109 lines
3.4 KiB
Markdown
109 lines
3.4 KiB
Markdown
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# UWB 定位原理
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#### 1.三维坐标转二维坐标
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在割草机器人项目中，割草机器人目前只考虑二维平面的定位。但是UWB测量的距离是三维距离，所以我们根据机器人的高度`carH`和`UWB`标签的高度`UwbH`计算出水平距离`dxy`。
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```cpp
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// dxy^2 = di^2 - (UwbH = carH)^2
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for(int i=0; i<3; i++){
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this->d[i] = sqrt(this->d[i] * this->d[i] - (AnchorPos[i][2] - CARHEIGHT) *
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(AnchorPos[i][2] - CARHEIGHT));
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}
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```
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<img src="./Image/UWB高度修正.png" alt="" style="zoom:50%;" />
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#### 2.多项式拟合
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UWB的定位是存在波动的，所以会根据UWB计算距离的规律对计算的距离进行多项式拟合，可以起到滤波提高精度作用。下面的计算实际是收集不同实测距离下，UWB的实际输出距离，利用3次多项式拟合得到的结果。
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下面的计算跟标签的位置以及高度无关，主要跟UWB的硬件设备的特性有关。
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```cpp
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d[0] = ((((4.9083e-07 * d[0]) - 4.6166e-04) * d[0]) + 1.0789) * d[0] + 5.4539;
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d[1] = ((((-4.1679e-07 * d[1]) + 5.0999e-04) * d[1]) + 0.7930) * d[1] + 29.8296;
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d[2] = ((((2.3514e-07 * d[2]) - 1.8277e-04) * d[2]) + 0.9935) * d[2] + 9.8852;
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```
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#### 3.位置求解
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UWB位置求解采用如下图示：
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<img src="./Image/UWB位置示意图.png" alt="" style="zoom:50%;" />
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UWB的定位可以用下面公式描述, 其中$(x,y)$是割草机器人上面的UWB的位置，另外三个坐标点是3个UWB标签的位置，可以有如下的公式。
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$$
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d_1^2 = (x_1 - x)^2 + (y_1 - y)^2 \space\space\space\space\space (1)\\
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d_2^2 = (x_2 - x)^2 + (y_2 - y)^2 \space\space\space\space\space(2)\\
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d_3^2 = (x_3 - x)^2 + (y_3 - y)^2 \space\space\space\space\space(3)\\
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$$
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$(2)-(1)$以及$(3)-(2)$消去二次项，可得：
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$$
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d_1^2 - d_2^2 = \left[ -2(x_1 - x_2)x + x_1^2 - x_2^2 \right] + \left[ -2(y_1 - y_2)y + y_1^2 - y_2^2 \right] \\
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d_1^2 - d_3^2 = \left[ -2(x_1 - x_3)x + x_1^2 - x_3^2 \right] + \left[ -2(y_1 - y_3)y + y_1^2 - y_3^2 \right]
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$$
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整理为矩阵形式：
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$$
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-2 \begin{bmatrix}
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x_1 - x_2 & y_1 - y_2 \\
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x_1 - x_3 & y_1 - y_3
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\end{bmatrix}
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\begin{bmatrix}
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x \\
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y
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\end{bmatrix}
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=
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\begin{bmatrix}
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(d_1^2 - d_2^2) - (x_1^2 - x_3^2) - (y_1^2 - y_3^2) \\
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(d_1^2 - d_3^2) - (x_1^2 - x_3^2) - (y_1^2 - y_3^2)
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\end{bmatrix}
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$$
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整理可得：
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$$
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\begin{align*}
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A &= -2\cdot \begin{bmatrix}
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x_1 - x_2 & y_1 - y_2 \\
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x_1 - x_3 & y_1 - y_3
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\end{bmatrix}\\
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b &= \begin{bmatrix}
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(d_1^2 - d_2^2) - (x_1^2 - x_2^2) - (y_1^2 - y_2^2) \\
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(d_1^2 - d_3^2) - (x_1^2 - x_3^2) - (y_1^2 - y_3^2)
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\end{bmatrix}\\
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X &= \begin{bmatrix}
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x\\
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y
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\end{bmatrix}
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\end{align*}
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$$
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矩阵A对应的代码：
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```cpp
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for(int i=0; i<2; i++){
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A.mat[i][0] = -2*(this->AnchorPos[0][0]-this->AnchorPos[i+1][0]);
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A.mat[i][1] = -2*(this->AnchorPos[0][1]-this->AnchorPos[i+1][1]);
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}
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```
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矩阵b对应的代码：
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```cpp
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for(int i=0; i<2; i++)
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{
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b.mat[i][0] = (this->d[0]*this->d[0]-this->d[i+1]*this->d[i+1])\
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- (this->AnchorPos[0][0]*this->AnchorPos[0][0]-this->AnchorPos[i+1][0]*this->AnchorPos[i+1][0])
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- (this->AnchorPos[0][1]*this->AnchorPos[0][1]-this->AnchorPos[i+1][1]*this->AnchorPos[i+1][1]);
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}
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```
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那么，上述矩阵可以通过$X=(A^T\cdot A)^{-1}A^T*b$ 求解UWB的位置。
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```cpp
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Mat AT=~A;
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uwbPos=(AT*A)%AT*b;
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this->uwb_data_.x_ = uwbPos.mat[0][0];
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this->uwb_data_.y_ = uwbPos.mat[1][0];
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```
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