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Control of Complex Maneuvers for a Quadrotor UAV using Geometric Methods on SE(3)
Taeyoung Lee, Melvin Leok, N. Harris McClamroch
TL;DR
Complex quadrotor maneuvers are difficult for controllers based on Euler angles, which become singular during complex rotations. The paper develops geometric tracking controllers for attitude, position, and velocity modes on SE(3); its examples show recovery from an initially upside-down attitude and maneuvers using transitions between modes.
Problem
Euler-angle-based quadrotor controllers have singularities during complex rotations, restricting their ability to achieve complex flight maneuvers.
Source: §I · p. 1Method
The paper develops coordinate-free geometric controllers on SE(3) for tracking attitude, center-of-mass position, or center-of-mass velocity, with position control directing thrust through a desired attitude.
Source: §I · p. 1, §V · pp. 5–6Results
The examples show recovery from an initially upside-down attitude and a prescribed complex acrobatic maneuver performed by switching between flight modes.
Source: §VII · pp. 8–9Takeaways & Limitations
The paper reports exponential stability for initial attitude errors below 90° and almost global exponential attractiveness for errors below 180°, and demonstrates complex maneuvers by switching controllers.
Source: §VIII · p. 10
Abstract
from arXiv · showhide
This paper provides new results for control of complex flight maneuvers for a quadrotor unmanned aerial vehicle (UAV). The flight maneuvers are defined by a concatenation of flight modes or primitives, each of which is achieved by a nonlinear controller that solves an output tracking problem. A mathematical model of the quadrotor UAV rigid body dynamics, defined on the configuration space $\SE$, is introduced as a basis for the analysis. The quadrotor UAV has four input degrees of freedom, namely the magnitudes of the four rotor thrusts; each flight mode is defined by solving an asymptotic optimal tracking problem. Although many flight modes can be studied, we focus on three output tracking problems, namely (1) outputs given by the vehicle attitude, (2) outputs given by the three position variables for the vehicle center of mass, and (3) output given by the three velocity variables for the vehicle center of mass. A nonlinear tracking controller is developed on the special Euclidean group $\SE$ for each flight mode, and the closed loop is shown to have desirable closed loop properties that are almost global in each case. Several numerical examples, including one example in which the quadrotor recovers from being initially upside down and another example that includes switching and transitions between different flight modes, illustrate the versatility and generality of the proposed approach.
I. INTRODUCTION
The paper addresses the difficulty of controlling complex quadrotor maneuvers with approaches based on Euler angles, which become singular during complex rotations. It develops coordinate-free geometric controllers on SE(3) for three tracking modes that can be combined into complex maneuvers.
I. INTRODUCTION
- Quadrotors use counter-rotating rotor pairs on a simple frame and have been developed for applications including surveillance and mobile sensor networks.The introduction also mentions educational projects and commercial products.Source: §I · p. 1
- Earlier quadrotor controllers based on Euler angles exhibit singularities during complex rotations, restricting their ability to achieve complex flight maneuvers.The introduction also notes that linear controllers are widely used to improve stability around an equilibrium.Source: §I · p. 1
- Geometric methods define controllers on the quadrotor’s SE(3) configuration manifold and avoid singularities and complexities from local coordinates.The proposed controllers achieve almost global asymptotic tracking of attitude, center-of-mass position, or center-of-mass velocity.Source: §I · p. 1
- The paper combines flight modes through a hybrid control architecture, with each mode tracking its specified outputs.A complex maneuver can concatenate modes such as hovering, reorientation, and time-varying position tracking.Source: §I · p. 1
II. QUADROTOR DYNAMICS MODEL
The model represents quadrotor configuration on SE(3), combining center-of-mass position with attitude. It treats rotor thrusts as directly controlled inputs and derives total thrust and moment inputs from the four rotor outputs.
II. QUADROTOR DYNAMICS MODEL
- Four identical rotors at the vertices of a square frame generate thrust and torque normal to the rotor plane.The body-fixed frame is centered at the vehicle’s center of mass, with its third axis normal to the rotor plane.Source: §II · pp. 1–2
- The quadrotor configuration manifold is SE(3), the semidirect product of center-of-mass position in R^3 and attitude in SO(3).SO(3) consists of rotation matrices R with R^T R = I and det R = 1.Source: §II · p. 2
- The model assumes directly controlled rotor thrusts, thrust perpendicular to the rotor plane, and torque proportional to each rotor’s thrust.Rotor and propeller dynamics are not included, though the control system can be extended to include linear rotor dynamics.Source: §II · p. 2
- For nonzero rotor distance d and torque coefficient cτf, the rotor thrusts can be recovered from total thrust f and moment M, which are treated as control inputs.The stated 4 × 4 mapping is invertible under those conditions.Source: §II · p. 2
III. GEOMETRIC TRACKING CONTROL OF A QUADROTOR UAV
Because four control inputs cannot asymptotically track both the quadrotor’s attitude and position, the paper assigns distinct outputs to separate flight modes. Complex maneuvers are formed by switching among those modes.
III. GEOMETRIC TRACKING CONTROL OF A QUADROTOR UAV
- Four inputs allow asymptotic tracking of at most four outputs, so the quadrotor cannot track both attitude and position simultaneously.This motivates flight modes defined by separate specified output sets.Source: §III · p. 2
- The attitude, position, and velocity modes respectively track vehicle attitude, center-of-mass position, and center-of-mass velocity.Each mode is associated with its own nonlinear controller.Source: §III · pp. 2–3
- A complex maneuver concatenates flight modes and specifies switching conditions and commanded outputs as functions of time.The paper describes switching among controllers for maneuvers involving large-angle transitions between modes.Source: §III · p. 3
IV. ATTITUDE CONTROLLED FLIGHT MODE
The attitude-mode controller uses feedback on attitude and angular velocity to track a commanded attitude, with exponential stability on a region covering almost all initial attitudes. A thrust choice can additionally track altitude, although translation remains only partially controlled in this mode.
IV. ATTITUDE CONTROLLED FLIGHT MODE
- The moment-vector controller uses feedback on attitude and angular velocity together with the commanded attitude, angular velocity, and angular acceleration.The attitude command is a smooth function of time.Source: §IV · p. 3
- The attitude tracking error equilibrium is exponentially stable under the stated initial-condition bound for the controller with positive gains.The proposition also provides bounds on the tracking error over time.Source: §IV · p. 3
- Exponential stability is guaranteed for almost all initial attitude errors below 180°, excluding a two-dimensional manifold of 180° errors.The stated region of attraction can be enlarged for angular velocity by choosing a larger attitude gain kR.Source: §IV · p. 4
- Choosing thrust for altitude tracking preserves the attitude-mode conclusions and makes altitude asymptotically track its command.The thrust magnitude is not required for attitude tracking itself.Source: §IV · p. 4
- Because translation is only partially controlled in attitude mode, the paper identifies it as most suitable for short periods in which an attitude maneuver is completed.The translational motion depends on the selected thrust controller.Source: §IV · p. 4
V. POSITION CONTROLLED FLIGHT MODE
The position-controlled mode constructs a desired thrust vector and corresponding attitude to track a commanded position. Its closed loop is exponentially attractive for almost all initial attitude errors, with an additional choice that constrains the asymptotic direction of the first body-fixed axis.
V. POSITION CONTROLLED FLIGHT MODE
- The controller forms a required thrust vector from position and velocity errors, gravity, and commanded acceleration, then aligns the vehicle attitude with that vector.When attitude tracking is exact, the thrust magnitude matches the vector’s norm; it decreases with larger attitude error to improve convergence properties.Source: §V · p. 5
- The position-mode tracking errors are exponentially attractive for almost all initial attitude errors, combining attitude convergence with the tighter position-mode region of attraction.The position-mode stability result requires an initial attitude error below 90°; the attitude controller can first reduce errors outside that region, provided they remain within its own region of attraction.Source: §V · pp. 5–6
- The desired attitude’s third axis sets the thrust direction, while a free choice of its first axis reflects the four inputs available for tracking a three-dimensional position command.The computed attitude is the attitude required for exact position-command tracking, and the vehicle attitude asymptotically converges to it.Source: §V · p. 6
- Projecting a desired first-axis direction onto the plane normal to the desired third axis makes the first body-fixed axis converge to that projection, within a plane set by commanded acceleration.The desired third axis asymptotically follows the direction of gravity minus commanded acceleration.Source: §V · p. 6
- When commanded acceleration is zero, choosing the desired first-axis direction in the horizontal plane makes the first body-fixed axis converge to it, allowing heading specification.The position-mode conclusions remain in force under the stated assumptions.Source: §V · p. 6
VI. VELOCITY CONTROLLED FLIGHT MODE
The velocity-controlled mode uses a nonlinear controller to track commanded velocity, with a computed attitude coordinating thrust and rotational control. Its stability guarantees depend on initial attitude error, and an optional construction specifies the asymptotic direction of the first body-fixed axis.
VI. VELOCITY CONTROLLED FLIGHT MODE
- The controller achieves almost-global asymptotic tracking of the commanded velocity vector.Its thrust magnitude and moment depend on translational velocity and the commanded velocity and acceleration, while a computed attitude aligns thrust with the tracking objective.Source: §VI · pp. 6–7
- With initial attitude error below 90°, the closed-loop tracking errors are exponentially stable.The stated condition applies to the exponential-stability result for the velocity-controlled mode.Source: §VI · p. 7
- The velocity-tracking errors are exponentially attractive under the broader almost-global condition described for this mode.The text relates almost-global exponential attractiveness to Ψ(R(0), Rc(0)) < 2.Source: §VI · p. 7
- A specified first-axis direction makes the first body-fixed axis asymptotically lie in the plane spanned by b1d and ge3 − ˙vd.When ˙vd = 0 and b1d is horizontal, the axis asymptotically converges to b1d.Source: §VI · p. 7
VII. NUMERICAL RESULTS ILLUSTRATING COMPLEX FLIGHT MANEUVERS
Numerical cases demonstrate attitude and position tracking, including recovery from an initially inverted attitude, and a maneuver that switches among flight modes. In the multi-mode case, prescribed rotations and tracking segments demonstrate complex aerobatic flight, while a selected thrust magnitude keeps position close to a target during aggressive attitude motion.
Case (I): Attitude Controlled Flight Mode
- In the attitude-controlled case, attitude and altitude tracking errors converge exponentially despite a large initial tracking error.The maneuver rotates about e3 sinusoidally while changing altitude.Source: §VII · pp. 7–8, Figure 5 · p. 8
Case (II): Position Controlled Flight Mode
- The position-controlled hovering case recovers from an initially upside-down attitude, with position error converging exponentially after the attitude error enters the specified region of attraction.The initial attitude error is 178°, and it enters that region at 0.88 seconds.Source: §VII · p. 8
Case (III): Transition Between Several Flight Modes
- Switching among five flight modes demonstrates a complex maneuver with 720° and 360° rotations and prescribed first-axis directions during tracking segments.The selected thrust magnitude keeps position close to xc during attitude maneuvers, but does not guarantee asymptotic convergence to xc; tests found it better than fixed thrust for keeping position stationary.Source: §VII · pp. 8–9, Figure 7 · p. 9
- During the multi-mode case, velocity error converges exponentially when the initial attitude-error function is below 1.With ||˙vd|| < g, the first body-fixed axis stays close to the plane spanned by e1 and e3.Source: §VII · pp. 8–9
VIII. CONCLUSIONS
The paper presents a global quadrotor model and tracking controllers for three flight modes, formulated on SE(3) to avoid attitude-representation singularities and ambiguities. The reported stability ranges and switched-mode simulations support complex aerobatic maneuvers.
VIII. CONCLUSIONS
- Controllers for three flight modes are formulated intrinsically on SE(3), avoiding Euler-angle singularities and quaternion ambiguities in representing attitude.The paper reports exponential stability for initial attitude errors below 90° and almost-global exponential attractiveness for errors below 180°.Source: §VIII · p. 10
- Switching among the proposed flight-mode controllers demonstrates complex acrobatic maneuvers in numerical examples.Source: §VIII · p. 10
A. Properties of the Hat Map
- The hat map identifies so(3) with R3 using the vector cross product, and its inverse is the vee map.The passage introduces the vee map as the inverse of the hat map.Source: A. Properties of the Hat Map · p. 10
B. Proof of Proposition 1
The proof establishes exponential stability of the attitude tracking error using error dynamics and a Lyapunov argument. It also confines the attitude to a sublevel set where the attitude error function remains positive-definite.
B. Proof of Proposition 1
- The proof derives attitude error dynamics for Ψ, eR, and eΩ from the attitude kinematics and summarizes them in equations (61), (62), and (64).It then uses a Lyapunov candidate to analyze those dynamics.Source: B. Proof of Proposition 1 · p. 10
- Under the stated choice of c2, W2 and M21 and M22 are positive-definite, supporting the Lyapunov stability argument.The proof bounds the Lyapunov function using these matrices.Source: B. Proof of Proposition 1 · p. 11
- The zero equilibrium of the attitude tracking error is exponentially stable, and the attitude error function decreases exponentially while remaining below 2.The proof also shows that the attitude remains in the sublevel set L2.Source: B. Proof of Proposition 1 · p. 11
C. Proof of Proposition 2
The proof reduces altitude tracking to a second-order linear error system and establishes exponential stability for positive gains. The rotational dynamics remain decoupled from translation.
C. Proof of Proposition 2
- Because rotational and translational dynamics are decoupled and the control moment and assumptions match Proposition 1, its conclusions carry over.Source: C. Proof of Proposition 2 · p. 11
- Taking the dot product of the translational dynamics with e3 and substituting the control yields the altitude error dynamics.The resulting equation is a second-order linear system.Source: C. Proof of Proposition 2 · p. 11
- The altitude error system is exponentially stable for positive kx and kv.The proof obtains this second-order linear system from the altitude component of the translational dynamics.Source: C. Proof of Proposition 2 · p. 11
D. Proof of Proposition 3
The proof establishes exponential stability of the complete tracking errors under the proposition’s conditions. It also bounds the attitude and velocity errors used in the stability analysis.
D. Proof of Proposition 3
- The attitude remains in L1, with attitude error less than 90°.This follows from the stated bound on Ψ.Source: D. Proof of Proposition 3 · p. 11
- The velocity tracking error is uniformly bounded, using a bound that makes its derivative negative when its norm exceeds the stated threshold.Source: D. Proof of Proposition 3 · p. 12
- Under condition (32), the zero equilibrium of the complete tracking errors is exponentially stable.The Lyapunov matrices and candidate are positive-definite under conditions (30) and (31).Source: D. Proof of Proposition 3 · p. 12
E. Proof of Proposition 4
The proof shows exponential attractiveness for the tracking errors under the stated initial-condition and proposition assumptions. It also derives the desired attitude-rate expressions and establishes exponential stability results for Propositions 6 and 7.
E. Proof of Proposition 4
- For initial conditions satisfying (34) and (35), the tracking error reaches Proposition 3’s exponential-stability region in finite time and is bounded during the transition.The complete tracking error is then exponentially attractive.Source: E. Proof of Proposition 4 · p. 13
F. Expression for Ωc, ˙Ωc for Proposition 5
- The desired angular velocity and its derivative are obtained from Rc and its first and second time derivatives.The construction defines A and C from tracking errors, gravity, desired acceleration, and the desired heading direction.Source: F. Expression for Ωc, ˙Ωc for… · p. 13
G. Proof of Proposition 6
- Under condition (48), the zero equilibrium of the complete tracking errors is exponentially stable.Condition (47) makes the stated Lyapunov matrices and candidate positive-definite.Source: G. Proof of Proposition 6 · p. 14
H. Proof of Proposition 7
- For initial conditions satisfying (50) and (51), the tracking error reaches Proposition 6’s exponential-stability region with bounded errors and is exponentially attractive.The proof also bounds the velocity error using the stated condition on eR.Source: H. Proof of Proposition 7 · p. 14