Energy-based Limit Cycle Compensation for Dynamically Balancing Wheeled Inverted Pendulum Machines

In this paper we present an energy-based algorithm to minimize limit cycles in dynamically balancing wheeled inverted pendulum (IP) machines. Because the algorithm is not based on absolute values of parameters, the performance is robust and accounts for mechanical reconfiguration and wear. The effects of phenomena such as drive-train friction, rolling friction, backlash and sensor bandwidth are well known, causing either limit cycles or instabilities in IP balancing machines and yet compensation or control design to mitigate these effects are not well known. The effects of these non-linearities can be observed in the energy behavior of IP balancing machines, hence, as a broader goal we seek to establish an energy-based framework for the investigation of non-linearities in this class of machines. We successfully demonstrate the effectiveness of our algorithm on a two-wheeled IP balancing machine, “Charlie”, developed in our laboratory. As an example we show a reduction in the amplitude of limit cycles over a 10 second period from 220 degrees in wheel angle and 15 degrees in pitch to 9.9 degrees and 1.3 degrees respectively. INTRODUCTION While the inverted pendulum (IP) has received wide attention over the past half century, the dynamically balancing wheeled IP machine is a relatively recent development [1, 2, 3]. Aspects of control theory concerned with balancing such a machine are well known, however the effect of non-linearities on balancing performance is not well understood. Mechanical non∗Address all correspondence to this author. linearities such as backlash in gear trains and friction in the drive mechanism have a profound effect on the dynamics of a balancing machine. In this paper we focus on non-linearities that generate limit cycle behavior in wheeled IP machines such as rolling resistance and drivetrain friction. To analyze the effect of these non-linearities we look at energy flowing in and out of the feedback control system. The energy model is an intuitive and powerful method of predicting stability and designing controllers for dynamical systems. While in this paper we use an energy-based observer to detect and correct limit cycles while balancing, we believe that this energy-based approach is well suited to detection and mitigation of other nonlinear behaviors in wheeled IP balancing machines. It is well known that friction can cause limit cycles in feedback control systems, a detailed description of the nature of these limit cycles can be found in work done by Olssen [4]. In particular the author details the effect of friction in a pendulum cart system and the resulting limit cycles. Campbell et.al. [5] describe limit cycles caused by stick-slip friction between the cart and track and synthesize a controller to stabilize the system. Papers [6,7,8,9] all deal with techniques to stabilize an inverted pendulum in the presence of friction. Armstrong-Hélouvry et.al. [10] detail an exhaustive survey of friction models, friction compensation techniques complete with standard practices in industry to combat problems caused by friction. A passivity-based compensator for friction finds a mention in work done by Astrom [11]. From the literature we see that both a friction observer and a dynamic friction model have been proposed for the design of an accurate compensator. However this process is not simple and also difficult to implement on hardware regardless of the friction This article has been accepted for publication DSCC2013,Stanford, CA but has not been fully edited. Content may change prior to final publication. This article has been accepted for publication DSCC2013,Stanford, CA but has not been fully edited. Content may change prior to final publication.