TOTAL ENERGY SHAPING CONTROL OF MECHANICAL SYSTEMS: SIMPLIFYING THE MATCHING EQUATIONS VIA COORDINATE CHANGES

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1 TOTAL ENERGY SHAPING CONTROL OF MECHANICAL SYSTEMS: SIMPLIFYING THE MATCHING EQUATIONS VIA COORDINATE CHANGES Giuseppe Viola,1 Romeo Ortega,2 Ravi Banavar Jose Angel Acosta,3 Alessanro Astolfi, Dipartimento i Informatica, Sistemi e Prouzione, Università i Roma Tor Vergata, Via el Politecnico, 1, Roma, Italy (astolfi,viola@isp.uniroma2.it) Plateau u Moulon, Gif-sur-Yvette, France (ortega@lss.supelec.fr) Systems an Control Engineering, I. I. T. - Bombay, 101 A, ACRE Builing Powai, Mumbai, , Inia (banavar@iitb.ac.in) Depto. e Ingeniería e Sistemas y Automática, Escuela Superior e Ingenieros, Camino e los Descubrimientos s/n., Sevilla, Spain (jaar@esi.us.es) Electrical Engineering Department, Imperial College, Exhibition Roa, Lonon, SW7 2AZ, UK (a.astolfi@ic.ac.uk) Abstract: Total Energy Shaping is a controller esign methoology that achieves (asymptotic) stabilization of mechanical systems enowing the close loop system with a Lagrangian or Hamiltonian structure with a esire energy function that qualifies as Lyapunov function for the esire equilibrium. The success of the metho relies on the possibility of solving two PDEs which ientify the kinetic an potential energy functions that can be assigne to the close loop. Particularly troublesome is the PDE associate to the kinetic energy which is nonlinear an non homogeneous an the solution, that efines the esire inertia matrix, must be positive efinite. In this paper we prove that we can eliminate or simplify the forcing term in this PDE moifying the target ynamics an introucing a change of coorinates in the original system. Furthermore, it is shown that, in the particular case of transformation to the Lagrangian coorinates, the possibility of simplifying the PDEs is etermine by the interaction between the Coriolis an centrifugal forces an the actuation structure. The example of a penulum on a cart is use to illustrate the results. Copyright c 2006 IFAC. Keywors: Nonlinear control, passivity-base control, energy-shaping, mechanical systems. 1. INTRODUCTION 1 The work of Giuseppe Viola was partially sponsore by the Control Training Site programme of the European Commission, contract number: HPMT-CT Corresponing author. 3 The work of J. A. Acosta was supporte by the Spanish Ministry of Science an Technology uner grants DPI an DPI Stabilization of uneractuate mechanical systems shaping their energy function, but preserving the systems structure, has attracte the atten-

2 tion of control researchers for several years. While fully actuate mechanical systems amit an arbitrary shaping of the potential energy by means of feeback, an therefore stabilization to any esire equilibrium, this is in general not possible for uneractuate systems. In certain cases this problem can be overcome by also moifying the kinetic energy of the system. The iea of total energy shaping was first introuce in (Ailon an Ortega, 1993) with the two main approaches being now: the metho of controlle Lagrangians (Bloch et al., 2000) an interconnection an amping assignment passivity base control (IDA PBC) (Ortega et al., 2002), see also the closely relate work (Fujimoto an Sugie, 2001). In both cases stabilization (of a esire equilibrium) is achieve ientifying the class of systems Lagrangian for the first metho an Hamiltonian for IDA PBC that can possibly be obtaine via feeback. The conitions uner which such a feeback law exists are calle matching conitions, an consist of a set of nonlinear partial ifferential equations (PDEs). In case these PDEs can be solve the original control system an the target ynamic system are sai to match. A lot of research effort has been evote to the solution of the matching equations. In (Bloch et al., 2000) the authors give a series of conitions on the system an the assignable inertia matrices such that the PDEs can be solve. Also, techniques to solve the PDEs have been reporte in (Blankenstein et al., 2002),(Auckly et al., 2000) an some geometric aspects of the equations are investigate in (Lewis, 2004). The case of uneractuation egree one systems has been stuie in etail in (Auckly an Kapitanski, 2002) an (Acosta et al., 2005). In the latter we prove that, if the inertia matrix an the force inuce by the potential energy (on the unactuate coorinate) are inepenent of the unactuate coorinate, then the PDEs can be explicitly solve. In (Mahinrakar et al., 2006) explicit solutions are also given for a class of two egrees of freeom systems, that inclues the interesting Acrobot example. In this paper we pursue the investigation aime at proviing constructive solutions to the PDEs. We concentrate our attention on the PDE associate to the kinetic energy which is nonlinear an non homogeneous an whose solution, that efines the esire inertia matrix, must be positive efinite. We stuy the possibility of eliminating the forcing term in this PDE. Our main contribution is the proof that it is possible to achieve this objective re parametrizing the target ynamics an introucing a change of coorinates in the original system. The class of coorinate changes that yiels an homogenous PDE is a solution of another PDE similar to the kinetic energy PDE but this time without the requirement of positive efiniteness. Furthermore, it is shown that, in the particular case of transformation to the Lagrangian coorinates, the possibility of simplifying the PDEs is etermine by the interaction between the Coriolis forces an the actuation structure. We illustrate the result with the example of the penulum on a cart. Notation: Unless inicate otherwise, all vectors in the paper are column vectors, even the graient of a scalar function: ( ) = ( ) when clear from the context the subinex in will be omitte. To simplify the expressions, the arguments of all functions will be omitte, an will be explicitly written only the first time that the function is efine. Caveat: This is an abrige version of the paper, where, for space reasons, all proofs have been omitte. The full version of the paper is available upon request to the authors. 2. BACKGROUND ON IDA PBC AND PROBLEM FORMULATION In (Blankenstein et al., 2002) an (Chang et al., 2002), it has been shown that the PDEs of the controlle Lagrangian metho an IDA PBC are the same, therefore, in the sequel we will restrict our attention to IDA PBC for which a brief review is now presente. IDA PBC was introuce in (Ortega et al., 2002) to regulate the position of uneractuate mechanical systems of the form Σ : q = ṗ 0 In q H 0 + u,(1) I n 0 p H G(q) where q R n, p R n are the generalize position an momenta, respectively, u R m an G R n m with rank G = m < n, H(q, p) = 1 2 p M (q)p + V (q) (2) is the total energy with M = M > 0 the inertia matrix, an V the potential energy. The main result of (Ortega et al., 2002) is the proof that for all matrices M (q) = M (q) Rn n an functions V (q) that satisfy the PDEs 0 = G { q (p M p) M M q (p M p) +2J 2 M p} (3) 0 = G { V M M V }, (4) for some J 2 (q, p) = J 2 (q, p) R n n an a full rank left annihilator G (q) R (n m) n of G, i.e., G G = 0 an rank(g ) = n m, the system (1) in close loop with the IDA PBC u = û(q, p), where

3 û(q, p) = (G G) G ( q H M M q H +J 2 M p), (5) takes the Hamiltonian form q 0 M M Σ : = q H ṗ M M, J 2 p H (6) where the new total energy function is H (q, p) = 1 2 p M (q)p + V (q). Further, if M is positive efinite in a neighborhoo of q R n an q = arg min V (q), then (q, 0) is a stable equilibrium point of (6) with Lyapunov function H. Clearly, the success of IDA PBC relies on the possibility of solving the PDEs (3) an (4). Particularly troublesome is the kinetic energy (KE) PDE (3), which is nonlinear an non homogeneous, an whose solution must be positive efinite. In this brief note we investigate the possibility of eliminating the forcing term G q (p M p) in this PDE via coorinate changes an reparametrization of the target ynamics. The presence of the forcing term introuces a quaratic term in M in the KE PDE that reners very ifficult its solution even with the help of the free skew symmetric matrix J 2. To reveal this eleterious effect let us eliminate the epenence on p an re-write the PDE in the equivalent form, see (Ortega et al., 2002) an (Acosta et al., 2005) for further etails, n (G k M M e i ) M q i (G k e i )M M q i M = J (q)a k (q) + A k (q)j (q), (7) for k = 1,..., n m, where e i R n is the i th vector of the n imensional Eucliean basis, J (q) α 1 (q).. α 2 (q).... α no (q) R n no, is a free matrix, α i R n, i = 1,..., n o n 2 (n) an we have efine the row vectors G k R1 n, G (G 1 )... (G n m), ( ) A k W 1 G ( ) k,..., Wno G k R n no, with the W i R n n skew symmetric matrices with elements 1s an 0s. For instance, for n = 3 we get W , W , W (8) 0 0 Remark 1. We bring to the reaers attention the fact that the target ynamics Σ is parameterize by the triple {M, V, J 2 }. Re parameterization of the target ynamics is a key step introuce in this paper. Remark 2. As shown in (Acosta et al., 2005), with the efinitions above, the free matrix J 2 can be written as n o J 2 = p M α iw i, (9) an the terms G k J 2, that appear in (3), become G k J 2 = p M J A k. Equation (7) is obtaine factoring (3) in the form p M M p, taking the symmetric part of the matrices J A k an setting the expression in brackets, which is inepenent of p, equal to zero. Remark 3. In (Acosta et al., 2005) we prove that, if n m = 1 an the inertia matrix an the force inuce by the potential energy (on the unactuate coorinate) are inepenent of the unactuate coorinate, then the PDEs can be explicitly solve. The first assumption on the inertia matrix implies, precisely, that the forcing term G q (p M p) = 0, which is essential for the construction of the solutions (see Proposition 3). 3. GENERATING AN HOMOGENEOUS KINETIC ENERGY PDE Our strategy to eliminate the forcing term in the KE PDE consists of two steps. First, we express system (1) in the new coorinates (q, p), with p = T (q) p, where T R n n is full rank, yieling: where an Σ : = q p 0 T T F T G q H p H u, (10) F 22 = T S(q, p) S (q, p) T, H(q, p) = 1 2 p T (q)m (q)t (q) p + V (q) S(q, p) = q (T (q) p). (11) It s worth noting that these equations can be seen as a particular form of the Boltzmann-Hamel equations (see (Whittaker, 1988)), in which p is the vector of quasi-velocities, relate to the velocity vector q by means of the relation p = T (q)m(q) q. Secon, we efine new target ynamics, in the coorinates (q, p), as 0 F 12 q H Σ : = q p J, (12) 2 (q, p) F 12 p H

4 where H (q, p) = 1 2 p M (q) p + Ṽ(q), (13) F 12 = M (q)t (q) M (q), M R n n an J 2 = J 2 is free. The propose target ynamics are clearly compatible with the new system representation in the sense that the first n equations are alreay matche. In Section 4 we establish the connection between (12) an the target ynamic system (6) expresse in the new coorinates see also Remark 7. To state our main result we nee the following assumption. Assumption A The full rank matrix T is such that, for k = 1,..., n m, n T M e i G T k + T (e i G k q i q ) M T i +G k e i T M T q i = 0. (14) Proposition 1. Consier the system (1) an the partial change of coorinates p = T (q) p where T satisfies Assumption A. For all matrices M (q) = M (q) Rn n an functions Ṽ(q) that satisfy the PDEs G T M T M q ( p M p) 2 J 2 M p = 0 (15) G T M T M Ṽ = G V, (16) for some J 2 (q, p) = J 2 (q, p) R n n, the system (1) in close loop with the IDA PBC u = ˆũ(q, p), where ˆũ(q, ( p) = (G G) G q H + T p + T J 2 M p T M ) T M q H (17) takes, in the coorinates (q, p), the Hamiltonian form (12), (13). Proposition 2. Assumption A hols with T = M if an only if G (q)c(q, q) q = 0, (18) where C R n n is the matrix of Coriolis an centrifugal forces of the mechanical system (1). Now we state a slightly moifie version of a result reporte in (Acosta et al., 2005), which gives a constructive solution of the homogeneous KE PDE (15) for systems with uneractuation egree one. Proposition 3. Consier equation (15). Suppose that n m = 1, M oes not epen on the unactuate coorinate, an the matrices G an T are function of a single element of q, say q r, r {1,..., n}. Then, for all esire locally positive efinite inertia matrices of the form qr M (q r ) = T (µ)g(µ)ψ(µ)g (µ)t T (µ)µ qr + M 0 (19) where the matrix function Ψ = Ψ R (n) (n) an the constant matrix M 0 = ( M 0 ) > 0 R n n, may be arbitrarily chosen, there exists a matrix J 2 such that the KE PDE (15) hols in a neighborhoo of q r. Remark 4. Comparing (3) with (15) we notice the absence of the forcing term in the latter therefore, the PDE that nees to be solve is (in principle) simpler. As it will be shown in Proposition 4, this simplification has been achieve without moifying the potential energy PDE (4), but it is subject to the conition of fining a matrix T satisfying Assumption A. Remark 5. It is interesting to compare the original KE PDE (7) an the aitional PDE that nees to be solve (14). At first glance, it may be argue that (14) is as complicate as, if not more complicate than, (7). Notice, however, that the terms e i G k in (14) are matrices with only one non zero row, while G k M M e i in (7) is a scalar that mixes all the terms in M. Also, we stress the fact that, contrary to M that must be symmetric an positive efinite, the only conition on T is invertibility. For instance, in the example of the penulum on a cart system, a solution to (14) is trivially obtaine while no obvious solution for (7) is available. Remark 6. To establish Proposition 1 we in t nee to use the transforme system ynamics (10) in Hamiltonian form. This is consistent with the fact that IDA PBC is applicable to systems in the general form ẋ = f(x) + g(x)u; see e.g., (Ortega an Garcia-Canseco, 2004). Remark 7. Clearly, changing coorinates of the system an the target ynamics oes not affect the matching conitions an, consequently, the PDEs will be equivalent. The subtlety here is that, as inicate in Remark 1, the target ynamic system Σ is parameterize by the triple {M, V, J 2 }, while the system Σ is parameterize by the triple { M, Ṽ, J 2 }. Along the same lines, feeback actions of the form u = α(q, p) + β(q, p)v, with β R m m full rank, will not affect the PDEs that live in Ker G. Inee, for a system of the form ẋ = f(x)+g(x)u an target y-

5 (q, p) Σ û Σ (q, p) {M, V, J 2 } triple {M, V, J 2 } solves the original matching equations (3),(4), where 5 (q, p) Φ Σ ˆũ Ψ, Φ Σ (q, p) { M, Ṽ, J 2 } Fig. 1. Diagrammatic escription of the systems transformations. namics ẋ = F (x) H the matching equations are g f = g F H, with g g = 0, inepenently of the feeback action. 4. SOLVING THE ORIGINAL PDES Proposition 1 establishes that, solving the new PDEs (15), (16), the system Σ escribe by equations (1) in close loop with the IDA PBC (17) takes, in the coorinates (q, p), the Hamiltonian form Σ escribe by equations (13). Three natural questions arise. What are the ynamics of the close loop system in the original coorinates (q, p)? What is the relationship between the solutions of the new matching problem { M, Ṽ, J 2 } an the solutions of the original matching problem {M, V, J 2 }? What is the relationship between the original matching controller û(q, p) an the new one ˆũ(q, p)? The answers to these questions are given in the proposition below. The rationale of the proposition is best explaine referring to Fig. 1. The connections between the noes Σ, Σ an Σ are given by Proposition 1. It remains to establish the connection with the original target ynamics noe Σ. Towars this en, we write Σ, escribe by equations (6), in the new coorinates an prove the existence of a bijective mapping Ψ : { M, Ṽ, J 2 } {M, V, J 2 }, that makes the transforme system equal to Σ that is, with the same structure matrix an the same Hamiltonian function. 4 This proves that Σ an Σ match an, consequently, the corresponing parameters {M, V, J 2 } solve the PDEs an efine the control û(q, p). Proposition 4. The triple { M, Ṽ, J 2 } solves the new matching equations (15),(16) if an only if the 4 The mapping Ψ can also be erive computing the Poisson brackets of the coorinates (q, p), as one in (Blankenstein et al., 2002) to establish the equivalence between controlle Lagrangians an IDA PBC. M = T M T V = Ṽ J 2 (q, p) = T J 2 (q, T p)t +S(q, T p)m T M T T M T M S (q, T p) (20) an S(q, p) is given in (11). Furthermore, the control (5) that matches Σ to Σ is obtaine as û(q, p) = ˆũ(q, T (q)p), with ˆũ efine in (17). 5. THE PENDULUM ON A CART EXAMPLE The ynamic equations of the penulum on a cart are given by (1) with n = 2, m = 1, an 1 b cos q M(q 1 ) = 1, V (q b cos q 1 c 1 ) = a cos q 1, G = e 2, a = g l, b = 1 l, c = M + m ml 2 where q 1 enotes the penulum angle with the upright vertical, q 2 the cart position, m an l are, respectively, the mass an the length of the penulum, M is the mass of the cart an g is the gravity acceleration. The equilibrium to be stabilize is the upwar position of the penulum with the cart place in any esire location, which correspons to q 1 = 0 an an arbitrary q 2. Noting that G = e 1, the KE PDE (7) takes the form 2 (e 1 M M e i ) M q i 0 α = 1 (q) α 1 (q) 2α 2 (q) M M M q 1, (21) with α i being free functions. Using these functions we can solve two of the three equations above, so it remains only one PDE to be solve. To simplify the expression of this equation we make M function only of q 1 leaing to the ODE (cm 11 bm 12 cos q 1 ) m 11 q 1 = 2b sin q 1 c b 2 cos 2 q 1 b cos q 1 (cm m 2 12) (c + b 2 cos 2 q 1 )m 11 m 12, where m ij (q 1 ) is the ij element of the matrix M. Even using m 12 as a egree of freeom fining a solution to this ODE is a aunting task. To simplify the PDE we procee, then, to apply the technique propose in the paper. By computing the Coriolis an centrifugal forces matrix using 5 For simplicity, we have omitte the argument q in the functions that epen only on q.

6 the Christoffel symbols of the secon kin (Kelly et al., 2005), it is easy to see that the conition G C q = 0 is satisfie. Therefore, we propose to take T = M an search for a solution of the homogeneous PDE (15), which becomes: G M M q ( p M p) 2 J 2 M p = 0. (22) It is easy to see that the hypotheses of Proposition 3 are satisfie with r = 1. By selecting k sin µ Ψ(µ) = m 3 b 2 cos 2 µ, kb 2 M 0 = 3 cos3 q 1 kb 2 cos2 q 1 kb 2 cos2 q 1 k cos q 1 + m 0 22 with k > 0 an m free parameters, it follows that a solution is provie by kb 2 M = 3 cos3 q 1 kb 2 cos2 q 1 kb, 2 cos2 q 1 k cos q 1 + m 0 22 J 2 = p α1 0 1 M. α 2 0 Such a solution is positive efinite an boune for all q 1 ( π 2, π 2 ). Regaring the potential energy, it can be seen that a solution of the PDE (16) is given by Ṽ = 3a kb 2 cos 2 q 1 + P 2 q 2 q b ln(sec q 1 + tan q 1 ) + 6m0 22 tan q 1, kb with P > 0 arbitrary an q 2 the cart position to be stabilize. The asymptotic analysis of the close loop system an some simulations may be foun in (Acosta et al., 2005), where the problem has been solve by using Lagrangian coorinates an carrying out a partial feeback linearization. 6. CONCLUSIONS In this paper we have investigate a way to simplify the solution of the matching equations of IDA-PBC for a class of uneractuate mechanical systems. We have shown that it is possible to transform the KE PDE into an homogeneous one by using coorinate transformations an a reparametrization of the target ynamics. This can be achieve provie that a solution of another PDE, involving the coorinates transformation matrix, can be foun. This new PDE is similar to the kinetic energy PDE, but without the requirement of positive efiniteness of its solutions. Moreover, it has been shown that in the particular case of transformation to the Lagrangian coorinates, the possibility of simplifying the PDEs is etermine by the interaction between the Coriolis an centrifugal forces an the actuation structure. The propose technique has been successfully applie to the penulum on a cart. REFERENCES Acosta, J. A., R. Ortega, A. Astolfi an A. M. Mahinrakar (2005). Interconnection an amping assignment passivity base control of mechanical systems with uneractuation egree one. IEEE Trans. Automat. Contr.. Ailon, A. an R. Ortega (1993). An observerbase controller for robot manipulators with flexible joints. Syst.& Cont. Letters 21, Auckly, D. an L. Kapitanski (2002). On the λ equations for matching control laws. SIAM J. Control an Optimization 41, Auckly, D., L. Kapitanski an W. White (2000). Control of nonlinear uneractuate systems. Comm. Pure Appl. Math. 3, Blankenstein, G., R. Ortega an A.J. van er Schaft (2002). The matching conitions of controlle lagrangians an interconnection assigment passivity base control. Int J of Control 75, Bloch, A., N. Leonar an J. Marsen (2000). Controlle lagrangians an the stabilization of mechanical systems. IEEE Trans. Automat. Contr.. Chang, D.E., A.M. Bloch, N.E. Leonar, J.E. Marsen an C.A. Woolsey (2002). The equivalence of controlle lagrangian an controlle hamiltonian systems for simple mechanical systems. ESAIM: Control, Optimisation, an Calculus of Variations 45, Fujimoto, K. an T. Sugie (2001). Canonical transformations an stabilization of generalize hamiltonian systems. Systems an Control Letters 42, Kelly, R., V. Santibanez an A. Loria (2005). Control of robot manipulators in joint space. Springer Verlag. Lonon. Lewis, A. (2004). Notes on energy shaping. 43r IEEE Conf Decision an Control, Dec 14 17, 2004, Paraise Islan, Bahamas. Mahinrakar, A. D., A. Astolfi, R. Ortega an G. Viola (2006). Further constructive results on interconnection an amping assignment control of mechanical systems: The acrobot example. American Control Conference, Minneapolis, USA, June Ortega, R. an E. Garcia-Canseco (2004). Interconnection an amping assignment passivity base control: A survey. European J of Control 10, Ortega, R., M. Spong, F. Gomez an G. Blankenstein (2002). Stabilization of uneractuate mechanical systems via interconnection an amping assignment. IEEE Trans. Automat. Contr. AC 47, Whittaker, E.T. (1988). A treatise on the analytical ynamics of particles an rigi boies. Cambrige University Press.