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Design, Modeling and Stabilization of a Moment Exchange Based Inverted Pendulum

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Design, Modeling and Stabilization of a Moment Exchange Based Inverted Pendulum Preprints of the 15th IFAC Symposium on System Identification Saint-Malo, France, July 6-8, 2009 Design, Modeling and Stabilization of a Moment Exchange Based Inverted Pendulum Jordan Meyer, Nathan Delson and Raymond A. de Callafon Dept. of Mechanical and Aerospace Engineering, University of California, San Diego, 9500 Gilman Drive 0411, La Jolla, CA 92093 Abstract: This paper summarizes the mechanical and control design concepts of an inverted or unstable pendulum where stabilization is achieved by a moment exchange generated by a controlled symmetric rotation of a rotational inertia attached to the pendulum. The proposed design of the pendulum has a fixed bottom rotation or point of support as opposed to the usual vertically or horizontally moving point of support to stabilize the pendulum, allowing for small form factor desktop design of an inverted pendulum experiment. The symmetry of the rotational inertia allows for stabilization of the pendulum without the need to control the position of the mass attached to the pendulum. The paper reviews the design considerations, dynamic modeling, system identification and control design strategy to stabilize the pendulum. 1. INTRODUCTION ratio to maximize the angle from which the pendulum can be brought to an upright position. For demonstration and evaluation of intricate concepts behind automatic control, the upside-down or inverted T pendulum Acheson and Mullin (1993) has been an ex- p = tensively used application in both research and teaching ¢ ¢ c ¢ ¢ of control system design. The classical inverted pendulum ¢ ¢ ¢r α1 ¢ r as described for example in Landau and Lifshitz (1976) ¢ ~¢ ¢ ¢ ?F is constraint to move on a vertical plane and under the ¢ ¢ ¢ ¢ m Tp influence of a (destabilizing) gravity force, while the point ¢ ¢ ¢ ¢ b R of support can be subjected to horizontal or vertical forces. ¢ ¢ ¢ ¢ # Stability studies using vertical oscillations of the point of ¢ ¢ ¢ ¢ ? support date back to Stephenson (1908), but the classical ¢ ¢ ¢ ¢ r ¢ a ¢ ¢ Fr } inverted pendulum still serves as a benchmark for many ? F control algorithms Aracil and Gordillo (2004). The devel- r r "!m α1 + α2 opment of new (non)linear control design methodologies Fig. 1. Schematics (left) and actual device (right) of for the classical and more complex inverted pendulums is inverted pendulum with rotating inertia for moment still an active research area, see e.g. Cheng et al. (2005); exchange stabilization Xu and Yu (2004); Casavola et al. (2004); Lundberg and Roberge (2003); Alonso et al. (2002). To illustrate the main idea behind the moment exchange inverted pendulum, consider the (inverted) pendulum de- In most of the control algorithms for the stabilization of picted in Figure 1. The pendulum consists of an inverted the classical inverted pendulum, horizontal forces on the rod (pendulum) connected to a rotating inertia. By ac- point of support are provided by a cart mechanism. By the celerating the rotating inertia, a moment exchange can controlled movement of the cart, stabilization of a single be generated between the rod and the inertia. The mo- or even multiple inverted pendulums can be achieved Shen ment exchange generates a moment that could be used et al. (2005). Unfortunately, horizontal movement of the to stabilize the inverted pendulum or dampen out the point of support of the pendulum requires a relative large motion of a stable pendulum. Due to the moment exchange horizontal surface for the operating range of the inverted the pendulum does not need a moving base or cart for pendulum. In addition, either belt driven motors, linear stabilization, allowing a much simpler table-top design. actuators or a controllable cart is required to stabilize the pendulum. 2. PENDULUM DYNAMICS The objective of this paper is to summarize the main design, modeling and identification concepts behind an 2.1 Equations of motion inverted pendulum that can be stabilized via a moment exchange with a rotating inertia. Such a inverted pendu- With the configuration depicted in Figure 1, the absolute lum design can be operated on a much smaller footprint angular rotation α1(t) of the pendulum rod and the and operates with a fixed rotational point of support. This relative rotation α2(t) of the rotating inertia can be paper also shows an optimization of moment exchange gear described by 462 a a Ip α¨1(t)=Frlb sin α1(t)+Fmlc sin α1(t) − Tp(t) rotating inertia Im with respect to point a.Towritedown c (1) Im(¨α1(t)+α¨2(t)) = Tp(t) analytic expressions for the moment of inertia we can use a the parallel axis theorem In (1) the following variables are used: Ip indicates the a b 2 b 2 2 angular momentum of the inverted pendulum around I = I + mr (βl) = I + β mrl c r r r the bottom rotation point a. Im indicates the angular momentum of the rotating inertia around its center and and assume a particular shape for the pendulum rod. connection point c. F is the gravitational force due to the The parameter 0 <β<1 is obtained from (2) and r parametrizes the center of gravity of the pendulum rod. pendulum rod mass acting at the center of gravity at point b b adistancel from point a along the pendulum. F is the The moment of inertia Ir of the pendulum rod around the b m point b is given by Ib = 1 m l2 + 1 m r2 or Ib = 1 m l2 + is the gravitational force due to the rotating inertia located r 12 r 4 r r 12 r 1 2 at point c adistancelc from point a along the pendulum. 12 mrr in case the pendulum rod with mass mr and length Tp(t) is a torque or moment exchange generated internally l is respectively a homogeneous cylinder with a radius r between the pendulum rod and the rotating inertia and or a homogeneous beam with a width r. For the different will be used to stabilize the pendulum. configurations of the pendulum rod, the expression of the b inertia Ir of the pendulum rod around the point b can be The symmetry of the rotating mass simplifies the equa- generalized to tions of motion in a number of ways; the centrifugal forces b 2 2 all pass through the pendulum pivot point and there- Ir = µmrl + νmrr (4) fore do not effect the dynamics of the system, there is and in case the pendulum rod is not homogeneous, the no gravitational component dependent on α2,andthere parameters 0 <µ,ν<1 can also be used to model the in- are no Coriolis forces Asada and Slotine (1986). Physical ertial contributions respectively due to mass distribution, properties such as length and diameter of the pendulum length and width dimension of the pendulum rod. are mostly determined by design considerations. In order Using the parallel axis theorem, Ia of the rotating inertia to obtain numerical values for the coefficients of the differ- m with respect to point a can be written as ential equation given in (1), we parametrize the coefficients a c 2 c 2 2 in terms of the design parameters: Im = Im + mm (γl) = Im + γ mml Definition 1. The physical design parameters of the in- c where the moment of inertia Im around the (center) point verted pendulum are characterized by the masses mr, mm, c of the rotating inertia is given by the dimensions l, r and R that are defined as follows: c 1 2 I = mmR • The total mass mr and mm of respectively the pen- m 2 dulum rod and the rotating inertia. assuming a cylindrically shaped rotating inertia with mass • The total length l and the width or radius r of the m and radius R. The parameter 0 <γ<1 is again pendulum rod. m obtained from (2) and models the location of the rotating • The radius R of the cylindrical rotating inertia where inertia along the pendulum rod. R reflects the maximum value allowed by space con- straints. It should be noted that mm indicates the total mass of the rotating inertia, e.g. mm would also include the mass of Gravitational forces and inertial constants can be ex- the rotor of a servo motor used to generate the moment pressed in terms of the above defined physical design pa- exchange torque. With the addition of a servo motor, the rameters. The gravitational forces acting on the pendulum assumption of a cylindrically shaped rotating inertia would are given by not be viable. In that case, the moment of inertia of the 2 Fr = mrg, Fm = mmg, g =9.81m/s rotating inertia with a radius R with respect to point c can be parametrized by where mr and mm denote respectively the mass of the c 2 pendulum rod and the rotating inertia. In addition, the Im = τmmR (5) center of gravity located at distances l and l can be b c where 0 <τ<1 can be used to model any cylindrically described by shaped rotating inertia with outer radius R and possibly lb = βl, β ∈ (0, 1),lc = γl, γ ∈ (0, 1) (2) connected to a servo motor. Combining the results yields where l is the known length of the pendulum rod and the the total inertia of the inverted pendulum free parameters 0 <β,γ<1 model the center of gravity a a a I = I +I =(β2 +µ)m l2 +νm r2 +γ2m l2 +τm R2 of the pendulum rod and the location of the pure moment p r m r r m m generated by the rotating inertia. This definition rewrites (6) the total gravity force contribution in (1) as Combining the analytic expressions for the gravitational F l sin α (t)+F l sin α (t)= forces and inertial constants allows the differential equa- r b 1 m c 1 (3) mpgl sin α1(t),mp = mrβ + mmγ tion in (1) to be written in the physical design parameters In case the mass of pendulum rod is evenly distributed, of the inverted pendulum.
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