Riccati Observers for the Nonstationary PnP Problem

This paper revisits the problem of estimating the pose (position and orientation) of a body in 3-D space with respect to (w.r.t.) an inertial frame by using 1) the knowledge of source points positions in the inertial frame, 2) the measurements of the body angular velocity expressed in the body's frame, 3) the measurements of the body translational velocity, either in the body frame or in the inertial frame, and 4) source points bearing measurements performed in the body frame. An important difference with the much studied static Perspective-n-Point problem addressed with iterative algorithms is that body motion is not only allowed but also used as a source of information that improves the estimation possibilities. With respect to the probabilistic framework commonly used in other studies that develop extended Kalman filter solutions, the deterministic approach here adopted is better suited to point out the observability conditions, that involve the number and disposition of the source points in combination with body motion characteristics, under which the proposed observers ensure robust estimation of the body pose. These observers are here named Riccati observers because of the instrumental role played by the continuous Riccati equation in the design of the observers and in the Lyapunov stability and convergence analysis that we develop independently of the well-known complementary (either deterministic or probabilistic) optimality properties associated with Kalman filtering. The set of these observers also encompasses extended Kalman filter solutions. Another contribution of this study is to show the importance of using body motion to improve the observers performance and, when this is possible, of measuring the body translational velocity in the inertial frame rather than in the body frame to allow for the body pose estimation from a single source point taken as the origin of the inertial frame. This latter possibility finds a natural extension in the Simultaneous Localisation And Mapping (SLAM) problem in Robotics.