Adaptively Prolated Trees (APT*)

• Adaptively Prolated Trees (APT*) is a sampling-based optimal motion planner that replaces spherical neighbor regions with force-aligned prolate ellipsoids to enhance exploration and refinement.
• It refines predecessor methods by dynamically adjusting local neighborhood shapes and batch sizes based on obstacle information and informed-set contraction.
• Empirical results show that APT* achieves faster initial solutions and lower path costs compared to FIT* and FDIT*, especially in high-dimensional planning challenges.

Adaptively Prolated Trees (APT) is a single-query, sampling-based asymptotically optimal motion planner proposed as an extension of Force Direction Informed Trees (FDIT). Its defining idea is to replace the standard spherical local neighbor region around each tree vertex with a force-aligned prolate ellipsoid whose anisotropy depends on obstacle information and on the current planning phase, as inferred from the size of the informed set. In parallel, APT* uses an adaptive batch-size mechanism driven by informed-set hypervolume contraction, with the stated aim of improving both the speed of obtaining an initial feasible path and the rate of subsequent cost refinement (Zhang et al., 27 Aug 2025).

1. Planning formulation and placement within optimal motion planning

APT* is formulated for the standard optimal motion-planning problem in a continuous state space $X \subseteq \mathbb{R}^n$, with obstacle region $X_{\text{obs}}$, free space

$X_{\text{free}} = cl(X \setminus X_{\text{obs}}),$

start state $\mathbf{x}_{\text{start}}$, and goal region $X_{\text{goal}} \subset X_{\text{free}}$. A path is a continuous map $\sigma : [0,1] \mapsto X$, and the objective is to compute a minimum-cost collision-free path

$\begin{split} \sigma^* &= \arg \min_{\sigma \in \Sigma} \left\{ c(\sigma) \middle| \sigma(0) = \mathbf{x}_{\text{start}}, \sigma(1) \in \mathbf{x}_{\text{goal}}, \right. \ &\qquad\qquad \left. \forall t \in [0, 1], \sigma(t) \in X_{\text{free}} \right\}. \end{split}$

Within sampling-based optimal planning, APT* belongs to the family of planners that build or search implicit random geometric graphs, alongside RRT, BIT, AIT, EIT, FIT, and FDIT. It is especially close to informed and batch-based methods: once a feasible solution exists, it restricts sampling to the informed subset induced by the current incumbent cost, and it allocates sampling effort through a batch-size mechanism that changes during planning. Its contribution is therefore presented not as a new planning paradigm, but as a refinement of local neighborhood definition and of sample allocation over time (Zhang et al., 27 Aug 2025).

The paper emphasizes two intended operating regimes. Before a solution is found, the planner behaves like an informed method with effectively full-domain sampling. After a solution is found, it exploits the informed subset $X_{\hat f}$, whose geometry is a prolate hyperellipsoid between start and goal. The shrinkage of this informed set becomes the central signal for both adaptive batching and the modulation of local anisotropy.

2. Lineage, predecessor methods, and terminological clarification

The most direct predecessor of APT* is FDIT*. FDIT* already treated vertices as electric charges and used a Coulomb-like virtual force to define an elliptical nearest-neighbor region instead of an isotropic Euclidean one. APT* argues that the effective amount of prolation should vary over the course of planning: too much prolation too early may reduce exploration and commit to local structure prematurely, whereas too little prolation too late wastes problem-specific information and slows optimization. On that basis, APT* adds two coupled modules on top of FDIT*: adaptive batch sizing derived from informed-set hypervolume, and nonlinear charge modulation so that force magnitude and ellipsoidal eccentricity change during planning (Zhang et al., 27 Aug 2025).

The method also inherits specific design elements from earlier informed planners. From Informed RRT*, it inherits restriction of sampling to an admissible ellipsoidal informed subset once a solution exists. From BIT*, it inherits batch-wise informed graph construction and search. From AIT* and EIT*, it inherits adaptive informed-search ideas. From FIT*, it inherits adaptive batch-size planning, which APT* extends and couples to neighborhood geometry. Experimentally, the baselines are RRT-Connect, Informed RRT*, BIT*, AIT*, EIT*, FIT*, and FDIT*.

A recurrent source of confusion is nomenclature. “APT*” in this context denotes “Adaptively Prolated Trees,” not “Adaptively Informed Trees.” The paper "Adaptively Informed Trees (AIT*): Fast Asymptotically Optimal Path Planning through Adaptive Heuristics" contains no planner named APT* and addresses a different mechanism based on adaptive heuristics and asymmetric bidirectional search (Strub et al., 2020).

3. Algorithmic pipeline

APT* starts from a standard optimal-planning initialization: the tree is initialized with $\mathbf{x}_{\text{start}}$, the current best solution cost is set to infinity until a path is found, a sample set $X_{\text{samples}}$ is maintained, and planner parameters are set, including the RGG normalization constant $X_{\text{obs}}$0, minimum and maximum batch sizes, rewiring factor, charge range $X_{\text{obs}}$1, and nonlinearity constants (Zhang et al., 27 Aug 2025).

Sampling and search are then organized around the informed set $X_{\text{obs}}$2. The paper refers to an admissible estimate $X_{\text{obs}}$3 and to the induced informed set $X_{\text{obs}}$4. After the first feasible solution, the planner restricts sampling to this subset and uses its hypervolume as an explicit control signal. The paper gives the Lebesgue measure of the $X_{\text{obs}}$5-dimensional hyperellipsoid as $X_{\text{obs}}$6, where $X_{\text{obs}}$7 is the current solution cost, $X_{\text{obs}}$8 is the Euclidean distance from start to goal, and $X_{\text{obs}}$9 is the gamma function. As the incumbent cost decreases, the informed subset contracts, and that contraction is used both to focus sampling and to update the next batch size.

The full planning loop follows the standard informed-tree pattern. Samples are drawn in batches, nearby vertices are identified, local collision checking is performed, edges are inserted or rewired to improve cost, and the procedure iterates until a time or iteration budget is exhausted. The distinctive modification is that “nearby” is no longer defined solely by an isotropic Euclidean ball. APT* computes a force-guided ellipsoidal neighborhood, and uses that neighborhood in both search and rewiring.

The planner is presented as an anytime asymptotically optimal method. When the time budget expires, it returns the best incumbent path found so far. If allowed to continue, it keeps refining the solution through additional samples and rewiring. Solution extraction is standard: the path is recovered from the goal-connected vertex through parent pointers back to the start.

4. Adaptive batch sizing

Adaptive batch sizing is one of the two principal additions that distinguish APT* from FDIT*. The stated logic is phase-dependent: large batches early help discover an initial solution quickly, while smaller batches later reduce unnecessary edge checks and computation during refinement. The batch size is updated by

$X_{\text{free}} = cl(X \setminus X_{\text{obs}}),$0

with $X_{\text{free}} = cl(X \setminus X_{\text{obs}}),$1 a decay factor (Zhang et al., 27 Aug 2025).

The decay signal is derived from informed-set contraction. The paper defines

$X_{\text{free}} = cl(X \setminus X_{\text{obs}}),$2

where $X_{\text{free}} = cl(X \setminus X_{\text{obs}}),$3 is the informed-set volume at the first feasible solution and $X_{\text{free}} = cl(X \setminus X_{\text{obs}}),$4 is the informed-set volume for the current incumbent. Thus $X_{\text{free}} = cl(X \setminus X_{\text{obs}}),$5: it is near $X_{\text{free}} = cl(X \setminus X_{\text{obs}}),$6 when no improvement has yet occurred, and decreases as the path improves.

A sigmoid smoothing is then applied:

$X_{\text{free}} = cl(X \setminus X_{\text{obs}}),$7

followed by a logarithmically smoothed decay factor

$X_{\text{free}} = cl(X \setminus X_{\text{obs}}),$8

The pseudocode labeled “APT* - Adaptive batch-size” effectively sets

$X_{\text{free}} = cl(X \setminus X_{\text{obs}}),$9

then recomputes the batch size only when the incumbent cost improves or when the first solution is found. At that point it updates $\mathbf{x}_{\text{start}}$0, stores the initial informed-set volume if necessary, computes the current informed-set volume, forms $\mathbf{x}_{\text{start}}$1, smooths it, computes $\mathbf{x}_{\text{start}}$2, and returns

$\mathbf{x}_{\text{start}}$3

The significance of this mechanism in APT* is broader than sample-count control. The paper explicitly couples batch size to charge modulation, so adaptive batching also controls the degree of local prolation. In that sense, batch size functions as the global phase variable for the planner.

5. Force-aligned prolate neighborhood construction

The core geometric module of APT* is an elliptical or prolate $\mathbf{x}_{\text{start}}$4-nearest-neighbor mechanism. It begins with a standard asymptotically valid radius based on informed-set measure:

$\mathbf{x}_{\text{start}}$5

where $\mathbf{x}_{\text{start}}$6 is a normalization constant, $\mathbf{x}_{\text{start}}$7 is dimension, $\mathbf{x}_{\text{start}}$8 is the informed-set measure, and $\mathbf{x}_{\text{start}}$9 is the $X_{\text{goal}} \subset X_{\text{free}}$0-dimensional unit ball (Zhang et al., 27 Aug 2025).

APT* then deforms the isotropic neighbor ball into a hyperellipsoid aligned with a resultant virtual force. If $X_{\text{goal}} \subset X_{\text{free}}$1 is the total force at the current vertex $X_{\text{goal}} \subset X_{\text{free}}$2, the principal axis is

$X_{\text{goal}} \subset X_{\text{free}}$3

Using Gram–Schmidt or QR, additional orthonormal directions $X_{\text{goal}} \subset X_{\text{free}}$4 are constructed to form

$X_{\text{goal}} \subset X_{\text{free}}$5

The semiaxes are

$X_{\text{goal}} \subset X_{\text{free}}$6

and with

$X_{\text{goal}} \subset X_{\text{free}}$7

the elliptical-RNN region is

$X_{\text{goal}} \subset X_{\text{free}}$8

This quadratic-form test replaces isotropic Euclidean neighborhood membership with an anisotropic predicate.

The force itself is inherited from FDIT*. Samples and vertices are treated as electric charges. Valid samples in free space contribute attractive forces, while invalid samples in obstacle space contribute repulsive forces. The paper first gives an $X_{\text{goal}} \subset X_{\text{free}}$9-dimensional Coulomb-like law,

$\sigma : [0,1] \mapsto X$0

with sign positive for valid vertices and negative for invalid ones. It then separates the components:

$\sigma : [0,1] \mapsto X$1

$\sigma : [0,1] \mapsto X$2

and

$\sigma : [0,1] \mapsto X$3

Algorithm 1 adds an iterative filtering loop based on the fraction of invalid neighbors inside the current ellipsoid:

$\sigma : [0,1] \mapsto X$4

The neighborhood is repeatedly refined while $\sigma : [0,1] \mapsto X$5, after which all invalid neighbors are removed. The resulting eRNN is therefore not only force-aligned but also explicitly filtered to keep the obstacle-content ratio below $\sigma : [0,1] \mapsto X$6.

The phrase “adaptively prolated” refers precisely to this mechanism. What is prolated is the local nearest-neighbor region around a vertex. In standard RRT* or BIT*-style planning that region is an $\sigma : [0,1] \mapsto X$7-ball of radius $\sigma : [0,1] \mapsto X$8; in APT* it is stretched into a prolate hyperellipsoid whose major axis is oriented by $\sigma : [0,1] \mapsto X$9. The paper states that zero force corresponds to spherical neighborhoods, while stronger force yields higher eccentricity. Early in planning, dense sampling is associated with smaller charge and more circular neighborhoods; later, sparse sampling is associated with larger charge and more elliptical neighborhoods.

Charge is therefore no longer fixed. Algorithm 3, “APT* - Calculate vertex charge,” sets

$\begin{split} \sigma^* &= \arg \min_{\sigma \in \Sigma} \left\{ c(\sigma) \middle| \sigma(0) = \mathbf{x}_{\text{start}}, \sigma(1) \in \mathbf{x}_{\text{goal}}, \right. \ &\qquad\qquad \left. \forall t \in [0, 1], \sigma(t) \in X_{\text{free}} \right\}. \end{split}$0

with normalization constants

$\begin{split} \sigma^* &= \arg \min_{\sigma \in \Sigma} \left\{ c(\sigma) \middle| \sigma(0) = \mathbf{x}_{\text{start}}, \sigma(1) \in \mathbf{x}_{\text{goal}}, \right. \ &\qquad\qquad \left. \forall t \in [0, 1], \sigma(t) \in X_{\text{free}} \right\}. \end{split}$1

It computes a biased normalized batch variable and then applies a nonlinear map. Five nonlinear mappings are compared: exponential (APT*-E), polynomial (APT*-P), logarithmic (APT*-L), iteration-based (APT*-I), and tanh (APT*-T). The best-performing variant is reported to be the tanh-based method. The paper also gives a Taylor or Bernoulli expansion for the tanh-based charge schedule,

$\begin{split} \sigma^* &= \arg \min_{\sigma \in \Sigma} \left\{ c(\sigma) \middle| \sigma(0) = \mathbf{x}_{\text{start}}, \sigma(1) \in \mathbf{x}_{\text{goal}}, \right. \ &\qquad\qquad \left. \forall t \in [0, 1], \sigma(t) \in X_{\text{free}} \right\}. \end{split}$2

with Bernoulli numbers

$\begin{split} \sigma^* &= \arg \min_{\sigma \in \Sigma} \left\{ c(\sigma) \middle| \sigma(0) = \mathbf{x}_{\text{start}}, \sigma(1) \in \mathbf{x}_{\text{goal}}, \right. \ &\qquad\qquad \left. \forall t \in [0, 1], \sigma(t) \in X_{\text{free}} \right\}. \end{split}$3

and truncation order $\begin{split} \sigma^* &= \arg \min_{\sigma \in \Sigma} \left\{ c(\sigma) \middle| \sigma(0) = \mathbf{x}_{\text{start}}, \sigma(1) \in \mathbf{x}_{\text{goal}}, \right. \ &\qquad\qquad \left. \forall t \in [0, 1], \sigma(t) \in X_{\text{free}} \right\}. \end{split}$4.

6. Empirical performance, theoretical status, and limitations

The evaluation uses Planner Arena, PDT, MoveIt, OpenRAVE, and OMPL implementations. Synthetic benchmarks comprise two families: Dividing Wall-gaps (DW), described as narrow-gap, wall-separated environments, and Random Rectangles (RR), described as axis-aligned hyperrectangles randomly generated in the state space. These are tested in $\begin{split} \sigma^* &= \arg \min_{\sigma \in \Sigma} \left\{ c(\sigma) \middle| \sigma(0) = \mathbf{x}_{\text{start}}, \sigma(1) \in \mathbf{x}_{\text{goal}}, \right. \ &\qquad\qquad \left. \forall t \in [0, 1], \sigma(t) \in X_{\text{free}} \right\}. \end{split}$5, $\begin{split} \sigma^* &= \arg \min_{\sigma \in \Sigma} \left\{ c(\sigma) \middle| \sigma(0) = \mathbf{x}_{\text{start}}, \sigma(1) \in \mathbf{x}_{\text{goal}}, \right. \ &\qquad\qquad \left. \forall t \in [0, 1], \sigma(t) \in X_{\text{free}} \right\}. \end{split}$6, and $\begin{split} \sigma^* &= \arg \min_{\sigma \in \Sigma} \left\{ c(\sigma) \middle| \sigma(0) = \mathbf{x}_{\text{start}}, \sigma(1) \in \mathbf{x}_{\text{goal}}, \right. \ &\qquad\qquad \left. \forall t \in [0, 1], \sigma(t) \in X_{\text{free}} \right\}. \end{split}$7, with each benchmark run $\begin{split} \sigma^* &= \arg \min_{\sigma \in \Sigma} \left\{ c(\sigma) \middle| \sigma(0) = \mathbf{x}_{\text{start}}, \sigma(1) \in \mathbf{x}_{\text{goal}}, \right. \ &\qquad\qquad \left. \forall t \in [0, 1], \sigma(t) \in X_{\text{free}} \right\}. \end{split}$8 times with varying seeds. Additional validation is reported on a dual-WAM manipulator in a cage environment in $\begin{split} \sigma^* &= \arg \min_{\sigma \in \Sigma} \left\{ c(\sigma) \middle| \sigma(0) = \mathbf{x}_{\text{start}}, \sigma(1) \in \mathbf{x}_{\text{goal}}, \right. \ &\qquad\qquad \left. \forall t \in [0, 1], \sigma(t) \in X_{\text{free}} \right\}. \end{split}$9 with $X_{\hat f}$0 runs and a $X_{\hat f}$1-second time budget, and on two real-world tasks on a DARKO base-manipulator platform—kitchen printer tool manipulation and shelf container insertion or rearrangement—each run $X_{\hat f}$2 times (Zhang et al., 27 Aug 2025).

Reported metrics include minimum, median, and maximum initial solution time, minimum, median, and maximum initial cost, final cost, success rate, and convergence curves over time, with path-length minimization as the objective. In the nonlinear charge comparison over $X_{\hat f}$3 runs, APT*-T is reported as best overall, especially in median initial time and median initial cost. Three truncation orders are given for the tanh method—$X_{\hat f}$4, $X_{\hat f}$5, and $X_{\hat f}$6—with an explicit tradeoff: larger $X_{\hat f}$7 gives better path quality but more computation, whereas smaller $X_{\hat f}$8 is faster but lower quality.

On the DW and RR benchmarks, the paper reports consistent improvements in median initial solution time over FIT* and FDIT*. In DW-$X_{\hat f}$9, FIT* has $\mathbf{x}_{\text{start}}$0, FDIT* has $\mathbf{x}_{\text{start}}$1, and APT* has $\mathbf{x}_{\text{start}}$2, corresponding to improvements of $\mathbf{x}_{\text{start}}$3 over FIT* and $\mathbf{x}_{\text{start}}$4 over FDIT*. In DW-$\mathbf{x}_{\text{start}}$5, the respective median initial times are $\mathbf{x}_{\text{start}}$6, $\mathbf{x}_{\text{start}}$7, and $\mathbf{x}_{\text{start}}$8, for improvements of $\mathbf{x}_{\text{start}}$9 and $X_{\text{samples}}$0. In RR-$X_{\text{samples}}$1, FIT* has $X_{\text{samples}}$2, FDIT* has $X_{\text{samples}}$3, and APT* has $X_{\text{samples}}$4, giving improvements of $X_{\text{samples}}$5 and $X_{\text{samples}}$6. The paper further states that APT* usually attains lower median initial and final costs, especially in higher dimensions.

In the cage-$X_{\text{samples}}$7 dual-arm benchmark, APT* reports $X_{\text{samples}}$8, $X_{\text{samples}}$9, $X_{\text{obs}}$00, and success $X_{\text{obs}}$01. FIT* reports $X_{\text{obs}}$02 with success $X_{\text{obs}}$03, and FDIT* reports $X_{\text{obs}}$04 with success $X_{\text{obs}}$05. The paper highlights around $X_{\text{obs}}$06 improvement over FIT* and $X_{\text{obs}}$07 over FDIT* in median initial time on this task.

For the real-world kitchen tool printing task under a $X_{\text{obs}}$08 s budget and $X_{\text{obs}}$09 trials, FIT* reports success $X_{\text{obs}}$10 with median cost $X_{\text{obs}}$11, FDIT* reports success $X_{\text{obs}}$12 with median cost $X_{\text{obs}}$13, and APT* reports success $X_{\text{obs}}$14 with average or reported best cost $X_{\text{obs}}$15. For the shelf container rearrangement task under a $X_{\text{obs}}$16 s budget and $X_{\text{obs}}$17 trials, FIT* reports success $X_{\text{obs}}$18 with median cost $X_{\text{obs}}$19, FDIT* reports success $X_{\text{obs}}$20 with median cost $X_{\text{obs}}$21, and APT* reports success $X_{\text{obs}}$22 with median cost $X_{\text{obs}}$23. These results are used in the paper to support the claim that the method is not limited to synthetic point-mass spaces and is applicable to real manipulation tasks with narrow passages and precise insertion constraints.

The theoretical posture of APT* is more circumscribed than the title alone suggests. The title and abstract explicitly claim asymptotically optimal motion planning, and the method uses an $X_{\text{obs}}$24 radius of the standard connectivity or optimality form based on informed-set measure and $X_{\text{obs}}$25. The planner is also framed as an extension of asymptotically optimal informed planners. However, the supplied text does not include a formal theorem statement, proof, or proof sketch specifically establishing asymptotic optimality or probabilistic completeness. The strongest precise summary from the available material is therefore that APT* is intended as an asymptotically optimal planner built on an RGG-consistent radius rule, while the excerpt itself provides indirect rather than standalone formal support.

Several limitations and sensitivities are also explicit. The method introduces a comparatively large parameter set, including $X_{\text{obs}}$26, $X_{\text{obs}}$27, $X_{\text{obs}}$28, the sigmoid slope, $X_{\text{obs}}$29, $X_{\text{obs}}$30, $X_{\text{obs}}$31, $X_{\text{obs}}$32, the ellipsoid scaling factor $X_{\text{obs}}$33, and the tanh truncation order $X_{\text{obs}}$34. The virtual-force and ellipsoidal-neighbor calculations add computational overhead relative to Euclidean nearest-neighbor methods. The paper does not provide a formal big-$X_{\text{obs}}$35 analysis, but it identifies the dominant costs: naïve nearest-neighbor processing is linear in $X_{\text{obs}}$36 per processed vertex, force accumulation is linear in $X_{\text{obs}}$37, and direct ellipsoidal membership testing through the quadratic form is $X_{\text{obs}}$38 per point, whereas adaptive-batch updates are effectively constant-time per update. A further conceptual sensitivity is the scheduling of prolation itself: the authors explicitly note that prolating neighbors too early may lead to local optimality, while prolating too late may forfeit problem-specific information. A plausible implication is that APT*’s gains depend on the quality of that phase coupling rather than on prolate geometry alone.