二次规划ST速度优化

$$
minimize \frac{1}{2} \cdot x^T \cdot H \cdot x + f^T \cdot x 
\\
s.t. LB \leq x \leq UB
\\
A_{eq}x = b_{eq}
\\
Ax \leq b
$$

$$
s = f_i(t) 
  = a_{0i} + a_{1i} \cdot t + a_{2i} \cdot t^2 + a_{3i} \cdot t^3 + a_{4i} \cdot t^4 + a_{5i} \cdot t^5
$$

$$
cost_1 = \sum_{i=1}^{n} \Big( w_1 \cdot \int\limits_{0}^{d_i} (f_i')^2(s) ds + w_2 \cdot \int\limits_{0}^{d_i} (f_i'')^2(s) ds + w_3 \cdot \int\limits_{0}^{d_i} (f_i^{\prime\prime\prime})^2(s) ds \Big)
$$

$$
cost_2 = \sum_{i=1}^{n}\sum_{j=1}^{m}\Big(f_i(t_j)- s_j\Big)^2
$$

$$
cost_3 = \sum_{i=1}^{n}\sum_{j=1}^{o}\Big(f_i(t_j)- s_j\Big)^2
$$

$$
cost = cost_1 + cost_2 + cost_3
$$

$$
A_{eq}x = b_{eq}
$$

$$
\begin{vmatrix}  1 & t_j & t_j^2 & t_j^3 & t_j^4&t_j^5 \\ \end{vmatrix} 
\cdot 
\begin{vmatrix}  a_k \\ b_k \\ c_k \\ d_k \\ e_k \\ f_k  \end{vmatrix} 
> 
\begin{vmatrix}  1 & t_{j-1} & t_{j-1}^2 & t_{j-1}^3 & t_{j-1}^4&t_{j-1}^5 \\ \end{vmatrix}  
\cdot 
\begin{vmatrix}  a_{k} \\ b_{k} \\ c_{k} \\ d_{k} \\ e_{k} \\ f_{k}  \end{vmatrix}
$$

$$
\begin{vmatrix}  1 & t_j & t_j^2 & t_j^3 & t_j^4&t_j^5 \\ \end{vmatrix} 
\cdot 
\begin{vmatrix}  a_k \\ b_k \\ c_k \\ d_k \\ e_k \\ f_k  \end{vmatrix} 
> 
\begin{vmatrix}  1 & t_{j-1} & t_{j-1}^2 & t_{j-1}^3 & t_{j-1}^4&t_{j-1}^5 \\ \end{vmatrix}  
\cdot 
\begin{vmatrix}  a_{l} \\ b_{l} \\ c_{l} \\ d_{l} \\ e_{l} \\ f_{l}  \end{vmatrix}
$$

$$
f_k(t_k) = f_{k+1} (t_0)
$$

$$
\begin{vmatrix} 
 1 & t_k & t_k^2 & t_k^3 & t_k^4&t_k^5 \\
 \end{vmatrix} 
 \cdot 
 \begin{vmatrix} 
 a_{k0} \\ a_{k1} \\ a_{k2} \\ a_{k3} \\ a_{k4} \\ a_{k5} 
 \end{vmatrix} 
 = 
\begin{vmatrix} 
 1 & t_{0} & t_{0}^2 & t_{0}^3 & t_{0}^4&t_{0}^5 \\
 \end{vmatrix} 
 \cdot 
 \begin{vmatrix} 
 a_{k+1,0} \\ a_{k+1,1} \\ a_{k+1,2} \\ a_{k+1,3} \\ a_{k+1,4} \\ a_{k+1,5} 
 \end{vmatrix}
$$

$$
\begin{vmatrix} 
 1 & t_k & t_k^2 & t_k^3 & t_k^4&t_k^5 &  -1 & -t_{0} & -t_{0}^2 & -t_{0}^3 & -t_{0}^4&-t_{0}^5\\
 \end{vmatrix} 
 \cdot 
 \begin{vmatrix} 
 a_{k0} \\ a_{k1} \\ a_{k2} \\ a_{k3} \\ a_{k4} \\ a_{k5} \\ a_{k+1,0} \\ a_{k+1,1} \\ a_{k+1,2} \\ a_{k+1,3} \\ a_{k+1,4} \\ a_{k+1,5}   
 \end{vmatrix} 
 = 0
$$

$$
f'_k(t_k) = f'_{k+1} (t_0)
\\
f''_k(t_k) = f''_{k+1} (t_0)
\\
f'''_k(t_k) = f'''_{k+1} (t_0)
$$

$$
Ax \leq b
$$

$$
\begin{vmatrix} 
 1 & t_0 & t_0^2 & t_0^3 & t_0^4&t_0^5 \\
  1 & t_1 & t_1^2 & t_1^3 & t_1^4&t_1^5 \\
 ...&...&...&...&...&... \\
 1 & t_m & t_m^2 & t_m^3 & t_m^4&t_m^5 \\
 \end{vmatrix} \cdot \begin{vmatrix} a_i \\ b_i \\ c_i \\ d_i \\ e_i \\ f_i \end{vmatrix} 
 \leq 
 \begin{vmatrix}
 l_{lb,0}\\
 l_{lb,1}\\
 ...\\
 l_{lb,m}\\
 \end{vmatrix}
$$

$$
\begin{vmatrix} 
 1 & t_0 & t_0^2 & t_0^3 & t_0^4&t_0^5 \\
  1 & t_1 & t_1^2 & t_1^3 & t_1^4&t_1^5 \\
 ...&...&...&...&...&... \\
 1 & t_m & t_m^2 & t_m^3 & t_m^4&t_m^5 \\
 \end{vmatrix} \cdot \begin{vmatrix} a_i \\ b_i \\ c_i \\ d_i \\ e_i \\ f_i \end{vmatrix} 
 \leq
 -1 \cdot
 \begin{vmatrix}
 l_{ub,0}\\
 l_{ub,1}\\
 ...\\
 l_{ub,m}\\
 \end{vmatrix}
$$

$$
f'(t_j) \geq v_{lb,j}
$$

$$
\begin{vmatrix}  
0& 1 & t_0 & t_0^2 & t_0^3 & t_0^4 \\  
0 & 1 & t_1 & t_1^2 & t_1^3 & t_1^4 \\ 
...&...&...&...&...&... \\ 
0& 1 & t_m & t_m^2 & t_m^3 & t_m^4 \\ 
\end{vmatrix} 
\cdot 
\begin{vmatrix} 
a_i \\ b_i \\ c_i \\ d_i \\ e_i \\ f_i 
\end{vmatrix}  
\geq  
\begin{vmatrix} v_{lb,0}\\ v_{lb,1}\\ ...\\ v_{lb,m}\\ \end{vmatrix}
$$

$$
f'(t_j) \leq v_{ub,j}
$$

$$
\begin{vmatrix} 
 0& 1 & t_0 & t_0^2 & t_0^3 & t_0^4 \\
 0 & 1 & t_1 & t_1^2 & t_1^3 & t_1^4 \\
 ...&...&...&...&...&... \\
 0 &1 & t_m & t_m^2 & t_m^3 & t_m^4 \\
 \end{vmatrix} \cdot \begin{vmatrix} a_i \\ b_i \\ c_i \\ d_i \\ e_i \\ f_i \end{vmatrix} 
 \leq
 \begin{vmatrix}
 v_{ub,0}\\
 v_{ub,1}\\
 ...\\
 v_{ub,m}\\
 \end{vmatrix}
$$