A core capability of intelligent systems is the ability to quickly learn new tasks by drawing on prior experience. Gradient (or optimization) based meta-learning has recently emerged as an effective approach for few-shot learning. In this formulation, meta-parameters are learned in the outer loop, while task-specific models are learned in the inner-loop, by using only a small amount of data from the current task. A key challenge in scaling these approaches is the need to differentiate through the inner loop learning process, which can impose considerable computational and memory burdens. By drawing upon implicit differentiation, we develop the implicit MAML algorithm, which depends only on the solution to the inner level optimization and not the path taken by the inner loop optimizer. This effectively decouples the meta-gradient computation from the choice of inner loop optimizer. As a result, our approach is agnostic to the choice of inner loop optimizer and can gracefully handle many gradient steps without vanishing gradients or memory constraints. Theoretically, we prove that implicit MAML can compute accurate meta-gradients with a memory footprint that is, up to small constant factors, no more than that which is required to compute a single inner loop gradient and at no overall increase in the total computational cost. Experimentally, we show that these benefits of implicit MAML translate into empirical gains on few-shot image recognition benchmarks.

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Meta-Learning with Implicit Gradients

Multirotor unmanned aerial vehicles (UAVs) are rapidly gaining popularity for many applications. However, safe operation in partially unknown, unstructured environments remains an open question. In this paper, we present a continuous-time trajectory optimization method for real-time collision avoidance on multirotor UAVs. We then propose a system where this motion planning method is used as a local replanner, that runs at a high rate to continuously recompute safe trajectories as the robot gains information about its environment. We validate our approach by comparing against existing methods and demonstrate the complete system avoiding obstacles on a multirotor UAV platform.

Continuous-time trajectory optimization for online UAV replanning

High-speed trajectory planning through unknown environments requires algorithmic techniques that enable fast reaction times while maintaining safety as new information about the operating environment is obtained. The requirement of computational tractability typically leads to optimization problems that do not include the obstacle constraints (collision checks are done on the solutions) or use a convex decomposition of the free space and then impose an ad-hoc time allocation scheme for each interval of the trajectory. Moreover, safety guarantees are usually obtained by having a local planner that plans a trajectory with a final “stop” condition in the freeknown space. However, these two decisions typically lead to slow and conservative trajectories. We propose FASTER (Fast and Safe Trajectory Planner) to overcome these issues. FASTER obtains high-speed trajectories by enabling the local planner to optimize in both the free-known and unknown spaces. Safety guarantees are ensured by always having a feasible, safe back-up trajectory in the free-known space at the start of each replanning step. Furthermore, we present a Mixed Integer Quadratic Program formulation in which the solver can choose the trajectory interval allocation, and where a time allocation heuristic is computed efficiently using the result of the previous replanning iteration. This proposed algorithm is tested extensively both in simulation and in real hardware, showing agile flights in unknown cluttered environments with velocities up to 3.6 $\mathrm{m}/\mathrm{s}$.

FASTER: Fast and Safe Trajectory Planner for Flights in Unknown Environments

Action anticipation, intent prediction, and proactive behavior are all desirable characteristics for autonomous driving policies in interactive scenarios. Paramount, however, is ensuring safety on ...

On infusing reachability-based safety assurance within planning frameworks for human–robot vehicle interactions:

Quadcopters have been used as hovering encountered-type haptic devices in virtual reality. We suggest that quadcopters can facilitate rich haptic interactions beyond force feedback by appropriating physical objects and the environment. We present HoverHaptics, an autonomous safe-to-touch quadcopter and its integration with a virtual shopping experience. HoverHaptics highlights three affordances of quadcopters that enable these rich haptic interactions: (1) dynamic positioning of passive haptics, (2) texture mapping, and (3) animating passive props. We identify inherent challenges of hovering encountered-type haptic devices, such as their limited speed, inadequate control accuracy, and safety concerns. We then detail our approach for tackling these challenges, including the use of display techniques, visuo-haptic illusions, and collision avoidance. We conclude by describing a preliminary study (n = 9) to better understand the subjective user experience when interacting with a quadcopter in virtual reality using these techniques.

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Beyond The Force: Using Quadcopters to Appropriate Objects and the Environment for Haptics in Virtual Reality

Quadrotor flight has typically been limited to sparse environments due to numerical complications that arise when dealing with large numbers of obstacles. We hypothesized that it would be possible to plan and robustly execute trajectories in obstacle-dense environments using the novel Iterative Regional Inflation by Semidefinite programming algorithm (IRIS), mixed-integer semidefinite programs (MISDP), and model-based control. Unlike sampling-based approaches, the planning algorithm first introduced by Deits theoretically guarantees non-penetration of the trajectories even with small obstacles such as strings. We present experimental validation of this claim by aggressively flying a small quadrotor (34g, 92mm rotor to rotor) in a series of indoor environments including a cubic meter volume containing 20 interwoven strings, and present the control architecture we developed to do so.

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Aggressive quadrotor flight through cluttered environments using mixed integer programming

Reach-avoid problems involve driving a system to a set of desirable configurations while keeping it away from undesirable ones. Providing mathematical guarantees for such scenarios is challenging but have numerous potential practical applications. Due to the challenges, analysis of reach-avoid problems involves making trade-offs between generality of system dynamics, generality of problem setups, optimality of solutions, and computational complexity. In this paper, we combine sum-of-squares optimization and dynamic programming to address the reach-avoid problem, and provide a conservative solution that maintains reaching and avoidance guarantees. Our method is applicable to polynomial system dynamics and to general problem setups, and is more computationally scalable than previous related methods. Through a numerical example involving two single integrators, we validate our proposed theory and compare our method to Hamilton-Jacobi reachability. Having validated our theory, we demonstrate the computational scalability of our method by computing the reach-avoid set of a system involving two kinematic cars.

Reach-Avoid Problems via Sum-or-Squares Optimization and Dynamic Programming

Reach-avoid games are excellent proxies for studying many problems in robotics and related fields, with applications including multi-robot systems, human-robot interactions, and safety-critical systems. However, solving reach-avoid games is difficult due to the conflicting and asymmetric goals of agents, and trade-offs between optimality, computational complexity, and solution generality are commonly required. This paper seeks to find attacker strategies in reach-avoid games that reduce computational complexity while retaining solution quality by using a receding horizon strategy. To solve for the open-loop strategy fast enough to enable a receding horizon approach, the problem is formulated as a mixed-integer second-order cone program. This formulation leverages the use of sums-of-squares optimization to provide guarantees that the strategy is robust to all possible defender policies. The method is demonstrated through numerical and hardware experiments.

Reach-Avoid Games Via Mixed-Integer Second-Order Cone Programming

We introduce an algorithm for synthesizing and verifying piecewise linear Lyapunov functions to prove global exponential stability of piecewise linear dynamical systems. The Lyapunov functions we synthesize are parameterized by feedforward neural networks with leaky ReLU activation units. To train these neural networks, we design a loss function that measures the maximal violation of the Lyapunov conditions in the state space. We show that this maximal violation can be computed by solving a mixed-integer linear program (MILP). Compared to previous learning-based approaches, our learning approach is able to certify with high precision that the learned neural network satisfies the Lyapunov conditions not only for sampled states, but over the entire state space. Moreover, compared to previous optimization-based approaches that require a pre-specified partition of the state space when synthesizing piecewise Lyapunov functions, our method can automatically search for both the partition and the Lyapunov function simultaneously. We demonstrate our algorithm on both continuous and discrete-time systems, including some for which known strategies for partitioning of the Lyapunov function would require introducing higher order Lyapunov functions.

Benoit Landry

Papers

Beyond The Force: Using Quadcopters to Appropriate Objects and the Environment for Haptics in Virtual Reality

Aggressive quadrotor flight through cluttered environments using mixed integer programming

Reach-Avoid Problems via Sum-or-Squares Optimization and Dynamic Programming

Reach-Avoid Games Via Mixed-Integer Second-Order Cone Programming

Counter-example guided synthesis of neural network Lyapunov functions for piecewise linear systems