Markus Hehn

A flying inverted pendulum

We extend the classic control problem of the inverted pendulum by placing the pendulum on top of a quadrotor aerial vehicle. Both static and dynamic equilibria of the system are investigated to find nominal states of the system at standstill and on circular trajectories. Control laws are designed around these nominal trajectories. A yaw-independent description of quadrotor dynamics is introduced, using a ‘Virtual Body Frame’. This allows for the time-invariant description of curved trajectories. The balancing performance of the controller is demonstrated in the ETH Zurich Flying Machine Arena testbed. Development potential for the future is highlighted, with a focus on applying learning methodology to increase performance by eliminating systematic errors that were seen in experiments.

Quadrocopter Trajectory Generation and Control

Abstract An algorithm is presented that allows the calculation of flight trajectories for quadrocopters. Trajectory feasibility constraints regarding the vehicle dynamics and input constraints are derived. They are then used in the planning algorithm to guarantee the feasibility of generated trajectories. The translational degrees of freedom of the quadrotor are decoupled, and time-optimal trajectories are found for each degree of freedom separately. The trajectory generation is fast enough to be performed online. Control inputs are calculated from the generated trajectory, and used to achieve closed-loop control similar to model predictive control. The trajectory generation and tracking performance is demonstrated in the ETH Zurich Flying Machine Arena testbed. Experimental results show good performance, with unmodeled aerodynamic effects causing trajectory deviations when decelerating from high speeds. Development potential for the future is highlighted, focusing on improving the performance and correcting for aerodynamic effects.

Cooperative quadrocopter ball throwing and catching

This paper presents a method for enabling a fleet of circularly arranged quadrocopters to throw and catch balls with a net. Based on a first-principles model of the net forces, nominal inputs for all involved vehicles are derived for arbitrary target trajectories of the net. Two algorithms that generate open-loop trajectories for throwing and catching a ball are also introduced. A set of throws and catches is demonstrated in the ETH Zurich Flying Machine Arena testbed.

Performance benchmarking of quadrotor systems using time-optimal control

Frequently hailed for their dynamical capabilities, quadrotor vehicles are often employed as experimental platforms. However, questions surrounding achievable performance, influence of design parameters, and performance assessment of control strategies have remained largely unanswered. This paper presents an algorithm that allows the computation of quadrotor maneuvers that satisfy Pontryagin’s minimum principle with respect to time-optimality. Such maneuvers provide a useful lower bound on the duration of maneuvers, which can be used to assess performance of controllers and vehicle design parameters. Computations are based on a two-dimensional first-principles quadrotor model. The minimum principle is applied to this model to find that time-optimal trajectories are bang-bang in the thrust command, and bang-singular in the rotational rate control. This paper presents a procedure allowing the computation of time-optimal maneuvers for arbitrary initial and final states by solving the boundary value problem induced by the minimum principle. The usage of the computed maneuvers as a benchmark is demonstrated by evaluating quadrotor design parameters, and a linear feedback control law as an example of a control strategy. Computed maneuvers are verified experimentally by applying them to quadrocopters in the ETH Zurich Flying Machine Arena testbed.

The Flying Machine Arena as of 2010

The Flying Machine Arena (FMA) is an indoor research space built specifically for the study of autonomous systems and aerial robotics. In this video, we give an overview of this testbed and some of its capabilities. We show the FMA infrastructure and hardware, which includes a fleet of quadrocopters and a motion capture system for vehicle localization. The physical components of the FMA are complemented by specialized software tools and components that facilitate the use of the space and provide a unified framework for communication and control. The flexibility and modularity of the experimental platform is highlighted by various research projects and demonstrations.

Quadrocopter pole acrobatics

We present the design of a system that allows quadrocopters to balance an inverted pendulum, throw it into the air, and catch and balance it again on a second vehicle. Based on first principles models, a launch condition for the pole is derived and used to design an optimal trajectory to throw the pole towards a second quadrocopter. An optimal catching instant is derived and the corresponding position is predicted by simulating the current position and velocity estimates forward in time. An algorithm is introduced that generates a trajectory for moving the catching vehicle to the predicted catching point in real time. By evaluating the pole state after the impact, an adaptation strategy adapts the catch maneuver such that the pole rotates into the upright equilibrium by itself. Experimental results demonstrate the performance of the system.

A computationally efficient algorithm for state-to-state quadrocopter trajectory generation and feasibility verification

An algorithm is proposed allowing for the rapid generation and evaluation of quadrocopter state interception trajectories. These trajectories are from arbitrary initial states to final states defined by the vehicle position, velocity and acceleration with a specified end of time. Sufficient criteria are then derived allowing trajectories to be tested for feasibility with respect to thrust and body rates. It is also shown that the range of a linear combination of the vehicle state can be solved for in closed form, useful e.g. for testing that the position remains within a box. The algorithm is applied by revisiting the problem of finding a trajectory to hit a ball towards a target with a racket attached to a quadrocopter. The trajectory generator is used in a model predictive control like strategy, where thousands of trajectories are generated and evaluated at every controller update step, with the first input of the optimal trajectory being sent to the vehicle. It is shown that the method can generate and evaluate on the order of one million trajectories per second on a standard laptop computer.

Feasiblity of motion primitives for choreographed quadrocopter flight

This paper describes a method for checking the feasibility of quadrocopter motions. The approach, meant as a validation tool for preprogrammed quadrocopter performances, is based on first principles models and ensures that a desired trajectory respects both vehicle dynamics and motor thrust limits. We apply this method towards the eventual goal of using parameterized motion primitives for expressive quadrocopter choreographies. First, we show how a large class of motion primitives can be formulated as truncated Fourier series. We then show how the feasibility check can be applied to such motions by deriving explicit parameter constraints for two particular parameterized primitives. The predicted feasibility constraints are compared against experimental results from quadrocopters in the ETH Flying Machine Arena.

Real-Time Trajectory Generation for Quadrocopters

This paper presents a trajectory generation algorithm that efficiently computes high-performance flight trajectories that are capable of moving a quadrocopter from a large class of initial states to a given target point that will be reached at rest. The approach consists of planning separate trajectories in each of the three translational degrees of freedom, and ensuring feasibility by deriving decoupled constraints for each degree of freedom through approximations that preserve feasibility. The presented algorithm can compute a feasible trajectory within tens of microseconds on a laptop computer; remaining computation time can be used to iteratively improve the trajectory. By replanning the trajectory at a high rate, the trajectory generator can be used as an implicit feedback law similar to model predictive control. The solutions generated by the algorithm are analyzed by comparing them with time-optimal motions, and experimental results validate the approach.

A Computationally Efficient Motion Primitive for Quadrocopter Trajectory Generation

A method is presented for the rapid generation and feasibility verification of motion primitives for quadrocopters and similar multirotor vehicles. The motion primitives are defined by the quadrocopters initial state, the desired motion duration, and any combination of components of the quadrocopters position, velocity, and acceleration at the motions end. Closed-form solutions for the primitives are given, which minimize a cost function related to input aggressiveness. Computationally efficient tests are presented to allow for rapid feasibility verification. Conditions are given under which the existence of feasible primitives can be guaranteed a priori . The algorithm may be incorporated in a high-level trajectory generator, which can then rapidly search over a large number of motion primitives which would achieve some given high-level goal. It is shown that a million motion primitives may be evaluated and compared per second on a standard laptop computer. The motion primitive generation algorithm is experimentally demonstrated by tasking a quadrocopter with an attached net to catch a thrown ball, evaluating thousands of different possible motions to catch the ball.