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Lecture 1: Quadrotor
- 1. Chapter 3 Quadrotor / Quadcopter
- 2. The First Manned Quadrotor • Quadrotor is a kind of unmanned aerial vehicle (UAV) • 29/9/1907: Louis Bréguet & Jacques Bréguet, under the guidance of Professor Charles Richet, demonstrated the first flying quadrotor named Bréguet-Richet Gyroplane No. 1
- 3. Advantages of quadrotor • Quadrotor is a rotary wing UAV • Its advantages over fixed wing UAVs: – Vertical Take Off and Landing (VTOL) – Able to hover – Able to make slow precise movements. – Four rotors provide a higher payload capacity – More flexible in maneuverability through an environment with many obstacles, or landing in small areas.
- 5. Quadrotor structure • Front motor (Mf) (+x) • Back motor (Mb) (-x) • Right motor (Mr) (+y) • Left motor (Ml) (-y) • Mfand Mb rotates CW • Mr and Mlrotates CCW • This arrangement can overcome torque effect to prevent on the spot spinning of the structure • Each spinning motor provides – thrust force (T) for lifting – torque () for rotating
- 6. Basic movements X (North) Y (East) Z (Down)
- 7. Reference Frames • There are a few reference frames to model the kinematics and dynamics of a quadrotor – Inertia Frame (Global frame), Fi – Vehicle Frame, Fv – Vehicle-1 Frame, Fv1 – Vehicle-2 Frame, Fv2 – Body Frame (Local frame), Fb
- 8. The inertia frame • For the context of quadrotor, the Earth is a flat surface • The starting position of the quadrotor is the origin of the global frame or the inertia frame (Fi) • Fi: x-y-z axis is right hand system with x pointing to North, y pointing to East and z pointing to Down, it is also known as the NED system X (North) Y (East) Z (Down)
- 9. The Vehicle Frame • Fv is the vehicle frame • It is the inertia frame, Fi, linear shifted to the centre of gravity (COG) for the quadrotor • The coordinates of the COG for the quadrotor wrt Fi is (xc, yc, zc). Xi (North) Yi (East) Zi (Down) Xv Yv Zv F v F i
- 10. The Vehicle-1 Frame Xv Yv Zv F v F v1 Zv Zv1 Yv1 Xv1
- 11. The Vehicle-2 Frame Xv 2 Zv2Zv F v1 F v2 Yv1 Yv2Yv1 Xv1
- 12. The Body Frame Xv 2 Zv2 Zb F v2 F b Xv2 Xb Yv2Yb
- 13. Vehicle Frame Body Frame
- 14. , , are known as Euler angles. They are measured from different frames (Roll in Fv2 frame, Pitch in Fv1 frame, Yaw in Fv frame)
- 15. Gimbal Lock • This is a fundamental problem when using sensors to sense Euler angles • When pitch angle is 90 degrees, roll and yaw rotation give the same sensor readings • Information for 1 dimension is lost and the actual configuration of the rigid body is not correctly sensed • Solution: – Avoid 90 degree pitch when using Euler angle sensor
- 16. Quadrotor State Variables • Positions in Fi : pn, pe, h • Velocities in Fb: u, v, w • Angular velocities in Fb: p, q, r • Euler angles: – Yaw angle in Fv: ψ – Pitch angle in Fv1: θ – Roll angle in Fv2: ϕ
- 17. Quadrotor Kinematics − −+ +− = = − w v u CCCSS CSSSCCCSSSSC SSCSCSCCSSCC w v u R h p p dt d v be n θφθφθ ψφψθφψφψθφψθ ψφψθφψφψθφψθ ( ) ( ) ( ) ( ) ( ) ( ) −= − − = + + = r q p CCCS SC TCTS CCS CSC S r q p RRRRRR r q p v v v v b v v v b v b v θφθφ φφ θφθφ θφφ θφφ θ ψ θ φ ψ θ φ ψ ψθφθθφ φ φ 0 0 1 and 0 0 01 0 0 0 0 0 0 12 12 2 122 • Relating position (Fi) and velocities (Fb) in the same frame (Fi): • Relating angular velocities (Fb) to Euler angle rates (Fv, Fv1, Fv2)
- 18. Equation of Coriolis pp dt d p dt d b bi ×+= ω • Inertia frame, Fi looking at Body frame, Fb • Vector p is moving in Fb and Fb is rotating and translating with respect to Fi • Time derivative of p as seen from Fi is obtained using equation of Coriolis:
- 19. Quadrotor Dynamics • Equation of Coriolis: • m is the mass • vector v is the velocities • vector ωb is the angular velocities in the body frame • vector f is the applied forces • In body coordinates: + − − − = z y x f f f m pvqu rupw qwrv w v u 1 fv dt vd m dt vd m b bi = ×+= ω
- 20. Rotational Motion • Equation of Coriolis for rotational motion: • vector h is angular momentum, h = Jωb • J is symmetric inertia matrix • vector m is the applied torque • Substitutes into equation of Coriolis: • Angular acceleration is hence given by: mhh dt d h dt d b bi =×+= ω = = = ψ θ φ τ τ τ ω m r q p I I I J b z y x ;; 00 00 00 { } { } { } +− +− +− = = × + zyx yxz xzy IpqII IprII IqrII r q p r q p J r q p r q p J ψ θ φ ψ θ φ τ τ τ τ τ τ )( )( )(
- 21. Summary of Equation Set − −+ +− = − w v u CCCSS CSSSCCCSSSSC SSCSCSCCSSCC h p p e n θφθφθ ψφψθφψφψθφψθ ψφψθφψφψθφψθ −= r q p CCCS SC TCTS θφθφ φφ θφθφ ψ θ φ 0 0 1 { } { } { } +− +− +− = zyx yxz xzy IpqII IprII IqrII r q p ψ θ φ τ τ τ )( )( )( + − − − = z y x f f f m pvqu rupw qwrv w v u 1
- 22. Thrust Force and Gravity Force • fx , fy , fz are total forces acting on the body frame, Fb • there are two components: – quadrotor thrust force (produced by propeller) – gravity force • Total thrust in Fb: T = Tf + Tb +Tl + Tr • Gravity force in Fi : (0,0,mg) • In Fb: − + − = + − = φθ φθ θ CmgC SmgC mgS Tmg RR Tf f f v i b v z y x 0 0 0 0 0 0
- 23. Torque / Moment • Roll : τϕ = l (Tl - Tr) • Pitch : τθ = l (Tf- Tb) • Yaw : τψ = τr+ τl- τf- τb • The drag of the propellers produces a yawing torque on the body of the quadrotor (Newton's 3rd Law) • The direction of the torque is int he opposite direction to the motion of the propeller • The thrust and torque of each motor is controlled by its angular speed in rpm: – Ti = kf ωi 2 – τi = km ωi 2 • i can take the value 1 (front motor), 2 (right motor), 3 (back motor) and 4 (left motor)
- 24. Simplified model • Use vehicle 1 frame for position estimate • Small Euler angles (sin, tan -> 0) • Ignore Coriolis terms (qr, pr, pq) −− −+ +− = w v u CCCSS CSSSCCCSSSSC SSCSCSCCSSCC h p p e n θφθφθ ψφψθφψφψθφψθ ψφψθφψφψθφψθ − + − = = + − − − = φθ φθ θ CC SC S g T m f f f m f f f m pvqu rupw qwrv w v u z y x z y x 0 0 111 = −= r q p r q p CCCS SC TCTS θφθφ φφ θφθφ ψ θ φ 0 0 1 = = z y x I I I r q p ψ θ φ τ τ τ ψ θ φ − −= = w v u CCCSS SC SCSSC w v u RR p p p v b v v z y x θφθφθ φφ θφθφθ 021 2 − +− −− + = − −= θφ ψφψθφ ψφψθφ θφθφθ φφ θφθφθ CC CSSSC SSCSC m T gw v u CCCSS SC SCSSC p p p z y x 0 0 0 − + − = + − = φθ φθ θ CmgC SmgC mgS Tmg RR Tf f f v i b v z y x 0 0 0 0 0 0 = = z y x I I I r q p ψ θ φ τ τ τ ψ θ φ − +− −− + = θφ ψφψθφ ψφψθφ CC CSSSC SSCSC m T gp p p z y x 0 0
- 25. State Estimates • States to be measured or estimated: – p, q, r (from sensors) – ,θ,ψϕ – dot p, dot q, dot r – px, py, pz (from sensors) – u, v, w – dot u, dot v, dot w • From rate gyroscopes, we can get (p,q,r) • Integrating and differentiating (p,q,r) to get ( ,θ,ψ) and angular accelerationϕ • From position sensor (usually external camera), we get (px, py, pz) • Differentiating position to get (u,v,w) • From accelerometer, we get T/m, toggether with Euler angles, we can get position acceleration = = z y x I I I r q p ψ θ φ τ τ τ ψ θ φ − +− −− + = θφ ψφψθφ ψφψθφ CC CSSSC SSCSC m T gp p p z y x 0 0 = −= r q p r q p CCCS SC TCTS θφθφ φφ θφθφ ψ θ φ 0 0 1 − −= w v u CCCSS SC SCSSC p p p z y x θφθφθ φφ θφθφθ 0 = ∫ ∫ ∫ rdt qdt pdt ψ θ φ
- 26. Case study: From rest to hover in z • A quadrotor is resting at its vehicle frame • There is no rotational movement • It starts to climb to a certain height and hovers • From rest, the thrust is incrased • T > mg, a is positive vertically, v increases, h increases • T < mg, a is negative vertically, v decreases to zero, h increases • T = mg, a is zero, v is zero, h maintains • Exercise: From hovering to the ground
- 27. Case study: From hover to x and hover • From hovering, • Negative pitch, ax is positive , vx increases, x increases • Positive pitch, ax is negative, vx decreases to zero, x increases • No pitch, T = mg, a is zero, vx is zero, x is maximum • Exercise: How to maintain h during these operation?
- 28. Jerk free planning • Jerk is the time derivative of acceleration • Physically, it is sudden start or stop • Maximum force is upon the quadrotor with jerk • Exercise: Qualitatively design a jerk profile such that quadrotor is climbing up to a height to hover with jerk-free movement. 3 3 2 2 dt zd dt vd dt ad jz ===