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Observability Analysis and Online Calibration of Visual-Inertial-Wheel Odometry for 4WIS4WID Mobile Robots

Branimir Ćaran, Vladimir Milić, Bojan Šekoranja, Bojan Jerbić

arXiv:2609.38462v1cs.RO

TL;DR

4WIS4WID robots offer flexible planar motion but bring many wheel parameters whose online calibration and observability have been little explored. The paper develops a tightly coupled VIWO system with a redundant wheel model and online calibration. Its analysis shows that lateral constraints restore steering-offset observability while identifying additional odometry-frame gauge freedoms and motion-dependent degeneracies.

  • Problem

    4WIS4WID robots have sixteen intrinsic parameters, while their online calibration and observability remain largely unexplored and prior work held calibrated kinematics fixed.

    Source: §I · pp. 1–2

  • Method

    A tightly coupled VIWO framework uses preintegration and a redundant 2D wheel-odometry model with longitudinal rolling and lateral no-slip constraints for online calibration.

    Source: §I · p. 2, §VI · p. 8

  • Results

    Lateral no-slip constraints restore steering-offset observability lost in a drive-only model, while the analysis identifies three additional odometry-frame gauge directions beyond standard VINS freedoms.

    Source: §I · p. 2, §VI · p. 8

  • Takeaways & Limitations

    The augmented map remains well-conditioned for any steering configuration, including lateral translation where the reduced map can lose rank.

    Source: A. 4WIS4WID Kinematic Model · p. 3, B. Necessity of the Lateral Constraints · p. 6

    Calibration is motion-dependent: without rotation, wheel positions produce no signal, and a restrictive degeneracy excludes straight, lateral, and constant-radius circular trajectories.

    Source: C. Degenerate Motions · p. 6

Abstract

from arXiv · show

In this paper, we present a visual-inertial-wheel odometry (VIWO) framework with online calibration for four-wheel independently steered and driven (4WIS4WID) mobile robots. We derive a 2D odometry model directly from the four driving velocities and steering angles, using both the longitudinal rolling constraints and the lateral no-slip constraints of all wheels. A preintegration model and analytical Jacobians are developed for efficient filtering and calibration. An observability analysis of the linearized VIWO system shows that a drive-only model makes all steering offsets unobservable, whereas the proposed redundant model restores their observability. The analysis also identifies four standard VINS unobservable directions and three additional directions associated with the arbitrary placement of the odometry reference frame. Furthermore, we characterize several degenerate motions, including zero yaw rate, constant steering, and a non-rolling wheel, and derive the corresponding excitation conditions for the thirteen wheel intrinsics that remain after fixing the odometry frame reference. The performance of the proposed system has been demonstrated in both simulation and real-world experiments on a 4WIS4WID mobile robot.

I. INTRODUCTION

The paper addresses the largely unexplored online calibration and observability of 4WIS4WID robots, whose freely moving instantaneous center of rotation comes with sixteen intrinsic parameters. It develops a tightly coupled VIWO framework and analyzes how its redundant wheel model affects calibration observability.

I. INTRODUCTION

  • 4WIS4WID online calibration and observability remain largely unexplored despite the platform’s sixteen intrinsic parameters and prior work that held calibrated kinematics fixed.Independent steering allows arbitrary planar ICR placement, increasing calibration and observability challenges when wheel measurements are coupled with visual and inertial sensing.Source: §I · pp. 1–2
  • The proposed tightly coupled VIWO framework performs online calibration for 4WIS4WID robots, focusing analysis on their kinematic observability and coupling with VIO.The paper uses a preintegration-based framework for the robot’s kinematic parameters.Source: §I · p. 2
  • Lateral no-slip constraints restore observability of steering offsets that remain unobservable in a drive-only model.The analysis also identifies three motion-independent SE(2) gauge freedoms for odometry-frame placement and motion-dependent degeneracies affecting the remaining wheel intrinsics.Source: §I · p. 2

II. 4WIS4WID VISUAL-INERTIAL-WHEEL ODOMETRY

  • The framework is validated in simulation and real-world experiments, with improved pose estimation accuracy reported against fixed kinematic parameters.The filter augments the MSCKF state with odometry-related calibration parameters while retaining the standard inertial and visual estimation structure.Source: §I · p. 2, §II · p. 2

A. 4WIS4WID Kinematic Model

The kinematic model uses each wheel’s drive and steering measurements to constrain the planar body twist. It combines longitudinal rolling and lateral no-slip constraints in a redundant least-squares model.

A. 4WIS4WID Kinematic Model

  • Each wheel’s rolling speed and steering direction provide two scalar constraints, yielding eight constraints on the planar body twist across four wheels.The constraints separate into a longitudinal component matching measured wheel speed and a lateral component requiring no sideways slip.Source: A. 4WIS4WID Kinematic Model · pp. 2–3
  • The stacked wheel constraints estimate the body twist in a least-squares sense, with the residual measuring encoder readings unexplained by any rigid-body twist.The residual spans a five-dimensional subspace and is zero for a kinematically exact, slip-free configuration.Source: A. 4WIS4WID Kinematic Model · p. 3
  • The augmented map requires full column rank, and its pseudo-inverse is evaluated numerically because the wheel layout does not permit the symmetric-layout closed form.The model uses wheel-frame constraints to separate the distinct roles of longitudinal and lateral measurements.Source: A. 4WIS4WID Kinematic Model · p. 3
  • Removing lateral rows gives a rolling-only reduced map that makes steering offsets unobservable under every motion and loses rank during lateral motion.This reduced model is exact when the no-slip assumption holds.Source: A. 4WIS4WID Kinematic Model · p. 3

B. Odometry Preintegration

Wheel encoder measurements are preintegrated between camera frames to provide relative odometry measurements for the filter. The method linearizes these measurements with respect to wheel intrinsics and accumulates their sensitivities efficiently.

B. Odometry Preintegration

  • High-rate wheel measurements are preintegrated between consecutive camera frames into a relative 2D pose, with yaw evolving as body velocity is integrated into position.The increment is expressed in the starting integration frame and reset at each new camera frame.Source: B. Odometry Preintegration · p. 3
  • Online calibration estimates wheel intrinsics and odometry-IMU spatial extrinsics, while retaining the temporal offset in the filter state; camera-IMU calibration is assumed known.The odometry-IMU temporal-offset model is adopted from.Source: B. Odometry Preintegration · p. 3

C. Odometry Measurement with Intrinsics

  • The preintegrated measurement is linearized about current intrinsic estimates, and sensitivities are accumulated alongside each sub-interval increment.The integration Jacobian is factored from the kinematics-specific twist Jacobians to keep the sixteen-parameter case tractable.Source: C. Odometry Measurement with Intrinsics · pp. 3–4
  • Wheel-position parameters affect the twist through its yaw-rate column, and the redundant map lets both position components influence the twist independently.Under the reduced rolling-only map, wheel position coordinates enter only through one scalar combination; three unobservable intrinsics are fixed by convention.Source: C. Odometry Measurement with Intrinsics · pp. 3–4
  • A singularity-free position increment uses mid-interval yaw and an arc factor, so it remains valid for zero-yaw-rate motion.The increment is obtained from the exact interval integral without the 1/ω singularity of direct evaluation.Source: C. Odometry Measurement with Intrinsics · p. 4

D. Odometry Measurement with Extrinsics

The preintegrated wheel measurement is a 2D relative pose, whereas the filter estimates 3D IMU poses. Their relation uses odometry-IMU spatial extrinsics and temporal alignment, with three gauge freedoms arising from the arbitrary odometry-frame placement.

D. Odometry Measurement with Extrinsics

  • Relating the 2D preintegrated odometry pose to the filter’s 3D IMU poses requires odometry-IMU spatial extrinsics and a temporal offset.The measurement function and Jacobians are adopted unchanged from.Source: D. Odometry Measurement with Extrinsics · p. 4
  • Because odometry-frame placement is arbitrary, a planar rigid transformation of the wheel array can be absorbed by a compensating change in the extrinsics and IMU pose.This introduces three motion-independent unobservable directions corresponding to SE(2) freedom.Source: D. Odometry Measurement with Extrinsics · pp. 4–5

E. Odometry Measurement Update

The update equates the preintegrated odometry measurement with the state-based prediction, then linearizes both to form a residual. It propagates measurement noise and screens candidate updates for inconsistency.

E. Odometry Measurement Update

  • The update equates the preintegrated measurement with the state-based prediction and linearizes both sides to form the residual measurement model.The preintegrated measurement carries the intrinsic Jacobian and accumulated covariance.Source: E. Odometry Measurement Update · p. 5
  • The intrinsic Jacobian block is 3 × 16 for this platform, compared with 3 × 3 for differential drive.The remaining Jacobian blocks are structurally identical to the differential-drive case.Source: E. Odometry Measurement Update · p. 5
  • The accumulated covariance propagates driving, steering, and lateral-slip noise, while consistency gating rejects intervals affected by unmodelled effects such as wheel slip.The estimator screens candidate measurements with a three-degree-of-freedom Mahalanobis distance test and can gate preintegration on the residual e⊥.Source: E. Odometry Measurement Update · p. 5

III. OBSERVABILITY ANALYSIS

The analysis builds an observability matrix from visual and wheel measurements and treats a state perturbation as unobservable when it leaves all measurements unchanged. It reuses an inertial state-transition construction that does not depend on the wheel kinematic model.

III. OBSERVABILITY ANALYSIS

  • The observability matrix stacks visual and wheel measurement Jacobians, and a direction n is unobservable when Mn = 0, leaving the measurements unchanged under that perturbation.Such a direction cannot be resolved by an estimator and has unbounded corresponding covariance.Source: §III · p. 5
  • The analysis reuses a transition map that unifies inertial propagation with clone creation and marginalization; its calibration-parameter block is the identity because calibration parameters are constant.The transition is independent of the kinematic model because wheel measurements do not enter inertial propagation or sliding-window bookkeeping; its calibration dimension changes from I13 to I26.Source: §III · p. 5

A. Structure of the Wheel Measurement Jacobian

At the true state, the analysis assumes exact calibration and no slippage, then examines how each wheel’s intrinsics affect the recovered twist. It derives the excitation needed to identify those parameters.

A. Structure of the Wheel Measurement Jacobian

  • At the true state, exact calibration and no slippage make the second term of the sensitivity expression vanish.The reduction is used in the ensuing observability analysis.Source: A. Structure of the Wheel Measurement… · p. 5
  • Each wheel’s four intrinsics affect the recovered twist only through two columns of the per-wheel sensitivity block.The block’s rank is at most two, so identifying all four intrinsics requires at least two distinct steering configurations.Source: A. Structure of the Wheel Measurement… · p. 5
  • The sensitivity row space must span R4 over time; radius, steering-offset, and wheel-position sensitivities require wheel rate, wheel speed, and yaw rate, respectively.The wheel-position sensitivities scale with yaw rate.Source: A. Structure of the Wheel Measurement… · p. 5

B. Necessity of the Lateral Constraints

The drive-only measurement map makes steering offsets unobservable, whereas retaining lateral constraints restores their information. The analysis also identifies unobservable directions from global VINS freedoms and the arbitrary placement of the odometry frame.

B. Necessity of the Lateral Constraints

  • The drive-only map is exact under no slip but makes all steering offsets unobservable, so it is inadmissible for calibration.At the true state, the lateral-slip term vanishes and the steering-offset columns are zero.Source: B. Necessity of the Lateral Constraints · p. 6
  • Retaining lateral rows restores steering-offset information through a coefficient −vj, which is non-zero whenever that wheel is rolling.The rolling measurements alone do not test whether the steering angles are mutually consistent with a single ICR.Source: B. Necessity of the Lateral Constraints · p. 6
  • The augmented map remains well-conditioned when all steering angles are equal and its pseudo-inverse stays defined in lateral translation.The reduced map’s first two columns become parallel in that steering configuration.Source: B. Necessity of the Lateral Constraints · p. 6
  • Seven motion-independent directions are unobservable: four standard VINS directions and three associated with arbitrary placement of the odometry frame on the robot body.The three frame-placement directions hold under general excitation.Source: B. Necessity of the Lateral Constraints · p. 6
  • Fixing xw1, yw1, and δo1 sets the odometry-frame gauge, leaving thirteen wheel intrinsics to estimate online; residual error in the fixed values is absorbed by xW E without bias.The wheel geometry is observable only up to the choice of odometry frame.Source: B. Necessity of the Lateral Constraints · p. 6

C. Degenerate Motions

Zero yaw rate makes all wheel positions unobservable, while constant steering leaves two intrinsics per wheel unobservable regardless of trajectory duration. These degeneracies constrain which motions can calibrate the wheel parameters.

C. Degenerate Motions

  • With ω ≡ 0, all eight wheel positions are unobservable because they enter only through yaw-rate moment arms.This includes straight-line and pure lateral motion.Source: C. Degenerate Motions · p. 6
  • With constant steering and nonzero yaw rate, two of each wheel’s four intrinsics remain unobservable, regardless of trajectory duration.The fixed steering configuration fixes the ICR in the body frame, leaving a fixed two-dimensional row space.Source: C. Degenerate Motions · p. 6
  • Straight-line, lateral, and constant-radius circular motions are excluded as calibration trajectories, and increasing speed or duration does not resolve the constant-steering degeneracy.Scaling yaw rate does not change the observability row space.Source: C. Degenerate Motions · p. 6

IV. SIMULATION VALIDATION

The simulation evaluates the predicted calibration convergence and degeneracies across six motion profiles, using perturbed initialization and specified sensor and encoder settings.

IV. SIMULATION VALIDATION

  • Six simulated trajectories test calibration and degeneracy behavior after fixing xw1, yw1, and δo1 to remove the odometry-frame gauge freedom.The profiles are ICR-sweep, straight-line, lateral, circular, stationary, and figure-eight motion; camera, IMU, and wheel-encoder rates are 10, 200, and 100 Hz.Source: §IV · p. 7

A. Calibration and Degeneracy Validation

In the real-robot experiment, calibrated VIWO has lower reported trajectory errors than the uncalibrated VIWO, VIO, and wheel odometry. Simulation and Monte-Carlo results also validate calibration behavior and characterize consistency, while the authors identify no-slip assumptions and limited validation scope as future-work boundaries.

A. Calibration and Degeneracy Validation

  • In simulation, estimated calibration parameters converge within reported bounds, and the predicted straight-line, lateral, and constant-steering degeneracies are reproduced.Wheel radii converge faster initially than wheel positions; the six-run error plot is illustrative, while statistical accuracy and consistency are reported over 20 runs.Source: A. Calibration and Degeneracy Validation · p. 7, Figure 2 · p. 7

B. Estimation Accuracy and Consistency

  • With calibration enabled, pose NEES is 3.56 and 3.72 against a reference value of 6; disabling calibration raises it to 10.39, while 170.75 reflects divergence in 5 of 20 runs.The 170.75 value is a mean affected by those divergent runs; the median is 2.03.Source: B. Estimation Accuracy and Consistency · p. 7, Table II · p. 7

V. EXPERIMENTAL VALIDATION

  • The real experiment used a 4WIS4WID robot with eight Dynamixel motors, a RealSense T265 camera-IMU, and OptiTrack position ground truth.The T265 intrinsic and extrinsic parameters were calibrated using Kalibr.Source: §V · p. 7
  • 0.212/1.64 meters/degrees is the reported RMSE for calibrated VIWO, compared with 0.343/3.04 for uncalibrated VIWO, 0.566/2.06 for VIO, and 0.774/17.46 for wheel odometry.The experiment used an OptiTrack system for ground-truth robot position.Source: §V · p. 7, Figure 3 · p. 8

VI. CONCLUSION AND FUTURE WORK

  • The paper concludes that the system was validated in simulation and on a real 4WIS4WID robot, while vertical-surface and longer-distance validation remain future work.The authors expect calibration errors to have a stronger effect over longer travelled distances.Source: §VI · p. 8
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