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

1.1. Related work

Closed loop uncertainty prediction and belief-space planning approaches are well established, though typically assume that a specified estimator model is consistent with the planning model [1], [2]. This model uncertainty is centered on the modeled posterior mean and zero-mean estimator error, despite the potential existence of conditionally nonzero bias in the deployed estimator. Approaches like LQG-MP provide a framework for predicting joint dispersion between truth and the published estimates of a Kalman filter [3], though they do not model the planner-conditioned error of externally deployed estimators when the information sets of the planner and estimator differ.

Observation intermittency provides another source of uncertainty that complicates planning problems [4], [5], [6]. Relevant treatments include scalar information-approximation approaches, like smoothing sensing-discontinuities [7] and measurement-probability information weighting [8], and approaches which retain acquisition randomness [9], [10]. While it is established that sensing geometry and random acquisition affect future uncertainty, accepted and rejected corrections yield different predicted error means, which contribute to additional covariance, changing the prospective task risk.

Various planning approaches consider modular estimators and estimator quality. Some employ learned future-error costs [11] while others propagate perception uncertainty explicitly [12], [13]. Instead of propagating an error cost, we propagate conditional estimator-error, modeling the first and second moments of estimator error conditioned on the planner’s information.

Our work builds upon prior analyses of estimator-controller information asymmetry [14], [15] and established machinery for mixture moment matching [16], with which we develop a model of planner-conditioned estimator error, its effect on stochastic correction outcomes, a closed moment propagation of that error through correction, and a task risk-metric which enables a planner to reason about trajectory impacts on estimator performance.

1.2. Contributions

We propose a belief-space planning approach which derives a model of the first and second moments of estimator error. This model is used to propagate the estimator error along candidate trajectories. The planner predicts the probability that usable corrections will be available to the estimator and moment-matches the first two error moments over both correction and no-correction outcomes at each step. This predicted error is then projected into task coordinates, from which the planner computes a task-weighted estimation risk metric. These predictions are then used by the receding-horizon planner to select trajectories.

In this work, we make the following claims:

• We show that, under different information sets, even estimators that are minimum mean-squared error (MMSE)-optimal under their own information may have a nonzero planner-conditioned error mean, as established in Proposition 1 and evaluated empirically through the error prediction tests of Section 7.3;

• We develop a moment-matched recursion for estimator-error propagation under intermittent corrections, capturing the additional covariance induced by separation between the accepted and rejected error means that probability-based noise inflation omits, while recovering the familiar modified-Riccati recursion in the zero-mean limit;

• We formulate a task-weighted planner that penalizes predicted estimator-error moments along candidate trajectories, identify error models for three estimator architectures, and evaluate closed-loop ship-deck landing of a tilt-rotor VTOL in simulation.

We find that estimator error contains substantial planner-predictable structure when the estimator is ignorant of task-relevant information the planner holds. Prediction gains vary with forecast horizon and operating regime, though in 16 paired closed-loop trials, the proposed planner consistently improved median task performance compared to traditional covariance-only belief-space planning.

3.1. Planner information

The planner does not require access to the details of an estimator’s internal belief representation or propagation mechanisms in order to predict its error. However, it does require the estimator to provide a mechanism by which to initialize the error moment predictors, a model for between-correction error flow, the effect of accepted corrections on the estimator’s predictions, the probability that a correction is accepted by the estimator, and the range over which the estimator’s behavior is considered valid.

3.2. Initialization and compensation

At the beginning of planning time $c$, the estimator-error moments are initialized using the set of all previously published estimator information $(m_c,P_c) = \mathcal I_E(I_{0:c})$ using some declared initialization procedure $\mathcal I_E$. The planner uses this initialized error prediction to compensate the estimator’s published state, yielding a compensated planner-side state estimate $\check x_c$, such that

$$\check x_c = \hat x_c \oplus m_c.$$(4) where $\oplus$ is the retraction operator on the relevant manifold.

3.3. Estimator-error propagation between corrections

We model estimator-error propagation between observation-based corrections over integration interval $\delta$ as $$e_{k+1}^- = F e_k^+ + \delta\left( b+B\phi_k+\Lambda_z z_k \right) + \eta_k, \qquad F=I+\delta A,$$(5) where $A$ and $b$ characterize autonomous error evolution, $\phi_k$ collects candidate-dependent features available to the planner, and $z_k$ represents the retained temporally correlated residual modes. The remaining residual satisfies $$\mathbb E[\eta_k\mid\mathcal F_k]=0.$$

Let $r_k\in[0,1]^{n_r}$ denote correction recency, with $\bar r_k = \mathbb E[r_k\mid\mathcal F_k],$ and $\Sigma_k^{rr} = \operatorname{Cov}(r_k\mid\mathcal F_k)$. Its contribution to the remaining flow uncertainty is $$\begin{aligned} \operatorname{Cov}(\eta_k\mid\mathcal F_k) = \delta\Bigg[ Q_0 + \sum_{j=1}^{n_r} \Big( (1-\bar r_{j,k})Q_j^{(1)} \\ + \big[ (1-\bar r_{j,k})^2 + \Sigma^{rr}_{jj,k} \big] Q_j^{(2)} \Big) \Bigg], \end{aligned}$$(6) where $Q_0$ is the baseline residual uncertainty, $Q_j^{(1)}$ is the uncertainty which increases linearly with the staleness of correction channel $j$, and $Q_j^{(2)}$ is the residual uncertainty that increases quadratically with staleness. The quadratic term uses the full second-moment of stochastic recency, since $\mathbb{E}[(1-r)^2|\mathcal F_k] = (1-\bar r_j)^2 + \Sigma_{jj,k}^{rr}$

The colored residual is represented by the stable autoregressive model $z_{k+1} = R_z z_k+\epsilon_k^z$ for $\rho(R_z)<1,$ with centered innovation $\epsilon_k^z$, where $\rho(R_z)<1$ is a constraint to ensure the stable decay of the latent residual state. Between corrections, recency evolves according to $$r_{k+1}^- = D_r r_k^+.$$(7)

Define the augmented recency-error state $s_k = [ e_k^\top, r_k^\top, z_k^\top ]^{\top}.$ For a fixed candidate command sequence, the between-correction estimator-error propagation model can then be written in affine form as $$s_{k+1}^- = \mathcal A_k s_k^+ + c_k + \omega_k,$$(8) where $\mathbb E[\omega_k\mid\mathcal F_k]=0,$ and $\operatorname{Cov}(\omega_k\mid\mathcal F_k)=\Omega_k.$ Its first two moments propagate as $$\begin{aligned} \mu_{k+1}^- &= \mathcal A_k\mu_k^+ + c_k, \\ \Sigma_{k+1}^- &= \mathcal A_k\Sigma_k^+\mathcal A_k^\top + \Omega_k, \end{aligned}$$(9) and the estimator-error moments $m_k$ and $P_k$ are the $e$-blocks of $\mu_k$ and $\Sigma_k$, respectively.

3.4. Stochastic corrections

Let $a_{k|c}\in\{0,1\}$ denote whether a usable correction occurs at rollout step $k$. We model $a_{k|c}\sim\operatorname{Bernoulli}(q_{k|c})$ with $$q_{k|c} = \mathcal Q( \bar x_{k|c}, r_{k|c}, I_c ).$$(10) Here $\mathcal Q$ is a planner-side correction-probability model accounting for sensing cadence, trajectory-dependent sensing geometry, and conditional estimator acceptance.

Define the error of the accepted-correction branch as $$e^{\mathrm{acc}} = Je^- + \epsilon^c, \qquad \mathbb{E}[\epsilon^c|\mathcal F] = 0, \quad \operatorname{Cov}(\epsilon^c|\mathcal F) = W.$$(11) for error-transition map $J$ and remaining correction error $\epsilon^c$. For linear updates, $J = I-KH$, where $K$ is the correction gain and $H$ is the measurement Jacobian. When observations are accepted, the recency component associated with an accepted measurement type is reset to $r^+_j=1$, while it continues decaying in the no-observation branch.

4.1. Rollout assumptions

We declare the following assumptions under which we formulate our moment-matching approach.

4.2. Moment matching over correction outcomes

Under the stochastic-acceptance model of Section 3.4, let event $a_k = 1$ denote correction acceptance at time $k$. Likewise, $a_k = 0$ denotes rejection. The first and second moments of estimator-error are defined for each branch as $$\begin{aligned} m_{\mathrm{rej}} &= \mathbb{E}[e^+\mid a=0,\mathcal F], & P_{\mathrm{rej}} &= \operatorname{Cov}(e^+\mid a=0,\mathcal F), \\ m_{\mathrm{acc}} &= \mathbb{E}[e^+\mid a=1,\mathcal F], & P_{\mathrm{acc}} &= \operatorname{Cov}(e^+\mid a=1,\mathcal F). \end{aligned}$$ An exact first-two-moment closure of the branches arises from the laws of total expectation and covariance, yielding the posterior error correction model:

$$\begin{aligned} m^+ &= (1-q)m_{\mathrm{rej}} + q m_{\mathrm{acc}}, \\ P^+ &= (1-q)P_{\mathrm{rej}} + qP_{\mathrm{acc}} \\ &\quad+ q(1-q) (m_{\mathrm{acc}}-m_{\mathrm{rej}}) (m_{\mathrm{acc}}-m_{\mathrm{rej}})^\top . \end{aligned}$$(12) The first two covariance terms model within-branch uncertainty, and the final term models the spread between the branch means. This final covariance term is zero precisely when the accepted and rejected branch means coincide and $0<q<1$.

4.3. Effect of predictable estimator error

We apply this mixture to the error model established in Section 3.4. In particular, define the $a_k=0$ branch moments $m_{\mathrm{rej}}=m^-$ and $P_{\mathrm{rej}}=P^-$. The $a_k=1$ branch is defined by (11), with resulting branch moments $m_{\mathrm{acc}}=Jm^-$ and $P_{\mathrm{acc}}=JP^-J^\top+W$.

Therefore, in general, scalar inflation alone cannot reconstruct the stochastic-acceptance uncertainty of (12).

Figure 2 details and data

Possible futures after stochastic corrections

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4.4. Zero-mean limit

Setting $m^-=0$ in Proposition 2 gives $$\begin{aligned} m^+&=0, \\ P^+ &= (1-q)P^- + q\left( JP^-J^\top +W \right). \end{aligned}$$(15)For a linear-Gaussian correction with $J=I-KH$, $W=KRK^\top$, $S^c=0$, $P^- = V^-$, and $K = V^-H^\top \left( HV^-H^\top+R \right)^{-1},$ (15) reduces to $$P^+ = P^- - qP^-H^\top \left( HP^-H^\top+R \right)^{-1} HP^- .$$(16)

This is precisely a modified-Riccati update, using the familiar intermittent-measurement correction [4]. Our proposed propagation therefore contains the conventional zero-mean recursion as the special case for $m^- = 0$, with additional terms appearing only when the rejected and accepted branches have different predicted means.

4.5. Comparison with probability-based noise inflation

For the stochastic correction, define the exact expected correction map $\bar J \triangleq (1-q)I+qJ = I-qKH$, hence $m^+ = \bar Jm^-$. Under noise-inflated update $R\mapsto R/q$, define $\widetilde J\triangleq I-\widetilde KH$. Probability-based noise inflation $R\mapsto R/q$ gives $$\widetilde K = qP^-H^\top \left( qHP^-H^\top+R \right)^{-1}.$$(17) For $0<q<1$, $P^-\succ0$, $R\succ0$, and full-row-rank $H$, $$\widetilde K\neq qK, \qquad \widetilde J\neq\bar J.$$(18) For (18), equality $\widetilde K=qK$ would imply $(1-q)HP^-H^\top=0$, which contradicts $HP^-H^\top\succ0$. Its covariance update is similarly $$\widetilde P^+ = \widetilde J P^-\widetilde J^\top + \widetilde K \left(\frac{R}{q}\right) \widetilde K^\top,$$(19) which is independent of $m^-$. In contrast, $$P^+(m^-)-P^+(0) = q(1-q) (J-I)m^- (m^-)^\top (J-I)^\top .$$(20)

Therefore, since noise inflation generally mismatches the expected mean update and its covariance update is bias-independent, it cannot represent bias-dependent recursion for an arbitrary $m^-$.

4.6. Recursive rollout closure

The proposed stochastic-correction recursion remains closed for the augmented error-recency state $\chi_k = [e_k, r_k]^\top$. The corresponding branch-conditioned moments are $(\mu_{\chi,\mathrm{rej}},\Sigma_{\chi,\mathrm{rej}})$ and $(\mu_{\chi,\mathrm{acc}},\Sigma_{\chi,\mathrm{acc}}),$ which propagate according to $$\begin{aligned} \mu_\chi^+ &= (1-q)\mu_{\chi,\mathrm{rej}} + q\mu_{\chi,\mathrm{acc}}, \\ \Sigma_\chi^+ &= (1-q)\Sigma_{\chi,\mathrm{rej}} + q\Sigma_{\chi,\mathrm{acc}} \\ &\quad+ q(1-q) (\mu_{\chi,\mathrm{acc}}-\mu_{\chi,\mathrm{rej}}) (\mu_{\chi,\mathrm{acc}}-\mu_{\chi,\mathrm{rej}})^\top . \end{aligned}$$(21)

5.1. Task projection

Define the projection of estimator-error into task-relevant coordinates as $$\delta y_k \simeq G_k e_k,$$(22) where $G$ is the Jacobian which maps estimator-coordinates into task-space. Let $\tau_i,i\in [1,n_y]$ be the set of tolerances on each task axis, which defines the successful task-completion set. The task-normalized cost metric can be constructed as:

$$M = \operatorname{diag} \left( \tau_1^{-2}, \ldots, \tau_{n_y}^{-2} \right).$$(23)

5.2. Task-weighted estimation risk

Compensation of the current physical nominal does not set $m_k=0$, and so $m_k$ remains the predicted mean of the external estimator error, determining future correction outcomes and their task-space effect.

Projecting the planner-conditioned error-moments into task-space enables the computation of task-normalized estimation risk. Let the expected task-normalized estimation risk metric be defined as $\mathcal R_E \triangleq \mathbb{E}[\lVert G_ke_k\rVert^2_M | \mathcal F_k]$. Hence $$\mathcal R_E(m_k,P_k) = (G_km_k)^\top M(G_km_k) + \operatorname{tr} \left( MG_kP_kG_k^\top \right).$$(24) This risk metric can be decomposed, such that $\mathcal R_E = \mathcal R_m + \mathcal R_P$ for the predictable-bias risk $\mathcal R_m = \lVert G_km_k\rVert^2_M$ and covariance risk $\mathcal R_P = \operatorname{tr}(MG_kP_kG_k^\top)$. This identity is exact for the supplied moments $(m,P)$ and fixed, $\mathcal F_k$-measurable projection $G_k$, requiring only finite second moments.

5.3. Receding-horizon planning objective

Let $\mathbf u_c = \left( u_{c|c}, \ldots, u_{c+N-1|c} \right)$ be the set of candidate controls from which the planner chooses. The planner evaluates candidates through open-loop recursion. Define the planner’s compensated state belief $\bar x_{\ell+1|c} = f(\bar x_{\ell|c},u_{\ell|c})$ and corresponding correction-probabilities evolution as $q_{\ell|c} = \mathcal Q(\bar x_{\ell|c}, r_{\ell|c}, I_c)$. Through bias-prediction, the first and second moments propagate according to $(m_{\ell+1|c},P_{\ell+1|c}) = \mathcal B(m_{\ell|c},P_{\ell|c},u_{\ell|c},q_{\ell|c})$.

Define the estimator-risk cost given the selected control as $$J_E(\mathbf u_c) = \sum_{k=c}^N\mathcal R_E(m_{k|c},P_{k|c})$$(25) Candidate controls affect $J_E$ through both error flow and $q$

We define the optimal candidate selection as $$\mathbf u_c^\star = \arg\min_{\mathbf u_c\in\mathbb U_c} \left[ J_{\mathrm{nom}}(\mathbf u_c) + w_E J_E(\mathbf u_c) \right].$$(26) where $J_{\mathrm{nom}}$ prices the error-compensated mean state. The first segment of the optimal policy $u_c=u_{c|c}^\star$ is applied, and the replanning closes the loop at the next estimator report.

Figure 3 details and data

Compare candidate commands at a shared rollout time

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5.4. Cross-entropy optimization

We address this planning problem using a Cross-Entropy-based receding-horizon formulation [18], [19].

When the estimator outputs information $I_k$, error moments $(m_c, P_c)$ are initialized, and the planner compensates the estimates according to (4). Candidate command sequences are then sampled from a distribution. For each command tape, physical nominal dynamics and estimator-error moments are propagated, with which $\mathcal Q$, $G_k$, $M$, and $\mathcal R_\mathrm{E}$ are evaluated. The objective of (26) is then evaluated, and the minimum-cost trajectory candidate is selected. The first command segment of this trajectory is applied, the candidate distribution is shifted for the next sampling pass, and the algorithm is repeated from the next public report.

6.1. Initialization and mean-error identification

At update $k$, let $\nu_k$ be the estimator’s public innovation. Initializing $m_c$ from a causal affine map of the most recent $L$ innovations available at planning time $c$, the coefficients $a_0$ and $A_i^\nu$ are fit offline on development data, to satisfy the equation: $$m_c = a_0 + \sum_{i=0}^{L-1} A_i^\nu \nu_{c-i},$$(27)

Define the between-correction rate-of-change of the estimator error

$$\Delta_k \triangleq \frac{ e_{k+1}^- - e_k^+ }{\delta}.$$(28)

as the regression target for parameter fitting, excluding correction jumps from the fit. These correction jumps are handled separately by $J$, $W$, and $S^c$.

The rate-model coefficients are fit to this target accordingly:

$$\Delta_k = A e_k^+ + b + B\phi_k + \Lambda_z z_k + \varepsilon_k^{\mathrm{fit}},$$(29) which directly implies (5) under forward Euler. In particular, $Ae_k^+$ is the model dependence on current estimator error, $b$ is the autonomous drift coefficient, $B\phi_k$ is the dependence predictable from planner-available features (e.g. command-related and recency-related features), $\Lambda_zz_k$ is the temporally correlated residual structure, and $\epsilon_k$ is the remaining, unexplained fitting residual.

6.2. Residual covariance

After fitting the mean error dynamics model, we form the one-step residual $\widehat\eta_k = e_{k+1}^- - F e_k^+ - \delta \left( b+B\phi_k+\Lambda_z z_k \right),$ by subtracting the model’s predicted error increment from the observed between-correction transition. The recency covariance terms are fit to the covariance of this residual, modeled by (6). The $Q^{(i)}$ terms are fit using between-correction transitions from development data, targeting the empirical second moments $\hat \eta_k \hat\eta_k^\top$ and regressors $1$, $(1-r_{j,k})$, and $(1-r_{j,k})^2$ subject to positive-semidefiniteness constraints.

6.3. Correction and acceptance identification

For accepted-correction events, $J_k$ is taken directly from the estimator’s public update map when available. Otherwise, it is identified from paired pre- and post- correction errors by weighted least squares $\widehat J = \arg\min_J \sum_{k} \left| e_k^{\mathrm{acc}}-Je_k^- \right|_2^2$. Using $\widehat\epsilon_k^c = e_k^{\mathrm{acc}} - \widehat J_k e_k^-,$ the correction-noise covariance and pre-correction cross-covariance are estimated from the residual second moments. These moments are estimated within the same operating regimes used by the selected error model.

The parameters of the correction-probability mapping $\mathcal Q$ are fit from the binary labels $a_k$ using the candidate-correction samples $\widehat\theta_q = \arg\min_{\theta_q} - \sum_{k} [ a_k\log q_k(\theta_q) + (1-a_k)\log\left(1-q_k(\theta_q)\right)]$. Known cadence or geometric components of $\mathcal Q$ are supplied analytically. Only unidentified components are estimated from data.

7.1. Shipboard task, dynamics, sensing, and success

Define $W$, $B$, and $D$ as the world, VTOL-body, and deck frames, respectively, and let $R_{AB}$ map coordinates from frame $B$ into frame $A$. The deck-relative state is defined by $$\begin{aligned} p &=R_{DW}(p_B^W-p_D^W), & R &=R_{DW}R_{WB}, \\ v &=R_{DW}(v_B^W-v_D^W)-\omega_{WD}^{D}\times p, & \omega &=R\omega_{WB}^{B}-\omega_{WD}^{D}. \end{aligned}$$(30) Thus, $v=\dot p$ includes the rotating-deck-frame contribution. We denote the resulting relative state by $\zeta_D=(p,R,v,\omega)$.

The aircraft state is $x_A=(p_B^W,q_{WB},v_B^W,\omega_{WB}^{B},\theta,T)\in\mathbb R^{20}$, where $\theta\in\mathbb R^4$ contains the joint positions and $T\in\mathbb R^3$ the thrust states. Its dynamics are $\dot x_A=f_{T1}(x_A,u^{\mathrm{act}})$, with four commanded joint rates and three throttle inputs; the thrust states have time constant $\tau_T=0.040\,\mathrm{s}$.

The planner defines the task-relative coordinates using the deck-relative coordinates above, with $$y= \begin{bmatrix} p-p_*\\ \operatorname{Log}(RR_*^\top)\\ v-v_*\\ \omega \end{bmatrix}, \quad R_*=I.$$(31) The planner-facing estimator-error coordinates use the corresponding 12-dimensional local error $e = [\delta p^\top, \delta \phi^\top, \delta v^\top, \delta\omega^\top]^\top$ where $\delta\phi = \operatorname{Log}(R\hat R^\top)$.

Simulation terminates at first physical contact between the aircraft and the ship, or planner-declared abort. Success is then evaluated using the declared acceptable landing footprint and relative landing attitude, velocities, and angular rates.

7.2. Evaluated estimators and planners

We consider three estimator architectures: E1, a truth-driven controlled-error emulator, providing a biased truth-driven estimator to the planner, for mechanism validation, E2, an IMU-driven error-state EKF with aircraft/deck navigation, gyroscope and accelerometer biases, ignorant of commands, and E3, a two-mode Gaussian sum of the inertial EKFs, for which public covariance includes between-mode spread. These estimator backends differ in how they obtain their error response coefficients, with closed-form derivation for E1 and offline fitting for the filters. Anchors are initialized using a learned affine map of the four most recent causal 12D innovations, with separately fitted residual covariance $P_0$.

We additionally evaluate three planner variants: P1, our formulation, which compensates for predicted estimator error and penalizes an estimator risk metric, P2, a traditional belief-space planner for which covariance trace is the only penalized uncertainty term, and P3, a nominal-only planner. The planners are tuned identically, differing only in additional cost function terms.

7.3. Predictive error model identification and evaluation

We first evaluate whether the estimator-error model predicts error unavailable to a zero-mean model, and then whether its predicted moments are calibrated against subsequently realized estimator error.

The identified contracts exhibit strongly horizon and operating regime-dependent behavior. We therefore report prediction error by forecast horizon and trajectory regime rather than relying only on pooled loss, and discuss one of the qualified E2 error models as an illustrative case. In this model, the task-scaled mean-prediction loss was 8.16% lower than the zero-mean predictor when pooled over two experimental datasets. However, this improvement was not uniform with horizon. Over evaluation horizons of 0.25, 0.5, 1.0, and 2.0 s, the error reductions were 1.61%, 7.61%, 12.58%, and 6.39% respectively. Through the closed-loop experiments, E1 exhibited similarly horizon-dependent error dynamics to those of the identified E2 and E3 models. Along horizons of less than 0.25s, the E1 model reduced squared task-scaled prediction loss, relative to a zero-mean baseline, by 75.99%, which dropped to a 17.98% reduction between 0.25 and 0.5s, before an increase to 76.39% reduction for horizons longer than 2s.

Prediction quality also depended strongly on trajectory regime. In a separate E2 evaluation, the selected model reduced pooled prediction loss by 2.33%, but this aggregate improvement was concentrated in an operating regime with large velocities and task-relative offsets, where loss decreased by 5.52%. The same model performed worse than the zero-mean predictor during near and intermediate-range operating regimes, by 3.0% and 1.01% respectively. Hence, an error model that appears beneficial when averaged across a task distribution will not necessarily improve prediction in every portion of the maneuver. This regime dependence is particularly relevant in receding-horizon planning, because the distribution visited by the planner changes systematically as the vehicle progresses from approach to terminal alignment. These experiments show that estimator-error prediction is neither globally stationary nor characterized adequately by a single pooled accuracy or calibration metric. Predictive abilities can strengthen or reverse with horizon, transfer unevenly between regimes, and coexist with poor covariance calibration. We therefore interpret the planner-conditioned error model as a qualified local predictive model, whose claimed validity is tied to the trajectory regimes, forecast horizons, and source populations on which its mean and covariance has been evaluated.

7.4. Closed-loop landing performance

Figure 4 details and data

Replay one matched landing trial

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Across 16 matched trial groups using E1, P1 achieved 8/16 physical captures, compared with 6/16 for both P2 and P3. In the paired P1-P2 comparison, P1 was uniquely successful on five trials while P2 was uniquely successful on three. Notably, P1 obtained lower component-level capture gauges than P2 in pitch and pitch rate in 11/16 trials and in longitudinal velocity in 12/16 trials. Terminal task gauge also favored planner P1, with median gauges of 0.99, 1.60, and 1.87 for P1, P2, and P3 respectively. The corresponding mean gauges were 3.69, 9.71, and 3.57, respectively, although the P2 performance mean was strongly influenced by an isolated extreme failure. Thus, the matched trials show a modest observed capture advantage and improved typical terminal task state for P1, while the available cohort remains too small to establish a difference in capture probability. Beyond 2s, P2 underpredicted realized estimator error by a factor of 5.73, whereas P1 produced an observed-to-predicted ratio of 1.31.