Generalized Lever Arm Calculations For Multi-Instrument Kalman Filters

T.A. Miller and M. Stark

Abstract:	Combined GPS and INS instrumentation frequently uses a Kalman Filter, embodied by the state vector, to integrate the data and provide a best possible solution. The state vector is the navigational data (like position and attitude) of a specific coordinate frame (the navigation point) associated with a point on a vehicle. But instruments are not generally co-located with the navigation point nor are they generally aligned with the attitude axes of the vehicle. Typically, a lever arm translation vector and skew matrix are used to describe the relation of each instrument’s frame of reference to the navigation point of interest and its coordinate frame of reference. In an extended Kalman Filter, the mathematical relationships of an instrument’s measurements to the state vector are the sensitivity equations, ( ) X h Z = , where Z is a vector of expected measurements calculated from the state vector, X , and h is the vector sensitivity equation. There seems to be two genres of methods to handle lever arms for distributed, multiple instruments. One such method is to modify h for each instrument to include the lever arm effects on that instrument. This requires an additional modification of the Jacobian ( ) X H matrix algorithms as well. We can call this first method the Instrument Method because h and ( ) X H depend primarily on the instrument model. This paper promotes a second method, which is to map the state, X, into the instrument’s frame of reference. Then the h algorithm is completely unchanged by the lever arm matrix. The H algorithm is only changed by multiplication by a lever arm matrix, L , before being used by the Kalman Filter. We can call this second method the State Method because L depends only on the structure of the state vector. The advantages of the State Method are that a) the code for lever arm modifications can be streamlined into a single software object, and b) the lever arm code is independent of the instruments so lever arm modifications to new instruments or new instrument models are handled transparently, and c) different instruments use the same set of physical assumptions. This paper describes the fundamental physical and mathematical reasoning behind the derivations, defines the general form of the lever arm matrix, defines a methodology for implementation, and provides examples for a state vector used in combined GPS/INS calculations.
Published in:	Proceedings of the 18th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2005) September 13 - 16, 2005 Long Beach Convention Center Long Beach, CA
Pages:	38 - 47
Cite this article:	Miller, T.A., Stark, M., "Generalized Lever Arm Calculations For Multi-Instrument Kalman Filters," Proceedings of the 18th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2005), Long Beach, CA, September 2005, pp. 38-47.
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