PaperlayerView exact revision on arXiv

Source-linked AI summary

AI-IMU Dead-Reckoning

Martin Brossard, Axel Barrau, Silvère Bonnabel

arXiv:1904.06064v1cs.ROstat.ML

TL;DR

The paper addresses accurate dead-reckoning for wheeled vehicles using only an IMU, motivated by the need for localization during exteroceptive-sensor failures. It combines an invariant extended Kalman filter with vehicle-motion pseudo-measurements and neural-network covariance adaptation. On KITTI, it reports accurate estimation of vehicle motion and IMU biases, including a 1.11% translational error in one ablation sequence, while competing with image-based and LiDAR methods.

  • Problem

    Accurate IMU-only dead-reckoning is difficult because vehicle-motion constraint confidence varies with maneuvers and low-cost IMUs drift rapidly.

  • Method

    The method uses an invariant extended Kalman filter with vehicle-motion pseudo-measurements and deep neural networks that dynamically adapt covariance noise parameters.

  • Results

    1.11% translational error was achieved by the full method on highway sequence 01, while the method competed with image-based methods on average.

  • Takeaways & Limitations

    IMU-only dead-reckoning can produce accurate position, velocity, orientation, and IMU-bias estimates without another sensor.

Abstract

from arXiv · show

In this paper we propose a novel accurate method for dead-reckoning of wheeled vehicles based only on an Inertial Measurement Unit (IMU). In the context of intelligent vehicles, robust and accurate dead-reckoning based on the IMU may prove useful to correlate feeds from imaging sensors, to safely navigate through obstructions, or for safe emergency stops in the extreme case of exteroceptive sensors failure. The key components of the method are the Kalman filter and the use of deep neural networks to dynamically adapt the noise parameters of the filter. The method is tested on the KITTI odometry dataset, and our dead-reckoning inertial method based only on the IMU accurately estimates 3D position, velocity, orientation of the vehicle and self-calibrates the IMU biases. We achieve on average a 1.10% translational error and the algorithm competes with top-ranked methods which, by contrast, use LiDAR or stereo vision. We make our implementation open-source at: https://github.com/mbrossar/ai-imu-dr

I. INTRODUCTION

The paper develops IMU-only dead-reckoning for wheeled vehicles, motivated by sensor failures and the drift of low- and medium-cost IMUs. Its approach combines vehicle-motion assumptions, Kalman filtering, and neural-network adaptation of filter noise parameters.

  • Accurate vehicle dynamics estimates can validate imaging sensors, correlate sensor feeds, and support emergency stops when imaging systems fail.
  • The method learns dynamic covariance-noise adaptation with deep neural networks because lateral slip varies between turns and straight-line motion.
  • A Kalman filter combines IMU outputs with wheeled-vehicle kinematic assumptions to estimate position, orientation, velocity, IMU biases, and uncertainty.
  • The approach uses only IMU signals, is evaluated on KITTI, and competes with top-ranked LiDAR and stereo-camera methods.
  • The framework is also presented as applicable to railway vehicles, GNSS coupling, IMU self-calibration, and speedometer-like uses in path reconstruction and map matching.
  • Low- and medium-cost IMUs suffer scale-factor, axis-misalignment, and random-walk errors that cause rapid localization drift.

B. Problem Modelling

The paper models dead-reckoning using three frames: the static world, IMU, and an ideal car frame estimated online. Given an initial known IMU configuration, the task is real-time estimation of IMU and car variables.

  • B. Problem Modelling: The model distinguishes the static world frame, IMU frame, and ideal car frame.The car frame is attached to the vehicle and estimated online.
  • B. Problem Modelling: The car frame’s orientation relative to the IMU frame is denoted Rc.
  • B. Problem Modelling: The IMU dead-reckoning problem starts from a known initial IMU configuration.
  • B. Problem Modelling: The objective is to estimate the IMU and car variables in real time.

III. KALMAN FILTERING WITH PSEUDO-MEASUREMENTS

The method uses an Extended Kalman Filter with loose equality constraints represented as pseudo-measurements. A neural network learns the filter’s process and pseudo-measurement noise parameters.

  • III. KALMAN FILTERING WITH PSEUDO-MEASUREMENTS: The EKF represents system evolution with a nonlinear discrete-time dynamical law and Gaussian process noise.The process noise has zero mean and covariance Qn.
  • III. KALMAN FILTERING WITH PSEUDO-MEASUREMENTS: Loose constraints h(xn) ≈ 0 are converted into fictitious observations with pseudo-measurement yn = 0.
  • III. KALMAN FILTERING WITH PSEUDO-MEASUREMENTS: The pseudo-measurement covariance Nn controls confidence in the constraint: larger Nn means lower confidence.
  • III. KALMAN FILTERING WITH PSEUDO-MEASUREMENTS: Starting from an initial Gaussian state belief, the EKF alternates propagation and update steps.Propagation advances the estimate and covariance; updating incorporates the pseudo-measurement.
  • III. KALMAN FILTERING WITH PSEUDO-MEASUREMENTS: A neural network learns both noise parameter matrices Qn and Nn rather than fixing them manually.

A. Defining the Dynamical Model f(·)

The dynamical model augments standard IMU states with an approximately constant car frame and uses soft vehicle-motion constraints. A neural network adapts their uncertainty because constraint validity changes with maneuvers.

  • A. Defining the Dynamical Model f(·): The car frame is rigidly attached to the vehicle and represents an unknown point where pseudo-measurements are most advantageously made.
  • A. Defining the Dynamical Model f(·): The car and IMU frames are rigidly attached to the same vehicle, so their misalignment is treated as approximately constant.
  • A. Defining the Dynamical Model f(·): Small Gaussian noises model temporal variations in the car-frame parameters caused by dampers and shock absorbers.
  • A. Defining the Dynamical Model f(·): The model generates scalar pseudo-observations by treating lateral and vertical car-frame velocities as roughly null.The level arm between the car and IMU frames is denoted n ∈ R3.
  • A. Defining the Dynamical Model f(·): These soft velocity constraints suit forward-moving cars and indoor wheeled robots, and outperform strictly null constraints in the cited setting.The uncertainty in Nn is deliberately nonzero.
  • A. Defining the Dynamical Model f(·): The constraints are expressed in the car frame, so they do not impose planar horizontal motion in the world frame.
  • A. Defining the Dynamical Model f(·): Constraint validity varies by maneuver, with lateral velocity much larger in turns than on straight lines.The AI-based noise adapter uses past and present IMU measurements to dynamically set Nn.

C. The Invariant Extended Kalman Filter (IEKF)

The IEKF fuses IMU measurements with wheeled-vehicle kinematic pseudo-measurements, while a deep-learning adapter dynamically selects their measurement covariance from raw IMU signals.

  • C. The Invariant Extended Kalman Filter (IEKF): The IEKF combines IMU measurements with pseudo-measurements derived from wheeled-vehicle dynamics to refine state estimates.The filter produces real-time state estimates and associated covariance.
  • C. The Invariant Extended Kalman Filter (IEKF): The noise parameter adapter converts raw IMU windows directly into covariance matrices for the filter update without using state estimates.A CNN computes the covariance Nn+1 at each instant.
  • C. The Invariant Extended Kalman Filter (IEKF): The adapter’s computed covariances optimize localization accuracy and may differ substantially from the actual statistical covariance of the pseudo-measurements.The process-noise covariance Q is fixed during operation and only its entries are learned during training.
  • C. The Invariant Extended Kalman Filter (IEKF): The adapter is designed to limit over-fitting, maintain independence from state estimates, support interpretability, and remain trainable.Its architecture uses a relatively small CNN-like network rather than a recurrent design.
  • C. The Invariant Extended Kalman Filter (IEKF): The adapter can inflate covariance by up to 10^β or reduce it by up to 10^-β relative to its initial values.When inactive, it recovers the initial covariance.

B. Implementation Details

The implementation specifies initial uncertainty and noise settings, then uses a compact two-layer temporal CNN to adapt measurement covariance.

  • B. Implementation Details: The implementation uses specified initial values for the error covariance P0, noise covariance Q, and measurement covariance parameters.The initial covariance assumes perfect prior knowledge of initial yaw, position, and zero vertical speed.
  • B. Implementation Details: The adapter is a two-layer 1D temporal CNN with kernel size 5 and dilation parameters 1 and 3, giving an input window N = 15.A fully connected layer outputs zlat and zup, with ReLU activations between layers.
  • B. Implementation Details: β = 3 allows each covariance element to be 10^3 higher or lower than its original value.The scaling range is applied relative to the initial covariance values.

C. Training

Training optimizes relative translation error using KITTI IMU data and learned adapter parameters, with regularization and sequence sampling designed for continual training.

  • C. Training: Training optimizes the relative translation error averaged over sub-sequences ranging from 100 m to 800 m.The loss is computed from filter estimates rather than from the adapter’s covariance values directly.
  • C. Training: The trainable parameters include 6210 adapter parameters plus 12 parameters from P0 and Q, optimized with Adam at learning rate 10^-4.Training repeatedly samples data, computes filter estimates and gradients, and updates the parameters.
  • C. Training: The training procedure uses continual training for 400 epochs, with dropout and data augmentation to reduce over-fitting.The authors motivate this setup for online training with accurate ground-truth poses.
  • C. Training: KITTI provides IMU, LiDAR, camera, and centimeter-accurate ground-truth pose data across environments including urban areas, highways, and streets.The benchmark includes 22 odometry sequences, with publicly available ground truth and raw synchronized IMU data for 11.
  • C. Training: The proposed method achieves performance comparable to image-based methods while using only IMU information.The table comparison includes LiDAR-based IMLS and stereo-based ORB-SLAM2 methods.

1) Relative Translation Error (trel):

The proposed IMU-only method is evaluated against LiDAR, stereo-vision, and direct IMU integration methods using relative trajectory errors. It competes with image-based methods while avoiding the rapid drift seen with direct inertial integration, including after vehicle stops.

  • The evaluation compares LiDAR, stereo-vision, direct IMU integration, and the proposed IMU-only approach.
  • The evaluation protocol trains the noise adapter without the sequence being evaluated, then runs IMU-based methods on the full raw sequence.
  • LiDAR and visual methods generally perform well, with LiDAR slightly better than its visual counterpart.
  • The proposed method competes on average with image-based methods, whereas direct IMU integration produces rapid drift, especially on longer sequences.
  • The proposed method appears unaffected by vehicle stops in sequence 07, while direct IMU integration broadly drifts after the car stops.
  • 1.11% translational error is achieved on highway sequence 01, compared with 1.94% without covariance adaptation and frame-alignment considerations.

C. Results on Sequences 00, 02 and 05

The method remains robust on sequences with logging gaps, while ablations show that dynamic covariance adaptation and frame alignment improve turning performance. The adapter increases measurement uncertainty during bends to accommodate motion-dependent violations of vehicle assumptions.

  • Discussion: The method’s performance reflects an IEKF, pseudo-measurements with neural covariance adaptation, and estimation of loose IMU-to-car-frame misalignment.The IEKF can be replaced by a conventional EKF if needed.
  • Ablation results: The full method achieves 1.11% translational error on highway sequence 01, versus 1.65% without alignment and 1.94% without covariance adaptation.All methods have the same rotational error, rrel = 0.12 deg /m.
  • Discussion: During the bend from t = 90 s to t = 110 s, the adapter increases covariance values, including a 10^2 inflation of the zero-velocity measurement covariance.The inflated covariance is used to enhance filter performance rather than represent actual statistical uncertainty.

APPENDIX A

The appendix represents the vehicle state and uncertainty using Lie-group structure, then propagates that state and covariance through the motion model and process-noise covariance.

  • State representation: The state embeds selected variables in SE2(3), while the bias vector is treated as an element of R6 with standard addition.The rotation belongs to SO(3), and the car-to-IMU level arm is represented as a vector in R3.
  • Linearized error: The linearized error concatenates the corresponding errors from the state’s component Lie groups.This construction combines group-specific error representations into the filter error state.
  • Linearized error: The Right IEKF maps each linearized error to the state with the corresponding Lie-group exponential map and right-multiplies the state elements.The state uncertainty is modeled as a zero-mean Gaussian with covariance Pn ∈R21×21.
  • Prediction step: The filter propagates the state and covariance using the vehicle dynamics, Jacobians Fn and Gn, and the Riccati equation.The process-noise covariance Qn is the classical covariance matrix used in the model.
  • Measurement model: The measurement vector stacks motion information and is assigned covariance Nn+1 before the IEKF update.The resulting measurement covariance is defined as cov(yn+1).

3) Update Step:

The update step forms a motion-information measurement with assessed uncertainty, then applies the IEKF correction using the measurement Jacobian and associated covariance updates.

  • Update step: The measurement vector yn+1 is assigned covariance Nn+1 = cov(yn+1) before updating the state.The assessed uncertainty is modeled as a zero-mean Gaussian variable.
  • Update step: The IEKF computes the updated state x̂+n+1 and covariance P+n+1 from the measurement information.The update follows the IEKF methodology.
  • Update step: The update consists of the Kalman gain, state innovation, state update, and covariance update.These operations are given by equations (30)–(36).
  • Measurement Jacobian: The measurement Jacobian Hn+1 is defined with respect to the linearized error.The matrix A selects the first two rows of the right part of the preceding expression, while B = RcT.
  • Lie-group formulation: SE2(3) provides the Lie-group representation underlying the state and its exponential-map-based uncertainty updates.The group is described as an extension of SE(3), with uncertainties mapped into its Lie algebra.
Loading 1904.06064v1…