Elsevier

Automatica

Volume 74, December 2016, Pages 222-229
Automatica

Brief paper
Energy shaping for position and speed control of a wheeled inverted pendulum in reduced space

https://doi.org/10.1016/j.automatica.2016.07.045Get rights and content

Abstract

The paper deals with the energy-based stabilization and speed control of a wheeled inverted pendulum, which is an underactuated, unstable mechanical system subject to nonholonomic constraints. We use the method of Controlled Lagrangians for the stabilization of an equilibrium characterized by the length of the driven path, the orientation, and the pitch angle. The approach is systematic and very intuitive, for it is physically motivated. Based on the stabilization results, we design a speed control law. After the presentation of the model under nonholonomic constraints in Lagrangian representation, we provide an elegant solution to the matching equations for kinetic and potential energy shaping for the considered system. Simulations show the applicability of the method, and the comparison with a linear controller emphasizes its performance.

Introduction

The wheeled inverted pendulum (WIP)–and its well-known commercial version, the Segway (Segway, 2016)–has gained interest for human assistance or transportation in the past several years due to its high maneuverability and simple construction (Li, Yang, & Fan, 2013). A WIP–shown from the side in Fig. 1 (left)–consists of a vertical body with two coaxial driven wheels mounted on the body. The actuation of both wheels in the same direction generates a forward (or backward) motion; opposite wheel velocities lead to a turning motion around the vertical axis. Mobile robotic systems based on the WIP, like the intelligent two wheeled road vehicle B2 presented in Baloh and Parent (2003), or the novel and more car-like Segway PUMA and Chevrolet En-V, are being developed to be used as new personal urban transportation systems (General Motors, 2010, PUMA, 2016). Some institutes have also developed their own WIPs for research purposes, e.g., JOE (Grasser, D’Arrigo, Colombi, & Rufer, 2002) and InPeRo (Nasrallah, Michalska, & Angeles, 2007), to give only two examples. These systems can be further used as service robots like KOBOKER (Lee & Jung, 2011) or moving information platforms like the Ballbot mObi (mObi, 2016).

The stabilization and tracking control for the WIP is challenging: The system is underactuated, the upward position of the body represents an unstable equilibrium that needs to be stabilized by feedback, and, in addition, the system motion is restricted by nonholonomic (non-integrable) constraints (Bloch, 2003) and is, thus, not smoothly stabilizable at a point, as proven by Brockett (Brockett, 1983). Many researchers around the world have put great effort into designing stabilization and tracking control laws for the WIP, particularly using linearized models (Grasser et al., 2002, Li et al., 2013, Muralidharan and Mahindrakar, 2014). During the last decade, however, a strong focus is set on the nonlinear model and nonlinear control laws (Kausar et al., 2012, Muralidharan et al., 2009, Nasrallah et al., 2007, Pathak et al., 2005). For a very complete overview of the existing work on modeling and control of WIPs until 2012, the reader is referred to Chan, Stol, and Halkyard (2013).

Existing methods often do not exploit the mechanical structure of the system, feature a cumbersome design procedure, or lack robustness due to a partial feedback linearization. A solution can be provided by energy shaping methods. These control techniques, like the method of Controlled Lagrangians, or Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC), have been successfully used for the stabilization of underactuated mechanical systems (Chang et al., 2002, Ortega et al., 2002) as well as the speed control for electromechanical systems (Ortega, Loria, Nicklasson, & Sira-Ramirez, 1998). These methods are attractive, since they shape the energy of the system but preserve its physical structure and, thus, appear natural. The idea of shaping the energy has been also extended to the stabilization of nonholonomic mechanical systems (Blankenstein, 2002, Maschke and van der Schaft, 1994). However, for the asymptotic stabilization of a desired configuration, a discontinuous or time-varying control law is required (Astolfi, 1996, Brockett, 1983). This can be achieved via energy shaping by assigning non-smooth potential functions (Fujimoto, Sakai, & Sugie, 2012).

In this paper, which generalizes and completes the results of the conference paper (Delgado & Kotyczka, 2015), we present a feasible and elegant solution of the energy shaping problem for position and speed control of the WIP. Instead of considering the six-dimensional manifold Q̃, which represents the configuration space of the WIP, we restrict our analysis to a lower dimensional space Q, on which the system evolves unconstrained. Based on the WIP’s nonlinear model, we design a passivity-based stabilizing and speed controller (for constant speed references) in the reduced space Q. Since the closed-loop mechanical-type energy is used as Lyapunov function, the framework is remarkably intuitive, for it is physically motivated. The controller is thereafter parametrized applying local linear dynamics assignment (LLDA), a method used to fix design parameters in nonlinear passivity-based control by making use of the linearized model (Kotyczka, 2013). Note that the simplicity of controller design, in turn, requires appropriate planning of the trajectories in the reduced space Q. The applicability and performance of the developed controllers is shown with a series of simulations. To sum it up, the novelty of the paper is the systematic and integrated design of a position and speed controller for the wheeled inverted pendulum system in a single, energy-based framework. An emphasis is put on the structural advantages of the approach.

Convention: For compactness, we will use the notation s(α)=sinα, and c(α)=cosα. When obvious from the context, arguments are omitted for simplicity.

Section snippets

Modeling

In a simple mechanical system with k nonholonomic constraints, the n-dimensional manifold Q̃ is the configuration space, its tangent bundle TQ̃ is the velocity phase space and a smooth non-integrable distribution DTQ̃ characterizes the constraints. The Lagrangian L is a map L:TQ̃R and is defined as the kinetic energy minus the potential energy L=KV. A curve q(t) is said to satisfy the constraints if q̇(t)Dq, for all qQ̃ and all times t. The constraint distribution D is assumed to be

Energy-based controller design

This section discusses the systematic design of an energy-based controller that is capable of asymptotically stabilizing an admissible equilibrium ξQ (an equilibrium is called admissible5  if GξV|ξ=0). Based thereon, a second controller is derived, which is capable of stabilizing a constant forward and turning velocity. We will put focus on the Lagrangian case, for velocities are more

Simulations

In this section, we consider five different simulation scenarios denoted by σi, i=1,,5, and three different controllers, namely, the energy shaping speed control laws of Proposition 4 (ES) and Proposition 6 (ES*), and a linear state feedback (LIN). For ES and ES*, k2=1, kp, and kd,2 have been chosen such that, locally, the yawing dynamics have closed-loop eigenvalues {1,6.2}. The parameters k1, k3, γ, and kd,1, and the function Π1(Φ(s,α))=12μ(Φ(ss,α))2 are chosen such that the remaining

Conclusion and further work

This paper discusses the design and parametrization of an energy shaping position and speed controller for a wheeled inverted pendulum. We have shown that the matching problem for the position stabilization in reduced coordinates can be solved in an elegant way. Moreover, the solution of the latter directly leads–with some minor assumptions–to the speed controller, such that it can be implemented without further effort. Simulations emphasize the performance of the approach: The transient

Sergio Delgado was born in Bogotá, Colombia. He received the Dipl.-Ing. degree in mechanical engineering and the Dr.- Ing. degree in automatic control from Technische Universität München (TUM) in 2010 and 2016, respectively. His research interests include energy-based nonlinear control and nonholonomic systems.

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    Sergio Delgado was born in Bogotá, Colombia. He received the Dipl.-Ing. degree in mechanical engineering and the Dr.- Ing. degree in automatic control from Technische Universität München (TUM) in 2010 and 2016, respectively. His research interests include energy-based nonlinear control and nonholonomic systems.

    Paul Kotyczka received the Dipl.-Ing. degree in electrical engineering and the Dr.-Ing. degree in automatic control from Technische Universität München (TUM) in 2005 and 2010, respectively. Since 2011 he has been Akademischer Rat (assistant professor) at the Institute of Automatic Control at TUM, where he leads the energy based modeling and control group. Since September 2015 he is a Marie Skłodowska-Curie Fellow at Laboratoire d’Automatique et de Génie des Procédés (LAGEP), Université Claude Bernard Lyon 1. His research interests include physical port-based modeling of finite- and infinite-dimensional systems, discretization and nonlinear control.

    The material in this paper was partially presented at the 5th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non Linear Control (LHMNLC 2015), July 4–7, 2015, Lyon, France. This paper was recommended for publication in revised form by Associate Editor Jun-ichi Imura under the direction of Editor Toshiharu Sugie.

    1

    The second author is on leave of absence from Institute of Automatic Control (TUM) with a European Commission Marie-Skłodowska-Curie Fellowship (Project reference 655204, EasyEBC).

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