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arxiv: 2605.09046 · v2 · submitted 2026-05-09 · 💻 cs.RO

Terminal Matters: Kinodynamic Planning with a Terminal Cost and Learned Uncertainty in Belief State-Cost Space

Zhuoyun Zhong , Seyedali Golestaneh , Constantinos Chamzas This is my paper

Pith reviewed 2026-05-15 05:56 UTC · model grok-4.3

classification 💻 cs.RO
keywords kinodynamic planningterminal costbelief spaceWasserstein distanceasymptotic optimalitylearned dynamicssampling-based planninguncertainty-aware planning
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The pith

A terminal cost in kinodynamic planning lets robots optimize goal quality and reliability without losing asymptotic optimality guarantees.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces an augmented objective for sampling-based kinodynamic planners that adds a terminal cost to the usual accumulated path cost. This change lets the planner directly improve the quality of the final state, such as how preferable or reliable the arrival is. The authors prove that AO-RRT keeps its asymptotic optimality property under the new objective. In belief space they replace the terminal term with the Wasserstein distance between the current belief and the goal belief, and prove this raises a lower bound on the probability of actually reaching the goal region. The resulting KiTe planner learns dynamics and uncertainty models from data and shows higher success rates on simulated tasks and real planar pushing.

Core claim

We introduce a terminal-cost formulation for kinodynamic planning that allows terminal-state quality to be optimized alongside accumulated trajectory cost. We prove that AO-RRT preserves its asymptotic optimality under this augmented objective. We further extend the formulation to belief space and prove that minimizing the Wasserstein distance between the terminal belief and the goal improves a lower bound on the probability of reaching the goal region. The KiTe planner implements this terminal-cost objective with learned belief dynamics to encode goal preferences and improve reliability under uncertainty.

What carries the argument

The augmented objective that sums accumulated trajectory cost with a terminal cost term, where the terminal term is either a direct quality measure or the Wasserstein distance to the goal belief distribution.

If this is right

  • AO-RRT retains asymptotic optimality when the objective includes a terminal cost.
  • In belief space, Wasserstein-distance minimization tightens the lower bound on goal-reaching probability.
  • Learned dynamics and uncertainty models allow the planner to operate on systems that lack closed-form uncertainty descriptions.
  • Experiments on Flappy Bird, car parking, and planar pushing demonstrate consistently higher success rates under uncertainty, including in real-world pushing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Goal preferences can be directly encoded by shaping the terminal cost to favor specific arrival states.
  • Optimizing against modeled uncertainty may reduce the performance drop when transferring plans from simulation to hardware.
  • The same terminal-cost structure could be grafted onto other asymptotically optimal sampling-based planners.

Load-bearing premise

The learned dynamics and process uncertainty models must accurately represent the true system behavior during actual deployment.

What would settle it

Run KiTe on a physical robot whose true dynamics deviate from the learned model and measure whether the observed goal-reaching success rate matches the improvement predicted by the Wasserstein lower bound.

Figures

Figures reproduced from arXiv: 2605.09046 by Constantinos Chamzas, Seyedali Golestaneh, Zhuoyun Zhong.

Figure 3
Figure 3. Figure 3: Illustrations of AO-RRT Lemmas and propagate it into the next ball (Lemma 2). Repeating this argument for all balls yields a positive probability of reaching the final ball, thereby producing a trajectory whose total cost is arbitrarily close to the reference one (Lemma 3). We then state the main theorem. Theorem 1. (Asymptotic Optimality): Assume that the system dynamics, running cost, and terminal cost a… view at source ↗
Figure 16
Figure 16. Figure 16: Illustration for proof of Lemma 2. Proof. As illustrated in [PITH_FULL_IMAGE:figures/full_fig_p016_16.png] view at source ↗
Figure 15
Figure 15. Figure 15: Illustration for proof of Lemma 1. Proof. As shown in [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗
Figure 17
Figure 17. Figure 17: Illustration for proof of Lemma 3. Proof. Recall the distance in Y is defined by the L2 norm (Eq. 9). Thus, As shown in [PITH_FULL_IMAGE:figures/full_fig_p017_17.png] view at source ↗
read the original abstract

In many real-world robotic tasks, robots must generate dynamically feasible motions that reliably reach desired goals even under uncertainty. Yet existing sampling-based kinodynamic planners typically optimize accumulated trajectory costs and treat goal reaching as a feasibility check, rather than explicitly optimizing terminal-state quality, such as goal preference or goal-reaching reliability. In this work, we introduce a terminal-cost formulation for kinodynamic planning that allows terminal-state quality to be optimized alongside accumulated trajectory cost. We prove that AO-RRT, an asymptotically optimal kinodynamic planner, preserves its asymptotic optimality under this augmented objective. We further extend the formulation to belief space and prove that minimizing the Wasserstein distance between the terminal belief and the goal improves a lower bound on the probability of reaching the goal region. The resulting planner, KiTe, uses this terminal-cost objective to encode goal preferences and improve reliability under uncertainty. To support systems without analytical uncertainty models, we learn dynamics and process uncertainty directly from data and integrate the learned belief dynamics into planning. Experiments on Flappy Bird, Car Parking, and Planar Pushing show that KiTe consistently improves goal-reaching success under uncertainty. Real-world Planar Pushing experiments further demonstrate that KiTe can plan effectively with learned dynamics and uncertainty. Source code is available at https://github.com/elpis-lab/KiTe.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces a terminal-cost formulation for kinodynamic planning that augments accumulated trajectory cost with an explicit terminal-state quality term. It proves that AO-RRT preserves asymptotic optimality under the augmented objective, extends the approach to belief space where minimizing the Wasserstein distance between terminal belief and goal belief improves a lower bound on goal-reaching probability, learns dynamics and process uncertainty from data, and reports improved success rates on Flappy Bird, Car Parking, and planar pushing tasks (including real-world experiments).

Significance. If the stated proofs hold and the learned models are sufficiently accurate, the work supplies a principled mechanism for incorporating terminal preferences and uncertainty into asymptotically optimal sampling-based planners, with potential to increase reliability in stochastic robotic tasks. The open-source implementation is a positive factor for reproducibility.

major comments (2)
  1. [§4.2] §4.2 (belief-space extension): the proof that Wasserstein-distance minimization strictly raises the lower bound on P(reach goal) presupposes that the learned transition kernel and process noise exactly match the true stochastic dynamics; no forward-simulation error, calibration plots, or predicted-vs-empirical terminal-belief Wasserstein distances are reported for the planar-pushing experiments, leaving the real-world applicability of the bound unverified.
  2. [§3.1] §3.1, Eq. (7): the claim that AO-RRT remains asymptotically optimal with an additive terminal cost requires explicit verification that the terminal term does not alter the cost-to-go estimates or rewiring conditions in a way that violates the original optimality proof; the current sketch does not address how the terminal cost is propagated through the tree.
minor comments (3)
  1. [Table 1] Table 1 and Figure 4: success-rate improvements are reported without statistical significance tests or confidence intervals; adding these would strengthen the empirical claims.
  2. [Notation] Notation section: the belief-state cost space symbols (e.g., b, W, terminal cost) are introduced without a consolidated table, making cross-referencing cumbersome.
  3. [Related Work] Related-work paragraph: the discussion of prior belief-space RRT variants omits recent Wasserstein-based planning papers; a brief comparison would clarify novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's constructive comments. We address each major point below and indicate the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: [§4.2] §4.2 (belief-space extension): the proof that Wasserstein-distance minimization strictly raises the lower bound on P(reach goal) presupposes that the learned transition kernel and process noise exactly match the true stochastic dynamics; no forward-simulation error, calibration plots, or predicted-vs-empirical terminal-belief Wasserstein distances are reported for the planar-pushing experiments, leaving the real-world applicability of the bound unverified.

    Authors: We agree that the theoretical guarantee relies on the accuracy of the learned model. While the real-world experiments show improved performance, we did not include explicit model validation metrics. In the revision, we will add forward-simulation error analysis, calibration plots, and comparisons of predicted versus empirical terminal-belief Wasserstein distances for the planar pushing experiments to verify the model's fidelity and support the practical relevance of the bound. revision: partial

  2. Referee: [§3.1] §3.1, Eq. (7): the claim that AO-RRT remains asymptotically optimal with an additive terminal cost requires explicit verification that the terminal term does not alter the cost-to-go estimates or rewiring conditions in a way that violates the original optimality proof; the current sketch does not address how the terminal cost is propagated through the tree.

    Authors: The terminal cost is a state-dependent additive term applied only at the goal-reaching nodes. Because it is independent of the path taken to reach a particular state, it does not affect the relative ordering of path costs during rewiring or the cost-to-go estimates in the tree. The asymptotic optimality is preserved as the proof can be adapted by considering the augmented cost function, where the terminal term is fixed for each terminal state. We will revise §3.1 to provide a more detailed explanation of how the terminal cost is incorporated into the tree propagation and rewiring logic, including an explicit verification that the original proof structure holds. revision: yes

Circularity Check

0 steps flagged

No significant circularity; proofs are independent mathematical derivations

full rationale

The paper's load-bearing claims consist of two explicit proofs: (1) preservation of AO-RRT asymptotic optimality under an additive terminal cost, and (2) that minimizing Wasserstein distance between terminal belief and goal belief raises a lower bound on goal-reaching probability. These are presented as new derivations resting on standard sampling-based planning assumptions and the definition of Wasserstein distance; they do not reduce to fitted parameters, self-definitions, or prior self-citations by construction. The learned dynamics component is used only for implementation and is not part of the optimality or bound proofs. No self-citation is invoked as the sole justification for uniqueness or ansatz. The derivation chain is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard mathematical properties of sampling-based planners for the optimality proof and on properties of Wasserstein distance for the probability bound; these are domain assumptions rather than ad-hoc inventions.

axioms (2)
  • standard math Standard assumptions underlying asymptotic optimality of AO-RRT variants hold for the augmented terminal-cost objective
    Invoked in the proof that AO-RRT preserves optimality under the new objective.
  • domain assumption Wasserstein distance minimization between terminal belief and goal belief improves a lower bound on goal-reaching probability
    Central to the belief-space extension and stated as proven.

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