It’s not simple for a robot to find its way out of a maze. Imagine machines trying to navigate through a children’s playroom to reach the kitchen, with various toys scattered across the floor and furniture blocking some potential paths. This disordered maze requires the robot to calculate the most optimal path to the goal without colliding with obstacles. What’s the bot to do?
The Graphs of Convex Sets (GCS) Trajectory Optimization algorithm from MIT Computer Science and Artificial Intelligence Laboratory (CSAIL) presents a scalable, collision-free motion planning system for these robot navigation needs. approach combines graph search (a method for finding discrete paths through a network) and convex optimization (an effective method for optimizing continuous variables to minimize a given cost) and can quickly find paths through maze-like environments while optimizing the robot’s trajectory. GCS can map collision-free trajectories in as many as 14 dimensions (and potentially more) to improve the way machines work in tandem in warehouses, libraries, and homes.
The CSAIL-led project consistently finds shorter paths in less time than comparable planners, demonstrating the GCS’s ability to plan efficiently in convoluted environments. In demonstrations, the system skillfully steered two robot arms holding a cup around a shelf, optimizing both for the shortest time and path. The duo’s synchronized movement resembled a dance routine, swinging around the edge of a shelf without dropping objects. In subsequent configurations, the researchers removed shelves and the robots swapped spray paint positions and passed sugar bowls to each other.
The success of these real-world tests shows the algorithm’s potential to support in areas such as manufacturing, where two robot arms working in tandem can remove an item from a shelf. Similarly, the duo could support put away books in a home or library while avoiding other items nearby. While these types of problems have previously been solved with sampling-based algorithms that can struggle in high-dimensional spaces, GCS uses swift convex optimization and can efficiently coordinate the work of multiple robots.
“Robots are great at repetitive, planned motion in applications such as automotive manufacturing and electronics assembly, but they struggle to generate real-time motion in new environments or tasks. Previous state-of-the-art motion planning methods use a ‘hub and spoke’ approach, using pre-computed graphs of a finite number of fixed configurations that are known to be safe. During operation, the robot must strictly adhere to this roadmap, which often leads to inefficient robot motions. Motion planning using Graph-of-Convex-Sets (GCS) allows robots to easily adapt to different configurations within pre-computed convex regions — allowing the robot to “turn corners” when planning its motion. In doing so, GCS allows the robot to quickly calculate plans in sheltered areas, very efficiently, using convex optimization. This paper presents a novel approach that has the potential to dramatically boost the speed and efficiency of the robot’s movements and its ability to adapt to modern environments,” says David MS Johnson, co-founder and CEO of Dexai Robotics.
GCS also proved useful in simulation demonstrations, where the team considered how a quadrotor 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 obstacles while also taking into account the quadrotor’s prosperous dynamics.
The MIT team’s recipe for success involves combining two key ingredients: graph search and convex optimization. The first component, GCS, crawls graphs by exploring their nodes, computing various properties in each node to find hidden patterns and identify the shortest path to a goal. Like the graph search algorithms used to calculate distances in Google Maps, GCS creates different trajectories to reach each point on its course to a goal.
By combining graph search and convex optimization, GCS can find paths through convoluted environments while optimizing the robot’s trajectory. GCS accomplishes this by graphing various points in the surrounding area, then calculating how to reach each of them on the way to the destination. This trajectory takes into account various angles to ensure the robot avoids collisions with the edges of obstacles. The resulting motion plan allows the machines to squeeze through potential obstacles, precisely maneuvering through each turn in the same way a driver avoids accidents on a narrow street.
GCS was originally proposed in 2021 document as a mathematical framework for finding shortest paths in graphs, where traversing an edge required solving a convex optimization problem. By precisely navigating through every vertex in gigantic graphs and high-dimensional spaces, GCS had clear potential for robot motion planning. In a follow-up paper, sixth-year MIT PhD student Tobia Marcucci and his team developed an algorithm that applies their framework to convoluted planning problems for robots moving in high-dimensional spaces. The 2023 paper was featured on the cover of last week’s issue, while the group’s initial work has been accepted for publication in the Society for Industrial and Applied Mathematics (SIAM).
While the algorithm does a great job of navigating through tight spaces without collisions, there’s still room for improvement. The CSAIL team notes that GCS could eventually support solve more convoluted problems where robots need to interact with their environment, such as pushing or moving objects out of the way. The team is also exploring applications of GCS trajectory optimization to task planning and robot movement.
“I’m very excited about this application of GCS to motion planning. But this is just the beginning. This framework is deeply connected to many fundamental results in optimization, control, and machine learning, giving us new possibilities for solving problems that are both continuous and combinatorial,” says Russ Tedrake, MIT professor, principal investigator of CSAIL, and co-author of a modern paper on the work. “There’s still a lot of work to be done!”
Marcucci and Tedrake wrote the paper with former CSAIL research assistant Mark Petersen; MIT electrical engineering and computer science (EECS) 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, a former MIT CSAIL postdoctoral fellow, and Pablo Parrilo, an MIT EECS professor. The group’s work was supported in part by Amazon.com Services, the National Defense Science and Engineering Graduate Fellowship Program of the Department of Defense, the National Science Foundation, and the Office of Naval Research.