超Grassmannian流形上的超Plücker映射

■表示■-维向量空间中的■-维子空间构成的超Grassmannian流形。本文研究超Grassmannian流形上的超Plücker映射,刻画了■和■上的超Plücker映射的齐次坐标矩阵。

基于Plücker直线的LiDAR点云配准法

高精度的LiDAR点云配准是实现点云数据整体性和保证空间目标三维表面拓扑重建的关键,本文提出了基于Plücker直线的LiDAR点云配准模型,利用Plücker直线表示LiDAR待配准点云与基准点云间的同名直线,根据同名Plücker直线重合的几何拓扑关系,建立Plücker直线共线条件方程,再用最小二乘法确定待配准LiDAR点云与基准点云间的相对位姿参数。结果表明,Plücker直线共线条件配准模型几何约束性较强,配准精度较高。

基于Plücker直线的机载LiDAR点云与航空影像的配准

机载LiDAR点云与航空影像的配准是地物提取的关键。本文提出了基于Plücker直线的机载LiDAR点云与航空影像的配准模型。基于三维线空间线变换Plücker直线方程,确定点云和影像中的同名直线间的相对位姿关系,建立Plücker直线共面条件方程,通过影像上的Plücker直线在三维线空间中的螺旋运动,实现像点到其对应LiDAR点的坐标变换。结果表明,Plücker直线共面条件配准模型简洁,将旋转和平移统一处理,避免了两者间的耦合误差,提高了配准精度,为获取高质量的地球三维空间信息提供了技术支持。

Calibration method for multi-line structured light vision sensor based on Plücker line

For the rapid calibration of multi-line structured light system,a method based on Plücker line was proposed.Most of the conventional line-structured light calibration methods extract the feature points and transform the coordinates of points to obtain the plane equation.However,a large number of points lead to complicated operation which is not suitable for the application scenarios of multi-line structured light.To solve this issue,a new calibration method was proposed that applied the form of Plücker matrix throughout the whole calibration process,instead of using the point characteristics directly.The advantage of this method is that the light plane equation can be obtained quickly and accurately in the camera coordinate frame.Correspondingly a planar target particularly for calibrating multi-line structured light was also designed.The regular lines were transformed into Plücker lines by extending the two-dimensional image plane and defining a new image space.To transform the coordinate frame of Plücker lines,the perspective projection mathematical model was re-expressed based on the Plücker matrix.According to the properties of the line and plane in the Plücker space,a linear matrix equation was efficiently constructed by combining the Plücker matrices of several coplanar lines so that the line-structured light plane equation could be furtherly solved.The experiments performed validate the proposed method and demonstrate the significant improvement in the calibration accuracy,when the test distance is 1.8 m,the root mean square(RMS)error of the three-dimensional point is within 0.08 mm.

A Line Complex-Based Evolutionary Algorithm for Many-Objective Optimization

In solving many-objective optimization problems(MaO Ps),existing nondominated sorting-based multi-objective evolutionary algorithms suffer from the fast loss of selection pressure.Most candidate solutions become nondominated during the evolutionary process,thus leading to the failure of producing offspring toward Pareto-optimal front with diversity.Can we find a more effective way to select nondominated solutions and resolve this issue?To answer this critical question,this work proposes to evolve solutions through line complex rather than solution points in Euclidean space.First,Plücker coordinates are used to project solution points to line complex composed of position vectors and momentum ones.Besides position vectors of the solution points,momentum vectors are used to extend the comparability of nondominated solutions and enhance selection pressure.Then,a new distance function designed for high-dimensional space is proposed to replace Euclidean distance as a more effective distancebased estimator.Based on them,a novel many-objective evolutionary algorithm(MaOEA)is proposed by integrating a line complex-based environmental selection strategy into the NSGAⅢframework.The proposed algorithm is compared with the state of the art on widely used benchmark problems with up to 15 objectives.Experimental results demonstrate its superior competitiveness in solving MaOPs.

Plücker Coordinate System of Screws and the Study of Generating Gears

After an introductiou of the Plucker coordinate geometry, a discussion is made of the expression of screws in Plucker coordiuates and the addition of screws’ As a result, the geometry or geuerating gears is re-studied and a formula is derived for calculating the axode leugth or hypoid gears’

Crum定理的一个新证明(英文)

导出了新的Plücker关系式,利用Wronskian技巧和Plücker关系式证明著名的Crum定理.给出了Crum定理一个新的简洁证明.在附录中证明了三个plücker关系式.

直线特征约束下利用Plücker坐标描述的LiDAR点云无初值配准方法

经典的基于点状特征匹配的地面激光雷达（light detection and ranging,LiDAR）点云配准算法实现过程中,点状特征的提取精度对算法运行结果的影响通常较大;基于迭代运算的LiDAR点云配准算法计算量大,对未知参数的初值依赖程度较高,在求解大转角刚体变换参数时算法不稳定。对此,提出了一种线状特征约束下基于Plücker直线坐标描述的LiDAR点云配准算法。立足于经典的向量代数与对偶四元数的相关理论与方法,分析并确定了Plücker直线坐标与对偶四元数之间的相互转换关系以及模型描述方法;以LiDAR点云配准前后同名线状特征的Plücker直线坐标相等为约束条件,构建了线状特征约束下基于Plücker直线坐标描述的刚体变换模型;立足于最小二乘基本准则,通过目标函数的极值化分析实现了线状特征约束下地面LiDAR点云配准参数的直接求解。实验结果表明,所构建的基于Plücker直线坐标描述的地面LiDAR点云配准模型,无需事先确定变换参数的初值,避免了多元函数的线性化过程,解除了参数结果对于迭代初值的依赖,理论上克服了迭代法在求解大转角相似变换参数时的算法不稳定问题。此外,较之单纯基于点状特征匹配的LiDAR点云配准算法,该算法可以有效地增强LiDAR点云配准过程的约束,达到提高配准质量的目的。

考虑重投影尺度因子的特征直线三维重建

文章首先用Plücker坐标表示3D直线;然后用重投影几何误差作为线特征三角法的目标函数,实现算法的最大似然估计;最后用线性和非线性两种方法分别实现空间直线的重建。在重投影时,考虑了每条直线投影时的尺度因子λ,提高了重建精度。真实图像和仿真数据的实验结果表明,该方法能够准确有效地实现直线三维重建。

基于图优化的单目线特征SLAM算法

提出了基于图优化的单目线特征同时定位和地图构建(SLAM)的方法.首先,针对主流视觉SLAM算法因采用点作为特征而导致构建的点云地图稀疏、难以准确表达环境结构信息等缺点,采用直线作为特征来构建地图.然后,根据现有线特征的SLAM算法都是基于滤波器的SLAM框架、存在线性化及更新效率的问题,采用基于图优化的SLAM解决方案以提高定位精度及地图构建的一致性和准确性.将线特征的Plücker坐标和Cayley参数化方式相结合,一方面采用Plücker坐标便于线性投影计算,另一方面采用Cayley参数化方式有利于线特征参数的非线性优化.仿真实验结果显示:所提出算法的位姿估计误差平方和与均方根误差分别是里程计位姿估计的2.5%和10.5%,是基于EKF线特征SLAM算法估计位姿误差的22.4%和33%,重投影误差仅为45.5像素;实际图像实验中的位姿估计误差平方和为958 cm^2,均方根误差为3.9413 cm,从而证明了所提出算法的有效性和准确性.

螺旋定理在曲柄摇杆机构中的应用研究

理清现有机构的演化脉络是归纳机构创新方法的重要途径之一,但机构演化的脉络在目前主要还是停留在图、文等定性描述的层面上,因此,为提升到精确、定量的层面上,可以把最原始、最典型的曲柄摇杆机构到曲柄滑块机构的演化过程作为研究对象。依据螺旋定理,得出四杆机构转动副与移动副的Plücker坐标表达式,并在此基础上基于对偶矩阵,以矩阵的形式定量地阐述曲柄摇杆机构向曲柄滑块机构的演化过程。这既有助于对机构演化脉络的认识从定性层面提升到定量层面,又为机构的演化脉络、创新方法的研究提供了一种新的路径,同时,也为计算机提供了分析并演绎机构演化过程的新手段。

Closed form of the inverse Plücker correspondence in line geometry

By mapping the homogeneous coordinates of two points in space to the Plücker coordinates of the line they determine, any transformation of type SL(4) upon points in space is mapped to a transformation of type SO0(3, 3), the latter being the connected component containing the identity of the special orthogonal transformation group of the linear space spanned by Plücker coordinates. This is the classical Plücker correspondence, two-to-one and onto. It has important applications in line geometry and projective transformations.While the explicit form of Plücker correspondence is trivial to present, its inverse in explicit form, which is also important in application, is not found in the literature. In this paper, we present a simple and unified formula for the inverse of the Plücker correspondence.

MOTOR METHOD OF ROBOT END EFFECTOR PATH GENERATION

According to the theory that rotation transformation from the basecoordinate to an arbitrary one can be described by a single equivalent rotation about agiven rotational axis,it is proposed that the orientation of robot can be described by usingthe vector of equivalent angular displacement.And the concept of pose motor and itsPlücker linear coordinate are established and used to describe the position and orientationof robot.Based on the fact that there is a one-to-one correspondence between the ori-ented straight lines in three-dimensional dual space,the interpolation plannings corre-sponding to the pose motors are developed in dual space.A new algorithm of robot CPmotion trajectory planning is presented.The correctness of the theory and derivation isconformed by an example.

The Pfaffian Technique: A (2 + 1)-Dimensional Korteweg de Vries Equation

The (2 + 1)-dimensional Korteweg de Vries (KdV) equation, which was first derived by Boiti et al., has been studied by various distinct methods. It is known that this (2 + 1)-dimensional KdV equation has rich solutions, such as multi-soliton solutions and dromion solutions. In the present article, a unified representation of its N-soliton solution is given by means of pfaffian. We’ll show that this (2 + 1)-dimensional KdV equation is nothing but the Plücker identity when its &#964-function is given by pfaffian.

一种高精度线结构光视觉传感器现场标定方法

针对现有线结构光视觉传感器标定方法存在的局限性,提出一种不需要求解光平面标定点的标定方法。根据光条图像求解平面靶标上光条在摄像机坐标系下的Plücker矩阵。在视觉传感器前合适位置将平面靶标摆放多次,联立所有光条空间直线的Plücker矩阵,求解光平面在摄像机坐标系的平面方程。最后通过非线性优化方法得到光平面方程在最大似然准则下的最优解。在标定过程中,所有光条点都参与光平面参数的计算过程,因此该方法标定结果精度高、稳健性强。实验证明,与现有方法相比该方法标定精度提高30%左右。

点线特征约束下基于单位四元数描述的LiDAR点云配准算法

针对经典的基于直线特征约束的LiDAR点云配准算法对于直线方向的高度依赖性问题,立足于直线特征的Plücker坐标描述方法,以点到线的距离为零作为点云配准的基本约束,以单位四元数作为描述空间旋转变换的基本算子,构建了点线特征约束下基于Plücker直线坐标描述的LiDAR点云配准的基本模型;立足于最小二乘法的基本准则,以配准后同名特征之间的距离平方和最小作为约束条件,根据Lagrange乘数法组建了相应的目标函数,推导并给出了求解配准参数的迭代计算模型.通过实测数据对算法进行了验证.结果表明:Plücker直线坐标的引入使得点与直线间距离的计算变得更加简洁、高效;利用单位四元数来实现空间旋转变换的描述,一方面,减少了描述参数的冗余,另一方面,对于配准模型的构建,较之于经典的基于向量代数的描述方法更为简洁;此外,较之于基于向量代数描述的LiDAR点云配准算法,四元数的引入降低了算法对迭代初值的依赖性,对任意选择的初始参数,该算法均能够得到正确的迭代结果.

面向线结构光测量的直线空间变换光平面标定方法

为提高标定精度,提出一种基于直线空间变换的光平面标定方法。首先利用互相关模板与5点滑动平均法提取激光条纹中心,然后采用正交回归法拟合图像中的光条直线方程。通过平面单应性变换获得靶标面光条直线方程,进一步再将靶标坐标系中的直线方程转换成Plücker矩阵形式。根据位姿转换关系得到相机坐标系下的直线方程,并建立超静定线面共面约束方程组,使用奇异值分解(SVD)求解光平面方程参数。所提方法测量的标准台阶块长度方均根(RMS)为0.065mm,平均误差小于0.030mm,圆柱直径的测量平均误差与RMS小于0.050mm。结果表明,利用光条自身整体信息拟合光平面,所提方法可实现较高的标定精度且标定过程简单,同时避免对每个特征点进行单独求解。

直线簇约束下的地面LiDAR点云配准方法

高精度的地面LiDAR点云配准是空间目标三维表面拓扑重建的关键,针对待配准LiDAR点云和基准LiDAR点云存在位置、姿态和比例缩放差异的问题,提出了基于直线簇的地面LiDAR点云配准方法。首先,根据直线间相交、平行和异面的拓扑关系,分别对待配准和基准LiDAR点云的直线进行聚簇,构建直线簇;然后,分别将同名直线用Plücker坐标表示,通过待配准LiDAR点云的直线簇在空间中的螺旋缩放运动,使其与基准LiDAR点云的直线簇比例尺一致,且同名Plücker直线重合,构建基于直线簇的共线条件方程,实现了比例因子和相对位姿一体化解算。实验结果表明,直线簇的螺旋缩放增强了配准方程的几何约束性,提高了抗噪声能力,实现了高精度的地面LiDAR点云配准。

Holomorphic 2-spheres in Grassmann manifolds

A generalized Frenet formula for holomorphic 2-spheresS 2 in Grassmann manifoldsG(k,n) is introduced and a generalized Pliicker formula is achieved. By use of these formulae some pinching theorems for Gaussian curvature are obtained which generalize the corresponding results about holomorphic and minimalS 2 inCP n-1. Einsteinian holomorphic subbundle of the trivial bundleS 2 ×C n is investigated.