Control of the Inertia Wheel Pendulum by Bounded Torques

Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 5 Seville, Spain, December -5, 5 ThC6.5 Control of the Inertia Wheel Pendulum by Bounded Torques Victor Santibañez, Rafael Kelly and Jesus Sandoval Abstract A control system for driving the inertia wheel pendulum to the upward position has been reported recently in the literature based on Interconnection and Damping Assignment Passivity based Control. An important feature of this controller is that it swings up and balances the pendulum in the upward position without switching between two different controllers. This paper, enhances the control system by the practical capability of maintaining the control action (applied actuator torque) inside prescribed limits. The perfomance of the proposed controller is illustrated via simulations, which have been compared with those reported in the literature. Index Terms Underactuated system, bounded control, asymptotic stability, passivity. I. INTRODUCTION Interconnection and Damping Assigment Passivity based Control (IDA PBC), introduced recently in 3, is a control design methodology that assigns a desired dynamic in closed-loop. An important feature of the IDA PBC technique is the systematic design procedure to yield an energy function in closed-loop which qualifies as a Lyapunov function. The Inertia Wheel Pendulum is an underactuated mechanical system consisting of a physical pendulum with a symmetric disk attached to the tip, which is free to spin about an axis parallel to the axis of rotation of the pendulum (see Figure ). Underactuated nature is because it has two degrees of freedom and only one actuator located at the disk. Based on the IDA PBC technique, 3 designed a controller that almost global resolved the problem of stabilization without switching the upward position of the inertia wheel pendulum. However, this controller may demand a high input torque that makes difficult a practical implementation. The goal of this paper is to propose an IDA PBC controller that provides bounded torque within desired bounds. This paper is organized as follows. Section II summarizes the inertia wheel pendulum model and control problem formulation. In Sections III and IV, we introduce the proposed controller and our stability analysis, respectively. Simulation on an inertia wheel pendulum are given in Section V. Finally, we offer some concluding remarks in Section VI. Work partially supported by CONACyT and COSNET, Mexico. V. Santibañez is with the Instituto Tecnológico de La Laguna, Apdo. Postal 49, Adm., Torreón, Coahuila, 7, MEXICO, vsantiba@itlalaguna,edu,mx R. Kelly and J. Sandoval are with the Dirección de Telemática, CICESE, Apdo. Postal 454, Adm., Ensenada, B.C., 8, MEXICO, jsandova@cicese.mx Fig.. Inertia wheel pendulum II. MODEL AND CONTROL PROBLEM FORMULATION The inertia wheel pendulum depicted in Figure, may be modelled as a two-degrees-of-freedom serial mechanism, the pendulum forms the first link and the rotating disk forms the second link. We assume that the center of mass of the disk is coincident with its axis of rotation and measure the angle of the pendulum clockwise from the vertical. Under these assumptions the Euler Lagrange s equations of motion can be written as 4: m + θ θ + m3 sin(q ) u () θ and θ are the joint positions of the pendulum and the disk, respectively, m 3 (m gl c + m gl ), and u is the control input torque acting between disk and pendulum, m m lc + m l + I and I. The remaining parameters are shown in Table I. Next, we introduce a global change of coordinates q q θ θ, () which can be expressed equivalently as θ q. (3) θ q -783-9568-9/5/$. 5 IEEE 866

TABLE I Parameters Description Notation Units Length of the pendulum l m Distance at the center of mass l c m of the pendulum Mass of the pendulum m kg Mass of the disk m kg Moment of inertia of the pendulum I kg.m Moment of inertia of the disk I kg.m Gravity acceleration g m/s This leads to a simplified model m q m3 sin(q + ) u, (4) q The model (4) can be written through Hamilton s equations of motion as p q m d q dt p m 3 sin(q ) u, (5) u with a Hamiltonian function given by H(q, p) p + p + m 3 cos(q ), (6) m q T q q and p T p T m q q are the generalized positions and momenta respectively. We can now formulate the control problem under actuator torque constraints addressed in this work. Consider the inertia wheel pendulum model (5). Assume the actuator is able to supply a known maximum torque u max such that u(t) u max. (7) We also assume that the maximum torque satisfies the following condition: u max >m 3 (8) Formally, the control objective can be expressed as to find a control law such that lim dist(q (t), Γ) and u(t) u max, (9) t dist(q (t), Γ) denote the smaller distance between q and every element in Γ{, 4π, π,, π, 4π, }. III. CONTROLLER DESIGN A. Previous work Based on IDA PBC methodology, the following control law was proposed in 3, u γ sin(q )+k p (q + γ q ) + k v k ( q + γ q ) }{{}}{{} u es u di () γ a a +a m 3, γ m(a+a3) (a +a ), k m(a+a) a a 3 a >, () and k p and k v are positive arbitrary constants. The remaining constant a, a and a 3 must satisfy a >, a a 3 a >, a + a <, u max m 3 > a a +a, () As it was pointed out in 3, the control law () is the sum of two terms u u es + u di u es γ sin(q )+k p (q + γ q ) (3) u di k v k ( q + γ q ). The term u es is designed to achieve the energy shaping and u di injects the damping to the closed loop system. The damping injection term u di can be written in terms of the generalized coordinates q and momenta p as u di k v k 3 ( + k 4 p ) (4) k 3 a+a a a 3 a >, k 4 a+a3 a +a. (5) Hence, the control law () can be rewritten as u γ sin(q )+k p (q + γ q ) + k v k 3 ( + k 4 p ) }{{}}{{} u es u di. (6) Closed loop system: A detailed stability analysis in Lagrangian formulation of the equilibria of the closed loop system formed by the control law (6) and the Hamiltonian system (5), and given by (7), which is shown in the next page, was presented in. It is easy to note that the equilibrium set is q nπ E q p γ nπ (8) n IN.WecanseefromFigurethatq with n even correspond to the desired upward position, while the ones with odd n are with pendulum hanging. 867

d dt q q p p m m 3 sin(q ) γ sin(q )+k p (q + γ q )+k v k 3 ( + k 4 p ) γ sin(q )+k p (q + γ q )+k v k 3 ( + k 4 p ). (7) B. Proposed controller By bearing in mind, the actuator torque constraints control problem, addressed in this work, we can modify the control law (6) in order to have bounded all its terms. By following similar steps to those in 3, we can propose the new energy shaping term u es γ sin(q )+k p tanh(q + γ q ) with γ and γ given in () and k p a positive constant suitable selected. This new u es is the result of proposing the following desired potential energy function: V d (q) m m 3 a + a cos(q )+k lncosh(q + γ q ). (9) Besides, because we require bounded control action, we propose the damping injection term u di as: u di k v tanh(k 3 ( + k 4 p )) () which still preserves the passivity property of the closed loop system (5). The constants k 3 and k 4 are given in (5) and k v is a properly selected positive constant. Therefore, we propose the following control law: u γ sin(q )+k p tanh(q + γ q ) }{{} u es + k v tanh(k 3 ( + k 4 p )). }{{} u di The positive constants k p and k v must be chosen sufficiently small. More specifically, they must satisfy u max γ >k p + k v. It is worth noticing that all terms in () are bounded; furthermore, the control law is bounded by u γ + k p + k v u max. () IV. STABILITY ANALYSIS A feature of the IDA PBC methodology is that the desired energy function H d (q, p) qualifies as a Lyapunov function candidate V (q, p). Hence, to carry out the stability analysis we propose V (q, p) H d (q, p) a 3 a p + a + m m 3 cos(q ) a + a + k lncosh(q + γ q ) (3) a a 3 a. The closed-loop system is obtained combining the control law () and the open loop system (5), which yields (4), shown in the next page. Notice that the equilibrium set is q nπ E q p γ nπ (5) n IN. We have omitted, due to paper length reasons, the proof that verify the local positive definiteness of V (q, p) forn even. The time derivate of function (3) along the trajectories of the closed-loop equation (4) becomes V (q, p) a 3p ṗ a ṗ a p ṗ + a ṗ m m 3 sin(q ) q a + a +k tanh(q + γ q ) q + γ q k v k 3 ( + k 4 p ) tanh(k 3 ( + k 4 p )) (6) which is a negative semidefinite function, because by design k v >. Because the closed-loop equation (4) is autonomous, we can use the LaSalle s invariance principle to demonstrate that the equilibrium for n even in the set E are asymptotically stable (). To this end, let us define the set Ω as Ω {q IR, p IR : V (q, p) } {q IR & p IR : k 3 + k 4 p tanh(k 3 + k 4 p ) }. In accordance with LaSalle s invariance principle, we need to determine the largest invariant set in Ω. To this 868

d dt q q p p m m 3 sin(q ) γ sin(q )+k p tanh(q + γ q )+k v tanh(k 3 ( + k 4 p )) γ sin(q )+k p tanh(q + γ q )+k v tanh(k 3 ( + k 4 p )). (4) end, we have that any trajectory in Ω must satisfy k 3 (t)+k 4 p (t) (7) and in its turn, k 3 ṗ (t)+k 4 ṗ (t). (8) Further, (7) can be written as k q (t)+γ q (t) (9) with k and γ defined in (). Integrating (9), we obtain the following: q (t)+γ q (t) k (3) for any constant k. Byusingṗ and ṗ from (4) in (8), yields k 3 sin(q (t))γ + k 4 m 3 γ +k p tanh(q (t)+γ q (t)) k 4 +k v tanh(k 3 ( (t)+k 4 p (t))) k 4. (3) Considering (7) and (3), (3) becomes k 3 sin(q (t))γ + k 4 m 3 γ +k p tanh(k) k 4 (3) which imply that q (t) must be constant and as consequence q (t). Next, from (9) we have that q (t) q (t). Besides, because (7) and (9) are equivalents, we have q (t) p (t). (33) q (t) (t) Incorporating (33) in (4), we obtain m 3 sin(q (t)) γ sin(q (t)) +k p tanh(q (t)+γ q (t)) +k v tanh(k 3 ( (t)+k 4 p (t))), (34) γ sin(q (t)) + k p tanh(q (t)+γ q (t)) +k v tanh(k 3 ( (t)+k 4 p (t))), (35) whose solutions are q (t) nπ (36) q (t) γ nπ with n IN. This development and particularly the conclusions (33) and (36) allow to establish that the largest invariant set in Ω is E. From LaSalle s invariance principle, we conclude that the equilibria in the set E for n even are asymptotically stable (see similar details in ). rad 4 3 q q 3 5 5 5 3 35 4 t sec Fig.. Joint positions q and q : Proposed control law (). So we have proven the following: Proposition. The inertia wheel pendulum (5) in closedloop with the bounded control law () has an infinite number of isolated locally asymptotically stable equilibria, which belong to equilibria set (5) with n IN even. Furthermore, the applied torque is bounded by u u max. V. SIMULATIONS In this section we present the simulation results obtained on the inertia wheel pendulum by using the parameters given in 3. These correspond to m 3, m.,.. We suppose the maximum torque that can supply the actuator is u max 45 Nm. The remaining parameters were selected as a,a.5 and a 3 6, which satisfy () and in accordance with () yields γ 3, γ 4.5 andk.66. The gains were chosen as k p 3.75 and k v. Substituting these values in the control law (), we have u <γ + k p + k v 43.75 Nm <u max, (37) The rest initial configuration was q () q () p () () T 3.4 T. Figures and 3 shown joint positions q and q with control laws () and (6), respectively. A simple observation shows that Figures and 3 are very similar and both converge toward the equilibrium point q q p T T. The applied bounded and non-bounded control inputs u, are shown in Figures 4 and 5, respectively. Notice that the bounded torque clearly evolve inside the prescribed 869

3 rad q q 5 5 5 3 35 4 t sec 5 4 3 3 4 5 Nm 5 u 5.3 Nm 4 5 5 5 3 35 4 t sec Fig. 3. Joint positions q and q : Original control law (6). Fig. 5. Applied non-bounded torque u: Original control law (6). 3 4 Nm 3 4 u 37.55 Nm 5 5 5 3 35 4 t sec 5 Van der Schaft A. (999). L Gain and passivity techniques in nonlinear control, Springer, Berlin. John Wiley & Sons, Inc., New York. Fig. 4. Applied bounded torque u: Proposed control law (). limit given in (37) with a maximum torque of 37.55 Nm, while the non-bounded torque achieves values greater than u max 45 Nm. VI. CONCLUDING REMARKS Motivated by practical matter on IDA PBC, this paper has proposed a control scheme with bounded torque for the inertia wheel pendulum. The proposed controller was designed following the IDA PBC methodology, but now taking care in the actuator torque restriction. Conditions on the controller gains are provided to guarantee that the control action remains within the prescribed limit. VII. ACKNOWLEDGMENTS The third author would like to thank the Instituto Tecnológico de La Paz for its financial support. References Kelly R. (4). Analysis of the IDA PBC control of the inertia wheel pendulum: Lagrangian formulation (in Spanish). XI Congreso Latinoamericano de Control Automático, La Habana, Cuba, May. Khalil, H. K. (996). Nonlinear systems analysis, Prentice- Hall, Upper Saddle River. 3 Ortega, R., M. W. Spong, F. Gómez-Estern and G. Blankenstein (). Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment. IEEE Transactions On Automatic Control. Vol. 47, No. 8, pp. 3-33. 4 Spong M. & Vidyasagar M. (989). Robot dynamics and control, 87