High-Low Gain Redesign for a 4 DOF Spherical Inverted Pendulum Guangyu Liu ∗ Lorenzo Marconi ∗∗ ∗ VRL, NICTA and Department of Electrical & Electronic Engineering, The University of Melbourne, Victoria, 3010, Australia, (e-mail: liug@ee.unimelb.edu.au) ∗∗ CASY-DEIS, University of Bologna, Italy. (e-mail: lmarconi@deis.unibo.it) Abstract: We revisit a previous high-low gain control idea for a 4 DOF spherical inverted pendulum using a different approach, inspired by a nested saturation tool proposed by Marconi and Isidori, that provides explicit tuning rules to deal with certain bounded external disturbances. The update controller is a robust, decentralized and “global” controller. 1. INTRODUCTION The pendulum is a cylindrical beam with the length 2L and the mass m attached to a horizontal plane via a universal joint that is driven by a planar control force F △ =(F x ,F y ) and sliding in the plane (see Fig. 1). The system has four degrees of freedom with the generalized coordinates q △ =(x, y, δ, ε) with the translation ones: (x, y) and a pair of Euler angles (δ, ε). The whole upper space denoted by U △ =  (q, ˙ q) ∈ R 8 |(δ, ε) ∈ ( − π 2 , π 2 ) × ( − π 2 , π 2 ) is defined as the “global” region. The benchmark problem is motivated by several practical problems: vector thrusted rockets hovering in the air, personal transporters (Segway), jugglers’ balancing problems and laboratory test-benches. Our aim is to design F such that, for any (q(0), ˙ q(0)) ∈ U , (q(t), ˙ q(t)) → 0 as t →∞. To achieve a “global” stability region, one could use strate- gies that switched between a local (or non-local) stabilizing controller and a swing-up strategy (see Albouy and Praly [2000], Shiriaev [2004]). See Liu et al [2007b] for a way- point tracking design with switching (see also Liu et al [2008a] for exact output tracking). Here, we assume that the pendulum is already swung up above the horizontal plane. Several non-local continuous stabilizing controllers (no switching) were proposed for the system Bloch et al [2001], Liu et al [2008b, 2006]. The controller of controlled Lagrangians Bloch et al [2001] (see [Liu , 2006b, Chapter 7] for details) yielded some non-local “bounded” stabilizing region but it suffered poor robustness using the parameters we attempted (see Liu et al [2007a]). A “semi-global” decentralized stabilizing controller was proposed in Liu et al [2006] (see also Liu et al [2008c]) based on Lyapunov theory of singular perturbed systems. Although the robustness was guaranteed by an associated Lyapunov function, it might be deteriorated when a larger domain of attraction was attempted. In Liu et al [2008b], a “global” high-low gain control idea that improved Liu et al [2005] was proposed for the pendulum through identifying some appropriate upper triangular form, where a high- gain controller was used to regulate angular dynamics and a low gain controller was used to regulate the rest of δ - mg x F y F L 2 ) , , ( 0 y x O X Y Z ε δ - mg x F y F L 2 ) , , ( 0 y x O X Y Z ε C Fig. 1. The spherical inverted pendulum the dynamics by applying the nested saturation tool Teel [1996]. However, the tuning rules are implicitly dealing with the disturbance. In this paper, we redesign the high and low gain controller Liu et al [2008b] inspired by a robust nested saturation procedure in [Isidori et al , 2003, Appendix C] and Mar- coni & Isidori [2001] (see Arcak et al [2001], Kaliora & Astofi [2004] for different approaches) such that it pro- vides explicit tuning rules for the design parameters at the presence of certain bounded disturbances. The controller is decentralized based on the structure of two interconnected chains of integrators Liu et al [2006, 2008c]) and yields a “global” domain of attraction inherit form Liu et al [2008b]. The effectiveness of the controller is evaluated through computer simulations. The paper is organized as follows. In Section 2, we recall the model and the decoupled dynamics in Liu et al [2006, 2008c]. In Section 3, we present our main result. Some simulations are given in Section 4. Final observation is given in Section 5. Notations: For a piecewise-continuous function u(t): [0, ∞) → R m , define ‖u(·)‖ a = lim sup t→∞ {max 1≤i≤m |u i (t)|} the asymptotic “norm” of u(·). The set of u(t), en- Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008 978-1-1234-7890-2/08/$20.00 © 2008 IFAC 5921 10.3182/20080706-5-KR-1001.1863