基于jerk能量最小化的空间五次G2插值曲线构造

Construction of spatial quintic G2 interpolating curves by jerk energy minimization

光顺曲线具有简单的曲率轮廓,在计算机辅助设计、轨道规划和相关应用领域有重要的需求.首先,针对给定的G2数据,构造一条含4个自由参数的空间五次Bézier曲线使其在端点满足特定曲率和Frenet标架.然后,使用jerk能量近似表示曲线的曲率变化,定义目标函数为jerk能量和曲线长度的加权组合,提出基于jerk能量最小化的空间五次G2插值曲线构造方法.最后,将目标函数简化为二元四次多项式,使用投影牛顿法求解约束优化问题.与之前基于应变能最小化的方法相比,实例表明该方法能够生成更光顺的曲线和更好地处理空间曲线.

Fair curves possess simple curvature profiles and secure significant demands in computer aided design,trajectory planning and related application fields.First,for given G2 data,a spatial quintic Bézier curve with four free parameters is constructed to satisfy specified curvatures and Frenet frames at endpoints.Then,by defining the objective function as a weighted combination of jerk energy(used to approximate curvature variation of a curve)and curve length,a construction method is proposed for obtaining spatial quintic G2 interpolating curves by jerk energy minimization.Finally,the objective function after simplification is converted into a bivariate quartic polynomial,and the constrained optimization problem is solved through the projected Newton method.Compared to the previous method based on strain energy minimization,numerical examples demonstrate that,in the new method,curves with better fairness can be generated and the case of spatial curves can be handled.