机器人中的数值优化进阶|【二】三次样条曲线推导（中）

机器人中的数值优化|【自用二】三次样条曲线推导

接之前，由于
$$c_i=3({\eta }_{i+1}-{\eta }_i)-2D_i-D_{i+1}$$
因此有
c = 3 [ − 1 1 0 0 . . . 0 0 − 1 1 0 . . . 0 0 0 − 1 1 . . . 0 . . . . . . 0 0 0 . . . − 1 1 ] n × ( n + 1 ) η − [ 2 1 0 0 . . . 0 0 2 1 0 . . . 0 0 0 1 1 . . . 0 . . . . . . 0 0 0 . . . 2 1 ] n × ( n + 1 ) D c = 3\begin{bmatrix} -1 & 1 & 0 & 0 & ... & 0 \\ 0 & -1 & 1 & 0 & ... & 0 \\ 0 & 0 & -1 & 1 & ... & 0 \\ ... & ... \\ 0 & 0 & 0 & ... & -1 & 1 \end{bmatrix}_{n\times(n+1)} \eta - \begin{bmatrix} 2 & 1 & 0 & 0 & ... & 0 \\ 0 & 2 & 1& 0 & ... & 0 \\ 0 & 0 & 1 & 1 & ... & 0 \\ ... & ... \\ 0 & 0 & 0 & ... & 2 & 1 \end{bmatrix}_{n\times(n+1)} D c=3 ​−100...0​1−10...0​01−10​001...​.........−1​0001​ ​n×(n+1)​η− ​200...0​120...0​0110​001...​.........2​0001​ ​n×(n+1)​D
$$c\in {\mathbb{R}}^{(n\times 2)}$$
令
[ − 1 1 0 0 . . . 0 0 − 1 1 0 . . . 0 0 0 − 1 1 . . . 0 . . . . . . 0 0 0 . . . − 1 1 ] n × ( n + 1 ) = P \begin{bmatrix} -1 & 1 & 0 & 0 & ... & 0 \\ 0 & -1 & 1 & 0 & ... & 0 \\ 0 & 0 & -1 & 1 & ... & 0 \\ ... & ... \\ 0 & 0 & 0 & ... & -1 & 1 \end{bmatrix}_{n\times(n+1)} = P ​−100...0​1−10...0​01−10​001...