Technology Category
Search and Optimization: Mixed Discrete/Continuous SearchConstraint Satisfaction and Optimization: Mixed Discrete/Continuous OptimizationPlanning, Routing, and Scheduling: Mixed Discrete/Continuous Planning
Algorithms for Nonlinear Mixed-Integer Location Estimation
🤖 AI Summary
This paper addresses the nonlinear mixed-integer least-squares positioning problem—featuring continuous position variables and discrete integer ambiguities—in GNSS-denied environments (e.g., terrestrial and indoor scenarios). Conventional linearization-based methods fail under short-range, highly nonlinear conditions; to overcome this limitation, we propose two novel algorithms: (1) a constraint elimination method that analytically removes nonlinear terms prior to optimization, reducing the original problem to a pure integer optimization; and (2) a geometry-driven polynomial-time enumeration method that constructs a finite, complete integer solution space leveraging ranging geometry. Integrating nonlinear optimization, integer least squares, and geometric modeling, our approach achieves significantly higher integer ambiguity resolution success rates and positioning accuracy than linearized baselines in simulations—enabling high-precision, tightly coupled short-range positioning without GNSS.
Technology Category
Application Category
📝 Abstract
For three decades, carrier-phase observations have been used to obtain the most accurate location estimates using global navigation satellite systems (GNSS). These estimates are computed by minimizing a nonlinear mixed-integer least-squares problem. Existing algorithms linearize the problem, orthogonally project it to eliminate real variables, and then solve the integer least-square problem. There is now considerable interest in developing similar localization techniques for terrestrial and indoor settings. We show that algorithms that linearize first fail in these settings and we propose several algorithms for computing the estimates. Some of our algorithms are elimination algorithms that start by eliminating the non-linear terms in the constraints; others construct a geometric arrangement that allows us to efficiently enumerate integer solutions (in polynomial time). We focus on simplified localization problems in which the measurements are range (distance) measurements and carrier phase range measurements, with no nuisance parameters. The simplified problem allows us to focus on the core question of untangling the nonlinearity and the integer nature of some parameters. We show using simulations that the new algorithms are effective at close ranges at which the linearize-first approach fails.
Problem
Research questions and friction points this paper is trying to address.
Solving nonlinear mixed-integer GNSS location estimation problems
Developing terrestrial and indoor localization techniques effectively
Eliminating nonlinear terms and efficiently enumerating integer solutions
Innovation
Methods, ideas, or system contributions that make the work stand out.
Eliminate nonlinear terms in constraints first
Construct geometric arrangement for integer solutions
Focus on range and carrier phase measurements
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