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Path Invariance of a Quadrotor System under Cyber Attacks with Theoretical Guarantees
Hamza Mahmood, Usman Ali, Adeel Akhtar
TL;DR
Quadrotor path-following controllers have not combined forward path invariance with adversarial-attack handling. This paper uses transverse feedback linearization to stabilize a path-following manifold under a single attacked rotor, and proves that the manifold is asymptotically stable and forward-invariant.
Problem
Prior path-invariance controllers do not address adversarial attacks, while bounded-distance set invariance does not ensure that a system stays on the desired curve once it reaches it.
Source: §I · p. 1Method
Results
Theorem 1 states that the path-following manifold is asymptotically stable and forward-invariant under the specified adversarial attack.
Source: A. Internal Dynamics · p. 5Takeaways & Limitations
The paper’s guarantee concerns strict following of a desired path with one rotor fully controlled by an adversarial agent.
Source: §I · p. 1, A. Quadrotor under Adversarial Attack · p. 3
Abstract
from arXiv · showhide
This paper presents a path-following controller for a quadrotor system to guarantee safe maneuvers, in terms of forward path invariance, in the presence of cyber-physical attacks. We assume that an adversarial agent can control any one of the rotors through a false data injection (FDI) type of attack. A feedback controller is designed using transverse feedback linearization which guarantees that the system follows a class of smooth curves under FDI attacks. Our proposed controller is computationally efficient, with a closed-form analytical expression, that not only mitigates the effect of bounded malicious signal but also ensures mission success. We provide theoretical guarantees of forward path invariance under FDI attacks with realistic assumptions and demonstrate the effectiveness of our approach through simulation.
I. INTRODUCTION
The paper targets forward-invariant path following under cyber-physical attacks, a setting not addressed by earlier path-following controllers. It proposes a controller for a quadrotor with one attacked rotor and gives theoretical guarantees for the resulting path-following manifold.
I. INTRODUCTION
- Earlier trajectory-tracking controllers may not keep a quadrotor on its trajectory after it reaches it, while prior path-invariance controllers do not consider adversarial attacks.The paper distinguishes path invariance from staying within a bound of a desired point or trajectory.Source: §I · p. 1
- The attack model gives an adversary control of one rotor through a false data-injection signal, while the controller uses the remaining actuators to follow a given curve.The method does not assume an a priori attack model; it uses the instantaneous resulting motor speed.Source: §I · p. 1
- The controller treats path following as stabilization of the path’s maximal control-invariant subset and uses transverse feedback linearization.The attacked system is not differentially flat and has uncontrolled internal dynamics, yet the paper reports strict following of the desired path.Source: §I · p. 1
- The stated contributions include a well-defined vector relative degree, a diffeomorphism to a partial linear system, and asymptotic stability and invariance under an attack on one motor.The paper says the controller is computationally efficient and has a closed-form expression.Source: §I · pp. 1–2
A. Notation
This section establishes notation for vectors, distances, function composition, and Lie derivatives.
A. Notation
- A vector with components x1 through xn is written as x = col(x1, . . . , xn).The point-to-set distance is introduced as the infimum of the distances from the point to elements of the set.Source: A. Notation · p. 2
- For maps h and s, their composition is denoted by s ◦ h.The notation specifies a map from A to C when h maps A to B and s maps B to C.Source: A. Notation · p. 2
- The section denotes Lie derivatives using L_g λ and L_g L_f λ for smooth maps f, g, and λ.The latter is written as the Lie derivative of L_f λ along g.Source: A. Notation · p. 2
II. PRELIMINARIES
The preliminaries introduce regular curves and parameterizations used to describe desired paths.
II. PRELIMINARIES
- A regular parameterized curve σ has a nonzero derivative at every parameter value.The section states that regularity is equivalent to having a unit-speed reparameterization.Source: §II · p. 2
- Every regular curve can be parameterized with unit speed, and the paper assumes this parameterization in later sections.Under that parameterization, the derivative has unit norm everywhere.Source: §II · p. 2
- A curve σ is L-periodic when σ(λ + L) = σ(λ) for all λ; a nonconstant curve periodic for some nonzero L is closed.Source: §II · p. 2
III. MATHEMATICAL MODEL OF A QUADROTOR
The mathematical model uses inertial and body-fixed frames, Euler angles, and a state-space description of quadrotor motion. It relates the four rotor forces to thrust and torque inputs through an invertible linear map.
III. MATHEMATICAL MODEL OF A QUADROTOR
- The model distinguishes a fixed inertial frame from a body-fixed frame attached to the quadrotor’s center of gravity.Orientation is defined relative to the inertial frame.Source: §III · p. 2
- The model represents orientation using ZYX Euler angles for yaw, pitch, and roll, with the stated coordinate representation singular at θ = ±π/2.The domain specified for the angles requires cos θ > 0.Source: §III · p. 2
- The attitude kinematics are written as Φ̇ = Y⁻¹(Φ)Ω, with Ω denoting the body rates.The model also defines position and velocity in the inertial frame.Source: §III · p. 2
- The four rotor forces are nonnegative, twice differentiable, and bounded above, and an invertible linear map relates them to thrust and torque inputs.The input vector consists of combined thrust and three body-axis torques; the state includes attitude, body rates, position, and velocity.Source: §III · pp. 2–3
A. Quadrotor under Adversarial Attack
The model assumes one rotor is fully controlled by an adversary injecting a false-data signal, while the other three inputs remain secure. The attack is smooth and bounded, and its instantaneous value is assumed detectable.
A. Quadrotor under Adversarial Attack
- The adversary completely controls the second motor’s thrust force, while the other three control inputs remain secure.The second rotor is selected without loss of generality as the vulnerable input.Source: A. Quadrotor under Adversarial Attack · p. 3
- The adversarial signal is a smooth, bounded map taking values from zero to a finite upper bound.An attack-detection mechanism is assumed to know its instantaneous value.Source: A. Quadrotor under Adversarial Attack · p. 3
- Under attack on the second motor, the remaining thrust forces can be represented using combined thrust and two torque inputs.The passage identifies these as uf, τq, and τr.Source: A. Quadrotor under Adversarial Attack · p. 3
- The model assumes nonzero combined thrust and excludes Euler-angle gimbal lock; the latter singularity is attributed to local parameterization.The authors describe these conditions as mild and note that gimbal lock is an artifact of Euler angles on SO(3).Source: A. Quadrotor under Adversarial Attack · p. 3
IV. PROBLEM FORMULATION
The problem is to make a quadrotor converge to a specified smooth curve and then remain on it. Because the full state-space lift of the curve may not be invariant, the goal is to stabilize a suitable subset.
IV. PROBLEM FORMULATION
- The desired path is a regular smooth curve with a unit-speed parameterization, and it may be closed or open.A smooth map represents the curve as a zero-level set in the output space.Source: §IV · p. 3
- The path is lifted to a state-space set Γ, and the controller must make the output converge to the curve while rendering Γ or a subset asymptotically stable and forward invariant.The problem is posed for initial conditions near Γ and under the stated model and adversarial signal assumptions.Source: §IV · p. 3
- The full set Γ generally cannot be made forward invariant, so the objective is to stabilize a subset Γ⋆ instead.Approaching the curve in output space is equivalent to the system state approaching Γ.Source: §IV · p. 3
- Path invariance is necessary for set stability, so asymptotic stability of the target set entails forward invariance.The section presents the path-following controller after stating this relationship.Source: §IV · p. 3
V. PATH-FOLLOWING CONTROLLER DESIGN
The controller targets the largest controlled-invariant subset of the path lift and uses transverse feedback linearization to stabilize it. The design establishes local coordinate and relative-degree properties, then applies linear feedback to the resulting partial linear system.
V. PATH-FOLLOWING CONTROLLER DESIGN
- The design stabilizes the largest controlled-invariant subset Γ⋆ of the path lift, called the path-following manifold, to obtain attraction and invariance.The full set Γ generally cannot be stabilized; making Γ⋆ attractive and invariant solves the stated problem.Source: §V · p. 3
- Because the initial virtual output has only two component functions and its decoupling matrix is not full rank, the design adds an output function and delays thrust input with two integrators.The extended inputs include the second derivative of combined thrust and two torque inputs.Source: §V · pp. 3–4
- The virtual output function used to obtain a well-defined relative degree is smooth when its tubular neighborhood around the curve is sufficiently small.The design first selects a function in a tubular neighborhood of the desired path.Source: §V · p. 4
- The extended system has vector relative degree {4, 4, 4} on Γ⋆ where x13 is nonzero, and the attack signal does not appear in the decoupling matrix.The determinant is nonzero under the stated thrust and geometric conditions.Source: §V · p. 4
- Under Assumption 1, the coordinate map T is a local diffeomorphism near each point of Γ⋆, completing coordinates for the two-dimensional internal dynamics.Two additional real-valued functions are chosen to complete the transformation.Source: §V · p. 4
- The transformed system consists of three decoupled integrator chains and two-dimensional internal dynamics, and the ξ-subsystem receives linear feedback.The controller is designed to make the resulting closed-loop linear system Hurwitz; the section also addresses boundedness of the internal state under bounded attacks.Source: §V · p. 5
A. Internal Dynamics
The main result establishes asymptotic stability and forward invariance of the path-following manifold under a bounded adversarial signal. The proof uses transformed coordinates and bounded internal dynamics; the controller also follows a desired speed profile along the path.
A. Internal Dynamics
- With bounded control inputs and Assumption 1, the internal-state derivatives are bounded and µ2 is bounded.The proof invokes bounded body rates, thrust forces, and adversarial signal, and the assumption that the system stays away from singularities.Source: A. Internal Dynamics · p. 5
- The path-following manifold Γ⋆ is asymptotically stable and forward-invariant under the stated attack signal and feedback conditions.The feedback transformation and controllers are specified in the theorem’s assumptions.Source: A. Internal Dynamics · p. 5
- The feedback makes the transformed ξ-subsystem exponentially stable, which establishes stability and forward invariance of Γ⋆.The coordinate transformation is a local diffeomorphism near Γ⋆.Source: A. Internal Dynamics · p. 5
- The controller makes the quadrotor follow a desired speed profile along the path despite bounded adversarial attacks.The path is forward-invariant by construction.Source: A. Internal Dynamics · p. 6
VI. NUMERICAL SIMULATION
The simulation tests a quadrotor with one attacked rotor, bounded thrust inputs, and parametric uncertainty. It reports accurate path following, actuator-respecting inputs, and bounded body rates under a bounded adversarial signal.
VI. NUMERICAL SIMULATION
- The simulation uses a non-closed desired curve and assumes nonnegative rotor thrusts bounded between 0 and 8 N for the secure rotors.The quadrotor has a weight-to-thrust ratio of 1:4.Source: §VI · p. 6
- All plotted inputs remain bounded and well within the actuators’ 8 N limit under the bounded adversarial signal.The signal is plotted against time in Figure 2a.Source: §VI · p. 6
- The bounded adversarial agent is accompanied by bounded linear and angular velocities and bounded body rates in Figure 2b.Source: §VI · p. 6
- Despite parametric uncertainties, the quadrotor follows the given path accurately when one rotor is completely controlled by an adversarial agent.The simulation assumes a bounded attack on rotor two and 10% inertia and 2% mass uncertainty.Source: §VI · p. 6