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 & Linearized 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 & Linearized 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 & Linearized 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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Linearized Subsystem Model Control

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

Step Response

(Negative) Step Response

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Linearized Subsystem 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 Linearized Subsystem Model

• Decoupled LTI Controller – Block Diagram

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PID Control of Linearized Subsystem Model

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

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Step Response

Step Response

PID Control of Linearized Subsystem 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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CS491/691 C. Papachristos

Short Period

mode:

Due to step input of elevator or vertical gust, aircraft will oscillate in pitch for short period of time (damped)

Phugoid

mode:

Due to kinetic/potential energy differential (e.g. receiving headwind) with almost no change in AoA, aircraft will accelerate, pitch up and climb until it decelerates, then accelerate, pitch down and descend, etc.

(damped usually –can be unstable– � but slow and easy to correct)

Control of Fixed-Wing Aircraft Model

• Increasing Complexity when dealing with Aerodynamics-driven flight

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Dutch�Roll�mode:

(combination of sideslip, yaw, roll)

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Stable but lightly damped

Aperiodic�Roll�mode:

(always stable)

Spiral�mode:�(during coordinated bank turn)

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Unstable for increasing radius turn

Control of Fixed-Wing Aircraft Model

• Increasing Complexity when dealing with Aerodynamics-driven flight

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CS491/691 C. Papachristos

Dutch Roll 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