Reinforcement learning (RL) algorithms for continuous control tasks require accurate sampling-based action selection. Many tasks, such as robotic manipulation, contain inherent problem symmetries. However, correctly incorporating symmetry into sampling-based approaches remains a challenge. This work addresses the challenge of preserving symmetry in sampling-based planning and control, a key component for enhancing decision-making efficiency in RL. We introduce an action sampling approach that enforces the desired symmetry. We apply our proposed method to a coordinate regression problem and show that the symmetry aware sampling method drastically outperforms the naive sampling approach. We furthermore develop a general framework for sampling-based model-based planning with Model Predictive Path Integral (MPPI). We compare our MPPI approach with standard sampling methods on several continuous control tasks.

Controlling robots through natural language instructions in open-vocabulary scenarios is pivotal for enhancing human-robot collaboration and complex robot behavior synthesis. However, achieving this capability poses significant challenges due to the need for a system that can generalize from limited data to a wide range of tasks and environments. Existing methods rely on large, costly datasets and struggle with generalization. This paper introduces Grounded Equivariant Manipulation (GEM), a novel approach that leverages the generative capabilities of pre-trained vision-language models and geometric symmetries to facilitate few-shot and zero-shot learning for open-vocabulary robot manipulation tasks. Our experiments demonstrate GEM's high sample efficiency and superior generalization across diverse pick-and-place tasks in both simulation and real-world experiments, showcasing its ability to adapt to novel instructions and unseen objects with minimal data requirements. GEM advances a significant step forward in the domain of language-conditioned robot control, bridging the gap between semantic understanding and action generation in robotic systems.