Estimation of homography dynamics on the special linear group

Visual servo control schemes use visual information obtained by one or multiple cameras as the primary measurement to regulate the motion of a robot [21, 11, 12, 6]. In the last decade, a number of visual servo control schemes have been proposed that extract homography image transformations between images of a planar scenes and use these as the primary visual information for a servo control problem [17, 7, 8]. A homography can be decomposed to explicitly reconstruct the pose (the translation and the rotation in Cartesian space) of the camera [9, 18] and the associated servo control task undertaken in Cartesian space [25, 16, 2, 24]. Alternatively, the control task can be defined in both image and Cartesian space; the rotation error is estimated explicitly and the translation error is expressed in the image [17, 7, 22, 8]. The resulting visual servo control algorithms are stable and robust [16] and do not depend on tracking of individual image features. Some recent work has been done on direct servo control of the homography matrix [1], an approach which offers considerable advantages in situations where the homography decomposition is ill-conditioned. A key component of this work is the identification of the group of homographies as a Lie-group isomorphic to the special linear group SL(3), an observation that has been known for some time in the computer vision community but had not been exploited before in the visual servo community. In all cases, the performance of the closed-loop system depends on the quality of the homography estimates used as input to controller. In the case of visual servo control applications, the homographies must be computed in real-time with minimal computational overhead. Moreover, in such applications the homographies vary continuously and usually smoothly. It is natural, then, to consider using a dynamical observer (or filter) process in the closed-loop system to achieve temporal smoothing and averaging of the homography measurements. Such a process will reduce noise in the homography estimates, smoothing resulting closed-loop inputs and leading to improved performance, especially in visual servo applications. There has been a surge of interest recently in nonlinear observer design for systems with certain invariance properties [23, 5, 10, 15, 3] that have mostly been applied to applications in robotic vehicles [19, 20]. From these foundations there is an emerging framework for observer design for invariant systems on Lie groups [13, 4, 14]. The special linear group structure of the homographies [1] makes the homography observer problem an ideal application of these recent developments in observer theory. In this chapter, we exploit the special linear Lie-group structure of the set of all homographies to develop a dynamic observer to estimate homographies on-line. The proposed homography observer is based on constant velocity invariant kinematics on the Lie group. We assume that the velocity is unknown and propose an integral extension of the nonlinear observer to obtain estimates for both the homography and the velocity. We prove the existence of a Lyapunov function for the system, and use this to prove almost global stability and local exponential stability around the desired equilibrium point. The proposed algorithm provides high quality temporal smoothing of the homography data along with a smoothed homography velocity estimate. The estimation algorithm has been extensively tested in simulation and on real data. The chapter is organised into five sections followed by a short conclusion. The present introduction section is followed by Section 2 that provides a recap of the Lie group structure of the set of homographies. The main contribution of the paper is given in Section 3. Sections 4 and 5 provide an experimental study with simulated and real data.