A Comparison of Bayesian Prediction Techniques for Mobile Robot Trajectory Tracking

Jose Luis Peralta-Cabezas; Miguel Torres-Torriti; Marcelo Guarini-Hermann

cs.ROFeb 17, 2026

A Comparison of Bayesian Prediction Techniques for Mobile Robot Trajectory Tracking

Authors: Jose Luis Peralta-Cabezas, Miguel Torres-Torriti, Marcelo Guarini-Hermann

Organizations: Dept. of Electrical Engineering, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Casilla 306-22, Santiago, Chile

Abstract

This paper presents a performance comparison of different estimation and prediction techniques applied to the problem of tracking multiple robots. The main performance criteria are the magnitude of the estimation or prediction error, the computational effort and the robustness of each method to non-Gaussian noise. Among the different techniques compared are the well known Kalman filters and their different variants (e.g. extended and unscented), and the more recent techniques relying on Sequential Monte Carlo Sampling methods, such as particle filters and Gaussian Mixture Sigma Point Particle Filter.

Figures & tables

Filter	Abrev.	Noise Assumption	Linearization Technique	Reference
Extended Kalman Filter	EKF	Gaussian	1 st Order Taylor	${\cite[cite]{[\@@bibref{}{KAL60}{}{}]},\cite[cite]{[\@@bibref{}{JAZ70}{}{}]}}$
Unscented Kalman Filter	UKF	Gaussian	None	[ JUL97 ]
Central Difference Kalman Filter	CDKF	Gaussian	2 nd Order Stirling	[ ITO00 ]
Divided Difference 1 st Order Kalman Filter	DD1	Gaussian	1 st Order Stirling	[ NOR00 ]
Divided Difference 2 nd Order Kalman Filter	DD2	Gaussian	2 nd Order Stirling	[ NOR00 ]
Square Root Unscented Kalman Filter	SRUKF	Gaussian	None	${\cite[cite]{[\@@bibref{}{MER04A}{}{}]},\cite[cite]{[\@@bibref{}{MER01}{}{}]}}$

Table 1: Summary of Bayesian prediction techniques compared in the present work.

Initial state estimate:	$\hat{x}_{0}=E\left[x_{0}\right]$
Initial state covariance:	$P_{0}=E\left[\left(x_{0}-\hat{x}_{0}\right)\left(x_{0}-\hat{x}_{0}\right)^{T}\right]$
Process noise covariance:	$Q_{0}=E\left[\left(v_{0}-\bar{v}_{0}\right)\left(v_{0}-\bar{v}_{0}\right)^{T}\right]$
Measurement noise covariace:	$R_{0}=E\left[\left(w_{0}-\bar{w}_{0}\right)\left(w_{0}-\hat{w}_{0}\right)^{T}\right]$

Algorithm 1 Extended Kalman Filter (EKF)

Symbol		Description
Robot Parameters
$\mathbf{o}_{1}$ , $\mathbf{o}_{2}$ , $\mathbf{o}_{3}$		rear, front-right and front-left wheels
$m$		robot’s mass
$I_{0}$		robot’s inertia moment
$r$		wheel radius
$L$		body radius

Table 2: Notation for the model equations of the three-wheel robot.

Figure 1: Robot coordinate frame $\left\{\mathbf{x^{r}},\mathbf{x^{r}}\right\}$ fixed to the robot’s body (local coordiantes) and the traction forces $f_{1}$ , $f_{2}$ and $f_{3}$ of the wheels.

Figure 2: World coordinate frame $\left\{\mathbf{x^{w}},\mathbf{y^{w}}\right\}$ fixed to the court (global coordinates) and two teams of RoboCup F-180 robots.

Figure 3: Forces acting on the robot and velocity components (see Table 2 for notation details).

Figure 4: PUC’s RoboCup F-180 system architecture.

Figure 5: Six different reference trajectories employed in the simulations.

Figure 6: Examples of the measured noisy trajectory corresponding to trajectory in the first row, second column of fig. 5 for different noise realizations.

	RMSE w/IF		RMSE wo/IF		$T_{c}/{T_{c}}_{min}$	$T_{c}/{T_{c}}_{min}$
Filter	Position [ $m$ ]	Heading [ $rad$ ]	Position [ $m$ ]	Heading [ $rad$ ]	w/IF	wo/IF
EKF	0.0048 b	0.2326	0.0229 b	0.2674	1.000	1.000
DD1	0.0434	0.1386	0.0491 c	0.1951	3.566	3.729
DD2	0.0342	0.1363	0.0491	0.1947	3.566	3.729
UKF	0.0379	0.0398 c	0.1101	0.1399 c	3.758	3.971
CDKF	0.0105 c	0.0518	0.1100	0.1398 b	3.727	3.960

Table 3: Results for the 1-frame ahead prediction.

	RMSE w/IF		RMSE wo/IF		$T_{c}/{T_{c}}_{min}$	$T_{c}/{T_{c}}_{min}$
Filter	Position [ $m$ ]	Heading [ $rad$ ]	Position [ $m$ ]	Heading [ $rad$ ]	w/IF	wo/IF
EKF	0.0234 b	0.2078	0.0478 b	0.2510	1.000	1.000
DD1	0.0548	0.1504	0.1232	0.1351	2.499	2.566
DD2	0.0658	0.1535	0.1232	0.1360 b	1.749	2.751
UKF	0.1187	0.0541	0.0617 c	0.2108	3.790	3.965
CDKF	0.0561	0.0325 a	0.0617	0.2115	3.594	3.756

Table 4: Results for the 4-frame ahead prediction.

	RMSE w/IF		RMSE wo/IF		$T_{c}/{T_{c}}_{min}$	$T_{c}/{T_{c}}_{min}$
Filter	Position [ $m$ ]	Heading [ $rad$ ]	Position [ $m$ ]	Heading [ $rad$ ]	w/IF	wo/IF
EKF	0.0611 c	0.2138	0.0828 b	0.2577	1.000	1.000
DD1	0.0884	0.1541	0.0928	0.2145	2.220	2.261
DD2	0.0582 b	0.1373	0.0927 c	0.2143	2.210	2.261
UKF	0.0812	0.0784	0.1549	0.1402 c	3.874	4.024
CDKF	0.1208	0.0400 a	0.1549	0.1395 b	3.561	3.696

Table 5: Results for the 8-frame ahead prediction.

Figure 7: Noise-free reference trajectory.

Figure 8: Measured trajectory.

Figure 9: Predicted trajectory 4-frames ahead using the EKF without the IF.

Figure 10: Predicted trajectory 4-frames ahead using the EKF with the IF.

Figure 11: Predicted trajectory 4-frames ahead using the PF.

Figure 12: Trajectory components

x^{w}

and

y^{w}

of a real F-180 robot predicted 4-frames ahead using the EKF with the IF.

Robot Trajectory Tracking

Robot trajectory tracking develops methods that make robots follow desired paths or motions accurately despite disturbances, uncertainty, and physical constraints.

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Denoising Particle Filters: Learning State Estimation with Single-Step Objectives

Lennart Röstel, Berthold Bäuml

Learning AI for Dextrous Robots Lab, Technical University of Munich, Germany.

Robot State Estimation Particle Filtering

Open paper AskPDF arXiv

Learning-based methods commonly treat state estimation in robotics as a sequence modeling problem. While this paradigm can be effective at maximizing end-to-end performance, models are often difficult to interpret and expensive to train, since training requires unrolling sequences of predictions in time. As an alternative to end-to-end trained state estimation, we propose a novel particle filtering algorithm in which models are trained from individual state transitions, fully exploiting the Markov property in robotic systems. In this framework, measurement models are learned implicitly by minimizing a denoising score matching objective. At inference, the learned denoiser is used alongside a (learned) dynamics model to approximately solve the Bayesian filtering equation at each time step, effectively guiding predicted states toward the data manifold informed by measurements. We evaluate the proposed method on challenging robotic state estimation tasks in simulation, demonstrating competitive performance compared to tuned end-to-end trained baselines. Importantly, our method offers the desirable composability of classical filtering algorithms, allowing prior information and external sensor models to be incorporated without retraining.

Abstract

Robot Trajectory Tracking

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