The main task is to implement the Rapidly Exploring Random Trees (RRT).

In RRTDubinsPlanner.py implement the Rapidly Exploring Random Tree (RRT) randomized sampling-based path planning algorithm according to the description and pseudocode presented in the Lecture 07. The algorithm should provide a collision free curvature constrained path through an environment represented by a geometrical map.

The RRTDubinslanner.plan function has the following prescription

import SamplingPlanner as sp
 
...
 
class RRTPlanner(sp.SamplingPlanner):
 
    ...
 
    def plan(self, environment, start, goal):
        """
        Method to plan the path
 
        Parameters
        ----------
        environment: Environment
            Map of the environment that provides collision checking 
        start: numpy array (4x4)
            start configuration of the robot in SE(3) coordinates
        goal: numpy array (4x4)
            goal configuration of the robot in SE(3) coordinates
 
        Returns
        -------
        list(numpy array (4x4))
            the path between the start and the goal Pose in SE(3) coordinates
        """
 
        #TODO: t3b-rrt - implement the curvature constrained RRT planner
 
        print (start)
        print (goal)
 
        assert not environment.check_robot_collision(start), 'start collision'
        assert not environment.check_robot_collision(goal), 'goal collision'
 
        path = [start,goal]
 
 
        return(path)

The path is contrained by the minimal turning radius RRTPlanner.turning_radius. The dubins package is to be used to compute the curvature constrained paths.

Although the curvature constrained path is effectively $SE(2)$, i.e., three degrees of freedom, $SE(3)$ is used to represent the individual poses. The pose $\mathbf{P} \in SE(3)$ is given as $$ \mathbf{P} = \begin{bmatrix} \mathbf{R} & \mathbf{T}\\ [0, 0, 0] & 1 \end{bmatrix}, $$ where $\mathbf{R} \in \mathcal{R}^{3\times3}$ is the rotation matrix for which $\mathbf{R}\cdot\mathbf{R} = \mathbf{I}$ and $\det(\mathbf{R})=1$. $\mathbf{T} \in \mathcal{R}^3$ is the translation vector Therefore, the individual poses are the rigid body transformations in the global reference frame. Hence, the position of the robot $r$ is given as the transformation of the robot base pose $\mathbf{r}_b$ in homogeneous coordinates, given as:$$ \begin{bmatrix} \mathbf{r}\\ 1 \end{bmatrix} = \mathbf{P}\cdot\begin{bmatrix} \mathbf{r}_b\\ 1\end{bmatrix}, $$ which can be also written as$$ \mathbf{r} = \mathbf{R}\cdot\mathbf{r}_b + \mathbf{T}. $$

The boundaries for individual configuration variables are given during the initialization of the planner in SamplingPlanner.limits variable as a list of lower-bound and upper-bound limit tuples, i.e. list( (lower_bound, upper_bound) ), for each of the variables $(x,y,z,\phi_x,\phi_y,\phi_z)$. The $(x,y,z,\phi_x,\phi_y,\phi_z)$ vector can be transformed to and from $SE(3)$ using the sp.te_2_se3 and sp.se3_2_te functions, respectively.

The individual poses shall not be further than SamplingPlanner.max_norm_translation, which distance should be computed along the curvature constrained path.

The collision checking is performed using the environment.check_robot_collision function that takes on the input an $SE(3)$ pose matrix. The collision checking function returns True if there is collision between the robot and the environment and False if there is no collision. Moreover, the SamplingPlanner.is_valid_edge can be used to check whether the connection from $SE(3)$ represented point start to $SE(3)$ represented point end is valid both in terms of the collision checker and maximum distance.

pip install dubins

On Linux (tested with Ubuntu 14.04, 16.04, 18.04)

1. Download the resource package and make the rapid library in environment/rapid directory

On MacOS

1. On line 7 of environment/rapid/Makefile change TARGET=librapid.so to TARGET=librapid.dylib
2. On line 11 of environment/rapid/Makefile change -soname to -install_name