Attitude Determination via SDP Relaxation
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Solution Overview
Problem
Current methods for attitude determination using multiple navigation antennas on a rigid body are inefficient due to the complexity of solving nonlinear programming problems, particularly in real-time applications where precise orientation calculations are required, such as in satellite navigation systems.
Innovation Solution
The use of Semi-Definite Programming (SDP) relaxation converts the attitude determination problem into a convex optimization form, allowing for efficient numerical algorithms to solve the quadratically constrained quadratic minimization problem and determine heading, pitch, and roll angles.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If nonlinear programming methods are used for attitude determination, then measurement precision can be achieved, but computational complexity and solution time increase significantly
Solution Approach 1:
The patent transforms the original nonlinear programming problem parameters into a different mathematical representation using semi-definite programming relaxation. By changing the problem formulation from nonlinear constraints to convex optimization with linear matrix inequalities, the computational complexity is reduced while maintaining the essential attitude determination functionality and precision.
2Measurement precision
If traditional optimization algorithms are used, then accurate attitude angles can be computed, but real-time processing capability is compromised
Solution Approach 1:
The patent replaces traditional iterative nonlinear optimization algorithms with a convex optimization approach based on semi-definite programming. This substitution transforms the computational mechanism from one requiring multiple iterative steps to a more efficient convex problem that can be solved faster, enabling real-time attitude determination while preserving accuracy.
3Measurement precision
If quadratic constrained quadratic minimization is solved directly, then optimal attitude solution is obtained, but numerical tractability decreases
Solution Approach 1:
The patent introduces an intermediary mathematical transformation using semi-definite programming relaxation as a bridge between the original quadratic constrained quadratic minimization problem and a numerically tractable convex optimization problem. This intermediary formulation maintains the optimality of the attitude solution while improving numerical solvability through linear matrix inequality constraints.
Data Source
AI summary
A method for determining attitude of an object having multiple GNSS antennas, the method including receiving GNSS signals from at least five satellites, wherein at least 2 of the five belong to a different satellite constellation than the other satellites; processing each of the GNSS signals to generate pseudorange code and carrier phase measurements; resolving carrier phase ambiguities for all the received GNSS signals; generating unbiased carrier phase measurements based on the resolving; determining the attitude, including heading, pitch, and roll angles ψ,θ,ϕ, respectively, by solving a quadratically constrained quadratic minimization problem through finding a minimum of a linear function subject to a linear matrix inequality constraint; and outputting the attitude.