Learning and Optimization with 3D Orientations

There exist numerous ways of representing 3D orientations. Each representation has both limitations and unique features. Choosing the best representation for one task is often a difficult chore, and there exist conflicting opinions on which representation is better suited for a set of family of tasks. Even worse, when dealing with scenarios where we need to learn or optimize functions with orientations as inputs and/or outputs, the set of possibilities (representations, loss functions, etc.) is even larger and it is not easy to decide what is best for each scenario. In this paper, we attempt to a) present clearly, concisely and with unified notation all available representations, and "tricks" related to 3D orientations (including Lie Group algebra), and b) benchmark them in representative scenarios. The first part feels like it is missing from the robotics literature as one has to read many different textbooks and papers in order have a concise and clear understanding of all possibilities, while the benchmark is necessary in order to come up with recommendations based on empirical evidence. More precisely, we experiment with the following settings that attempt to cover most widely used scenarios in robotics: 1) direct optimization, 2) imitation/supervised learning with a neural network controller, 3) reinforcement learning, and 4) trajectory optimization using differential dynamic programming. We finally provide guidelines depending on the scenario, and make available a reference implementation of all the orientation math described.