ScienceDirect IFAC-PapersOnLine 48-1 (2015) 328–333 Available online at www.sciencedirect.com 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2015.05.011 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Underactuated mechanical systems, nonholonomic constraints. 1. INTRODUCTION The Wheeled Inverted Pendulum (WIP) - and its com- mercial version, the Segway - has gained interest in the past several years due to its maneuverability and simple construction (see e.g. Grasser et al. [2002], Segway [2015, Jan]). Other robotic systems based on the WIP are becom- ing popular as well in the robotic community for human assistance or transportation as can be seen in the works of Li et al. [2012], Nasrallah et al. [2007], Baloh and Parent [2003]. A WIP consists of a vertical body with two coaxial driven wheels. The stabilization and tracking control for the WIP is chal- lenging: the system belongs to the class of underactuated mechanical systems, since the control inputs are less than the number of configuration variables: There are a total of two control variables τ 1 and τ 2 which are the torques ap- plied to rotate the wheels, and six configuration variables, namely, the x- and y- position of the WIP on the horizontal plane, the relative rotation angle of each of the wheels with respect to the body φ 1 and φ 2 , the orientation angle θ, and the tilting angle α. In addition, the system is restricted by nonholonomic (nonintegrable) constraints and is thus not smoothly stabilizable at a point as proven by Brockett [1983]. These constraints do not restrict the state space on which the dynamics evolve, but the motion direction at a given point: The rolling constraint impedes a sideways mo- tion, and the forward velocity of the WIP and its yaw rate are directly given by the angular velocity of the wheels. Wheeled robots have largely been considered as purely kinematic systems, due to the simplification in the motion and controllability analysis. The WIP, however, needs to be stabilized by dynamic effects, such that the complete dynamics need to be taken into account. In mechanical systems with nonholonomic constraints the configuration space Q is a finite dimensional smooth manifold, TQ is the tangent bundle - the velocity phase space - and a smooth (non-integrable) distribution D⊂ TQ defines the con- straints 1 . While traditional approaches like the Lagrange- d’Alembert equations lead to the equations of motion of nonholonomic mechanical systems (see, e. g., Pathak et al. [2005]), geometric approaches help to understand the structure and the intrinsic properties of the system. There is a lot of work regarding the modeling of nonholonomic systems, see for example Bloch [2003], Ostrowski [1999], Bloch et al. [1996], Bloch et al. [2009] and the references therein. These geometric tools help understand the mech- anism of locomotion, i. e., the way motion is generated by changing the shape of the mechanical system. Symmetries can be exploited to develop dynamical models in a reduced space. Roughly speaking, the Lagrangian L exhibits a symmetry if it does not depend on one configu- ration variable, lets say, q j . The variable q j is called cyclic. The Lagrangian is thus invariant under transformations in cyclic coordinates. Lie group action and symmetry reduc- tion has been successfully applied to model other types of nonholonomic mechanical systems in the differential geometric framework. See for example the works by Bloch et al. [1996], Ostrowski [1999], Gajbhiye and Banavar [2012]. As shown, e. g., by Ostrowski [1999], the resulting equations can be put in a simplified form containing apart from the reduced equations of motion, also the momentum and reconstruction equation, which describe the dynamics of the system along the group directions. That is, how the system translates and rotates in space due to the change in the shape variables. Bloch et al. [2009] further show the advantage of using the Hamel equations to obtain the reduced nonholonomic equations of motion: The momen- tum equation is in this case given in a body frame which appears to be more natural than in a spatial frame, for the latter is rarely conserved for systems with nonholonomic constraints. The derivation of the reduced nonholomonic equations can be done as well using the constrained La- grangian and a so-called Ehresmann connection which relates motion along the shape directions with the motion 1 The distribution D defines the admissible velocities * Technische Universit¨ at M¨ unchen, Boltzmannstr. 15, D-85748, Garching (Tel: +49-89-289 15679; e-mail: s.delgado@tum.de). ** Indian Institute of Technology Bombay, India (e-mail: {banavar,sneha}@sc.iitb.ac.in) Abstract: This paper develops the equations of motion in the reduced space for the wheeled inverted pendulum, which is an underactuated mechanical system subject to nonholonomic constraints. The equations are derived from the Lagrange-d’Alembert principle using variations consistent with the constraints. The equations are first derived in the shape space, and then, a coordinate transformation is performed to get the equations of motion in more suitable coordinates for the purpose of control. Reduced equations of motion for a wheeled inverted pendulum Sergio Delgado * , Sneha Gajbhiye ** , Ravi N. Banavar **