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车载重力测量采取的是相对重力测量的方式,属于动基座重力测量的一种,其中动基座重力测量的计算公式如下:
$$ \delta {{{g}}^n}{\text{ = }}{{\dot v}}_{}^n - {{C}}_b^n{{{f}}^b} + \left( {2{{\omega }}_{ie}^n + {{\omega }}_{en}^n} \right) \times {{v}}_{}^n - {{{\gamma }}^n} $$ (1) 式中:$ \delta {{{g}}^n} $为重力扰动矢量;$ \dot{ { v}}^{n} $为载体加速度;$ v^{n} $为载体速度;$ {{{f}}^b} $为加速度计的比力测量值;$ {C}_{b}^{n} $为载体坐标系(b系)和导航坐标系(n系)之间的方向余弦矩阵;${{ \omega}}_{i e}^{n} $为地球自转角速度在n系下的投影;$ {{\omega}}_{e n}^{n} $为导航坐标系相对于地球坐标系(e系)的转速在n系下的投影;${{ \gamma}}^{n} $为正常重力值。
在基于SINS/LDV组合系统的重力测量中,公式(1)中的参数分别由SINS和LDV测得:一类由SINS测得,如b系下的比力$ {{{f}}^b} $和方向余弦矩阵$ {C}_{b}^{n} $;另一类由LDV和SINS共同测得,如载体速度$ v^{n} $、加速度$ \dot{{v}}^{n} $、正常重力值$ {{{\gamma }}^n} = {[0{\text{ 0 }}\gamma {\text{]}}^{{\rm{T}}} } $以及科里奥利加速度$ \left( {2{{\omega }}_{ie}^n + {{\omega }}_{en}^n} \right) \times {{v}}_{}^n $。
文中,对于SINS/LDV组合车载重力测量方法的初步研究中只讨论重力标量测量,即重力异常,也即公式(1)中的天向分量:
$$ \begin{gathered} \delta {g_U} = {{\dot v}_U} - {f_U} - \left( {\frac{{{v_E}}}{{{R_N} + h}} + 2{\omega _{ie}}{\rm{cos}}L} \right) \cdot {v_E} {\text{ }} - \frac{{v_N^2}}{{{R_M} + h}} - \gamma \\ \end{gathered} $$ (2) 式中:${\dot v_U}$为载体天向加速度;$ {f_U} $为等效天向加速度计测量值;$ {v_E} $、$ {v_N} $为载体东向和北向速度;$ {\omega _{ie}} $为地球自转角速度;L、h分别为载体所在位置的纬度和高度;RM、RN分别为载体所在位置的子午圈半径和卯酉圈半径。
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由于二维激光多普勒测速仪提供的是载体对地速度,因此SINS/LDV组合重力测量的误差模型与传统的SINS/GNSS组合方式存在一定的差异,其多项误差源于SINS姿态误差与LDV速度误差的耦合。SINS/LDV组合重力测量误差模型为:
$$ \begin{split} {\rm{d}}\delta {{{g}}^n} =& {{C}}_b^n\delta {{{{\dot v}}}^b} - \left[ {{{{f}}^n} \times } \right]\phi - {{C}}_b^n\delta {{{f}}^b} + \\& {\text{ }}\left( {2{{\omega }}_{ie}^n + {{\omega }}_{en}^n} \right) \times \left( {{{C}}_b^n\delta {{{v}}^b} + \left[ {{{{v}}^n} \times } \right]{{\phi}} } \right) -\\& {\text{ }} \left[ {{{{v}}^n} \times } \right]\left( {2\delta {{\omega }}_{ie}^n + \delta {{\omega }}_{en}^n} \right) - \delta {{{\gamma }}^n} \end{split}$$ (3) 式中:$ {\rm{d}}\delta {{{g}}^n} $为重力测量误差;$ \delta {{{\dot v}}^b} $为加速度计算误差在b系下的投影;${{ \phi }}$为惯导的姿态误差;$\delta {{{f}}^b} $为比力测量误差;$ \delta {{{v}}^b} $为测速仪速度误差在b系下的投影;$ {{{v}}^n} $为理想情况下测速仪速度在n系中的投影;$ \delta {{\omega }}_{ie}^n $为地球自转角速度计算误差;$ \delta {{\omega }}_{en}^n $为n系相对e系的转速计算误差。
由公式(3)可以看出,SINS/LDV组合重力测量的误差主要来源于两个方面:(1) 与惯导系统相关的比力测量误差$ \delta {{{f}}^b} $和姿态计算误差$ {{\phi}} $;(2) 与测速仪相关的速度测量误差$ \delta {{{{v}}}^b} $、加速度计算误差$ \delta {{\dot {{v}}}^b} $以及位置计算误差。
同样,在进行系统误差模型分析时也只考虑天向分量。为了使重力标量测量精度优于1 mGal,需要分别对SINS、LDV以及测量方案进行约束,其中参考文献[12]对SINS/GNSS组合重力测量方案中的SINS进行了分析与约束,由于SINS在SINS/GNSS和SINS/LDV组合方案中的作用相同,因此参考文献[12]中对SINS的约束也适用于SINS/LDV组合方案。通过对天向误差模型进行解耦和分析,根据车载重力测量的实际速度情况,假设载体速度为20 m/s时,对LDV和测量方案进行如表1所示的约束。
表 1 系统指标
Table 1. System index
Index Accuracy Velocity measurement accuracy of LDV ≤1‰ Calculation accuracy of acceleration/mGal ≤1 Measurement time of single line/h ≤1.7 Horizontal accuracy/m ≤10 Height accuracy/m ≤3 其中,100 m的水平定位误差引起的重力测量误差约0.07 mGal,因此其对重力测量的影响可以忽略,但是为了提高系统的可靠性,表1中对水平定位精度进行了10 m的约束,可以适当放宽约束。
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单一的惯导系统误差随着时间累计,难以满足长时间的高精度导航和重力测量需求,激光多普勒测速仪具有精度高、实时性好等优点,可以通过测速仪提高惯导系统精度,进一步地,高精度的惯导姿态又可以提高测速仪在n系下的投影。为了实现测速仪与惯导系统的信息融合,通常采用卡尔曼滤波技术[13-14]。首先需要构建15维状态空间模型,构建状态方程如下:
$$ {{\dot X}}(t) = {{F}}(t){{X}}(t) + {{G}}(t){{W}}(t) $$ (4) 式中:$ {{F}}(t) $为系统状态转移矩阵;$ {{G}}(t) $为系统噪声转移矩阵;$ {{W}}(t) $为系统噪声矩阵;$ {{X}}(t) = {[ {\begin{array}{*{20}{c}} \phi & {\delta {{v}}}& {\delta {{p}}}& {{\varepsilon }}& \nabla \end{array}} ]^{\text{T}}} $为状态量,其中$ \phi $为姿态误差,$ \delta {{v}} $为速度误差,$ \delta {{p}} $为位置误差,$ {{\varepsilon }} $为陀螺漂移,$ \nabla $为加速度计零偏。
构建量测方程如下:
$$ {{Z}}(t) = {{H}}(t){{X}}(t) + {{V}}(t) $$ (5) 式中:$ {{V}}(t) $为量测噪声矩阵。以测速仪在n系下的投影为基准,将惯导系统速度误差作为量测量:
$$ {{Z}}(t) = \left[ {{{v}}_{{\text{SINS}}}^n - {{v}}_{{\text{LDV}}}^n} \right] $$ (6) 式中:$ {{v}}_{{\text{SINS}}}^n $为惯导系统速度;$ {{v}}_{{\text{LDV}}}^n $为测速仪速度。
量测矩阵:
$$ {{H}}(t) = \left[ {\begin{array}{*{20}{c}} {{{{0}}_{3 \times 3}}}&{{{{I}}_{3 \times 3}}}&{{{{0}}_{3 \times 9}}} \end{array}} \right] $$ (7) 为了抑制高度通道的发散,提高滤波器收敛速度,每次滤波结束之后需要将误差进行反馈补偿,补偿方式如下:
$$ \begin{gathered} {{C}}_b^n = \left( {{{I}} + \left( {\phi \times } \right)} \right){{\hat {{C}}}}_b^n \\ {{{v}}^n} = {{{{\hat {{v}}}}}^n} - \delta {{{v}}^n} \\ {{{p}}^n} = {{{{\hat {{p}}}}}^n} - \delta {{{p}}^n} \\ \end{gathered} $$ (8) 式中:$ {{\hat {{C}}}}_b^n $和$ {{C}}_b^n $分别为理想姿态矩阵和实际姿态矩阵;$ {{{\hat {{v}}}}^n} $和$ {{{v}}^n} $分别为理想速度和实际速度;$ {{{\hat {{p}}}}^n} $和$ {{{p}}^n} $分别为理想位置和实际位置。
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为了验证SINS/LDV组合重力测量系统的性能,选择位于湖南省长沙市的黑麋峰进行了单条测线的重复重力测量实验。实验设备选择笔者单位自研的捷联惯导系统和二维激光多普勒测速仪,其数据输出频率均为100 Hz。惯导系统内部的激光陀螺精度优于0.003 (°)/h,加速度计精度优于20 μg,测速仪测速精度优于1‰。实验车同时搭载GPS,以SINS/GPS组合车载重力测量的结果作为对比,其中GPS的单点水平定位精度为1.5 m,差分水平定位精度为0.4 m。实验设备安装如图2所示,惯导系统安装在实验车内部,为了减小杆臂误差,测速仪安装在惯导系统正下方的车底,GPS天线置于车顶。
实验轨迹和实验过程中的高度变化如图3~4所示,实验共六条重复测线,单条测线长约11 km,最大海拔变化约332.7 m。
在黑麋峰实验环境下,实验车姿态变化较为剧烈且GPS信号受遮挡严重,实验过程中的GPS工作状态、测速仪品质因子以及测速仪输出分别如图5~7所示。
GPS工作时采用GPGGA的数据输出格式,工作状态为1表示单点定位、2表示码差分以及4表示固定解,其中工作状态为4时定位精度最高。通过图5可以发现,GPS工作期间部分时间段下定位精度明显下降,由位置一次差分的速度和二次差分的加速度精度也会随之下降。
测速仪品质因子反映了测速仪测量的可靠性,当品质因子高于100时,数据输出稳定可靠。由图6可以看出,在实验车移动过程中,测速仪的品质因子普遍高于100,说明实验过程中测速仪比较稳定。进一步地,图7反映了实验过程中二维测速仪两个探头的对地速度,可以看出实验过程中测速仪探头输出稳定变化,受测量环境影响较小。
SINS/LDV组合导航结果如图8~9所示。可以看出,SINS/LDV组合重力测量系统的水平误差和高度误差最大分别为约17 m和2 m,其中高度误差满足表1中的系统指标,而水平误差超过了系统指标,其对重力测量的影响不大,因此认为水平误差在可接受范围内。
对比基于SINS/GNSS、SINS/LDV组合系统的重力异常计算,结果如图10~11所示。
图 10 SINS/LDV组合重力异常测量结果
Figure 10. Gravity anomaly measurement results of the integration of SINS/LDV
图 11 SINS/GNSS组合重力异常测量结果
Figure 11. Gravity anomaly measurement results of the integration of SINS/GNSS
由图10可以看出,基于SINS/LDV组合系统的六条测线一致性较好,由于黑麋峰实验环境海拔变化较大,重力异常变化也相对较大,符合预期。另外,如图11所示,由于卫星信号受遮挡的原因,SINS/GNSS组合系统的六条测线一致性相对较差,但是其重力异常变化范围仍与SINS/LDV组合系统一致。
此次实验中,由于没有建立外部基准点,只进行系统的内符合精度评估,其目的是验证系统的重复性和稳定性。两套系统的内符合精度评估结果如表2和表3所示。
表 2 SINS/LDV组合重力测量系统内符合精度
Table 2. Internal coincidence accuracy of SINS/LDV integrated gravimetry system
Maximum Minimum RMS of
single lineRMS of
total lineInternal
coincidence
accuracy/
mGalLine 1 1.08 −1.46 0.73 0.70 Line 2 0.98 −1.00 0.47 Line 3 1.18 −2.47 0.66 Line 4 2.37 −1.78 1.05 Line 5 0.83 −1.24 0.53 Line 6 1.26 −1.04 0.59 表 3 SINS/GNSS组合重力测量系统内符合精度
Table 3. Internal coincidence accuracy of SINS/GNSS integrated gravimetry system
Maximum Minimum RMS of
single lineRMS of
total lineInternal
coincidence
accuracy/
mGalLine 1 2.22 −1.57 1.03 1.53 Line 2 1.82 −3.05 1.30 Line 3 3.36 −2.92 1.75 Line 4 1.38 −2.77 1.02 Line 5 2.58 −4.73 2.46 Line 6 1.62 −2.59 1.05 由表2~3可以看出,SINS/LDV组合系统的单条测线内符合最大值为1.05 mGal,最小值为0.47 mGal,总的内符合精度为0.70 mGal;SINS/GNSS组合系统的单条测线内符合精度最大值为2.46 mGal,最小值为1.02 mGal,总的内符合精度为1.53 mGal。通过实验结果可知,在GNSS信号部分拒止的情况下,基于SINS/LDV组合重力测量系统的六条测线的内符合精度普遍优于基于SINS/GNSS组合系统的测线精度,并且其总内符合精度也相比SINS/GNSS组合系统提高了约54%,充分验证了SINS/二维LDV组合重力测量方案的有效性。
Strapdown vehicle autonomous gravimetry method based on two-dimensional laser Doppler velocimeter
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摘要: 传统的车载重力测量通常采用捷联惯导系统(Strapdown Inertial Navigation System, SINS)/全球导航卫星定位系统(Global Navigation Satellite System, GNSS)组合的方式,但是在如山谷、隧道以及高楼林立等特殊环境下,GNSS信号会受到遮挡,导致重力测量系统精度下降。针对特殊环境下传统车载重力测量方法精度下降的问题,提出了一种基于捷联惯导系统/二维激光多普勒测速仪(Laser Doppler Velocimeter, LDV)组合的车载重力测量方式,分析了系统重力测量原理和误差模型,设计了滤波器方案,通过车载重力测量实验对系统精度进行了验证。实验针对丛林遮蔽的山地环境下完成了六条重复测线重力测量,同时比对SINS/GNSS组合重力测量系统的测量精度,其中SINS/GNSS组合系统的单条测线内符合精度最大为2.46 mGal,最小为1.03 mGal,总内符合精度为1.53 mGal;SINS/LDV组合系统的单条测线内符合精度最大为1.05 mGal,最小为0.47 mGal,总内符合精度为0.70 mGal,其总内符合精度相比于SINS/GNSS组合系统提高了约53%。车载重力测量实验证明了SINS/LDV组合重力测量系统在卫星信号拒止环境下的有效性。
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关键词:
- 车载重力测量 /
- 二维激光多普勒测速仪 /
- 捷联惯导系统 /
- 组合导航 /
- 内符合精度
Abstract:Objective As one of the basic physical fields of the earth, gravity field reflects the distribution of underground materials and the changes of space and time. It has important value in resource exploration, military application and space science research. At present, the measurement methods of earth gravity field include aerial gravimetry, marine gravimetry, satellite gravimetry and ground gravimetry, et al. As an important method of gravity field measurement, ground gravimetry is mainly used for local fine construction of earth gravity field, which can be divided into ground static gravimetry and ground dynamic gravimetry. Due to the high cost and low efficiency of ground static gravimetry, ground dynamic gravimetry is usually adopted, namely ground vehicle gravimetry. At present, strapdown inertial navigation system (SINS)/global navigation satellite system (GNSS) integrated system is usually used in vehicle gravimetry, which lacks autonomy and has limited accuracy in the special environment where GNSS signal is blocked. To solve this problem, this paper proposes a strapdown vehicle autonomous gravimetry method based on two-dimensional laser Doppler velocimeter (LDV). Methods In this paper, a high-precision autonomous vehicle gravimetry method is designed. In order to improve the autonomy of the system, SINS/LDV integrated system is adopted in this paper, which does not need to rely on external signal sources. In order to ensure the measurement accuracy of the system, the LDV adopted by the system is two-dimensional, which is sensitive to the velocity of the vertical direction, so as to reduce the measurement error. The systematic errors are analyzed, and the constraints on the device accuracy and measurement scheme of LDV are proposed (Tab.1). The data processing flow of SINS/LDV integrated gravimetry system is also proposed. The system can finally realize the high-precision gravimetry in special environment. Result and Discussions The experiment was conducted in a special environment. During the experiment, the altitude changed greatly (Fig.4) and the GPS was seriously blocked (Fig.5). There are altogether 6 repeated lines in the experiment, each of which is about 11 km. During the experiment, the maximum horizontal error of SINS/LDV integrated navigation is about 17 m (Fig.8), and the maximum height error is about 2 m (Fig.9), which meets the requirements of system standard (Tab.1). Gravity anomaly were calculated according to the results of integrated navigation. The six lines based on SINS/LDV integrated system had a good consistency, and the maximum and minimum internal coincidence accuracy of a single survey line are 1.05 mGal and 0.47 mGal, and the total internal coincidence accuracy is 0.70 mGal (Tab.2). However, the consistency of the six lines based on SINS/GNSS integrated system are relatively poor. The maximum and minimum accuracy of internal coincidence of a single line are 2.46 mGal and 1.02 mGal, and the total internal coincidence accuracy is 1.53 mGal (Tab.3). The accuracy of SINS/LDV integrated system is generally better than that of SINS/LDV integrated system, and the total accuracy of SINS/LDV integrated system is about 54% higher than that of SINS/GNSS integrated system. Conclusions In this paper, a strapdown vehicle autonomous gravimetry method based on two-dimensional laser Doppler velocimeter is studied. The measurement principle and error model of the system are analyzed, and the corresponding index and data processing flow of the system are given. The vehicle gravimetry experiment shows that the consistency of the six lines in SINS/LDV integrated gravimetry system is high, while that of the six lines in SINS/GNSS integrated gravimetry system is relatively poor when the satellite signal is seriously blocked. Accordingly, the single internal coincidence accuracy of SINS/LDV integrated system is generally better than that of SINS/GNSS integrated system, and the total internal coincidence accuracy of SINS/LDV integrated system is nearly half higher than that of SINS/GNSS integrated system. The experimental results show that SINS/LDV integrated gravimetry system can realize gravimetry, and the gravimetry accuracy is higher than SINS/GNSS integrated gravimetry system in special measuring environment. The research of this paper provides technical support for the vehicle gravimetry in the environment when the GNSS signal is blocked, and the relevant results can be applied in geological exploration, gravity matching and the refinement of the earth's local gravity field. -
表 1 系统指标
Table 1. System index
Index Accuracy Velocity measurement accuracy of LDV ≤1‰ Calculation accuracy of acceleration/mGal ≤1 Measurement time of single line/h ≤1.7 Horizontal accuracy/m ≤10 Height accuracy/m ≤3 表 2 SINS/LDV组合重力测量系统内符合精度
Table 2. Internal coincidence accuracy of SINS/LDV integrated gravimetry system
Maximum Minimum RMS of
single lineRMS of
total lineInternal
coincidence
accuracy/
mGalLine 1 1.08 −1.46 0.73 0.70 Line 2 0.98 −1.00 0.47 Line 3 1.18 −2.47 0.66 Line 4 2.37 −1.78 1.05 Line 5 0.83 −1.24 0.53 Line 6 1.26 −1.04 0.59 表 3 SINS/GNSS组合重力测量系统内符合精度
Table 3. Internal coincidence accuracy of SINS/GNSS integrated gravimetry system
Maximum Minimum RMS of
single lineRMS of
total lineInternal
coincidence
accuracy/
mGalLine 1 2.22 −1.57 1.03 1.53 Line 2 1.82 −3.05 1.30 Line 3 3.36 −2.92 1.75 Line 4 1.38 −2.77 1.02 Line 5 2.58 −4.73 2.46 Line 6 1.62 −2.59 1.05 -
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