Author: Wenda Zhao

This week we have a guest blog post from Wenda Zhao, Ph.D. candidate at the Dynamic System Lab (with Prof. Angela Schoellig), University of Toronto Institute for Aerospace Studies (UTIAS). Enjoy!

Accurate indoor localization is a crucial enabling capability for indoor robotics. Small and computationally-constrained indoor mobile robots have led researchers to pursue localization methods leveraging low-power and lightweight sensors. Ultra-wideband (UWB) technology, in particular, has been shown to provide sub-meter accurate, high-frequency, obstacle-penetrating ranging measurements that are robust to radio-frequency interference, using tiny integrated circuits. UWB chips have already been included in the latest generations of smartphones (iPhone 12, Samsung Galaxy S21, etc.) with the expectation that they will support faster data transfer and accurate indoor positioning, even in cluttered environments.

A Crazyflie with an IMU and UWB tag flies through a cardboard tunnel. A vision-based  motion capture system would not be able to achieve this due to the occlusion.

In our lab, we show that a Crazyflie nano-quadcopter can stably fly through a cardboard tunnel with only an IMU and UWB tag, from Bitcraze’s Loco Positioning System (LPS), for state estimation. However, it is challenging to achieve a reliable localization performance as we show above. Many factors can reduce the accuracy and reliability of UWB localization, for either two-way ranging (TWR) or time-difference-of-arrival (TDOA) measurements. Non-line-of-sight (NLOS) and multi-path radio propagation can lead to erroneous, spurious measurements (so-called outliers). Even line-of-sight (LOS) UWB measurements exhibit error patterns (i.e., bias), which are typically caused by the UWB antenna’s radiation characteristics. In our recent work, we present an M-estimation-based robust Kalman filter to reduce the influence of outliers and achieve robust UWB localization. We contributed an implementation of the robust Kalman filter for both TWR and TDOA (PR #707 and #745) to Bitcraze’s crazyflie-firmware open-source project.


The conventional Kalman filter, a primary sensor fusion mechanism, is sensitive to measurement outliers due to its minimum mean-square-error (MMSE) criterion. To achieve robust estimation, it is critical to properly handle measurement outliers. We implement a robust M-estimation method to address this problem. Instead of using a least-squares, maximum-likelihood cost function, we use a robust cost function to downweigh the influence of outlier measurements [1]. Compared to Random Sample Consensus (RANSAC) approaches, our method can handle sparse UWB measurements, which are often a challenge for RANSAC.

From the Bayesian maximum-a-posteriori perspective, the Kalman filter state estimation framework can be derived by solving the following minimization problem:

Therein, xk and yk are the system state and measurements at timestep k. Pk and Rk denote the prior covariance and measurement covariance, respectively.  The prior and posteriori estimates are denoted as xk check and xk hat and the measurement function without noise is indicated as g(xk,0). Through Cholesky factorization of Pk and Rk, the original optimization problem is equivalent to

where ex,k,i and ey,k,j are the elements of ex,k and ey,k. To reduce the influence of outliers, we incorporate a robust cost function into the Kalman filter framework as follows:

where rho() could be any robust function (G-M, SC-DCS, Huber, Cauchy, etc.[2]).

By introducing a weight function for the process and measurement uncertainties—with e as input—we can translate the optimization problem into an Iteratively Reweighted Least Squares (IRLS) problem. Then, the optimal posteriori estimate can be computed through iteratively solving the least-squares problem using the robust weights computed from the previous solution. In our implementation, we use the G-M robust cost function and the maximum iteration is set to be two for computational reasons. For further details about the robust Kalman filter, readers are referred to our ICRA/RA-L paper and the onboard firmware (mm_tdoa_robust.c and mm_distance_robust.c).


We demonstrate the effectiveness of the robust Kalman filter on-board a Crazyflie 2.1. The Crazyflie is equipped with an IMU and an LPS UWB tag (in TDOA2 mode). With the conventional onboard extended Kalman filter, the drone is affected by measurement outliers and jumps around significantly while trying to hover. In contrast, with the robust Kalman filter, the drone shows a more reliable localization performance.

The robust Kalman filter implementations for UWB TWR and TDOA localization have been included in the crazyflie-firmware master branch as of March 2021 (2021.03 release). This functionality can be turned on by setting a parameter (robustTwr or robustTdoa) in estimator_kalman.c. We encourage LPS users to check out this new functionality.

As we mentioned above, off-the-shelf, low-cost UWB modules also exhibit distinctive and reproducible bias patterns. In our recent work, we devised experiments using the LPS UWB modules and showed that the systematic biases have a strong relationship with the pose of the tag and the anchors as they result from the UWB radio doughnut-shaped antenna pattern. A pre-trained neural network is used to approximate the systematic biases. By combining bias compensation with the robust Kalman filter, we obtain a lightweight, learning-enhanced localization framework that achieves accurate and reliable UWB indoor positioning. We show that our approach runs in real-time and in closed-loop on-board a Crazyflie nano-quadcopter yielding enhanced localization performance for autonomous trajectory tracking. The dataset for the systematic biases in UWB TDOA measurements is available on our Open-source Code & Dataset webpage. We are also currently working on a more comprehensive dataset with IMU, UWB, and optical flow measurements and again based on the Crazyflie platform. So stay tuned!


[1] L. Chang, K. Li, and B. Hu, “Huber’s M-estimation-based process uncertainty robust filter for integrated INS/GPS,” IEEE Sensors Journal, 2015, vol. 15, no. 6, pp. 3367–3374.

[2] K. MacTavish and T. D. Barfoot, “At all costs: A comparison of robust cost functions for camera correspondence outliers,” in IEEE Conference on Computer and Robot Vision (CRV). 2015, pp. 62–69.


The authors are with the Dynamic Systems Lab, Institute for Aerospace Studies, University of Toronto, Canada, and affiliated with the Vector Institute for Artificial intelligence in Toronto.

Feel free to contact us if you have any questions or suggestions:

Please cite this as:

author={W. {Zhao} and J. {Panerati} and A. P. {Schoellig}},
title={Learning-based Bias Correction for Time Difference of Arrival Ultra-wideband Localization of Resource-constrained Mobile Robots},
journal={IEEE Robotics and Automation Letters},

Accurate indoor localization is a crucial enabling technology for many robotic applications, from warehouse management to monitoring tasks. Ultra-wideband (UWB) localization technology, in particular, has been shown to provide robust, high-resolution, and obstacle-penetrating ranging measurements. Nonetheless, UWB measurements are still corrupted by non-line-of-sight (NLOS) communication and spatially-varying biases due to doughnut-shaped antenna radiation pattern. In our recent work, we present a lightweight, two-step measurement correction method to improve the performance of both TWR and TDoA-based UWB localization.  We integrate our method into the Extended Kalman Filter (EKF) onboard a Crazyflie and demonstrate a closed-loop position estimation performance with ~20cm root-mean-square (RMS) error.

A stylized depiction of our UWB indoor localization system and the schematics of the proposed estimation framework.


UWB measurement errors can be separated into two groups: (1) systematic bias caused by limitations in the UWB antenna pattern and (2) spurious measurements due to NLOS and multi-path propagation. We propose a two-step UWB bias correction approach exploiting machine learning (to address(1)) and statistical testing (to address (2)). The data-driven nature of our approach makes it agnostic to the origin of the measurement errors it corrects. 

(1) Neural Network Bias Correction

The doughnut-shaped antenna radiation pattern causes the relative poses of anchors and tags to have a noticeable impact on the received signal power, which leads to systematic, predictable biases.  To empirically demonstrate the systematic measurement errors resulting from varying the relative pose between anchors and tags, we placed two DWM1000 UWB anchors at a distance of 4m and collected both TWR and TDoA UWB range measurements for the UWB tag mounted on top of a Crazyflie spinning around its own z-axis.

Left: schematics of the ranges (∆p’s), azimuth (α’s) and elevation angles (β’s) defining the relative poses of tag T and anchors A0, A1 when collecting the systematic bias measurements. Right: the neural network’s inferred bias (in red) with respect to the tag’s varying azimuth angle towards anchor T0, αT0, plotted against the UWB raw measurements.

We choose to leverage the nonlinear representation power of neural networks to learn the systematic bias which only depends on anchor-tag relative poses. Considering the limited onboard computation power, we select a fully connected neural network with 50 neurons in each of two layers with ReLU activation. To represent the relative pose between the UWB tag and anchors, we select the relative distance ∆p and roll, pitch, and yaw angles of the quadcopter as the input features x for the network. As we used fixed anchors, we do not include their poses as inputs (this level of generalization is left for future work). Given sufficient training data, the spatially-varying measurement bias can be described by a nonlinear function b=f(x) captured by the trained neural network.

(2) Outlier (Spurious Measurements) Rejection

Besides our learning-based bias correction, we use a quadcopter’s dynamic model to filter inconsistent UWB range measurements. Given the estimated velocity v and maximum acceleration amax, we can compute the maximum distance dmax a quadcopter can cover during time ∆t. Based on this information, we can reject unattainable measurements before fusing them into the EKF by comparing the measurement innovation with dmax

Moreover, we use a statistical hypothesis test to further classify potential outlier measurements. Since the measurement innovation vector is assumed to be distributed according to a multivariate Gaussian distribution, the normalized sum of squares of its values should follow a Chi-square distribution. We use the Chi-square hypothesis test to determine whether a measurement innovation is likely coming from this distribution.

UWB measurement bias f (x) prediction performance of the trained neural network (in red) compared to the actual measurement errors (blue dots) as well as the role of model-based filtering (purple dots) and statistical validation (orange dots) in rejecting outlier measurement innovations (teal dots) during a 60” flight experiment.

Data Collection and Training

We use a Crazyflie 2.0 quadcopter and the Loco Positioning System (LPS)’s UWB DW1000 modules as our research platforms. Our calibration approach runs on the Crazyflie STM32 microcontroller within the FreeRTOS real-time operating system. We equipped a cuboid flying arena with 8 UWB anchors, one for each vertex. The anchor positions were measured using a Leica total station theodolite.

Left: three-dimensional plot of our flight arena showing the positions and poses of the eight UWB DW1000 anchors (each facing towards its own x-axis, i.e., the red versor). Right: two of the training trajectories we flew to collect the samples that we used to train our neural network-based bias estimator

For all experiments, the ground truth position of the Crazyflie was provided by 10 Vicon cameras. The neural network was trained using PyTorch. To perform inference on the Crazyflie’s microcontroller, we re-use PyTorch’s trained weights in a plain C re-implementation. Since the DW1000 modules in the LPS provide UWB measurements every 5ms, the neural network inference runs at 200Hz during flight as well. Our outlier rejection method is also implemented in plain C and merged with the onboard EKF.

Close-loop Position Estimation Performance

We demonstrate the position estimation and close-loop performance of the proposed methods by flying a Crazyflie quadcopter along planar and non-planar circular trajectories (which were not among the trajectories used for training). A comparison between the estimation error of (A) the UWB localization estimate enhanced with outlier rejections and (B) the estimated enhanced with both outlier rejection and neural network bias compensation is conducted in our experiments for both TWR and TDoA2 modes. We repeated all of our experiments 10 times with a target velocity of 0.375m/s. The quadcopter trajectories during these flight tests are displayed in the following plots.  

Flight paths and the tracking performance of our approach with (in blue) and without (in orange) the neural network bias correction for two reference trajectories (planar and non-planar circular orbits) and both UWB modes (TWR and TDoA).

The distributions of the RMS estimation errors are summarized into a box plot. TWR-based ranging results in better localization performance than TDoA. However, we observe that, with our neural network bias compensation, the average RMS error of TDoA localization is around 0.21m, which is comparable to that of TWR-based localization (~0.19m). Thanks to the neural network bias compensation, the average reduction in the RMS error is ~18.5% and 48% for TWR and TDoA, respectively. Most notably, this result suggests that bias compensation might help closing the performance gap between TWR- and TDoA-based localization.

Root mean square error (RMSE) of the quadcopter position estimate before (in orange) and after (in blue) the neural network calibration step for both TWR and TDoA ranging modes. Each pair of box plots refers to a planar reference trajectory (left of each pair) and a reference trajectory with varying z (right of each pair), showing a greater performance enhancement for the latter.


In this work, we presented a two-step methodology to improve UWB localization—for both TWR- and TDoA-based measurements. We used a lightweight neural network to model and compensate for pose-dependent and spatially-varying biases and an outlier rejection mechanism to filter spurious measurements. Through several real world flight experiments tracking different trajectories, we showed that we are able to improve localization accuracy for both TWR and TDoA, granting safer indoor flight. In our future work, we will include the anchors’ pose information to allow our method to further generalize to previously unobserved indoor environments, with different anchor configurations.


The authors are with the Dynamic Systems Lab, Institute for Aerospace Studies, University of Toronto, Canada, and affiliated with the Vector Institute for Artificial Intelligence in Toronto.

Feel free to contact us if you have any questions or ideas: Please cite this as:

  title={Learning-based Bias Correction for Ultra-wideband Localization of Resource-constrained Mobile Robots},
  author={Wenda Zhao and Abhishek Goudar and Jacopo Panerati and Angela P. Schoellig},
  journal={arXiv preprint arXiv:2003.09371},