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Rigid Body Localization via Gaussian Belief Propagation with Quadratic Angle Approximation

Niclas Führling, Hyeon Seok Rou, Giuseppe Abreu, David González G., Osvaldo Gonsa

TL;DR

The paper addresses range-based rigid body localization (RBL) in 3D by removing the need for an accurate orientation prior. It introduces a quadratic angle approximation for the rotation, yielding a linearized model suitable for Gaussian belief propagation (GaBP) and a bilinear GaBP estimator. A double-GaBP procedure with interference cancellation significantly improves rotation estimates, especially for large angle deviations, while translation performance remains on par with state-of-the-art methods. The approach achieves superior rotation RMSE without increasing asymptotic complexity, making it attractive for real-time 3D sensing in autonomous and ISAC contexts.

Abstract

Gaussian belief propagation (GaBP) is a technique that relies on linearized error and input-output models to yield low-complexity solutions to complex estimation problems, which has been recently shown to be effective in the design of range-based GaBP schemes for stationary and moving rigid body localization (RBL) in three-dimensional (3D) space, as long as an accurate prior on the orientation of the target rigid body is available. In this article we present a novel range-based RBL scheme via GaBP that removes the latter limitation. To this end, the proposed method incorporates a quadratic angle approximation to linearize the relative orientation between the prior and the target rigid body, enabling high precision estimates of corresponding rotation angles even for large deviations. Leveraging the resulting linearized model, we derive the corresponding message-passing (MP) rules to obtain estimates of the translation vector and rotation matrix of the target rigid body, relative to a prior reference frame. Numerical results corroborate the good performance of the proposed angle approximation itself, as well as the consequent RBL performance in terms of root mean square errors (RMSEs) in comparison to the state-of-the-art (SotA), while maintaining a low computational complexity

Rigid Body Localization via Gaussian Belief Propagation with Quadratic Angle Approximation

TL;DR

The paper addresses range-based rigid body localization (RBL) in 3D by removing the need for an accurate orientation prior. It introduces a quadratic angle approximation for the rotation, yielding a linearized model suitable for Gaussian belief propagation (GaBP) and a bilinear GaBP estimator. A double-GaBP procedure with interference cancellation significantly improves rotation estimates, especially for large angle deviations, while translation performance remains on par with state-of-the-art methods. The approach achieves superior rotation RMSE without increasing asymptotic complexity, making it attractive for real-time 3D sensing in autonomous and ISAC contexts.

Abstract

Gaussian belief propagation (GaBP) is a technique that relies on linearized error and input-output models to yield low-complexity solutions to complex estimation problems, which has been recently shown to be effective in the design of range-based GaBP schemes for stationary and moving rigid body localization (RBL) in three-dimensional (3D) space, as long as an accurate prior on the orientation of the target rigid body is available. In this article we present a novel range-based RBL scheme via GaBP that removes the latter limitation. To this end, the proposed method incorporates a quadratic angle approximation to linearize the relative orientation between the prior and the target rigid body, enabling high precision estimates of corresponding rotation angles even for large deviations. Leveraging the resulting linearized model, we derive the corresponding message-passing (MP) rules to obtain estimates of the translation vector and rotation matrix of the target rigid body, relative to a prior reference frame. Numerical results corroborate the good performance of the proposed angle approximation itself, as well as the consequent RBL performance in terms of root mean square errors (RMSEs) in comparison to the state-of-the-art (SotA), while maintaining a low computational complexity
Paper Structure (14 sections, 44 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 14 sections, 44 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Rigid Body Localization System Model
  3. Rigid Body System Model
  4. Measurement Model
  5. SotA Parameter-based System Model via Small Angle Approximation
  6. Proposed method
  7. System Model via Quadratic Angle Approximation
  8. Proposed Linear Model
  9. Bivariate GaBP for Rigid Body Parameter Estimation
  10. Performance Evaluation
  11. Numerical Results
  12. Complexity Analysis
  13. Conclusion
  14. Linear Model Derivation

Figures (5)

  • Figure 1: An illustration of the transformation of a rigid body, whose location has a 3D rotation $\boldsymbol{Q}$ and a translation $\boldsymbol{t}$ applied to the reference frame, as determined by equation \ref{['eq:basic_model_RB']}.
  • Figure 2: Illustration of the transformed rigid body surrounded by $M$ anchors, whose locations are known.
  • Figure 3: Various Sine and Cosine approximations
  • Figure 5: RMSE of the translation estimate of the proposed method and the SotA, over the range error $\sigma$.
  • Figure 7: RMSE of the rotation estimate of the proposed method and the SotA, over the range error $\sigma$.