Finding robot poses and trajectories represents a foundational aspect of robot motion planning. Despite decades of research, efficiently and robustly addressing these challenges is still difficult. Existing approaches are often plagued by various limitations, such as intricate geometric approximations, violations of collision constraints, or slow first-order convergence. In this paper, we introduce two novel optimization formulations that offer provable robustness, achieving second-order convergence while requiring only a convex approximation of the robot's links and obstacles. Our first method, known as the Explicit Collision Barrier (ECB) method, employs a barrier function to guarantee separation between convex objects. ECB uses an efficient matrix factorization technique, enabling a second-order Newton's method with an iterative complexity linear in the number of separating planes. Our second method, referred to as the Implicit Collision Barrier (ICB) method, further transforms the separating planes into implicit functions of robot poses. We show such an implicit objective function is twice-differentiable, with derivatives evaluated at a linear complexity. To assess the effectiveness of our approaches, we conduct a comparative study with a first-order baseline algorithm across six testing scenarios. Our results unequivocally justify that our method exhibits significantly faster convergence rates compared to the baseline algorithm.

翻译：寻找机器人位姿与轨迹是机器人运动规划的基础问题。尽管经过数十年研究，高效且鲁棒地解决这些挑战仍然困难。现有方法常受限于各种缺陷，例如复杂的几何近似、违反碰撞约束或缓慢的一阶收敛性。本文提出两种新颖的优化公式，具有可证明的鲁棒性，在仅需机器人连杆与障碍物的凸近似条件下，即可实现二阶收敛。第一种方法称为显式碰撞屏障法（ECB），采用屏障函数保证凸体之间的分离。ECB使用高效矩阵分解技术，可实现二阶牛顿法，其迭代复杂度与分离平面数量呈线性关系。第二种方法称为隐式碰撞屏障法（ICB），进一步将分离平面转化为机器人位姿的隐函数。我们证明该隐式目标函数是二次可微的，且导数计算具有线性复杂度。为评估方法有效性，我们在六个测试场景中与一阶基线算法进行对比研究。结果明确表明，相较于基线算法，我们的方法具有显著更快的收敛速度。