# Motion Planning around Obstacles with Convex Optimization

arXiv: https://arxiv.org/abs/2205.04422 · HTML: https://mlsanity.com/paper/2205.04422/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 May 2022.

## Built on by (most cited first, 27 of 27)

- [VoxPoser: Composable 3D Value Maps for Robotic Manipulation with Language Models](https://mlsanity.com/paper/2307.05973/) (2023)
- [Inference-time Scaling of Diffusion Models through Classical Search](https://mlsanity.com/paper/2505.23614/) (2025)
- [RAYEN: Imposition of Hard Convex Constraints on Neural Networks](https://mlsanity.com/paper/2307.08336/) (2023)
- [M2Diffuser: Diffusion-based Trajectory Optimization for Mobile Manipulation in 3D Scenes](https://mlsanity.com/paper/2410.11402/) (2024)
- [Pinet: Optimizing hard-constrained neural networks with orthogonal projection layers](https://mlsanity.com/paper/2508.10480/) (2025)
- [Empirical Analysis of Sim-and-Real Cotraining of Diffusion Policies for Planar Pushing from Pixels](https://mlsanity.com/paper/2503.22634/) (2025)
- [Robotic Table Wiping via Reinforcement Learning and Whole-body Trajectory Optimization](https://mlsanity.com/paper/2210.10865/) (2022)
- [Discrete-Guided Diffusion for Scalable and Safe Multi-Robot Motion Planning](https://mlsanity.com/paper/2508.20095/) (2025)
- [Constant-time Motion Planning with Anytime Refinement for Manipulation](https://mlsanity.com/paper/2311.00837/) (2023)
- [Accelerating Search-Based Planning for Multi-Robot Manipulation by Leveraging Online-Generated Experiences](https://mlsanity.com/paper/2404.00143/) (2024)
- [A^* for Graphs of Convex Sets](https://mlsanity.com/paper/2407.17413/) (2024)
- [Space-Time Graphs of Convex Sets for Multi-Robot Motion Planning](https://mlsanity.com/paper/2503.00583/) (2025)
- [GHOST: Solving the Traveling Salesman Problem on Graphs of Convex Sets](https://mlsanity.com/paper/2511.06471/) (2025)
- [X-Safe: Embodiment-Agnostic Safeguarding for Robot Arm Manipulation Policies via Action Masking](https://mlsanity.com/paper/2606.22278/) (2026)
- [Hierarchical Deep Learning for Intention Estimation of Teleoperation Manipulation in Assembly Tasks](https://mlsanity.com/paper/2403.19770/) (2024)
- [BOMP: Bin-Optimized Motion Planning](https://mlsanity.com/paper/2411.00221/) (2024)
- [Toward Generalist Neural Motion Planners for Robotic Manipulators: Challenges and Opportunities](https://mlsanity.com/paper/2603.24318/) (2026)
- [Search-Based Spatiotemporal and Multi-Robot Motion Planning on Graphs of Space-Time Convex Sets](https://mlsanity.com/paper/2607.00444/) (2026)
- [From Distances to Trajectories: Real-Time Signed Distance Function Mapping and Distance-Accelerated Motion Planning for UAVs](https://mlsanity.com/paper/2607.19306/) (2026)
- [V*: An Efficient Motion Planning Algorithm for Autonomous Vehicles](https://mlsanity.com/paper/2508.06404/) (2025)
- [Constant-Time Planning for Chaining Collision-free Motion to Manipulation Behaviors](https://mlsanity.com/paper/2512.00939/) (2025)
- [Approximately Optimal Global Planning for Contact-Rich SE(2) Manipulation on a Graph of Reachable Sets](https://mlsanity.com/paper/2601.10827/) (2026)
- [Multi Graph Search for High-Dimensional Robot Motion Planning](https://mlsanity.com/paper/2602.12096/) (2026)
- [Interventional Causal Circuits for Safe Robot Action Testing and Failure Recovery](https://mlsanity.com/paper/2607.14826/) (2026)
- [RLMM-Flow: A Flow-based Mobile Manipulation Framework with Latent-Space Reinforcement Learning](https://mlsanity.com/paper/2607.26460/) (2026)
- [Accelerating Human-Aware Robot Trajectory Generation via Diffusion and Consistency Distillation](https://mlsanity.com/paper/2608.03159/) (2026)
- [Unified Branch-and-Bound Search for the Steiner Traveling Salesman Problem on Graphs of Convex Sets](https://mlsanity.com/paper/2608.21319/) (2026)

## Abstract (arXiv)

Trajectory optimization offers mature tools for motion planning in high-dimensional spaces under dynamic constraints. However, when facing complex configuration spaces, cluttered with obstacles, roboticists typically fall back to sampling-based planners that struggle in very high dimensions and with continuous differential constraints. Indeed, obstacles are the source of many textbook examples of problematic nonconvexities in the trajectory-optimization problem. Here we show that convex optimization can, in fact, be used to reliably plan trajectories around obstacles. Specifically, we consider planning problems with collision-avoidance constraints, as well as cost penalties and hard constraints on the shape, the duration, and the velocity of the trajectory. Combining the properties of B'ezier curves with a recently-proposed framework for finding shortest paths in Graphs of Convex Sets (GCS), we formulate the planning problem as a compact mixed-integer optimization. In stark contrast with existing mixed-integer planners, the convex relaxation of our programs is very tight, and a cheap rounding of its solution is typically sufficient to design globally-optimal trajectories. This reduces the mixed-integer program back to a simple convex optimization, and automatically provides optimality bounds for the planned trajectories. We name the proposed planner GCS, after its underlying optimization framework. We demonstrate GCS in simulation on a variety of robotic platforms, including a quadrotor flying through buildings and a dual-arm manipulator (with fourteen degrees of freedom) moving in a confined space. Using numerical experiments on a seven-degree-of-freedom manipulator, we show that GCS can outperform widely-used sampling-based planners by finding higher-quality trajectories in less time.