Many robotic applications—from drone navigation to surgical automation—require planning trajectories that respect directional constraints, which are inherently non-Euclidean. Trajectory planning in robotics often involves orientation or directional constraints that naturally reside on Riemannian manifolds. In particular, when robot motion is governed by orthogonality constraints—such as in directional alignment, pose smoothing, or coordinated orientation—the optimization variables lie on the Stiefel manifold. Traditional Euclidean optimization methods struggle with such constraints, leading to suboptimal or infeasible trajectories. This article proposes a Riemannian optimization framework tailored to the Stiefel manifold for robot trajectory planning. By leveraging the intrinsic geometry of the manifold, we design efficient algorithms that compute gradient updates via Riemannian geometry and enforce feasibility through retraction mappings. The proposed method enables smooth and constraint-respecting trajectory generation, particularly in scenarios where robot poses or motion directions are orthonormal by design. We validate our approach through simulated path planning tasks involving orientation-aware constraints. Compared to baseline projection-based or unconstrained methods, our approach achieves better convergence behavior, improved geometric consistency, and enhanced applicability in robotics. The results demonstrate that manifold-aware optimization not only improves theoretical soundness but also provides practical benefits for engineering systems.
Ma et al. (Sun,) studied this question.