Algorithm 2 Elliptical path generation

Input: Initial and desired configuration of the robot

Output: List of angles, W

1.Apply Equation (1) to determine a,b,h,k based on initial and final configuration such that:

$\{\begin{array}{l}\frac{{\left(x_r-h\right)}^2}{a^2}+\frac{{\left(y_r-k\right)}^2}{b^2}=1\\ \frac{\left(x_r-h\right)}{a^2}+\frac{\left(y_r-k\right)·m_r}{b^2}=0\end{array},\hspace{1em}\hspace{1em}\{\begin{array}{l}\frac{{\left(x_d-h\right)}^2}{a^2}+\frac{{\left(y_d-k\right)}^2}{b^2}=1\\ \frac{\left(x_d-h\right)}{a^2}+\frac{\left(y_d-k\right)·m_d}{b^2}=0\end{array}$

where (xr, yr) is the robot’s center coordinate of the head of initial configuration, mr is the robot’s tangent of the orientation of initial configuration and (xd, yd) is the center coordinate of the head of desired goal configuration, md is the tangent of the orientation of the desired configuration

2.Current configuration, Ci = Initial configuration, I   3. While Ci ≠ Desired Configuration, D: Execute set of angles to follow ellipse to generate a new set of configurations {C} Select new configuration, N from {C} which closely satisfies the ellipse Add angle of N to list of angles, W Ci = N   4.Return W