Quadcopters
Presented by: Andrew Depriest
What is a quadcopter?
● Helicopter - uses rotors for lift and propulsion
● Quadcopter (aka quadrotor) - uses 4 rotors
Parrot AR.Drone 2.0
History
1907 - Breguet-Richet Gyroplane
● Louis Breguet and Prof. Charles Richet
● First rotary wing aircraft to lift off the ground
● Lifted only a few feet while tethered
1920 - Oehmichen No.2
● 2nd of 6 designs by Etienne Oehmichen
● 4 rotors and 8 propellers (stabilize and steer)
● Completed a 1km closed circuit flight
History
1922 - de Bothezat helicopter
● Dr. George de Bothezat and Ivan Jerome
● US Air Service
● Highest altitude of ~5m
● Demonstrated feasibility
1956 - Convertawings Model A Quadrotor
● Intended prototype for larger civil and military copters
● Controlled by varied rotor thrust
● First to demonstrate successful forward flight
History
1958 - Curtiss-Wright VZ-7
● Curtiss-Wright company for US Army
● Performed well during tests
● Didn’t meet Army standards
Modern
● Bell Boeing Quad TiltRotor
● Aermatica Spa's Anteos
● AeroQuad and ArduCopter
● Parrot AR.Drone
● Nixie
Uses
● Research - evaluate new ideas
o Cheap
o Variety of sizes
o Maneuverability
● Military & Law Enforcement
o Surveillance and reconnaissance
o Search and rescue
● Commercial
o Aerial imagery
o Package delivery
How it works
Rotors produce:
● Thrust
● Torque
● Drag force
Control input:
● Angular Velocity
Modelling and control of quadcopter
Teppo Luukkonen - Aalto University in Espoo, Finland
“Present the basics of quadcopter modelling and control as to form a
basis for further research and development”
● Study the mathematical model of the quadcopter dynamics
● Develop proper methods for stabilisation and trajectory control of
the quadcopter
“The challenge ... is that the quadcopter has six degrees of freedom but
there are only four control inputs”
Mathematical Model
Quadcopter:
● Position
● Pitch, roll, yaw
● Pose
Body Frame:
● Linear Velocity
● Angular Velocity
Body-to-Inertial Frame:
● Rotation matrix
o orthogonal
o R-1 = RT
Inertial-to-body
Mathematical Model (cont’d)
Transformation matrices (angular vel.)
● inertial-to-body
● body-to-inertial
Symmetric structure
● Inertia matrix is diagonal
Lift force - lift constant and angular vel.
Torque - drag constant and angular vel.
● inertia moment term small, omitted
Roll = -2nd rotor, +4th rotor
Pitch = -1st rotor, +3rd rotor
Yaw = +/-(+1st, +3rd, -2nd, -4th)
More math (summarized)
Newton-Euler equations
● Quadcopter is assumed rigid body
● Force for accel. of mass + centrifugal force = gravity + thrust
● Body frame
o External torque = ang. accel. + centripetal + gyroscopic forces
● In inertial frame
o Centrifugal is nullified
o Angular accels. calculated using transformation matrix and it’s
time derivative
More math (summarized)
Euler-Lagrange equations
● Lagrangian = Translational + rotational energies - potential energy
● Euler-Lagrange equations
o Linear and angular components independent
● Jacobian matrix, Coriolis term, aerodynamical effects ….
Too much math.
Model Simulation
● Used MATLAB 2010
● Initial stable state
● Params used:
Stabilisation
PID controller used
● Simple structure
● Easy implementation
● General form
o Proportional - uses diff. between desired and present positions
o Integral - uses diff. between desired and present attitudes
o Derivative - uses diff. between desired and present positions
● Specific form - PD controller
o Torque calculated taking into account gravity, mass, and
moment of inertia
Stabilisation Simulation
Note: The PD only stabilizes hover (altitude and attitude)
It does not consider accel. in the x and y axis
Starting z=1
Desired z=0
What’s left (more math)
● Trajectory control
o Have desired trajectory
o Generate linear accelerations to accomplish it
o Derive the roll, pitch, and thrust values for those
● Heuristic model for trajectory generation
o Jerk and jounce have to be reasonable (3rd and 4th derivatives
of position)
o Symmetry on acceleration and deceleration
● Integrated PD controller
o Take into account possible deviations in attitude
Conclusion
● Simulation proved the model to be realistic
● Simulation also proved the PD controller to be efficient in stabilising the
altitude and attitude
o x and y positions were not considered, they varied due to deviation of
pitch and roll angles
● Proposed heuristic method produced good trajectories using parameters to
generate jounce, and using jounce to derive position, it’s other derivatives,
torque, etc
● Integrated PD operated well to take into account unmodelled disturbances
like wind, but could perform poorly depending on parameters used
● These were simulations, some aerodynamics were omitted, localization
was trivialized, so effects of imprecise measurements and knowledge
needs to be further studied
References
● Luukkonen, Teppo. "Modelling and control of quadcopter."
Independent research project in applied mathematics, Espoo
(2011).
● <http://en.wikipedia.org/wiki/Quadcopter>
