Funnel Libraries for Real-Time Robust Feedback Motion Planning (1601.04037v3)

Abstract: We consider the problem of generating motion plans for a robot that are guaranteed to succeed despite uncertainty in the environment, parametric model uncertainty, and disturbances. Furthermore, we consider scenarios where these plans must be generated in real-time, because constraints such as obstacles in the environment may not be known until they are perceived (with a noisy sensor) at runtime. Our approach is to pre-compute a library of "funnels" along different maneuvers of the system that the state is guaranteed to remain within (despite bounded disturbances) when the feedback controller corresponding to the maneuver is executed. We leverage powerful computational machinery from convex optimization (sums-of-squares programming in particular) to compute these funnels. The resulting funnel library is then used to sequentially compose motion plans at runtime while ensuring the safety of the robot. A major advantage of the work presented here is that by explicitly taking into account the effect of uncertainty, the robot can evaluate motion plans based on how vulnerable they are to disturbances. We demonstrate and validate our method using extensive hardware experiments on a small fixed-wing airplane avoiding obstacles at high speed (~12 mph), along with thorough simulation experiments of ground vehicle and quadrotor models navigating through cluttered environments. To our knowledge, these demonstrations constitute one of the first examples of provably safe and robust control for robotic systems with complex nonlinear dynamics that need to plan in real-time in environments with complex geometric constraints.

• The paper introduces a funnel-based motion planning framework that computes invariant sets using sums-of-squares programming to guarantee safety despite disturbances.
• The methodology enables real-time selection and composition of motion plans, allowing robots to maintain safe trajectories in complex, uncertain environments.
• Simulations on fixed-wing airplanes, ground vehicles, and quadrotors validate the approach, demonstrating its robust performance in practical applications.

Funnel Libraries for Real-Time Robust Feedback Motion Planning

The advancement of robotic motion planning in uncertain and unstructured environments has seen significant progress with the introduction of "Funnel Libraries for Real-Time Robust Feedback Motion Planning" by Majumdar and Tedrake. In this work, the authors address the intricate challenge of enabling robots to autonomously navigate complex environments while accounting for uncertainties in dynamics and environmental conditions. This paper delineates a solution framework that leverages pre-computed "funnels" — invariant sets determined through regional analysis around nominal trajectories — for real-time validation and composition of motion plans, potentially transforming how robustness in motion planning is perceived.

Summary and Contributions

This paper presents an innovative approach to motion planning by pre-computing a library of "funnels" around different system maneuvers. These funnels represent regions where the system is guaranteed to remain, despite bounded disturbances or uncertainties. By utilizing convex optimization techniques, especially sums-of-squares (SOS) programming, the work exploits the tools from Lyapunov theory to compute robust control strategies that are safe with respect to environmental disturbances.

The authors make distinguishable contributions through this work:

1. Funnel Computation Framework: A rigorous method to compute funnels using SOS programming, which forms an inner approximation of the reachable states around nominal trajectories.
2. Real-Time Composition: A strategy for real-time selection and composition of motion plans relying on the pre-computed funnels, ensuring safe navigation and obstacle avoidance in unknown environments.
3. Validation on Multiple Platforms: The methodology is validated via simulations on a fixed-wing airplane, a ground vehicle, and a quadrotor, demonstrating applicability across diverse robotic platforms.
4. Geometric Conditions and Safety Guarantees: Exploration of geometric conditions on the environment that extend the assurance of finding a collision-free funnel, serving as a promising strategy for perpetual flight in environments like forests.

Numerical Results and Applications

The numerical experiments, especially those involving a ground vehicle model navigating through an environment populated with obstacles, showcase the efficacy of this approach. The planner's ability to maintain robust and safe trajectories provides a stark contrast to traditional trajectory-based methods, marking qualitative and quantitative improvements in handling disturbances and environmental uncertainty. Furthermore, hardware experiments on a small fixed-wing airplane demonstrate this approach's capability to handle real-world non-linearities and dynamic constraints effectively. Notably, the tests showed that the system retains robustness while executing complex maneuvers that typically demand significant agility.

Theoretical and Practical Implications

From a theoretical standpoint, this paper extends the frontier of real-time robust motion planning by integrating formal verification methodologies, such as SOS programming, to ensure safety and performance guarantees. This strengthens the bridge between control theory and robotic applications, facilitating further exploration into autonomously operating systems with uncertainties in both dynamics and environmental interactions.

Practically, the implications for robotics are profound. The framework allows systems to adapt to real-world variability, potentially reducing the need for over-engineered systems that rely on overly conservative safety measures. The adaptive use of pre-computed funnel libraries enhances the system's capability to operate safely outside of controlled environments, paving the way for future developments in autonomous vehicles, UAVs, and robotic systems accessing complex, unstructured environments.

Future Directions

While this approach is pioneering, several avenues remain open for research. Enhancements in computational efficiency for real-time funnel computation, more sophisticated integration with perception systems, and extending the methodology to handle a broader range of real-world dynamics and environments can further solidify this framework's applicability. Additionally, incorporating robust real-time planning adjustments based on online feedback systems represents a potential area for significant advancement.

In summary, Majumdar and Tedrake's "Funnel Libraries for Real-Time Robust Feedback Motion Planning" provide a foundational step towards resilient autonomous systems capable of reasoning amidst uncertainty and complexity. Their work is a template that future researchers and practitioners can build upon to unlock the full potential of autonomous robotics in our everyday world.