Automatically generating agile whole-body motions for legged and humanoid robots remains a fundamental challenge in robotics. While numerous trajectory optimization approaches have been proposed, there is no clear guideline on how the choice of floating-base space parameterization affects performance, especially for agile behaviors involving complex contact dynamics. In this paper, we present a comparative study of different parameterizations for direct transcription-based trajectory optimization of agile motions in legged systems. We systematically evaluate several common choices under identical optimization settings to ensure a fair comparison. Furthermore, we introduce a novel formulation based on the tangent space of SE(3) for representing the robot's floating-base pose, which, to our knowledge, has not received attention from the literature. This approach enables the use of mature off-the-shelf numerical solvers without requiring specialized manifold optimization techniques. We hope that our experiments and analysis will provide meaningful insights for selecting the appropriate floating-based representation for agile whole-body motion generation.
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