Aerial Robotics

Control Introduction

C. Papachristos

Robotic Workers (RoboWork) Lab

University of Nevada, Reno

CS-491/691

Control

• Control Theory deals with dynamical systems, aiming to provide control models and laws to:
• Ensure stability
• Optimize response and control effort
• Deal with delays and minimize overshoots

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• Model-based control synthesis allows to do this systematically across diverse systems and settings

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• Start with a simple “practical control” approach

CS491/691 C. Papachristos

Newton-Euler Dynamics:

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Next, we append�Forces and Moments

“Practical” Multirotor Control

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“Practical” Multirotor Control

Forces (in Body Frame):

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Moments (in Body Frame):

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Underactuated Dynamical System Control

• Under-actuation requires the generation of rotational motion, to induce translational one
• Pitch allows to induce longitudinal motion
• Roll allows to induce lateral motion
• Thrust allows to induce vertical motion

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Decoupled System Model Control

“Simplified” Decoupled Dynamics Model through Linearization

• Derivation of Translational model:

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• At hovering “trim” conditions:

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• Which yields:

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Decoupled System Model Control

“Simplified” Decoupled Dynamics Model through Linearization

• Derivation of Rotational model:

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• At hovering “trim” conditions:

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Decoupled System Model Control

“Simplified” Decoupled Dynamics Model through Linearization

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• Based on these assumptions, the Newton-Euler�formulation of the robot’s dynamics for the pitch /�longitudinal subsystems are yielded as:

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Decoupled System Model Control

• Behavior (“Open-Loop”) of such a Decoupled Dynamical System:

Step Response

(Negative) Step Response

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Decoupled System Model Control

• Decoupled System Model Controller – Block Diagram

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PID Control

• Proportional-Integral-Derivative (PID) output feedback control

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• One of the most common control practices in industry
• Efficient, Robust, Intuitive control of hugely diverse processes and dynamical systems
• Often used as model-less controller
• Not optimal in control performance, but a good starting point for system stabilization

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PID Control of Decoupled System Model

• Decoupled LTI Controller – Block Diagram

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PID Control of Decoupled System Model

• Behavior (“Closed-Loop”) of such a PID-controlled Decoupled Dynamical System:

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Step Response

Step Response

PID Control of Decoupled System Model

• Proportional-Integral-Derivative (PID) limitations

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• The control margins of the aerial vehicle have limits and therefore the specific PID control design has to account for these

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• The integral term needs special caution in cases of critically stable or unstable systems (e.g. UAVs)

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• “Around” the hovering/trimmed-flight operating point, an aerial vehicle can be modeled as a linear system;�However, outside such regions it is a highly nonlinear system

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• A PID controller trimmed / tuned around a specific linearized version of the full system model naturally cannot maintain an equally good behavior for the entire operational envelope
• Techniques such as Gain scheduling can be employed to deal with this limitation

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• The entire premise of Roll/Pitch/Yaw-based control is operation “around” the hovering-flight operating point

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• As we have established, such an intuitive parametrization of rotation exhibits singularities

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• Can we do better by involving explicit model knowledge ?

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• Can we do better by involving full model knowledge ?

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• Can we do better by involving non-singular parametrizations of orientation ?

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Nonlinear Model Control

• Nonlinear Dynamics Controller:

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Geometric Control of Nonlinear System Model

Problem:

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Geometric Control of Nonlinear System Model

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Geometric Control of Nonlinear System Model

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Geometric Control of Nonlinear System Model

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where:

Geometric Control of Nonlinear System Model

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Control of Fixed-Wing Aircraft Model

• Increasing Complexity when dealing with Aerodynamics-driven flight
• “Simple” configurations still contain Coupled Dynamics effects:

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• Equations of motion after Linearization:

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• Resulting Dynamics contain effects of:
• Short Period Oscillation motion (high-damping, high-frequency)
• Phugoid motion (low-damping, low-frequency): “Apply an impulse deflection�on the elevator, will cause the pitch angle to increase, therefore the plane goes upwards.�This causes the velocity to decrease, and hence the lift is reduced. Slowly, the pitch angle�will decrease again, and therefore the plane will go downwards. This causes the velocity�to increase, which, in turn increases the lift. The pitch angle will again increase…”
• Aperiodic Roll motion (very high-damping, very fast)
• Spiral motion (marginally unstable in most cases, but very slow)
• Dutch roll motion: “Roll from yaw impulse that instantaneously increases lift of one wing…”

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• Different guidance principles exist, e.g.,: “Coordinated” vs “Flat” turns

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Flight control surfaces of Boeing 727

Flight control surface Deflection Angle

By Piotr Jaworski;

PioM EN DE PL (Poznań/Poland)

- Own work, CC BY-SA 3.0

“Symmetric” motion:

“Asymmetric” motion:

(see: https://www.aerostudents.com/courses/flight-dynamics/flight-dynamics.php)

Elevator

Tail

Aileron

Rudder

Control of Fixed-Wing Aircraft Model

• Increasing Complexity when dealing with Aerodynamics-driven flight

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Dutch Roll mode:

Aperiodic Roll mode:

Spiral mode:

Control of Fixed-Wing Aircraft Model

• Nonlinear Control of Fixed-Wing Model

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Paper:

F. Gavilan, J.A. Acosta, and R. Vasquez, “Control of the longitudinal flight dynamics of an UAV using adaptive backstepping," Proceedings of the 18^th World Congress of the International Federation of Automatic Control (IFAC), 2011, pp. 1892-1897

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Longitudinal Equations of Motion:

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with:

Elevator Angle

Control of Fixed-Wing Aircraft Model

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where:

with:

References

States

Controls

for:

for:

Time for Questions !

CS-491/691

CS491/691 C. Papachristos