Control of Complex Maneuvers for a Quadrotor UAV using Geometric Methods on SE(3)

I. INTRODUCTION

• Earlier linear controllers target equilibrium stability,,,,, while hybrid-mode approaches require reachability analyses to guarantee safety and performance.The paper identifies complex aerobatic maneuvers as a relatively little-studied control problem.Source: §I · p. 1
• Euler-angle controllers have singularities during complex rotations,, while quaternion controllers can suffer ambiguity, discontinuity, or unwinding,,.These representation issues can restrict maneuvering or make control sensitive to noise and disturbances.Source: §I · p. 1
• The proposed geometric controllers track attitude, position, or velocity in a hybrid architecture, with dynamics modeled globally on SE(3).The paper presents these modes as primitives that can be combined for complex maneuvers.Source: §I · p. 1
• The controllers have almost-global tracking properties and are designed to remain robust when switching between flight modes.The proposed system extends geometric control to an underactuated rigid body with coupled translation and rotation, and avoids coordinate singularities and quaternion ambiguities.Source: §I · pp. 1–2

II. QUADROTOR DYNAMICS MODEL

• Four rotors at the square-frame vertices generate thrust normal to the rotor plane, described in inertial and body-fixed reference frames.The body-fixed frame originates at the center of mass; its first two axes lie in the rotor plane and its third is normal to it.Source: §II · p. 2, Figure 1 · p. 2
• The quadrotor configuration comprises center-of-mass location and attitude, placing it on SE(3), the semidirect product of R3 and SO(3).The rotation matrix maps body-fixed vectors into the inertial frame.Source: §II · p. 2
• The model assumes directly controlled rotor thrusts along the negative body b3 axis and rotor torques proportional to thrust.The paper notes that the control system can be extended to include linear rotor dynamics.Source: §II · p. 2
• When rotor spacing and the thrust-to-torque constant are nonzero, the rotor-thrust allocation matrix is invertible, so total thrust and moment can serve as control inputs.The allocation recovers the four individual propeller thrusts from those inputs.Source: §II · pp. 2–3

III. GEOMETRIC TRACKING CONTROLS

• The four-input vehicle cannot asymptotically track both attitude and position simultaneously, motivating separate flight modes.Each mode specifies outputs whose exact tracking defines that mode.Source: §III · p. 3
• The attitude mode achieves asymptotic attitude tracking, while the position mode tracks the center-of-mass position.The velocity mode achieves asymptotic tracking of the center-of-mass velocity.Source: §III · p. 3
• A maneuver can concatenate modes and specify switching conditions and commanded outputs over time.The paper gives hovering, reorientation, and time-varying position tracking as an example sequence.Source: §III · p. 3
• Earlier hybrid architectures were sensitive to switching because each mode had a limited region of attraction and required reachability analyses for safety and performance.The proposed system is described as robust to switching because each mode has almost-global stability properties.Source: §III · p. 3

IV. ATTITUDE CONTROLLED FLIGHT MODE

• The controller tracks a time-varying attitude command using geometric attitude and angular-velocity errors on SO(3).The attitude-mode moment controller is specified for positive gains k_R and k_Ω.Source: §IV · pp. 3–4
• The attitude error function is locally positive-definite when the rotation angle from the command is below 180°.The cited sublevel set Ψ(R, R_d) < 2 almost covers SO(3).Source: §IV · p. 3
• For initial attitudes within its stated region, the attitude controller makes the zero tracking-error equilibrium exponentially stable.The region of attraction almost covers SO(3), excluding attitudes exactly 180° from the command; increasing k_R can enlarge the estimated region.Source: §IV · p. 4
• Choosing thrust to track an altitude command preserves the attitude-mode conclusions and makes altitude asymptotically track its command.The thrust choice uses positive gains and assumes the stated altitude-command conditions.Source: §IV · p. 4
• Because translation is only partially controlled in attitude mode, the paper identifies it as most suitable for short attitude maneuvers.The thrust magnitude can instead be chosen for an additional translational objective.Source: §IV · p. 4

V. POSITION CONTROLLED FLIGHT MODE

• A backstepping controller chooses a computed attitude whose thrust axis follows a position-tracking direction, while thrust magnitude achieves position tracking.The moment control makes the vehicle attitude asymptotically track the computed attitude.Source: §V · p. 5
• For initial attitude error below 90°, the position-mode tracking-error equilibrium is exponentially stable, and the attitude controller brings larger errors into its region of attraction.Combining the attitude and position-mode results gives almost global exponential attractiveness when Ψ(R(0), Rc(0)) < 2.Source: §V · p. 5
• Almost all initial attitude errors yield exponential attractiveness of the position-tracking closed loop; the excluded attitudes form a two-dimensional manifold in SO(3).The result concerns the combined position, velocity, attitude, and angular-velocity tracking errors.Source: §V · p. 6
• The controller projects a chosen direction b1d onto the plane normal to b3c, constraining the first body-fixed axis to the plane spanned by b1d and ge3 − ¨xd asymptotically.The third body-fixed axis converges to the direction ge3 − ¨xd.Source: §V · p. 6
• When commanded acceleration is zero, choosing b1d in the horizontal plane makes the first body-fixed axis converge to b1d, specifying the vehicle’s heading.Proposition 5 retains the conclusions of Propositions 3 and 4 while adding this asymptotic-direction property.Source: §V · p. 6

VI. VELOCITY CONTROLLED FLIGHT MODE

• The velocity-mode controller uses commanded velocity and a computed attitude, with positive gains and an orthogonal b1c chosen to ensure Rc ∈ SO(3).The closed-loop structure and stability properties are similar to those of the position-controlled mode.Source: §VI · pp. 6–7
• For Ψ(R(0), Rc(0)) < 1, the velocity-mode tracking-error equilibrium is exponentially stable; for Ψ(R(0), Rc(0)) < 2, it is exponentially attractive.The errors are (ev, eR, eΩ).Source: §VI · p. 7
• The explicit propositions and proofs for velocity-controlled flight are relegated to [22] because of page limitations.Source: §VI · p. 7

Case (I): Position Controlled Flight Mode

• At 0.88 seconds, the attitude error enters the stated region of attraction, after which position tracking error converges exponentially.The initial attitude error is 178°; the example begins with the vehicle upside down.Source: §VII · p. 7, Figure 4 · p. 7

Case (II): Transition Between Several Flight Modes

• In the initial velocity mode, velocity tracking error converges exponentially, while the first body-fixed axis approaches the plane specified by the desired direction and acceleration.Because the desired acceleration is small relative to gravity, the axis remains close to the plane spanned by e1 and e3.Source: §VII · p. 8
• During the 720° attitude maneuver, the selected thrust magnitude keeps position close to [8, 0, 0], but does not guarantee convergence to that point.Altitude tracking results do not apply because the third body-fixed axis becomes horizontal during the maneuver.Source: §VII · p. 8
• With zero desired acceleration and horizontal b1d, the position modes yield convergence of the first body-fixed axis to b1d; the maneuver also keeps position near xc = [6, 0, 0] during an attitude segment.The final position mode specifies b1d = e2, so the first body-fixed axis points left of the flight path.Source: §VII · p. 8
• The five-mode maneuver combines attitude, velocity, and position tracking to perform the prescribed complex aerobatic maneuver.It includes a 720° rotation about e2 and a 360° rotation about e1.Source: §VII · pp. 7–8, Figure 5 · p. 8

VIII. CONCLUSIONS

• The controllers provide almost-global exponential attractiveness for initial attitude errors below 180° and exponential stability below 90°.The numerical examples demonstrate recovery from an upside-down attitude and complex maneuvers using mode switching.Source: §VIII · p. 8
• The global quadrotor model and three tracking controllers are formulated intrinsically on SE(3), avoiding Euler-angle singularities and quaternion ambiguities..Source: §VIII · p. 8
• Switching among the three flight-mode controllers is demonstrated to produce complex acrobatic maneuvers.Source: §VIII · p. 8

A. Properties of the Hat Map

• The hat map identifies so(3) with R3 using the vector cross product, with x-hat y equal to x cross y.Source: A. Properties of the Hat Map · p. 8

B. Proof of Proposition 1

• The Lyapunov argument bounds the attitude error function so that R(t) remains in the sublevel set L2, where Ψ is positive-definite.The proof establishes the bound by showing the relevant Lyapunov quantity is non-increasing.Source: B. Proof of Proposition 1 · pp. 9–10
• The attitude tracking error dynamics are derived from the attitude kinematics and control moment, yielding the error equations used in the stability proof.The proof first formulates the error dynamics for Ψ, eR, and eΩ.Source: B. Proof of Proposition 1 · p. 9
• For a positive c2 satisfying the stated conditions, the zero equilibrium of the attitude tracking error is exponentially stable and Ψ decreases exponentially.The proof uses positive-definiteness and decrescence of the Lyapunov function.Source: B. Proof of Proposition 1 · p. 10

C. Proof of Proposition 2

• Because the rotational dynamics are decoupled from translation and the control moment assumptions match Proposition 1, its conclusions carry over.Source: C. Proof of Proposition 2 · p. 10
• Altitude tracking is analyzed by projecting the translational dynamics onto the inertial e3 axis.Source: C. Proof of Proposition 2 · p. 10
• The resulting second-order linear altitude-error system is exponentially stable for positive kx and kv.Source: C. Proof of Proposition 2 · p. 10

• The velocity tracking error is uniformly bounded, and the proof uses Lyapunov analysis to establish exponential stability.Source: D. Proof of Proposition 3 · p. 10
• Within L1, the attitude error is less than 90° throughout the proof.L1 is the set of attitudes satisfying Ψ(R, R_c) < 1.Source: D. Proof of Proposition 3 · p. 10
• The Lyapunov analysis restricts the proof to D to bound the third-order term k_x||e_R||||e_x||||e_v|| without the more complicated analysis needed by an alternative bound.D also imposes ||e_x|| < e_xmax; the alternative bound would require a smaller initial attitude error.Source: D. Proof of Proposition 3 · pp. 10–11
• Under conditions (31)–(33), the complete tracking-error equilibrium is exponentially stable, with a conservative region of attraction given by a sublevel set of V contained in D and satisfying (34).The proof establishes negative definiteness of the Lyapunov derivative for the complete system.Source: D. Proof of Proposition 3 · pp. 11–12

• The attitude and angular-velocity error z2 decreases exponentially and enters Proposition 3’s region of attraction in finite time t*.Source: E. Proof of Proposition 4 · p. 12
• The position and velocity error z1 is bounded on [0, t*], enabling the complete tracking error to remain uniformly bounded before exponential decrease.Source: E. Proof of Proposition 4 · p. 12
• For initial conditions satisfying (36) and (37), the tracking-error equilibrium is exponentially attractive.The proof first establishes bounded tracking errors until entry into the exponential-stability region.Source: E. Proof of Proposition 4 · p. 12