You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
Efficient planning in dynamic and uncertain environments is a fundamental challenge in robotics. In the context of trajectory optimization, the feasibility of paths can change as the environment evolves. Therefore, it can be beneficial to reason about multiple possible paths simultaneously. We build on prior work that considers graph-based trajectories to find solutions in multiple homotopy classes concurrently. Specifically, we extend this previous work to an online setting where the unreachable (in time) part of the graph is pruned and the remaining graph is reoptimized at every time step. As the robot moves within the graph on the path that is most promising, the pruning and reoptimization allows us to retain candidate paths that may become more viable in the future as the environment changes, essentially enabling the robot to dynamically switch between numerous homotopy classes. We compare our approach against prior work without the homotopy switching capability and show improved performance across several metrics in simulation with a 2D robot in multiple dynamic environments under noisy measurements and execution.

Cited Abstract(s):
[1]: Motion planning as trajectory optimization requires generating trajectories that minimize a desired objective function or performance metric. Finding a globally optimal solution is often intractable in practice: despite the existence of fast motion planning algorithms, most are prone to local minima, which may require re-solving the problem multiple times with different initializations. In this work we provide a novel motion planning algorithm, GPMP-GRAPH, that considers a graph-based initialization that simultaneously explores multiple homotopy classes, helping to contend with the local minima problem. Drawing on previous work to represent continuous-time trajectories as samples from a Gaussian process (GP) and formulating the motion planning problem as inference on a factor graph, we construct a graph of interconnected states such that each path through the graph is a valid trajectory and efficient inference can be performed on the collective factor graph. We perform a variety of benchmarks and show that our approach allows the evaluation of an exponential number of trajectories within a fraction of the computational time required to evaluate them one at a time, yielding a more thorough exploration of the solution space and a higher success rate.
In this work, we present a novel trajectory optimization algorithm, Planning Online by Switching Homotopies ( ), to handle such scenarios. We build on GPMP-GRAPH [1] where multiple trajectories are inter-connected and represented as a factor graph upon which probabilistic inference is performed to optimize the entire graph. However, instead of retaining the same optimized trajectory from the initial time step for execution as GPMP-GRAPH does, POSH maintains and updates the entire graph at every time step. Specifically, at any time step the graph is pruned to remove unreachable (in time) states and is then reoptimized considering changes in the environment to find the new optimal trajectory. This grants our algorithm the unique ability to dynamically switch between different homotopy classes as illustrated in Fig. and allows it to better contend with dynamic environments as demonstrated by our experiments.