As a parametric motion representation, B\'ezier curves have significant applications in polynomial trajectory optimization for safe and smooth motion planning of various robotic systems, including flying drones, autonomous vehicles, and robotic manipulators. An essential component of B\'ezier curve optimization is the optimization objective, as it significantly influences the resulting robot motion. Standard physical optimization objectives, such as minimizing total velocity, acceleration, jerk, and snap, are known to yield quadratic optimization of B\'ezier curve control points. In this paper, we present a unifying graph-theoretic perspective for defining and understanding B\'ezier curve optimization objectives using a consensus distance of B\'ezier control points derived based on their interaction graph Laplacian. In addition to demonstrating how standard physical optimization objectives define a consensus distance between B\'ezier control points, we also introduce geometric and statistical optimization objectives as alternative consensus distances, constructed using finite differencing and differential variance. To compare these optimization objectives, we apply B\'ezier curve optimization over convex polygonal safe corridors that are automatically constructed around a maximal-clearance minimal-length reference path. We provide an explicit analytical formulation for quadratic optimization of B\'ezier curves using B\'ezier matrix operations. We conclude that the norm and variance of the finite differences of B\'ezier control points lead to simpler and more intuitive interaction graphs and optimization objectives compared to B\'ezier derivative norms, despite having similar robot motion profiles.

翻译：作为参数化运动表示方法，贝塞尔曲线在多机器人系统（包括飞行无人机、自动驾驶车辆和机械臂）的安全平滑运动规划中，于多项式轨迹优化领域具有重要应用。优化目标作为贝塞尔曲线优化的核心要素，显著影响最终生成的机器人运动轨迹。已知标准物理优化目标（如最小化总速度、加速度、加加速度和急动度）可转化为贝塞尔曲线控制点的二次优化问题。本文提出基于图论的统一视角，利用根据控制点相互作用图拉普拉斯算子导出的贝塞尔控制点共识距离，来定义和理解贝塞尔曲线优化目标。除阐明标准物理优化目标如何定义贝塞尔控制点间的共识距离外，我们还引入基于有限差分和微分方差构建的几何与统计优化目标作为替代性共识距离。为比较这些优化目标，我们在绕具有最大间隙与最小长度的参考路径自动构建的凸多边形安全走廊上实施贝塞尔曲线优化。通过贝塞尔矩阵运算，我们给出了贝塞尔曲线二次优化的显式解析公式。研究结论表明：相较于贝塞尔导数范数，贝塞尔控制点有限差分的范数与方差能生成更简洁直观的相互作用图和优化目标，同时保持相近的机器人运动曲线特征。