$i$ is the index of segment that contains the $s_0$.
so we get the equality constraint as
so we get the equality constraint as
$$
\begin{vmatrix}
1 & s_0 & s_0^2 & s_0^3 & s_0^4&s_0^5 \\
...
...
@@ -205,7 +205,7 @@ $$
### 2.3 Joint smoothness constraints
This constraint is to make the spline joint smooth. Let's assume two segment $seg_k$ and $seg_{k+1}$ are connected, and the accumulated s of segment $seg_k$ is $s_k$. The we can get the constraint euqation as
This constraint is to make the spline joint smooth. Let's assume two segment $seg_k$ and $seg_{k+1}$ are connected, and the accumulated s of segment $seg_k$ is $s_k$. The we can get the constraint equation as
$$
f_k(s_k) = f_{k+1} (s_0)
$$
...
...
@@ -251,7 +251,7 @@ $$
### 2.4 Sampled points for boundary constraint
Evenly sample m points along the path, and check the obstacle boundary at those points. convert the constraint into QP inequality constraints
Evenly sample m points along the path, and check the obstacle boundary at those points. Convert the constraint into QP inequality constraints
$$
Ax \leq b
$$
...
...
@@ -273,7 +273,7 @@ $$
$$
similarly, for upper boundary $l_{ub,j}$, we get the ineuqality constraints as
similarly, for upper boundary $l_{ub,j}$, we get the inequality constraints as
This constraint is to make the spline joint smooth. Let's assume two segment $seg_k$ and $seg_{k+1}$ are connected, and the accumulated s of segment $seg_k$ is $s_k$. The we can get the constraint euqation as
This constraint is to make the spline joint smooth. Let's assume two segment $seg_k$ and $seg_{k+1}$ are connected, and the accumulated s of segment $seg_k$ is $s_k$. The we can get the constraint equations as
$$
f_k(t_k) = f_{k+1} (t_0)
$$
...
...
@@ -162,7 +162,7 @@ $$
$$
similarly, for upper boundary $l_{ub,j}$, we get the ineuqality constraints as
similarly, for upper boundary $l_{ub,j}$, we get the inequality constraints as
