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Geometric Nonlinear PID Control of a Quadrotor UAV on SE(3)
Farhad Goodarzi, Daewon Lee, Taeyoung Lee
TL;DR
Quadrotor tracking must handle attitude-representation singularities or ambiguity, uncertain coupled dynamics, and limited prior stability analysis. The paper develops nonlinear PID controllers directly on SE(3), including a new integral design, and proves almost-global asymptotic stability with bounded integral terms while demonstrating tracking in simulation.
Problem
Existing quadrotor controllers face attitude singularities or ambiguity, controller complexity, incomplete stability analysis, and simplified dynamic models.
Method
The paper develops nonlinear PID controllers on SE(3) for coupled translational and rotational dynamics, using integral terms that incorporate angular-velocity error and bounded integral action.
Results
The proposed attitude and position tracking systems achieve almost-global asymptotic attractiveness or stability under translational and rotational uncertainties, with uniformly bounded integral terms.
Takeaways & Limitations
The approach provides geometric quadrotor tracking with rigorous stability analysis while avoiding Euler-angle singularities, quaternion unwinding, and discontinuous control concerns.
Abstract
from arXiv · showhide
Nonlinear PID control systems for a quadrotor UAV are proposed to follow an attitude tracking command and a position tracking command. The control systems are developed directly on the special Euclidean group to avoid singularities of minimal attitude representations or ambiguity of quaternions. A new form of integral control terms is proposed to guarantee almost global asymptotic stability when there exist uncertainties in the quadrotor dynamics. A rigorous mathematical proof is given. Numerical example illustrating a complex maneuver, and a preliminary experimental result are provided.
I. INTRODUCTION
The paper addresses quadrotor tracking limitations caused by Euler-angle singularities, quaternion ambiguity, controller complexity, incomplete stability analysis, and simplified dynamics. It proposes nonlinear PID control directly on SE(3) for uncertain, fully coupled quadrotor dynamics.
- I. INTRODUCTION: Euler-angle controllers involve complicated trigonometric expressions and singularities that restrict complex rotational maneuvers.
- I. INTRODUCTION: Quaternion controllers avoid singularities but can represent one attitude by antipodal points, creating ambiguity and possible unwinding.
- I. INTRODUCTION: Prior quadrotor tracking approaches are limited by complex controller structures, missing stability proofs, planar-motion simplifications, or timescale-separation assumptions.
- I. INTRODUCTION: The proposed nonlinear PID controllers track attitude and position commands despite uncertainties in translational and rotational dynamics.
- I. INTRODUCTION: The control system uses the full six-degrees-of-freedom coupled model on SE(3), with Lyapunov analysis and robustness to unstructured uncertainties.
III. ATTITUDE CONTROLLED FLIGHT MODE
The attitude flight mode defines geometric tracking errors on SO(3) and uses a nonlinear PID moment controller without cancelling the angular-velocity cross term. Its integral design yields almost-global asymptotic stability under uncertainty, with local exponential stability.
- A. Attitude Tracking Errors: Attitude tracking is formulated using an error function and error vectors on SO(3), avoiding Euler-angle singularities and quaternion unwinding.
- A. Attitude Tracking Errors: The attitude error function is positive-definite about the desired attitude, while its undesired critical points correspond to 180-degree rotations.
- B. Attitude Tracking Controller: The controller combines proportional, derivative, and integral terms, while avoiding cancellation of the angular-velocity cross term to simplify its structure.
- B. Attitude Tracking Controller: Integrating angular-velocity error alongside attitude error supports exponential stability in the presence of rotational disturbances.
- B. Attitude Tracking Controller: The zero attitude-error equilibrium is almost globally asymptotically stable, with a globally uniformly bounded integral term and local exponential stability.
- B. Attitude Tracking Controller: Attitude tracking does not require specifying thrust magnitude, so this mode is suited to short attitude maneuvers while translation is only partially controlled.
IV. POSITION CONTROLLED FLIGHT MODE
The position-controlled flight mode constructs tracking errors, integral terms, saturation, and a computed attitude so the quadrotor can follow arbitrary smooth position commands.
- The controller accepts an arbitrary smooth position tracking command xd(t) ∈R3 and defines position and velocity tracking errors.
- The saturation function satσ clips scalar inputs to [−σ, σ] and applies element by element to vector inputs.
- The position controller computes Rc(t) ∈ SO(3) and Ωc so attitude dynamics follow the commanded attitude and angular velocity.
- Integral control terms are introduced for both attitude and position tracking dynamics, with bounded-command and uncertainty assumptions.
B. Position Tracking Controller
The position tracking controller combines thrust and moment commands with nonlinear PID terms, while its stability results cover both small and larger initial attitude errors.
- The position-controlled flight mode uses control expressions for thrust magnitude and moment vector.
- The computed attitude aligns the quadrotor thrust axis with the direction required for position tracking, while the moment command drives attitude toward Rc.
- When the initial attitude error is below 90°, Proposition 3 establishes exponential stability under the stated parameter conditions.
- The zero equilibrium of tracking errors is exponentially stable, and the integral terms ei and eI remain uniformly bounded.
- For initial attitude errors with 1 ≤ Ψ(R(0), Rc(0)) < 2, the tracking errors are attractive and converge to zero as t →∞.
C. Direction of the First Body-Fixed Axis
The construction of Rc leaves a one-dimensional choice in its first body-fixed axis, which is used to control the axis’s asymptotic direction.
- Rc is built with a specified third column b3c and a first column b1c chosen orthogonal to it, leaving one degree of freedom.
- Choosing b1c properly constrains the asymptotic direction of the first body-fixed axis.
- The method selects b1c as the normalized projection of a desired direction b1d onto the plane perpendicular to b3c.
V. NUMERICAL EXAMPLE
The numerical example evaluates a complex flipping maneuver under disturbances, comparing control without and with the proposed integral terms, and includes preliminary hardware attitude tracking.
- Numerical setup: The simulation uses specified quadrotor parameters, translational and rotational disturbances, and controller gains for the maneuver.The parameters include J = [0.43, 0.43, 1.02]×10−2 kgm2 and m = 0.755 kg, with disturbances ∆x and ∆R.
- Numerical setup: The desired maneuver combines a 360◦ rotation about the second body-fixed axis with a 90◦ heading change about the vertical e3 axis.The maneuver starts from hover and combines pitching with yawing motion.
- Simulation results: Without integral terms, the simulation exhibits steady-state errors in both attitude tracking and position tracking.These results are shown in Figure 2.
- Simulation results: The proposed integral terms eliminate the steady-state error while maintaining good tracking performance during the flipping maneuver.The resulting controlled maneuver is illustrated in Figure 4.
- Simulation results: The geometric controller uses two flight modes and large regions of attraction to generate complex maneuvers without time-consuming planning efforts.The paper describes this unified maneuver generation as another contribution.
- Experimental result: A preliminary hardware experiment reports good attitude tracking for combined rolling and pitching commands with a 2-second period.Position-tracking experiments were ongoing.
A. Proof of Proposition 2
The proof derives attitude-error dynamics and Lyapunov bounds, establishes stability under specified domains, and rules out undesired equilibria to obtain almost-global convergence.
- Lyapunov analysis: The proof begins by deriving error dynamics and defining a Lyapunov function to establish boundedness conditions for tracking errors.The analysis then evaluates the Lyapunov function and its derivative along controlled solutions.
- Lyapunov analysis: The attitude Lyapunov bounds are analyzed using z2 = [∥eR∥, ∥eΩ∥]T and positive-definite matrix conditions on the domain D2.D2 is defined by Ψ(R, Rd) < ψ2 < 2.
- Stability result: The zero equilibrium is Lyapunov stable, with eR and eΩ converging to zero as t →∞ under the stated matrix conditions.The integral-state equilibrium includes the disturbance-dependent offset ∆R/kI.
- Stability result: The undesired attitude equilibria are unstable because a neighborhood exists where W2 > 0 and its derivative is positive.This instability excludes their stable manifolds from the desired equilibrium’s region of attraction.
- Stability result: The desired attitude equilibrium is almost globally asymptotically stable, since the excluded stable manifolds have measure zero.The result is with respect to eR and eΩ.
- Stability result: Restricting the Lyapunov analysis to D2 yields local exponential stability with respect to eR and eΩ.The conclusion follows from the positive definiteness of M22 under the condition on c2.
B. Proof of Proposition 3
The proof constructs coupled translational and rotational Lyapunov analyses, bounds their interaction within a specified domain, and establishes exponential stability of tracking errors with bounded integral terms.
- Lyapunov analysis: The translational proof derives position and velocity error dynamics, then combines them with rotational stability analysis through a complete-system Lyapunov candidate.The coupling appears through attitude-tracking effects in the translational dynamics.
- Domain condition: The domain D1 ensures the attitude-dependent quantity used in the velocity-error dynamics is well-defined and positive.This follows from the relation between the attitude error function and the cosine of the eigen-axis rotation angle.
- Stability result: The zero tracking-error equilibrium is exponentially stable for ex, ev, eR, and eΩ, while the integral terms ei and eI remain uniformly bounded.The result follows from the positive-definite Lyapunov terms and the imposed integral-term condition.
- Stability bound: The Lyapunov derivative bounds the translational–rotational coupling and is rendered negative definite when the stated matrix conditions hold.The proof uses positive definiteness of the matrices in the bounds for the Lyapunov function and its derivative.
C. Proof of Proposition 4
The proof establishes attractiveness by first showing attitude errors enter the required region in finite time, then proving the remaining translational errors stay bounded until that time.
- Attractiveness: Attitude tracking errors asymptotically decrease to zero and enter the region required for the translational result after a finite time t∗.The proof then applies the preceding proposition to establish attractiveness, while separately bounding z1 before t∗.