雅可比矩阵及其应用

Last updated on：a year ago

定义

重积分

$$\phi(x) = (\phi_1(x_1,…,x_m),…,\phi_m(x_1,…,x_m))$$

矩阵微分

$1\times m$行向量偏导算子记为

$$D_\pmb{x} = \frac{\partial}{\partial \pmb{x}^T} = [\frac{\partial}{\partial x_1},…,\frac{\partial}{\partial x_m}]$$

$$D_\pmb{x} f\pmb{x} = \frac{\partial f\pmb{x}}{\partial \pmb{x}^T} = [\frac{\partial f\pmb{x}}{\partial x_1},…,\frac{\partial f\pmb{x}}{\partial x_m}]$$

$m\times 1$列向量偏导算子即梯度算子记作$$\nabla_\pmb{x}$$，定义为

$$\nabla_\pmb{x} = \frac{\partial}{\partial \pmb{x}} = [\frac{\partial}{\partial x_1},…,\frac{\partial}{\partial x_m}]^T$$

$$\nabla_\pmb{x} f\pmb{x} = \frac{\partial f\pmb{x}}{\partial \pmb{x}} = [\frac{\partial f\pmb{x}}{\partial x_1},…,\frac{\partial f\pmb{x}}{\partial x_m}]^T$$

其他定义

$$\pmb {J} ={\begin{bmatrix}{\dfrac {\partial \pmb {f} }{\partial x_{1}}}&\cdots &{\dfrac {\partial \pmb {f} }{\partial x_{n}}}\end{bmatrix}}={\begin{bmatrix}\nabla ^{\mathrm {T} }f_{1}\\vdots \\nabla ^{\mathrm {T} }f_{m}\end{bmatrix}}={\begin{bmatrix}{\dfrac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{1}}{\partial x_{n}}}\\vdots &\ddots &\vdots \{\dfrac {\partial f_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{m}}{\partial x_{n}}}\end{bmatrix}}$$

应用

$${\frac{\partial f_i}{\partial x_j}} = \frac{f_i (\pmb(x)+he_j)-f_i (\pmb(x))}{h}$$

参考资料

[2] 皮埃尔·科尔梅，分析与代数原理（及数论）：第二版，第二卷，高等教育出版社，p317-318（重积分）

[3] 张贤达，矩阵分析与应用：第2版，清华大学出版社，p143-161（矩阵微分）

[4] Waldron, K.J., Wang, S.L. and Bolin, S.J., 1985. A study of the Jacobian matrix of serial manipulators.