Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination by Fusing Vision and Inertial Measurements

This paper analyzes the problem of data fusion when the adopted sensors are a monocular camera and inertial sensors (i.e. one tri-axial accelerometer and one tri-axial gyrometer). The paper provides two main contributions. The first is the analytical derivation of all the observable modes, i.e. all the physical quantities that can be estimated by only using the information in the sensor data acquired during a short time interval. Specifically, the observable modes are the vehicle speed and attitude, the absolute scale and the bias affecting the inertial measurements. This holds even in the case when the camera only observes a single point feature. The analytical derivation of the previous observable modes is based on a non standard observability analysis, which fully accounts the system non linearities. The second contribution is the analytical derivation of closed-form solutions which analytically express all the previous observable modes in terms of the sensor measurements collected during a very short time interval. This allows introducing a very simple and powerful new method able to simultaneously estimate all the observable modes without the need of any initialization or a priori knowledge. Both the observability analysis and the derivation of the closed-form solutions are carried out in several different contexts, including the case of biased and unbiased inertial measurements, the case of a single and multiple features, and in presence and absence of gravity. The performance of the proposed approach is evaluated via extensive Monte Carlo simulations and by using two distinct real data sets. Key-words: Sensor Fusion, Inertial Sensors, Vision, Non linear Observability, Aerial Robotics in ria -0 05 69 08 3, v er si on 5 13 A pr 2 01 1 Résumé : Ce document considère le problème de la fusion sensorielle dans le cas très importante des capteurs inertielles et vison. Ce document contient deux contributions. La première est la dérivation analytique de tous les modes observables, c’est á dire les quantités physiques qu’on peut estimer a partir des données sensorielles. Plus précisément, les modes observables sont: la vitesse et l’orientation du véhicule, le facteur d’échelle et le bias qui est dans les données inertielles. Ces quantités physiques sont observables même dans le cas ou seulement une feature est disponible. La deuxième contribution est la dérivation d’une formule analytique qui permet d’exprimer les modes observables précédents en fonction des données sensorielles (de la camera et des capteurs inertielles) prises dans un intervalle de temps très court. Cela permet de déterminer les modes observables sans la nécessite d’aucune connaissance a priori et sans aucune initialisation. Le document contient simulations et données réels pour tester la méthode d’estimation proposée. Mots-clés : Fusion Sensoriel, Capteurs inertiels, Vision, Observabilité, Robotique Aérienne in ria -0 05 69 08 3, v er si on 5 13 A pr 2 01 1 Closed-Form Solutions for IMU & Vision 3