Receding Horizon Trajectory Optimization for Simultaneous Signal Landscape Mapping and Receiver Localization

Z.M. Kassas, J.A. Bhatti, T.E. Humphreys

Abstract:	A new navigation paradigm, termed opportunistic navigation (OpNav), has been proposed to improve navigation robustness in Global Navigation Satellite Systems (GNSS)-challenged environments. This paradigm aims to extract positioning and timing information from ambient radio frequency “signals of opportunity” (SOPs) [1]. OpNav treats all signals as potential SOPs, from conventional GNSS signals to communications signals never intended for use as timing or positioning sources. The OpNav estimation problem is similar to the simultaneous localization and mapping (SLAM) problem in robotics [2]. Both imagine an agent, which starting with incomplete knowledge of its location and surroundings, simultaneously builds a map of its environment and locates itself within that map. In traditional SLAM, the map that gets constructed as the robot moves through the environment is composed of landmarks with associated positions. OpNav extends this concept to radio signals, with SOPs playing the role of landmarks. In contrast to a SLAM environmental map, the OpNav signal landscape is dynamic and more complex. In pseudorange-only OpNav, the receiver must estimate simultaneously with its own states, the states of each SOP, namely the position and velocity of the transmitter, time offset from a reference time base, rate of change of time offset, and a set of parameters that characterize the oscillator stability. Metaphorically, the signal landscape map can be thought of as a “jello map”, with the jello firmer as the oscillators are more stable. In collaborative OpNav (COpNav), multiple receivers share information to construct and continuously refine a global signal landscape. The observability of COpNav environments comprising multiple receivers with velocity random walk dynamics making pseudorange measurements on multiple SOPs was analyzed and the degree of observability, also known as estimability, of the various states was quantified in [3]. While observability is a Boolean property, i.e. it asserts whether a system is observable or not, it does not specify which trajectory is best for information gathering, and consequently estimability. This is the subject of this paper. To this end, the receiver dynamics is modified so to allow for controlled maneuvering. Optimizing an observer's path in tracking problems has been studied extensively [4]. In such problems, the observer, who has perfect knowledge about its own states, is tracking a mobile target. The trajectory optimization objective is to prescribe trajectories for the observer to maintain good estimates of the target's states. In SLAM, the problem of trajectory optimization is more involved, due to the coupling between the localization accuracy and map quality [5]. Trajectory optimization in OpNav environments can be thought of as a hybrid of: (i) optimizing an observer's path in tracking problems and (ii) optimizing the robot's path in SLAM. First, due to the dynamical nature of the clock error states, the SOP's state-space is non-stationary, which makes the problem analogous to tracking non-stationary targets. Second, the similarity to SLAM is due to the coupling between the receiver localization accuracy and signal landscape fidelity. An initial receiver trajectory optimization study was conducted in [6]. The following problem was considered. A receiver with no a priori knowledge about its own states is dropped in a planar OpNav environment. Assuming the receiver can prescribe its own trajectory in the form of velocity commands, what motion planning strategy should the receiver adopt to build a high-fidelity map of the OpNav signal landscape while simultaneously localizing itself within this map in space and time? To answer this question, first, the minimum conditions under which the OpNav environment is fully-observable were established, and the need for receiver maneuvering to achieve observability was highlighted. It was shown that the environment is fully-observable if the initial states of at least one “anchor” SOP are known. An optimal closed-loop information-theoretic greedy strategy was proposed for receiver motion planning. Three information measures were compared: D-optimality, A-optimality, and E-optimality. It was shown that all such strategies outperformed a receiver moving randomly as well as in a pre-defined trajectory. Among these information measures, the D-optimality outclassed the A-optimality and E-optimality criteria. The present paper's contribution is to extend the work of [6] in two ways. First, it generalizes the receiver dynamics model from a simple first-order model to a second-order model, in which the receiver commands its acceleration. It is shown that with such higher-order model, as long as the receiver's initial state is known or the initial state of at least one SOP is known, then the OpNav environment is fully-observable. Subsequently, in contrast to the findings in [3], it is concluded that a receiver with controlled maneuvers requires less a priori knowledge about the OpNav environment for complete observability. The second contribution is to extend the greedy motion planning strategy to a multi-step look-ahead trajectory optimization strategy. It is well-established in the literature that multi-step look-ahead strategies, also known as receding horizon, outperform greedy strategies. However, receding horizon strategies come at the cost of increased computational burden. This paper investigates the tradeoffs associated with extending the planning horizon on the increased computational burden versus the achieved improvement in the fidelity of the signal landscape map and receiver localization. References: [1] Pesyna, K.M., Kassas, Z.M., Bhatti, J.A., and Humphreys, T.E., “Tightly-Coupled Opportunistic Navigation for Deep Urban and Indoor Positioning,” Proceedings of ION GNSS, Portland, September, 2011. [2] Durrant-Whyte, H., and Bailey, T., “Simultaneous Localization and Mapping: Part I,” IEEE Robotics & Automation Magazine, vol. 13, no. 2, pp. 99-110, June, 2006. [3] Kassas, Z.M., and Humphreys, T.E., “Observability and Estimability of Collaborative Opportunistic Navigation with Pseudorange Measurements,” Proceedings of ION GNSS, Nashville, September, 2012. [4] Ponda, S., Kolacinski, R., and Frazzoli, E., “Trajectory Optimization for Target Localization Using Small Unmanned Aerial Vehicles,” Proceedings of AIAA Guidance, Navigation, and Control Conference, Chicago, August, 2009. [5] Leung, C., Huang, S., Kwok, N., and Dissanayake, G., “Planning Under Uncertainty Using Model Predictive Control for Information Gathering,” Robotics and Autonomous Systems, Vol. 54, No. 11, 2006, pp. 898-910. [6] Kassas, Z.M., and Humphreys, T.E., “Motion Planning for Optimal Information Gathering in Opportunistic Navigation Systems,” AIAA Guidance, Navigation, and Control Conference, Boston, August, 2013, submitted and under review.
Published in:	Proceedings of the 26th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2013) September 16 - 20, 2013 Nashville Convention Center, Nashville, Tennessee Nashville, TN
Pages:	1962 - 1969
Cite this article:	Kassas, Z.M., Bhatti, J.A., Humphreys, T.E., "Receding Horizon Trajectory Optimization for Simultaneous Signal Landscape Mapping and Receiver Localization," Proceedings of the 26th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2013), Nashville, TN, September 2013, pp. 1962-1969.
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