HW 4: Path Planning with RRTs

CS 545: Introduction to Robotics 
HW 4: Path Planning with RRTs
Submit all answers in a PDF named hw4 {USCNetID}.pdf (example: hw4 ttrojan.pdf).
In this assignment we’ll explore a fundamental problem in robotics: path planning. That
is, the problem of moving a robot from an initial configuration A to a goal configuration
B, while avoiding obstacles along the way.
You can think of this as a literal path planning problem where a mobile base must
navigate to a goal, or in a more abstract application, such as finding a sequence of joint
angles that will move an arm from one position to another.
This problem has two primary issues that make it difficult to apply shortest-path algorithms from graph theory, like Djikstra’s or A-star:
1. The search space is continuous: we don’t have a predefined set of discrete nodes to
search through. Any kind of discretized planning will require us to create our own
nodes.
2. There may not be an accurate measure of distance. For example, how do we determine the distance between two configurations for a robot arm? Do we look at the
end effector pose? The joint angles? There is no clear answer.
1 RRT
One method to solve this problem is an algorithm called Rapidly-exploring Random
Trees (RRT). The algorithm itself is quite simple, and only has 1 hyperparameter to tune!
Consider a 2D search problem where we must find a path from a starting point Nstart
to a goal point Ngoal, while avoiding the obstacles. To do so, we must build a path (a
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CS 545: Introduction to Robotics Fall 2021
sequence of points) connecting Nstart to Ngoal. We’ll be assuming a euclidean distance
function for our implementation.
Figure 1: A 2D RRT search space with no obstacles (left) and a single obstacle (right).
Nstart is in green, while Ngoal is in red.
An RRT is an iteratively built tree with clever use of random sampling that is likely
(though not guaranteed) to build one such path from Nstart to Ngoal.
We first randomly sample a point in the search space. Let’s call it Psample.
We then compute the direction vector between the closest node Nclose in our search tree
and Psample.
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CS 545: Introduction to Robotics Fall 2021
Now we create a new node that is a fixed distance δ away from Nclose along our direction
vector, which we’ll call Nnew.
If Nnew is not in collision with an obstacle, we add it to our search tree.
Figure 2: A valid Nnew will not collide with any obstacles (left). A Nnew that does collide
(right) should be discarded.
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CS 545: Introduction to Robotics Fall 2021
After every new addition, we check if that point is within δ of the goal position. If so,
we connect to the goal and draw our path.
Figure 3: Once Nnew is close enough to Ngoal, connect the two nodes and trace the path
from Nstart to Ngoal.
Note that the path we build is quite convoluted. RRT will only build a valid path, not
necessarily the shortest path! The algorithm provides no proof of optimality, but it is
relatively fast to compute compared to other path planning algorithms.
The example above is restricted to a 2-dimensional search problem, but the code you will
write will be able to work across N-dimensional search spaces.
2 RRT
Implement the empty methods in the RRT, CollisionBox, and CollisionSphere classes
in src/rrt.py and src/collision.py. The skeleton code lays out the basic structure,
alongside a Node implementation you may find helpful as a data structure. You are free
to modify the RRT class as needed, but you MUST implement the methods provided.
3 Testing
Unit tests are provided to help you check and bugfix your implementation. Look through
the tests in test/test rrt.py to understand what each unit test is looking for.
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CS 545: Introduction to Robotics Fall 2021
You can run unit tests for your RRT with the following commands:
cd code/rrt
# Run all tests for RRT
nose2 test.test_rrt
# Run a single test class
nose2 test.test_rrt.TestRRTInit
# Run a single test case
nose2 test.test_rrt.TestRRTInit.test_rrt_init_basic
The TestRRTBuild test case will run an end-to-end test of your RRT. Make sure you’ve
passed this final test case before moving on.
In the PDF, include a screenshot showing that all tests pass with nose2 test.test rrt.
4 Visualization
To get a visual understanding of what your RRT is doing, we have implemented a special
case of the RRT for 2-dimensional search problems with a visualization method.
Finish the implementation in src/planar rrt.py and examine the unit test in
test/test planar rrt.py. You can change the input values to experiment with your
PlanarRRT, but first try out the default configuration. You should see a visualization
similar to the ones illustrated in this document!
# Run the PlanarRRT
nose2 test.test_planar_rrt
In the PDF, include an image of the visualization that you generate when you run the
above command with the default configuration.
5 Questions
In the PDF file, answer each of the following questions in several sentences.
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CS 545: Introduction to Robotics Fall 2021
1. Why is the path returned by the RRT not guaranteed to be optimal (i.e. not the
shortest feasible path)?
2. What effect will increasing δ have on the performance of the RRT?
3. What effect will increasing the bounds of the search space have on the performance
of the RRT? How about increasing the number of dimensions of the search space?
4. Why is it important to have a relatively small δ?
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