Frenet-Serret-Based Trajectory Prediction

TL;DR

This work addresses real-time 3D trajectory prediction from position measurements without assuming target maneuvers or sensor noise characteristics. It introduces two AISE-based predictors: AISE/va, which estimates $\hat{\mathbf{v}}_k$ and $\hat{\mathbf{a}}_k$ to extrapolate $\hat{p}_{k+l}$ via a second-order Taylor expansion, and AISE/FS, which builds a Frenet-Serret (FS) frame from velocity, acceleration, and jerk to propagate a 3D trajectory using FS dynamics. The paper combines adaptive input and state estimation with real-time numerical differentiation and FS kinematics, and demonstrates through parabolic and helical examples that AISE/FS achieves higher accuracy than AISE/va and traditional filters (e.g., $\alpha$-$\beta$-$\gamma$). The results suggest that FS-based trajectory prediction is particularly effective for complex maneuvers, with potential impact on guidance, navigation, control, and autonomous systems where accurate real-time predictions are critical.

Abstract

Trajectory prediction is a crucial element of guidance, navigation, and control systems. This paper presents two novel trajectory-prediction methods based on real-time position measurements and adaptive input and state estimation (AISE). The first method, called AISE/va, uses position measurements to estimate the target velocity and acceleration. The second method, called AISE/FS, models the target trajectory as a 3D curve using the Frenet-Serret formulas, which require estimates of velocity, acceleration, and jerk. To estimate velocity, acceleration, and jerk in real time, AISE computes first, second, and third derivatives of the position measurements. AISE does not rely on assumptions about the target maneuver, measurement noise, or disturbances. For trajectory prediction, both methods use measurements of the target position and estimates of its derivatives to extrapolate from the current position. The performance of AISE/va and AISE/FS is compared numerically with the $α$-$β$-$γ$ filter, which shows that AISE/FS provides more accurate trajectory prediction than AISE/va and traditional methods, especially for complex target maneuvers.

Figures (7)

• Figure 1: $\vect{r}_{\rmo_{\rm T}/\rmo_{\rmE}}$ is the physical position vector between $\rmo_{\rm T}$ and $\rmo_\rmE$.
• Figure 2: Block diagram of AISE.
• Figure 3: Block diagram of AISE/va and AISE/FS trajectory prediction methods.
• Figure 4: Example \ref{['eg:traj_extra_parabola']}: Trajectory prediction for a parabolic trajectory using AISE/FS. (a) The purple line shows the predicted trajectory with horizon $\ell = 100$ steps. (b) Zoom of (a).
• Figure 5: Example \ref{['eg:traj_extra_parabola']}: Trajectory prediction for a parabolic trajectory using AISE/FS. Estimates of $\Tilde{\kappa},$$\Tilde{\tau}$, and $u$ are computed using AISE.