求解测地线边界值问题的均匀B样条配点法

A Uniform B-Spline Collocation Method for Solving the Boundary Value Problem of Geodesic Curves

曲面上的测地线在图像处理、机器人学、数控加工等领域有着广泛的应用.针对参数曲面,提出了采用二阶、三阶均匀B样条配点法求解测地线边界值问题.首先建立了测地线边界值问题的离散控制方程,在给定初始逼近后,该方程可采用近似牛顿迭代法进行求解;然后推导了B样条配点法的计算精度;最后在Matlab2016b软件环境下,结合球面、圆环面以及样条曲面对所提出的方法进行了验证.实验考察了不同方法的计算精度、时间成本以及计算效率,结果表明三阶B样条配点法获得相同计算精度所需的时间通常更少,即该方法在计算效率方面具有优势.当计算测地线长度误差为0.01%时,三阶B样条配点法相较于Chen的近似测地线法节省时间成本5.5%~26.9%.对于二阶B样条配点法,其计算效率与Kasap的中心差分法相当.所提出的方法计算结果为连续曲线,无需额外的插值函数,一个潜在的应用场景为自动铺带轨迹规划.

Geodesics on surfaces are widely used in image processing,robotics,NC machining,etc.In this work,the second and third order uniform B-spline collocation methods are proposed to solve the BVP(boundary value problem)of geodesics on parametric surfaces.Firstly,the discrete governing equation of geodesics is established,which can be solved by the approximate Newton iteration method after an initial approximation is provided.Then the accuracy of the B-spline collocation method is deduced.Finally,the proposed method is verified on the spherical surface,torus surface and spline surface in Matlab 2016b software environment.The computational ac-curacy,time cost and efficiency of different methods are investigated.The experimental results show that the third order B-spline collocation method generally requires less time to obtain the same accuracy,which indicates the method has advantages in computational efficiency.When the length error of the geodesic is 0.01%,the third or-der B-spline collocation method saves 5.5%to 26.9%of the time cost compared with Chen’s geodesic-like me-thod.For the second order B-spline collocation method,the computational efficiency is comparable to Kasap’s cen-tral difference method.The proposed method ensures continuous geodesic curves,and no additional interpolation function is needed.A potential application of this method is trajectory planning for automated tape placement.