Geometric descent algorithms for attitude determination using GPS

Abstract This paper describes a set of numerical optimization algorithms for solving the GPS-bascd attitude determination problem. We pose the problem as one of minimizing the function tr (Θ N Θ T Q - 2Θ W ) with respect to Θ ∈ SO (3), where N, Q, and W are given 3×3 matrices. The method of steepest descent and Newton's method are generalized to SO (3) by taking advantage of its Lie group structure. Analytic solutions to the line search procedure are also derived. Results of numerical experiments for the class of geometric descent algorithms proposed here are presented.

[1]  M. Chu.  On the Continuous Realization of Iterative Processes , 1988 .

[2]  Yaakov Oshman,et al.  Minimal-Parameter Attitude Matrix Estimation from Vector Observations , 1998 .

[3]  F. Markley.  Attitude determination using vector observations and the singular value decomposition , 1988 .

[4]  Daniele Mortari.  Energy Approach Algorithm for Attitude Determination from Vector Observations , 1995 .

[5]  W. Boothby.  An introduction to differentiable manifolds and Riemannian geometry , 1975 .

[6]  U. Helmke,et al.  Optimization and Dynamical Systems , 1994, Proceedings of the IEEE.

[7]  G. Wahba.  A Least Squares Estimate of Satellite Attitude , 1965 .

[8]  K. S. Arun,et al.  Least-Squares Fitting of Two 3-D Point Sets , 1987, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[9]  Daniele Mortari,et al.  Euler-q Algorithm for Attitude Determination from Vector Observations , 1998 .

[10]  Bradford W. Parkinson,et al.  Global positioning system : theory and applications , 1996 .

[11]  Penina Axelrad,et al.  Spacecraft attitude estimation using the Global Positioning System - Methodology and results for RADCAL , 1996 .

[12]  R. Brockett,et al.  A new formulation of the generalized Toda lattice equations and their fixed point analysis via the momentum map , 1990 .

[13]  김중곤.  Numerical optimization on the rotation group , 1998 .

[14]  R. Hanson,et al.  Analysis of Measurements Based on the Singular Value Decomposition , 1981 .

[15]  Penina Axelrad,et al.  Attitude Estimation Algorithms for Spinning Satellites Using Global Positioning System Phase Data , 1997 .

[16]  Itzhack Y. Bar-Itzhack,et al.  Polar decomposition for attitude determination from vector observations , 1992 .

[17]  M. Shuster.  A survey of attitude representation , 1993 .

[18]  B. Kolman,et al.  A survey of Lie groups and Lie algebras with applications and computational methods , 1972 .

[19]  James R. Wertz,et al.  Spacecraft attitude determination and control , 1978 .

[20]  R. Brockett.  Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems , 1991 .

[21]  Zexiang Li,et al.  Geometric algorithms for workpiece localization , 1998, IEEE Trans. Robotics Autom..

[22]  Bradford W. Parkinson,et al.  Two Studies of High Performance Attitude Determination Using GPS: Generalizing Wahba's Problem for High Output Rates and Evaluation of Static Accuracy Using a Theodolite , 1992 .

[23]  R. Brockett.  Least squares matching problems , 1989 .