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在该像旋补偿机构中,柔节相当于一个弹簧,受力时柔性单元发生弹性形变,内环带动镜筒正反向旋转,外环固定不动,外力取消后,像旋补偿机构恢复原态。八个柔性单元完全一样,放大单个柔性单元机构如图5所示,主要由三部分组成,左右是完全对称的半径为$ r $的四分之一圆,中间部分由恒定宽度为$ t $、长度为$ l $的直梁组成。
因此,在计算像旋补偿机构时,通过对单个柔性单元的受力分析进行计算,就可以计算出整体像旋补偿的受力变形。如图6所示,在对单个柔性单元进行受力分析时,假设一端固定,另一端为自由端,在主要转动平面内作用在柔性单元自由端的主要有$ {{F}}_{{x}} $、$ {{F}}_{{y}} $和$ {{M}}_{{z}} $三个载荷,在此三个载荷的作用下产生三个方向的位移$ {\;{\mu }}_{{x}} $、$ {\;{\mu }}_{{y}} $和$ {{\theta }}_{{z}} $。设$ {F}={\left[ {{F}}_{{x}},{{F}}_{{y}},{{M}}_{{z}}\right]}^{{\rm{T}}} $,$ {U}={\left[ \;{{\mu }}_{{x}},\;{{\mu }}_{{y}},{{\theta }}_{{z}}\right]}^{{\rm{T}}} $,根据力学知识和卡氏第二定理[11,19]可得:
$$ U=CF $$ (1) 式中:C为柔性单元的柔度矩阵。用$ {{C}}_{{a},{b}} $表示载荷b引起的在a方向的柔度,其被定义为:
$$ C=\left[\begin{array}{ccc}{C}_{x,{F}_{x}}& 0& 0\\ 0& {C}_{y,{F}_{y}}& {C}_{y,{M}_{z}}\\ 0& {C}_{z,{F}_{y}}& {C}_{z,{M}_{z}}\end{array}\right] $$ (2) 其中,各柔度表达式为:
$$ \left\{\begin{array}{c} C_{x, F_{x}}=\dfrac{1}{E w} \displaystyle\int_{0}^{L} \dfrac{{\rm{d}} x}{t(x)} \\ C_{y, F_{y}}=\dfrac{12}{E w} \displaystyle\int_{0}^{L} \dfrac{x^{2} {\rm{d}} x}{t(x)^{3}} \\ C_{y, M_{z}}=\dfrac{12}{E w} \displaystyle\int_{0}^{L} \dfrac{x {\rm{d}} x}{t(x)^{3}} \\ C_{z, F_{y}}=\dfrac{12}{E w} \displaystyle\int_{0}^{L} \dfrac{x {\rm{d}} x}{t(x)^{3}} \\ C_{z, M_{z}}=\dfrac{12}{E w} \displaystyle\int_{0}^{L} \dfrac{{\rm{d}} x}{t(x)^{3}} \end{array}\right. $$ (3) 式中:E为柔性单元所用材料的弹性模量;w为柔性单元的纵向宽度;t(x)为柔性单元切口厚度函数。在文中所选的柔性单元中,t(x)表达式为:
$$ {t}\left(x\right)=\left\{\begin{array}{c}t+2\left[r-\sqrt{x\left(2r-x\right)}\right]\\ t\\ t+2\left[r-\sqrt{{r}^{2}-(x-l-r{)}^{2}}\right]\end{array}\right.\begin{array}{cc}& x\in \left[0,r\right)\\ & x\in \left[r,l+r\right]\\ & x\in \left(l+r,l+2r\right]\end{array} $$ (4) -
理想情况下,柔性单元工作运动时转动中心固定不动,但实际工作过程中,柔性铰链在受到载荷影响下会产生微小的弹性变形,因此,使用柔性铰链回转中心的偏移量作为转动精度的评价。与柔性铰链柔度定义相似,将柔性铰链回转中心位移量设为$ {{\nu }}_{{x}} $和 $ {{\nu }}_{{y}} $,设$ {V}={\left[ {{\nu }}_{{x}},{{\nu }}_{{y}}\right]}^{{\rm{T}}} $,可以得到:
$$ {V}=C^{0} {F} $$ (5) 式中:$ {C}^{0} $为柔性单元的精度矩阵。用$ {C}_{a,{b}}^{0} $表示载荷b引起的在a方向柔性铰链的精度,当忽略平面外载荷时其被定义为:
$$ \left[C^{0}\right]=\left[\begin{array}{ccc} C_{x,{F_{x}}}^{0} & 0 & 0 \\ 0 & C_{y, F_{y}}^{0} & C_{y{,} M_{z}}^{0} \end{array}\right] $$ (6) 其中,各精度的具体表达式为:
$$ \left\{\begin{array}{l} C_{x, F_{x}}^{0}=\dfrac{1}{E w} \displaystyle\int_{0}^{L} \dfrac{{\rm{d}} x}{t(x)} \\ C_{y, F_{y}}^{0}=\dfrac{12}{E w} \displaystyle\int_{0}^{L} \dfrac{x^{2} {\rm{d}} x}{t(x)^{3}} \\ C_{y, M_{z}}^{0}=\dfrac{12}{E w}\left(\displaystyle\int_{0}^{L} \dfrac{x {\rm{d}} x}{t(x)^{3}}-L \displaystyle\int_{0}^{L} \dfrac{{\rm{d}} x}{t(x)^{3}}\right) \end{array}\right. $$ (7) -
将公式(4)代入公式(3)中可以得到柔性单元在各个方向上的柔度函数,则在主要的工作方向上柔度为:
$$ \begin{split} {C}_{z,{M}_{z}}=& \frac{12}{Ew}\left[\frac{l}{{t}^{3}}+\frac{2r\left(6{r}^{2}+4rt+{t}^{2}\right)}{{t}^{2}\left(2r+t\right){\left(4r+t\right)}^{2}}+\right.\\ & \left. \frac{6r\left(2r+t\right)}{{\left(4r+t\right)}^{{5}/{2}}\cdot {t}^{{5}/{2}}}\arctan\sqrt{1+4\dfrac{r}{t}}\right] \end{split} $$ (8) 在主要的工作方向上精度为:
$$ {C}_{x,{F}_{x}}^{0}=\dfrac{1}{2Ew}\left[{l}/{{t}}+\dfrac{2\left(2{r}/{t}+1\right)}{\sqrt{1+4{r}/{t}}}\arctan\sqrt{1+4{r}/{t}}-{{\pi }}/{2}\right] $$ (9) 其中,假定l+2r=20 mm,w初始设计为20 mm,柔度和精度主要与圆角r和厚度t相关,柔度与r、t的关系如图7所示,精度与r、t的关系如图8所示。
从图7、图8可以看出柔性单元柔度和精度与设计参数的对应关系。柔性单元圆角r越大,直梁厚度t越小,该结构的柔度越大;反之,r越小,t越大,该结构的柔度越小。对于柔性单元的精度,r和t越小,柔性单元的精度越高。
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设像旋补偿工作过程中压电驱动器所需要带动镜筒转动的角度为$ {\theta } $,相机镜筒的半径(相机补偿机构内圈半径)为R,根据力学知识和参考文献[20]可以得到该像旋补偿机构单个柔性单元所需要的驱动力$ {{F}}_{{y}} $和$ {{M}}_{{z}} $,其中:
$$ F_{y}=\frac{R C_{z, M_{z}}+C_{z, F_{y}}}{C_{z, M_{z}} \cdot C_{y, F_{{y}}}-C_{z, F_{y}}^{2}} \theta $$ (10) $$ M_z=\frac{R C_{y, M_z}+C_{y, F_y}}{C_{z, M_z} \cdot C_{y, F_y}-C_{y, M_z}^2} \theta $$ (11) 由公式(3)可知$ {C}_{z,{F}_{y}}={C}_{y,{M}_{z}} $,因为整个像旋补偿机构是完全对称的,所以得出整个机构所需要的总体力矩$ {M}={N}\times \left(R\times {F}_{y}+{M}_{z}\right) $。将公式(10)和(11)代入M可得:
$$ {M}={N} \theta \cdot \frac{R^2 C_{z, M_z}+2 R C_{y, M_z}+C_{y, F_y}}{C_{z, M_z} \cdot C_{y, F_y}-C_{y, M_z}^2} $$ (12) 同时可以计算单个柔性单元的最大应力$ {\sigma } $,如公式(13)所示:
$$ \sigma=6 k_{b} \frac{M_{z}+\left(-F_{y}\right) l}{w t^{2}}+k_{t} \frac{F_{{x}}}{w t} $$ (13) 式中:$ {k}_{b}、{k}_{t} $均为应力集中系数。
将像旋补偿机构看作质量刚度系统,设系统的转动惯量为J,相应的转动刚度为K,系统工作方向上的谐振频率为f(基频),f可以近似表示为:
$$ {f}=\frac{1}{2 \pi} \sqrt{\frac{K}{J}}=\frac{1}{2 \pi} \sqrt{\frac{M}{J \theta}} $$ (14) -
根据系统设计要求及相关技术指标,像旋补偿系统的像旋补偿角度$ {\theta }=\pm {2}{{{'}}} $,像旋补偿机构内圈半径按照R=135 mm设计。为避免发生共振,要求像旋补偿机构的基频远高于控制系统伺服带宽$ {f}_{c} $(60 Hz),一般要求系统的谐振频率$ {f} $为控制带宽$ {f}_{c} $的2~4倍。为满足空间探测的复杂环境要求,该像旋补偿机构实际应用时选用TC4材料,E=110 GPa,材料疲劳应力为660 MPa,则柔性机构中的最大应力不大于安全系数取2时的许用应力。结合空间相机实际的工作背景,取初始设计的尺寸t=0.5 mm,r=5 mm,w=20 mm,l=10 mm。
根据以上技术要求, 可以得到如下的约束条件:
$$ \begin{split} &{f} \geqslant 120\;{{\rm{Hz}}}\\ & {\sigma } \leqslant 330\;{{\rm{MPa}}} \end{split}$$ -
根据系统的设计要求,可以得到具体的优化设计模型如下:
$$\begin{split} & {f}_{{{\rm{obj}}}}=\min \left({M}_{(r,l,t,w)}\right)=\min({M}_{\left(x\right)}) \\ & \text{}\text{suppose:}{x}_{1}=r,{x}_{2}=l,{x}_{3}=t,{x}_{4}=w \\ & x={\left({x}_{1},{x}_{2},{x}_{3},{x}_{4}\right)}^{\text{T}} \end{split} $$ 约束条件:
$$ \left\{\begin{array}{l} {[2,1,0,0 ; 2,0,1,0] \leqslant x \leqslant [20 ; 13]} \\ {[4.5 ; 8 ; 0.25 ; 15] \leqslant x \leqslant[5.5 ; 10 ; 0.75 ; 25]} \\ -M+J \theta(240 \pi)^{2} \leqslant 0 \\ \sigma-3.3 \times 10^{8} \leqslant 0 \end{array}\right. $$ -
遗传算法(Genetic Algorithm, GA)是模拟达尔文生物进化论中的适者生存的生物进化过程的一种计算模型,是一种通过模拟自然进化过程搜索最优解的方法。遗传算法是模仿自然界生物进化机制发展起来的随机全局搜索和优化方法,它通过将实际模型转化为数学模型,将寻找最优解的过程对应地转化为生物进化过程中的选择、交叉和变异,通过不断的迭代寻找最优解。遗传算法的本质是一种在数学模型内搜索寻找最优解的方法,它可以在在搜索过程中自动获取和积累有关搜索空间的知识,并自适应地控制搜索过程以求得最佳解[21]。遗传算法的基本运算过程如图9所示。
将之前得到的数学模型代入Matlab优化工具箱的遗传算法中,可以得到最优化求解结果X=(5.491, 9.001, 0.518, 17.88)T,考虑到实际加工要求和系统设计要求,对所得的结果近似求解,可以得到结果X=(5.5, 9, 0.5, 18)T。因此,设计得到的柔性单元的最终设计尺寸为t=0.5 mm,r=5.5 mm,w=18 mm,l=9 mm。
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在完成像旋补偿机构的优化设计之后,使用UG将优化完成的像旋补偿机构进行重新三维建模,之后通过Hypermesh软件对其进行有限元前处理,具体的三维结构图和有限元网格划分分别如图10(a)和图10(b)所示。
然后对像旋补偿机构有限元静力学仿真分析,假定像旋补偿机构底部通过螺栓进行完全固定,在机构负载为450 N的状态下,两侧压电陶瓷驱动器总驱动力为115 N。对像旋补偿机构内环两测对称的各施加57.5 N的外力,对此时机构的位移和应力进行仿真分析。
仿真分析结果位移变量如图11 (a)所示,对柔性单元的放大变形量如图11 (b)所示,此时平面内最大位移分量为77.5 μm,与理论计算模型误差为1.79%,远小于5%,符合系统的设计要求。仿真结果的应力变形如图12(a)所示,最大应力出现在柔性单元的直梁位置(如图12(b)所示),此时最大应力为65 MPa,远远小于许用应力330 MPa ,符合系统的设计需求,该像旋补偿机构具有较高的稳定性和安全性。
在对像旋补偿机构完成静力学仿真之后搭建实验平台,对其进行实验验证分析,主要测试流程:将像旋补偿机构外环固定在气浮平台上,内环施加有垂直向下的45 kg的负载(如图13所示)。为了避免实验误差,需要对像旋补偿机构进行多处位置测量,在像旋补偿机构内环外侧边缘做四处测试记号,分别记录在施加0~150 N外力(以15 N为间隔选取10个点)情况下各处测试记号的位移。为了提升实验精度,实验取四处数据的平均值。
图 10 (a) 优化后的像旋补偿机构;(b)像旋补偿机构有限元模型
Figure 10. (a) Optimized image rotation compensation mechanism; (b) Finite element model of image rotation compensation mechanism
主要的实验数据记录如表1所示。
表 1 力-位移实验测试数据
Table 1. Test data of force-displacement experiment
Outside force/N First place Second place Third place Fourth place Average value/μm 15 13 12 13 13 12.75 30 20 20 21 20 20.25 45 29 28 29 31 29.25 60 39 39 39 37 38.50 75 47 47 48 49 47.75 90 59 57 57 58 57.75 105 68 67 68 66 67.25 120 76 78 77 79 77.50 135 87 86 88 87 87.00 150 98 97 97 95 96.75 绘制相应的力-位移(转角)曲线(如图14所示),可以看出,在工作区间内,像旋补偿机构具有很好的线性度,即该像旋补偿机构具有良好的性能。然后计算相应的转角,并将计算结果与有限元仿真数据进行对比,计算误差。
根据实验数据可知,所设计的像旋补偿机构具有较好的刚度,在工作范围内得到的系统具有良好的线性度。由表2可知,实验数据与理论计算结果误差最大为2.5%,均小于5%,符合要求,验证了理论推导的可行性,验证说明了像旋补偿机构的合理性,符合系统设计需求。
表 2 力-位移实验测试数据与仿真数据对比
Table 2. Comparison of force-displacement test data and simulation data
Outside force/
NExperimental data/
μmSimulation data/
μmError 15 12.75 12.92 1.3% 30 20.25 20.22 0.16% 45 29.25 29.70 1.5% 60 38.50 39.33 2.1% 75 47.75 48.95 2.5% 90 57.75 58.74 1.7% 105 67.25 68.46 1.8% 120 77.50 78.19 0.88% 135 87.0 87.92 1.1% 150 96.75 97.50 0.78% 然后对像旋补偿机构进行有限元模态分析, 在约束状态下,对系统的前六阶模态进行仿真分析,可以得到系统的前六阶模态振型如图15所示,固有频率如表3所示。
表 3 系统的前六阶模态固有频率
Table 3. The first six modes natural frequencies of the system
Order Natural frequency/Hz First order 252.41 Second order 1400.20 Third order 1400.41 Fourth order 1707.10 Fifth order 1707.1 Sixth order 1769.51 由图15和表3可以看出,像旋补偿机构的固有频率满足系统的设计需求,像旋补偿机构具有较高的控制带宽,系统具有较高的控制精度。
Image rotation compensation mechanism of large field of view space camera and its optimization design
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摘要: 在对空间碎片进行分析研究中,大视场的天基目标探测载荷成为提高探测效率的有效方式,在实际成像过程中,由于曝光时间长,加上卫星自身在轨姿态的运动、二维转台的转动等因素,会导致空间相机在图像上产生像旋。通过齐次坐标变换法对像旋进行分析,得出相机存在±2′的像旋,并利用柔性单元和压电驱动器设计了一种新型无摩擦、无磨损、免润滑的像旋补偿机构,带动空间相机反方向的旋转对像旋进行补偿。然后对柔性单元应力和系统固有频率进行优化设计,推导柔度公式和精度公式,并对结果进行有限元仿真分析。结果表明:仿真结果与理论计算模型的最大相对误差均小于 5% ,该机构能够实现大视场空间相机的像旋补偿,并且具有较高的精度。Abstract:
Objective Space-based target detection is the main way to observe space debris. In recent years, with the gradual increase of space debris, it is difficult for small field of view space cameras to meet the observation needs, and the use of large field of view space cameras is increasing. During the observation of space debris, due to the orbital motion of the satellite itself and the motion of the two-dimensional turntable, the image rotation will occur in the imaging of the large field of view space camera, especially when observing dim targets, the camera's exposure time will increase, and the generated image rotation will also increase. It seriously affects the accuracy of recognition and reduces the efficiency of large field of view space camera. Therefore, image rotation compensation must be carried out for large field of view space camera. Methods In order to determine the performance index of image rotation compensation, the imaging coordinate system of the system is established (Fig.1), and the image rotation of the system is calculated by the homogeneous coordinate change method. According to the performance index of image rotation compensation, a new type of image rotation compensation mechanism based on the inner and outer rings of symmetrical right straight circular flexure hinge is proposed (Fig.3). Then, the flexibility and accuracy formula of the flexible element of the image rotation compensation mechanism is deduced according to the second theorem of Cassegrain, and the relationship between the flexibility and the structure size is analyzed. Then, the image rotation compensation structure is optimized by genetic algorithm. Finally, the static and modal analysis of the image rotation compensation mechanism is carried out by simulation (Fig.11, 12, 14), and it is verified by experiments. Results and Discussions By analyzing and calculating the ± 2′ image rotation of a large field of view space camera, an image rotation compensation mechanism composed of eight completely symmetrical flexible elements is designed for the image rotation change. By analyzing it, the relationship between the flexibility and accuracy of the flexible element and the size of the flexible element is obtained (Fig.7-8). Through genetic algorithm, the final design size of the flexible element is t=0.5 mm, r=5.5 mm, w=18 mm, l=9 mm. The simulation analysis results show that the maximum displacement component in the plane is 77.5 μm. The error with theoretical calculation model is 1.79%, far less than 5%, which meets the design requirements of the system; The maximum stress is 65 MPa, which is far less than the allowable stress of 330 MPa, which meets the design requirements of the system. The image rotation compensation mechanism has high stability and safety. Through experimental verification, the experimental value and theoretical error of the image rotation compensation mechanism are also less than 5%, and the image rotation compensation mechanism has good linearity in the working range (Fig.14). The results of modal analysis (Tab.3) show that all modes of the system meet the design requirements. Conclusions For the image rotation generated by the large field of view space camera during imaging, the image rotation angle generated by the camera is calculated by the homogeneous coordinate transformation method to be ± 2′, and then a set of image rotation compensation mechanism based on the flexible element is designed by this technical index, the mathematical model of the image rotation compensation mechanism is established, and the flexibility matrix and precision matrix of the flexible element of the image rotation compensation mechanism are derived; Then, according to the derived formula and the stress and fundamental frequency of the system, the image rotation compensation mechanism is optimized by genetic algorithm. Finally, the image rotation compensation mechanism is determined to be composed of the inner and outer rings connected by 8 straight beam fillet flexible elements. When the total force is 115 N, the camera is compensated with ± 2′ image rotation, and the maximum displacement of the inner ring is 77.5 μm. Then the first six natural frequencies of the system are verified by the finite element simulation, which meet the system design requirements, and the system is verified by experiments. According to the experimental results, the system has good linearity, and the error between the experimental results and the simulation results is less than 5%, which verifies the reliability of the system. -
表 1 力-位移实验测试数据
Table 1. Test data of force-displacement experiment
Outside force/N First place Second place Third place Fourth place Average value/μm 15 13 12 13 13 12.75 30 20 20 21 20 20.25 45 29 28 29 31 29.25 60 39 39 39 37 38.50 75 47 47 48 49 47.75 90 59 57 57 58 57.75 105 68 67 68 66 67.25 120 76 78 77 79 77.50 135 87 86 88 87 87.00 150 98 97 97 95 96.75 表 2 力-位移实验测试数据与仿真数据对比
Table 2. Comparison of force-displacement test data and simulation data
Outside force/
NExperimental data/
μmSimulation data/
μmError 15 12.75 12.92 1.3% 30 20.25 20.22 0.16% 45 29.25 29.70 1.5% 60 38.50 39.33 2.1% 75 47.75 48.95 2.5% 90 57.75 58.74 1.7% 105 67.25 68.46 1.8% 120 77.50 78.19 0.88% 135 87.0 87.92 1.1% 150 96.75 97.50 0.78% 表 3 系统的前六阶模态固有频率
Table 3. The first six modes natural frequencies of the system
Order Natural frequency/Hz First order 252.41 Second order 1400.20 Third order 1400.41 Fourth order 1707.10 Fifth order 1707.1 Sixth order 1769.51 -
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