Optimal Measurement Selection For Any-time Kalman Filtering With Processing Constraints † Nima Moshtagh, Lingji Chen, Raman Mehra * Abstract— In an embedded system with limited processing resources, as the number of tasks grows, they interfere with each other through preemption and blocking while waiting for shared resources such as CPU time and memory. The main task of an Any-time Kalman Filter (AKF) is real-time state estima- tion from measurements using available processing resources. Due to limited computational resources, the AKF may have to select only a subset of all the available measurements or use out- of-sequence measurements for processing. This paper addresses the problem of measurement selection needed to implement AKF on systems that can be modeled as double-integrators, such as mobile robots, aircraft, satellites etc. It is shown that a greedy sequential selection algorithm provides the optimal selection of measurements for such systems given the processing constraints. I. I NTRODUCTION This paper addresses the problem of measurement se- lection as part of an Any-time Kalman Filter (AKF) for obtaining the best state estimate of systems that can be modeled as a double-integrator, such as mobile robots, aircrafts, satellites etc [9]. State estimators are typically designed with fixed measurement and propagation update step sizes, nominally equal to the real-time interval. However, this may increasingly become difficult to achieve in practice as the complexity of the estimator in terms of number of states and measurements as well as the number of other resource requesting tasks on a processor grows. For instance, TPF-I [19] (NASA’s first space-based mission to directly observe planets outside our own solar system) uses multiple smaller telescopes that need to collaborate with each other and maintain extremely precise formations. Within a processor, tasks interfere with each other through preemption and blocking when waiting for shared resources such as CPU time and memory. The execution times of the tasks themselves may be data dependent or may vary due to hardware features such as caches. To achieve good perfor- mance in embedded applications with limited resources, the constraints of implementation platform must be taken into account at design-time. Therefore, the achievable estimation accuracy depends not only on the algorithms, but also on their actual implementation and communication related delays. Typically, the algorithms are implemented on a real-time multi-tasking processor that allocates on-board computa- tional resources to multiple tasks and functions according † This work is supported in part by NASA, Jet Propulsion Laboratory, contract # NNC08CA34C. * The authors are with Scientific Systems Company, Inc. 500 W. Cummings Park, Suite 3000, Woburn, MA 01801. nmoshtagh, chen, rmk@ssci.com to some scheduling policy. The processor’s task scheduler may induce delays that were unaccounted for at design-time and may sometimes preempt measurement processing and estimation tasks in favor of other tasks. Hence, estimation accuracy and in general the performance of any embedded algorithm can be significantly lower than expected during execution. A direct consequence of the preemption of the measure- ment processing is that the estimator may have to select only a subset of all the available measurements for processing, or update the state using out-of-sequence measurement. Thus, an AKF is needed to select and process such measurements stored in the buffer. The focus of this paper is on the measure- ment selection problem. There are two issues to be addressed in this problem; one is the minimum subset that is necessary to maintain observability of the state, and the second is the selection of the best set in terms of information extraction for estimation if more than the minimum is available. Our goal is to provide algorithms to help optimize the measurement selection processing. The algorithm can imple- ment a computationally expensive, but optimal measurement selection search, and it can be used as a benchmark for evaluation purposes. In Section III the measurement selection problem is formulated as an optimization problem. In the most general case the size of the problem grows exponen- tially with the desired number of measurements. But, we show that in the absence of the process noise the optimization problem can be converted into a convex problem [1] and easily solved to find the global solution. In the presence of the process noise, however, the problem is not convex anymore, and a greedy sequential selection (GSS) algorithm is presented in Section IV that finds a selection polynomially in the number of desired measurements. Based on extensive simulations, and comparison with optimal solutions, we have conjectured that GSS algorithm actually produces the optimal set of measurements for a double-integrator model, in the presence of process noise. In Section V we study the problem Fig. 1. The structure of the Any-time Kalman Filter (AKF) with measurement selection. Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009 ThBIn4.3 978-1-4244-3872-3/09/$25.00 ©2009 IEEE 5074