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arxiv: 2605.01860 · v1 · submitted 2026-05-03 · 💻 cs.RO

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Optimizing Trajectory-Trees in Belief Space: An Application from Model Predictive Control to Task and Motion Planning

Camille Phiquepal , Marc Toussaint
Authors on Pith no claims yet

Pith reviewed 2026-05-10 14:44 UTC · model grok-4.3

classification 💻 cs.RO
keywords trajectory treesbelief space planningmodel predictive controltask and motion planningpartially observable systemsrobotic planning under uncertainty
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The pith

Trajectory-trees optimized in belief space capture observation-dependent contingencies that sequential paths miss.

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

The paper shows that planning arborescent trajectories instead of single sequences lets a robot branch its plans where beliefs are expected to split due to future observations. This dependency is handled directly by optimizing the tree in belief space rather than assuming one fixed future. The approach is demonstrated first on model predictive control for driving, where a single-branch tree lowers costs, and then extended to task and motion planning by layering decision trees at the task level with trajectory trees at the motion level. Real-time feasibility comes from a distributed optimization method that exploits the tree's decomposability. Readers care because many robotic tasks involve hidden states or uncertain observations where the best action depends on what will be learned later.

Core claim

Computing arborescent trajectories (trajectory-trees) instead of sequential trajectories for partially observable robotic planning problems allows the optimal course of action to depend on future observations by branching where the belief state evolves into multiple distinct scenarios. In MPC this is realized as PO-MPC with a single branching point, solved via the Distributed Augmented Lagrangian algorithm for real-time use on linear and nonlinear driving examples. In TAMP the PO-LGP planner combines logic-geometric programming with task-level decision trees and motion-level trajectory trees, scaling to larger problems by treating optimized explorative policies as macro-actions.

What carries the argument

Trajectory-trees (arborescent trajectories) in belief space, which branch at predicted observation points to model multiple contingencies instead of one forward evolution.

If this is right

  • A tree with one branching point already reduces control cost relative to a single trajectory by incorporating the value of future information.
  • The decomposable structure of the tree formulation permits parallel solving via Distributed Augmented Lagrangian, meeting MPC timing limits.
  • At the task level, decision trees combined with motion-level trajectory trees extend logic-geometric programming to partially observable settings.
  • Explorative policies optimized as macro-actions let the approach scale beyond very small belief spaces.

Where Pith is reading between the lines

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

  • The same branching structure could be applied to multi-agent settings where each agent maintains beliefs about the others.
  • Replacing hand-crafted branching predictions with learned models of observation likelihoods would reduce the manual modeling burden.
  • In deployed systems the tree representation naturally supplies contingency plans that can be swapped in when an unexpected observation arrives.

Load-bearing premise

Belief-state evolution and branching points can be predicted accurately enough in advance that the resulting tree optimization stays tractable under real-time constraints.

What would settle it

In the autonomous driving MPC experiments, run the sequential-trajectory baseline and the single-branch tree optimizer on identical noisy-observation trials and check whether the tree version produces measurably lower total control cost or fails to finish within the MPC time budget.

Figures

Figures reproduced from arXiv: 2605.01860 by Camille Phiquepal, Marc Toussaint.

Figure 1
Figure 1. Figure 1: Example of partially observable TAMP problem: The blocks’ colors are initially not visible. The robot must look (b) and react to observations to reach the goal state (c), defined by a given color order. for several initial block configurations; the blocks’ colors need to be observed. In [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example of partially observable MPC Problem: a pedestrian is detected (a), whose intention is uncertain. The control policy must account for 2 cases: The pedestrian may walk along the street (b) or cross (c). Conversely, ignoring the eventuality that a pedestrian might cross is unsafe. To overcome those limitations, we consider multiple modalities or state hypotheses, and compute trajectory-trees in belief… view at source ↗
Figure 3
Figure 3. Figure 3: Branching point: The trajectory-tree branches where the belief state evolution is anticipated to diverge into multiple possible outcomes. Optimizing trajectory-trees has different implications depending on the use-case: In Model Predictive Control (MPC), a key requirement is that optimization must be sufficiently rapid to be executed in real time. To maintain tractability, we adopt a trajectory￾tree struct… view at source ↗
Figure 4
Figure 4. Figure 4: Illustrative example of trajectory-tree in belief space: The trajectory stages (in black) represent the system evolution under an applied control u. The belief update stages (in blue) correspond to probabilistic branching of the belief state. From a geometrical persective, the motion forms a trajectory-tree. Trajectory-trees represent the evolution of the hybrid belief state with probabilistic branching oc… view at source ↗
Figure 5
Figure 5. Figure 5: Example of tree structures: In PO-MPC 5a the trajectory-tree assumes full observability after the first branching (similarly to Q-MDP). In PO-LGP for TAMP 5b, Look actions provide observations, resulting in a belief state update. In the following, we use the term continuous trajectory￾tree to denote the continuous motions that compose a trajectory-tree. When the context is clear, we may refer to the contin… view at source ↗
Figure 7
Figure 7. Figure 7: ) which is the part executed by the controller until the next planning cycle happens. This trunk spans a time interval which we call the branching horizon. Beyond the branching horizon, the trajectory-tree evolves into |H| branches, each corresponding directly to a specific state hypothesis m ∈ H. The full trajectory-tree spans a total time interval, which we refer to as the prediction horizon, following s… view at source ↗
Figure 6
Figure 6. Figure 6: PO-MPC Control loop: trajectory-trees are optimized with respect to a belief distribution over multiple hypotheses provided by a perception module. Belief space inference is therefore decoupled from planning. At each planning cycle, a trajectory-tree is optimized based on the current belief state estimate. Such architecture in which perception components output multiple hypotheses or modalities, each assoc… view at source ↗
Figure 8
Figure 8. Figure 8: Decomposition into subproblems: each branch defines an optimization subproblem. The controls before the branching horizon are shared across branches, inducing coupling. variable z˜. Consequently, the subproblems cannot be optimized independently. However, this coupling is weak, as the non-anticipativity constraint only applies over the branching horizon, which is a small portion of the full trajectory-tree… view at source ↗
Figure 9
Figure 9. Figure 9: Execution flow for |H| = 2. The costly steps (Newton minimizations) are parallelized. the gradients. As [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Example of trajectory-tree when braking: a.1, a.2, and a.3 anticipate that a pedestrian crosses. a.4 corresponds to the free road scenario. c. is obtained with single hypothesis MPC assuming that the closest pedestrian crosses. in the worst case (see red curve in [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Influence of the belief state on the braking within the branching horizon: Low crossing probabilities (e.g. a/, b/) lead to a more optimistic trajectory-tree. § [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Avoidance of 2 uncertain obstacles: Each trajectory corresponds to a different combination of object presences. State space and kinematics: Optimization is performed in SE(2). No slippage is assumed, this is enforced by the following non-holonomic constraint: x˙(t)cos(θ) − y˙(t) sin(θ) = 0, ∀t, (9a) where x, y, θ is the vehicle pose. Partially observable discrete state: The discrete states represent the … view at source ↗
Figure 13
Figure 13. Figure 13: Decomposed (D-AuLa) vs. undecomposed (AuLa) optimization: D-AuLa scales exponentially better in the pedestrian case 13a. In the slalom case 13b, it is faster by a factor 2.75. D-AuLa scales nearly linearly with respect to the number of branches. This can be understood easily; one optimizes as many subproblems as branches in the tree, but the size of each subproblem does not change. In the pedestrian examp… view at source ↗
Figure 14
Figure 14. Figure 14: Example of hypotheses: The continuous part represents the normal vector of the colored face. It corresponds to a logical fact in the symbolic part indicating which side is colored. The symbolic state also contains the blocks’ colors. There are 72 hypotheses (asuming each side can be the colored side). • S is a finite set of symbolic states • X is the continuous state space • A is a finite set of symbolic … view at source ↗
Figure 15
Figure 15. Figure 15: Decision tree for the block stacking problem with 2 blocks: The square nodes are action nodes, circular nodes are observation nodes. At the node (4) the color of the block is observed leading to two contingencies. The node (14) in green fulfills the goal condition. The nodes in gray are not yet expanded. Decision tree vs. decision graph: Some nodes in the decision tree may be symbolically equivalent like … view at source ↗
Figure 16
Figure 16. Figure 16: Candicate policy for the stacking problem with 2 blocks: A policy is a subset of the decision tree 15. Observation nodes with only one outcoming edge are omitted for clarity. Candidate policies form the symbolic part of a trajectory￾tree as defined in Section 3. The procedure for extracting candidate policies from the decision tree is detailed in Section 5.2.2. 5.1.4 Trajectory-Trees The candidate policie… view at source ↗
Figure 17
Figure 17. Figure 17: TAMP solver: After the decision tree expansion, candidate policies are generated using dynamic programming and are optimized piecewise to inform about the actual action costs. The final trajectory-tree is re-optimized jointly. of actions defines an optimization problem on a trajectory￾tree. We use ψ to denote a continuous trajectory-tree. The cost and constraints functions are defined by the symbolic acti… view at source ↗
Figure 18
Figure 18. Figure 18: Illustration of the structure implied by the T-KOMO formulation. c, g and h represent 2 nd order functions applying on 3 consecutive configurations. formulation (11) for a fixed candidate policy π and over a discretized representation of the trajectory-tree. The cost terms ci(ψi−k:i) are weighted by the probability to reach a given configuration of the tree, which leads to trajetory-trees more optimized t… view at source ↗
Figure 19
Figure 19. Figure 19: Comparison of Hessian matrices: When optimizing a trajectory-tree, the Hessian (a) is not banded-symmetric, unlike in the sequential trajectory case (b). This reflects the branching illustrated in [PITH_FULL_IMAGE:figures/full_fig_p020_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Example of Look action: The robot is incentivized to place its sensor at a distance ddesired from the object to observe. The angle w.r.t. the surface should remain smaller than αmax (14a) and the side center shall be centered in the sensor’s field of view (14b). on the relative pose between the sensor and the object to observe. It does not determine, by itself, if the sensor or the object should be moved … view at source ↗
Figure 21
Figure 21. Figure 21: Policies obtained for the problems Baxter-A, Baxter-B and Franka-A. (a) Start of Franka-C×A’ (b) Start pose for explorative policy (Franka-A’) (c) Look action detecting no color (d) Look action identifying the block (e) End of Franka-A, block is identified and re-arranged (f) Goal state of Franka A×C [PITH_FULL_IMAGE:figures/full_fig_p022_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Examples of configurations for Franka C×A’: Blocks colors are unknown at the start 22a. Blocks are brought to a fixed position 22b. From there, the explorative policy Franka-A’ observes the sides 22c, 22d and rearranges the block 22e. The goal state is 22f. all contingencies. Planning for each variation is carried out five times and we report on the average planning times and iterations [PITH_FULL_IMAGE:… view at source ↗
Figure 23
Figure 23. Figure 23: Evolution of the hypothesized cost of the candidate policies: Over the iterations, the cost estimate of the candidate policies is refined by the result of the piecewise optimization. Some actions may be infeasible leading to infinite costs, such that any choice of c0 still allows the algorithm to iterate enough to generate candidate policies circumventing infeasible actions, within the limits of decision … view at source ↗
Figure 24
Figure 24. Figure 24: Influence of c0 on the trajectory cost of π ⋆ and the total number of iterations: A low c0 (optimistic) leads to better trajectory-trees at the expense of the number of iterations. For the problem Franka-A, optimal policies are found when c0 is below 1.0. 5.3.4 Influence of Motion Planning Failures In the problem Baxter-D the logic definition is relaxed to make the Look actions available also when the rob… view at source ↗
Figure 26
Figure 26. Figure 26: Illustration of the optimization decomposition schemes With the D-AuLa solver, the joint optimization can be decomposed into sub-optimization problems, as shown in [PITH_FULL_IMAGE:figures/full_fig_p024_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: shows the planning times observed with variations of the Baxter problems. The measurements for the belief state sizes of one, two, and six are obtained with the problems Baxter-A, Baxter-B and Baxter-C from the Table. 3. Additional data points for belief state sizes of three, four, and five were measured by removing hypotheses from the initial belief state of the Baxter-C problem, thereby enabling a finer… view at source ↗
Figure 28
Figure 28. Figure 28: Geometrical View of the KKT conditions: The constraint gradients weighted by the Lagrange multipliers cancel out the gradient of the cost function c. The green area indicates where the inequality constraints g is satisfied, while the equality constraints h is satisfied along the blue contour. A.2 Consensus Form of the Alternating Direction Method of Multipliers (ADMM) The Consensus ADMM algorithm solves o… view at source ↗
Figure 29
Figure 29. Figure 29: Policy for problem Baxter-C. With 3 unknown blocks, the robot must observe 2 times, resulting in 6 possible contingencies. C.2 Planned Policy for Franka-C×A’ 0 1 grasp block_2 2 red 2.1 2.2 green 2.3 blue 3 place on tableR) 16 place on tableR 33 place on tableR 5 green 5.1 5.2 blue 6 place on block_2 10 place on tableR 18 red 18.1 18.2 blue 19 place on tableR 25 place on tableR 35 35.1 red 35.2 green 36 p… view at source ↗
Figure 30
Figure 30. Figure 30: Policy for problem Franka-C×A’. The high level policy on the top of the image contains macro-actions (indicated with a bold circle). The macro-actions expand into low level exploration policies with 5 branching points. Prepared using sagej.cls [PITH_FULL_IMAGE:figures/full_fig_p041_30.png] view at source ↗
read the original abstract

This paper explores the benefits of computing arborescent trajectories (trajectory-trees) instead of commonly used sequential trajectories for partially observable robotic planning problems. In such environments, a robot infers knowledge from observations, and the optimal course of action depends on these observations. \revise{Trajectory-trees, optimized in belief space, naturally capture this dependency by branching where the belief state is expected to evolve into multiple distinct scenarios, such as upon receiving an observation. Unlike sequential trajectories, which model a single forward evolution of the system, trajectory-trees capture multiple possible contingencies.} First, we focus on Model Predictive Control (MPC) and demonstrate the benefits of planning tree-like trajectories. We formulate the control problem as the optimization of a tree with a single branching (PO-MPC). This improves performance by reducing control costs through more informed planning. To satisfy the real-time constraints of MPC, we develop an optimization algorithm called Distributed Augmented Lagrangian (D-AuLa), which leverages the decomposability of the PO-MPC formulation to parallelize and accelerate the optimization. We apply the method to both linear and non-linear MPC problems using autonomous driving examples. Second, we address Task And Motion Planning (TAMP), and introduce a planner (PO-LGP) reasoning on decision trees at task level, and trajectory-trees at motion-planning level. This approach builds upon the Logic-Geometric-Programming Framework (LGP) and extends it to partially observable problems. The experiments show the method's applicability to problems with a small belief state size, and scales to larger problems by optimizing explorative policies, which are used as macro-actions in an overarching task plan.

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

3 major / 3 minor

Summary. The paper claims that optimizing trajectory-trees in belief space outperforms sequential trajectories for partially observable robotic planning by explicitly capturing observation-dependent contingencies. It introduces PO-MPC, a single-branch tree formulation for model predictive control solved in real time via the Distributed Augmented Lagrangian (D-AuLa) method and demonstrated on linear/non-linear autonomous driving examples; it also presents PO-LGP, which extends the Logic-Geometric-Programming (LGP) framework to partially observable TAMP by combining task-level decision trees with motion-level trajectory-trees and using pre-optimized explorative policies as macro-actions to scale to larger belief spaces.

Significance. If the experimental claims hold and the fixed branching structures prove robust, the work would offer a practical bridge between receding-horizon MPC and high-level TAMP under uncertainty, with the D-AuLa parallelization providing a concrete algorithmic advance for real-time belief-space optimization. The extension of LGP to PO settings and the use of macro-actions for scalability are potentially valuable contributions to the robotics planning literature.

major comments (3)
  1. [Abstract / PO-MPC section] Abstract and PO-MPC formulation: the central claim that 'trajectory-trees, optimized in belief space, naturally capture this dependency by branching where the belief state is expected to evolve into multiple distinct scenarios' rests on the pre-chosen branching points remaining representative online. With only a single branching point used in PO-MPC, the formulation does not address how the tree handles deviations when actual observations produce belief trajectories outside the predicted branch (common in noisy non-linear driving); this structural assumption is load-bearing and requires either a robustness analysis or explicit comparison against receding-horizon sequential MPC under mismatched observations.
  2. [PO-LGP and TAMP experiments] PO-LGP description and experiments: the approach relies on pre-optimized explorative policies as macro-actions for larger belief spaces. If real observations cause the belief state to evolve differently from the precomputed branches, the overarching task plan may lose its contingency benefits; the manuscript should include sensitivity experiments or bounds showing that the combined task-motion plan remains effective under such deviations, as this directly affects the scalability claim.
  3. [D-AuLa algorithm description] D-AuLa solver: while the method exploits decomposability for parallelization to meet MPC real-time constraints, the paper must clarify any approximations or convergence guarantees when the augmented Lagrangian is applied to the non-convex belief-space tree optimization; without this, it is difficult to assess whether the reported performance gains in the driving examples are due to the tree structure or to solver-specific relaxations.
minor comments (3)
  1. [Abstract] The abstract states that the method 'improves performance by reducing control costs' but does not preview any quantitative metrics (e.g., cost reduction percentages or timing); adding a brief results summary would strengthen the abstract.
  2. [Introduction / Preliminaries] Notation for belief states, branching points, and the distinction between decision trees and trajectory-trees should be introduced with a small illustrative figure early in the paper to aid readability.
  3. [Experiments] Ensure that all experimental figures clearly label the planned tree branches versus executed paths under actual observations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We address each major comment below and have revised the manuscript to strengthen the presentation and address the concerns where possible.

read point-by-point responses

  1. Referee: [Abstract / PO-MPC section] Abstract and PO-MPC formulation: the central claim that 'trajectory-trees, optimized in belief space, naturally capture this dependency by branching where the belief state is expected to evolve into multiple distinct scenarios' rests on the pre-chosen branching points remaining representative online. With only a single branching point used in PO-MPC, the formulation does not address how the tree handles deviations when actual observations produce belief trajectories outside the predicted branch (common in noisy non-linear driving); this structural assumption is load-bearing and requires either a robustness analysis or explicit comparison against receding-horizon sequential MPC under mismatched observations.

    Authors: We agree that the representativeness of the chosen branching point is important. In the PO-MPC formulation, the tree is re-optimized at every receding-horizon step using the updated belief, which inherently adapts to observed deviations within the horizon. The single branching point is placed at the time step where the belief is predicted to split most significantly. To directly address the concern about mismatched observations, we have added an explicit comparison in the experiments section against receding-horizon sequential MPC under increased observation noise in the non-linear driving example, demonstrating that the tree structure still yields lower costs by planning for contingencies. We have also expanded the discussion on branching-point selection. revision: yes

  2. Referee: [PO-LGP and TAMP experiments] PO-LGP description and experiments: the approach relies on pre-optimized explorative policies as macro-actions for larger belief spaces. If real observations cause the belief state to evolve differently from the precomputed branches, the overarching task plan may lose its contingency benefits; the manuscript should include sensitivity experiments or bounds showing that the combined task-motion plan remains effective under such deviations, as this directly affects the scalability claim.

    Authors: We acknowledge that deviations in belief evolution could affect the contingency benefits. The explorative policies are designed to reduce uncertainty across likely scenarios, and the task-level decision tree permits higher-level replanning. To strengthen the scalability claim, we have added sensitivity experiments in the revised PO-LGP section by perturbing the observation model parameters and reporting success rates and plan costs, showing that the macro-action approach maintains effectiveness for the tested belief-space sizes. revision: yes

  3. Referee: [D-AuLa algorithm description] D-AuLa solver: while the method exploits decomposability for parallelization to meet MPC real-time constraints, the paper must clarify any approximations or convergence guarantees when the augmented Lagrangian is applied to the non-convex belief-space tree optimization; without this, it is difficult to assess whether the reported performance gains in the driving examples are due to the tree structure or to solver-specific relaxations.

    Authors: We thank the referee for this point on solver properties. D-AuLa decomposes the tree optimization into parallel subproblems using the augmented Lagrangian, with consensus constraints enforced across branches. While the underlying problem is non-convex and we do not claim global convergence, the method uses local optimality with warm-starting from prior MPC iterations. We have added a dedicated paragraph in the D-AuLa section clarifying these aspects, including the practical convergence behavior observed and a note that performance improvements are validated by direct comparison to a sequential baseline solved with the same optimizer, isolating the contribution of the tree structure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces PO-MPC (single-branch tree optimization with D-AuLa solver) and PO-LGP (extending LGP to belief-space trees with explorative policies as macro-actions). These are new algorithmic formulations applied to driving and TAMP examples. The claim that trees capture contingencies by branching at predicted belief scenarios follows directly from the stated optimization objective and tree structure definition, without any quoted reduction of outputs to fitted inputs or self-defined quantities by construction. Prior LGP citation provides the base framework but is not load-bearing for the partial-observability extension or the reported performance gains.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no concrete free parameters, axioms, or invented entities can be extracted; the work appears to rely on standard assumptions from belief-space planning and optimization literature.

pith-pipeline@v0.9.0 · 5605 in / 1114 out tokens · 33542 ms · 2026-05-10T14:44:33.961261+00:00 · methodology

discussion (0)

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Reference graph

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