https://hal-mines-paristech.archives-ouvertes.fr/hal-01671724v2Barrau, AxelAxelBarrauSafran TechBonnabel, SilvereSilvereBonnabelCAOR - Centre de Robotique - Mines Paris - PSL (École nationale supérieure des mines de Paris) - PSL - Université Paris sciences et lettresLinear observation systems on groups (I)HAL CCSD2018[SPI] Engineering Sciences [physics][SPI.AUTO] Engineering Sciences [physics]/AutomaticBonnabel, Silvere2018-02-08 16:24:502022-10-18 11:02:092018-02-09 13:51:36enPreprints, Working Papers, ...https://hal-mines-paristech.archives-ouvertes.fr/hal-01671724v2application/pdf2The present paper is an accessible digest, along with extensions, of previous work by the authors. We propose an unifying and versatile framework for a class of discrete time systems whose state is an element of a group, that we call linear observation systems on groups. Those systems strictly mimic linear systems in the sense that + is replaced with group multiplication , and linear maps by endomorphisms. Generalized linear observers on groups, which are the group counterpart of linear observers (and known as invariant observers on groups), are shown to share some important properties with linear observers, namely the fact the estimation error equation is autonomous. We then prove that, linear observation systems are in fact the only ones such that the error equation is autonomous, and relate them to group-affine systems we have previously introduced in continuous time. We also introduce a family of groups called SE_K(D), and leverage it to prove many non-linear discrete-time systems of navigation and robotics (including Simultaneous Localization And Mapping) are in fact linear observation systems on SE_K(D).