It’s not easy for a robot to find its way out of a maze. Imagine the machines trying to cross a child’s playroom to get to the kitchen, with various toys scattered on the floor and furniture blocking some possible paths. This messy maze requires the robot to calculate the optimal journey to its destination without hitting any obstacles. What should the bot do?
The Graphs of Convex Set (GCS) Trajectory Optimization algorithm by MIT Computer Science and Artificial Intelligence Laboratory (CSAIL) researchers presents a scalable collision-free motion planning system for these robotic navigation needs. The getting closer combines graph search (a method of finding discrete paths in a network) and convex optimization (an efficient method for optimizing continuous variables so that a given cost is minimized) and can quickly find paths through maze-like environments while simultaneously optimizes the robot’s trajectory. GCS can map collision-free trajectories in up to 14 dimensions (and possibly more), aiming to improve how machines work together in warehouses, libraries and households.
The CSAIL-led project consistently finds shorter paths in less time than comparable planners, demonstrating GCS’s ability to efficiently plan in complex environments. In demonstrations, the system deftly guided two robotic arms holding a mug around a shelf while optimizing for the shortest time and path. The duo’s synchronized movement resembled a partner dance routine, swinging around the edges of the bookcase without falling objects. In subsequent setups, the researchers removed the shelves and the robots swapped spray paint locations and gave each other a box of sugar.
The success of these real-world tests shows the potential of the algorithm to help in fields such as manufacturing, where two robotic arms working in parallel could take an object off a shelf. Likewise, this duo could help place books in a house or library, avoiding the other objects nearby. While problems of this nature were previously tackled with sampling-based algorithms, which can struggle in high-dimensional spaces, GCS uses fast convex optimization and can efficiently coordinate the work of multiple robots.
“Robots excel at repetitive, pre-planned movements in applications such as automotive or electronics assembly, but struggle with creating real-time movement in new environments or tasks. Previous state-of-the-art motion design methods use a “node and radius” approach, using precomputed graphs of a finite number of stable configurations, which are known to be safe. In operation, the robot must strictly adhere to this roadmap, often leading to inefficient robot movements. Motion design using Graph-of-Convex-Sets (GCS) allows robots to easily adapt to different configurations within pre-computed convex regions — allowing the robot to “turn the corner” as it makes its movement plans. In this way, GCS allows the robot to quickly compute designs in safe regions very efficiently using convex optimization. This work presents a new approach that has the potential to dramatically improve the speed and efficiency of robot movements and their ability to adapt to new environments,” says David MS Johnson, co-founder and CEO of Dexai Robotics.
The GCS also thrived in simulation demonstrations, where the team tested how a quad could fly through a building without hitting trees or failing to enter doors and windows at the right angle. The algorithm optimized the path around the obstacles while taking into account the rich dynamics of the quad.
The recipe behind the MIT team’s success involves the marriage of two key ingredients: graph search and convex optimization. The first component of GCS searches graphs by exploring their nodes, computing different properties on each one to find hidden patterns and identify the shortest path to reach the goal. Like the graph search algorithms used to calculate distance in Google Maps, GCS creates different trajectories to reach each point on its way to its destination.
By combining graph search and convex optimization, GCS can find paths through complex environments and simultaneously optimize the robot’s trajectory. GCS accomplishes this goal by plotting different points in the surrounding area and then calculating how each one will arrive on the way to its final destination. This trajectory represents different angles to ensure that the robot avoids colliding with the edges of its obstacles. The resulting motion pattern allows the machines to push through potential obstacles, maneuvering precisely around each turn in the same way a driver avoids accidents on a narrow road.
GCS was originally proposed in a Document 2021 as a mathematical framework for finding shortest paths in graphs where traversing an edge required solving a convex optimization problem. Accurately moving at every vertex in large graphs and high-dimensional spaces, GCS had clear potential in robotic motion design. In a follow-up paper, MIT sixth-year doctoral student Tobia Marcucci and his team developed an algorithm by applying their framework to complex design problems for robots moving in high-dimensional spaces. The 2023 article appeared on its cover Robotics Science last week, while the team’s initial paper was accepted for publication in the Society for Industrial and Applied Mathematics (SIAM) Journal of Optimization.
While the algorithm excels at navigating tight spaces without collisions, there is still room for growth. The CSAIL team notes that GCS could eventually help with more involved problems where robots need to interact with their environment, such as pushing or sliding objects out of the way. The group is also investigating applications of GCS trajectory optimization for task planning and robot motion.
“I am very excited about this application of GCS in motion planning. But this is only the beginning. This framework is deeply connected to many key results in optimization, control, and machine learning, giving us new leverage on problems that are both continuous and combinatorial,” says Russ Tedrake, MIT professor, CSAIL principal investigator, and co-author on a new paper for the project. “There’s a lot of work to be done!”
Marcucci and Tedrake co-authored the paper with former CSAIL graduate research assistant Mark Petersen. MIT Electrical Engineering and Computer Science (EECS), CSAIL, and aeronautics and astronautics graduate student David von Wrangel SB ’23. The more general Graph of Convex Sets framework was developed by Marcucci and Tedrake in collaboration with Jack Umenberger, former postdoc at MIT CSAIL, and Pablo Parrilo, professor of EECS at MIT. The team’s work was supported, in part, by Amazon.com Services, the Department of Defense’s National Defense Science and Engineering Graduate Fellowship Program, the National Science Foundation, and the Office of Naval Research.