Abstract--Neuro-symbolic reasoning systems face fundamental challenges in maintaining semantic coherence while satisfying physical and logical constraints. Building upon our previous work on Ontology Neural Networks, we present an enhanced framework that integrates topological conditioning with gradient stabilization mechanisms. The approach employs Forman-Ricci curvature to capture graph topology, Deep Delta Learning for stable rank-one perturbations during constraint projection, and Covariance Matrix Adaptation Evolution Strategy for parameter optimization. Experimental evaluation across multiple problem sizes demonstrates that the method achieves mean energy reduction to 1.15 compared to baseline values of 11.68, with 95 percent success rate in constraint satisfaction tasks. The framework exhibits seed-independent convergence and graceful scaling behavior up to twenty-node problems, suggesting that topological structure can inform gradient-based optimization without sacrificing interpretability or computational efficiency. Integrating symbolic reasoning with neural learning remains a central challenge in artificial intelligence. While neural networks excel at pattern recognition and gradient-based optimization, they often struggle to maintain explicit constraints or provide interpretable intermediate representations. The opacity of deep neural representations makes it difficult to verify whether learned policies respect domain knowledge or physical laws. Conversely, symbolic systems offer logical transparency and formal guarantees but lack the flexibility to learn from noisy, incomplete data or adapt to distributional shifts.

Safe multi-agent motion planning (MAMP) under task-induced constraints is a critical challenge in robotics. Many real-world scenarios require robots to navigate dynamic environments while adhering to manifold constraints imposed by tasks. For example, service robots must carry cups upright while avoiding collisions with humans or other robots. Despite recent advances in decentralized MAMP for high-dimensional systems, incorporating manifold constraints remains difficult. To address this, we propose a manifold-constrained Hamilton-Jacobi reachability (HJR) learning framework for decentralized MAMP. Our method solves HJR problems under manifold constraints to capture task-aware safety conditions, which are then integrated into a decentralized trajectory optimization planner. This enables robots to generate motion plans that are both safe and task-feasible without requiring assumptions about other agents' policies. Our approach generalizes across diverse manifold-constrained tasks and scales effectively to high-dimensional multi-agent manipulation problems. Experiments show that our method outperforms existing constrained motion planners and operates at speeds suitable for real-world applications. Video demonstrations are available at https://youtu.be/RYcEHMnPTH8 .

We collect teleoperation data for a constrained bimanual pick-and-place task. Then, we perturb these demonstrations to generate three additional datasets that still accomplish the task, but contain increasing constraint violation. We train a policy on each of these datasets and analyze task success and constraint adherence. Lastly, we collect demonstrations for the same task on hardware, train a policy, and evaluate its performance on similar metrics. Abstract-- Diffusion policies have shown impressive results in robot imitation learning, even for tasks that require satisfaction of kinematic equality constraints. However, task performance alone is not a reliable indicator of the policy's ability to precisely learn constraints in the training data. T o investigate, we analyze how well diffusion policies discover these manifolds with a case study on a bimanual pick-and-place task that encourages fulfillment of a kinematic constraint for success. We study how three factors affect trained policies: dataset size, dataset quality, and manifold curvature. Our experiments show diffusion policies learn a coarse approximation of the constraint manifold with learning affected negatively by decreases in both dataset size and quality. On the other hand, the curvature of the constraint manifold showed inconclusive correlations with both constraint satisfaction and task success.

Despite the promise of scientific machine learning (SciML) in combining data-driven techniques with mechanistic modeling, existing approaches for incorporating hard constraints in neural differential equations (NDEs) face significant limitations. Scalability issues and poor numerical properties prevent these neural models from being used for modeling physical systems with complicated conservation laws. We propose Manifold-Projected Neural ODEs (PNODEs), a method that explicitly enforces algebraic constraints by projecting each ODE step onto the constraint manifold. This framework arises naturally from semi-explicit differential-algebraic equations (DAEs), and includes both a robust iterative variant and a fast approximation requiring a single Jacobian factorization. We further demonstrate that prior works on relaxation methods are special cases of our approach. PNODEs consistently outperform baselines across six benchmark problems achieving a mean constraint violation error below $10^{-10}$. Additionally, PNODEs consistently achieve lower runtime compared to other methods for a given level of error tolerance. These results show that constraint projection offers a simple strategy for learning physically consistent long-horizon dynamics.

Safety is a critical requirement for the real-world deployment of robotic systems. Unfortunately, while current robot foundation models show promising generalization capabilities across a wide variety of tasks, they fail to address safety, an important aspect for ensuring long-term operation. Current robot foundation models assume that safe behavior should emerge by learning from a sufficiently large dataset of demonstrations. However, this approach has two clear major drawbacks. Firstly, there are no formal safety guarantees for a behavior cloning policy trained using supervised learning. Secondly, without explicit knowledge of any safety constraints, the policy may require an unreasonable number of additional demonstrations to even approximate the desired constrained behavior. To solve these key issues, we show how we can instead combine robot foundation models with geometric inductive biases using ATACOM, a safety layer placed after the foundation policy that ensures safe state transitions by enforcing action constraints. With this approach, we can ensure formal safety guarantees for generalist policies without providing extensive demonstrations of safe behavior, and without requiring any specific fine-tuning for safety. Our experiments show that our approach can be beneficial both for classical manipulation tasks, where we avoid unwanted collisions with irrelevant objects, and for dynamic tasks, such as the robot air hockey environment, where we can generate fast trajectories respecting complex tasks and joint space constraints.

Motion planning is a fundamental problem in robotics that involves generating feasible trajectories for a robot to follow. Recent advances in parallel computing, particularly through CPU and GPU architectures, have significantly reduced planning times to the order of milliseconds. However, constrained motion planning especially using sampling based methods on GPUs remains underexplored. Prior work such as pRRTC leverages a tracking compiler with a CUDA backend to accelerate forward kinematics and collision checking. While effective in simple settings, their approach struggles with increased complexity in robot models or environments. In this paper, we propose a novel GPU based framework utilizing NVRTC for runtime compilation, enabling efficient handling of high complexity scenarios and supporting constrained motion planning. Experimental results demonstrate that our method achieves superior performance compared to existing approaches.

Robot foundation models hold the potential for deployment across diverse environments, from industrial applications to household tasks. While current research focuses primarily on the policies' generalization capabilities across a variety of tasks, it fails to address safety, a critical requirement for deployment on real-world systems. In this paper, we introduce a safety layer designed to constrain the action space of any generalist policy appropriately. Our approach uses ATACOM, a safe reinforcement learning algorithm that creates a safe action space and, therefore, ensures safe state transitions. By extending ATACOM to generalist policies, our method facilitates their deployment in safety-critical scenarios without requiring any specific safety fine-tuning. We demonstrate the effectiveness of this safety layer in an air hockey environment, where it prevents a puck-hitting agent from colliding with its surroundings, a failure observed in generalist policies.

We propose a method for improving the prediction accuracy of learned robot dynamics models on out-of-distribution (OOD) states. We achieve this by leveraging two key sources of structure often present in robot dynamics: 1) sparsity, i.e., some components of the state may not affect the dynamics, and 2) physical limits on the set of possible motions, in the form of nonholonomic constraints. Crucially, we do not assume this structure is known a priori, and instead learn it from data. We use contrastive learning to obtain a distance pseudometric that uncovers the sparsity pattern in the dynamics, and use it to reduce the input space when learning the dynamics. We then learn the unknown constraint manifold by approximating the normal space of possible motions from the data, which we use to train a Gaussian process (GP) representation of the constraint manifold. We evaluate our approach on a physical differential-drive robot and a simulated quadrotor, showing improved prediction accuracy on OOD data relative to baselines.

Constrained Motion Planning (CMP) aims to find a collision-free path between the given start and goal configurations on the kinematic constraint manifolds. These problems appear in various scenarios ranging from object manipulation to legged-robot locomotion. However, the zero-volume nature of manifolds makes the CMP problem challenging, and the state-of-the-art methods still take several seconds to find a path and require a computationally expansive path dataset for imitation learning. Recently, physics-informed motion planning methods have emerged that directly solve the Eikonal equation through neural networks for motion planning and do not require expert demonstrations for learning. Inspired by these approaches, we propose the first physics-informed CMP framework that solves the Eikonal equation on the constraint manifolds and trains neural function for CMP without expert data. Our results show that the proposed approach efficiently solves various CMP problems in both simulation and real-world, including object manipulation under orientation constraints and door opening with a high-dimensional 6-DOF robot manipulator. In these complex settings, our method exhibits high success rates and finds paths in sub-seconds, which is many times faster than the state-of-the-art CMP methods.

In order for a bimanual robot to manipulate an object that is held by both hands, it must construct motion plans such that the transformation between its end effectors remains fixed. This amounts to complicated nonlinear equality constraints in the configuration space, which are difficult for trajectory optimizers. In addition, the set of feasible configurations becomes a measure zero set, which presents a challenge to sampling-based motion planners. We leverage an analytic solution to the inverse kinematics problem to parametrize the configuration space, resulting in a lower-dimensional representation where the set of valid configurations has positive measure. We describe how to use this parametrization with existing algorithms for motion planning, including sampling-based approaches, trajectory optimizers, and techniques that plan through convex inner-approximations of collision-free space.