T R R ECHNICAL ESEARCH EPORT Controllability of Lie-Poisson Reduced Dynamics by V. Manikonda, P.S. Krishnaprasad T.R. 97-59 ISR INSTITUTE FOR SYSTEMS RESEARCH Sponsored by the National Science Foundation Engineering Research Center Program, the University of Maryland, Harvard University, and Industry Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. 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SUPPLEMENTARY NOTES The original document contains color images. 14. ABSTRACT see report 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF ABSTRACT OF PAGES RESPONSIBLE PERSON a. REPORT b. ABSTRACT c. THIS PAGE 27 unclassified unclassified unclassified Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18 (cid:3) Controllability of Lie-Poisson Reduced Dynamics Vikram Manikonda and P.S. Krishnaprasad fvikram, [email protected] Department of Electrical Engineering and Institute for Systems Research University of Maryland, College Park, MD 20742 Abstract In this paper we discuss controllability of Lie-Poisson reduced dynamics of a class of mechanical systems. We prove conditions (boundednessof coadjoint orbits and existence of a radially unbounded Lyapunov function) under which the drift vector (cid:12)eld (of the reduced system) is weakly positively Poisson stable (WPPS). TheWPPSnatureofthedriftvector(cid:12)eldalongwiththeLiealgebrarankcondition is used to show controllability of the reduced system. We discuss the dynamics, Lie-Poisson reduction, and controllability of hovercraft, spacecraft and underwater vehicles, all treated as rigid bodies. 1 Introduction A geometric approach to the study of mechanical systems has had a profound in(cid:13)u- ence in recent years on our understanding of dynamics and control aspects. Playing an essential role in this are recent developments in reduction theory (cf. [1]), i.e. the exploitation of invariance of the controlled dynamics to a group of transformations (the symmetry group). The existence of a symmetry group permits dropping of the dynamics to a lower dimensional (reduced) space. Lagrangian reduction [2, 3] involves dropping the Euler-Lagrange equations to the quotient of the velocity phase space given by the symmetry group while Hamiltonian reduction involves projecting the Poisson bracket to the reduced (quotient) space which also inherits a Poisson structure (attributed to Lie and Berezin-Kirillov-Kostant-Souriau, see the work of Weinstein for historical remarks [4]). In particular, if the con(cid:12)guration space of the system can be identi(cid:12)ed with a Lie (cid:3) (cid:3) group G, a left invariant Hamiltonian on T G gives rise to reduced dynamics on T G=G (cid:3) This research was supported in parts by grants from the National Science Foundation’s Engineer- ing Research Centers Program: NSFD CDR 8803012 and by the Army Research O(cid:14)ce under Smart Structures URI Contract No. DAAL03-92-G0121. 1 (cid:3) which is isomorphic to G the dual of the Lie algebra of G. The reduced bracket is now the Lie-Poisson bracket. The complete dynamics can then be reconstructed from the reduced system. The geometric phase [5] associated with a trajectory of the reduced system describes the motion of the complete (lifted) system. There has also been recent progress in the area of control in the presence of nonholonomic constraints [6, 7, 3, 8]. For a large class of mechanical systems, the con(cid:12)guration space can be identi(cid:12)ed with a Lie group G. Often the dynamics of such systems are G-invariant and hence they (cid:3) can be reduced to obtain a set of Lie-Poisson reduced dynamics on T G=G. Examples of such systems include hovercraft, spacecraft and underwater vehicles modeled as rigid bodies. The design and control of autonomous versions of these vehicles has been of much recent interest. For example the amphibious versatility of hovercraft has given them a role in specialized applications including search and rescue, emergency medical services, ice breaking, Arctic o(cid:11)-shoreexploration, andrecreationalactivities[9]. Certain environmental aspects (such as ice-roughness, Arctic rubble (cid:12)elds etc.) also provide a niche for operations by hovercraft. Similarly a growing industry in underwater vehicles for deep sea explorations has led to the demand for more versatile, robust and high performance autonomous vehicles that can cope with actuator failures, disturbances, exploit sensor based local navigation etc. In this paper we discuss the controllability of the Lie-Poisson reduced dynamics of a class of mechanical systems which include as examples hovercraft, spacecraft and bottom heavy underwater vehicles. In each case we identify the con(cid:12)guration space with a Lie group G. The G-invariance of the Hamiltonian and the forcing term (control) is used to (cid:3) (cid:3) obtain a set of Lie-Poisson equations on T G=G which is isomorphic to G the dual of the Lie algebra of G. We show that depending on the existence of a radially unbounded Lyapunov type function, the driftvector (cid:12)eld(ofthe reduced system) isweakly positively Poisson stable (WPPS). The WPPS nature of the drift vector (cid:12)eld along with the Lie algebra rank condition is used to show controllability of the reduced system. The paper is organized as follows. In section 2 we present a brief overview of Lie-Poisson reduction. In section 3 we present our main result on controllability of Lie-Poisson reduced dynamics. In section 4 we discuss in some detail, the dynamics, reduction and reduced space controllabilityof the hovercraft, the spacecraft and the underwater vehicle. Conclusions and future work is discussed in section 5. 2 Lie-Poisson Reduction Recall(cf. [1,10])thatifGisasymmetrygroupactingonaPoissonmanifoldM, thenthe quotient manifold M=G inherits a Poisson structure so that whenever P;Q : M=G ! IR correspond to G invariant functions, P;Q : M ! IR, their Poisson bracket fP;QgM=G e e corresponds to the G-invariant function fP;QgM. If H : M ! IR is a G-invariant e e Hamiltonian, then H descends to H : M=G ! IR and determines the reduced dynamics f 2 on M=G. The solutions of the reduced Hamiltonian system on M=G are projections of 4 (cid:3) the solutions of the complete system de(cid:12)ned on M. In particular if M = T G and (cid:3) G acts on itself by left translations then M=G (cid:17) G , the dual of the Lie algebra of G, (cid:3) (cid:3) a left invariant Hamiltonian on T G gives rise to reduced dynamics on G(cid:0) (the space (cid:3) G associated with the minus Poisson structure). The reduced bracket is now the minus Lie-Poisson bracket, f(cid:1);(cid:1)g(cid:0) de(cid:12)ned in its coordinate free from by fF;Hg(cid:0)(x) = (cid:0) < x;[rF(x);rH(x)] > : b b Let fX1;(cid:1)(cid:1)(cid:1);Xrg and fX1;(cid:1)(cid:1)(cid:1);Xrg be a basis for the Lie algebra G and the dual basis (cid:3) b i (cid:3) r b G respectively, i.e. < Xi;Xj >= (cid:14)j. Any (cid:22) 2 G(cid:0) can be expressed as (cid:22) = i=1(cid:22)iXi 1 (cid:3) and the Lie-Poisson bracket of two di(cid:11)erentiable functions P;Q 2 C (G ) is given by P r k k@F @H fF;Hg(cid:0)((cid:22)) = (cid:0) cij(cid:22) i j (1) i;j;k=1 @(cid:22) @(cid:22) X k wherecij;i;j;k = 1;(cid:1)(cid:1)(cid:1);rarethestructureconstantsofG relativetoabasisfX1;(cid:1)(cid:1)(cid:1);Xrg. Equivalently (1) can be written as T fF;Hg(cid:0)((cid:22)) = rF (cid:3)((cid:22))rH (2) where r k k [(cid:3)((cid:22))]ij = (cid:0) cij(cid:22) k=1 X The rank of Poisson tensor (cid:3) determines the nontrivial Casimirs of (cid:22). The Lie-Poisson reduced dynamics can now be expressed (cid:22)_i = f(cid:22)i;Hg(cid:0); i = 1;(cid:1)(cid:1)(cid:1);r (3) f where H is the reduced Hamiltonian. In thefrest of this paper we will denote the Lie-Poisson reduced dynamics with the following notation. (cid:22)_ = f(cid:22);Hg(cid:0) (4) f The ith component on the right hand side being f(cid:22)i;;Hg(cid:0) , i = 1;(cid:1)(cid:1)(cid:1);r. (cid:3) The induced symplectic foliation by Lie-Poisson brackfet on G has a particularly nice interpretation in terms of the dual to the adjoint representation of the underlying Lie group G on the Lie algebra G. This is given by the following theorem, which appears to be due to Kirillov[11, 12], Arnold [13], Kostant [14] and Souriau [15], though similar ideas (cid:12)rst appear in the work of Lie, Borel and Weil. 3 (cid:3) (cid:3) Theorem 1 Let G be a connected Lie group with coadjoint representation Ad G on G . (cid:3) (cid:3) Then the orbits of Ad G are immersed submanifolds of G and are precisely the leaves (cid:3) of the symplectic foliation induced by the Lie-Poisson bracket on G . Moreover, for each (cid:3) (cid:3) g 2 G, the coadjoint map Ad G is a Poisson mapping on G preserving the leaves of the foliation. As(cid:13)owsof(4)remainoncoadjointorbitsonwhichtheystarted,thegeometryofcoadjoint orbits plays an important role in understanding the dynamics of the Lie-Poisson reduced equations. 3 Poisson Stability and Controllability The state space of a large class of mechanical systems such as hovercraft, spacecraft underwater vehicle etc. can be identi(cid:12)ed with a Lie group G. The Hamiltoniandynamics (cid:3) (de(cid:12)ned on T G) of these systems subject to external forces can be written in the form of a control system as m x_ = f(x)+ gi(x)ui (5) i=1 X (cid:3) where x 2 T G, f(x) = fx;Hg and u = (u1;(cid:1)(cid:1)(cid:1)um). (H is the Hamiltonian de(cid:12)ned on (cid:3) T G). Here we do not assume that gi are Hamiltonian vector (cid:12)elds. Often we observe that the vector (cid:12)elds f and gi’s are G-invariant. This allows us to drop the the vector (cid:3) (cid:3) (cid:12)elds f and gi from T G to T G=G and the reduced dynamics take the form m (cid:22)_ = f((cid:22))+ gi((cid:22))ui (6) i=1 X e e f (cid:3) (cid:3) where(cid:22) 2 T G=G,f andgi aretheprojectionsoff andg onT G=G. Fromthediscussion in section 2 we know that f = f(cid:22);Hg(cid:0) where H is the reduced Hamiltonian. Hence (6) e e can be written as e f f m (cid:22)_ = f(cid:22);Hg(cid:0) + gi((cid:22))ui (7) i=1 X f e f Studying controllabilityof systems of the form (7) or of more general systems of the form m n x_ = f(x)+ gi(x)ui; x 2 IR u = (u1;(cid:1)(cid:1)(cid:1)um) 2 U (8) i=1 X 4 is usually a hard problem. We know that if a system of the form (8) satis(cid:12)es the Lie algebra rank condition (LARC) then it is locally accessible, and in addition if f = 0 then LARC implies that the system is controllable. (See appendix for details.) While the kinematic equations of motion can often be written as a drift free system, once dynamics are included LARC does not imply controllability. Proving controllability is usually much harder than proving accessibility. In [16] su(cid:14)cient conditions are given, in terms of a \group action", that a locally accessible system is also locally reachable. In [17]su(cid:14)cient conditionsforthe controllabilityofa conservative dynamicalpolysystem on a compact Riemannian manifold are presented. More recently this result was extended by [18] to dynamical polysystems where the drift vector (cid:12)eld was required to be weakly positively Poisson stable. We extend this result to Lie-Poisson reduced dynamics. We prove conditions under which the reduced dynamics are controllable. Before we present our results we introduce some de(cid:12)nitions and related theorems regarding Poisson stable systems. We follow the development in [18, 19, 20, 21, 22, 23]. X Let X be a smooth complete vector (cid:12)eld on M and let (cid:30)t ((cid:1)) denote its (cid:13)ow. De(cid:12)nition: A point p 2 M is called positively Poisson stable for X if for all T > 0 and X any neighborhood Vp of p, there exists a time t > T, such that (cid:30)t (p) 2 Vp. The vector (cid:12)eld X is called positively Poisson stable if the set of Poisson stable points for X is dense in M. De(cid:12)nition: A point p 2 M is called nonwandering point of X if for all T > 0, any X neighborhood Vp of p, there exists a time t > T such that (cid:30)t (Vp) Vp 6= (cid:30), where X X (cid:30)t (Vp) = f(cid:30)t (q) j q 2 Vpg. T One should observe here that though it is a su(cid:14)cient condition that the nonwandering set of a positively Poisson stable vector (cid:12)eld is the entire manifold M, there could exist weakerconditionsunderwhichthenonwanderingsetisM. Thisgivesrisetothede(cid:12)nition of a weakly positively Poisson stable (WPPS). De(cid:12)nition: The vector (cid:12)eld X is called weakly positively Poisson stable if its nonwan- dering set is M. The following theorem on controllability is due to Kuang-Yow Lian et. all. [18]. Earlier versions of this theorem and the corollary that follows, where the hypothesis required f to be only Poisson Stable, are due to Lobry [17], Bonnard and Crouch [24]. Theorem 2 If the system m m x_ = f(x)+ gi(x)ui; u = (u1;(cid:1)(cid:1)(cid:1)um) 2 U (cid:26) IR i=1 X where U contains fu j juij (cid:20) Mi 6= 0;i;(cid:1)(cid:1)(cid:1);mg is such that f is a weakly positively Poisson stable vector (cid:12)eld, then the system is controllable if the accessibility LARC is satis(cid:12)ed. 5 Corollary 1 If the system m x_ = f(x)+ gi(x)ui; u = (u1;(cid:1)(cid:1)(cid:1)um) 2 U i=1 X is such that f is a weakly positively Poisson stable vector (cid:12)eld, and accessibility LARC is satis(cid:12)ed, then the system with controls constrained by ui 2 f(cid:0)Mi;Mig;Mi > 0; i = 1;(cid:1)(cid:1)(cid:1);m is controllable. In the setting of Lie Poisson reduced dynamics we can make the following observation. Theorem 3 Let G be a Lie group that acts on itself by left (right) translations. Let (cid:3) H : T G ! IR be a left (right) invariant Hamiltonian. Then, (cid:3) (cid:3) (i) If G is a compact group, the coadjoint orbits of G = T G=G are bounded and the Lie- Poisson reduced Hamiltonian vector (cid:12)eld XH de(cid:12)ned by XH((cid:22)) = f(cid:22);Hg(cid:0)(+) is WPPS. (ii) If G is a noncompact group then the Lie-Poisson reduced Hamiltfonian vector (cid:12)eld e e (cid:3) X is WPPS if there exists a function V : G ! IR such that V((cid:22)) is bounded below, H _ V((cid:22)) ! 1 as (cid:22) ! 1 and V = 0 along trajectories of the system. e (cid:3) (cid:3) Here H = HjG(cid:3) is the restriction of H to the quotient manifoldG = T G=G and f(cid:1);(cid:1)g(cid:0)(+) (cid:3) (cid:3) is the induced minus (plus) Lie-Poisson bracket on the quotient manifold G = T G=G. f (cid:3) (cid:3) (cid:3) Proof: (i) The map (cid:21) : T G ! G(cid:0) is a Poisson map, and the Poisson manifold G(cid:0) is symplectically foliated by co-adjoint orbits, i.e. it is a disjoint union of symplectic leaves (cid:3) that are just the co-adjoint orbits. Any Hamiltonian system on G(cid:0) leaves invariant the symplectic leaves and hence restricts to a canonical Hamiltonian system on a leaf. To (cid:3) study the dynamics of a particular system with initial condition (cid:22)(0) 2 G(cid:0), we therefore restrict attention to the co-adjoint orbit through (cid:22)(0). By hypothesis, each co-adjoint orbit is compact. The (cid:13)ow starting at (cid:22)(0) preserves the symplectic volume measure on the orbit. Hence by the Poincar(cid:19)e Recurrence Theorem, we know that for almost every (cid:3) X point p 2 G(cid:0) and any neighborhood Vp of p there exists a time t > T such that (cid:30)t (p) returns to Vp i.e. XH is WPPS. (cid:3) (ii)Let D = f(cid:22) j V((cid:22)) (cid:20) Eg and Orb((cid:1)) denote the coadjoint through (cid:22)(0) in G(cid:0). Then e the integral curve of X starting at (cid:22)(0) lies entirely in the set S = D\Orb((cid:1)). Since S H (cid:3) 2 closed and bounded in G(cid:0), it is compact in Orb((cid:1)), and hence as before XH is WPPS. e In many situations the function H(cid:30) = H + (cid:30)(Ci) where H is the reduced Hamiltonian e and Ci are the Casimirs are a good choice for V((cid:1)). f f Remark: In our present setting of Lie-Poisson reduced dynamics, WPPS conditions in theorem2canbeveri(cid:12)edwhenever thehypotheses oftheorem3hold. OnceWWPSofthe drift vector (cid:12)eld has been established theorem 2 can be used to conclude controllability. 6 4 Examples In this section we discuss the controllability of the Lie-Poisson reduced dynamics of hovercraft, spacecraft and the underwater vehicles. These systems satisfy conditions of theorem3. Thekinematicsanddynamicsoftheseexamplescanalsobefoundin[1,25,26] for completeness we present necessary details here. 4.1 Hovercraft - Planar Rigid Body with a Thruster In this section we discuss the dynamics of a planar rigid body with a thruster. The r r r con(cid:12)guration of the system is shown in Figure 1. Let fe1;e2;e3g be an inertial frame of b b b reference (cid:12)xed atOandfe1;e2;e3gbeabodyframe(cid:12)xedontherigidbodyB atitscenter r r of mass. Since the rigid body is restricted to move in the e1e2 plane, a typical material b r b point q in the rigid body is then represented in the inertial frame as q = Rq +r where R is an element of SO(2), the special orthogonal group of 2(cid:2)2 matrices and r = (x;y) is a vector from O to the center of mass of B. Hence at any instant, the con(cid:12)guration X(t) of B can be uniquely identi(cid:12)ed by the pair (R;r) or equivalently as an element of SE(2), the Special Euclidean group of 2(cid:2)2 matrices. Recall 4 R r 2 SE(2) = f j R 2 SO(2);r 2 IR g 0 1 ! b Let us assume that the thruster is mounted at the point C de(cid:12)ned by the vector d in r body coordinates and d in the inertial frame of reference. The thrusters exert a force r f in inertial coordinates such that the line of action of the force passes through C and b makes an angle (cid:30) with the vector d . We now derive the equations of motion of a rigid r body subject to a force f along a speci(cid:12)ed line of action. 4.1.1 Symmetry and Reduction We assume fornow thatthe rigidbody (which willlaterbe approximatedtoa hovercraft) has su(cid:14)cient lift and can glide on the surface with no friction. The Lagrangian L : TSE(2) ! IR for this case is simply the kinetic energy, i.e. L(R;r;R_;r_) = 1I(cid:10)2 + mkr_k2 (9) 2 2 where m is the total mass, I is the moment of inertia of B in the body frame, (cid:10) is the scalar body angular velocity about the center of mass. The corresponding Hamiltonian 7 eb 3 eb 2 er 3 b d l c eb 1 r j r f dr er 2 o er 1 Figure 1: Planar rigid body with thruster is given by 2 1 2 kpk H = (cid:5) + (10) 2I 2m where (cid:5) = I(cid:10) is the body angular momentum and p = mr_ is the spatial linear momen- tum. Collecting together the Newton-Euler balance laws one can write the dynamics in spatial (inertial) variables (R;r;(cid:25);p) as R_ = R(cid:10) (11-a) r_ = p=m (11-b) b r r (cid:25)_ = d (cid:2)f (11-c) r p_ = f (11-d) where 0 (cid:0)1 (cid:10) = (cid:10);: 1 0 ! b We observe that the lifted action of G on TSE(2) de(cid:12)ned by (cid:3) (cid:30)g : TSE(2) ! TSE(2) (R;r;R_;r_) 7! (R(cid:22)R;R(cid:22)r+r(cid:22);R(cid:22)R_;R(cid:22)r_) 8