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Quadrotor UAV Modeling and Control

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Aniket Shirsat
Aniket Shirsat

The document describes the modeling and control of a quadrotor unmanned aerial vehicle (UAV). It first outlines various modeling assumptions. It then derives the nonlinear model, including kinematic and dynamic equations, by defining reference frames, rotational matrices, forces and moments. Linear models are developed by linearizing the nonlinear model. Finally, it discusses control design, analysis and simulation for the quadrotor.

Engineering
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Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Modeling and Control of Quadrotor UAV Aniket Shirsat April 2, 2015
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Table of contents 1 Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics 2 Linear Models State Space Model Nominal Model Parameters Linear Model Analysis 3 Control Design Analysis Simulation 4 Conclusion
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Assumptions Aircraft is a rigid body.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Assumptions Aircraft is a rigid body. Propellers are rigid.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Assumptions Aircraft is a rigid body. Propellers are rigid. Quadrotor frame is symmetrical.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Assumptions Aircraft is a rigid body. Propellers are rigid. Quadrotor frame is symmetrical. Mass center and geometric center coincide.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Assumptions Aircraft is a rigid body. Propellers are rigid. Quadrotor frame is symmetrical. Mass center and geometric center coincide. Motor inertia small and neglected.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Assumptions Aircraft is a rigid body. Propellers are rigid. Quadrotor frame is symmetrical. Mass center and geometric center coincide. Motor inertia small and neglected.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Reference Frames Necessity of different frames Actuator inputs and forces act on the body. (Body frame) IMU, accelerometers measures quantities in the body frame. GPS measures position in the inertial frame. (Inertial frame) Model development is carried out in the inertial frame.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Kinematics Position of the quadrotor in the inertial frame η =[XYZ] Attitude of the quadrotor is represented by φ : Roll ,θ : Pitch, ψ : Yaw Angular rates in the body frame are ν =[pqr]
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Rotation Matrix Rotation matrix sequence to go from earth to body is the Yaw-Pitch-Roll euler angle sequence Rotation about z axis by ψ. Rψ =    cos(ψ) sin(ψ) 0 − sin(ψ) cos(ψ) 0 0 0 1    (1)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Rotation Matrix Rotate about the new y-axis positive Pitch(θ). Rθ =    cos(θ) 0 − sin(θ) 0 1 0 sin(θ) 0 cos(θ)    (2)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Rotation Matrix Rotate about the new x axis positive Roll(φ). Rφ =    1 0 0 0 cos(φ) sin(φ) 0 − sin(φ) cos(φ)    (3)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Rotation Matrix The complete rotation matrix is RBody Earth = Rotz,ψ ∗ Roty,θ ∗ Rotz,φ (4)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Angular Velocity The angular velocity matrix is given by ΩBody Earth = ˙φ + Rφ ˙θ + RφRθ ˙ψ = Ω ˙η (5) where Ω =    1 0 − sin(θ) 0 cos(φ) sin(φ) cos(θ) 0 − sin(φ) cos(φ) cos(θ)    (6)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Euler rates The euler rates are given by    ˙φ ˙θ ˙ψ    = Ω−1    p q r    (7) where Ω−1 =    1 tan(θ) sin(φ) tan(θ) cos(φ) 0 cos(φ) − sin(φ) 0 sin(φ) cos(θ) cos(φ) cos(θ)    (8)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Frames representation Inertial and body reference frame for quadrotor Figure: Inertial and Body reference frame for a Quadrotor
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Forces Motor force fi = kω2 i (9) Thrust T = 4 i=1 fi = k 4 i=1 ω2 i (10) Thrust in the body frame TB =    0 0 T    (11)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Moments Roll Moment Figure: Roll moment about the x axis Roll torque τφ = lk(ω2 4 − ω2 2) (12)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Pitch Moment Pitch moment Figure: Pitch moment about the y axis Pitch Torque τθ = lk(ω2 3 − ω2 1) (13)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Yaw Moment Yaw Moment Figure: Yaw movement about the z axis Yaw Torque τψ = b(ω2 1 + ω2 3 − ω2 2 − ω2 4) (14)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Moment of Inertia Schematic Figure: Schematic for inertia calculation Since the quadrotor is assumed to be symmetrical Ixx = Iyy The inertia matrix is I =    Ixx 0 0 0 Iyy 0 0 0 Izz    (15)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Moment of Inertia Moment of inertia about the x- axis is Ixx = 2MR2 5 + 2ml2 (16) Moment of inertia about y- axis is Iyy = 2MR2 5 + 2ml2 (17) Moment of inertia about z- axis is Izz = 2MR2 5 + 4ml2 (18)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Euler- LaGrange Forulation Lagrangian L is L(q, ˙q) = Etrans + Erot − Epot (19) Translational Acceleration f = RTB = m¨ξ + mg    0 0 1    (20)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Equation of Motion Angular Acceleration τ = τB = J¨η+ d dt (J) ˙η− 1 2 ∂ ∂η ( ˙ηT J ˙η) = J¨η+C(η, ˙η) ˙η (21) which can be rearranged to give ¨η = J−1 (τB − C(η, ˙η) ˙η) (22)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Longitudinal state space model State Space equation ˙XLong = ALong XLong + BLong ULong (23) State Transition Matrix ALong =      Z˙z 0 Zθ 0 0 X˙x Xθ 0 0 0 0 1 0 0 Θθ Θ˙θ      (24)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Longitudinal state space model Input matrix BLong =      ZT 0 XT 0 0 0 0 Θτθ      (25) States Xlong = ˙z ˙xθ ˙θ T (26) Inputs ULong = Tτθ T (27)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Nominal parameters Parameter Value Unit m 0.468 kg Ixx 4.856e-3 kg.m2 Iyy 4.856e-3 kg.m2 Izz 8.801e-3 kg.m2 Ax 0.25 Ay 0.25 Az 0.25 g 9.81 m/sec2 l 0.225 m
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Forward flight Longitudinal Dynamics : Nominal Plant P(s) = g11 s+a g12 s2(s+a) g21 s+a g22 s2(s+a) (28)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Forward and Vertical flight Equilibrium Thrust Variation Thrust varies significantly with vertical velocity.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis Equilibrium τθ variation τθ is always zero.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis Expression for θeq θeq = arctan( Ax ˙xeq mg + Az ˙zeq ) (29) θeq variation Varies significantly with forward velocity as compared to vertical velocity.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis Gain 11 : Thrust to ˙Z Gradual decease with increasing forward velocity, Little impact of vertical velocity.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis Gain 12 : Thrust to ˙X At Hover it is zero, validating the Nominal plant is decoupled.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis Gain 21 : τθ to ˙Z At hover, it is zero thus ensuring that the model is decoupled.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis Gain 22 Significantly affected by the vertical climb velocity. Little impact of forward velocity.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of mass Mass = 1.5 kg
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of mass Mass = 3 kg The effect of mass on Thrust is more significant at low forward flight speeds.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of mass Gain: Thrust to ˙Z Mass increase causes the gain to decrease significantly and is more significant at low forward flight speeds.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of mass Gain: Thrust to ˙X Mass increase causes the gain to decrease and its effect is more significant with high flight velocities.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of mass Gain : τtheta to ˙Z Mass increase causes the gain to increase very gradually but its effect becomes more significant at high forward velocities.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis Gain 22 : τtheta to ˙X Mass increase causes the gain to increase significantly but it remains unchanged with forward velocities.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of mass Pole Zero map with change in mass As mass increases the poles move towards the origin thereby decreasing the stability of the system.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of Length Gain: Thrust to ˙Z No impact of the gain.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis Gain: Thrust to ˙X No impact on gain.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of Length Gain: τtheta to ˙Z Gain remains unaffected.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of Length Gain: τtheta to ˙X Length variation does not affect the system properties in forward flight.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of Length Pole Zero map: Impact of length
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Control design : Controller for vertical velocity Transfer function : Thrust to ˙Z P11 = g11 (s + a) (30) Transfer function: τθ to ˙X P22 = g22 s2(s + a) (31)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Control design Controller : Thrust to ˙Z Use a PID structure K11 = g(s + z) s (32)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Control design Time Domain Specifications: Ts ≤ 5 sec and % Mp ≤ 10% CL Poles : s=-1± j1
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Control design : Controller for forward velocity Controller: τθ to ˙X Use a Lead lag structure K22 = s2+2ζωz s+ω2 z s2+2ζωps+ω2 p p z s+z s+p (33)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Control design where ωz = ωm(ζ tan(φm) + ζ2 tan(φm)2 + 1) (34) ωp = ωm(−ζ tan(φm) + ζ2 tan(φm)2 + 1) (35) p z = 1 + sin(φm) 1 − sin(φm) (36)
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Control design φm : desired phase lead / Phase Margin at the desired unity gain frequency. ωm: desired unity gain frequency. z : Zero of the single lead- lag compensator. p : Pole of the single lead- lag compensator.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Analysis Sensitivity S = [I + L]−1 (37) Sensitivity : PM 60 deg As ζ increases |So|peak decreases . As ω increases |So|peak decreases significantly.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Analysis Complimentary Sensitivity T = L[I + L]−1 (38) Complimentary Sensitivity : PM 60 deg As ζ increases |To|peak decreases. As ωg increases |To|peak decreases initially and again increases with ω .
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Analysis Reference to Control Action Tru = KS = K[I + PK]−1 (39) Reference to control action : PM 60 deg As ζ increases |KS|peak remains unaffected. As ωg increases |KS|peak increases significantly.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Analysis Disturbance to output Tdoy = PS = P[I + PK]−1 (40) Disturbance to output : PM 60 deg As ζ increases |PS|peak remains relatively unaffected. As ωg increases |PS|peak decreases significantly.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Analysis Overshoot : PM 60 deg As ζ increases % Mp decreases gradually. As ωg increases % Mp decreases significantly.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Analysis Settling Time: PM 60 deg
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Analysis % Mp ≤ 20 % & Ts ≤ 6 sec =⇒ =⇒ ωg ≥ 3 rad/sec and ζ ≥ 0.8
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Simulation Plant : τθ to ˙X
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Simulation Controller : τθ to ˙X As ωg ↑ |K| ↑.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Simulation Open Loop (L) : τθ to ˙X
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Simulation Sensitivity : τθ to ˙X As ωg ↑ , |So| ↓ at low frequencies.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Simulation Complimentary Sensitivity : τθ to ˙X As ωg ↑ , |To| rolls off at higher ωg .
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Simulation Reference to control action : τθ to ˙X As ωg ↑ , |KS|peak ↑ .
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Simulation Disturbance to output : τθ to ˙X As ωg ↑ , |PS|peak ↓ .
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Simulation For ˙Z = 1m/s
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Simulation For ˙X = 1m/s
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Conclusion A modeling procedure for the longitudinal dynamics of the quadrotor. A control methodology for designing controller using classical control techniques. Future work: Include a procedure for accounting aerodynamic model. Optimize the controller for large forward velocities ( ˙X = 10 ∼ 20m/s). Design a Multi-variable controller for aggressive flight maneuvers.
Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Further Reading Randal W Beard. Quadrotor dynamics and control. Brigham Young University, 2008. Teppo Luukkonen. Modelling and control of quadcopter. Independent research project in applied mathematics, Espoo, 2011. Armando Rodriguez. Analysis and Design of Multivariable Feedback Control Systems. CONTROL3D,L.L.C., Tempe, AZ, 2002. Brian L Stevens and Frank L Lewis. Aircraft control and simulation.

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Quadrotor UAV Modeling and Control

  • 1. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Modeling and Control of Quadrotor UAV Aniket Shirsat April 2, 2015
  • 2. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Table of contents 1 Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics 2 Linear Models State Space Model Nominal Model Parameters Linear Model Analysis 3 Control Design Analysis Simulation 4 Conclusion
  • 3. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Assumptions Aircraft is a rigid body.
  • 4. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Assumptions Aircraft is a rigid body. Propellers are rigid.
  • 5. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Assumptions Aircraft is a rigid body. Propellers are rigid. Quadrotor frame is symmetrical.
  • 6. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Assumptions Aircraft is a rigid body. Propellers are rigid. Quadrotor frame is symmetrical. Mass center and geometric center coincide.
  • 7. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Assumptions Aircraft is a rigid body. Propellers are rigid. Quadrotor frame is symmetrical. Mass center and geometric center coincide. Motor inertia small and neglected.
  • 8. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Assumptions Aircraft is a rigid body. Propellers are rigid. Quadrotor frame is symmetrical. Mass center and geometric center coincide. Motor inertia small and neglected.
  • 9. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Reference Frames Necessity of different frames Actuator inputs and forces act on the body. (Body frame) IMU, accelerometers measures quantities in the body frame. GPS measures position in the inertial frame. (Inertial frame) Model development is carried out in the inertial frame.
  • 10. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Kinematics Position of the quadrotor in the inertial frame η =[XYZ] Attitude of the quadrotor is represented by φ : Roll ,θ : Pitch, ψ : Yaw Angular rates in the body frame are ν =[pqr]
  • 11. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Rotation Matrix Rotation matrix sequence to go from earth to body is the Yaw-Pitch-Roll euler angle sequence Rotation about z axis by ψ. Rψ =    cos(ψ) sin(ψ) 0 − sin(ψ) cos(ψ) 0 0 0 1    (1)
  • 12. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Rotation Matrix Rotate about the new y-axis positive Pitch(θ). Rθ =    cos(θ) 0 − sin(θ) 0 1 0 sin(θ) 0 cos(θ)    (2)
  • 13. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Rotation Matrix Rotate about the new x axis positive Roll(φ). Rφ =    1 0 0 0 cos(φ) sin(φ) 0 − sin(φ) cos(φ)    (3)
  • 14. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Rotation Matrix The complete rotation matrix is RBody Earth = Rotz,ψ ∗ Roty,θ ∗ Rotz,φ (4)
  • 15. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Angular Velocity The angular velocity matrix is given by ΩBody Earth = ˙φ + Rφ ˙θ + RφRθ ˙ψ = Ω ˙η (5) where Ω =    1 0 − sin(θ) 0 cos(φ) sin(φ) cos(θ) 0 − sin(φ) cos(φ) cos(θ)    (6)
  • 16. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Euler rates The euler rates are given by    ˙φ ˙θ ˙ψ    = Ω−1    p q r    (7) where Ω−1 =    1 tan(θ) sin(φ) tan(θ) cos(φ) 0 cos(φ) − sin(φ) 0 sin(φ) cos(θ) cos(φ) cos(θ)    (8)
  • 17. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Frames representation Inertial and body reference frame for quadrotor Figure: Inertial and Body reference frame for a Quadrotor
  • 18. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Forces Motor force fi = kω2 i (9) Thrust T = 4 i=1 fi = k 4 i=1 ω2 i (10) Thrust in the body frame TB =    0 0 T    (11)
  • 19. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Moments Roll Moment Figure: Roll moment about the x axis Roll torque τφ = lk(ω2 4 − ω2 2) (12)
  • 20. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Pitch Moment Pitch moment Figure: Pitch moment about the y axis Pitch Torque τθ = lk(ω2 3 − ω2 1) (13)
  • 21. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Yaw Moment Yaw Moment Figure: Yaw movement about the z axis Yaw Torque τψ = b(ω2 1 + ω2 3 − ω2 2 − ω2 4) (14)
  • 22. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Moment of Inertia Schematic Figure: Schematic for inertia calculation Since the quadrotor is assumed to be symmetrical Ixx = Iyy The inertia matrix is I =    Ixx 0 0 0 Iyy 0 0 0 Izz    (15)
  • 23. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Moment of Inertia Moment of inertia about the x- axis is Ixx = 2MR2 5 + 2ml2 (16) Moment of inertia about y- axis is Iyy = 2MR2 5 + 2ml2 (17) Moment of inertia about z- axis is Izz = 2MR2 5 + 4ml2 (18)
  • 24. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Euler- LaGrange Forulation Lagrangian L is L(q, ˙q) = Etrans + Erot − Epot (19) Translational Acceleration f = RTB = m¨ξ + mg    0 0 1    (20)
  • 25. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Equation of Motion Angular Acceleration τ = τB = J¨η+ d dt (J) ˙η− 1 2 ∂ ∂η ( ˙ηT J ˙η) = J¨η+C(η, ˙η) ˙η (21) which can be rearranged to give ¨η = J−1 (τB − C(η, ˙η) ˙η) (22)
  • 26. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Longitudinal state space model State Space equation ˙XLong = ALong XLong + BLong ULong (23) State Transition Matrix ALong =      Z˙z 0 Zθ 0 0 X˙x Xθ 0 0 0 0 1 0 0 Θθ Θ˙θ      (24)
  • 27. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Longitudinal state space model Input matrix BLong =      ZT 0 XT 0 0 0 0 Θτθ      (25) States Xlong = ˙z ˙xθ ˙θ T (26) Inputs ULong = Tτθ T (27)
  • 28. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Nominal parameters Parameter Value Unit m 0.468 kg Ixx 4.856e-3 kg.m2 Iyy 4.856e-3 kg.m2 Izz 8.801e-3 kg.m2 Ax 0.25 Ay 0.25 Az 0.25 g 9.81 m/sec2 l 0.225 m
  • 29. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Forward flight Longitudinal Dynamics : Nominal Plant P(s) = g11 s+a g12 s2(s+a) g21 s+a g22 s2(s+a) (28)
  • 30. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Forward and Vertical flight Equilibrium Thrust Variation Thrust varies significantly with vertical velocity.
  • 31. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis Equilibrium τθ variation τθ is always zero.
  • 32. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis Expression for θeq θeq = arctan( Ax ˙xeq mg + Az ˙zeq ) (29) θeq variation Varies significantly with forward velocity as compared to vertical velocity.
  • 33. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis Gain 11 : Thrust to ˙Z Gradual decease with increasing forward velocity, Little impact of vertical velocity.
  • 34. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis Gain 12 : Thrust to ˙X At Hover it is zero, validating the Nominal plant is decoupled.
  • 35. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis Gain 21 : τθ to ˙Z At hover, it is zero thus ensuring that the model is decoupled.
  • 36. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis Gain 22 Significantly affected by the vertical climb velocity. Little impact of forward velocity.
  • 37. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of mass Mass = 1.5 kg
  • 38. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of mass Mass = 3 kg The effect of mass on Thrust is more significant at low forward flight speeds.
  • 39. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of mass Gain: Thrust to ˙Z Mass increase causes the gain to decrease significantly and is more significant at low forward flight speeds.
  • 40. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of mass Gain: Thrust to ˙X Mass increase causes the gain to decrease and its effect is more significant with high flight velocities.
  • 41. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of mass Gain : τtheta to ˙Z Mass increase causes the gain to increase very gradually but its effect becomes more significant at high forward velocities.
  • 42. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis Gain 22 : τtheta to ˙X Mass increase causes the gain to increase significantly but it remains unchanged with forward velocities.
  • 43. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of mass Pole Zero map with change in mass As mass increases the poles move towards the origin thereby decreasing the stability of the system.
  • 44. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of Length Gain: Thrust to ˙Z No impact of the gain.
  • 45. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis Gain: Thrust to ˙X No impact on gain.
  • 46. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of Length Gain: τtheta to ˙Z Gain remains unaffected.
  • 47. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of Length Gain: τtheta to ˙X Length variation does not affect the system properties in forward flight.
  • 48. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Model Analysis : Impact of Length Pole Zero map: Impact of length
  • 49. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Control design : Controller for vertical velocity Transfer function : Thrust to ˙Z P11 = g11 (s + a) (30) Transfer function: τθ to ˙X P22 = g22 s2(s + a) (31)
  • 50. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Control design Controller : Thrust to ˙Z Use a PID structure K11 = g(s + z) s (32)
  • 51. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Control design Time Domain Specifications: Ts ≤ 5 sec and % Mp ≤ 10% CL Poles : s=-1± j1
  • 52. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Control design : Controller for forward velocity Controller: τθ to ˙X Use a Lead lag structure K22 = s2+2ζωz s+ω2 z s2+2ζωps+ω2 p p z s+z s+p (33)
  • 53. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Control design where ωz = ωm(ζ tan(φm) + ζ2 tan(φm)2 + 1) (34) ωp = ωm(−ζ tan(φm) + ζ2 tan(φm)2 + 1) (35) p z = 1 + sin(φm) 1 − sin(φm) (36)
  • 54. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Control design φm : desired phase lead / Phase Margin at the desired unity gain frequency. ωm: desired unity gain frequency. z : Zero of the single lead- lag compensator. p : Pole of the single lead- lag compensator.
  • 55. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Analysis Sensitivity S = [I + L]−1 (37) Sensitivity : PM 60 deg As ζ increases |So|peak decreases . As ω increases |So|peak decreases significantly.
  • 56. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Analysis Complimentary Sensitivity T = L[I + L]−1 (38) Complimentary Sensitivity : PM 60 deg As ζ increases |To|peak decreases. As ωg increases |To|peak decreases initially and again increases with ω .
  • 57. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Analysis Reference to Control Action Tru = KS = K[I + PK]−1 (39) Reference to control action : PM 60 deg As ζ increases |KS|peak remains unaffected. As ωg increases |KS|peak increases significantly.
  • 58. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Analysis Disturbance to output Tdoy = PS = P[I + PK]−1 (40) Disturbance to output : PM 60 deg As ζ increases |PS|peak remains relatively unaffected. As ωg increases |PS|peak decreases significantly.
  • 59. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Analysis Overshoot : PM 60 deg As ζ increases % Mp decreases gradually. As ωg increases % Mp decreases significantly.
  • 60. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Analysis Settling Time: PM 60 deg
  • 61. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Analysis % Mp ≤ 20 % & Ts ≤ 6 sec =⇒ =⇒ ωg ≥ 3 rad/sec and ζ ≥ 0.8
  • 62. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Simulation Plant : τθ to ˙X
  • 63. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Simulation Controller : τθ to ˙X As ωg ↑ |K| ↑.
  • 64. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Simulation Open Loop (L) : τθ to ˙X
  • 65. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Simulation Sensitivity : τθ to ˙X As ωg ↑ , |So| ↓ at low frequencies.
  • 66. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Simulation Complimentary Sensitivity : τθ to ˙X As ωg ↑ , |To| rolls off at higher ωg .
  • 67. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Simulation Reference to control action : τθ to ˙X As ωg ↑ , |KS|peak ↑ .
  • 68. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Controller Simulation Disturbance to output : τθ to ˙X As ωg ↑ , |PS|peak ↓ .
  • 69. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Simulation For ˙Z = 1m/s
  • 70. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Linear Simulation For ˙X = 1m/s
  • 71. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Conclusion A modeling procedure for the longitudinal dynamics of the quadrotor. A control methodology for designing controller using classical control techniques. Future work: Include a procedure for accounting aerodynamic model. Optimize the controller for large forward velocities ( ˙X = 10 ∼ 20m/s). Design a Multi-variable controller for aggressive flight maneuvers.
  • 72. Modeling and Control of Quadrotor UAV Aniket Shirsat Non-Linear Model Modeling Assumptions Reference Frames Kinematics Dynamics Linear Models State Space Model Nominal Model Parameters Linear Model Analysis Control Design Analysis Simulation Conclusion Further Reading Randal W Beard. Quadrotor dynamics and control. Brigham Young University, 2008. Teppo Luukkonen. Modelling and control of quadcopter. Independent research project in applied mathematics, Espoo, 2011. Armando Rodriguez. Analysis and Design of Multivariable Feedback Control Systems. CONTROL3D,L.L.C., Tempe, AZ, 2002. Brian L Stevens and Frank L Lewis. Aircraft control and simulation.