Continuous-Time Gaussian Process Motion Planning via Probabilistic Inference (1707.07383v3)

Abstract: We introduce a novel formulation of motion planning, for continuous-time trajectories, as probabilistic inference. We first show how smooth continuous-time trajectories can be represented by a small number of states using sparse Gaussian process (GP) models. We next develop an efficient gradient-based optimization algorithm that exploits this sparsity and GP interpolation. We call this algorithm the Gaussian Process Motion Planner (GPMP). We then detail how motion planning problems can be formulated as probabilistic inference on a factor graph. This forms the basis for GPMP2, a very efficient algorithm that combines GP representations of trajectories with fast, structure-exploiting inference via numerical optimization. Finally, we extend GPMP2 to an incremental algorithm, iGPMP2, that can efficiently replan when conditions change. We benchmark our algorithms against several sampling-based and trajectory optimization-based motion planning algorithms on planning problems in multiple environments. Our evaluation reveals that GPMP2 is several times faster than previous algorithms while retaining robustness. We also benchmark iGPMP2 on replanning problems, and show that it can find successful solutions in a fraction of the time required by GPMP2 to replan from scratch.

• The paper introduces a probabilistic inference framework that formulates motion planning as MAP estimation using Gaussian process representations.
• It exploits sparse GP models with a tridiagonal precision matrix to enable efficient trajectory interpolation and optimization.
• It demonstrates significant efficiency gains and rapid incremental replanning with GPMP2 and iGPMP2 compared to traditional methods.

Continuous-time Gaussian Process Motion Planning via Probabilistic Inference

The paper introduces a probabilistic inference approach for motion planning that leverages continuous-time representations of trajectories using Gaussian processes (GPs). This method provides a novel perspective on planning problems by viewing them as inference on factor graphs, thus allowing for efficient and robust trajectory optimization. The core contribution of this work is the development of the Gaussian Process Motion Planner (GPMP) and its more refined successor, GPMP2, which exploit the sparsity inherent in GP representations to offer computational advantages over traditional planning algorithms.

Key Contributions

1. Gaussian Processes for Trajectory Representation: The authors utilize GP models to represent smooth continuous-time trajectories with a reduced set of parameters, or support states. This enables an efficient exploration of trajectory space while maintaining smoothness and compliance with constraints.
2. Sparse GP Models: By adopting a linear time-varying stochastic differential equation (LTV-SDE), the paper ensures that the inverse of the covariance matrix of the GP—a precision matrix—is sparse and tridiagonal. This property is pivotal as it allows for computational efficiencies during both trajectory interpolation and optimization.
3. Probabilistic Inference on Factor Graphs: The authors reformulate the motion planning problem as a maximum a posteriori (MAP) inference problem on a factor graph. This formulation not only represents the trajectory optimization problem concisely but also benefits from existing algorithms for efficient inference such as those used in SLAM.
4. Incremental Replanning with iGPMP2: To address dynamic environments or changes in planning conditions, the incremental variant iGPMP2 allows rapid replanning by updating only parts of the trajectory that are affected. This is made possible through the use of Bayes trees, enabling the system to efficiently incorporate new information without recomputing the entire trajectory from scratch.

Notable Results

• Efficiency Gains: The paper demonstrates that GPMP2 is significantly faster than previous algorithms, including both sampling-based and trajectory optimization-based motion planners. It achieves this while maintaining robustness and the ability to interpolate trajectories efficiently.
• Improved Success Rates: For the tasks tested, GPMP2 shows competitive success rates compared to established methods like RRT-Connect and TrajOpt, with the additional benefit of much lower processing times.
• Replanning Performance: iGPMP2 performs replanning tasks in a fraction of the time required to plan from scratch, showcasing its capability for real-time applications in dynamic environments.

Implications and Future Directions

The implications of this work are significant for robotics, where efficient and reliable motion planning is crucial. By leveraging the duality between optimization and inference, the authors provide a framework that can be adapted to include constraints dynamically, potentially extending to more complex robotic systems and environments.

Future research directions could explore the integration of more sophisticated learning-based obstacle representations, thereby potentially enabling the planner to adapt to previously unseen environments or complex tasks autonomously. Additionally, expanding the scalability of this approach to multi-agent systems or higher-dimensional robots and exploring online learning of model parameters for real-time adaptation remains an exciting prospect.

In conclusion, the paper advances the state of the art in motion planning by marrying continuous-time trajectory optimization with probabilistic inference, presenting a method that is both theoretically elegant and practically efficient.