【四足机器人那些事】雅各比矩阵

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【四足机器人那些事】雅各比矩阵

一、雅各比

雅各比矩阵式多元形式的导数，假设有以下3个函数，每个函数有6个自变量：

y 1 = f 1 ( x 1 , x 2 , x 3 ) y 2 = f 2 ( x 1 , x 2 , x 3 ) y 3 = f 3 ( x 1 , x 2 , x 3 ) \begin{aligned} &y_1 = f_1(x_1, x_2, x_3) \\ \\ &y_2 = f_2(x_1, x_2, x_3) \\\\ &y_3 = f_3(x_1, x_2, x_3) \end{aligned} ​y1​=f1​(x1​,x2​,x3​)y2​=f2​(x1​,x2​,x3​)y3​=f3​(x1​,x2​,x3​)​

写成矩阵的形式：

Y = F ( x ) Y = F(x) Y=F(x)

其中：

Y = [ y 1 y 2 y 3 ] X = [ x 1 x 2 x 3 ] Y = \begin{bmatrix}y_1\\y_2\\y_3\end{bmatrix} \quad X = \begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix} Y=⎣⎡​y1​y2​y3​​⎦⎤​X=⎣⎡​x1​x2​x3​​⎦⎤​

计算 y i y_i yi​的微分与 x i x_i xi​的微分的函数：

δ y 1 = ∂ f 1 ∂ x 1 δ x 1 + ∂ f 1 ∂ x 2 δ x 2 + ∂ f 1 ∂ x 3 δ x 3 δ y 2 = ∂ f 2 ∂ x 1 δ x 1 + ∂ f 2 ∂ x 2 δ x 2 + ∂ f 2 ∂ x 3 δ x 3 δ y 3 = ∂ f 3 ∂ x 1 δ x 1 + ∂ f 3 ∂ x 2 δ x 2 + ∂ f 3 ∂ x 3 δ x 3 \begin{aligned} &\delta y_1 = \frac{\partial f_1}{\partial x_1}\delta x_1 +\frac{\partial f_1}{\partial x_2}\delta x_2 + \frac{\partial f_1}{\partial x_3}\delta x_3 \\\\ &\delta y_2 = \frac{\partial f_2}{\partial x_1}\delta x_1 +\frac{\partial f_2}{\partial x_2}\delta x_2 + \frac{\partial f_2}{\partial x_3}\delta x_3 \\\\ &\delta y_3 = \frac{\partial f_3}{\partial x_1}\delta x_1 +\frac{\partial f_3}{\partial x_2}\delta x_2 + \frac{\partial f_3}{\partial x_3}\delta x_3 \\\\ \end{aligned} ​δy