as n [7]. We wll assume that the vehcle moves n the xz-plane and s submerged n ar. The equatons o moton, n local coordnates, are gven by (60-(65. b = ν cosθ + ν 3 sn θ (60 b 3 = ν 3 cosθ ν sn θ (6 θ = Ω (62 ν = ν 3 Ω m+mν 3 m+m ν ν 3 = ν Ω m+mν m+m ν 3 + ϕ ν m+m ν + ϕ ν 3 m+m ν 3 M ν 3 M ν Ω = ν ν 3 + τ Ω I I (63 (64 (65 Moreover, we assume M ν = M ν 3, hence we wrte m = m+m ν and I = I b + J Ω 2. REMARK 3.6. Kelley s strct necessary condton or the sngular control τ Ω to be optmal holds. Indeed, t s an easy computaton to show that λ (4 Ω = A 4 + τ Ω B 4 where B 4 = λ ν 3 ϕ ν3 + λ ν ϕ ν mi 2. Snce along a strct τ Ω -sngular arc the controls ϕ ν and ϕ ν3 are bang, B 4 = λ ν 3 + λ ν mi 2 s strctly negatve: B 4 < 0. Analyss o the τ Ω -sngular arcs ollows the procedure descrbed n [26]. Frst, we put the Hamltonan system (36 nto a sem-canoncal orm. We assume that ϕ ν and ϕ ν3 are bang. Snce a τ Ω -sngular arc s o ntrnsc order two, the our rst coordnates o the new system (κ,ξ are κ =(κ,κ 2,κ 3,κ 4 where κ = λ Ω /I,κ 2 = λ Ω /I =( λ θ + λ ν ν 3 λ ν3 ν /I,κ 3 = λ Ω /I = ( λ ν ϕ ν + λ ν ϕ ν3 /(mi,κ 4 = λ (3 Ω /I = ((λ b cosθ λ b3 sn θ + Ωλ ν3 ϕ ν3 (λ b sn θ + λ b3 cosθ λ ν Ωϕ ν /(mi. To completely dene a new coordnate system we need to nd ξ such that the Jacoban D(κ,ξ/D(χ,λ s 4

15 o ull ran. We suggest ξ = b, ξ 5 = λ b cosθ λ b3 sn θ ξ 2 = b 3, ξ 6 = λ b sn θ + λ b3 cosθ (66 ξ 3 = θ, ξ 7 = λ θ ξ 4 = ν, ξ 8 = λ ν The correspondng D(κ,ξ/D(χ,λ s then o ull ran and the canoncal Hamltonan system s κ = κ 2, κ 2 = κ 3, κ 3 = κ 4 κ 4 = Ω(2ξ 5 + 2ξ 6 + Ωλ ν3 Ωξ 8 /(mi (ξ 8 ϕ ν + λ ϕ3 ϕ ν3 τ Ω /(mi 2 ξ = ξ 4 cosξ 3 + ν 3 snξ 3, ξ2 = ν 3 cosξ 3 ξ 4 snξ 8, ξ3 = Ω ξ 4 = Ων 3 + ϕ ν /m, ξ5 = Ωξ 6, ξ6 = Ωξ 5 ξ 7 = ξ 4 ξ 6 ν 3 ξ 5, ξ8 = ξ 5 λ ν3 Ω (67 where λ ν3 = (ξ 8 ϕ ν3 miκ 3 /ϕ ν ν 3 Ω = (ξ 7 + λ ν3 ξ 4 Iκ 2 /ξ 8 = (miκ 4 ξ 5 ϕ ν3 + ξ 6 ϕ ν /(λ ν3 ϕ ν3 + ξ 8 ϕ ν Snce we were able to reduce our system to a sem-canoncal orm, along wth Remar 3.6, Kelley s condton holds; t s now possble to apply the results rom [26]. Here, the authors descrbe the behavor o all extremals n the vcnty o the sngular manold S = {(χ,λ κ = 0, =,,4}. In partcular, we can conclude that or each pont (χ 0,λ 0 n S there exsts a 2-dmensonal ntegral manold o the Hamltonan system such that the behavor o the solutons nsde ths manold s smlar to that o the chatterng arcs n the Fuller problem (also we have the exstence o untwsted chatterng arcs. To be more specc, there s a one-parameter amly o solutons o system (67 whch reach (χ 0,λ 0 n a nte tme, however there are nntely many swtchng tmes or the τ Ω control and the swtchng tmes ollow a geometrc progresson. It s mportant to notce that ths result does not mply the optmalty o such traectores nor does t mply (assumng ϕ ν,ϕ ν3 are constants that every uncton between a τ Ω -sngular and a τ Ω bang-bang traectory ncludes chatterng n the control. In order or such a uncton to have chatterng, the control must be dscontnuous, [2]. Ths s realzed at the uncton where the angular velocty vanshes (.e. Ω = 0. In [] the reader can see an example o a chatterng uncton computed n the non-symmetrc case. 3.5 Tme Optmal Traectores In ths subsecton, we dsplay an example o a tme optmal traectory or a submerged rgd body mmerged n a real lud contanng sngular arcs. The ntal conguraton s set to the orgn and we wsh to reach a nal conguraton η T = (6,4,,0,0,0, wth both conguratons beng at rest. The expermental values o the hydrodynamc coecents and the bounds on the control that we assume (68 5

16 0 5 γ ν 0 γ Ω γ ν2 0 γ Ω γ ν γ Ω t (s t (s Fgure : Tme Optmal thrust strategy endng at η T = (6,4,,0,0,0. or these smulatons can be ound n [6]. Fgure shows the tme optmal strategy wth a drect method beng used or the numercal computatons. The tme or ths traectory s 25.85s. The structure s mostly bang-bang, except or the τ Ω3 control whch contans sngular arcs. These sngular arcs depend on our choce o ntal and nal conguratons. In ths case, orentaton s the ey to optmalty; rst orent, then move. We orent the vehcle such that we can use the maxmum avalable translatonal thrust to realze the moton, and the vehcle needs to mantan ths orentaton over the entre traectory. Sngular arcs do not appear n τ Ω and τ Ω2 because ther ull power s needed to oset the rghtng moments. The translatonal controls ϕ ν,ν 2,ν 3 are used to ther ull extent, as one would expect or a tme optmal translatonal dsplacement. 4 Decouplng Vector Felds In terms o ane derental geometry, Proposton 3.0 has mportant consequences. Indeed, there s a relaton between our result and the exstence o decouplng vector elds. Ths s what we establsh n ths secton. Through ths secton we mae the ollowng addtonal assumptons. We consder a rgd body movng n ar. Moreover, we assume the center o gravty C G concde wth the center o buoyancy C B. Snce we already assumed the vehcle to be neutrally buoyant t mples that there are no restorng orces or moments actng on the vehcle. To ease the notaton n ths secton, we wll use m = m+m ν and = I b + J Ω. As we wll see, our results depend on the symmetres o the rgd body, hence we ntroduce some termnology. DEFINITION 4.. We call our system netcally unque all the egenvalues n the 6

17 netc energy metrc G are dstnct. The above denton express that or a netcally unque system the added mass (m + M ν and added nerta (I b + J Ω coecents are all dstnct. Snce the added mass We reer to I, U as the translatonal control vector elds and I, V the rotatonal control vector elds. Fnally, to ease notaton, we wll use m = m+m ν and = I b + J Ω. In the sequel we denote by I the set o nput vector elds to our system: I = {I,...,I }. Hence, we are consderng under-actuated systems 6 γ (tγ (t = = σ (tĩ (γ(t, (69 wth < 6, {Ĩ,...,Ĩ } an ndependent subset o I, and σ,...,σ the correspondng controls; see (6. We dene I = {Ĩ,...,Ĩ } and I = SpanI. We note here that under our assumptons, I s dagonal, and thus each I, =,...,6, s a sngle degree o reedom nput to the system. We rst gve a classcaton o the decouplng vector elds wth respect to the number degrees o reedom we can nput to the system; a one-nput system can be controlled n only one degree o reedom. More detals on ths classcaton can be ound n [2]. In ths secton, unless stated otherwse we do not mpose any bounds on the control. Let us rst ntroduce some termnology. Suppose we have a general ane-connecton control system gven by γ (tγ (t = a= u a (tz a (γ(t, (70 where u (t,...,u (t are measurable controls and {Z,...,Z } s a set o locally dened ndependent vector-elds on the conguraton space Q whose mages le n a ran- smooth dstrbuton Z T Q. In [3] the authors ntroduce the noton o nematc reductons. Decouplng vector elds are nematc reductons o ran one. More precsely, we have the ollowng denton. DEFINITION 4.2. A decouplng vector eld or an ane-connecton control system s a vector eld V on Q havng the property that every reparametrzed ntegral curve or V s a traectory or the ane-connecton control system. More precsely, let γ : [0,S] Q be a soluton or γ (s = V(γ(s and let s : [0,T] [0,S] satsy s(0 = s (0 = s (T = 0, s(t = S, s (t > 0 or t (0,T, and (γ s : [0,T] T Q s absolutely contnuous. Then γ s : [0,T] Q s a traectory or the ane-connecton control system. Addtonally, an ntegral curve o V s called a nematc moton or the aneconnecton control system. A necessary and sucent condton or V to be a decouplng vector eld or the ane-connecton control system (70 s that both V and V V are sectons o Z [3, p. 426]. Notce that Z = T Q (.e. (70 s ully-actuated then every vector eld s a 7

18 decouplng vector eld, and Z has ran = (.e. (70 s sngle-nput then V s a decouplng vector eld and only both V and V V are multples o Z. In the under-actuated settng, decouplng vector elds are ound by solvng a system o homogeneous quadratc polynomals n several varables. Gven a vector eld V, we must have that V = a= ha Z a snce V Span{Z,...,Z }. Now, snce V V Span{Z,...,Z } we want V V = h a Z a h b Z b 0 (mod Z. (7 Startng wth the mddle o the above equaton, we get that h a Z a h b Z b = h a Za h b Z b = h a Za (h b Z b = h a [Z a (h b Z b + h b Za Z b ] h a h b Za Z b (mod Z (72 2 h a h b ( Za Z b + Zb Z a (mod Z. Thus, we are concerned wth calculatng Za Z b or a,b {,...,} to nd the coecents h,...,h such that V s decouplng. In the case o a rgd body submerged n a real lud, the control system s a orced ane-connecton control system and the exstng theory does not apply. Frst, we wll consder a rgd body movng n an deal lud,.e. we neglect the drag orces and momentum. Moreover, n the rest o ths secton we assume that the center o buoyancy and the center o gravty o our submerged rgd body concde; there s no restorng moment. Under these assumptons, the equatons o moton developed n Secton 2 can be represented as a ully-actuated ane-connecton control system γ (tγ (t = 6 = σ (ti (γ(t. In ths case there are no quadratc polynomals to solve and every let-nvarant vector eld s a decouplng vector eld. However, the stuaton s not as straghtorward n the under-actuated scenaro; practcally speang, the case o an actuator alure. In ths stuaton, the body may be unable to apply a orce or torque n one or more o the sx DOF, lmtng the vehcle s controllablty. Ths s an nterestng case because t s lely that an underwater vehcle loses actuator power or one reason or another but stll needs to move. For example, we would le the vehcle to be able to return home n a dstressed stuaton. Decouplng vector elds gve possble traectores or the return home whch the vehcle s able to realze n an under-actuated condton. In [2], the authors consder an under-actuated stuaton that ders rom the ones we are consderng here (they assume three body-xed control orces that are appled at a pont derent rom the center o gravty. Let us rst dscuss the degenerate stuaton o one sngle DOF nput vector eld, =. Ths can also be vewed as a loss o 5 DOF stuaton. Clearly the only possble moton or the body s a moton along or about a sngle prncple axs o nerta. DEFINITION 4.3. The vector elds I, {,,6}, are called the pure motons. Ther ntegral curves correspond to a moton along or about a sngle prncple axs o nerta; that moton s ether purely translatonal or purely rotatonal. 8

19 Snce n our case I I = 0, the decouplng vector elds o the sngle nput system γ (tγ (t=i (γ(tσ (t are multples o the nput vector eld I. Generc sngle-nput ane-connecton control systems have no decouplng vector elds snce a generc vector eld wll not satsy the condton that Z Z Span{Z}. However, a vector eld Z does satsy Z Z Span{Z}, then va a reparameterzaton we get Z Z = 0. Geometrcally, we reer to Z as auto-parallel; the ntegral curves o Z are geodescs or the connecton. Suppose now that we use two nput vector elds; = 2. A calculaton o the terms G( I wth I Fx, {,...,6} where <. Let V = h I I I shows the ollowng (see also Equaton (72. + h I and ε be the standard permutaton symbol. We have: h h (( (m m I +3, U and U \{, } h h (( + ( I V V, V and V \{, } h h (ε ( m m I U and V and U \{, 3} 0 = +3 (mod {I } (73 We can deduce that gven a netcally unque two-nput system I2 = {Ĩ,Ĩ 2 } n whch both nputs do not act upon the same prncple axs o nerta, a vector eld V s decouplng and only V I and has all but one o ts components equal to zero. In partcular, t has the orm V = h Ĩ or V = h 2 Ĩ 2. These are pure motons. I both nputs act on the same prncple axs o nerta ( U, I2 = {I 3+ }, every vector eld V I s decouplng snce V V I. I we loosen the netcally unque assumpton and let m = m or, U or = l,l V, then every vector eld V I s decouplng. Ater ntroducng some addtonal termnology, we wll summarze the results or the other nput systems n a theorem. DEFINITION 4.4. A vector eld V s called an axal moton t s o the orm V = h I + h +3 I +3 where U. We use the term axal motons snce these are a translaton and rotaton actng on the same prncple axs o nerta. These can be seen as an extenson o the pure motons. DEFINITION 4.5. A vector eld V s called a coordnate moton t s o the orm V = h I + h I + h I where = or 4, = 2 or 5 and = 3 or 6. We choose the term coordnate moton snce all three axes o the coordnate rame represented. THEOREM 4.6. Under our assumptons on a submerged rgd body n an deal lud we have the ollowng characterzaton or the decouplng vector elds n terms o the number o degrees o reedom we can nput to the system. 9

20 Case : Sngle-nput system, I = {Ĩ }. The decouplng vector elds are multples o Ĩ ; these are pure motons. Case 2: Two-nput system, I2 = {Ĩ,Ĩ 2 } n whch both nputs do not act upon the same prncple axs o nerta. Then, or a netcally unque system, a vector eld V I s decouplng and only V has all but one o ts components equal to zero. In partcular, t has the orm V = h Ĩ or V = h 2 Ĩ 2 ; these are pure motons. I the nput vector elds act on the same prncpal axs o nerta, then every vector eld n I s decouplng. Case 3: Three-nput system.. Three Translatonal Inputs: I3 = {I,I 2,I 3 }. For a netcally unque system, a vector eld V I s decouplng and only V has all but one o ts components equal to zero. In partcular, t has the orm V = h I or U ; these are the pure translatonal motons. Assumng exactly two o the m s are equal, we get the axal motons as addtonal decouplng vector elds: V = h I + h I, where m = m and m m. I m = m = m, then every vector eld V I s decouplng snce V V I. 2. Three Rotatonal Inputs: I3 = {I 4,I 5,I 6 }. In ths case VV I or all V I, thus each vector eld V I s decouplng. 3. Mxed Translatonal and Rotatonal Inputs. Suppose we have a netcally unque three nput system such that the nputs are not all translatonal or all rotatonal but represents motons along three dstnct axs. Then, every vector eld V I s decouplng. I the three nput system s such that t represents an axal moton plus another nput vector eld, the decouplng vector elds are the axal motons, V = h I the pure motons, V = h I + h +3 I +3 or U, and where and +3. The remars about the symmetres n the case o three translatonal nput are vald n ths case also. Case 4: Four nput system.. Three Translaton, One Rotaton: I4 = {I,I 2,I 3,I } where V. For a netcally unque system the decouplng vector elds are the axal motons V = h 3 I 3 +h I or the coordnate motons V = h I + h I + h I wth, U,, 3. I m 3 = m or U and 3, then V = h I + h 3 I 3 + h I s also a decouplng vector eld. I m = m 2 = m 3, then every vector eld V I s a decouplng vector eld. 2. Three Rotatons, One Translaton: I4 = {I 4,I 5,I 6 } where U. Then the decouplng vector elds are the axal motons V = h I + h +3 I +3 or the coordnate motons V = h 4 I 4 + h 5 I 5 + h 6 I Two Translatons, Two Rotatons. For a netcally unque system, two prncple axes are repeated: I4 = {I } where, U, +3,I +3 20

21 then the decouplng vector elds are ether the pure motons V = h a I a or a {,,+3, + 3} or the axal motons V = h a I a + h a+3 I a+3 where a = or a =. I m = m, then addtonal decouplng vector elds or the system are the axal motons plus a multple o the other translatonal nput vector eld: V = h I + h I + h I where = +3 or = + 3. And, =, then addtonal decouplng vector elds or the system are o the orm V = h +3 I +3 + h +3I +3. For a netcally unque system, one prncple axs s repeated: I4 = {I +3,I +3 } where,, U, then the decouplng vector elds are the axal motons V = h I or the coordnate motons V = h I + h I + h +3 I +h +3 I I = then h or h +3 must be zero, and addtonal decouplng vector elds are the axal motons plus a multple o the other rotatonal nput vector eld: V = h I Case 5: Fve nput system. + h +3 I +3 + h +3I +3.. Three Translatons, Two Rotatons: I5 = {I,I 2,I 3,I } where, V, and let V such that or. For a netcally unque system the decouplng vector elds are: (a The axal motons plus a multple o a translatonal nput V = h a I a + h a+3 I a+3 + h 3I 3 where a U ( 3. (b The coordnate motons V = h a I a + h b I b + h 3 I 3 where a,b I5 ( 3. (c The motons dened by V = h 3 I 3 + h bi b 3. (d The pure moton V = h 3 I 3 Assumng that m 3 = m 3, addtonal decouplng vector elds are gven where b I 5 ( by V = h a I a + h I + h I + h I where a = 3 or a = 3 and U { 3, 3}. Assumng that 3 = 3, addtonal decouplng vector elds are gven by V = h I + h 2 I 2 + h 3 I 3 + h a I a where a = or a =. 2. Two Translatons, Three Rotatons: I5 = {I 4,I 5,I 6 } where, U, and let U such that or. For a netcally unque system the decouplng vector elds are: (a The axal motons plus a multple o a rotatonal nput V = h a I a + h a+3 I a+3 + h +3I +3 where a V (+3. (b The coordnate motons V = h a I a + h b I b + h +3 I +3 where a,b I5 (+3. (c The motons dened by V = h +3 I +3 + h bi b where b I5 (+ 3. (d The pure moton V = h +3 I +3 Loosenng the netc unqueness assumpton does not provde any addtonal decouplng vector elds n ths case. 2

22 Case 6: Sx nput system. Every vector eld s decouplng. The maor applcaton o computng vector elds s the desgn o traectores or our system. Ths s addressed n the ollowng secton. 4. Moton Plannng Our ultmate goal s to use the aorementoned theory to desgn traectores or our mechancal system. Based on Theorem 4.6 we gve partal answers to the moton plannng problem or the under-actuated scenaros consdered n the prevous secton. Notce rst that the system s ully actuated, the moton plannng problem o connectng any two conguratons at rest s trval. The smple soluton s the concatenaton o pure motons. At the other extreme, wth only one nput vector eld the rgd body s restrcted to movement n only one degree o reedom. An nterestng queston s the mnmal number o nputs whch we need n order to solve the moton plannng problem usng exclusvely decouplng vector elds. Snce we consder ran-one nematc reductons n ths paper, we wll ollow [2] or the dentons and termnology. DEFINITION 4.7. A submerged rgd body s sad to be nematcally controllable every pont n the conguraton space SE(3 s reachable va a sequence o nematc motons. We are nterested n motons such that the ntal and nal conguratons are at rest. As mentoned beore, we can reparameterze each moton to satsy boundary constrants on the controls, and to begn and end at rest. Hence, n what ollows, we assume that each nematc moton starts and ends at rest. The man obectve o ths secton s to determne how many nput vector elds, each controllng one degree o reedom, are needed to provde enough decouplng vector elds or nematc controllablty. We begn wth the ollowng obvous lemma. LEMMA 4.8. I a vehcle s nematcally controllable, t cannot be controlled by only translatonal motons or only rotatonal motons. COROLLARY 4.9. A submerged rgd body s not nematcally controllable there s only a sngle nput control vector eld: I. The same s true there are only two nput control vector elds I2 = {I } wth, U or, V. Proo. The rst statement above ollows rom Lemma 4.8 snce wth only one nput vector eld, you would have only translatonal or rotatonal moton. For the second statement, usng Lemma 4.8 we need only consder the combnatons o one translatonal moton and one rotatonal moton as nput vector elds. Upon examnaton, t s obvous that two nput control vector elds restrcts the moton o the rgd body to a 2-D plane. The ollowng remar wll be useul or our next results. REMARK 4.0. Gven any two translatonal control vector elds {I ther Le bracet vanshes: [I elds {I eld I, V,,. },, U, ] = 0. Gven two dstnct rotatonal control vector },, V, ther Le bracet produces the thrd rotatonal control vector 22

23 THEOREM 4.. Assume that our system s netcally unque. I the set o decouplng vector elds contan only one translatonal control vector eld and one rotatonal control vector eld, the nematc motons o the rgd body are restrcted to a plane n R 3. Thus, a submerged rgd body wth only two control vector elds {I nematcally controllable. } s not Proo. I, U or, V then we are done by Lemma 4.8. Thus, suppose the two nputs are I and I where U and V. Now consder L = [I ]. For = +3, L = 0 snce both nputs act on the same axs. I +3, then L = I where U and ( 3. Thus, the movement or a two nput system s dened by M = span{i } where, U, V and ( 3. Ths denes a plane n R 3. THEOREM 4.2. Assume that our system s netcally unque. I the set o decouplng vector elds contans at least one translatonal control vector eld and two dstnct rotatonal control vector elds, then the sugmerged rgd body s nematcally controllable. Proo. Assume that the decouplng vector elds or our system contan the vector elds I where U,, V and < <. An easy computaton shows that {I,[I ], [I ],[[I ],[I ]]} are sx lnearly ndependent vectors whch span R 6. Thus, there exsts a path between any two zero velocty conguratons through the concatenaton o ntegral curves o decouplng vector elds or whch each segment s reparameterzed to start and end at zero velocty. REMARK 4.3. A specal case o the prevous theorem are the coordnate motons V = h I + h I + h I, where U and, V. We wll now relate Theorem 4.6 to Theorem 4.2 by dscussng the controllablty o each case. Obvously the ollowng cases are nematcally controllable: cases 4.2, 4.3, 5., 5.2 and 6 snce ther nematc motons contan at least one translatonal control vector eld and two dsctnct rotatonal control vector elds. By Lemma 4.8 we now that cases, 3. and 3.2 are not nematcally controllable snce ther nput control vector elds are ether purely translatonal or purely rotatonal. Case 2 was dscussed n Corollary 4.9 and Theorem 4. and are not nematcally controllable. Now, we consder case 3.3. Frst notce that the nematc motons are generated by two translatons and one rotaton then t s not nematcally controllable. Indeed, assume that the set o decouplng vector elds s generated by {I } where, U, V. Assume rst that, and are dstnct and not congruent mod 3. Then, [I ] = 0, [I ] = I, [I ] = I and [I ] = 0. Thus we get that the movement s dened by M = span{i }, and the vehcle clearly can not reach η nal = (0,0,0,φ 0,θ 0,ψ 0 rom the orgn. Assume now that and = + 3. Then [I ] = 0, [I ] = 0 and [I ] = I κ where κ U and κ. Snce the bracet wth I κ wth I or I s zero by Remar 4.0, the only combnaton let s [I κ ] = I. Thus we get that the movement s dened by the nematc motons assocated to M = span{i κ }. Here agan, the vehcle,i 23

24 can not reach η nal = (0,0,0,φ 0,θ 0,ψ 0 rom the orgn. We can conclude that n case 3.3 the rgd body s nematcally controlable and only there s at least one translatonal control vector eld and two dstnct rotatonal control vector elds. Smlar computatons to those above wll show that case 4. s not nematcally controllable because there s only one rotatonal control vector eld. The above dscusson mples the sucent condton or Theorem 4.2. Practcally, ths analyss s mportant to real AUV s. Suppose you have a ullyactuated submersble whch controls heave, roll and ptch wth one set thrusters, V. Whle surge, sway, and yaw are controlled wth another set o thrusters, H. Suppose you lost control o one set o these thrusters. Wth the above analyss we can see that losng V would lmt the moton o the vehcle to a plane. However, losng the H set would not aect the controllablty o the vehcle and t would reman nematcally controllable. Thus, n the desgn process o the vehcle, we could save money by requrng that robustness need only be mplemented onto hal the system; the V set o thrusters. Also, or energy conservaton, t may be better to use only one set o thrusters to save battery le. Ths nowledge can save tme and money or the desgner and end-user ale. Here we demonstrate a practcal applcaton to summarze the results o ths secton. Suppose we have the above mentoned vehcle and we are only allowed to use the V set o thrusters. Suppose that we want to start at the orgn and end at η nal = (a,b,c,0,0,90 where a,b,c R. Note that we do not mpose a nal φ or θ; snce we have drect control on these, upon reachng η nal we can ust apply control drectly to φ and θ to reach (a,b,c,φ 0,θ 0,90. We also assume that postve b values are n the drecton o gravty. The basc dea to realze ths dsplacement s use the ptch and roll controls to pont the bottom o the vehcle n the drecton o η nal and then use pure heave or the translatonal dsplacement. Upon reachng (a,b,c,φ,θ,ψ we can do ptch and roll movements to realze η nal. For ths example, the vehcle needs c 2 + a 2 + b 2 unts usng pure to ptch tan ( b c and roll tan ( a c then translate heave, and then undo the roll and ptch. The vehcle s now at (a,b,c,0,0,0. To reach η nal apply a 90 ptch ollowed by a 90 roll ollowed by a 90 ptch. Ths concatenaton results n a 90 yaw, and the moton s realzed. It should be clear that any other rotatonal conguraton s possble as well. 5 Decouplng Vector Felds and Tme Optmalty We are now ready to descrbe the relatonshp between the exstence o decouplng vector elds and Proposton 3.0. For an ane connecton control system dened on a conguraton space Q, t s easy to very that [vlt(x,[s,vlt(x]] = vlt(2 X X, where X s any vector eld dened on T Q and S s the geodesc spray o the connecton. Applyng ths equaton to: γ γ (t = 6 = σ (ti (γ(t, (74 24