Physical Interpretation of Rigidity for Bearing Formations: Application to Mobility and Singularity Analyses

Abstract

Research on formation control and cooperative localization for multi-robot systems has been an active field over the last years. A powerful theoretical framework for addressing formation control and localization, especially when exploiting onboard sensing, is that of formation rigidity (mainly studied for the cases of distance and bearing measurements). Rigidity of a formation depends on the topology of the sensing/communication graph but also on the spatial arrangement of the robots, since special configurations ('singularities' of the rigidity matrix), which are hard to detect in general, can cause a rigidity loss and prevent convergence of formation control/localization algorithms based on formation rigidity. The aim of this paper is to gain additional insights into the internal structure of bearing rigid formations by considering an alternative characterization of formation rigidity using tools borrowed from the mechanical engineering community. In particular, we show that bearing rigid graphs can be given a physical interpretation related to virtual mechanisms, whose mobility and singularities can be analyzed and detected in an analytical way by using tools from the mechanical engineering community (Screw theory, Grassmann geometry, Grassmann-Cayley algebra). These tools offer a powerful alternative to the evaluation of the mobility and singularities typically obtained by numerically determining the spectral properties of the bearing rigidity matrix (which typically prevents drawing general conclusions). We apply the proposed machinery to several case formations with different degrees of actuation, and discuss known (and unknown) * Address all correspondence to this author. singularity cases for representative formations. The impact on the localization problem is also discussed. IntroductionFormation control and cooperative localization for multi-robot teams has been a topic of extensive research over the last years [1-7]. Among the many challenges, considerable efforts are still devoted to devising decentralized formation controllers and/or localization schemes based on only local (onboard) sensing and communication. When considering these sensing/communication requirements, one has to face several challenges related to the decentralized design of the control/estimation algorithms, as well as the constraints arising from the use of onboard sensors. For instance, typical sensors only measure part of the relative pose among robot pairs (e.g., a distance or a bearing vector), while knowledge of the full relative pose is often needed for implementing formation control schemes. Also, a sensor provides measurements naturally expressed in the body-frame of the robot carrying it, and therefore one also faces the need of letting the robots agree over some common shared frame where to express all the individually collected measurements for then communicating them to the other robots in the group.In this context, a powerful theoretical framework for addressing decentralized formation control/localization from onboard ...