The proposed deep learning-based reachability framework can be extended to higher-dimensional legged robots by leveraging several strategies. First, the architecture of the neural network used to approximate the Hamilton-Jacobi (HJ) value function can be scaled up to accommodate the increased complexity of the state space. This may involve using deeper networks with more hidden layers and neurons to capture the intricate dynamics of higher-dimensional systems. Additionally, incorporating advanced techniques such as convolutional layers or recurrent neural networks could help in learning spatial and temporal dependencies more effectively.
Second, the training process can be enhanced by utilizing more sophisticated sampling methods to generate training data that covers a wider range of the state space. Techniques such as domain randomization can be employed to simulate various conditions and perturbations, ensuring that the model learns to generalize across different scenarios. Furthermore, the use of parallel computing resources can significantly speed up the training process, allowing for the handling of larger datasets and more complex models.
Third, the framework can be adapted to include multi-modal inputs that represent different aspects of the robot's dynamics, such as contact forces, joint torques, and environmental interactions. This holistic approach would enable the model to learn a more comprehensive representation of the robot's behavior.
Lastly, to address the curse of dimensionality, a hierarchical approach can be implemented where the dynamics are decomposed into simpler sub-problems. Each sub-problem can be solved independently, and their solutions can be integrated to form a complete control strategy for the higher-dimensional robot. This modular design not only simplifies the learning process but also enhances the scalability of the reachability framework.