Total Energy Shaping of a Class of Underactuated Port-Hamiltonian Systems using a New Set of Closed-Loop Potential Shape Variables*

Transcription

1 51st IEEE Conference on Decision an Control December Maui Hawaii USA Total Energy Shaping of a Class of Uneractuate Port-Hamiltonian Systems using a New Set of Close-Loop Potential Shape Variables* Christopher Renton 1 Yik Ren Teo 2 an Tristan Perez 3 Abstract This paper proposes a metho for esigning setpoint regulation controllers for a class of uneractuate mechanical systems in Port-Hamiltonian System (PHS) form. A new set of potential shape variables in close loop is propose which can replace the set of open loop shape variables the configuration variables that appear in the kinetic energy. With this choice the close-loop potential energy contains free functions of the new variables. By expressing the regulation objective in terms of these new potential shape variables the esire equilibrium can be assigne an there is freeom to reshape the potential energy to achieve performance whilst maintaining the PHS form in close loop. This complements contemporary results in the literature which preserve the openloop shape variables. As a case stuy we consier a robotic manipulator mounte on a flexible base an compensate for the motion of the base while positioning the en effector with respect to the groun reference. We compare the propose control strategy with special cases that correspon to other energy shaping strategies previously propose in the literature. I. INTRODUCTION A mechanical system with configuration variables q Q an conjugate momenta p L/ q P where L is the Lagrangian is calle fully actuate if the forces τ F prouce by actuator configuration is such that im F = im P. If im F < im P the system is sai to be uneractuate [1]. For fully-actuate mechanical systems set-point regulation can be achieve by reshaping the potential energy such that it attains its minimum at the esire equilibrium. A positive efinite potential energy function is chosen for the close-loop system an the control law can be foun by matching the ynamics of the open-loop an esire closeloop systems [2]. If the mechanical system is moelle as a Port Hamiltonian System (PHS) 1 the more general metho of Interconnection an Damping Assignment Passivity Base Control (IDA-PBC) allows not only potential energy shaping but also kinetic energy shaping that is total energy shaping [4]. The matching conitions must satisfy a PDE in terms of both kinetic an potential energy an the solution of *This work has been supporte by Boeing Research an Technology Australia (BR&TA) through a collaborative research grant. 1 C. Renton is with the School of Engineering University of Newcastle Australia christopher.renton at uon.eu.au 2 Y. Teo is with the School of Engineering University of Newcastle Australia yikren.teo at uon.eu.au 3 T. Perez is with the School of Engineering University of Newcastle Australia Tristan.Perez at newcastle.eu.au an the Centre for Ships an Ocean Structures (CeSOS) Norwegian University of Science an Technology Tronheim Norway. 1 Also referre to as Port-Controlle Hamiltonian Systems in the literature [3]. this PDE provies the control law [4]. For uneractuate mechanical systems this PDE amits solutions only for a class of achievable total energy in close loop. Such a PDE in general is ifficult to solve. As iscusse in [5 p. 21] shape variables are configuration variables that appear in the open-loop kinetic energy. The remaining set of configuration variables are calle external variables. Uneractuate mechanical systems can be classifie accoring to which shape variables have actuation [5]. In [6] an [7] the control of a class of mechanical systems with unactuate shape variables 2 is consiere. The controllers are esigne to preserve the open-loop shape variables in the close-loop kinetic energy. A constructive metho is propose in [8] to solve the matching PDE for systems with uneractuation egree one (im F = im P 1) an more recently in [9] this approach is extene to reuce the problem of solving the non-homogeneous kinetic energy PDE to a simpler problem of fining a transformation of the momentum state. Both approaches in [8] [9] also choose to preserve the shape variables in the close loop. In this paper we consier a class of mechanical systems with unactuate external variables with stable ynamics. For this class of systems potential energy shaping can be applie. This implements a partial state-feeback that is the external variables o not appear in the control law. The stability of the close loop system relies on the passivity of the unactuate variables [2]. A performance issue arises if the ynamics in the unactuate variables are relatively slow since there is no irect control authority in these channels. Hence one can attempt to shape the total energy giving aitional freeom to the esigner. The performance of these controllers however may be limite if the open-loop shape variables are preserve in the close loop. Our moest contribution is therefore a metho to esign set-point regulation controllers for the above mentione class of uneractuate mechanical systems in PHS form. We o so by proposing a new set of potential shape variables in close loop. Here the meaning of the wor potential is twofol. On the one han it means there is an option to use these variables as shape variables in the close loop. On the other han even though these variables are relate to the kinetic energy they can be use to shape the potential energy. We show that the close-loop potential energy contains free functions of these variables. By expressing the regulation objective in terms of the potential shape variables we can 2 With some abuse of terminology we will refer to a configuration variable q j as unactuate when no generalise force acts on the erivative of its conjugate momentum namely ṗ j = ( L/ j )/t /12/$ IEEE 463

2 assign the esire equilibrium an have the freeom to reshape the potential energy to achieve performance whilst maintaining the PHS form in close loop. As a case stuy we consier a robotic manipulator mounte on a flexible base an compensate for motion of the base while positioning the en effector with respect to the groun reference. We compare our control strategy with special cases that correspon to other energy shaping strategies previously propose in the literature. II. A CLASS OF PORT HAMILTONIAN SYSTEMS We consier the following moel of a simple mechanical system: [ṗ ] {[ ] [ ]} [ ] H(pq) [ ] I D G = + τ. (1) q I p H(pq) The variable q R n gives the configuration of the system an p R n is the conjugate momentum. We consier the control input τ R m (generalise forces). For typical uneractuate systems m < n. We further assume that the configuration variables are partitione as follows: [ ] qe q = (2) q s where q s Q s = R m is the vector of shape variables an q e Q e = R n m is the vector of external variables. The parameter D = D T R n n is the positive efinite amping an G R n m with rank(g) = m is the input coupling matrix. For the above class of systems the shape variables are fully actuate an we will assume [ ] (n m) m G =. (3) I m m If G is not in the form (3) then in some cases it may be possible to transform the control variables to obtain the esire form. The Hamiltonian function H(p q) is given by H(p q) = 1 2 pt M 1 (q s )p +V(q) (4) }{{} T (pq s) where M(q s ) = M T (q s ) > is the mass matrix which etermines the open-loop kinetic energy T (p q s ) an V(q) is the open-loop potential energy. III. SET-POINT REGULATION CONTROL FOR UNDERACTUATED MECHANICAL SYSTEMS Let us efine z(q) Z as the potential shape variables the properties of this transformation are efine in Section III-D an follow from the set of conitions for stability. We then consier set-point regulation of z to a esire equilibrium point z. A. Close-loop PHS Let the esire close-loop PHS be [ṗ ] { [ ] M = (z)m 1 (q s ) q M 1 (q s )M (z) [ ] } [ ] H D (5) p H where D = D T > is the esire amping an the closeloop Hamiltonian is given by H = 1 2 pt M 1 (z)p +V (q) (6) }{{} T (pz) where M (z) = M T (z) > is use to shape the kinetic energy T (p z). The potential energy V (q) is use to assign the esire close-loop equilibrium at which the potential energy attains its minimum. B. Matching To fin a control law τ we must match the ṗ an q equations in (1) an (5). The q equation is alreay matche by construction of (5) since q = H p = M 1 (q s )p = M 1 (q s )M (z)m 1 (z)p The matching of ṗ yiels = M 1 (q s )M (z) H p. ṗ = DM 1 (q s )p H + Gτ (7) = D M 1 (z)p M (z)m 1 (q s ) H (8). Since (8) consists of n equations an m < n unknown control forces we nee to satisfy the following aitional n m constraints to fin a solution for τ : { G DM 1 (q s )p D M 1 (z)p + H M (z)m 1 (q s ) H } = (9) where G is any full-rank left annihilator of G that is G G = an rank(g ) = n m. If (9) is satisfie the control law is given by τ = ( G T G ) { 1 G T DM 1 (q s )p D M 1 (z)p + H M (z)m 1 (q s ) H }. (1) We can separate the matching equation (9) into powers of p uner the assumption that D an D are inepenent 464

3 of p. Thus a particular solution of (9) is obtaine by solving the following equations: { V G M (z)m 1 (q s ) V } = (11a) G { DM 1 (q s )p D M 1 (z)p} = (11b) { T G M (z)m 1 (q s ) T } =. (11c) The objective is to choose M (z) V (q) an D to satisfy (11). C. Main result We consier choosing z to shape the total energy. Since the shape variables are actuate G T = then (11c) can be expresse as follows: G M (z)m 1 (q s ) T z T =. (12) z This homogeneous PDE is solve for any z(q) = [z 1 (q)... z m (q)] T that satisfies G M (z)m 1 (q s ) z i = (13) i {1... m}. Thus matching the kinetic energy has been reuce to fining m functions of q from which we can form the shape variables for the close-loop system. Note that (13) is simpler to solve than (12) since the latter involves the partial erivatives of M 1 (z). To solve (13) we propose a particular z an then try to solve an algebraic equation in M (z). The potential energy PDE (11a) can be separate into partial erivatives of external an shape variables G { V M (z)m 1 (q s ) [ V ] e M (z)m 1 (q s ) [ ] } V = (14) s which is satisfie by V (q) = V e (q) + V s (q s ) where V e (q) an V s (q s ) are solutions of { [ V Ve ]} G M (z)m 1 (q s ) e = (15a) [ ] G M (z)m 1 (q s ) V s =. (15b) s Equation (15a) can be simplifie by choosing so that G = [ I (n m) (n m) (n m) m ] (16) V e e = Γ 1 (q) V e (17) where Γ(q) is the upper-left (n m) (n m) block of M (z)m 1 (q s ). The matrix Γ(q) is invertible since it is the (n m) th -orer leaing principal submatrix of a prouct of two square full rank matrices. Equation (17) shows that the equilibrium points in the unactuate coorinates cannot be move 3 as previously reporte in [8] for systems with uneractuation egree one where n m = 1 q e an Γ(q) are scalars an (17) may be integrate irectly 1 V V e (q) = q e. (18) Γ(q) e We can now esign M (z) an thus Γ(q) so that V e (q) in (17) is positive efinite in the external coorinates q e in a neighbourhoo of its equilibrium which we enote q e. This also suggests the possibility of extening this approach to problems with unstable open-loop ynamics in the unactuate coorinates. Since the prouct of two positive efinite matrices M (z)m 1 (q s ) an thus Γ(q) is not necessarily positive efinite (or even symmetric) it may be possible to esign M (z) to prouce a positive-efinite potential V e (q) even for a non-positive-efinite V(q). This is beyon the scope of the current paper. Equation (15b) can be expresse in terms of z as follows: G M (z)m 1 (q s ) T z [ ] V z z = (19) which is satisfie for any free function V z (z) since z alreay satisfies (13). Therefore a solution for the closeloop potential energy is given by V (q) = V e (q) + V z (z(q)) (2) where V e (q) is a solution to (15a) an V z (z) is any free function of z. We can then choose a function V z (z) so it is minimise at the esire equilibrium z an reners V (q) positive efinite in a neighbourhoo of z. Damping injection can be achieve by fining a matrix D which satisfies the algebraic constraints in (11b) that is G D M 1 (z) = G DM 1 (q s ). (21) This etermines the first n m rows (an ue to symmetry the first n m columns) of D. D. Stability Since the solution of V e (q) from (17) is minimise at q e = q e an we choose V z (z) to be minimise at z = z we can show that V (q) is minimise at q e = q e an z = z. Proposition 1: Given V e (q) > q e q e V z (z) > z z an assuming that T z s is non-singular then V (q) = V e (q) + V z (z) is minimise at q e = q e an z = z. Proof: The stationary points of V (q) are foun by setting V = V e + T z V z z =. Then separating the 3 although the stability of the existing equilibrium points may be moifie as shown in [6] an [7] for unstable systems. 465

4 partial erivatives into q e an q s components it follows from (17) that V = Γ 1 (q) V + T z V z = e e e z (22) V = T z V z =. s s z (23) Equation (23) is satisfie if an only if z = z since T z s is full-rank. Equation (22) is satisfie for q e = q e an z = z. To show there are no other stationary points consier some q e q e an z = z that satisfies (22). Then Γ 1 (q) V e = for some q e q e but this is a contraiction since V e = if an only if q e = q e an Γ(q) is full-rank. Therefore V (q) is only stationary for q e = q e an z = z. To establish that V (q) is minimise we note that it must be positive efinite since it is the sum of two positive efinite functions. The conition that T z s is full rank requires z to be a function of all the elements of q s. This is not particularly restrictive since if this were not the case we coul not assign any particular equilibrium point in all the q s coorinates. Note also that z may epen on elements of q e but there is no ifficulty in having the equilibrium specifie by q e = q e an z = z which appears overetermine. The reason for this is that no choice of z which solves (13) can moify the location of the unactuate equilibrium points; so there is no conflict. We can use this feature to our avantage as we show in the following section since z being epenent on the unactuate coorinates allows us to shape ynamics in q e. We can now show that the close-loop system is asymptotically stable. Proposition 2: Consier the ynamics of the system (1) in close-loop with the control law (1) an M (z) V (q) an D satisfy (11) an the conitions given in Proposition 1 hol. Then the close-loop system can be written as the PHS (5) which has an asymptotically stable equilibrium point at p = q e = q e an z = z. Proof: Using H as a Lyapunov caniate function which is minimise at p = q e = q e an z = z we can compute its erivative with respect to time along the solutions of (5) an obtain Ḣ (p q) = T H p ṗ + T H q (24) = T H p D H p which establishes stability. Asymptotic stability follows by applying the Invariance Principle. Since Ḣ = = p = then from (5) we have ṗ = M (z)m 1 (q s ) V which is zero only for q e = q e an z = z. Therefore the largest invariant set containe within {p q R n Ḣ = } is p = q e = q e an z = z. IV. CASE STUDY We consier the control of a robotic manipulator mounte on a flexible base. This is shown in Figure 1. The robot task consists of moving a heavy tool to a particular position in the workspace. The base has 1 DOF q 1 an the robot arm has 2 DOFs which correspon to rotation of the arm q 2 an extension of the arm q 3. Control torque τ m acts to rotate the arm an the control force F A acts to exten the arm. Due to this configuration an the weight of the tool the control system shoul be esigne to compensate for motion of the base while positioning the en effector with respect to the groun reference. Fig. 1. Robotic manipulator mounte on a flexible base. The open-loop mass matrix M(q s ) is given by M(q s ) = m B + m T m T (q 3 + l) cos q 2 m T sin q 2 m T (q 3 + l) cos q 2 m T (q 3 + l) 2. m T sin q 2 m T (25) The open-loop potential energy V(q) is given by V(q) = 1 2 k Aq k Bq (m B + m T )gq 1 + m T g(q 3 + l) sin q 2. (26) The input coupling matrix G an its left-annihilator G are given by G = 1 G = [ 1 ]. (27) 1 Let the close-loop mass matrix M (z) be parameterise as follows: a 1 a 2 a 3 M (z) = a 2 a 4 a 5 (28) a 3 a 5 a 6 an let the close-loop amping D be parameterise as follows: D = b 1 b 2 b 3 b 2 b 4 b 5. (29) b 3 b 5 b 6 466

5 The regulation task is to position the tool that is x [x y] T x [x y ] T where x = (q 3 + l) cos q 2 y = q 1 + (q 3 + l) sin q 2 (3a) (3b) an x an y are the coorinates of the esire position in the workspace. A. Total energy shaping preserving shape variables We esign a controller using total energy shaping that preserves shape variables in the close loop an recover the special case of potential energy shaping. This controller correspons to a particular case of the controllers that can be esigne with the metho propose in [4]. Substituting z = q s into (13) an assuming et M(q) = m B m 2 T (q 3+l) 2 we have the constraints on the elements of M (z) (a 3 sin q 2 a 1 )m T (q 3 + l) cos q 2 +a 2 (m B + m T cos 2 q 2 ) = (31a) (a 2 cos q 2 a 1 (q 3 + l))m T sin q 2 +a 3 (q 3 + l)(m B + m T sin 2 q 2 ) =. (31b) By choosing a 1 > as a esign parameter a 2 an a 3 can be expresse in terms of a 1 a 2 = a 1 m T (q 3 + l) cos q 2 m B + m T (32a) a 3 = a 1 m T sin q 2 m B + m T (32b) an a 4 a 5 an a 6 must be chosen to ensure M (z) >. We will choose a 5 = to simplify the esign. Then the choice a 4 = a2 2 a 1 + δ an a 6 = a2 3 a4 δ + α leaves δ > an α > as esign parameters which guarantees M (z) >. From (18) an (2) we have V (q) = m B + m T a 1 ( ) 1 2 k Bq1 2 + (m B + m T )gq 1 + V z (q 2 q 3 ) (33) where V z ( ) is a free function. Since V z cannot epen on q 1 we choose V z (q 2 q 3 ) = 1 2 k x ((q 3 + l) cos q 2 x ) k y ( q 1 + (q 3 + l) sin q 2 y ) 2 (34) where q 1 (m B+m T )g k B is the steay-state equilibrium position of q 1 an k x k y > are free parameters. We have inclue q 1 to provie steay-state gravity compensation in the absence of q 1. Substituting (29) into (21) etermines the first row an column of D b 1 = b B( a 2 cos q 2 + (q 3 + l)(a 1 a 3 sin q 2 )) (35a) b 2 = b B(a 2 (q 3 + l) a 4 cos q 2 ) (35b) b 3 = b B(a 3 a 6 sin q 2 ). m B (35c) Then b 4 b 5 an b 6 must be chosen to satisfy D >. We will choose b 5 = to simplify the esign. Then the choice b 4 = b2 2 b 1 + γ an b 6 = b2 3 b4 γ + β leaves γ > an β > as esign parameters which guarantees D >. Note that if we set a 1 = m B + m T a 4 = m T (q 3 + l) 2 a 5 = an a 6 = m T we have M (z) = M(q) an we recover the special case of potential energy shaping. To show stability we note that V e (q) > V z (z) > an T z s = I which satisfies the conitions in Proposition 1. Since M(q) is singular at q 3 + l = care shoul be taken. Assuming we have an initial conition q 3 () + l > an target q3 + l > we can expect asymptotic stability by applying Proposition 2 since the trajectory of the states will not excite the singularity. B. Total energy shaping with potential shape variables We esign a controller using total energy shaping by proposing potential shape variables which are compatible with the regulation task (3). Substituting z = x into (13) an assuming et M(q) we have the following constraints on the elements of M (z): a 3 (q 3 + l) cos q 2 a 2 sin q 2 = a 3 (q 3 + l) sin q 2 a 2 cos q 2 =. (36a) (36b) The solution to (36) is given by a 2 = a 3 =. Then a 1 a 4 a 5 an a 6 are free parameters subject to a 1 a 4 > an a 4 a 6 > a 2 5 which guarantees M (z) >. From (18) an (2) we have V (q) = m B + m T a 1 ( ) 1 2 k Bq1 2 + (m B + m T )gq 1 + V z (x y) (37) where V z ( ) is a free function. Now we can choose V z (x y) = 1 2 k x (x x ) k y (y y ) 2 (38) where k x k y > are free parameters. Substituting (29) into (21) etermines the first row an column of D b 1 = a 1b B m B (39a) b 2 = b B(a 4 cos q 2 + a 5 (q 3 + l) sin q 2 ) (39b) b 3 = b B(a 5 cos q 2 + a 6 (q 3 + l) sin q 2 ). (39c) 467

6 Then b 4 b 5 an b 6 must be chosen to satisfy D >. We will choose b 5 = to simplify the esign. Then the choice b 4 = b2 2 b 1 + γ an b 6 = b2 3 b4 γ + β leaves γ > an β > as esign parameters which guarantees D >. Note that for this choice of z T z s is only full rank for q 3 + l however this also correspons to the singularity in M(q s ) so we o not expect any reuction in the region of attraction compare to the previous case. We shoul note however that while the equilibrium is unique in z coorinates there are two solutions in q s coorinates. These correspon to the cases of the extene arm q 3 + l > an the inverte arm q 3 + l < an the corresponing angles q 2 which iffer by π. Unlike the esign in the previous section where the equilibrium in q s is specifie assuming that q 3 () + l > an target q3 + l > is not sufficient to prevent the state trajectory from crossing the singularity. This may be overcome by constraining q 3 + l > however this is beyon the scope of the current paper. C. Simulation Results The plant parameters corresponing to the manipulator shown in Figure 1 are given in the Appenix. The flexibility of the base is exaggerate to appreciate the effect of the controllers. Three control esigns are consiere: i) Potential energy shaping that is M (z) = M(q s ) an z = q s ii) Total energy shaping preserving open-loop shape variables that is z = q s iii) Total energy shaping with potential shape variables z = x. The controller parameters given in Table I. The initial conition was p() = q() = an the target was locate at x = 4 m an y = 3 m. Figures 2 an 3 show the simulation results. We can see that for the esigns (i) an (ii) the q 2 an q 3 states converge quickly to their steay-state values an the control action remains small thereafter. The natural base oscillation however causes large errors in the tool position which the controller is not informe about. While aitional freeom is available in the esign (ii) it was still not possible to improve the transient response significantly over that achieve by only shaping potential energy. The controller for case (iii) can be seen to be using the q 2 an q 3 states to actively compensate for motion in the base. The result is the tool error quickly converges to zero while the controller continues to prouce control forces which actively cancel the base motion at the tool position until the natural motion of the base ecays ue to the issipative forces in the base. V. CONCLUSIONS We have shown that by reucing the kinetic energy matching PDE (over the unknown elements of M ) to one involving potential shape variables z we have both simplifie the esign proceure for total energy shaping an mae it easier to esign amissible close-loop potential energy Position Error (m) Position Error (m) Position Error (m) Tool Error Error in X Error in Y Motor Torque (N.m) Motor Torque (N.m) Motor Torque (N.m) Control Action Motor Torque Actuator Force Fig. 2. Tool positioning error (left) an control action (right) for potential energy shaping (top) total energy shaping with z = q s (mile) an z = x (bottom). Momentum (N.s) Momentum (N.s) Momentum (N.s) Momenta p1 p2 p Angular Momentum (N.m.s) Angular Momentum (N.m.s) Angular Momentum (N.m.s) Position (m) Position (m) Position (m) Configuration Variables q1 q2 q3 Fig. 3. State variables: momenta (left) an isplacements (right) for potential energy shaping (top) total energy shaping with z = q s (mile) an z = x (bottom) Actuator Force (N) Actuator Force (N) Actuator Force (N) Angular Position (eg) Angular Position (eg) Angular Position (eg) 468

7 functions for this class of uneractuate mechanical systems. Once z has been chosen we are able to choose the various free functions of z in the elements of M (z) subject to algebraic constraints rather than having to irectly solve the kinetic energy PDE (11c). In a case stuy we have shown how this controller can be use for compensating vibration of a robotic manipulator mounte on a flexible unactuate base. The propose controller shows improvement in performance relative to controllers base on potential energy shaping an total energy shaping preserving shape variables. [8] J. Acosta R. Ortega A. Astolfi an A. Mahinrakar Interconnection an amping assignment passivity-base control of mechanical systems with uneractuation egree one Automatic Control IEEE Transactions on vol. 5 no. 12 pp [9] G. Viola R. Ortega R. Banavar J. Acosta an A. Astolfi Total energy shaping control of mechanical systems: simplifying the matching equations via coorinate changes Automatic Control IEEE Transactions on vol. 52 no. 6 pp APPENDIX The plant parameters are given by m B = 1 kg m T = 1 kg l = 1 m k A = N/m k B = 15 N/m an the openloop amping matrix is given by b B D = b M (4) b A where b B = 3 Ns/m b M = 1 Nms/ra an b A = 5 Ns/m. The controller parameters are given in Table I. TABLE I CONTROLLER PARAMETERS. Parameter PE TE (z = q s) TE (z = x) a 1 m B + m T 5 5 a 4 m T (q 3 + l) 2 5 a 6 m T 1 α N/A 5 N/A δ N/A 1 N/A b 1 3 b 4 45 b 5 2 γ N/A 8 8 β k x k y Expression given in Section IV REFERENCES [1] F. Bullo an A. D. Lewis Geometric control of mechanical systems. Springer 25 vol. 49. [2] R. Ortega A. Loría P. J. Nicklasson an H. J. Sira-Ramírez Passivitybase Control of Euler-Lagrange Systems: Mechanical Electrical an Electromechanical Applications. Springer Verlag [3] A. van er Schaft L2-gain an passivity techniques in nonlinear control. Lonon: Springer-Verlag 2. [4] R. Ortega A. Van Der Schaft B. Maschke an G. Escobar Interconnection an amping assignment passivity-base control of portcontrolle hamiltonian systems Automatica vol. 38 no. 4 pp [5] R. Olfati-Saber Nonlinear control of uneractuate mechanical systems with application to robotics an aerospace vehicles Ph.D. issertation Massachusetts Institute of Technology 2. [6] F. Gomez-Estern R. Ortega F. Rubio an J. Aracil Stabilization of a class of uneractuate mechanical systems via total energy shaping in Decision an Control 21. Proceeings of the 4th IEEE Conference on vol. 2. IEEE 21 pp [7] R. Ortega M. Spong F. Gómez-Estern an G. Blankenstein Stabilization of a class of uneractuate mechanical systems via interconnection an amping assignment Automatic Control IEEE Transactions on vol. 47 no. 8 pp