Reduced-Order Kalman Filtering with Relative Measurements
David S. Bayard · Journal of Guidance Control and Dynamics · 2009
In this paper, the symbol zk is used to denote a relative measurement, shown in Eq. (1), and yk is used to denote a conventional measurement, shown in Eq. (2). As an example, let x 2 R denote the three-dimensional position of a helicopter. Consider the situation of Fig. 2where a scanning lidar measures the vector offset f x noise (dashed lines) to a feature at location f at two different timesm and k, form< k. Denote the measurements as m f xm noise and k f xk noise, respectively. If f is known, then a conventional Kalman filter can be used to apply measurements m and k separately at their respective times m and k. However, if f is unknown, the situation becomes more complicated. Assume that a lidar preprocessing system is capable of recognizing that the feature at f is common to the two measurements m and k. Forming the difference as zk m k xk xm noise gives a relative measurement of the desired form Eq. (1). In the time interval between m and k it is possible that conventional (i.e., nonrelative) measurements are also available. For instance, these can be of the position type (beacons, satellite-based metrology, camera-based landmark tracking, etc.) or of the velocity type (velocimeters, Doppler radar, etc.). Such additional measurements play the role of the intervening measurements in the current formulation. Many extensions of this application are possible. For example, full pose measurements and camera-derived bearing angles are used as relative measurements [2–4]. Because the relative measurement zk depends on an earlier state xm, it does not directly fit into the standard Kalman filter mechanization.One knownmethod to handle a relativemeasurement is to replicate the state xm at time m to give a new augmented-state vector A m xm; xm T (cf. [7]). The replicated part of the state is given constant dynamics, and the augmented system is propagated until time k at which time the relative measurement can be incorporated using a linear measurement of the form