Power-based Safety Layer for Aerial Vehicles in Physical Interaction using Lyapunov Exponents

Authors: Eugenio Cuniato, Nicholas Lawrance, Marco Tognon, Roland Siegwart · IROS 2022

Licences

Open Access
Paper: CC BY 4.0

Everything below draws on the full paper under its CC BY 4.0 licence; its figure is credited to the authors.

TL;DR

Tracking how fast control errors diverge lets a tiltrotor drone allow power while errors shrink and dissipate it when they grow, so it stops safely when a pushed cart is pulled away.

Why it matters

The layer needs no energy budget and leaves the nominal controller in place, so it can be added to pose or wrench tracking that already works. The authors show analytically that the pose and force setpoints stay inside the resulting safe sets, so the constraint does not make the task unreachable, and they state that the method applies to any fully actuated mechanical system.

Problem

Fully actuated aerial robots that push on surfaces can become unstable after impacts, gusts or loss of contact, and some controllers then damage the platform. Energy-tank safety layers limit the system's energy and only implicitly its power, and they need an energy budget that is hard to choose: too little ends the task early, too much no longer ensures safety.

Proposed solution

The authors estimate the Largest Lyapunov Exponent (LLE) of the closed-loop pose and wrench error dynamics for each degree of freedom by low-pass filtering a pointwise convergence rate. A power limit derived from the LLE (positive while errors converge, dissipative while they diverge) is enforced with control barrier functions in a quadratic program on a jerk-level augmented system, placed on top of the existing hybrid position/force controller.

Key insights

The LLE of the error dynamics works as a live danger signal: negative while errors converge, and positive within 0.2 s after a pushed surface is removed.
Constraining power flow directly, instead of stored energy, removes the need to pick an energy-tank budget and allows both positive and dissipative power limits.
Moving the control input to the jerk level turns the commanded wrench into a state, so barrier-function constraints can be applied to a wrench controller driven by sensor readings and the input stays continuous when switching controllers.
Setting the error acceleration to zero in the LLE estimate enlarges the pose safe set, mostly where position and velocity errors point the same way; scaling the power limit by a computed factor below 1 fits it back inside the original set.
The damping gain kλ trades how quickly the vehicle stops against how much power is allowed when no dissipation is needed.

Main results

Free flight on a figure-eight with a 14 N virtual disturbance for 1 s and linear inputs limited to 10 N: without the layer the vehicle overshoots at 0.5 m/s in y and 0.25 m/s in x and generates up to 4 W of power.
With the layer enabled, the maximum velocity-error overshoots are more than halved and power spikes peak at 0.5 W, while position errors take about 2 s longer to converge.
While inputs were saturated, power briefly broke the barrier-function limit (two spikes in y around 4 s) and returned inside the bounds once saturation ended.
Wrench tracking of a 3 N to 5 N push on a cart: the LLE settles at −0.2 s⁻¹ (versus −0.5 s⁻¹ for pose tracking); when the cart is pulled away at about 30 s, the LLE rises above zero within 0.2 s and the vehicle dissipates energy and stops instead of accelerating.
Safe-set analysis for the example gains: without acceleration measurement, the allowed maximum velocity roughly doubles, from about 0.25 m/s to about 0.5 m/s.
Raisim simulations of the wrench experiment: a higher kλ makes the vehicle dissipate kinetic energy faster and stop sooner, and allows more power while the LLE is negative.

Limitations

Attitude and angular velocity came from a motion capture system; the authors state that outdoor use would need on-board state estimation with an IMU and/or optical sensors.
The LLE is computed with the error acceleration set to zero, which the authors show enlarges the safe set and roughly doubles the allowed velocity unless the power limit is scaled down.
When control inputs saturate, power can exceed the imposed limit for short periods, as the authors observe in the free-flight test.
The free-flight disturbances were virtual software forces rather than physical impacts, and the effect of kλ was studied only in simulation.
The formulation relies on a fully actuated platform with feedback linearization, and only one platform was tested; human-robot interaction tasks are left to future work.

Field impact

It won the IROS 2022 Best Paper Award on Safety, Security, and Rescue Robotics in memory of Motohiro Kisoi and has 25 citations. It was published in IEEE Robotics and Automation Letters (DOI 10.1109/LRA.2022.3176959). The paper does not state a release of code, data or hardware designs.

Abstract

As the performance of autonomous systems increases, safety concerns arise, especially when operating in non-structured environments. To deal with these concerns, this work presents a safety layer for mechanical systems that detects and responds to unstable dynamics caused by external disturbances. The safety layer is implemented independently and on top of already present nominal controllers, like pose or wrench tracking, and limits power flow when the system’s response would lead to instability. This approach is based on the computation of the Largest Lyapunov Exponent (LLE) of the system’s error dynamics, which represent a measure of the dynamics’ divergence or convergence rate. By actively computing this metric, divergent and possibly dangerous system behaviors can be promptly detected. The LLE is then used in combination with Control Barrier Functions (CBFs) to impose power limit constraints on a jerk controlled system. The proposed architecture is experimentally validated on an Omnidirectional Micro Aerial Vehicle (OMAV) both in free flight and interaction tasks.

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