用约束迹的半定规划提升机器人估计与标定的准确性
Certifiably Optimal Estimation and Calibration in Robotics via Trace-Constrained Semi-Definite Programming
- 通过固定迹的半定规划松弛非凸问题，实现全局最优解
- 设计梯度优化方法将解投影到低秩近似，提升实际可用性
- 适用于位姿估计、手眼标定等任务，适合追求精度的机器人研究者
机器人中的许多非凸问题可通过半定规划（SDP）转化为凸问题，并求得全局最优解。然而，这些解的实际性能高度依赖于将其舍入为秩-1矩阵的过程，这一过程常难以实现。本文聚焦于迹约束的半定规划（TCSDP），其中决策变量为固定迹的半正定矩阵。我们提出一种基于梯度的精化方法，将松弛后的解投影至秩-1、低成本候选解。同时，针对旋转、平移等常见机器人量，给出了迹约束的SDP松弛形式，并构建了模块化虚拟机器人抽象，简化跨场景建模。实验表明，该框架可广泛应用于机器人任务，我们在透视n点（PnP）估计、手眼标定及双机器人系统标定中展示了其有效性。
原文摘要 · Abstract (English)
Many nonconvex problems in robotics can be relaxed into convex formulations via Semi-Definite Programming (SDP) that can be solved to global optimality. The practical quality of these solutions, however, critically depends on rounding them to rank-1 matrices, a condition that can be challenging to achieve. In this work, we focus on trace-constrained SDPs (TCSDPs), where the decision variables are Positive Semi-Definite (PSD) matrices with fixed trace values. We show that the latter can be used to design a gradient-based refinement procedure that projects relaxed SDP solutions toward rank-1, low-cost candidates. We also provide fixed-trace SDP relaxations for common robotic quantities, such as rotations and translations, and a modular virtual robot abstraction that simplifies modeling across different problem settings. We demonstrate that our trace-constrained SDP framework can be applied to many robotics tasks, and we showcase its effectiveness through simulations in Perspective-n-Point (PnP) estimation, hand-eye calibration, and dual-robot system calibration.
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