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激光雷达是针对大尺寸产品进行形貌特征扫描的数字化测量设备,是一种球坐标测量系统。图 1所示为激光雷达的测量模型。当测量系统对P点进行坐标测量时,球坐标测量系统输出坐标为P(r,θ,ϕ)。
测得的斜距r、俯仰角θ、水平角ϕ经坐标转换为直角坐标(x,y,z),见下式:
$ \left\{\begin{array}{l} x=r \sin \theta \cos \phi \\ y=r \sin \theta \sin \phi \\ z=r \cos \theta \end{array}\right. $
(1) 同理,在对产品进行3维外形扫描时,也是依据此原理进行坐标转换,从而得到外形特征。
由(1)式可以看出,激光雷达的测量主要是依据测量距离和测量角度来完成测量任务,因此,系统自身的距离误差和角度误差是影响测量精度的关键[4]。
已知激光雷达系统的距离误差和角度误差均服从一个数字期望为μ、标准差为σ的正态分布,且误差范围在(μ-2σ, μ+2σ)内的极限距离误差为10μm+2.5μm/m,极限角度误差为6.8μm/m。现利用MATLAB软件对空间中的一点进行蒙特卡洛仿真测量实验。选取水平角为0°、俯仰角为45°的方向,距离为30m的一点进行1000次模拟测量,如图 2所示。
由系统误差值和图 2可以分析得出,测距误差是近似于线性的传递且对精度影响较小,而测角误差对测量结果的不确定度影响较大。因此,为了满足激光雷达测量系统对提高精度的需求,在测量场配置过程中,尤其需要注意对测角误差的控制。
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根据测量任务的不同,激光雷达的数据采集方式主要有两种,分别是单点坐标测量和外形扫描测量。
(1) 单点坐标测量是利用激光雷达测量系统配套的不锈钢球进行辅助测量,如图 5所示。当测量系统对转站点进行测量时,需在被测产品上布置具有磁性的基准靶座,将工具球放置在靶座上即可进行测量。在测量前设置好工具球尺寸,对半球体进行螺旋扫描,扫描的响应时间为2s,而普通的表面点响应时间仅为0.2s。这是由于测量工具球是对整个半球进行扫描后,选取反射能量最大的一点,即为垂直入射点,此点通过补偿工具球半径,即可得到被测点的坐标值。
(2) 外形扫描测量是激光雷达测量系统的优势所在,大多数测量系统采用人工手持扫描仪的方式进行产品的外形测量,这种方法不仅耗费人力,更重要的是人工测量的位置有限,一些大型产品的高度和宽度都使得手臂的伸展受到限制。而激光雷达通过软件与系统的结合使得激光照射到的部分均可以自动测量。扫描主要包括周界线扫描和矩形盒扫描两种方式。
周界线扫描主要分为闭合式和开放式两种,对于外形自动扫描来说,常用闭合周界线扫描进行测量,如图 6所示。通过在配套的测量软件中选取3个以上的测量引导点,即构成空间中的测量区域,在软件中设置测量行距和列距,将区域内网格化。从起始点开始沿着轮廓逐行扫描,到达轮廓边界时跳至下一行,直到测量区域内均已测量完成。
矩形盒扫描通过设置扫描盒的宽度、高度、中心位置和扫描间距,对盒空间内进行扫描。是对大尺寸形状不规则产品常用的外形自动扫描的方法,测量景深为±200mm。矩形盒扫描也分为3种方式,依据对测量精度和效率的不同需求选取不同的测量方式[17-19],表 1中为对比。
Table 1. Comparison of accuracy and response speed of rectangular box scanning method
scanning method precision/mm response speed/(points·s-1) vision box scan 0.1~0.2 125~1000 quick meter scan 0.05 20 meter box scan 0.025 0.5~2 -
经过多站位雷达的测量后,需要对测量数据进行统一空间测量网,即转站处理,通常采用奇异值分解法来完成坐标系转换,是工程上常用的坐标配准算法。
坐标转换方程为:
$ \boldsymbol{P}=\left[\begin{array}{ll} \boldsymbol{R} & \boldsymbol{T} \\ 0 & 1 \end{array}\right] \boldsymbol{Q} $
(2) 式中, P为参考坐标系,Q为待转换坐标系,R为旋转矩阵,T为平移矩阵。在配准两坐标系时,应将公共点坐标重心化,即首先确定旋转矩阵。则:
$ \left\{\begin{array}{l} \boldsymbol{P}_{\mathrm{g}}=\frac{\sum \boldsymbol{P}}{n_{\boldsymbol{P}}} \\ \boldsymbol{Q}_{\mathrm{g}}=\frac{\sum \boldsymbol{Q}}{n_{\boldsymbol{Q}}} \end{array}\right. $
(3) 式中,∑P为坐标系内nP个点的坐标和,∑Q为坐标系内nQ个点的坐标和,Pg和Qg分别代表参考坐标系和待转换坐标系的重心坐标。重心化后的测量点坐标为:
$ \left\{\begin{array}{l} \boldsymbol{P}_{\mathrm{h}}=\boldsymbol{P}-\boldsymbol{P}_{\mathrm{g}} \\ \boldsymbol{Q}_{\mathrm{h}}=\boldsymbol{Q}-\boldsymbol{Q}_{\mathrm{g}} \end{array}\right. $
(4) 已知目标函数为:
$ \begin{gathered} \sum\left\|\boldsymbol{P}_{\mathrm{h}}-\boldsymbol{R} \boldsymbol{Q}_{\mathrm{h}}\right\|^2= \\ \sum\left(\boldsymbol{P}_{\mathrm{h}}^{\mathrm{T}} \boldsymbol{P}_{\mathrm{h}}+\boldsymbol{Q}_{\mathrm{h}}^{\mathrm{T}} \boldsymbol{Q}_{\mathrm{h}}-2 \boldsymbol{P}_{\mathrm{h}}^{\mathrm{T}} \boldsymbol{R} \boldsymbol{Q}_{\mathrm{h}}\right) \end{gathered} $
(5) 为了使目标函数最小,设H=∑QhPhT,即求取trace(RH)最大值。则对矩阵H进行奇异值分解:
$ \boldsymbol{H}=\boldsymbol{U} \boldsymbol{D} \boldsymbol{V}^{\mathrm{T}} $
(6) 式中, U和V为正交单位矩阵,D为对角矩阵。再计算最佳旋转矩阵R为:
$ \boldsymbol{R}=\boldsymbol{V} \boldsymbol{U}^{\mathrm{T}} $
(7) 平移矩阵T即为:
$ \boldsymbol{T}=\boldsymbol{P}_{\mathrm{h}}-\boldsymbol{R} \boldsymbol{Q}_{\mathrm{h}} $
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为了证明基于激光雷达测量系统的测量场规划研究的正确性,设计两组实验进行对比分析。实验中分别采用两种测量系统对产品进行测量场的建立,数据处理方法均使用奇异值分解法,验证系统测量场规划配置的实用性。
第1次测量实验利用工程中应用较多且精度较高的3维扫描系统PRO CMM光学跟踪仪,如图 7所示。依据测量人员的经验对测量产品进行测量站位的布置和测量点的选取。
第2次测量实验利用激光雷达测量系统,依据测量场系统配置原则先进行测量场规划,后根据规划结果建立测量场,图 8为测量场示意图。
验证测量精度的参量主要包括点最大误差、点平均误差和全局均方根(root mean square, RMS)。实验数据如表 2所示。
Table 2. Comparison of experimental accuracy
accuracy parameter the first experiment accuracy/mm the second experiment accuracy/mm point maximum error 0.1572 0.0749 point average error 0.0328 0.0094 global RMS 0.0411 0.0143 在大尺寸测量任务中,测量点的误差通常限制在0.1mm以内,这就使第1次实验中部分点由于误差过大而需要剔除。而全局均方根代表了整个测量场的精度,大尺寸测量任务中通常要求保证在0.05mm以内,虽然两次实验的均方根精度均满足测量要求,但第1次实验的全局均方根是第2次的近3倍,误差趋于公差边界。
由两次实验的数据对比可以分析得出,第2次测量实验的整体测量精度明显优于第1次,同时也验证了激光雷达测量系统测量场规划方法的理论正确性和实际应用性。
激光雷达测量系统的测量场规划研究
Research on the measurement field planning of lidar measurement system
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摘要: 为了解决大尺寸空间测量场建立过程中会出现测量效率无法保证、精度不达标,以及测量过程中任务经常有遗漏等问题,对测量场的建立进行了规划研究。以激光雷达测量系统为载体,采用蒙特卡洛仿真方法对系统的测量模型进行分析,并对测量场全周期的建立过程,即系统站位配置、转站点布置优化、数据采集方式选择以及测量数据的预处理等进行了理论分析和实验验证。结果表明,经过规划的激光雷达测量场精度可以达到0.05mm以内,单点测量精度达到0.1mm以内,符合大尺寸空间测量场精度要求,实现了对测量场的规划评估。此研究对于大尺寸测量领域具有一定的意义。Abstract: In order to solve the problems that the measurement efficiency cannot be guaranteed, the precision is not up to the standard, and the task is often omitted during the measurement process, the establishment of the measurement field was planned and studied. Taking lidar measurement system as the carrier, the measurement model of the system was analyzed by monte carlo simulation method, and the establishment process of the whole period of the measurement field, including station configuration, station layout optimization, data acquisition method selection and measurement data preprocessing, was theoretically analyzed and experimentally verified. The results show that the precision of the planned lidar measurement field can reach less than 0.05mm, and the single point measurement precision can reach less than 0.1mm, which conforms to the accuracy requirements of the large-size space measurement field and realizes the planning evaluation of the measurement field. This research has certain significance in the field of large-scale measurement.
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Table 1. Comparison of accuracy and response speed of rectangular box scanning method
scanning method precision/mm response speed/(points·s-1) vision box scan 0.1~0.2 125~1000 quick meter scan 0.05 20 meter box scan 0.025 0.5~2 Table 2. Comparison of experimental accuracy
accuracy parameter the first experiment accuracy/mm the second experiment accuracy/mm point maximum error 0.1572 0.0749 point average error 0.0328 0.0094 global RMS 0.0411 0.0143 -
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