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空间向量之间的夹角怎么求,空间向量夹角的计算公式

时间:2023-05-04 21:48:26 阅读:219741 作者:3628

有些设备正常工作时需按合适的方位安装,比如GPS天线必须朝向天空才能保证信号最佳,温湿度传感器监测口必须朝向被测目标才能及时响应。软件需求是在安装角度异常时提醒用户改变位置。那么设备如何感知当前方位呢?需要一颗加速度传感器硬件支持,辅以算法实现。

1、重力加速度

根据物理常识,地面上任何物体静止时都受到1g的重力加速度,且方向是竖直向下。

因为倾斜角的不同,1g的加速度按向量分解到xyz三轴:
acc_x=1g.sinθ.cosϕ
acc_y=-1g.sinθ.sinϕ
acc_z=1g.cosϕ
符号.代表相乘,读取加速度传感器的xyz三轴数据的细节,不是本文考虑的范畴。

因为传感器的位数精度和量程不同,同样的1g,读寄存器的数值不同,为统一后文描述,假设数值255对应1g的加速度。物体水平静止时加速度值理想情况是(0,0,255),反向水平放置是(0,0,-255)。这个数值等比例的缩小或放大,不影响角度的判断。

2、空间向量夹角

假设期望的正确安装方式下三轴是(x0,y0,z0),实际三轴数据是(x1,y1,z1)。那如何得出当前偏差的角度呢?

已知两空间向量的坐标为a=(x1,y1,z1),b=(x2,y2,z2),则两向量的夹角余弦cosθ公式为:


在实际应用中,翻转角度为181度时,按179度处理。本文只考虑0-180度的应用,180度以上的需要额外再计算翻转。

根据空间向量夹角余弦,再反余弦得出在0-180度的角度,即可判断设备安装角度是否正确。

3、代码实现 #include "math.h"#include "stdio.h"#define PI 3.1415926typedef struct{ unsigned short x; unsigned short y; unsigned short z;}sensor_data_struct;static sensor_data_struct ref={0,0,255};static sensor_data_struct test={0,180,180};//计算当前向量与参考向量的夹角float get_angle(sensor_data_struct data){ float cosine; float temp,angle; cosine=(data.x*ref.x+data.y*ref.y+data.z*ref.z)/ ((sqrt(data.x*data.x+data.y*data.y+data.z*data.z))* (sqrt(ref.x*ref.x+ref.y*ref.y+ref.z*ref.z))); temp=acos(cosine); angle=(temp*180)/PI; return angle;}int main(int argc, char *argv[]){ float angle; printf("reference vector (%d,%d,%d)rn",ref.x,ref.y,ref.z); printf("test vector (%d,%d,%d)rn",test.x,test.y,test.z); angle=get_angle(test); printf("angle = %f'rn",angle); return 0;}

reference vector (0,0,255)
test vector (0,180,180)
angle = 45.000004’

4、优化改进

求解角度使用的三角函数,部分单片机可能不支持;对角度的精度,使用整形即可。基于这个条件,可以建立cosθ的数组表,以1度-2度-3度…180度步进,按如下代码生成数组表:

void creat_table(void){ float i; for(i=0;i<180;i++) { if((unsigned char )i%9==0) { printf("rn"); } printf("%f,",cos(i*PI/180));//角度转弧度再传入 }}

根据代码生成数组表后,查找余弦表,数组的下标即为角度。

static const float cos_table[180]={ 1.000000,0.999848,0.999391,0.998630,0.997564,0.996195,0.994522,0.992546,0.990268, 0.987688,0.984808,0.981627,0.978148,0.974370,0.970296,0.965926,0.961262,0.956305, 0.951057,0.945519,0.939693,0.933580,0.927184,0.920505,0.913545,0.906308,0.898794, 0.891007,0.882948,0.874620,0.866025,0.857167,0.848048,0.838671,0.829038,0.819152, 0.809017,0.798636,0.788011,0.777146,0.766044,0.754710,0.743145,0.731354,0.719340, 0.707107,0.694658,0.681998,0.669131,0.656059,0.642788,0.629320,0.615661,0.601815, 0.587785,0.573576,0.559193,0.544639,0.529919,0.515038,0.500000,0.484810,0.469472, 0.453991,0.438371,0.422618,0.406737,0.390731,0.374607,0.358368,0.342020,0.325568, 0.309017,0.292372,0.275637,0.258819,0.241922,0.224951,0.207912,0.190809,0.173648, 0.156434,0.139173,0.121869,0.104528,0.087156,0.069756,0.052336,0.034900,0.017452, 0.000000,-0.017452,-0.034899,-0.052336,-0.069756,-0.087156,-0.104528,-0.121869,-0.139173, -0.156434,-0.173648,-0.190809,-0.207912,-0.224951,-0.241922,-0.258819,-0.275637,-0.292372, -0.309017,-0.325568,-0.342020,-0.358368,-0.374607,-0.390731,-0.406737,-0.422618,-0.438371, -0.453990,-0.469472,-0.484810,-0.500000,-0.515038,-0.529919,-0.544639,-0.559193,-0.573576, -0.587785,-0.601815,-0.615661,-0.629320,-0.642788,-0.656059,-0.669131,-0.681998,-0.694658, -0.707107,-0.719340,-0.731354,-0.743145,-0.754710,-0.766044,-0.777146,-0.788011,-0.798635, -0.809017,-0.819152,-0.829038,-0.838671,-0.848048,-0.857167,-0.866025,-0.874620,-0.882948, -0.891007,-0.898794,-0.906308,-0.913545,-0.920505,-0.927184,-0.933580,-0.939693,-0.945519, -0.951057,-0.956305,-0.961262,-0.965926,-0.970296,-0.974370,-0.978148,-0.981627,-0.984808, -0.987688,-0.990268,-0.992546,-0.994522,-0.996195,-0.997564,-0.998630,-0.999391,-0.999848,};unsigned short get_angle(sensor_data_struct data){ float cosine; unsigned short i; cosine=(data.x*ref.x+data.y*ref.y+data.z*ref.z)/ ((sqrt(data.x*data.x+data.y*data.y+data.z*data.z))* (sqrt(ref.x*ref.x+ref.y*ref.y+ref.z*ref.z))); for(i=0;i<180;i++) { if(cos_table[i]<cosine)//查表 { return i; } } return 180;//error}int main(int argc, char *argv[]){ unsigned short angle;//改成整形 printf("reference vector (%d,%d,%d)rn",ref.x,ref.y,ref.z); printf("test vector (%d,%d,%d)rn",test.x,test.y,test.z); angle=get_angle(test); printf("angle = %d'rn",angle); return 0;}

reference vector (0,0,255)
test vector (0,180,180)
angle = 46’

查表得出46度,因为查表以及浮点的精度,所以角度误差±1度,但这个不影响业务逻辑。

5、小节

1、空间向量夹角公式可在基于xyz三轴的传感器中应用。
2、针对范例中的应用,两个向量的参数必须是在静止情况下采样,根据向量模进行过滤,否则角度计算错误。
3、未考虑大于180度的翻转。

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