Dynamic Tracking Error Overview

• Dynamic tracking error is a time-varying discrepancy defined in context-specific frameworks, such as Lie groups in control or Euclidean metrics in reinforcement learning.
• It is applied across diverse domains including geometric control, mobile robotics, visual SLAM, and online optimization to evaluate and regulate system performance.
• The design of dynamic tracking error preserves structural properties and stability through coordinate choices that maintain energy, momentum, and convergence guarantees.

Dynamic tracking error denotes a time-varying discrepancy between an actual evolving process and a desired, estimated, or benchmarked one, but its precise meaning is domain dependent. In geometric control, it is often an intrinsically defined error on a full phase space, including configuration and momentum, chosen so that the error dynamics preserve plant structure (Hampsey et al., 2024). In reinforcement learning for mobile manipulation, it is the instantaneous Euclidean distance between a moving target object and the end-effector (Wang et al., 2020). In dynamic-object SLAM and visual tracking, it is commonly expressed through pose, speed, or photometric reprojection error of moving objects relative to camera motion (Wadud et al., 2022). In online optimization, it is the distance between an iterate and the current optimizer of a time-varying objective (Syed et al., 2023). Across these settings, the common theme is that the error is not static: it evolves with system dynamics, reference motion, uncertainty, and the chosen representation.

1. Dynamic tracking error as a family of domain-specific concepts

Dynamic tracking error does not admit a single universal definition. The supplied literature instead shows a family of closely related constructions, each tied to the state representation and decision objective of its field.

In geometric mechanics and tracking control, the term refers to an error state that lives on the same geometric object as the plant, rather than to coordinate subtraction. For fully actuated Euler–Poincaré systems on a matrix Lie group, the error is defined on the semidirect-product phase space $\mathbf G\ltimes \mathfrak g^\ast$, with configuration and momentum parts chosen so that the error dynamics are themselves “Euler–Poincaré-like” (Hampsey et al., 2024). A closely related underwater-vehicle formulation defines the error on $\SE(3)\ltimes \mathbb{R}^6$, again to preserve Hamiltonian or Lie–Poisson structure (Hampsey et al., 4 Jun 2026).

In kinematic and error-state control formulations, dynamic tracking error is often the deviation of the actual state from a moving reference expressed in convenient local coordinates. For a tractor–trailer system, the state is the deviation from a time-parameterized reference trajectory in body-fixed tractor and trailer frames (Kayacan et al., 2021). For a quadrotor, the error state consists of position error, velocity error, and a multiplicative orientation error expressed in exponential coordinates (Reich, 27 Jan 2025). For a unicycle-type mobile robot, the error state is $[e_1,e_2,e_3]^T$, where $e_1$, $e_2$, and $e_3$ are longitudinal, lateral, and heading errors in the robot frame, and the learning objective acts directly on error dynamics (Kayacan et al., 2021).

In perception and tracking systems, dynamic tracking error is typically tied to estimated object motion. DyOb-SLAM evaluates dynamic-object tracking mainly through object pose error and speed error, while also reporting camera pose error because object-motion estimation depends on camera pose (Wadud et al., 2022). DOT uses robust photometric reprojection residuals to estimate object motion and then converts residual motion into a dynamic/static decision (Ballester et al., 2020). Dynamic 3D Gaussians treats tracking error as the mismatch between persistent dynamic scene elements and true scene motion, evaluated after training by 2D and 3D trajectory benchmarks such as Median Trajectory Error (Luiten et al., 2023).

Other fields use simpler state discrepancies. In mobile manipulation with reinforcement learning, the core instantaneous tracking error is the Euclidean distance $d_t$ between object position and gripper position (Wang et al., 2020). In distributed multi-object tracking with mobile sensors, tracking error is inseparable from localization and frame-alignment uncertainty, because an object estimate can be wrong even when the detector is accurate if robots are geometrically misaligned (Peterson et al., 2023). In finance, tracking error is the volatility of active return relative to a policy benchmark, and “dynamic” means that the permitted or desired level of that volatility changes with the opportunity set and governance state (Alankar et al., 3 Mar 2026).

This suggests that dynamic tracking error is best understood not as one formula, but as a representational principle: the discrepancy must be defined on the variables that actually govern evolution, estimation, or decision quality in the problem at hand.

2. Structure-preserving dynamic errors in geometric control

A technically distinctive use of dynamic tracking error appears in geometric control of mechanical systems on Lie groups. Here, the objective is not merely to compare trajectories, but to do so in a way that respects symmetry, momentum transport, and topology.

For fully actuated mechanical systems whose configuration space is a matrix Lie group and whose dynamics are of Euler–Poincaré type, the state is posed on the cotangent bundle $T^\ast \mathbf G$, left-trivialized as $\mathbf G\times \mathfrak g^\ast$. The controlled extended Euler–Poincaré equations are

$\dot{Q} = Q U,\qquad \dot{P} = \ad_U^\ast P + \tau.$

The central construction is a semidirect-product group structure

$\SE(3)\ltimes \mathbb{R}^6$0

with multiplication

$\SE(3)\ltimes \mathbb{R}^6$1

which permits a globally defined right-invariant tracking error on the full phase space (Hampsey et al., 2024).

Given actual and desired trajectories $\SE(3)\ltimes \mathbb{R}^6$2 and $\SE(3)\ltimes \mathbb{R}^6$3, the error is defined by

$\SE(3)\ltimes \mathbb{R}^6$4

and becomes

$\SE(3)\ltimes \mathbb{R}^6$5

Thus the configuration error is

$\SE(3)\ltimes \mathbb{R}^6$6

while the momentum error is not $\SE(3)\ltimes \mathbb{R}^6$7, but the coadjoint-transported difference

$\SE(3)\ltimes \mathbb{R}^6$8

The associated dynamic input errors $\SE(3)\ltimes \mathbb{R}^6$9 are defined by transporting the input mismatch through the same symmetry action.

The main structural payoff is that the error system closes in the same extended Euler–Poincaré form: $[e_1,e_2,e_3]^T$0 This makes the error “dynamic” in a strict sense: it has its own plant-like configuration and momentum evolution, not just a kinematic mismatch (Hampsey et al., 2024).

The underwater-submersible formulation is closely analogous. For a fully actuated rigid body on $[e_1,e_2,e_3]^T$1 with added-mass coupling, the paper defines a right-invariant error on $[e_1,e_2,e_3]^T$2: $[e_1,e_2,e_3]^T$3 In components,

$[e_1,e_2,e_3]^T$4

$[e_1,e_2,e_3]^T$5

The translational and rotational momentum errors are therefore coupled by the semidirect-product structure. The resulting error dynamics remain Euler–Poincaré/Lie–Poisson, with a time-varying transformed inertia $[e_1,e_2,e_3]^T$6, and admit an energy-based tracking controller (Hampsey et al., 4 Jun 2026).

These constructions differ sharply from Euclidean subtraction. The literature explicitly motivates them by the fact that on Lie groups neither configurations nor cotangent vectors admit a globally meaningful subtraction, and that naïve differences ignore momentum transformation laws (Hampsey et al., 2024). A plausible implication is that dynamic tracking error, in this geometric sense, is fundamentally a choice of error coordinates that preserve the mechanics needed for Lyapunov, passivity, or Hamiltonian analysis.

3. Error-state and body-frame formulations for trajectory tracking

A second major line of work uses dynamic tracking error as a state variable for model-based control, typically by expressing the actual-minus-reference discrepancy in a moving local frame or on a tangent space.

For a tractor–trailer unmanned ground vehicle, the reference trajectory is time-based and the tracking error is defined in vehicle-attached coordinates: $[e_1,e_2,e_3]^T$7 The block-diagonal rotation matrix $[e_1,e_2,e_3]^T$8 maps tractor and trailer position errors into local body frames, yielding the error state

$[e_1,e_2,e_3]^T$9

The resulting nonlinear error dynamics are derived directly from the plant kinematics and linearized around the zero-error manifold to obtain a linear time-varying model

$e_1$0

which serves as the prediction model in a linear MPC (Kayacan et al., 2021). Here, dynamic tracking error means a local-coordinate deviation state whose evolution explicitly depends on reference velocity and yaw rates.

For quadrotor trajectory tracking, the paper formulates the true state as a nominal state plus an error state: $e_1$1 Position and velocity errors are additive, while orientation error is multiplicative: $e_1$2 The error state is

$e_1$3

with control error

$e_1$4

Linearization yields

$e_1$5

and the controller minimizes the infinite-horizon quadratic cost

$e_1$6

using LQR (Reich, 27 Jan 2025). The orientation error is therefore not a direct angle subtraction but a local Lie-algebra coordinate, chosen for linearization and real-time control.

Tracking-Error Learning Control for mobile robots goes further by using error dynamics themselves as learning objectives. The unicycle tracking errors in robot coordinates satisfy

$e_1$7

TELC defines the longitudinal dynamic error cost

$e_1$8

and the lateral dynamic error cost

$e_1$9

Feedforward coefficients are then updated by gradient descent on these costs so that the feedforward term gradually absorbs model mismatch and the feedback action approaches zero when the robot is on track (Kayacan et al., 2021). In this formulation, dynamic tracking error is explicitly a target transient error dynamics, rather than only a geometric position discrepancy.

These works share a common methodological choice: tracking is recast as regulation of a moving-error system. The specific state representation differs—body-frame coordinates, exponential coordinates, or learned error dynamics—but the technical role is the same.

4. Dynamic tracking error in perception, SLAM, and object tracking

In perception systems, dynamic tracking error usually refers to the discrepancy in estimated motion state of a moving object, or to the residual that remains after compensating for camera motion.

DyOb-SLAM evaluates dynamic-object tracking through object pose error and object speed error. Camera pose is estimated from static points using the reprojection residual

$e_2$0

while inter-frame object motion is estimated through

$e_2$1

For evaluation, the paper defines pose error using

$e_2$2

and speed error using

$e_2$3

with $e_2$4 (Wadud et al., 2022). Dynamic tracking error therefore means error in the estimated 6-DoF object pose and velocity magnitude over time, with camera-pose error treated as an upstream contributor.

DOT defines dynamic tracking error through direct photometric alignment. Camera motion is estimated by minimizing

$e_2$5

and object motion is estimated by minimizing

$e_2$6

Dynamic/static classification then uses the dynamic disparity

$e_2$7

together with pose-uncertainty entropy

$e_2$8

so that motion evidence is judged relative to observability (Ballester et al., 2020). Here, dynamic tracking error is essentially residual motion inconsistency after explaining the scene by camera motion and object motion.

Dynamic 3D Gaussians takes a different route: tracking is an emergent property of a persistent dynamic scene representation, not a directly supervised training loss. Each Gaussian has time-varying center and rotation but time-invariant scale, color, opacity, and background logit. Local rigidity is regularized by

$e_2$9

and long-term drift is controlled by

$e_3$0

Tracking quality is evaluated by 2D and 3D Median Trajectory Error, with reported full-model values of 2D MTE $e_3$1 and 3D MTE $e_3$2 cm across six PanopticSports sequences (Luiten et al., 2023).

A further variant appears in self-supervised tracker monitoring. SSLTrack does not estimate tracking error from ground truth during operation; instead it detects likely failure when the feature similarity between the last trusted reference crop and the current tracker-predicted crop drops below a threshold. This is a dynamic tracking error signal in the sense of online failure detection rather than geometric state deviation (Anjum et al., 2024).

Across these systems, tracking error is operationalized as pose error, speed error, photometric residual, trajectory error, or failure likelihood. The exact metric depends on whether the target of tracking is a physical motion state, an image projection, or the continued identity of a tracked region.

5. Tracking error in learning, optimization, and distributed estimation

Outside geometric control and vision, dynamic tracking error often appears as an evolving estimation or optimization discrepancy.

In multi-task reinforcement learning for mobile manipulation, the instantaneous tracking-related scalar is the Euclidean distance between object and gripper: $e_3$3 where “$e_3$4 is the Euclidean distance between the object and the gripper positions” (Wang et al., 2020). The observation explicitly contains the position difference vector and velocity difference vector between gripper and object, but the reward penalizes only Cartesian position error. The paper reports that unseen random dynamic trajectories achieve $e_3$5 m tracking error in simulation, while real-robot tracking errors on circle, square, and sine trajectories are $e_3$6 m, $e_3$7 m, and $e_3$8 m (Wang et al., 2020).

In distributed mobile multi-object tracking, MOTLEE shows that tracking error can arise from coordinate-frame inconsistency as much as from object dynamics. Measurements are transformed with localization-aware covariance

$e_3$9

and then into a neighbor’s frame with

$d_t$0

Association uses Mahalanobis distance, while fusion proceeds through a Kalman-Consensus Filter in information form (Peterson et al., 2023). The reported mobile experiment gives average MOTA $d_t$1 for MOTLEE, compared with $d_t$2 under ground-truth localization and $d_t$3 for a localization-ignorant baseline. This suggests that in distributed tracking, dynamic tracking error includes geometric inconsistency induced by localization drift, not just target-state estimation noise.

In online optimization, the tracking error state is explicitly

$d_t$4

where $d_t$5 is the optimizer of the current time-varying objective $d_t$6. For inexact OGD with absolute oracle error, the exact recursive bound is

$d_t$7

while for stochastic OGD with zero-mean additive noise it becomes

$d_t$8

The dynamic regret is then obtained from these tracking-error bounds through a smoothness argument (Syed et al., 2023). Here, dynamic tracking error is not a physical tracking quantity but the optimization gap to the current minimizer of a changing objective.

In recurrent models for symbolic state tracking, the paper on error control dynamics defines hidden-state tracking error along the state-separating subspace

$d_t$9

and proves that affine recurrent trackers cannot contract perturbations in $T^\ast \mathbf G$0 once they preserve symbolic states exactly. The resulting state-relevant error accumulates as

$T^\ast \mathbf G$1

Tracking remains readable only while the distinguishability ratio

$T^\ast \mathbf G$2

stays below a decoder-dependent threshold (Chung et al., 8 May 2026). This is a dynamic tracking error notion rooted in hidden-state drift and error control rather than explicit trajectory following.

These cases broaden the term substantially. Dynamic tracking error can be an RL reward signal, an uncertainty-aware distributed estimation discrepancy, an online-optimization iterate gap, or a hidden-state readability margin. What remains common is that the error evolves jointly with the system, the reference, and the mechanism used to correct or fuse information.

6. Stability, constraints, and recurrent misconceptions

A recurring misconception is that a dynamic tracking error should always converge globally to zero. The literature in the supplied set does not support that general statement.

In Lie-group control, global asymptotic stabilization is often topologically obstructed. The Euler–Poincaré paper proves local asymptotic stability at identity in general and identifies almost-global behavior in the $T^\ast \mathbf G$3 rigid-body example, where the unstable $T^\ast \mathbf G$4-rotation set prevents global asymptotic tracking (Hampsey et al., 2024). The underwater-submersible paper likewise proves asymptotic convergence locally around the identity error, not global convergence on $T^\ast \mathbf G$5 (Hampsey et al., 4 Jun 2026).

A second misconception is that tracking error is necessarily a simple subtraction of states or outputs. On Lie groups and cotangent bundles, the literature explicitly rejects naïve subtraction because configuration and momentum transform under group actions (Hampsey et al., 2024). In distributed tracking, even object-state differences are meaningless unless robot frames are properly aligned (Peterson et al., 2023). In recurrent symbolic tracking, the relevant error is not arbitrary hidden-state deviation but only the component lying in the symbolic subspace $T^\ast \mathbf G$6 (Chung et al., 8 May 2026).

A third misconception is that the same scalar metric can summarize tracking error across domains. The supplied papers instead use very different quantities: Euclidean distance $T^\ast \mathbf G$7 in RL (Wang et al., 2020), pose and speed error in SLAM (Wadud et al., 2022), photometric reprojection residual in visual tracking (Ballester et al., 2020), MOTA in distributed MOT (Peterson et al., 2023), $T^\ast \mathbf G$8 in online optimization (Syed et al., 2023), and tracking-error volatility relative to a benchmark in portfolio governance (Alankar et al., 3 Mar 2026). This suggests that “dynamic tracking error” is a role played by a discrepancy variable, not a single cross-disciplinary metric.

The literature also shows that dynamic tracking error is often inseparable from the mechanism that constrains it. In control, the error coordinates are chosen so that Lyapunov or Hamiltonian structure is preserved (Hampsey et al., 2024). In optimization, the contraction factor and disturbance terms determine a nonzero steady-state radius when drift or oracle errors persist (Syed et al., 2023). In SSLTrack, tracking error is operationally the signal that decides when human intervention is necessary (Anjum et al., 2024). In finance, dynamic tracking error is explicitly governed by a board-imposed ceiling

$T^\ast \mathbf G$9

so the dynamics of the error are as much about governance as about market opportunity (Alankar et al., 3 Mar 2026).

A plausible implication is that dynamic tracking error is best treated as a design object rather than merely an evaluation statistic. The strongest papers in this set do not just measure it; they shape its geometry, transport law, covariance, or admissible range so that tracking remains interpretable, controllable, or governable over time.