Graphs on Mars

Math is absolutely fundamental to a space mission! It permeates every aspect of the mission from spacecraft design, to launch, to calculating the trajectory, to communication, to landing, to operations and safe return. Algebra, geometry, calculus, statistics, probability, and many other areas of math are involved!

In 2021, a car-sized robotic rover called Perseverance landed on Mars as part of NASA's Mars 2020 mission. Since then, it has roamed the surface of Mars, with the goals of searching for signs of ancient microbial life on Mars, and collecting rock and soil samples for possible return to Earth. How does Perseverance navigate the rocky and difficult terrain of Mars in order to accomplish its goals? By using an area of math called graph theory. Let's dive into it!

What is Graph Theory?

Graph theory is a branch of mathematics that studies relationships between objects using a visual representation called a graph. Essentially, it's about modeling connections between things, where the "things" are represented as nodes (also called vertices) and the connections between them are represented as edges (also called links or lines). For example, in a map of cities and roads, the cities are the nodes, and the roads between them are the edges. That's a basic graph.

Credit: cglab

How does the Perseverance rover use Graph Theory?

Graph theory plays a crucial role in the operations of Perseverance on Mars, particularly in navigation. Perseverance's Enhanced AutoNav (ENav) software utilizes graph theory to represent the Martian terrain as a series of points or areas (nodes) connected by traversable paths (edges). Then, algorithms like Dijkstra's or A* are applied to identify the most efficient and safe routes for Perseverance to travel between its current location and its targets. This involves considering factors like:

• Slope and obstacles: Identifying areas with difficult terrain, obstacles like rocks or craters, and potential hazards like sand dunes to avoid or navigate safely.

• Scientific goals: Planning paths that allow Perseverance to access areas of high scientific interest, such as specific rock formations or locations for sample collection.

  
  

• Resource Management: Optimizing for factors like energy consumption or time efficiency.

  
  

• Re-planning: Updating the terrain model (and thus the graph) with new information, allowing for efficient re-planning of paths to adapt to unexpected situations and new discoveries.

(Expand the image below to see the animation.)

Perseverance landing and mission. Credit: NASA/JPL-Caltech

So, the next time you look up at the sky and see the red planet, know that at that very instant, math is being put to use on it!