tl;dr
RAM provides a fast, differentiable surrogate for pose reachability across morphologies, enabling rapid design and planning workflows.
Abstract
Many stages of the robotic lifecycle, from morphology synthesis to operation, rely fundamentally on the reachable workspace. However, current methods for approximating workspaces are slow, imprecise, or tied to a single morphology. We introduce Reachability Across Morphologies (RAM): a morphology-conditioned, implicit neural representation that acts as a fast, differentiable surrogate for pose reachability, generalising to unseen morphologies while inherently accounting for self-collisions. To train RAM, we publish a large-scale dataset of $3 \cdot 10^{10}$ samples generated solely from forward kinematics. Experiments show that our model achieves an $F_1\text{-score}$ of $86\%$ at nanosecond inference, outperforming the baseline by $14\%$ while reducing inference time by three orders of magnitude. We further demonstrate speed-ups of one and two orders of magnitude for gradient-based morphology and trajectory optimisation, respectively.
Highlights
RAM
A morphology-conditioned implicit neural representation that evaluates pose reachability in nanoseconds, generalises to unseen morphologies without retraining, and inherently accounts for self-collisions.
Large-scale Reachability Dataset
A public dataset of \(3\cdot10^{10}\) samples across \(3\cdot10^4\) unique robot morphologies, generated solely from forward kinematics with pose sampling restricted to geometrically plausible regions.
Accelerating Morphology &
Trajectory Optimisation
RAM's differentiability enables gradient-based morphology and trajectory optimisation at one and two orders of magnitude lower computational cost respectively compared to numerical inverse kinematics.
Summary
RAM Overview: We present RAM, a surrogate model for robot reachability. Trained on a diverse dataset of thirty billion samples (I), the morphology-conditioned implicit neural representation (II) generalises accurately to unseen morphologies (III). Due to its differentiability and efficient inference, RAM can substantially accelerate downstream robotics tasks by replacing inverse kinematics (IV).
Dataset
Samples
\(3\cdot10^{10}\)
Morphologies
\(3\cdot10^{4}\)
Dataset Size
202 GB
GPU Hours (H100)
360
Morphology
Serial kinematic chains of 5–7 revolute joints are sampled via hierarchical rejection sampling over Modified Denavit-Hartenberg parameters, yielding uniform coverage of the valid morphological space and covering operable, non-degenerate, conventional designs.
Pose
Poses are sampled uniformly from \(\mathcal{B}^3(t_0, r_\text{mov}) \rtimes SO(3)\), the geometrically plausible region centred at the first joint with moveable length \(r_\text{mov}\). This concentrates samples where the robot can actually reach, maximising the learning signal.
Reachability Label
\(SE(3)\) is discretised into cubic \(\mathbb{R}^3\) voxels and \(SO(3)\) Voronoi cells. Joint configurations are sampled via forward kinematics to mark cells reachable, achieving an \(F_1\)-score of 91% against analytical reference labels at the chosen resolution — a practical ceiling for RAM training.
Results
Workspace slices. A fixed-orientation cross-section through a representative workspace, showing ground-truth labels, RAM's prediction, and the GGIK baseline. RAM closely tracks the ground-truth boundary while GGIK produces a blurrier, over-smoothed approximation.
RAM vs. GGIK
Inference measured end-to-end from pose and morphology input to reachability output.
| Classifier | Training (h) | Inference (s) | \(F_1\) Random (%) | \(F_1\) Boundary (%) |
|---|---|---|---|---|
| RAM (ours) | 57 | \(5.7 \cdot 10^{-8}\) | \(86^{+1}_{-1}\) | \(78^{+3}_{-3}\) |
| GGIK | 756 | \(2.2 \cdot 10^{-2}\) | \(72^{+1}_{-1}\) | \(69^{+3}_{-2}\) |
RAM outperforms GGIK by 14% in \(F_1\)-score while reducing inference time by more than five orders of magnitude, training in 57 hours versus one month.
Generalisation. Evaluated out-of-distribution on morphologies with 1–4, 8, and 9 DoF, RAM maintains \(F_1\)-scores above 78% for four, eight, and nine DoF, demonstrating robust generalisation beyond its training range of 5–7 DoF. Performance drops substantially below four DoF, where workspace topology is fundamentally different from the training distribution.
Latent space. Across principal components of the LSTM latent space, DoF shows the highest Spearman correlation (86%), followed by the standard deviation of link lengths (58%), indicating the representation organises itself by kinematic complexity.
Applications
RAM's differentiability and nanosecond inference unlock gradient-based optimisation that was previously bottlenecked by numerical inverse kinematics.
Morphology Optimisation
Gradient-based optimisation of link lengths and offsets to maximise reachability of task poses. RAM reduces mean pose error by 78% (vs. 69% for IK baseline) while running consistently one order of magnitude faster. Both methods reduce self-collisions from 16% to 6%.
Trajectory Optimisation
Waypoints on a nominal trajectory are perturbed in \(SE(3)\) tangent space to improve reachability. RAM reduces pose error by 62% and self-collisions from 10% to 4%, while running more than two orders of magnitude faster than the implicit-differentiation IK baseline.
BibTeX
@misc{walter2026ramreachabilitymorphologies,
title={RAM: Reachability Across Morphologies},
author={Tim Walter and Xinyu Chen and Jonathan Külz and Matthias Althoff},
year={2026},
eprint={2606.09108},
archivePrefix={arXiv},
primaryClass={cs.RO},
url={https://arxiv.org/abs/2606.09108},
}