3 known landmarks are enough for solving planar bearing SLAM and fully reconstruct unknown inputs

Abstract

In this paper we show that for an observer moving in the plane with no other information than the measurement of relative bearing to three known landmarks, it is possible to completely reconstruct its position and velocity. In particular this applies to the case where no model of the vehicle, nor odometry or acceleration measurements are available. Furthermore, in the same hypotheses, the position of any further landmark can be reconstructed from its bearing only. These results are more general than what is currently known on nonlinear observability of the SLAM problem, which relies on known observer velocities. Our results are also more general than the 2D version of known structure-from-motion observability results, which assume unknown but constant velocities. The proposed method is used to build a nonlinear observer directly applicable to a range of problems from computer vision to autonomous visual navigation

Visual-inertial structure from motion: Observability vs minimum number of sensors

“…In particular, the extended state includes the unknown inputs together with their time derivatives up to a given order. Note that, this augmented state has already been considered in [1] where a sufficient condition for the state observability has been provided.…”

The Unicycle in Presence of a Single Disturbance: Observability Properties

“…Note that, in the trivial case analyzed by the previous lemma, the criterion provided below simplifies, since we do not need to compute the 2 , and we do not need to check that its differential belongs to the codistribution computed at every step of algorithm 2. In practice, we skip the steps 4 and 5 in the procedure below.…”

“…Very recently, also the case when some of the inputs are unknown and time dependent has been considered in the framework of mobile robotics [2,16]. In [2] the authors investigate the observability properties of a fundamental problem in the framework of mobile robotics (the bearing SLAM).…”

“…In [2] the authors investigate the observability properties of a fundamental problem in the framework of mobile robotics (the bearing SLAM). In [16] the author derives the observability properties in the case of robots equipped with visual and inertial sensors, when some of the inertial sensors are missing.…”

“…On the other hand, the problem of finding a simple analytic condition that allows us to check the weak local observability of the state is still open in the nonlinear case and for any number of UI. The methods introduced in [2,16] only provide sufficient conditions for the state observability. They are based on a suitable state extension.…”