Control Through Planner-Conditioned Estimator Error

Belief-space planning for VTOL ship-deck landing

Controls
Belief-Space Planning
State Estimation

A planner can predict parts of an estimator's error using information the estimator does not receive. This work accounts for that predictable error and intermittent corrections when planning VTOL landings on a moving ship deck.

Belief-Space Planning with Planner-Conditioned Estimator Error under Intermittent Observations

Peter M. Donley and Joseph L. Moore

Submitted to the American Control Conference (ACC); under review.

A VTOL approaches a ship deck while its camera loses sight of the deck during a flare. Below, planner-conditioned error predictions follow the realized error during the observation dropout. — Predictable estimator error during a deck landing (illustrative one-dimensional deck-relative model, with command-blind estimation) (a) The vehicle conducts a flare maneuver, pitching the deck out of the camera’s field of view between 2 and 5 s. (b) The estimator does not model the commanded flare, so its prediction accumulates error during the dropout while its modeled error distribution remains centered at zero with widening covariance $V$ (gray). The planner issued the command and knows the estimator is command-ignorant, so predicts the evolution of estimator-error conditioned on its own information $\mathcal F$ . Planner-predicted estimator error $m$ tracks the realized estimator error (black) at a much narrower covariance $P$ (blue). After visibility is restored, corrections (green) are accepted stochastically. In this example, accepted and uncorrected outcomes have different predicted error means, whose spread temporarily widens $P$ . Colors are consistent across figures, where gray represents estimator reports, blue represents planner-conditioned moments, green indicates correction acceptance, and shading for intervals with no available corrections.

Abstract

Belief-space planners with separately designed or off-the-shelf estimators may have access to state-relevant information the estimator does not observe. Consequently, even an estimator that minimizes mean-squared error under its own information can have a nonzero error mean when conditioned on planner information. When future corrections are intermittent and stochastic, differences between accepted and rejected error means introduce additional uncertainty terms to correction events that zero-mean assumptions ignore. In this paper, we propose a belief-space planning approach that predicts, propagates, and penalizes planner-conditioned estimator error along evaluated trajectories. A planner-conditioned estimator-error model combines affine error dynamics with a moment recursion that preserves stochastically-induced correction uncertainty. We describe a method by which to predict estimator error dynamics within specified operating regimes and incorporate predicted error moments into a task-weighted quadratic risk objective. We evaluate the approach in simulation for a tilt-rotor VTOL landing on a ship deck using receding-horizon planning. We find that conditioning on planner information reduced error-prediction loss for a command-blind EKF by 13.4% relative to a planner-ignorant model, with strongly regime and horizon-dependent forecasting abilities. In a small set of 16 paired closed-loop trials, the planner lowered the median terminal task gauge from 1.60 to 0.99, and predicted estimator error beyond 2s within a factor of 1.3, versus a 5.7-fold underprediction by covariance-only planning.

Figures from the paper

Error trajectories branch after stochastic corrections, showing the additional uncertainty created by different corrected and uncorrected error means. — Intermittent correction of biased error (Proposition 2) using a scalar example with $q=0.5$ and $J=0.2$ . Sensor loss leaves the estimator-error with a predictable mean $m^-\approx 2$ . Once observation is restored, each correction opportunity (ticks) is accepted with probability $q$ , so the possible futures branch into those already corrected (green) and those not yet corrected (orange). The exact moments (blue) cover both groups. Additional covariance spread $P$ is induced by $q(1-q)vv^\top$ , which vanishes for $m^- = 0$ . Noise inflation $R\mapsto R/q$ (dashed band) treats each opportunity as a weaker but certain correction, so it falls between the groups. After the first correction opportunity, its $2\sigma$ band contains under 2% of possible futures, against over 99% for exact moments. Ten representative paths of $2\times10^4$ simulations are drawn.

Two candidate braking trajectories have the same sensing schedule, but different predicted estimation risk because their commands occur at different times. — Candidate commands can affect estimator error despite identical sensing (illustrative one-dimensional task-relative model). Both candidates lose observability over the same interval (shaded), so their correction probabilities $q$ are identical. Both candidates also command the same total deceleration, with candidate A braking during sensing loss and B braking before. A covariance-only evaluation ( $m \equiv 0$ , dashed), typical of Gaussian belief-space planning approaches, predicts the same risk for both trajectories. The full model attributes predictable estimator error to A’s unobserved maneuvering, so A’s risk, in tolerance-normalized units according to (24), is dominated by $\mathcal R_m$ (dark), peaking at 2.5 $\times$ that of trajectory B.

Matched VTOL landing trajectories for three planners, with the proposed planner reaching the deck capture region in the illustrated trial. — Representative 2D projection of matched ship-deck landing trial. Curves show measured deck-relative truth trajectories for P1 (blue), P2 (orange), and P3 (purple). P1 reaches the capture region $\kappa = 0.73$ while P2 ( $\kappa = 1.14$ ) and P3 ( $\kappa = 15.03$ ) terminate outside it.

Control Through Planner-Conditioned Estimator Error

Abstract

Figures from the paper

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