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.

Robots hold great promise for performing repetitive or hazardous tasks, but achieving human-like dexterity, especially in contact-rich and dynamic environments, remains challenging. Rigid robots, which rely on position or velocity control, often struggle with maintaining stable contact and applying consistent force in force-intensive tasks. Learning from Demonstration has emerged as a solution, but tasks requiring intricate maneuvers, such as powder grinding, present unique difficulties. This paper introduces Diffusion Policies For Compliant Manipulation (DIPCOM), a novel diffusion-based framework designed for compliant control tasks. By leveraging generative diffusion models, we develop a policy that predicts Cartesian end-effector poses and adjusts arm stiffness to maintain the necessary force. Our approach enhances force control through multimodal distribution modeling, improves the integration of diffusion policies in compliance control, and extends our previous work by demonstrating its effectiveness in real-world tasks. We present a detailed comparison between our framework and existing methods, highlighting the advantages and best practices for deploying diffusion-based compliance control.

Handling orientations of robots and objects is a crucial aspect of many applications. Yet, ever so often, there is a lack of mathematical correctness when dealing with orientations, especially in learning pipelines involving, for example, artificial neural networks. In this paper, we investigate reinforcement learning with orientations and propose a simple modification of the network's input and output that adheres to the Lie group structure of orientations. As a result, we obtain an easy and efficient implementation that is directly usable with existing learning libraries and achieves significantly better performance than other common orientation representations. We briefly introduce Lie theory specifically for orientations in robotics to motivate and outline our approach. Subsequently, a thorough empirical evaluation of different combinations of orientation representations for states and actions demonstrates the superior performance of our proposed approach in different scenarios, including: direct orientation control, end effector orientation control, and pick-and-place tasks.