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Optimal Trajectory Generation for Improved Magnetic Navigation

Justin Kang, Teddy Herrera, Liraz Mudrik, Sean Kragelund, Alfonso Sciacchitano, Isaac Kaminer

arXiv:2609.27553v1eess.SYcs.SYmath.OC

TL;DR

Magnetic-navigation accuracy depends on the vehicle’s path through the geomagnetic field, making trajectory-dependent observability a practical concern. This paper optimizes trajectories using the PCRLB and a path-length penalty; across ten cases, optimized paths improved navigation outcomes over straight-line paths with a modest path-length increase.

Abstract

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Magnetic navigation has emerged as a promising alternative for navigation in Global Positioning System (GPS)-denied environments, leveraging geomagnetic field maps in conjunction with onboard magnetometer measurements. However, its performance is highly sensitive to trajectory-dependent observability, which limits its practical effectiveness under conventional flight paths. This paper proposes an optimal trajectory design framework for magnetic navigation that maximizes information content along the flight path. The trajectory generation problem is formulated as an optimal control problem that minimizes the posterior Cram\'{e}r--Rao lower bound on the position estimation error, subject to a penalty on path length. The resulting trajectories are non-intuitive and significantly enhance the observability of the navigation system. Simulation results demonstrate that the proposed optimal trajectories yield substantial reductions in estimation error compared to conventional straight-line trajectories, highlighting the critical role of trajectory design in enabling high-accuracy magnetic navigation. These findings suggest that trajectory optimization can substantially improve the viability of magnetic navigation as a robust alternative for aerospace applications in GPS-denied environments.

II. BACKGROUND

The background describes the geomagnetic field model and explains how field gradients shape the information available for position estimation. It uses the PCRLB trace to quantify trajectory-dependent uncertainty, with gradient direction as well as magnitude affecting observability.

II. BACKGROUND

  • The local North–East–Down frame is treated as inertial, and SI units are used unless otherwise stated.Source: §II · p. 2

A. Geomagnetic Field Model

B. Posterior Cram´er–Rao Lower Bound

III. PROBLEM FORMULATION

This section sets out a trajectory-optimization problem that balances estimation performance against path length. It then introduces the models, solution method, and filter used to evaluate navigation performance.

III. PROBLEM FORMULATION

  • The formulation seeks a path between specified endpoints while balancing path length against position-estimation uncertainty.Source: §III · pp. 3–4
  • The cost combines path length with estimation performance quantified by the PCRLB.Source: §III · p. 4
  • The section introduces a numerical solution method and a navigation filter for evaluating performance.Source: §III · p. 4

A. System Model

The system model uses planar vehicle kinematics and scalar magnetometer measurements of the geomagnetic field. Its trajectory-design assumptions simplify the motion model, while performance evaluation retains additional vehicle states.

A. System Model

  • For trajectory design, the vehicle is a point mass at constant altitude and airspeed, with heading as its sole degree of freedom.Source: A. System Model · p. 4
  • The simplified planar kinematics assume altitude-hold and airspeed-hold autopilot loops; performance is later evaluated with a six-state model retaining altitude, vertical rate, and speed.Source: A. System Model · p. 4
  • A scalar magnetometer measures total field intensity modeled as the WMM core field plus the EMAG2V3 anomaly field, with zero-mean Gaussian measurement noise.Source: A. System Model · p. 4
  • The model pairs continuous-time vehicle dynamics with discrete-time magnetic measurements.Source: A. System Model · p. 4

B. Trajectory Optimization Problem

The trajectory problem trades path length against estimation uncertainty, represented by the PCRLB. Its FIM propagation uses a noise-free design trajectory, and magnetic measurement information depends on the field gradient along that path.

B. Trajectory Optimization Problem

  • The path from a specified start to end balances shorter travel against the possibility that longer paths reach higher-gradient regions and reduce estimation error.Source: B. Trajectory Optimization Problem · p. 4

1. PCRLB for the Magnetic Navigation Problem

  • With process noise set to zero for trajectory design, the FIM recursion propagates prior information through the state transition and adds new measurement information.Source: §1 · p. 4
  • The FIM remains positive definite along candidate trajectories under the specified positive-definite initial information and the no-process-noise assumption.Source: §1 · p. 4
  • The measurement Jacobian is obtained from the field gradient at the vehicle’s position, and its heading component is zero because the scalar field measurement does not depend on heading.Source: §1 · p. 4

2. Optimal Control Problem

The optimal control formulation trades path length against terminal position-estimation uncertainty, subject to the vehicle dynamics and trajectory constraints. Its normalized form makes the estimation-accuracy versus path-length trade-off selectable through a single weight.

2. Optimal Control Problem

  • The objective balances path length against the terminal PCRLB trace using positive weights for the two terms.At constant speed, minimizing path length is equivalent to minimizing flight time.Source: §2 · pp. 4–5
  • The terminal PCRLB trace remains finite and differentiable on weak-gradient paths, where the bound stays at the prior-information level.Source: §2 · p. 5
  • A normalized weight wP in [0, 1] sweeps attainable accuracy–length trade-offs and supports selecting weights from a Pareto front.The normalized terms are dimensionless and of comparable magnitude.Source: §2 · p. 5
  • The problem is constrained by the vehicle dynamics, initial condition, terminal position, and control bounds.Source: §2 · p. 5
  • The terminal heading is free, as is final time; the FIM is computed recursively at discrete measurement times within the flight interval.Source: §2 · p. 5

C. Numerical Solution Method

The non-convex trajectory problem can stall when initialized in a flat magnetic-field region, where the terminal cost provides little descent direction. The method uses a gradient-aware A* seed before optimal-control refinement.

C. Numerical Solution Method

  • Gradient-weighted A* places the initial path where the cost is sensitive, after which DIDO performs optimal-control refinement.The method also resamples trajectories to obtain dynamically consistent paths.Source: C. Numerical Solution Method · p. 5

1. A* Gradient Search for Initialization

  • A* balances discrete path length against magnetic-field gradient magnitude, favoring paths through regions with greater field variation.Source: §1 · p. 5
  • Because A* ignores vehicle dynamics and optimizes local gradient rather than propagated PCRLB, a PID tracker makes its path flyable for use as DIDO’s initial guess.The following pseudospectral solution addresses the proxy and uncertainty-propagation limitations.Source: §1 · pp. 5–6

2. Pseudospectral Optimal Control Solution

DIDO solves the trajectory problem by pseudospectral collocation, with trajectories resampled for uniform-time navigation evaluation. The study compares optimized and straight-line paths across two coastal regions using EKF simulations.

2. Pseudospectral Optimal Control Solution

  • DIDO transcribes the continuous-time problem into nonlinear programming using Legendre–Gauss–Lobatto collocation, enforcing costs and constraints at collocation points.The PCRLB recursion is evaluated along the discretized trajectory; histories are interpolated and tracked to produce dynamically consistent paths on the magnetometer’s uniform sampling grid.Source: §2 · p. 6

D. Navigation Performance Evaluation

  • A six-state EKF fuses inertial, magnetometer, and altimeter measurements to evaluate navigation performance, while trajectory optimization uses a reduced three-state model.The six-state model captures INS drift effects from accelerometer and gyroscope biases.Source: D. Navigation Performance Evaluation · p. 6

A. Simulation Setup

  • Ten case studies across California and New York compare straight paths through low-gradient regions with DIDO-FIM paths through higher-gradient regions, using 1,000 Monte Carlo EKF runs per trajectory.The anomaly map is treated as exact, and the heading prior is uninformative.Source: A. Simulation Setup · p. 6

B. Trajectory Characteristics

DIDO-FIM trajectories follow higher-variability magnetic regions with modest added distance. Across the cases, they improve navigation performance, while filter divergence and the PCRLB’s limits qualify how results should be compared.

B. Trajectory Characteristics

  • Across all ten cases, optimized paths were smooth and dynamically feasible, following higher-gradient structure for 1–9% more distance, averaging 5%.The jagged A* Gradient paths served as initialization for the DIDO solver.Source: B. Trajectory Characteristics · pp. 6–7

C. Navigation Performance

  • Eight of ten straight-line cases diverged, while all ten DIDO-FIM cases remained stable with DRMS errors of 88–338 m.The same filter tuning was selected for information-rich optimized paths and applied unchanged to straight-line paths, so divergence is not a like-for-like accuracy comparison.Source: C. Navigation Performance · p. 7
  • In the two stable straight-line comparisons, DRMS fell from 251 m to 151 m in N2 and from 452 m to 88 m in N4.PCRLB trace also decreased from 3.76 m2 to 0.40 m2 in N2 and from 1.01 m2 to 0.50 m2 in N4.Source: C. Navigation Performance · pp. 7–8
  • DIDO-FIM paths had lower PCRLB traces than corresponding straight-line paths in all ten cases, with larger trace reductions tending to accompany larger DRMS improvements.For N1, PCRLB trace decreased from 3.15 to 0.29 m2.Source: C. Navigation Performance · p. 8
  • PCRLB is a lower bound, not an absolute EKF error prediction; its ranking applies to paths between the same endpoints, while terminal trace alone misses information timing.Long low-gradient stretches may allow filter drift before later informative measurements arrive.Source: C. Navigation Performance · p. 8
  • DRMS improvement ranges from 40% (N2: 251 m to 151 m) to over 99% (N1: 39,423 m to 147 m), for 1–9% longer paths.Among the two cases where both filters converged, DRMS fell 40–81% with 9–13 km of additional path length.Source: C. Navigation Performance · p. 8

V. CONCLUSION

The paper presents a PCRLB-based trajectory optimization framework for magnetic navigation and reports improved accuracy over straight-line paths across ten cases. Optimized paths increased distance modestly while reducing navigation error.

V. CONCLUSION

  • The framework minimizes the posterior Cramér–Rao lower bound on position error to maximize magnetic information along the flight path.The table reports navigation performance for all ten cases, with 1,000 Monte Carlo runs per case.Source: §V · p. 8, Table I · p. 9
  • For Case N5, the magnetic field gradient magnitude remains consistently higher along the DIDO-FIM path.Source: Figure 8 · p. 9
  • Across ten cases, DIDO-FIM reduced DRMS to 88–338 m versus an approximately 1 km INS-only baseline, with about 5% average path-length increase.In eight cases straight-line navigation performed worse than unaided dead reckoning; all ten optimized trajectories improved on it.Source: §V · p. 9, Table I · p. 9

A. EKF Implementation

The navigation filter is a six-state extended Kalman filter that combines inertial, altitude, and magnetic measurements. Its EKF implementation linearizes and discretizes the dynamics, then uses magnetic-anomaly innovations for measurement updates.

A. EKF Implementation

  • The navigation filter uses a six-state extended Kalman filter with position, velocity, and heading in its state vector.Its measurements comprise altitude and total magnetic field; the state also includes altitude.Source: A. EKF Implementation · p. 9
  • The state dynamics use heading rate and longitudinal and vertical specific forces obtained from the IMU.Source: A. EKF Implementation · p. 9
  • For discrete-time EKF implementation, the dynamics are linearized using Jacobians and discretized with a state transition matrix, discrete input matrix, and process-noise covariance.The process-noise covariance is obtained via the Van Loan method or Taylor series expansion; the discrete measurement model retains H and R.Source: A. EKF Implementation · p. 9
  • The scalar magnetometer measurement is converted to an anomaly measurement by subtracting the interpolated core field, and the anomaly innovation drives the EKF update.The core field is based on the estimated position and WMM; the anomaly is interpolated from the EMAG2V3 map.Source: A. EKF Implementation · pp. 9–10

B. Discrete-Time Information Recursion and Its Continuous-Time Limit

The appendix connects the discrete-time information recursion to the continuous-time bound, explaining its sampling assumptions and its behavior as the sampling interval shrinks.

B. Discrete-Time Information Recursion and Its Continuous-Time Limit

C. Pareto-Based Weight Selection for Case N5

For Case N5, a fixed weight sweep is used to estimate a path-length–PCRLB trade-off and select a solution by its Monte Carlo DRMS error.

C. Pareto-Based Weight Selection for Case N5

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