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SE(3)-DiffusionFields: Learning smooth cost functions for joint grasp and motion optimization through diffusion
Julen Urain, Niklas Funk, Jan Peters, Georgia Chalvatzaki
TL;DR
Robot manipulation requires combining learned task-specific objectives with hand-designed motion constraints, but decoupled grasp generation and trajectory planning can yield infeasible solutions. The paper learns smooth SE(3) diffusion cost functions for 6DoF grasping and uses them in a joint gradient-based grasp-and-motion optimizer. Across simulated and real manipulation tasks, the framework is reported to outperform traditional decoupled approaches, with reliable performance in a multi-objective setting.
Problem
Robot manipulation must jointly consider grasp poses, collisions, smoothness, and joint limits, while existing explicit generators decouple grasp selection from trajectory planning.
Method
The paper learns smooth SE(3) diffusion cost functions for 6DoF grasp poses and combines them with other differentiable costs in joint gradient-based optimization.
Results
The framework outperforms traditional decoupled optimization approaches and shows superior efficiency, adaptiveness, and success rates across the reported evaluations.
Takeaways & Limitations
SE(3)-DiffusionFields provides a joint grasp-and-motion optimization framework for generating diverse, high-quality grasp poses and robot trajectories.
Abstract
from arXiv · showhide
Multi-objective optimization problems are ubiquitous in robotics, e.g., the optimization of a robot manipulation task requires a joint consideration of grasp pose configurations, collisions and joint limits. While some demands can be easily hand-designed, e.g., the smoothness of a trajectory, several task-specific objectives need to be learned from data. This work introduces a method for learning data-driven SE(3) cost functions as diffusion models. Diffusion models can represent highly-expressive multimodal distributions and exhibit proper gradients over the entire space due to their score-matching training objective. Learning costs as diffusion models allows their seamless integration with other costs into a single differentiable objective function, enabling joint gradient-based motion optimization. In this work, we focus on learning SE(3) diffusion models for 6DoF grasping, giving rise to a novel framework for joint grasp and motion optimization without needing to decouple grasp selection from trajectory generation. We evaluate the representation power of our SE(3) diffusion models w.r.t. classical generative models, and we showcase the superior performance of our proposed optimization framework in a series of simulated and real-world robotic manipulation tasks against representative baselines.
I. INTRODUCTION
Robot manipulation requires jointly optimizing grasp and placing poses, trajectory smoothness, collision avoidance, and joint limits. The paper learns smooth SE(3) diffusion cost functions so grasp selection and motion planning can be optimized together.
- Manipulation tasks become multi-objective motion optimization problems involving grasping, placing, smoothness, collisions, and joint limits.
- Grasp pose selection and collision avoidance are expensive to model and are commonly approximated with learning-based approaches.
- Explicit grasp generators require separate optimization phases and can produce grasps that make trajectory optimization infeasible.
- The paper learns smooth data-driven SE(3) cost functions as diffusion models, exposing informative gradients throughout the workspace.
- SE(3) diffusion models represent multimodal grasp distributions while moving random samples toward low-cost regions through inverse diffusion.
- The proposed framework combines learned grasp costs with trajectory smoothness, collision avoidance, and other differentiable costs in one gradient-based optimization loop.
II. PRELIMINARIES
Diffusion models learn score fields from progressively noise-perturbed data and generate samples by reversing a Langevin diffusion process. The paper’s setting extends these ideas to constrained SE(3) pose representations.
- Diffusion models iteratively move noisy random samples toward a learned distribution rather than generating samples in one explicit step.
- Denoising Score Matching perturbs data across ordered Gaussian noise scales and trains a noise-conditioned vector field to estimate each score.
- Annealed Langevin MCMC generates samples by starting from a noisy distribution and simulating inverse diffusion from the largest to smallest noise scale.
- The learned score vectors point toward samples from the training distribution.
- SE(3) represents object or robot-link poses with rotation and translation on a smooth constrained Lie-group manifold.
- Derivatives of functions on SE(3) are six-dimensional vectors because distances and local changes are defined on the Lie-group geometry.
- Figure 2 visualizes iterative refinement of random SE(3) grasp-pose samples using inverse Langevin diffusion.
III. SE(3)-DIFFUSION FIELDS
This section adapts diffusion models to the non-Euclidean SE(3) space used in robot manipulation. It develops SE(3) grasp-pose models and integrates them into joint grasp-and-motion optimization.
- The method modifies diffusion-model design choices for SE(3), a crucial non-Euclidean space in robot manipulation.
- The section covers SE(3) diffusion modifications, a neural architecture and training procedure for 6DoF grasp distributions, and joint optimization integration.
A. From Euclidean diffusion to diffusion in SE(3)
The paper extends denoising score matching and Langevin sampling from Euclidean space to SE(3), ensuring diffusion updates remain on the pose manifold. It then converts the learned score representation into an energy field composable with other costs.
- An SE(3) diffusion model outputs a six-dimensional vector for any pose query, conditioned on the current noise scale.
- SE(3) denoising score matching perturbs training poses with Gaussian noise on the Lie group and trains the model to match the perturbed distribution’s score.
- Inverse Langevin sampling is adapted to preserve the SE(3) manifold throughout the diffusion process.
- Repeated inverse-diffusion updates move randomly sampled SE(3) poses toward the training-data distribution.
- The paper learns a scalar energy field rather than only a score vector, allowing sample quality evaluation and composition with other costs.
B. Architecture & training of Grasp SE(3)-DiffusionFields
Grasp SE(3)-DiF learns object-conditioned 6DoF grasp costs by jointly modeling object SDFs and denoising energies. Its architecture transforms grasp poses into object-frame point features, then decodes them into scalar energies.
- Scope: SE(3)-DiF focuses on object-conditioned cost functions that capture 6DoF grasp pose distributions while assuming access to object pose.Perception for encoding point clouds into object pose and shape is deferred to future work.
- Training: The model jointly learns the object’s Signed Distance Field and grasp energy using supervised SDF training and denoising score matching.Joint SDF and grasp-pose learning is reported to improve grasp-generation quality.
- Training: Training perturbs successful grasp poses with noise and learns energy to approximate the noisy log-probability distribution for each object.The energy is computed over perturbed grasp poses before applying the denoising score-matching loss.
- Architecture: Given a grasp pose, the model transforms fixed gripper points from the world frame into the object’s local frame before feature encoding.The point representation expresses grasp poses through N 3D points and conditions encoding on object shape and noise level.
- Architecture: The feature encoder outputs SDF predictions and additional features, which are flattened and passed to a decoder producing the scalar energy e.The encoder output has size N×(1 + ψ).
C. Grasp and motion optimization with diffusion models
The framework incorporates the learned SE(3) grasp energy into a weighted, differentiable trajectory objective alongside motion costs. It then jointly optimizes grasp pose and robot joint trajectories through an inverse diffusion process.
- Objective: Motion optimization minimizes a weighted sum of cost terms over a trajectory’s robot joint positions.The objective includes positive weights ω_j for individual costs c_j(τ).
- Joint optimization: The learned SE(3)-DiF grasp cost is combined with differentiable collision-avoidance and trajectory-smoothness costs in one objective.This replaces separate treatment of grasp sampling and trajectory planning with a joint optimization loop.
- Motivation: Joint optimization is hypothesized to improve sample efficiency because separately sampled grasps can be infeasible for the subsequent trajectory-planning problem.The claim is presented as a hypothesis about decoupled versus joint approaches.
- Joint-space integration: Forward kinematics maps robot configurations into end-effector poses so the SE(3) energy can be evaluated as a joint-space cost.The cost is defined as c(q_t, k) = E_θ(ϕ_ee(q_t), k).
- Inverse diffusion: The motion-generation problem is framed as an inverse diffusion process targeting q(τ|k) ∝ exp(−J(τ,k)).Particles represent complete trajectories and evolve from noise level k = L toward k = 1.
IV. EXPERIMENTAL EVALUATION
The experimental evaluation studies 6DoF grasp generation using an SE(3)-DiF trained on successful grasps from the Acronym mug dataset. It evaluates generated poses with success rate and distributional similarity.
- Experimental setup: The experimental section first evaluates SE(3)-DiF as a 6DoF grasp-pose generative model trained on the Acronym dataset.The dataset contains successful 6DoF grasp poses for ShapeNet objects, with the study focusing on mugs.
- Metrics: Generated grasps are evaluated using success rate and Earth Mover Distance between generated samples and the training-data distribution.EMD measures divergence between empirical probability distributions.
- Dataset: The mug collection contains approximately 90K successful 6DoF grasp poses from 90 different mugs.The mugs’ meshes are obtained from ShapeNet.
A. Evaluation of 6DoF grasp pose generation
The grasp-generation evaluation compares SE(3)-DiF with three models inspired by 6dof-GraspNet using success and diversity metrics. The authors report that SE(3)-DiF produces high-quality, diverse grasp poses and attribute baseline gradient failures to poor behavior far from good samples.
- Diversity: SE(3)-DiF outperforms all baselines significantly on EMD, for which lower values indicate greater similarity to the target distribution.The authors associate the result with better capture of the data distribution than VAE+Refine.
- Failure analysis: Classifier-only optimization has low success when initial grasp poses are far from good samples because its gradient field can contain near-zero-slope plateaus.Those plateaus prevent initial samples from moving toward high-probability regions.
- Conclusion: The authors conclude that SE(3)-DiF generates grasp poses that are both high-quality and diverse.The conclusion follows their reported success-rate and EMD comparisons.
B. Performance on grasp and motion optimization
The framework is evaluated on picking amidst clutter by integrating the learned grasp SE(3)-DiF as an additional cost in multi-objective trajectory optimization. Joint optimization is substantially more particle-efficient than the decoupled approach and benefits from smooth grasp-cost gradients.
- The evaluation measures success by whether the robot grasps the target object at the end of execution.The task is picking amidst clutter, with trajectories generated using the learned grasp SE(3)-DiF as an additional cost function.
- Figure 6 presents simulated and real robot environments for picking amidst clutter.
- 25 particles match the decoupled approach’s success rate with 800 particles.The efficiency gap is attributed to decoupled grasp poses violating environment constraints such as clutter or joint limits.
- Joint optimization iteratively improves entire trajectories with respect to all objectives, finding trajectories that satisfy the combined costs.The approach jointly considers grasping and motion objectives rather than generating grasp poses separately from trajectory optimization.
- The grasp SE(3)-DiF provides useful gradients for motion optimization, whereas the grasp classifier lacks smooth gradients across the whole space.With the classifier, motion optimization is unable to find solutions that move trajectories toward successful grasps.
C. Grasp and motion optimization on real robots
The framework is evaluated on a real-world pick-mug-and-place-on-shelf task under realistic operating conditions. The reported results indicate high performance without observed sim2real discrepancies, while the evaluation assumes full object-state knowledge.
- The real-world evaluation uses a pick-mug-and-place-on-shelf scenario driven by the 6DoF grasp diffusion model as an additional cost.
- The experiments include non-perfect state information caused by small calibration errors in externally retrieved mug poses.
- The experiments also include variations in the mug’s shape.
- The approach is reported as highly performant in real settings without suffering sim2real discrepancies.
- The evaluation assumes full object-state knowledge and does not rely on complex perception systems.The authors identify imperfect perception as a potential source of sim2real gaps in other settings.
V. RELATED WORK
The related-work discussion situates diffusion models in robotics across several task categories and places 6D grasp generation within a broader family of existing approaches.
- Diffusion models in Robotics: Diffusion models have been applied in robotics to text-conditioned scene rearrangement, decision-making, and controllable traffic generation.
- Diffusion models in Robotics: An earlier work integrated a diffusion process into a motion-planning problem.
- 6D grasp generation: 6D grasp pose generation has been addressed with methods ranging from classifiers to explicit samplers.
VI. CONCLUSION
SE(3)-DiF learns task-space data-driven cost functions that provide informative gradients for joint grasp and motion optimization. The framework combines learned and hand-designed costs into a single optimization objective, while accounting for SE(3) geometry during modeling and differentiation.
- VI. CONCLUSION: SE(3)-DiF represents task-space data-driven costs as diffusion models for robotic motion generation.The model is trained for 6DoF grasp pose distributions and produces informative gradients across SE(3).
- VI. CONCLUSION: The framework jointly optimizes grasp poses and trajectories by combining the learned grasp cost with smoothness, collision, and other differentiable costs.This forms a single smooth objective optimized through a gradient-based loop.
- VI. CONCLUSION: The method connects motion generation with inverse diffusion while generating diverse and high-quality 6DoF grasp poses.Diffusion training and sampling are adapted to SE(3), including manifold-aware perturbations and score matching.
- VI. CONCLUSION: The framework outperforms traditional decoupled optimization approaches, with experiments reporting superior efficiency, adaptiveness, and success rates.The training procedure combines diffusion-model learning with signed-distance-function representation learning for geometric reasoning.
- VI. CONCLUSION: SE(3) derivatives are computed using manifold-aware Jacobians, tangent-space mappings, and the chain rule rather than direct Euclidean addition.The formulation uses ⊕ and ⊖ operators appropriate to the domain and codomain, with left-Jacobian assumptions in this work.
B. Evaluation of SE(3)-DiffusionFields for robot grasp pose generation
The evaluation compares joint optimization of grasp poses and robot motion against separate or hierarchical procedures across simulated manipulation tasks. SE(3)-DiF is combined with differentiable task costs to optimize grasping and trajectories in a single loop.
- Baselines: The separate procedure first optimizes grasp poses without the current object pose or environmental constraints, then optimizes joint-space motion for fixed grasps.Its second-stage objective includes distance to the selected grasp pose and table-collision avoidance.
- Results: The split procedure produces many infeasible grasp poses, performs worse than joint SE(3)-DiF optimization, and drops particularly in the flipped-mug scenario.Using ten joint samples per grasp improves the split baseline over one sample, but remains inferior to the proposed method.
- Results: 0.88: the joint method’s single-particle success ratio remains at this high level while overall successful-particle ratios decline only slightly.The result supports reliable grasp-pose generation despite the multi-objective setting.
- Evaluation setup: The experiments evaluate SE(3)-DiF as a cost function for jointly optimizing grasp pose and trajectory in three robot tasks.The objective combines the learned grasp cost with trajectory smoothness, collision avoidance, and other heuristic costs.
- Picking with occlusions: Joint optimization outperforms hierarchical optimization across all evaluated occlusion cases and benefits from more initial particles.The authors attribute the difference to jointly updating trajectory and grasp costs rather than fixing a potentially infeasible grasp first.
- Pick and reorient: The pick-and-reorient experiment solves grasp selection and placement motion in one optimization loop, requiring the grasp to support collision-free reaching and the desired final pose.Success is measured over 100 trials with randomly initialized object and target poses.