How does a Quadrotor fly?
A journey from physics, mathematics, control
systems and computer science
towards a “Controllable...
Overview
1 Why Multi-rotors?
2 Structure and Physics of a Quadrotor
3 From Analysis to Driving:
How can I impose a movemen...
Part I
Why Multi-rotors?
Corrado Santoro How does a Quadrotor fly?
Flying Machines
“To fly” has been one of the dreams of the humans
But the story tells that building flying machines is not e...
Design and Implementation problems
Airplanes (fixed wing)
Wing profile and shape
Wing and stab size/area
Wing load
Position ...
Multi-rotors ...
are mechanically simple: they have n motors and n
propellers
do not require complex mechanical parts to c...
Part II
Structure and Physics of a Quadrotor
Corrado Santoro How does a Quadrotor fly?
Structure of a Quadrotor (Mechanics)
Four equal propellers generating four thrust forces
Two possible configurations: “+” a...
Structure of a Quadrotor (Electronics)
Corrado Santoro How does a Quadrotor fly?
Forces and Rotation speeds
ω1, ω2, ω3, ω4: rotation speeds of the propellers
T1, T2, T3, T4: forces generated by the prope...
Moments
M1, M2, M3, M4: moments generated by the forces
Mi = L × Ti
Corrado Santoro How does a Quadrotor fly?
Hovering Condition (Equilibrium)
1 Equilibrium of forces: 4
i=1 Ti = −mg
2 Equilibrium of directions: T1,2,3,4||g
3 Equili...
Reference Systems
There are two reference systems:
1 The inertial reference systems, i.e. the Earth frame
(xE , yE , zE )
...
Euler Angles
Three angles (φ, θ, ψ) define the transformation between the
two systems:
Roll, φ: angle of rotation along axi...
Angular Speeds
The derivative of (φ, θ, ψ) w.r.t. time are the angular/rotation
speeds ( ˙φ, ˙θ, ˙ψ) of the system:
˙φ, Ro...
Part III
From Analysis to Driving:
How can I impose a movement to my quadrotor?
Corrado Santoro How does a Quadrotor fly?
Hovering Condition (Equilibrium)
1 Equilibrium of forces: 4
i=1 Ti = −mg
2 Equilibrium of directions: T1,2,3,4||g
3 Equili...
Going Up and Down
1 No equilibrium of forces: 4
i=1 Ti = −mg
2 Equilibrium of directions: T1,2,3,4||g
3 Equilibrium of mom...
Yaw Rotation
1 Equilibrium of forces: 4
i=1 Ti = −mg
2 Equilibrium of directions: T1,2,3,4||g
3 Equilibrium of moments: 4
...
Roll Rotation
No equilibrium of moments: 4
i=1 Mi = 0
... by unbalancing propeller speeds as:
(ω1 + ω4) − (ω2 + ω3) = 0
As...
Roll Rotation and Translated Flight
Total thrust T = 4
i=1 Ti is decomposed in:
Lift Force:TL = T cos φ
Drag Force:TD = T ...
Pitch Rotation
No equilibrium of moments: 4
i=1 Mi = 0
... by unbalancing propeller speeds as:
(ω1 + ω2) − (ω3 + ω4) = 0
A...
Equations of Movement
We assume a common factor of proportionality k and F =
√
T
(we will see that such an assumption is n...
Equations of Movement




˙φ
˙θ
˙ψ
F



 =




k −k −k k
k k −k −k
k −k k −k
k k k k








ω1
ω2
ω3
...
Controlling Roll, Pitch and Yaw Rates, and Total Thrust




ω1
ω2
ω3
ω4



 = K−1




˙φ
˙θ
˙ψ
F



 =

...
Part IV
The ideal world and the real world:
Why we need Control Systems Theory!
Corrado Santoro How does a Quadrotor fly?
Can we really set the rotation rate of propellers??
Motor/Propeller Driving Schema
Drivers, motors and propellers are chos...
Can we really set the rotation rate of propellers??
Motor/Propeller Driving Schema
Same PWM signals applied different driv...
The “Real world” effect
Problem
We need to set ωi by




ω1
ω2
ω3
ω4



 = K−1




˙φ
˙θ
˙ψ
F




but we d...
The Mathematician/Physicists Solution
Solution ??
Let’s characterize each driver/motor/propeller chain and derive the
func...
The Computer Scientist/Engineer Solution
Solution ??
Let’s sperimentally tune:
an offset for each channel
a gain for each ...
The Control System Engineer Solution
Solution!!!! Use feedback!
1 Measure your variable through a sensor
2 Compare the mea...
Our Scenario
Our measures:
Actual angular velocities on the three axis ( ˙φM, ˙θM, ˙ψM )
They are measured through a 3-axi...
Using Feedback to Control the Quadrotor
The overall schema of the feedback controller is:
Corrado Santoro How does a Quadr...
Using Feedback to Control the Quadrotor
Algorithmically
while True do
On ∆T timer tick ;
( ˙φT , ˙θT , ˙ψT , F) = sample r...
Using Feedback to Control the Quadrotor
Algorithmically
while True do
On ∆T timer tick ;
( ˙φT , ˙θT , ˙ψT , F) = sample r...
The P.I.D. Controller
The most common used controller type is the
Proportional-Integral-Derivative controller, represented...
The P.I.D. Controller
PID Function
C(k) := Kpe(k) + Ki
k
j=0
e(j) ∆T + Kd
e(k) − e(k − 1)
∆T
Constants Kp, Ki , Kd regulat...
Part V
Rates and Angles:
Could I control the attitude?
Corrado Santoro How does a Quadrotor fly?
Rates are not Angles
The above schema controls rates:
suppose a roll angle of φ = 10o
but no roll rotation (rate), i.e. ˙φ...
Measuring Angles (instead of Rates): Gyros
First we must measure euler angles (φ, θ, ψ)!
We could do this by using Gyrosco...
Measuring Angles: Accelerometers
An accelerometer is a sensor measuring the acceleration over
the three axis (ax , ay , az...
Measuring Angles: Two sensors, No reliability!
Gyros
Drift
Approximate discrete integration
Accelerometers
Precise only if...
Sensor Fusion
Basic Algorithm
while True do
On ∆T timer tick ;
( ˙φ, ˙θ, ˙ψ) = sample gyro();
(ax, ay , az) = sample accel...
Sensor Fusion: Algorithms
The key is the filter function!
DCM (Direction Cosine Matrix)
Complementary filters
Kalman filters
...
Sensor Fusion: Algorithms
High computational load due to:
Rotations in the 3D space
Matrix calculations
May we reduce the ...
Representing Rotations in 3D
Direction Cosine Matrix
DCM =


cθcψ sφsθcψ − cφsψ cφsθcψ + sφsψ
cθsψ sφsθsψ + cφcψ cφsθsψ ...
Representing Rotations in 3D
Quaternions
A quaternion is a complex number with one real part and three
imaginary parts:
q ...
Rotations in 3D and Quaternions
Transformations from Euler angles to quaternion exist:
q → (φ, θ, ψ)
(φ, θ, ψ) → q
Rotatin...
So far so good: Controlling attitude
Attitude control is achieved using (once again) feedback
controllers.
We set the Targ...
Let’s remind the schema of Rate Controllers
Corrado Santoro How does a Quadrotor fly?
Complete Attitude Controller
Corrado Santoro How does a Quadrotor fly?
Control “loops”: Requirements
Two control loops in the schema
rate control (inner);
attitude control (outer);
Attitude con...
Finally, the overall algorithm
while True do
On Tr timer tick ;
( ˙φM, ˙θM, ˙ψM ) = sample gyro();
(ax, ay , az) = sample ...
Part VI
What about Altitude or GPS control?
Corrado Santoro How does a Quadrotor fly?
Let’s repeat the schema!
Do you need another kind of control? Repeat the schema!
Identify your source of measure m
Identif...
Altitude Control
HT = our target height
HM = measured height (from a sensor)
F = output variable to control (desired thrus...
GPS Control
LatT , LonT = our target position
LatM , LonT = measured position (from a GPS sensor)
φT , θT = target variabl...
Vision-based Control
while True do
On Tr timer tick ;
...;
if after H loops then
(∆X, ∆Y, ∆ψ) = identify target with camer...
Conclusions
It seems easy ....
Corrado Santoro How does a Quadrotor fly?
... but, where is the trick?
Are sensors reliable?
Sometimes, NO!
Noise due to mechanical vibrations (MEMS-IMU to be
filter...
Will Multi-rotors be the future of personal
transportation systems?
Where do I park my multi-rotor??
Corrado Santoro How d...
Demonstration Flight
First prototype: PROBLEMS!!!
DIY is fun but ...
The frame is not well balanced... but the control wil...
How does a Quadrotor fly?
A journey from physics, mathematics, control
systems and computer science
towards a “Controllable...
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How does a Quadrotor fly? A journey from physics, mathematics, control systems and computer science towards a "Controllable Flying Object"

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How does a Quadrotor fly? A journey from physics, mathematics, control systems and computer science towards a "Controllable Flying Object"

  1. 1. How does a Quadrotor fly? A journey from physics, mathematics, control systems and computer science towards a “Controllable Flying Object” Corrado Santoro ARSLAB - Autonomous and Robotic Systems Laboratory Dipartimento di Matematica e Informatica - Universit`a di Catania, Italy santoro@dmi.unict.it Keynote - L.A.P. 1 Course - Jan 10, 2014 Corrado Santoro How does a Quadrotor fly?
  2. 2. Overview 1 Why Multi-rotors? 2 Structure and Physics of a Quadrotor 3 From Analysis to Driving: How can I impose a movement to my quadrotor? 4 The ideal world and the real world: Why we need Control Systems Theory! 5 Rates and Angles: Could I control the attitude? 6 What about Altitude or GPS control? Corrado Santoro How does a Quadrotor fly?
  3. 3. Part I Why Multi-rotors? Corrado Santoro How does a Quadrotor fly?
  4. 4. Flying Machines “To fly” has been one of the dreams of the humans But the story tells that building flying machines is not easy! A basic and common component: the wing Two kind of “flying machines” (excluding rockets and balloons): 1 Fixed wing, i.e. airplanes 2 Rotating wing, i.e. helicopters Corrado Santoro How does a Quadrotor fly?
  5. 5. Design and Implementation problems Airplanes (fixed wing) Wing profile and shape Wing and stab size/area Wing load Position of the COG Motion is achieved by driving (mechanically) the mobile surfaces (aleirons, rudder, elevator) Helicopters (rotating wing, VTOL) Size and structure of the rotor Mechanical system to control motion inclination Yaw balancing system for the rotor at tail Position of the COG Motion is achieved by (mechanically) changing the inclination of the rotor and the pitch of the rotor wings Corrado Santoro How does a Quadrotor fly?
  6. 6. Multi-rotors ... are mechanically simple: they have n motors and n propellers do not require complex mechanical parts to control the flight can fly and move only by changing motor speed are controlled only by a electronic-/computer-based system Building them is simple!! Corrado Santoro How does a Quadrotor fly?
  7. 7. Part II Structure and Physics of a Quadrotor Corrado Santoro How does a Quadrotor fly?
  8. 8. Structure of a Quadrotor (Mechanics) Four equal propellers generating four thrust forces Two possible configurations: “+” and “×” Propellers 1 and 3 rotates CW, 2 and 4 rotates CCW Required to compensate the action/reaction effect (Third Newton’s Law) Propellers 1 and 3 have opposite pitch w.r.t. 2 and 4, so all thrusts have the same direction Corrado Santoro How does a Quadrotor fly?
  9. 9. Structure of a Quadrotor (Electronics) Corrado Santoro How does a Quadrotor fly?
  10. 10. Forces and Rotation speeds ω1, ω2, ω3, ω4: rotation speeds of the propellers T1, T2, T3, T4: forces generated by the propellers Ti ∝ ω2 i : on the basis of propeller shape, air density, etc. m: mass of the quadrotor mg: weight of the quadrotor Corrado Santoro How does a Quadrotor fly?
  11. 11. Moments M1, M2, M3, M4: moments generated by the forces Mi = L × Ti Corrado Santoro How does a Quadrotor fly?
  12. 12. Hovering Condition (Equilibrium) 1 Equilibrium of forces: 4 i=1 Ti = −mg 2 Equilibrium of directions: T1,2,3,4||g 3 Equilibrium of moments: 4 i=1 Mi = 0 4 Equilibrium of rotation speeds: (ω1 + ω3) − (ω2 + ω4) = 0 Violating one (or more) of these conditions implies to impose a certain movement to the quadrotor Corrado Santoro How does a Quadrotor fly?
  13. 13. Reference Systems There are two reference systems: 1 The inertial reference systems, i.e. the Earth frame (xE , yE , zE ) 2 The quadrotor reference system, i.e. the Body frame (xB, yB, zB) Corrado Santoro How does a Quadrotor fly?
  14. 14. Euler Angles Three angles (φ, θ, ψ) define the transformation between the two systems: Roll, φ: angle of rotation along axis xB||xE Pitch, θ: angle of rotation along axis yB||yE Yaw, ψ: angle of rotation along axis zB||zE They are called Euler Angles Corrado Santoro How does a Quadrotor fly?
  15. 15. Angular Speeds The derivative of (φ, θ, ψ) w.r.t. time are the angular/rotation speeds ( ˙φ, ˙θ, ˙ψ) of the system: ˙φ, Roll rate ˙θ, Pitch rate ˙ψ, Yaw rate Corrado Santoro How does a Quadrotor fly?
  16. 16. Part III From Analysis to Driving: How can I impose a movement to my quadrotor? Corrado Santoro How does a Quadrotor fly?
  17. 17. Hovering Condition (Equilibrium) 1 Equilibrium of forces: 4 i=1 Ti = −mg 2 Equilibrium of directions: T1,2,3,4||g 3 Equilibrium of moments: 4 i=1 Mi = 0 4 Equilibrium of rotation speeds: (ω1 + ω3) − (ω2 + ω4) = 0 As a consequence: ˙φ = 0 ˙θ = 0 ˙ψ = 0 φ = 0 θ = 0 ψ = 0 Corrado Santoro How does a Quadrotor fly?
  18. 18. Going Up and Down 1 No equilibrium of forces: 4 i=1 Ti = −mg 2 Equilibrium of directions: T1,2,3,4||g 3 Equilibrium of moments: 4 i=1 Mi = 0 4 Equilibrium of rotation speeds: (ω1 + ω3) − (ω2 + ω4) = 0 By increasing/decreasing the rotation speed of all the propellers we can: Go Up: 4 i=1 Ti > −mg Go Down: 4 i=1 Ti < −mg Euler angles and rates remain 0 Corrado Santoro How does a Quadrotor fly?
  19. 19. Yaw Rotation 1 Equilibrium of forces: 4 i=1 Ti = −mg 2 Equilibrium of directions: T1,2,3,4||g 3 Equilibrium of moments: 4 i=1 Mi = 0 4 No equilibrium of prop speeds: (ω1 + ω3) − (ω2 + ω4) = 0 As a consequence: ˙ψ = kY ((ω1 + ω3) − (ω2 + ω4)) ψ = ˙ψdt Corrado Santoro How does a Quadrotor fly?
  20. 20. Roll Rotation No equilibrium of moments: 4 i=1 Mi = 0 ... by unbalancing propeller speeds as: (ω1 + ω4) − (ω2 + ω3) = 0 As a consequence: ˙φ = kR((ω1 + ω4) − (ω2 + ω3)) φ = ˙φdt No equilibrium of directions: T1,2,3,4 not parallel to g Corrado Santoro How does a Quadrotor fly?
  21. 21. Roll Rotation and Translated Flight Total thrust T = 4 i=1 Ti is decomposed in: Lift Force:TL = T cos φ Drag Force:TD = T sin φ Avoiding diving implies TL = T cos φ = −mg thus in translated flight we need more power w.r.t. hovering or yawing. Corrado Santoro How does a Quadrotor fly?
  22. 22. Pitch Rotation No equilibrium of moments: 4 i=1 Mi = 0 ... by unbalancing propeller speeds as: (ω1 + ω2) − (ω3 + ω4) = 0 As a consequence: ˙θ = kP((ω1 + ω2) − (ω3 + ω4)) θ = ˙θdt Also in this case the total thrust is decomposed thus we need more power w.r.t. hovering or yawing. Corrado Santoro How does a Quadrotor fly?
  23. 23. Equations of Movement We assume a common factor of proportionality k and F = √ T (we will see that such an assumption is not a problem!): ˙φ = k((ω1 + ω4) − (ω2 + ω3)) = kω1 − kω2 − kω3 + kω4 ˙θ = k((ω1 + ω2) − (ω3 + ω4)) = kω1 + kω2 − kω3 − kω4 ˙ψ = k((ω1 + ω3) − (ω2 + ω4)) = kω1 − kω2 + kω3 − kω4 F = k((ω1 + ω2 + ω3 + ω4)) = kω1 + kω2 + kω3 + kω4 or, using matrices:     ˙φ ˙θ ˙ψ F     =     k −k −k k k k −k −k k −k k −k k k k k         ω1 ω2 ω3 ω4     Corrado Santoro How does a Quadrotor fly?
  24. 24. Equations of Movement     ˙φ ˙θ ˙ψ F     =     k −k −k k k k −k −k k −k k −k k k k k         ω1 ω2 ω3 ω4     = K     ω1 ω2 ω3 ω4     This equation gives the angular velocities of the quadrotor, given the speed of the propellers. But if we want to control the quadrotor we must understand how to set ωi in order to impose a certain rotation rate of axis in the body frame. Corrado Santoro How does a Quadrotor fly?
  25. 25. Controlling Roll, Pitch and Yaw Rates, and Total Thrust     ω1 ω2 ω3 ω4     = K−1     ˙φ ˙θ ˙ψ F     =     k k k k −k k −k k −k −k k k k −k −k k         ˙φ ˙θ ˙ψ F     Corrado Santoro How does a Quadrotor fly?
  26. 26. Part IV The ideal world and the real world: Why we need Control Systems Theory! Corrado Santoro How does a Quadrotor fly?
  27. 27. Can we really set the rotation rate of propellers?? Motor/Propeller Driving Schema Drivers, motors and propellers are chosen to be of the same type for the four arms. Software (firmware) controls PWM, but ... 1 Are the drivers really all the same? 2 Are the motors really all the same? 3 Are the propellers really all the same? 4 Is the COG placed at the center of the quadrotor? The answer is: In general, No!! Corrado Santoro How does a Quadrotor fly?
  28. 28. Can we really set the rotation rate of propellers?? Motor/Propeller Driving Schema Same PWM signals applied different driver/motor/propeller chains provoke different thrust forces, even if the components are of the same type! Corrado Santoro How does a Quadrotor fly?
  29. 29. The “Real world” effect Problem We need to set ωi by     ω1 ω2 ω3 ω4     = K−1     ˙φ ˙θ ˙ψ F     but we don’t have a direct control on ωi and propeller thrust Corrado Santoro How does a Quadrotor fly?
  30. 30. The Mathematician/Physicists Solution Solution ?? Let’s characterize each driver/motor/propeller chain and derive the functions: Ti = fi (PWMi ) Then, let’s invert the functions: PWMi = f−1 i (Ti ) But... Characterization is not so easy If we change a component, we must repeat the process There are unpredictable variables, e.g. air density, wind, etc. Corrado Santoro How does a Quadrotor fly?
  31. 31. The Computer Scientist/Engineer Solution Solution ?? Let’s sperimentally tune: an offset for each channel a gain for each channel until the system behaves as expected! But... Tuning is not so easy If we change a component, we must repeat the process There are unpredictable variables, e.g. air density, wind, etc. Corrado Santoro How does a Quadrotor fly?
  32. 32. The Control System Engineer Solution Solution!!!! Use feedback! 1 Measure your variable through a sensor 2 Compare the measured value with your desired set point 3 Apply the correction to the system on the basis of the error 4 Go to 1 Tuning is easy and, if the controller is properly designed ... it works no matter the components it works also in the presence of uncontrollable variables, e.g. air density, wind, etc. Corrado Santoro How does a Quadrotor fly?
  33. 33. Our Scenario Our measures: Actual angular velocities on the three axis ( ˙φM, ˙θM, ˙ψM ) They are measured through a 3-axis gyroscope! Our set-points: Desired angular velocities on the three axis ( ˙φT , ˙θT , ˙ψT ) They are given through the remote control Corrado Santoro How does a Quadrotor fly?
  34. 34. Using Feedback to Control the Quadrotor The overall schema of the feedback controller is: Corrado Santoro How does a Quadrotor fly?
  35. 35. Using Feedback to Control the Quadrotor Algorithmically while True do On ∆T timer tick ; ( ˙φT , ˙θT , ˙ψT , F) = sample remote control(); ( ˙φM, ˙θM, ˙ψM ) = sample gyro(); e ˙φ := ˙φT − ˙φM; e ˙θ := ˙θT − ˙θM; e ˙ψ := ˙ψT − ˙ψM; C ˙φ :=roll rate controller(e ˙φ); C ˙θ :=pitch rate controller(e ˙θ); C ˙ψ :=yaw rate controller(e ˙ψ); (pwm1, pwm2, pwm3, pwm4)T := K−1(C ˙φT , C ˙θT , C ˙ψT , F)T ; send to motors(pwm1, pwm2, pwm3, pwm4); end Corrado Santoro How does a Quadrotor fly?
  36. 36. Using Feedback to Control the Quadrotor Algorithmically while True do On ∆T timer tick ; ( ˙φT , ˙θT , ˙ψT , F) = sample remote control(); ( ˙φM, ˙θM, ˙ψM ) = sample gyro(); e ˙φ := ˙φT − ˙φM; e ˙θ := ˙θT − ˙θM; e ˙ψ := ˙ψT − ˙ψM; C ˙φ :=roll rate controller(e ˙φ); C ˙θ :=pitch rate controller(e ˙θ); C ˙ψ :=yaw rate controller(e ˙ψ); (pwm1, pwm2, pwm3, pwm4)T := K−1(C ˙φT , C ˙θT , C ˙ψT , F)T ; send to motors(pwm1, pwm2, pwm3, pwm4); end The key is in the controllers!! Corrado Santoro How does a Quadrotor fly?
  37. 37. The P.I.D. Controller The most common used controller type is the Proportional-Integral-Derivative controller, represented by the following function: PID Function C := xxx rate controller(e); That is: C(t) := Kpe(t) + Ki t 0 e(τ) dτ + Kd de(t) dt In a discrete world (at kth sampling instant): C(k) := Kpe(k) + Ki k j=0 e(j) ∆T + Kd e(k) − e(k − 1) ∆T Corrado Santoro How does a Quadrotor fly?
  38. 38. The P.I.D. Controller PID Function C(k) := Kpe(k) + Ki k j=0 e(j) ∆T + Kd e(k) − e(k − 1) ∆T Constants Kp, Ki , Kd regulate the behaviour of the controller: Kp drives the short-term action Ki drives the long-term action Kd drives the action on the basis of the “error trend” Constants Kp, Ki , Kd are tuned: Using a specific tuning method (Ziegler-Nichols) Sperimentally by means of “trial-and-error” Corrado Santoro How does a Quadrotor fly?
  39. 39. Part V Rates and Angles: Could I control the attitude? Corrado Santoro How does a Quadrotor fly?
  40. 40. Rates are not Angles The above schema controls rates: suppose a roll angle of φ = 10o but no roll rotation (rate), i.e. ˙φ = 0 and no roll rotation command (sticks set to center) ⇒ the quadrotor is not horizontal and performs a translated flight Could we control angles instead of rates? Corrado Santoro How does a Quadrotor fly?
  41. 41. Measuring Angles (instead of Rates): Gyros First we must measure euler angles (φ, θ, ψ)! We could do this by using Gyroscopes, Accelerometers, Magnetometers, but... Gyroscopes measure angular velocities which can be integrated in order to derive the angle α(t) = t 0 ˙α(τ)dτ, but: Numeric integration is affected by approximation errors Gyroscopes are affected by an offset, i.e. they give non-zero value when the measure should be zero Such an offset is not constant over time and depends on the temperature The estimated angle is not reliable! Corrado Santoro How does a Quadrotor fly?
  42. 42. Measuring Angles: Accelerometers An accelerometer is a sensor measuring the acceleration over the three axis (ax , ay , az). If the sensor is static sensed values are the projections of g vector in the sensor reference system Two functions (using arctan) determines pitch and roll: φ = tan−1 −ay −az θ = tan−1 ax√ a2 y +a2 z But if the object is moving (e.g. shaking) other accelerations appear The computed angles are not reliable! Corrado Santoro How does a Quadrotor fly?
  43. 43. Measuring Angles: Two sensors, No reliability! Gyros Drift Approximate discrete integration Accelerometers Precise only if sensor is not “shaking” We have two different source of the same information which are affected by two different error types. We can use both measures by fusing them in order to adjust the error and obtain a reliable information. Corrado Santoro How does a Quadrotor fly?
  44. 44. Sensor Fusion Basic Algorithm while True do On ∆T timer tick ; ( ˙φ, ˙θ, ˙ψ) = sample gyro(); (ax, ay , az) = sample accel(); (φ, θ, ψ) = (φ, θ, ψ) + ∆T( ˙φ, ˙θ, ˙ψ); ˆφ = tan−1 (−ay/ − az); ˆθ = tan−1 (ax/ a2 y + a2 z); (φ, θ, ψ) = fusion filter(φ, θ, ψ, ˆφ, ˆθ); end Corrado Santoro How does a Quadrotor fly?
  45. 45. Sensor Fusion: Algorithms The key is the filter function! DCM (Direction Cosine Matrix) Complementary filters Kalman filters Basic idea: Derive an error function e(t) = real(t) − estimated(t) Design a controller able to guarantee limt→∞ e(t) = 0 Corrado Santoro How does a Quadrotor fly?
  46. 46. Sensor Fusion: Algorithms High computational load due to: Rotations in the 3D space Matrix calculations May we reduce the load? Corrado Santoro How does a Quadrotor fly?
  47. 47. Representing Rotations in 3D Direction Cosine Matrix DCM =   cθcψ sφsθcψ − cφsψ cφsθcψ + sφsψ cθsψ sφsθsψ + cφcψ cφsθsψ − sφcψ −sθ sφcθ cφcθ   s = sin, c = cos This matrix is re-computed at each iteration!! Rotating a vector v = (x, y, z) implies the product DCM · v. Corrado Santoro How does a Quadrotor fly?
  48. 48. Representing Rotations in 3D Quaternions A quaternion is a complex number with one real part and three imaginary parts: q = q0 + q1i + q2j + q3k i, j, k = imaginary units i2 = j2 = k2 = ijk = −1 While Complex numbers can be used to represent rotations in 2D, Quaternions can be used to represent rotations in 3D. Corrado Santoro How does a Quadrotor fly?
  49. 49. Rotations in 3D and Quaternions Transformations from Euler angles to quaternion exist: q → (φ, θ, ψ) (φ, θ, ψ) → q Rotating a vector v using a quaternion implies the product qvq∗ where q∗ is the conjugate of q and v = {0, vx , vy , vz}. The overall fusion algorithm can be written using quaternion algebra, thus avoiding continuous sin, cos calculation. Quaternions avoid gimbal lock! The attitude can be easily obtained by using: q → (φ, θ, ψ) Corrado Santoro How does a Quadrotor fly?
  50. 50. So far so good: Controlling attitude Attitude control is achieved using (once again) feedback controllers. We set the Target (desired) Attitude (φT , θT , ˙ψT ) from remote controller. Current quad attitude (φM, θM, ˙ψM) is computed using sensor fusion. The error signals (differences) are sent to PID controllers whose output are the target rates for rate controllers. The basic model is “cascading controllers”: attitude controllers which drives rate controllers. Corrado Santoro How does a Quadrotor fly?
  51. 51. Let’s remind the schema of Rate Controllers Corrado Santoro How does a Quadrotor fly?
  52. 52. Complete Attitude Controller Corrado Santoro How does a Quadrotor fly?
  53. 53. Control “loops”: Requirements Two control loops in the schema rate control (inner); attitude control (outer); Attitude control “drives” rate control, thus rate control must have “enough time” to reach the desired target. Loops must have different dynamics, i.e. sampling time Tr = rate control sampling time Ta = attitude control sampling time Ta >> Tr , Ta = nTr , n ∈ N, n > 1 In our quad: Tr = 5ms, Ta = 50ms Corrado Santoro How does a Quadrotor fly?
  54. 54. Finally, the overall algorithm while True do On Tr timer tick ; ( ˙φM, ˙θM, ˙ψM ) = sample gyro(); (ax, ay , az) = sample accel(); (φM, θM) = fusion filter( ˙φM , ˙θM, ˙ψM, ax , ay , az); if after N loops then (φT , θT , ˙ψT , F) = sample remote control(); ˙φT :=roll controller(φM, φT ); ˙θT :=pitch controller(θM, θT ); end C ˙φ :=roll rate controller( ˙φM, ˙φT ); C ˙θ :=pitch rate controller( ˙θM, ˙θT ); C ˙ψ :=yaw rate controller( ˙ψM, ˙ψT ); (pwm1, pwm2, pwm3, pwm4)T := K−1(C ˙φT , C ˙θT , C ˙ψT , F)T ; send to motors(pwm1, pwm2, pwm3, pwm4); end Corrado Santoro How does a Quadrotor fly?
  55. 55. Part VI What about Altitude or GPS control? Corrado Santoro How does a Quadrotor fly?
  56. 56. Let’s repeat the schema! Do you need another kind of control? Repeat the schema! Identify your source of measure m Identify your target t Identify the variables to drive v Identify the sampling time Use a (PID) controller v = pid(t, m) Corrado Santoro How does a Quadrotor fly?
  57. 57. Altitude Control HT = our target height HM = measured height (from a sensor) F = output variable to control (desired thrust) MTr = altitude control sampling time, M > N while True do On Tr timer tick ; ...; if after M loops then HM = sample altitude sensor(); F :=altitude controller(HM, HT ); end ... end Corrado Santoro How does a Quadrotor fly?
  58. 58. GPS Control LatT , LonT = our target position LatM , LonT = measured position (from a GPS sensor) φT , θT = target variables to control (desired pitch and roll) GTr = GPS control sampling time, G > N while True do On Tr timer tick ; ...; if after G loops then (LatM, LonM) = sample gps(); φT :=gps lon controller(LonM, LonT ); θT :=gps lat controller(LatM, LatT ); end ... end Note: for a proper GPS navigation, a compass (with related yaw control) is mandatory. Corrado Santoro How does a Quadrotor fly?
  59. 59. Vision-based Control while True do On Tr timer tick ; ...; if after H loops then (∆X, ∆Y, ∆ψ) = identify target with camera(); φT :=x controller(∆X); θT :=y controller(∆Y); ˙ψT :=heading controller(∆ψ); end ... Corrado Santoro How does a Quadrotor fly?
  60. 60. Conclusions It seems easy .... Corrado Santoro How does a Quadrotor fly?
  61. 61. ... but, where is the trick? Are sensors reliable? Sometimes, NO! Noise due to mechanical vibrations (MEMS-IMU to be filtered by applying Fourier analysis) False positives due to wiring problems (Magnetometers, ADC, etc.) Are execution platforms reliable? Check it! Controllers need precise (real-time) timing DO NOT Windows to stabilize your quad!!! You can try with RT-Linux Is PID Tuning really easy? NO! You must learn it! ... and be sure to have a large set of propellers!! Are all those things fun? OF COURSE!!!! ¨⌣ Corrado Santoro How does a Quadrotor fly?
  62. 62. Will Multi-rotors be the future of personal transportation systems? Where do I park my multi-rotor?? Corrado Santoro How does a Quadrotor fly?
  63. 63. Demonstration Flight First prototype: PROBLEMS!!! DIY is fun but ... The frame is not well balanced... but the control will do the job Too many vibrations (many of them suppressed using Chebyshev filters) Wrong choice of motors (specs report a thurst of 400gr each, but ...) Wiring/Electronics problems Current spikes reset the ultrasonic sensor I2C sometimes locks (a watchdog intervenes and turn-off motors) Firmware problems Still working on the sensor fusion algorithm, since it is not satisfactory (we want more stability...) Corrado Santoro How does a Quadrotor fly?
  64. 64. How does a Quadrotor fly? A journey from physics, mathematics, control systems and computer science towards a “Controllable Flying Object” Corrado Santoro ARSLAB - Autonomous and Robotic Systems Laboratory Dipartimento di Matematica e Informatica - Universit`a di Catania, Italy santoro@dmi.unict.it Keynote - L.A.P. 1 Course - Jan 10, 2014 Corrado Santoro How does a Quadrotor fly?

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