Necessary and sufficient conditions for localization of multiple robot platforms

In this paper, we consider the problem of cooperatively localizing a formation of networked robots/vehicles in SE(2). First, we propose necessary and sufficient conditions to establish when a team of robots with heterogeneous sensors can be localized. We then show how these conditions are analogous to well-known results in the literature on kinematics of planar mechanisms. We show how localization is equivalent to solving a system of nonlinear closure equations. Depending on what sensors are available for each robot, the multirobot formation can be modeled as a sensing graph consisting of vertices representing robots and edges corresponding to sensory information. We establish conditions that must be satisfied by this graph and show how this graph influences estimates of positions and orientations of the robots in a team through experiments and simulations. INTRODUCTION In order for a team of mobile robots to navigate autonomously in some desired formations and further perform cooperative tasks, such as mapping, surveillance and target acquisition, they must be able to localize themselves in the formation as well as in a global reference frame [1, 2]. Therefore, how to estimate robots’ positions and orientations (poses) in a precise and efficient way is of particular interest. Our interest in this paper is localizing a team of heterogeneous robots in the two-dimensional Special Euclidean group, SE(2) [3], and in localizing targets with information obtained from heterogeneous sensors. Specifically, we are interested in conditions under which all robots in the formation can be localized in the environment, and in minimizing the relative and absolute uncertainty in the estimates. Our goal in this paper is to derive necessary and sufficient conditions for localizing a formation of three or more robots in SE(2) from distributed measurements and quantifying the quality of the resulting estimates. The adaptation of the sensing graph and formation geometry to improving these estimates is discussed in [4]. Our study of team localization has benefited from extensive research on parallel mechanisms in the past decades [5–9]. In the following sections, we will show that a multi-robot formation can be modeled as a closed kinematic chain. Measurements of features in the environment constrain the robot’s position and orientation in the world, in much the same way a linkage does. Similarly, measurements of one robot’s position and orientation by another robot constrains estimates of the relative position and orientation and can be thought of as a linkage connecting the two robots constraining their relative configurations. Thus a multi-robot platform with distributed and often redundant measurement information can be viewed as a mechanism with closed kinematic chains [10, 11]. The task of localizing the platform in SE(2) based on the measurements is analogous to the forward kinematic analysis problem of parallel mechanisms [12,13], whose goal it is to determine the platform’s position and orientation relative to the base given the sensed lengths of the linkages. Whether or not a set of measurements is sufficient to localize each robot in the formation is analogous to asking if a parallel kinematic chain is statically stable. Thus, a system of robots that cannot be localized with a given set of measurements can be thought of as a mechanism with closed kinematic chains 1 Copyright c © 2004 by ASME