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Observability Analysis of Joint Steering and Extrinsic Calibration

Subodh Mishra

arXiv:2609.05498v1cs.RO

TL;DR

Jointly estimating vehicle pose, LiDAR extrinsics, and steering bias raises the question of which motion patterns make those quantities locally distinguishable. The report analyzes the system using Lie-derivative-based observability and finds that combined straight and curved motion yields full local weak observability of all seven states.

  • Problem

    Joint estimation of motion state and calibration quantities from the same measurements requires determining which vehicle motions leave pose, sensor extrinsics, or steering bias indistinguishable.

    Source: §1 · p. 1

  • Method

    The report applies a Lie-derivative-based nonlinear observability analysis to a planar bicycle model with LiDAR extrinsics and steering-angle bias in the state.

    Source: §1 · p. 1, §3 · p. 3

  • Results

    Stationary, straight-line, and constant-curvature motion each retain unobservable directions, whereas combining straight and curved segments makes all seven states locally weakly observable.

    Source: 1) Straight-Line Motion: δm = b · p. 4, 3) Stationary Case: v = 0 · pp. 5–6, §4 · p. 7

  • Takeaways & Limitations

    Calibration trajectories should mix straight and curved segments rather than rely on a single repetitive motion pattern.

    Source: §4 · p. 7

    The analysis is structural and local; practical calibration accuracy and convergence also depend on numerical conditioning, measurement noise, model mismatch, and excitation.

    Source: §4 · p. 7

Abstract

from arXiv · show

This technical report studies the local weak observability of a planar bicycle-model vehicle when vehicle pose, planar LiDAR extrinsic calibration, and steering-angle bias are estimated jointly. A Lie-derivative-based nonlinear observability analysis is used to examine stationary, straight-line, constant-curvature, and combined straight-plus-arc motion. The resulting observability matrices and nullspaces describe how pose, LiDAR translation and yaw offsets, and steering bias become coupled under different motion primitives. Stationary motion and individual motion primitives retain unobservable directions, whereas the combination of straight and curved motion removes the identified degeneracies and yields full local weak observability of the seven-state system. The analysis provides a theoretical basis for selecting calibration trajectories that sufficiently excite both steering and sensor-extrinsic parameters.

1 Introduction

Joint calibration of vehicle motion state and slowly varying calibration quantities is useful when a mobile robot estimates them from the same measurements. This report examines how vehicle motion affects local observability of pose, LiDAR extrinsics, and steering bias.

1 Introduction

  • Observability depends on the vehicle motion as well as the sensing model.Motion that appears informative for navigation may still leave pose, sensor extrinsics, or steering bias indistinguishable.Source: §1 · p. 1
  • The report studies a planar bicycle-model vehicle with LiDAR translation and yaw extrinsics and steering-angle bias included in the state.It characterizes unobservable directions for stationary, straight-line, and constant-curvature motion, then examines combined primitives.Source: §1 · p. 1
  • The analysis uses a nonlinear observability matrix built from the measurement Jacobian and successive Lie derivatives, following Hermann and Krener.The report retains derivations to show dependence on state, steering input, and motion primitive.Source: §1 · pp. 1–2

2 Problem Setup and Full Observability Matrix

The full observability matrix is presented using abbreviations for relative angles and LiDAR translation offsets. Its displayed entries include trigonometric terms, vehicle displacement, speed, wheelbase, and steering-related quantities.

2 Problem Setup and Full Observability Matrix

  • The report introduces ∆ as θj − θi, ϕ as θoff − θi, and α as δm − b.These abbreviations define relative vehicle and extrinsic yaw angles and the steering input adjusted by bias.Source: §2 · p. 2
  • A and B are linear combinations of xoff and yoff weighted by sin ∆ and cos ∆.The trigonometric abbreviations are c∆, s∆, cϕ, and sϕ.Source: §2 · p. 2
  • The displayed full observability matrix contains rows with trigonometric terms, displacement differences, and speed- and steering-dependent terms.The supplied matrix entries include v, L, tan α, and sθ alongside pose and extrinsic terms.Source: §2 · p. 2

3 Nonlinear Observability Analysis

The report evaluates local observability for a seven-state vehicle, LiDAR-extrinsic, and steering-bias system under several motion conditions. Individual motion primitives retain distinct unobservable directions, while combined straight and curved segments yield full local weak observability.

Problem and Model

  • The measurement model uses LiDAR odometry pose relative to a reference pose, with state variables for vehicle pose, LiDAR extrinsics, and steering bias.The vehicle dynamics are represented by the continuous-time bicycle model.Source: §3 · p. 3

Nonlinear Observability Analysis

  • The analysis builds its observability matrix from each output gradient and the gradients of its first and second Lie derivatives.Evaluating it under different motion primitives tests which combinations of the seven states are locally distinguishable.Source: §3 · pp. 3–4

Straight-Line Motion

  • During straight-line motion, two unobservable directions couple vehicle position with LiDAR translation offsets, preventing their unique separation.The basis-vector components depend on trigonometric functions of θi, θj, and θoff.Source: 1) Straight-Line Motion: δm = b · p. 4

Constant-Curvature Motion

  • A constant-curvature arc retains one geometric nullspace direction coupling vehicle position, LiDAR translation, and extrinsic yaw.The steering bias does not participate in that direction and is observable under the stated constant-curvature motion.Source: 2) Constant-Curvature Motion (Arc): v… · p. 5

Stationary Case

  • With the vehicle stationary, motion-dependent information vanishes and the observability nullspace has four dimensions.Two additional unobservable modes are extrinsic yaw and steering bias; the latter remains unobservable because vehicle heading does not change when v = 0.Source: 3) Stationary Case: v = 0 · pp. 5–6

Combined Trajectory: Straight + Arc

  • Combining straight and curved segments makes all seven states locally weakly observable.The straight-motion and arc nullspaces intersect only at zero, and the combined observability matrix has full rank.Source: 4) Combined Trajectory: Straight + Arc · p. 7

4 Conclusion

The seven-state analysis finds characteristic unobservable directions under stationary, straight-line, and constant-curvature motion, while combining straight and curved segments yields full-rank observability. This is a structural, local result; practical estimation also depends on conditioning, noise, model mismatch, and excitation.

  • Stationary, straight-line, and constant-curvature motion each retain characteristic unobservable directions in the seven-state system.Limited excitation creates ambiguity between vehicle pose and sensor extrinsics, while steering bias becomes identifiable only with sufficient steering-dependent information.Source: §4 · p. 7
  • Combining straight and constant-curvature segments yields full-rank observability, motivating trajectories that mix both rather than repeat one motion pattern.The individual primitives leave different nullspace directions.Source: §4 · p. 7
  • The result is structural and local; practical calibration accuracy and convergence also depend on conditioning, noise, model mismatch, and excitation.Source: §4 · p. 7
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