首页> 中国专利> 一种基于PNP透视模型的合作目标位姿精度测量方法

一种基于PNP透视模型的合作目标位姿精度测量方法

页面导航

摘要
著录项
法律信息
说明书
相似文献

摘要

本发明公开了一种基于PNP透视模型的合作目标位姿精度测量方法，包括如下步骤：在合作目标外表面安装N个视觉标记点，采用相机对视觉标记进行标记图像的拍摄；对合作目标与相机之间的位置姿态的初始值进行预先设定；其中第i个视觉标记点的三维空间坐标和有效焦距的水平、垂直分量，主点坐标的水平、垂直分量为未确定性系统误差，误差极限已知；第i个视觉标记点对应标记图像中视觉标记点中心二维坐标为随机误差，对应的误差极限已知；量化解析各项参数对位姿误差的影响权重系数；计算合作目标与相机之间的位置姿态的未确定性系统误差分量和随机误差分量并加权获得目标位姿测量误差，完成对目标位姿测量精度的预估。

著录项

公开/公告号CN104729481A

专利类型发明专利
公开/公告日2015-06-24

原文格式PDF
申请/专利权人北京空间飞行器总体设计部;
展开▼

申请/专利号CN201510107422.3
发明设计人谭启蒙;李劲东;蔡伟;胡成威;高升;袁宝峰;陈磊;杜晓东;
展开▼

申请日2015-03-12
分类号G01C11/00(20060101);
代理机构11120 北京理工大学专利中心;
代理人高燕燕;仇蕾安
地址 100094 北京市海淀区友谊路104号
入库时间 2023-12-18 09:18:47

法律信息

法律状态公告日

法律状态信息

法律状态
2017-05-24

授权

授权
2015-07-22

实质审查的生效 IPC(主分类):G01C11/00 申请日:20150312

实质审查的生效
2015-06-24

公开

公开

说明书

技术领域

本发明属于光电测量领域，涉及一种基于PNP透视模型的合作目标位姿精度测量方法，尤其适用于空间合作目标三维位置姿态测量精度的量化估计方法。

背景技术

空间合作目标通常是安装有人工视觉标记等特征标识的观测目标，其中，人工视觉标记主要是指已知外形、尺寸、颜色、数量以及空间布局的视觉标记点集合。利用单目视觉相机包含人工视觉标记的目标图像信息，根据各标记点在目标坐标系下的三维空间坐标值及其在像平面中的特征点中心二维坐标值之间的一一对应关系约束，计算出的旋转矩阵和平移矢量，即可准确反映合作目标相对于相机之间的空间位置与姿态，上述测量过程应归结为PNP透视模型问题(Perspective-N-Point Problem)。因此，视觉标记不仅是单目视觉相机的观测对象，还是空间合作目标位姿测量的基准与解算依据。

在PNP透视模型中，N表示标记点的数目，该问题的求解条件要求标记点个数N不小于3。当N>5时，可以利用直接线性变换法(Direct Linear Transformation)求解最小二乘解作为初值，再代入非线性优化算法，解算出位姿参数的最优解；当N≤5时，PNP问题转化为求解多元高次非线性方程组，且解不唯一。

基于PNP透视模型的目标位姿测量精度的影响因素主要包括相机内参标定误差、标记点三维空间坐标值获取误差以及图像标记点中心二维坐标定位误差等参数。目前，围绕上述模型的目标位姿测量精度的合理量化预估的研究，主要集中于定性、定量分析单一误差因素对位姿测量精度的影响。例如：Larry Davis在《Predicting Accuracy in Pose Estimation for Marker-based Tracking》文献中，利用公式推导与仿真试验相结合，归纳出观测目标自身的设计尺寸越小，其三维空间坐标值获取误差越大，则位姿测量误差将越大。程凌、周前祥在《基于视觉方法的交会对接相对位姿测量》文献中，通过建立高精度的测量控制场以提高标记点三维空间坐标的精度定位，以进一步改善目标位姿的测量精度。朱枫等在《合作目标姿态对视觉位姿测量精度的影响分析》文献中，针对P3P 透视模型问题，量化分析目标的初始位姿对位姿测量精度的影响，仿真实验结果表明，当3个标记点构成的三角形所在平面与相机像平面垂直时位姿精度最高，而当这两个平面平行时位姿精度最差的结论。周静等在《摄像机标定参数误差对位姿测量精度的影响》文献中，从标定方法选择的角度，量化分析相机标定参数误差对P3P位姿精度的影响关系，公式推导与试验结果表明，测量距离方向位置精度主要取决于焦比误差和光轴方向平移量误差的影响，二者分别在远距离段和近距离段起主要作用；姿态角精度则主要受限于主点坐标误差和外姿态角误差。郝颖明在《基于点特征的位姿测量精度与鲁棒性研究》文献中，基于简化条件下的理论分析与一般条件下的误差统计分析，针对P3P透视模型，量化分析了6个位姿分量误差(即三个旋转角度和三个平移分量)与相机标定误差、目标模型误差、图像坐标检测误差之间的关系表达式，并通过直接仿真试验得出以下结论：(1)图像坐标检测误差和相机标定误差对位姿误差的影响远大于目标模型误差对位姿结果的影响，故后者可以忽略不计；(2)当其他因素均固定不变时，测量距离与位姿误差呈正比例关系；(3)在远距离处，图像坐标检测误差对位姿误差的影响占主导地位；(4)在近距离处，相机内参标定误差对位姿测量误差的影响占主导地位。

现有研究成果往往都是针对P3P透视模型开展位姿精度量化分析，仅专注于单一误差因素对位姿测量精度的影响权重。对于N>3的PNP透视模型的目标位姿测量精度量化估计等方面的研究也相对较少，尚未检索到相关文献。

此外，上述文献均未涉及如何将相机内参标定误差、标记点三维空间坐标值获取误差以及图像标记点中心二维坐标定位误差等多种误差因素有效融合在一起，并建立其与位姿精度的函数映射关系，因此，根本无法全面量化预估目标位姿精度的整体水平。

发明内容

有鉴于此，本发明提供了一种基于PNP透视模型的合作目标位姿精度测量方法，方法不受标记点个数N的约束限制，量化解析相机内参标定误差、视觉标记点三维空间坐标值获取误差、视觉标记点中心二维坐标定位误差等一种或多种因素对位姿误差的影响关系，实现测量系统设计初期对目标位姿测量精度水平的准确预估与客观评价。

为了达到上述目的，本发明的技术方案为：该方法包括如下步骤：

步骤(1)、在合作目标外表面安装N个视觉标记点，采用相机对视觉标记进行标记图像的拍摄。

步骤(2)、建立相机坐标系O_c-X_cY_cZ_c：相机的光心为原点O_c，光轴为Z_c轴，相机所拍摄图像平面的水平和垂直方向分别为X_c轴和Y_c轴；建立关于合作目标的目标坐标系O_W-X_WY_WZ_W：合作目标的质心为原点O_W，X_W、Y_W、Z_W与相机坐标系中的X、Y、Z轴相平行，且正方向均保持一致；其中第i个视觉标记点的三维空间坐标为(X_wi,Y_wi,Z_wi)及其对应标记图像中视觉标记点中心二维坐标为(u_i,v_i)；

对于合作目标与相机之间的位置姿态的初始值进行预先设定。

相机内参标定结果为：有效焦距的水平、垂直分量分别为f_x,f_y，主点坐标的水平、垂直分量分别为u₀,v₀。

步骤(3)、将步骤(1)中的参数f_x,f_y,u₀,v₀以及X_wi,Y_wi,Z_wi确定为未确定性系统误差，f_x,f_y,u₀,v₀,X_wi,Y_wi,Z_wi各自对应的误差极限已知，且分别为 e(f_x),e(f_y),e(u₀),e(v₀),e(X_wi),e(Y_wi),e(Z_wi)。

将u_i,v_i确定为随机误差，对应的误差极限已知，且分别为δ(u_i),δ(v_i)。

步骤(4)、量化解析各项参数对位姿误差的影响权重系数。

对于第i个视觉标记点，其位姿测量计算公式包括F_2i-1和F_2i，二者分别为：

$(\begin{matrix} F_{2 i - 1} (u_{i}, v_{i}, f_{x}, f_{y}, u_{0}, v_{0}, X_{wi}, Y_{wi}, Z_{wi}, α, β, γ, t_{x}, t_{z}) = (u_{i} - u_{0}) \cdot (- \sin β \cdot X_{wi} + \cos β \sin α \cdot Y_{wi} + \cos β \cos α \cdot Z_{wi} + t_{z}) \\ - f_{x} \cdot [\cos γ \cdot \cos β \cdot X_{wi} + (\cos γ \cdot \sin β \cdot \sin α - \sin γ) \cdot Y_{wi} + (\cos γ \cdot \sin β \cdot \cos α - \sin γ \cdot \sin α) \cdot Z_{wi} + t_{x}]; \\ F_{2 i} (u_{i}, v_{i}, f_{x}, f_{y}, u_{0}, v_{0}, X_{wi}, Y_{wi}, Z_{wi}, α, β, γ, t_{y}, t_{z}) = (v_{i} - v_{0}) \cdot (- \sin β \cdot X_{wi} + \cos β \sin α \cdot Y_{wi} + \cos α \cdot Z_{wi} + t_{z}) \\ - f_{y} \cdot [\sin γ \cdot \cos β \cdot X_{wi} + (\sin γ \cdot \sin β \cdot \sin α + \cos γ \cdot \cos α) \cdot Y_{wi} + (\sin γ \cdot \sin β \cdot \cos α + \cos γ \cdot \sin α) \cdot Z_{wi} + t_{y}]; \end{matrix})$

令：

$M = (\begin{matrix} \frac{\partial F_{1}}{\partial α} & \frac{\partial F_{1}}{\partial β} & \frac{\partial F_{1}}{\partial γ} & \frac{\partial F_{1}}{\partial t_{x}} & \frac{\partial F_{1}}{\partial t_{y}} & \frac{\partial F_{1}}{\partial t_{z}} \\ \frac{\partial F_{2}}{\partial α} & \frac{\partial F_{2}}{\partial β} & \frac{\partial F_{2}}{\partial γ} & \frac{\partial F_{2}}{\partial t_{x}} & \frac{\partial F_{2}}{\partial t_{y}} & \frac{\partial F_{2}}{\partial t_{z}} \\ . . . & . . . & . . . & . . . & . . . & . . . \\ \frac{\partial F_{2 i - 1}}{\partial α} & \frac{\partial F_{2 i - 1}}{\partial β} & \frac{\partial F_{2 i - 1}}{\partial γ} & \frac{\partial F_{2 i - 1}}{\partial t_{x}} & \frac{\partial F_{2 i - 1}}{\partial t_{y}} & \frac{\partial F_{2 i - 1}}{\partial t_{z}} \\ \frac{\partial F_{2 i}}{\partial α} & \frac{\partial F_{2 i}}{\partial β} & \frac{\partial F_{2 i}}{\partial γ} & \frac{\partial F_{2 i}}{\partial t_{x}} & \frac{\partial F_{2 i}}{\partial t_{y}} & \frac{\partial F_{2 i}}{\partial t_{z}} \\ . . . & . . . & . . . & . . . & . . . & . . . \\ \frac{{\partial F}_{2 N - 1}}{\partial α} & \frac{\partial F_{2 N - 1}}{\partial β} & \frac{\partial F_{2 N - 1}}{\partial γ} & \frac{\partial F_{2 N - 1}}{\partial t_{x}} & \frac{\partial F_{2 N - 1}}{\partial t_{y}} & \frac{\partial F_{2 N - 1}}{\partial t_{z}} \\ \frac{\partial F_{2 N}}{\partial α} & \frac{\partial F_{2 N}}{\partial β} & \frac{\partial F_{2 N}}{\partial γ} & \frac{\partial F_{2 N}}{\partial t_{x}} & \frac{\partial F_{2 N}}{\partial t_{y}} & \frac{\partial F_{2 N}}{\partial t_{z}} \end{matrix}) . X^{'} = \cdot (\begin{matrix} \frac{\partial α}{\partial f_{x}} & \frac{\partial α}{\partial f_{y}} & \frac{\partial α}{\partial u_{0}} & \frac{\partial α}{\partial v_{0}} & \frac{\partial α}{{\partial u}_{i}} & \frac{\partial α}{\partial v_{i}} & \frac{\partial α}{\partial X_{wi}} & \frac{\partial α}{\partial Y_{wi}} & \frac{\partial α}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \frac{\partial β}{\partial f_{x}} & \frac{\partial β}{\partial f_{y}} & \frac{\partial β}{\partial u_{0}} & \frac{\partial β}{\partial v_{0}} & \frac{\partial β}{\partial u_{i}} & \frac{\partial β}{\partial v_{i}} & \frac{\partial β}{\partial X_{wi}} & \frac{\partial β}{\partial Y_{wi}} & \frac{\partial β}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \frac{\partial γ}{\partial f_{x}} & \frac{\partial γ}{\partial f_{y}} & \frac{\partial γ}{\partial u_{0}} & \frac{\partial γ}{\partial v_{0}} & \frac{\partial γ}{\partial u_{i}} & \frac{\partial γ}{{\partial v}_{i}} & \frac{\partial γ}{\partial X_{wi}} & \frac{\partial γ}{\partial Y_{wi}} & \frac{\partial γ}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \frac{\partial t_{x}}{\partial f_{x}} & \frac{\partial t_{x}}{\partial f_{y}} & \frac{\partial t_{x}}{\partial u_{0}} & \frac{\partial t_{x}}{\partial v_{0}} & \frac{\partial t_{x}}{\partial u_{i}} & \frac{\partial t_{x}}{\partial v_{i}} & \frac{\partial t_{x}}{\partial X_{wi}} & \frac{\partial t_{x}}{\partial Y_{wi}} & \frac{\partial t_{x}}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \frac{\partial t_{y}}{\partial f_{x}} & \frac{\partial t_{y}}{\partial f_{y}} & \frac{\partial t_{y}}{\partial u_{0}} & \frac{\partial t_{y}}{\partial v_{0}} & \frac{\partial t_{y}}{\partial u_{i}} & \frac{\partial t_{y}}{\partial v_{i}} & \frac{\partial t_{y}}{\partial X_{wi}} & \frac{\partial t_{y}}{\partial Y_{wi}} & \frac{\partial t_{y}}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \frac{\partial t_{z}}{\partial f_{x}} & \frac{\partial t_{z}}{\partial f_{y}} & \frac{\partial t_{z}}{\partial u_{0}} & \frac{\partial t_{z}}{\partial v_{0}} & \frac{\partial t_{z}}{\partial u_{i}} & \frac{\partial t_{z}}{\partial v_{i}} & \frac{\partial t_{z}}{\partial X_{wi}} & \frac{\partial t_{z}}{\partial Y_{wi}} & \frac{\partial t_{z}}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \end{matrix})$

$Y^{'} = (\begin{matrix} \frac{\partial F_{1}}{\partial f_{x}} & \frac{\partial F_{1}}{\partial f_{y}} & \frac{\partial F_{1}}{\partial u_{0}} & \frac{\partial F_{1}}{\partial v_{0}} & \frac{\partial F_{1}}{\partial u_{i}} & \frac{\partial F_{1}}{\partial v_{i}} & \frac{\partial F_{1}}{\partial X_{wi}} & \frac{\partial F_{1}}{\partial Y_{wi}} & \frac{\partial F_{1}}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \frac{\partial F_{2}}{\partial f_{x}} & \frac{\partial F_{2}}{\partial f_{y}} & \frac{\partial F_{2}}{\partial u_{0}} & \frac{\partial F_{2}}{\partial v_{0}} & \frac{\partial F_{2}}{\partial v_{0}} & \frac{\partial F_{2}}{\partial v_{i}} & \frac{\partial F_{2}}{\partial X_{wi}} & \frac{\partial F_{2}}{\partial Y_{wi}} & \frac{\partial F_{2}}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \frac{\partial F_{2 i 1}}{\partial f_{x}} & \frac{\partial F_{2 i} 1}{\partial f_{y}} & \frac{\partial F_{2 i 1}}{\partial u_{0}} & \frac{\partial F_{2 i 1}}{\partial v_{0}} & \frac{\partial F_{2 i 1}}{\partial v_{0}} & \frac{\partial F_{2 i 1}}{\partial v_{i}} & \frac{\partial F_{2 i 1}}{\partial X_{wi}} & \frac{\partial F_{2 i 1}}{\partial Y_{wi}} & \frac{\partial F_{2 i 1}}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \frac{\partial F_{2 i}}{\partial f_{x}} & \frac{\partial F_{2 i}}{\partial f_{y}} & \frac{\partial F_{2 i}}{\partial u_{0}} & \frac{\partial F_{2 i}}{\partial v_{0}} & \frac{\partial F_{2 i}}{\partial v_{0}} & \frac{\partial F_{2 i}}{\partial v_{i}} & \frac{\partial F_{2 i}}{\partial X_{wi}} & \frac{\partial F_{2 i}}{\partial Y_{wi}} & \frac{\partial F_{2 i}}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \frac{\partial F_{2 N 1}}{\partial f_{x}} & \frac{\partial F_{2 N 1}}{\partial f_{y}} & \frac{\partial F_{2 N 1}}{\partial u_{0}} & \frac{{\partial F}_{2 N 1}}{\partial v_{0}} & \frac{\partial F_{2 N 1}}{\partial v_{0}} & \frac{\partial F_{2 N 1}}{\partial v_{i}} & \frac{\partial F_{2 N 1}}{\partial X_{wi}} & \frac{\partial F_{2 N 1}}{\partial Y_{wi}} & \frac{\partial F_{2 N 1}}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \frac{\partial F_{2 N}}{\partial f_{x}} & \frac{\partial F_{2 N}}{\partial f_{y}} & \frac{\partial F_{2 N}}{\partial u_{0}} & \frac{\partial F_{2 N}}{\partial v_{0}} & \frac{\partial F_{2 N}}{\partial v_{0}} & \frac{\partial F_{2 N}}{\partial v_{i}} & \frac{\partial F_{2 N}}{\partial X_{wi}} & \frac{\partial F_{2 N}}{\partial Y_{wi}} & \frac{\partial F_{2 N}}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \end{matrix}),$ 其中X′为6行×2N列的矩阵，自第 10列开始重复前9列内容；Y′为2N行×2N列的矩阵，且自第10列开始重复前9列内容。

求解方程M·X′＝-Y′中的X′，存在以下两种情况：

①当2×N＝6时，利用最小二乘法求解，得到最优解为：

X′＝(-1)·M^-1·Y′

②当2×N＞6时，利用最小二乘法求解，得到最优解为：

X′＝(-1)·(M^T·M)^-1·(M^T·Y′)。

步骤(5)、在随机误差u_i,v_i的影响下，合作目标与相机之间的位置姿态 [t_x,t_y,t_z,α,β,γ]^T的随机误差分量分别为δ(t_x),δ(t_y),δ(t_z),δ(α),δ(β),δ(γ)；根据如下公式进行计算：

$(\begin{matrix} δ (α) = \sqrt{Σ_{i = 1}^{N} [{(\frac{\partial α}{\partial u_{i}} \cdot δ (u_{i}))}^{2} + {(\frac{\partial α}{\partial v_{i}} \cdot δ (v_{i}))}^{2}]} \\ δ (β) = \sqrt{Σ_{i = 1}^{N} [{(\frac{\partial β}{\partial u_{i}} \cdot δ (u_{i}))}^{2} + {(\frac{\partial β}{\partial v_{i}} \cdot δ (v_{i}))}^{2}]} \\ δ (γ) = \sqrt{Σ_{i = 1}^{N} [{(\frac{\partial γ}{\partial u_{i}} \cdot δ (u_{i}))}^{2} + {(\frac{\partial γ}{\partial v_{i}} \cdot δ (v_{i}))}^{2}]} \\ δ (t_{x}) = \sqrt{Σ_{i = 1}^{N} [{(\frac{\partial t_{x}}{\partial u_{i}} \cdot δ (u_{i}))}^{2} + {(\frac{\partial t_{x}}{\partial v_{i}} \cdot δ (v_{i}))}^{2}]} \\ δ (t_{y}) = \sqrt{Σ_{i = 1}^{N} [{(\frac{\partial t_{y}}{\partial u_{i}} \cdot δ (u_{i}))}^{2} + {(\frac{\partial t_{y}}{\partial v_{i}} \cdot δ (v_{i}))}^{2}]} \\ δ (t_{z}) = \sqrt{Σ_{i = 1}^{N} [{(\frac{\partial t_{z}}{\partial u_{i}} \cdot δ (u_{i}))}^{2} + {(\frac{\partial t_{z}}{\partial v_{i}} \cdot δ (v_{i}))}^{2}]} \end{matrix}) .$

在未确定性系统误差f_x,f_y,u₀,v₀,X_wi,Y_wi,Z_wi影响下，合作目标与相机之间的位置姿态[t_x,t_y,t_z,α,β,γ]^T的未确定性系统误差分量分别为e(t_x),e(t_y),e(t_z),e(α),e(β),e(γ)；根据如下公式进行计算：

$(\begin{matrix} e (α) = \sqrt{{[\frac{\partial α}{\partial f_{x}} \cdot e (f_{x})]}^{2} + {[\frac{\partial α}{\partial f_{y}} \cdot e (f_{y})]}^{2} + {[\frac{\partial α}{\partial u_{0}} \cdot e (u_{0})]}^{2} + {[\frac{\partial α}{\partial v_{0}} \cdot e (v_{0})]}^{2} + Σ_{i = 1}^{N} {{[\frac{\partial α}{\partial X_{wi}} \cdot e (X_{wi})]}^{2} + {[\frac{\partial α}{\partial Y_{wi}} \cdot e (Y_{wi})]}^{2} + {[\frac{\partial α}{\partial Z_{wi}} \cdot e (Z_{wi})]}^{2}}} \\ e (β) = \sqrt{{[\frac{\partial β}{\partial f_{x}} \cdot e (f_{x})]}^{2} + {[\frac{\partial β}{\partial f_{y}} \cdot e (f_{y})]}^{2} + {[\frac{\partial β}{\partial u_{0}} \cdot e (u_{0})]}^{2} + {[\frac{\partial β}{\partial v_{0}} \cdot e (v_{0})]}^{2} + Σ_{i = 1}^{N} {{[\frac{\partial β}{\partial X_{wi}} \cdot e (X_{wi})]}^{2} + {[\frac{\partial β}{\partial Y_{wi}} \cdot e (Y_{wi})]}^{2} + {[\frac{\partial β}{\partial Z_{wi}} \cdot e (Z_{wi})]}^{2}}} \\ e (γ) = \sqrt{{[\frac{\partial γ}{\partial f_{x}} \cdot e (f_{x})]}^{2} + {[\frac{\partial γ}{\partial f_{y}} \cdot e (f_{y})]}^{2} + {[\frac{\partial γ}{\partial u_{0}} \cdot e (u_{0})]}^{2} + {[\frac{\partial γ}{\partial v_{0}} \cdot e (v_{0})]}^{2} + Σ_{i = 1}^{N} {{[\frac{\partial γ}{\partial X_{wi}} \cdot e (X_{wi})]}^{2} + {[\frac{\partial γ}{\partial Y_{wi}} \cdot e (Y_{wi})]}^{2} + {[\frac{\partial γ}{\partial Z_{wi}} \cdot e (Z_{wi})]}^{2}}} \\ e (t_{x}) = \sqrt{{[\frac{\partial t_{x}}{\partial f_{x}} \cdot e (f_{x})]}^{2} + {[\frac{\partial t_{x}}{\partial f_{y}} \cdot e (f_{y})]}^{2} + {[\frac{\partial t_{x}}{\partial u_{0}} \cdot e (u_{0})]}^{2} + {[\frac{\partial t_{x}}{\partial v_{0}} \cdot e (v_{0})]}^{2} + Σ_{i = 1}^{N} {{[\frac{\partial t_{x}}{\partial X_{wi}} \cdot e (X_{wi})]}^{2} + {[\frac{\partial t_{x}}{\partial Y_{wi}} \cdot e (Y_{wi})]}^{2} + {[\frac{\partial t_{x}}{\partial Z_{wi}} \cdot e (Z_{wi})]}^{2}}} \\ e (t_{y}) = \sqrt{{[\frac{\partial t_{y}}{\partial f_{x}} \cdot e (f_{x})]}^{2} + {[\frac{\partial t_{y}}{\partial f_{y}} \cdot e (f_{y})]}^{2} + {[\frac{\partial f_{y}}{\partial u_{0}} \cdot e (u_{0})]}^{2} + {[\frac{\partial t_{y}}{\partial v_{0}} \cdot e (v_{0})]}^{2} + Σ_{i = 1}^{N} {{[\frac{\partial t_{y}}{\partial X_{wi}} \cdot e (X_{wi})]}^{2} + {[\frac{\partial t_{y}}{\partial Y_{wi}} \cdot e (Y_{wi})]}^{2} + {[\frac{\partial t_{y}}{\partial Z_{wi}} \cdot e (Z_{wi})]}^{2}}} \\ e (t_{z}) = \sqrt{{[\frac{\partial t_{z}}{\partial f_{x}} \cdot e (f_{x})]}^{2} + {[\frac{\partial t_{z}}{\partial f_{y}} \cdot e (f_{y})]}^{2} + {[\frac{\partial t_{z}}{\partial u_{0}} \cdot e (u_{0})]}^{2} + {[\frac{\partial t_{z}}{\partial v_{0}} \cdot e (v_{0})]}^{2} + Σ_{i = 1}^{N} {{[\frac{\partial t_{z}}{\partial X_{wi}} \cdot e (X_{wi})]}^{2} + {[\frac{\partial t_{z}}{\partial Y_{wi}} \cdot e (Y_{wi})]}^{2} + {[\frac{\partial t_{z}}{\partial Z_{wi}} \cdot e (Z_{wi})]}^{2}}} \end{matrix})$

步骤(6)采用目标位姿随机误差分量与目标位姿未确定性系统误差分量依据如下公式相加获得目标位姿测量误差，完成对目标位姿测量精度的预估：

$(\begin{matrix} {(Δα)}_{total}^{2} = {[e (α)]}^{2} + \frac{1}{n} \cdot {[δ (α)]}^{2}; \\ {(Δβ)}_{total}^{2} = {[e (β)]}^{2} + \frac{1}{n} \cdot {[δ (β)]}^{2}; \\ {(Δγ)}_{total}^{2} = {[e (γ)]}^{2} + \frac{1}{n} \cdot {[δ (γ)]}^{2}; \\ {(Δ t_{x})}_{total}^{2} = {[e (t_{x})]}^{2} + \frac{1}{n} \cdot {[δ (t_{x})]}^{2}; \\ {(Δ t_{y})}_{total}^{2} = {[e (t_{y})]}^{2} + \frac{1}{n} \cdot {[δ (t_{y})]}^{2}; \\ {(Δ t_{z})}_{total}^{2} = {[e (t_{z})]}^{2} + \frac{1}{n} \cdot {[δ (t_{z})]}^{2}; \end{matrix})$ 其中n任意取值。

有益效果：

本发明与现有技术相比的优点在于：

(1)本发明提供的合作目标位姿测量精度预估方法不受PNP透视模型中的视觉标记点个数N的限制约束，可适用于N不小于3的任意正整数；

(2)本发明提供的合作目标位姿测量精度预估方法能够明确划分相机内参标定误差、标记点三维空间坐标值获取误差以及图像标记点中心二维坐标定位误差等因素的误差属性特征，针对不同属性的随机误差分量和未确定性系统误差分量分别进行误差合成计算；

(3)本发明提供的目标位姿精度预估方法不仅能够独立分析单一误差因素对位姿测量的影响，还能够有效融合相机内参标定误差、标记点三维空间坐标值获取误差以及图像标记点中心二维坐标定位误差等多种因素对目标位姿精度的合成影响；

(4)本发明提供的目标位姿精度预估方法能够实现在测量系统方案设计初期对目标位姿测量精度水平的准确预估与客观评价。

附图说明

图1为基于PNP透视模型的合作目标三维位姿单目视觉测量原理示意图；

图2为基于PNP透视模型的合作目标位姿精度预估流程图。

具体实施方式

下面结合附图并举实施例，对本发明进行详细描述。

实施例1、本发明的基于PNP透视模型的合作目标三维位姿单目视觉测量原理如图1所示，假设合作目标表面粘贴N个标记点，其中第i个标记点在目标坐标系下的三维空间坐标值及其在图像坐标系中的特征点中心二维坐标可分别表示为(X_wi,Y_wi,Z_wi)和(u_i,v_i)；利用单目视觉相机实施测量目标的位置姿态，相机内参主要包括水平、垂直有效焦距f_x,f_y、主点坐标u₀,v₀；上述合作目标相对于相机之间的空间位姿参量可定义为[t_x,t_y,t_z,α,β,γ]^T。对于第i个标记点而言，根据PNP 透视模型，可得到下列方程组形式：

$(\begin{matrix} \frac{u_{i} - u_{0}}{f_{x}} = \frac{\cos γ \cdot \cos β \cdot X_{wi} + (\cos γ \cdot \sin β \cdot \sin α - \sin γ) \cdot Y_{wi} + (\cos γ \cdot \sin β \cdot \cos α - \sin γ \cdot \sin α) \cdot Z_{wi} + t_{x}}{- \sin β \cdot X_{wi} + \cos β \cdot \sin α \cdot Y_{wi} + \cos β \cdot \cos α \cdot Z_{wi} + t_{z}} \\ \frac{v_{i} - v_{0}}{f_{y}} = \frac{\sin γ \cdot \cos β \cdot X_{wi} + (\sin γ \cdot \sin β \cdot \sin α + \cos γ \cdot \cos α) \cdot Y_{wi} + (\sin γ \cdot \sin β \cdot \cos α + \cos γ \cdot \sin α) \cdot Z_{wi} + t_{y}}{- \sin β \cdot X_{wi} + \cos β \cdot \sin α \cdot Y_{wi} + \cos β \cdot \cos α \cdot Z_{wi} + t_{z}} \end{matrix}) - - - (1)$

公式(1)进一步整理可得以下方程形式：

$(\begin{matrix} F_{2 i - 1} (u_{i}, v_{i}, f_{x}, f_{y}, u_{0}, v_{0}, X_{wi}, Y_{wi}, Z_{wi}, α, β, γ, t_{y}, t_{z}) = (u_{i} - u_{0}) \cdot (- \sin β \cdot X_{wi} + \cos β \sin α \cdot Y_{wi} + \cos α \cdot Z_{wi} + t_{z}) \\ - f_{x} \cdot [\cos γ \cdot \cos β \cdot X_{wi} + (\cos γ \cdot \sin β \cdot \sin α - \sin γ) \cdot Y_{wi} + (\cos γ \cdot \sin β \cdot \cos α - \sin γ \cdot \sin α) \cdot Z_{wi} + t_{x}]; \\ F_{2 i} (u_{i}, v_{i}, f_{x}, f_{y}, u_{0}, v_{0}, X_{wi}, Y_{wi}, Z_{wi}, α, β, γ, t_{y}, t_{z}) = (v_{i} - v_{0}) \cdot (- \sin β \cdot X_{wi} + \cos β \sin α \cdot Y_{wi} + \cos β \cos α \cdot Z_{wi} + t_{z}) \\ - f_{y} \cdot [\sin γ \cdot \cos β \cdot X_{wi} + (\sin γ \cdot \sin β \cdot \sin α + \cos γ \cdot \cos α) \cdot Y_{wi} + (\sin γ \cdot \sin β \cdot \cos α + \cos γ \cdot \sin α) \cdot Z_{wi} + t_{y}]; \end{matrix}) - - - (2)$

如图2所示，基于PNP透视模型的合作目标位姿测量精度预估流程具体如下：

(1)预先设定合作目标与相机之间的初始位置姿态[t_x,t_y,t_z,α,β,γ]^T，已知标记点个数，且各视觉标记点三维空间坐标为(X_wi,Y_wi,Z_wi)及其对应图像特征点中心二维坐标为(u_i,v_i)；获取相机内参标定结果，分别确定有效焦距与主点坐标为 f_x,f_y,u₀,v₀。

(2)合理划分上述影响位姿精度的各项参数的误差属性特征，并分别给出各项参数的误差极限阈值。其中，水平、垂直有效焦距与主点坐标等相机内参标定误差和视觉标记点三维空间坐标获取误差均属于未确定性系统误差，上述参数误差极限应分别表示为e(f_x),e(f_y),e(u₀),e(v₀),e(X_wi),e(Y_wi),e(Z_wi)；而标记图像特征点中心二维坐标的定位误差则属于随机误差，其极限阈值应表示为δ(u_i),δ(v_i)；上述各项参数误差均彼此独立、互不相关，倘若对上述某一项参数求一阶偏导数，则其余各参数对该参数的一阶偏导均等于零。最终，计算出的位姿分量误差可定义为既包括随机误差分量定义为 δ(t_x),δ(t_y),δ(t_z),δ(α),δ(β),δ(γ)，又包含未确定性系统误差分量依次定义为 e(t_x),e(t_y),e(t_z),e(α),e(β),e(γ)。

(3)量化解析各项参数对位姿误差的影响权重系数

现以相机水平有效焦距f_x为例予以解释，其他参数对位姿误差的影响权重系数的求解均与之相类似。

对公式(2)中的F_2i-1和F_2i分别进行关于f_x的一阶Taylor展开，整理可得出以下形式，

$(\begin{matrix} \frac{\partial F_{2 i - 1}}{\partial α} \cdot \frac{\partial α}{\partial f_{x}} + \frac{\partial F_{2 i - 1}}{\partial β} \cdot \frac{\partial β}{\partial f_{x}} + \frac{\partial F_{2 i - 1}}{\partial γ} \cdot \frac{\partial γ}{\partial f_{x}} + \frac{\partial F_{2 i - 1}}{\partial t_{x}} \cdot \frac{{\partial t}_{x}}{\partial f_{x}} + \frac{\partial F_{2 i - 1}}{\partial t_{y}} \cdot \frac{\partial t_{y}}{\partial f_{x}} + \frac{\partial F_{2 i - 1}}{\partial t_{z}} + \frac{\partial t_{z}}{\partial f_{x}} + \frac{\partial F_{2 i - 1}}{\partial f_{x}} = 0 \\ \frac{\partial F_{2 i}}{\partial α} \cdot \frac{\partial α}{\partial f_{x}} + \frac{\partial F_{2 i}}{\partial β} \cdot \frac{\partial β}{\partial f_{x}} + \frac{\partial F_{2 i}}{\partial γ} \cdot \frac{\partial γ}{\partial f_{x}} + \frac{\partial F_{2 i}}{\partial t_{x}} \cdot \frac{\partial t_{x}}{\partial f_{x}} + \frac{\partial F_{2 i}}{\partial t_{y}} \cdot \frac{\partial t_{y}}{\partial f_{x}} + \frac{\partial F_{2 i}}{\partial t_{z}} \cdot \frac{\partial t_{z}}{\partial f_{x}} + \frac{\partial F_{2 i}}{\partial f_{x}} = 0 \end{matrix}) - - - (3)$

将N个标记点分别代入公式(3)并整理成矩阵形式可得，

$(\begin{matrix} \frac{{\partial F}_{1}}{\partial α} & \frac{{\partial F}_{1}}{\partial β} & \frac{{\partial F}_{1}}{\partial γ} & \frac{{\partial F}_{1}}{{\partial t}_{x}} & \frac{{\partial F}_{1}}{{\partial t}_{y}} & \frac{{\partial F}_{1}}{{\partial t}_{z}} \\ \frac{{\partial F}_{2}}{\partial α} & \frac{{\partial F}_{2}}{\partial β} & \frac{{\partial F}_{2}}{\partial γ} & \frac{{\partial F}_{2}}{{\partial t}_{x}} & \frac{{\partial F}_{2}}{{\partial t}_{y}} & \frac{{\partial F}_{2}}{{\partial t}_{z}} \\ . . . & . . . & . . . & . . . & . . . & . . . \\ \frac{{\partial F}_{2 i - 1}}{\partial α} & \frac{{\partial F}_{2 i - 1}}{\partial β} & \frac{{\partial F}_{2 i - 1}}{\partial γ} & \frac{{\partial F}_{2 i - 1}}{{\partial t}_{x}} & \frac{{\partial F}_{2 i - 1}}{{\partial t}_{y}} & \frac{{\partial F}_{2 i - 1}}{{\partial t}_{z}} \\ \frac{{\partial F}_{2 i}}{\partial α} & \frac{{\partial F}_{2 i}}{\partial β} & \frac{{\partial F}_{2 i}}{\partial γ} & \frac{{\partial F}_{2 i}}{{\partial t}_{x}} & \frac{{\partial F}_{2 i}}{{\partial t}_{y}} & \frac{{\partial F}_{2 i}}{{\partial t}_{z}} \\ . . . & . . . & . . . & . . . & . . . & . . . \\ \frac{{\partial F}_{2 N - 1}}{\partial α} & \frac{{\partial F}_{2 N - 1}}{\partial β} & \frac{{\partial F}_{2 N - 1}}{\partial γ} & \frac{{\partial F}_{2 N - 1}}{{\partial t}_{x}} & \frac{{\partial F}_{2 N - 1}}{{\partial t}_{y}} & \frac{{\partial F}_{2 N - 1}}{{\partial t}_{z}} \\ \frac{{\partial F}_{2 N}}{\partial α} & \frac{{\partial F}_{2 N}}{\partial β} & \frac{{\partial F}_{2 N}}{\partial γ} & \frac{{\partial F}_{2 N}}{{\partial t}_{x}} & \frac{{\partial F}_{2 N}}{{\partial t}_{t}} & \frac{{\partial F}_{2 N}}{{\partial t}_{z}} \end{matrix}) \cdot (\begin{matrix} \frac{\partial α}{{\partial f}_{x}} \\ \frac{\partial β}{{\partial f}_{x}} \\ \frac{\partial γ}{{\partial f}_{x}} \\ \frac{{\partial t}_{x}}{{\partial f}_{x}} \\ \frac{{\partial t}_{y}}{{\partial f}_{x}} \\ \frac{{\partial t}_{z}}{{\partial f}_{x}} \end{matrix}) = (- 1) \cdot (\begin{matrix} \frac{{\partial F}_{1}}{{\partial f}_{x}} \\ \frac{{\partial F}_{2}}{{\partial f}_{x}} \\ . . . \\ \frac{{\partial F}_{2 i - 1}}{{\partial f}_{x}} \\ \frac{{\partial F}_{2 i}}{{\partial f}_{x}} \\ . . . \\ \frac{{\partial F}_{2 N - 1}}{{\partial f}_{x}} \\ \frac{{\partial F}_{2 N}}{{\partial f}_{x}} \end{matrix}) - - - (4)$

公式(4)可以简化为下列矩阵方程形式：

M·X＝-Y (5)

其中，列向量X的维数为(6×1)，即为代求未知参数向量；等号右侧的列向量Y的维数是(2N×1)，其中的元素可表示为以下形式：

公式(5)中，等号左侧的系数矩阵M的维数是(2N×6)，其中各项元素分量的计算公式具体如下：

$(\begin{matrix} \frac{{\partial F}_{2 i - 1}}{\partial α} = (u_{i} - u_{0}) \cdot (\cos γ \cdot \cos α \cdot Y_{wi} - \cos β \cdot \sin α \cdot Z_{wi}) \\ + f_{x} \cdot [- \cos α \cdot \sin β \cdot \cos γ \cdot Y_{wi} - (\cos γ \cdot \sin β \cdot \sin α + \sin γ \cdot \cos α) \cdot Z_{wi}]; \\ \frac{{\partial F}_{2 i}}{\partial α} = (v_{i} - v_{0}) \cdot (\cos β \cdot \cos α \cdot Y_{wi} - \cos β \cdot \sin α \cdot Z_{wi}) \\ - f_{y} \cdot [(\sin γ \cdot \sin β \cdot \cos α - \cos γ \cdot \sin α) \cdot Y_{wi} + (- \sin γ \cdot \sin β \cdot \sin α + \cos γ \cdot \cos α) \cdot Z_{wi}]; \\ \frac{{\partial F}_{2 i}}{\partial β} = (u_{i} - u_{0}) \cdot (- \cos β \cdot X_{wi} - \sin β \cdot \sin α \cdot Y_{wi} - \sin β \cdot \cos α \cdot Z_{wi}) \\ - f_{x} \cdot (- \cos γ \cdot \sin β \cdot X_{wi} + \cos γ \cdot \cos β \cdot \sin α \cdot Y_{wi} + \cos γ \cdot \cos α \cdot \cos β \cdot Z_{wi}); \\ \frac{{\partial F}_{2 i}}{\partial β} = (v_{i} - v_{0}) \cdot (- \cos β \cdot X_{wi} + \sin γ \cdot \cos β \cdot \sin α \cdot Y_{wi} - \sin β \cdot \cos α \cdot Z_{wi}) \\ - f_{y} \cdot [- \sin γ \cdot \sin β \cdot X_{wi} + \sin γ \cdot \cos β \cdot \sin α \cdot Y_{wi} + \sin γ \cdot \cos β \cdot \cos α \cdot Z_{wi}]; \\ \frac{{\partial F}_{2 i}}{\partial γ} = - f_{x} \cdot [\sin γ \cdot \cos β \cdot X_{wi} - \cos γ \cdot Y_{wi} + (- \sin γ \cdot \sin β \cdot \cos γ \cdot \sin α) \cdot Z_{wi}]; \\ \frac{{\partial F}_{2 i - 1}}{\partial γ} = - f_{y} \cdot [\cos γ \cdot \cos β \cdot X_{wi} + (\cos γ \cdot \sin β \cdot \sin α - \sin γ \cdot \cos α) \cdot Y_{wi} + (\cos γ \cdot \sin β \cdot \cos α - \sin γ \cdot \sin α) \cdot Z_{wi}]; \\ \frac{{\partial F}_{2 i - 1}}{{\partial t}_{x}} = - f_{x}; \frac{{\partial F}_{2 i}}{{\partial t}_{x}} = 0; \\ \frac{\partial F_{2 i - 1}}{{\partial t}_{y}} = 0; \frac{{\partial F}_{2 i}}{{\partial t}_{y}} = - f_{y}; \\ \frac{{\partial F}_{2 i - 1}}{{\partial t}_{z}} = u_{i} - u_{0}; \frac{{\partial F}_{2 i}}{{\partial t}_{z}} = v_{i} - v_{0}; \end{matrix})$

为了提高计算效率，现将公式（5）中的列向量X扩展成维数为（6×2N）的矩阵形式X’，将Y扩展为维数为（2N×2N）的矩阵形式Y’即令：

$X^{'} {= \cdot (\begin{matrix} \frac{\partial α}{{\partial f}_{x}} & \frac{\partial α}{{\partial f}_{y}} & \frac{\partial α}{{\partial u}_{0}} & \frac{\partial α}{{\partial v}_{0}} & \frac{\partial α}{{\partial u}_{i}} & \frac{\partial α}{{\partial v}_{i}} & \frac{\partial α}{{\partial X}_{wi}} & \frac{\partial α}{{\partial Y}_{wi}} & \frac{\partial α}{{\partial Z}_{wi}} & . . . & . . . & . . . & . . . \\ \frac{\partial β}{{\partial f}_{x}} & \frac{\partial β}{{\partial f}_{y}} & \frac{\partial β}{{\partial u}_{0}} & \frac{\partial β}{{\partial v}_{0}} & \frac{\partial β}{{\partial u}_{i}} & \frac{\partial β}{{\partial v}_{i}} & \frac{\partial β}{{\partial X}_{wi}} & \frac{\partial β}{{\partial Y}_{wi}} & \frac{\partial β}{{\partial Z}_{wi}} & . . . & . . . & . . . & . . . \\ \frac{\partial γ}{{\partial f}_{x}} & \frac{\partial γ}{{\partial f}_{y}} & \frac{\partial γ}{{\partial u}_{0}} & \frac{\partial γ}{{\partial v}_{0}} & \frac{\partial γ}{{\partial u}_{i}} & \frac{\partial γ}{{\partial v}_{i}} & \frac{\partial γ}{{\partial X}_{wi}} & \frac{\partial γ}{{\partial Y}_{wi}} & \frac{\partial γ}{{\partial Z}_{wi}} & . . . & . . . & . . . & . . . \\ \frac{\partial t_{x}}{{\partial f}_{x}} & \frac{{\partial t}_{x}}{{\partial f}_{y}} & \frac{\partial t_{x}}{{\partial u}_{0}} & \frac{{\partial t}_{x}}{{\partial v}_{0}} & \frac{{\partial t}_{x}}{{\partial u}_{i}} & \frac{{\partial t}_{x}}{{\partial v}_{i}} & \frac{{\partial t}_{x}}{{\partial X}_{wi}} & \frac{{\partial t}_{x}}{{\partial Y}_{wi}} & \frac{{\partial t}_{x}}{{\partial Z}_{wi}} & . . . & . . . & . . . & . . . \\ \frac{{\partial t}_{y}}{{\partial f}_{x}} & \frac{\partial t_{y}}{{\partial f}_{y}} & \frac{{\partial t}_{y}}{{\partial u}_{0}} & \frac{{\partial t}_{y}}{{\partial v}_{0}} & \frac{{\partial t}_{y}}{{\partial u}_{i}} & \frac{{\partial t}_{y}}{{\partial v}_{i}} & \frac{{\partial t}_{y}}{{\partial X}_{wi}} & \frac{{\partial t}_{y}}{{\partial Y}_{wi}} & \frac{{\partial t}_{y}}{{\partial Z}_{wi}} & . . . & . . . & . . . & . . . \\ \frac{{\partial t}_{z}}{{\partial f}_{x}} & \frac{{\partial t}_{z}}{{\partial f}_{y}} & \frac{{\partial t}_{z}}{{\partial u}_{0}} & \frac{{\partial t}_{z}}{{\partial v}_{0}} & \frac{{\partial t}_{z}}{{\partial u}_{i}} & \frac{{\partial t}_{z}}{{\partial v}_{i}} & \frac{{\partial t}_{z}}{{\partial X}_{wi}} & \frac{{\partial t}_{z}}{{\partial Y}_{wi}} & \frac{{\partial t}_{z}}{{\partial Z}_{wi}} & . . . & . . . & . . . & . . . \end{matrix})}^{'} Y^{'} = (\begin{matrix} \frac{\partial F_{1}}{\partial f_{x}} & \frac{\partial F_{1}}{\partial f_{y}} & \frac{\partial F_{1}}{\partial u_{0}} & \frac{\partial F_{1}}{\partial v_{0}} & \frac{\partial F_{1}}{\partial u_{i}} & \frac{\partial F_{1}}{\partial v_{i}} & \frac{\partial F_{1}}{\partial X_{wi}} & \frac{\partial F_{1}}{\partial Y_{wi}} & \frac{\partial F_{1}}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \frac{\partial F_{2}}{\partial f_{x}} & \frac{\partial F_{2}}{\partial f_{y}} & \frac{\partial F_{2}}{\partial u_{0}} & \frac{\partial F_{2}}{\partial v_{0}} & \frac{\partial F_{2}}{\partial v_{0}} & \frac{\partial F_{2}}{\partial v_{i}} & \frac{\partial F_{2}}{\partial X_{wi}} & \frac{\partial F_{2}}{\partial Y_{wi}} & \frac{\partial F_{2}}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \frac{\partial F_{2 i 1}}{\partial f_{x}} & \frac{\partial F_{2 i} 1}{\partial f_{y}} & \frac{\partial F_{2 i 1}}{\partial u_{0}} & \frac{\partial F_{2 i 1}}{\partial v_{0}} & \frac{\partial F_{2 i 1}}{\partial v_{0}} & \frac{\partial F_{2 i 1}}{\partial v_{i}} & \frac{\partial F_{2 i 1}}{\partial X_{wi}} & \frac{\partial F_{2 i 1}}{\partial Y_{wi}} & \frac{\partial F_{2 i 1}}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \frac{\partial F_{2 i}}{\partial f_{x}} & \frac{\partial F_{2 i}}{\partial f_{y}} & \frac{\partial F_{2 i}}{\partial u_{0}} & \frac{\partial F_{2 i}}{\partial v_{0}} & \frac{\partial F_{2 i}}{\partial v_{0}} & \frac{\partial F_{2 i}}{\partial v_{i}} & \frac{\partial F_{2 i}}{\partial X_{wi}} & \frac{\partial F_{2 i}}{\partial Y_{wi}} & \frac{\partial F_{2 i}}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \frac{\partial F_{2 N 1}}{\partial f_{x}} & \frac{\partial F_{2 N 1}}{\partial f_{y}} & \frac{\partial F_{2 N 1}}{\partial u_{0}} & \frac{{\partial F}_{2 N 1}}{\partial v_{0}} & \frac{\partial F_{2 N 1}}{\partial v_{0}} & \frac{\partial F_{2 N 1}}{\partial v_{i}} & \frac{\partial F_{2 N 1}}{\partial X_{wi}} & \frac{\partial F_{2 N 1}}{\partial Y_{wi}} & \frac{\partial F_{2 N 1}}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \frac{\partial F_{2 N}}{\partial f_{x}} & \frac{\partial F_{2 N}}{\partial f_{y}} & \frac{\partial F_{2 N}}{\partial u_{0}} & \frac{\partial F_{2 N}}{\partial v_{0}} & \frac{\partial F_{2 N}}{\partial v_{0}} & \frac{\partial F_{2 N}}{\partial v_{i}} & \frac{\partial F_{2 N}}{\partial X_{wi}} & \frac{\partial F_{2 N}}{\partial Y_{wi}} & \frac{\partial F_{2 N}}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \end{matrix}),$

，其中X′为6行×2N列的矩阵，自第10列开始重复前9列内容；Y′为2N行× 2N列的矩阵，且自第10列开始重复前9列内容；

此时，公式（5）随即可扩展成下列形式：

$(\begin{matrix} \frac{\partial F_{1}}{\partial α} & \frac{\partial F_{1}}{\partial β} & \frac{\partial F_{1}}{\partial γ} & \frac{\partial F_{1}}{\partial t_{x}} & \frac{\partial F_{1}}{\partial t_{y}} & \frac{\partial F_{1}}{\partial t_{z}} \\ \frac{\partial F_{2}}{\partial α} & \frac{\partial F_{2}}{\partial β} & \frac{\partial F_{2}}{\partial γ} & \frac{\partial F_{2}}{\partial t_{x}} & \frac{\partial F_{2}}{\partial t_{y}} & \frac{\partial F_{2}}{\partial t_{z}} \\ . . . & . . . & . . . & . . . & . . . & . . . \\ \frac{\partial F_{2 i - 1}}{\partial α} & \frac{\partial F_{2 i - 1}}{\partial β} & \frac{\partial F_{2 i - 1}}{\partial γ} & \frac{\partial F_{2 i - 1}}{\partial t_{x}} & \frac{\partial F_{2 i - 1}}{\partial t_{y}} & \frac{\partial F_{2 i - 1}}{\partial t_{z}} \\ \frac{\partial F_{2 i}}{\partial α} & \frac{\partial F_{2 i}}{\partial β} & \frac{\partial F_{2 i}}{\partial γ} & \frac{\partial F_{2 i}}{\partial t_{x}} & \frac{\partial F_{2 i}}{\partial t_{y}} & \frac{\partial F_{2 i}}{\partial t_{z}} \\ . . . & . . . & . . . & . . . & . . . & . . . \\ \frac{{\partial F}_{2 N - 1}}{\partial α} & \frac{\partial F_{2 N - 1}}{\partial β} & \frac{\partial F_{2 N - 1}}{\partial γ} & \frac{\partial F_{2 N - 1}}{\partial t_{x}} & \frac{\partial F_{2 N - 1}}{\partial t_{y}} & \frac{\partial F_{2 N - 1}}{\partial t_{z}} \\ \frac{\partial F_{2 N}}{\partial α} & \frac{\partial F_{2 N}}{\partial β} & \frac{\partial F_{2 N}}{\partial γ} & \frac{\partial F_{2 N}}{\partial t_{x}} & \frac{\partial F_{2 N}}{\partial t_{y}} & \frac{\partial F_{2 N}}{\partial t_{z}} \end{matrix}) . . (\begin{matrix} \frac{\partial α}{\partial f_{x}} & \frac{\partial α}{\partial f_{y}} & \frac{\partial α}{\partial u_{0}} & \frac{\partial α}{\partial v_{0}} & \frac{\partial α}{{\partial u}_{i}} & \frac{\partial α}{\partial v_{i}} & \frac{\partial α}{\partial X_{wi}} & \frac{\partial α}{\partial Y_{wi}} & \frac{\partial α}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \frac{\partial β}{\partial f_{x}} & \frac{\partial β}{\partial f_{y}} & \frac{\partial β}{\partial u_{0}} & \frac{\partial β}{\partial v_{0}} & \frac{\partial β}{\partial u_{i}} & \frac{\partial β}{\partial v_{i}} & \frac{\partial β}{\partial X_{wi}} & \frac{\partial β}{\partial Y_{wi}} & \frac{\partial β}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \frac{\partial γ}{\partial f_{x}} & \frac{\partial γ}{\partial f_{y}} & \frac{\partial γ}{\partial u_{0}} & \frac{\partial γ}{\partial v_{0}} & \frac{\partial γ}{\partial u_{i}} & \frac{\partial γ}{{\partial v}_{i}} & \frac{\partial γ}{\partial X_{wi}} & \frac{\partial γ}{\partial Y_{wi}} & \frac{\partial γ}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \frac{\partial t_{x}}{\partial f_{x}} & \frac{\partial t_{x}}{\partial f_{y}} & \frac{\partial t_{x}}{\partial u_{0}} & \frac{\partial t_{x}}{\partial v_{0}} & \frac{\partial t_{x}}{\partial u_{i}} & \frac{\partial t_{x}}{\partial v_{i}} & \frac{\partial t_{x}}{\partial X_{wi}} & \frac{\partial t_{x}}{\partial Y_{wi}} & \frac{\partial t_{x}}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \frac{\partial t_{y}}{\partial f_{x}} & \frac{\partial t_{y}}{\partial f_{y}} & \frac{\partial t_{y}}{\partial u_{0}} & \frac{\partial t_{y}}{\partial v_{0}} & \frac{\partial t_{y}}{\partial u_{i}} & \frac{\partial t_{y}}{\partial v_{i}} & \frac{\partial t_{y}}{\partial X_{wi}} & \frac{\partial t_{y}}{\partial Y_{wi}} & \frac{\partial t_{y}}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \frac{\partial t_{z}}{\partial f_{x}} & \frac{\partial t_{z}}{\partial f_{y}} & \frac{\partial t_{z}}{\partial u_{0}} & \frac{\partial t_{z}}{\partial v_{0}} & \frac{\partial t_{z}}{\partial u_{i}} & \frac{\partial t_{z}}{\partial v_{i}} & \frac{\partial t_{z}}{\partial X_{wi}} & \frac{\partial t_{z}}{\partial Y_{wi}} & \frac{\partial t_{z}}{\partial Z_{wi}} & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \end{matrix}) = (- 1) (\begin{matrix} \frac{{\partial F}_{1}}{{\partial f}_{x}} & \frac{{\partial F}_{1}}{{\partial f}_{y}} & \frac{{\partial F}_{1}}{{\partial u}_{0}} & \frac{\partial F_{1}}{\partial v_{0}} & \frac{\partial F_{1}}{\partial u_{i}} & \frac{\partial F_{1}}{\partial v_{i}} & \frac{\partial F_{1}}{\partial X_{wi}} & \frac{\partial F_{1}}{\partial Y_{wi}} & \frac{\partial F_{1}}{\partial Z_{wi}} & . . . & . . . & . . . & . . . \\ \frac{\partial F_{2}}{{\partial f}_{x}} & \frac{{\partial F}_{2}}{{\partial f}_{y}} & \frac{\partial F_{2}}{\partial u_{0}} & \frac{\partial F_{2}}{\partial v_{0}} & \frac{\partial F_{2}}{\partial v_{0}} & \frac{\partial F_{2}}{\partial v_{i}} & \frac{\partial F_{2}}{\partial X_{wi}} & \frac{\partial F_{2}}{\partial Y_{wi}} & \frac{\partial F_{2}}{\partial Z_{wi}} & . . . & . . . & . . . & . . . \\ . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . \\ \frac{\partial F_{2 i 1}}{\partial f_{x}} & \frac{\partial F_{2 i 1}}{\partial f_{y}} & \frac{\partial F_{2 i 1}}{\partial u_{0}} & \frac{\partial F_{2 i 1}}{\partial v_{0}} & \frac{\partial F_{2 i 1}}{\partial v_{0}} & \frac{\partial F_{2 i 1}}{\partial v_{i}} & \frac{\partial F_{2 i 1}}{\partial X_{wi}} & \frac{\partial F_{2 i 1}}{\partial Y_{wi}} & \frac{\partial F_{2 i 1}}{\partial Z_{wi}} & . . . & . . . & . . . & . . . \\ \frac{\partial F_{2 i}}{\partial f_{x}} & \frac{\partial F_{2 i}}{\partial f_{y}} & \frac{\partial F_{2 i}}{\partial u_{0}} & \frac{\partial F_{2 i}}{\partial v_{0}} & \frac{\partial F_{2 i}}{\partial v_{0}} & \frac{\partial F_{2 i}}{\partial v_{i}} & \frac{\partial F_{2 i}}{\partial X_{wi}} & \frac{\partial F_{2 i}}{\partial Y_{wi}} & \frac{\partial F_{2 i}}{\partial Z_{wi}} & . . . & . . . & . . . & . . . \\ . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . \\ \frac{\partial F_{2 N 1}}{\partial f_{x}} & \frac{\partial F_{2 N 1}}{\partial f_{y}} & \frac{\partial F_{2 N 1}}{\partial u_{0}} & \frac{\partial F_{2 N 1}}{\partial v_{0}} & \frac{\partial F_{2 N 1}}{\partial v_{0}} & \frac{\partial F_{2 N 1}}{\partial v_{i}} & \frac{\partial F_{2 N 1}}{\partial X_{wi}} & \frac{\partial F_{2 N 1}}{\partial Y_{wi}} & \frac{\partial F_{2 N 1}}{\partial Z_{wi}} & . . . & . . . & . . . & . . . \\ \frac{\partial F_{2 N}}{\partial f_{x}} & \frac{\partial F_{2 N}}{\partial f_{y}} & \frac{\partial F_{2 N}}{\partial u_{0}} & \frac{\partial F_{2 N}}{\partial v_{0}} & \frac{\partial F_{2 N}}{\partial v_{0}} & \frac{\partial F_{2 N}}{\partial v_{i}} & \frac{\partial F_{2 N}}{\partial X_{wi}} & \frac{\partial F_{2 N}}{\partial Y_{wi}} & \frac{\partial F_{2 N}}{\partial Z_{wi}} & . . . & . . . & . . . & . . . \end{matrix})$

公式(8)可进一步简化为矩阵方程形式：

M·X′＝-Y′ (9)

公式(8)和公式(9)中的系数矩阵M始终保持一致，其各项元素分量的计算公式参见公式(7)；公式(9)中的代求未知矩阵X’中的每一列即可表示各项参数对位姿误差的影响权重系数，而等号右侧的系数矩阵Y’的维数是 (6×2N)，其中的各项元素计算公式具体如下：

$(\begin{matrix} \frac{{\partial F}_{2 i - 1}}{{\partial f}_{x}} = - \cos γ \cdot \cos β \cdot X_{wi} - (\cos γ \cdot \sin β \cdot \sin α - \sin γ) \cdot Y_{wi} - (\cos γ \cdot \sin β \cdot \cos α - \sin γ \cdot \sin α) \cdot Z_{wi} - t_{x}; \frac{{\partial F}_{2 i}}{{\partial f}_{x}} = 0; \\ \frac{{\partial F}_{2 i - 1}}{{\partial f}_{y}} = 0; \frac{{\partial F}_{2 i}}{{\partial f}_{y}} = - \sin γ \cdot \cos β \cdot X_{wi} - (\sin γ \cdot \sin β \cdot \sin α + \cos γ \cdot \cos α) \cdot Y_{wi} - (\sin γ \cdot \sin β \cdot \cos α + \cos γ \cdot \sin α) \cdot Z_{wi} - t_{y}; \\ \frac{{\partial F}_{2 i - 1}}{{\partial u}_{0}} = \sin β \cdot X_{wi} - \cos β \cdot \sin α \cdot Y_{wi} - \cos β \cdot \cos α \cdot Z_{wi} - t_{z}; \frac{{\partial F}_{2 i}}{{\partial u}_{0}} = 0; \\ \frac{{\partial F}_{2 i - 1}}{{\partial v}_{0}} = 0; \frac{{\partial F}_{2 i}}{{\partial v}_{0}} = \sin β \cdot X_{wi} - \cos β \cdot \sin α \cdot Y_{wi} - \cos β \cdot \cos α \cdot Z_{wi} - t_{z}; \\ \frac{{\partial F}_{2 i - 1}}{{\partial u}_{i}} = - \sin β \cdot X_{wi} + \cos β \cdot \sin α \cdot Y_{wi} + \cos β \cdot \cos α \cdot Z_{wi} + t_{z}; \frac{{\partial F}_{2 i}}{{\partial u}_{i}} = 0; \\ \frac{{\partial F}_{2 i - 1}}{{\partial v}_{i}} = 0; \frac{{\partial F}_{2 i}}{{\partial v}_{i}} = - \sin β \cdot X_{wi} + \cos β \cdot \sin α \cdot Y_{wi} + \cos β \cdot \cos α \cdot Z_{wi} + t_{z}; \\ \frac{{\partial F}_{2 i - 1}}{{\partial X}_{wi}} = - (u_{i} - u_{0}) \cdot \sin β - f_{x} \cdot \cos γ \cdot \cos β; \frac{{\partial F}_{2 i}}{{\partial X}_{wi}} = - (v_{i} - v_{0}) \cdot \sin β - f_{y} \cdot \sin γ \cdot \cos β; \\ \frac{{\partial F}_{2 i - 1}}{{\partial Y}_{wi}} = (u_{i} - u_{0}) \cdot \cos β \cdot \sin α - f_{x} \cdot (\sin γ \cdot \sin β \cdot \sin α - \sin γ) \cdot Y_{wi}; \\ \frac{{\partial F}_{2 i}}{{\partial Y}_{wi}} = (v_{i} - v_{0}) \cdot \cos β \cdot \sin α - f_{y} \cdot (\sin γ \cdot \sin β \cdot \sin α + \cos γ \cdot \cos α); \\ \frac{{\partial F}_{2 i - 1}}{{\partial Z}_{wi}} = (u_{i} - u_{0}) \cdot \cos β \cdot \cos α - f_{x} \cdot (\cos γ \cdot \sin β \cdot \cos α - \sin γ \cdot \sin α) \cdot Y_{wi}; \\ \frac{{\partial F}_{2 i}}{{\partial Z}_{wi}} = (v_{i} - v_{0}) \cdot \cos β \cdot \cos α - f_{y} \cdot (\sin γ \cdot \sin β \cdot \cos α + \cos γ \cdot \sin α); \end{matrix}) - - - (10)$

对于公式(9)的求解，主要存在以下两种情况：

①当2×N＝6时，利用最小二乘法求解，可得最优解为：

X′＝(-1)·M^-1·Y′ (11)

②当2×N＞6时，利用最小二乘法求解，可得最优解为：

X′＝(-1)·(M^T·M)^-1·(M^T·Y′) (12)

(4)根据步骤(2)中，标记图像特征点中心二维坐标的定位误差属于随机误差，由其所引起的目标位姿随机误差分量合成计算公式如下：

$(\begin{matrix} δ (α) = \sqrt{Σ_{i = 1}^{N} [{(\frac{\partial α}{{\partial u}_{i}} \cdot δ (u_{i}))}^{2} + {(\frac{\partial α}{{\partial v}_{i}} \cdot δ (v_{i}))}^{2}]} \\ δ (β) = \sqrt{Σ_{i = 1}^{N} [{(\frac{\partial β}{{\partial u}_{i}} \cdot δ (u_{i}))}^{2} + {(\frac{\partial β}{{\partial v}_{i}} \cdot δ (v_{i}))}^{2}]} \\ δ (γ) = \sqrt{Σ_{i = 1}^{N} [{(\frac{\partial γ}{{\partial u}_{i}} \cdot δ (u_{i}))}^{2} + {(\frac{\partial γ}{{\partial v}_{i}} \cdot δ (v_{i}))}^{2}]} \\ δ (t_{x}) = \sqrt{Σ_{i = 1}^{N} [{(\frac{{\partial t}_{s}}{{\partial u}_{i}} \cdot δ (u_{i}))}^{2} + {(\frac{{\partial t}_{s}}{{\partial v}_{i}} \cdot δ (v_{i}))}^{2}]} \\ δ (t_{y}) = \sqrt{Σ_{i = 1}^{N} [{(\frac{{\partial t}_{y}}{{\partial u}_{i}} \cdot δ (u_{i}))}^{2} + {(\frac{{\partial t}_{y}}{{\partial v}_{i}} \cdot δ (v_{i}))}^{2}]} \\ δ (t_{z}) = \sqrt{Σ_{i = 1}^{N} [{(\frac{{\partial t}_{z}}{{\partial u}_{i}} \cdot δ (u_{i}))}^{2} + {(\frac{{\partial t}_{z}}{{\partial v}_{i}} \cdot δ (v_{i}))}^{2}]} \end{matrix}) - - - (13)$

(5)根据步骤(2)中，相机内参标定误差与标记点三维空间坐标获取误差均属于未确定性系统误差，由上述误差分量引起的目标位姿未确定性系统误差分量合成计算公式如下：

$(\begin{matrix} e (α) = \sqrt{{[\frac{\partial α}{{\partial f}_{x}} \cdot e (f_{x})]}^{2} + {[\frac{\partial α}{{\partial f}_{y}} \cdot e (f_{y})]}^{2} + {[\frac{\partial α}{{\partial u}_{0}} \cdot e (u_{0})]}^{2} + {[\frac{\partial α}{{\partial v}_{0}} \cdot e (v_{0})]}^{2} + Σ_{i = 1}^{N} {{[\frac{\partial α}{{\partial X}_{wi}} \cdot e (X_{wi})]}^{2} + {[\frac{\partial α}{{\partial Y}_{wi}} \cdot e (Y_{wi})]}^{2} + {[\frac{\partial α}{{\partial Z}_{wi}} \cdot e (Z_{wi})]}^{2}}} \\ e (β) = \sqrt{{[\frac{\partial β}{{\partial f}_{x}} \cdot e (f_{x})]}^{2} + {[\frac{\partial β}{{\partial f}_{y}} \cdot e (f_{y})]}^{2} + {[\frac{\partial β}{{\partial u}_{0}} \cdot e (u_{0})]}^{2} + {[\frac{\partial β}{{\partial v}_{0}} \cdot e (v_{0})]}^{2} + Σ_{i = 1}^{N} {{[\frac{\partial β}{{\partial X}_{wi}} \cdot e (X_{wi})]}^{2} + {[\frac{\partial β}{{\partial Y}_{wi}} \cdot e (Y_{wi})]}^{2} + {[\frac{\partial β}{{\partial Z}_{wi}} \cdot e (Z_{wi})]}^{2}}} \\ e (γ) = \sqrt{{[\frac{\partial γ}{{\partial f}_{x}} \cdot e (f_{x})]}^{2} + {[\frac{\partial γ}{{\partial f}_{y}} \cdot e (f_{y})]}^{2} + {[\frac{\partial γ}{{\partial u}_{0}} \cdot e (u_{0})]}^{2} + {[\frac{\partial γ}{{\partial v}_{0}} \cdot e (v_{0})]}^{2} + Σ_{i = 1}^{N} {{[\frac{\partial γ}{{\partial X}_{wi}} \cdot e (X_{wi})]}^{2} + {[\frac{\partial γ}{{\partial Y}_{wi}} \cdot e (Y_{wi})]}^{2} + {[\frac{\partial γ}{{\partial Z}_{wi}} \cdot e (Z_{wi})]}^{2}}} \\ e (t_{x}) = \sqrt{{[\frac{\partial t_{x}}{{\partial f}_{x}} \cdot e (f_{x})]}^{2} + {[\frac{\partial t_{x}}{{\partial f}_{y}} \cdot e (f_{y})]}^{2} + {[\frac{\partial t_{x}}{{\partial u}_{0}} \cdot e (u_{0})]}^{2} + {[\frac{\partial t_{x}}{{\partial v}_{0}} \cdot e (v_{0})]}^{2} + Σ_{i = 1}^{N} {{[\frac{\partial t_{x}}{{\partial X}_{wi}} \cdot e (X_{wi})]}^{2} + {[\frac{\partial t_{x}}{{\partial Y}_{wi}} \cdot e (Y_{wi})]}^{2} + {[\frac{\partial t_{x}}{{\partial Z}_{wi}} \cdot e (Z_{wi})]}^{2}}} \\ e (t_{y}) = \sqrt{{[\frac{\partial t_{y}}{{\partial f}_{x}} \cdot e (f_{x})]}^{2} + {[\frac{\partial t_{y}}{{\partial f}_{y}} \cdot e (f_{y})]}^{2} + {[\frac{\partial t_{y}}{{\partial u}_{0}} \cdot e (u_{0})]}^{2} + {[\frac{\partial t_{y}}{{\partial v}_{0}} \cdot e (v_{0})]}^{2} + Σ_{i = 1}^{N} {{[\frac{\partial t_{y}}{{\partial X}_{wi}} \cdot e (X_{wi})]}^{2} + {[\frac{\partial t_{y}}{{\partial Y}_{wi}} \cdot e (Y_{wi})]}^{2} + {[\frac{\partial t_{y}}{{\partial Z}_{wi}} \cdot e (Z_{wi})]}^{2}}} \\ e (t_{z}) = \sqrt{{[\frac{\partial t_{z}}{{\partial f}_{x}} \cdot e (f_{x})]}^{2} + {[\frac{\partial t_{z}}{{\partial f}_{y}} \cdot e (f_{y})]}^{2} + {[\frac{\partial t_{z}}{{\partial u}_{0}} \cdot e (u_{0})]}^{2} + {[\frac{\partial t_{z}}{{\partial v}_{0}} \cdot e (v_{0})]}^{2} + Σ_{i = 1}^{N} {{[\frac{\partial t_{z}}{{\partial X}_{wi}} \cdot e (X_{wi})]}^{2} + {[\frac{\partial t_{z}}{{\partial Y}_{wi}} \cdot e (Y_{wi})]}^{2} + {[\frac{\partial t_{z}}{{\partial Z}_{wi}} \cdot e (Z_{wi})]}^{2}}} \end{matrix}) - - - (14)$

(6)对于步骤(4)中的对于随机误差利用n次重复性测量具有抵偿性，n 任意取值。而步骤(5)中的系统误差则固定不变，这两类误差合成即可推导出基于PNP透视模型的目标位姿测量精度预估总误差的计算公式可写为以下形式：

综上，以上仅为本发明的较佳实施例而已，并非用于限定本发明的保护范围。凡在本发明的精神和原则之内，所作的任何修改、等同替换、改进等，均应包含在本发明的保护范围之内。

去获取专利，查看全文>

相似文献

专利
中文文献
外文文献

1. 一种基于PNP透视模型的合作目标位姿精度测量方法 [P] . 中国专利： CN104729481B . 2017.05.24
2. 一种基于PNP透视模型的合作目标位姿精度测量方法 [P] . 中国专利： CN104729481A . 2015-06-24
3. Model-based method and medical imaging management system for high precision measurement [P] . SE0400101L . 2005-07-21

机译：基于模型的高精度测量方法及医学影像管理系统
4. SEGMENT POSITION-ATTITUDE MEASURING DEVICE, POSITION-ATTITUDE MEASURING METHOD AND ASSEMBLING DEVICE [P] . 日本专利： JP2004353186A . 2004-12-16

机译：段位姿态测量装置，位姿测量方法及组装装置
5. Information processing apparatus, control method thereof, and computer readable storage medium that calculate an accuracy of correspondence between a model feature and a measurement data feature and collate, based on the accuracy, a geometric model and an object in an image [P] . 美国专利： US9914222B2 . 2018-03-13

机译：信息处理设备，其控制方法以及计算机可读存储介质，其计算模型特征与测量数据特征之间的对应精度并基于该精度对图像中的几何模型和对象进行核对