On the attitude estimation of nonholonomic wheeled mobile robots

Abstract

This paper addresses the attitude estimation problem of wheeled mobile robots (WMRs) with nonholonomic constraints moving in a plane. We consider the case that only the robot’s Cartesian position and linear velocity are available from sensor measurements. Based on the kinematic model, three attitude observers are proposed. The first observer is designed directly on the Special Orthogonal group $SO\left(2\right)$ and guarantees that for almost all initial conditions, the estimated attitude converges to the actual one with an exponential convergence rate. The second observer estimates the heading direction of the mobile robot, i.e., its dynamics evolves on the unit circle ${\mathbb{S}}^1$. The first and second observers rely on linear velocity measurements. On the other hand, the third observer presents a globally exponentially stable error dynamics and requires only Cartesian position measurements but does not evolve on ${\mathbb{S}}^1$. Additionally, a control-observer algorithm is proposed to solve the trajectory-tracking problem of WMRs. Finally, the performances of the proposed attitude observers and control law were evaluated by a set of experiments on a commercial WMR.

Introduction

In recent years, the problems of multi-robot coordination and swarm robotics have attracted the attention of the control and robotics communities. Many control approaches for swarm robotics that achieve an autonomous operation require measuring the robot’s pose (position and orientation) and, in some cases, its linear and angular velocity. Therefore, the robots must be equipped with a variety of sensors (e.g., accelerometers, gyroscopes, optical encoders, ultrasonic sensors, Global Positioning Systems (GPS), and vision systems), which increase the cost and complexity of the robotic swarm. One possible solution to reduce computational burden and costs is to design state observers. In some applications, the objective is to estimate the robot’s position relative to a particular coordinate frame (Hamel & Samson, 2017), while other tasks require orientation or attitude estimation (Crassidis, Markley, & Cheng, 2007).

The attitude estimation problem of rigid bodies in a three-dimensional space (spatial rotations) is not new, and several filtering methods and nonlinear observers can be found in the literature (Berkane et al., 2016, Khosravian and Namvar, 2011, Martin and Salaün, 2008, Sanyal et al., 2008). Extended Kalman filters have been successfully used to solve the attitude estimation problem (Lefferts et al., 1982, Markley, 2003). However, these filters are based on linearization methods and may fail for relatively large initial estimation errors. A novel Invariant Extended Kalman Filter (IEKF) for Lie groups is reported in Barrau and Bonnabel (2016). Complementary nonlinear filters with almost global convergence properties are presented in Mahony, Hamel, and Pflimlin (2008). The nonlinear observers are designed based on the body kinematics and evolve on the special orthogonal group $SO\left(3\right)$. Similar nonlinear observers on $SO\left(3\right)$ can be found in Berkane and Tayebi, 2018, Mahony et al., 2009 and Zlotnik and Forbes (2017). Globally exponentially stable attitude observer using single vector observations and a cascade observer with similar stability properties are presented in Batista et al., 2012a, Batista et al., 2012b, respectively. However, the dynamics of the observers do not preserve the structure of $SO\left(3\right)$. Quaternion-based nonlinear attitude observers are proposed in Guerrero-Castellanos, Madrigal-Sastre, Durand, Torres, and Munoz-Hernández (2013) and Thienel and Sanner (2003). A nonlinear orientation observer for aerial robotic vehicles is proposed in Hua, Ducard, Hamel, Mahony, and Rudin (2014). A lot of effort has been made, and novel solutions have been proposed to solve the attitude estimation problem. However, little attention has been paid to the particular problem of orientation estimation of WMRs with nonholonomic constraints moving in a plane (planar rotations). Although the approaches mentioned so far could be used to solve this problem, such solutions could be computationally demanding and somehow inefficient in the sense that the attitude of WMRs evolves on $SO\left(2\right)$ or the unit circle ${\mathbb{S}}^1$. In this work, we focus on the attitude estimation of WMRs with nonholonomic constraints described by the unicycle model. This problem is addressed in Velasco-Villa, Aranda-Bricaire, Rodríguez-Cortés, and González-Sierra (2012). In this work, the authors proposed an attitude observer based on the Immersion and Invariance technique (Astolfi, Karagiannis, & Ortega, 2007) combined with a kinematic tracking control law. In Noijen, Lambrechts, and Nijmeijer (2005), the authors proposed a control-observer algorithm for nonholonomic WMRs that estimates the orientation error. The observer was designed based on the kinematic model and a coordinate transformation of the position and orientation tracking errors. A similar approach is presented in Cui, Liu, and Lv (2017). A uniformly globally asymptotically stable orientation observer for vehicle platooning is presented in Bayuwindra, Lefeber, Ploeg, and Nijmeijer (2019). The observer proposed by the authors estimates not only the heading angle but also the Cartesian velocity of the vehicle. Recently, a globally exponentially attitude observer and tracking controller for WMRs is proposed in Pliego-Jiménez, Martinez-Clark, Cruz-Hernández, and Arellano-Delgado (2021). Nevertheless, the aforementioned observers do not evolve on $SO\left(2\right)$ or ${\mathbb{S}}^1$. A distributed attitude observer for nonholonomic agents is given in Van Tran and Ahn (2020).

This paper addresses the attitude estimation problem of WMRs moving in a plane, assuming that only the Cartesian position or the linear velocity is available from sensor measurements. The attitude observers were designed based on the kinematic model of the robot. The WMRs with nonholonomic constraints are underactuated systems where the position and attitude subsystems are highly coupled; this fact was also exploited in the observer design. Three attitude observers are proposed. The first observer is designed on $SO\left(2\right)$, and the second observer estimates the heading direction of the robot; therefore, it evolves on ${\mathbb{S}}^1$ and can be viewed as a simplified version of the first observer. These two observers require the robot’s linear velocity, and their error dynamics are almost globally exponentially stable. A globally exponentially observer that estimates the heading direction and requires only position measurements is proposed as an alternative solution. However, this observer does not preserve the unit norm constraint. Finally, a control-observer algorithm for trajectory tracking of WMRs is proposed to show a potential application of the first attitude observer.