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Estimation-Aware Trajectory Optimization with Set-Valued Measurement Uncertainties

Aditya Deole, Mehran Mesbahi

TL;DR

The paper addresses estimation-aware trajectory planning under state-dependent, set-valued measurement uncertainties modeled by ellipsoids $\mathcal{E}(\bm x,Q)$. It introduces a set-valued observability framework and proves that, for observability-regular maps, the degree of observability lower bound is concave with respect to the state trajectory, enabling convex-form optimization under suitable envelopes $\hat{\Lambda}$. A practical optimization approach combines this observability metric with trajectory constraints through trust-region sequential convex programming (SCvx), and is demonstrated on both a Dubins-car-like system with vision-based estimation and an Ego-Target Rendezvous problem using ML-based estimation. The results show that estimation-aware trajectories can significantly improve state-estimation performance (lower tracking error and reduced state covariance) at the cost of some task deviation, providing a principled method to balance estimation quality with mission objectives. The framework is general, extensible to sensor placement and multi-agent scenarios, and adaptable to real-time MPC by updating the uncertainty envelope as new data arrive.

Abstract

In this paper, an optimization-based framework for generating estimation-aware trajectories is presented. In this setup, measurement (output) uncertainties are state-dependent and set-valued. Enveloping ellipsoids are employed to characterize state-dependent uncertainties with unknown distributions. The concept of regularity for set-valued output maps is then introduced, facilitating the formulation of the estimation-aware trajectory generation problem. Specifically, it is demonstrated that for output-regular maps, one can utilize a set-valued observability measure that is concave with respect to the finite horizon state trajectories. By maximizing this measure, estimation-aware trajectories can then be synthesized for a broad class of systems. Trajectory planning routines are also examined in this work, by which the observability measure is optimized for systems with locally linearized dynamics. To illustrate the effectiveness of the proposed approach, representative examples in the context of trajectory planning with vision-based estimation are presented. Moreover, the paper presents estimation-aware planning for an uncooperative Target-Rendezvous problem, where an Ego-satellite employs an onboard machine learning (ML)-based estimation module to realize the rendezvous trajectory.

Estimation-Aware Trajectory Optimization with Set-Valued Measurement Uncertainties

TL;DR

The paper addresses estimation-aware trajectory planning under state-dependent, set-valued measurement uncertainties modeled by ellipsoids . It introduces a set-valued observability framework and proves that, for observability-regular maps, the degree of observability lower bound is concave with respect to the state trajectory, enabling convex-form optimization under suitable envelopes . A practical optimization approach combines this observability metric with trajectory constraints through trust-region sequential convex programming (SCvx), and is demonstrated on both a Dubins-car-like system with vision-based estimation and an Ego-Target Rendezvous problem using ML-based estimation. The results show that estimation-aware trajectories can significantly improve state-estimation performance (lower tracking error and reduced state covariance) at the cost of some task deviation, providing a principled method to balance estimation quality with mission objectives. The framework is general, extensible to sensor placement and multi-agent scenarios, and adaptable to real-time MPC by updating the uncertainty envelope as new data arrive.

Abstract

In this paper, an optimization-based framework for generating estimation-aware trajectories is presented. In this setup, measurement (output) uncertainties are state-dependent and set-valued. Enveloping ellipsoids are employed to characterize state-dependent uncertainties with unknown distributions. The concept of regularity for set-valued output maps is then introduced, facilitating the formulation of the estimation-aware trajectory generation problem. Specifically, it is demonstrated that for output-regular maps, one can utilize a set-valued observability measure that is concave with respect to the finite horizon state trajectories. By maximizing this measure, estimation-aware trajectories can then be synthesized for a broad class of systems. Trajectory planning routines are also examined in this work, by which the observability measure is optimized for systems with locally linearized dynamics. To illustrate the effectiveness of the proposed approach, representative examples in the context of trajectory planning with vision-based estimation are presented. Moreover, the paper presents estimation-aware planning for an uncooperative Target-Rendezvous problem, where an Ego-satellite employs an onboard machine learning (ML)-based estimation module to realize the rendezvous trajectory.
Paper Structure (15 sections, 1 theorem, 39 equations, 9 figures, 1 algorithm)

This paper contains 15 sections, 1 theorem, 39 equations, 9 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Set-valued uncertainty measurement model
  3. Observability with set-valued output uncertainty
  4. Observability metric for discrete time output-tubes
  5. Finite horizon Estimation-Aware Trajectory Planning
  6. Maximizing observability for a pre-defined nominal trajectory
  7. Observability based augmented trajectory design for satellite rendezvous
  8. Conclusions
  9. Appendix
  10. System dynamics for Dubins Car
  11. Trust Region Algorithm Implementation
  12. System Dynamics for Satellite Rendezvous
  13. SCvx implementation details
  14. Simulation setup
  15. Validation experiment Design

Key Result

Theorem IV.1

Given a system with a finitely observable state $\bar{\bm x}_0$, the lower bound on the degree of observability is a concave function with respect to the state sequence $\bm x_{0:T}$ if the output map is observability-regular.

Figures (9)

  • Figure 1: Representations of static and dynamics state to output maps for a given input sequence.
  • Figure 2: Representation of distinguishable and undistinguishable initial conditions using tube separation.
  • Figure 3: Comparative Analysis for Dubins car example with nominal trajectory and estimation-aware trajectories for increasing exploration bounds are presented.
  • Figure 4: Ego Agent trajectory for Target rendezvous with a keep-out zone.
  • Figure 5: Comparative analysis for satellite rendezvous example. Here estimation-aware trajectories and their error analysis for increasing observability condition are presented.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Definition II.1
  • Definition III.1
  • Definition III.2
  • Definition III.3
  • Definition IV.1
  • Theorem IV.1