FAAS & UC-FAS: Nonlinear Control Framework

• FAAS is a framework that transforms underactuated nonlinear systems into decoupled, fully actuated linear subsystems using smooth diffeomorphic transformations.
• It systematically cancels nonlinearities, enabling direct eigenstructure assignment with precise pole placement and eliminating the need for high-order derivative estimation.
• The UC-FAS extension validates the approach on 6-DOF quadrotors, demonstrating exponential tracking error decay and robust closed-loop stability under actuator constraints.

The Function–Actuator–Aligned Space (FAAS) paradigm, and its generalization as the Unidirectionally-Connected Fully-Actuated-System (UC-FAS), provides a principled framework for transforming underactuated nonlinear dynamical systems—such as quadrotors—into fully actuated, decoupled linear subsystems through systematic coordinate and input transformations. This exact cancellation of nonlinearities in the transformed space enables direct eigenstructure assignment for the closed-loop system, overcoming classical challenges associated with high-order derivative estimation and nonlinear feedback design in the control of multi-degree-of-freedom aerial vehicles (Ren et al., 14 Oct 2025).

1. Foundations: The Fully-Actuated-System (FAS) Approach

The FAS approach aims to recast an underactuated nonlinear system of the form

$$\dot x = f(x) + g_u(x)u,\quad x\in\mathbb{R}^n,\;u\in\mathbb{R}^m,\;m < n$$

into a fully actuated, linearizable normal form via a smooth diffeomorphic transformation $(x, u) \mapsto (z, v)$. In these coordinates, the dynamics can be expressed as

$z^{(m)} = v,$

facilitating pole placement and precise closed-loop eigenstructure assignment. The coordinate transformation is parameterized by smooth, invertible maps $\Phi(x)$ and $U(x)$ such that

$$z = \Phi(x), \quad v = U(x)u + \alpha(x, \dot{x}, \dots, u^{(k)}),$$

leading to the desired chain of linear derivatives. The process typically involves constructing chains of Lie derivatives and defining $v$ to absorb all nonlinearities, resulting in an input-affine, fully actuated model suitable for linear control design (Ren et al., 14 Oct 2025).

2. UC-FAS Model Derivation for 6-DOF Quadrotor Dynamics

2.1 Original Quadrotor Dynamics

• Translational dynamics (in the inertial frame):

$$\begin{aligned} \ddot x &= \frac{T}{m}(\cos \phi \sin \theta \cos \psi + \sin \phi \sin \psi), \ \ddot y &= \frac{T}{m}(\cos \phi \sin \theta \sin \psi - \sin \phi \cos \psi), \ \ddot z &= \frac{T}{m} \cos \phi \cos \theta - g. \end{aligned}$$

• Attitude kinematics:

$\dot\Phi = M(\phi,\theta)\Lambda,$

with $\Phi = [\phi\;\theta\;\psi]^T$ (Euler angles), $\Lambda = [p\;q\;r]^T$ (body rates), and $M(\phi, \theta)$ a configuration-dependent matrix.

• Attitude dynamics:

$\dot\Lambda = J^{-1}(R(\Phi)\Lambda + \tau),$

where $J$ is the inertia matrix and $\tau$ the vector of torque inputs (Ren et al., 14 Oct 2025).

2.2 Input–State Transformations to UC-FAS

• Attitude subsystem: Through differentiation and augmentation, the roll-pitch-yaw dynamics are decoupled into a second-order chain:

$$\ddot\Phi = \bar u, \quad \bar u = [\bar u_1\,\bar u_2\,\bar u_3]^T$$

• Vertical channel: A virtual input

$u_0 = \frac{T}{m}\cos\phi\cos\theta$

yields $\ddot z = u_0 - g$.

• Horizontal position: By repeated differentiation, the $x$ and $y$ axes are brought to fourth-order chains:

$$x^{(4)} = g_x(\cdot) + G_x(\cdot)[\bar u_1\,\bar u_2]^T, \quad y^{(4)} = g_y(\cdot) + G_y(\cdot)[\bar u_1\,\bar u_2]^T$$

where $G_x$, $G_y$ are full-rank under standard pitch/roll constraints.

The aggregated UC-FAS model thus consists of three decoupled chains:

• Vertical ($\ddot z$): second order
• Yaw ($\ddot\psi$): second order
• Horizontal ($x^{(4)}, y^{(4)}$): two fourth-order chains, with unidirectional connections via known nonlinear functions. Provided $|\phi|, |\theta| < \frac{\pi}{2}$ and $u_0 \neq 0$, the system is fully actuated in the transformed coordinates (Ren et al., 14 Oct 2025).

3. Controller Synthesis via Direct Eigenstructure Assignment

3.1 Elimination of Derivative Estimation

Classical FAS controllers require estimates of high-order derivatives (e.g., $\dot u_0$, $\ddot u_0$), demanding observers or differentiators. In the UC-FAS reformulation, all such derivatives manifest only in known feed-forward terms ($g_X$), so no online estimation or observer is needed.

3.2 Chain-by-Chain Pole Placement

Each transformed channel is a linear constant-coefficient ODE:

• Vertical: $\ddot z + a_{0,1}\dot z + a_{0,0}z = 0$
• Yaw: $\ddot\psi + a_{1,1}\dot\psi + a_{1,0}\psi = 0$
• Horizontal: $$X^{(4)} + a_{i,3}X^{(3)} + a_{i,2}X^{(2)} + a_{i,1}X^{(1)} + a_{i,0}X = 0$$, $i=1,2$

Design proceeds by selecting the desired closed-loop eigenvalues, then computing the feedback gains directly via the companion-form approach or via the parametric design lemma from the controllability Gramian. Gains are applied analytically, and since the transformed systems are decoupled chains, no cross-channel interaction limits pole placement (Ren et al., 14 Oct 2025).

3.3 Parametric Design

Selecting a target matrix (diagonal or Jordan form) and leveraging the system's controllability properties, one applies the design lemma in the paper to recover feedback matrices ensuring that the eigenstructure of each decoupled chain matches the desired specification.

4. Closed-loop Stability and Robustness Analysis

Each decoupled chain is a linear time-invariant system with closed-loop characteristic polynomials arbitrarily placed in the open left half-plane. Consequently, every channel is exponentially stable. The full UC-FAS model forms a cascade with feed-through nonlinearities depending only on known state and input signals; under the stated constraints, small-gain and backstepping principles ensure global exponential stability outside singularity manifolds.

No high-order observer is required, guaranteeing that robustness properties directly inherit from linear design margins. A block-diagonal quadratic Lyapunov function can be explicitly constructed to certify stability (Ren et al., 14 Oct 2025).

5. Implementation and Simulation Outcomes

A tracking control experiment demonstrates the UC-FAS approach on a 6-DOF quadrotor following a 3D spiral $(x, y, z)$ and sinusoidal yaw $\psi(t)$ over a 100-second interval. The design uses:

• $z$- and $\psi$-chains with double real poles at $-4$ and $-5$ ($[20, 9]$ gains)
• $x, y$-chains with quadruple negative real poles at $-5,-6,-7,-8$ (gains $[1680, 1066, 251, 26]$)
• Actuator saturations: $T\in[-100,100]$ N, $\tau_{\phi,\theta,\psi}\in[-0.5,0.5]$ N·m

Tracking errors decay exponentially. The 3D trajectory overlays the reference within millimeter precision; yaw tracking is phase-accurate. No overshoot is observed; time histories are smooth. The analytic, observer-free design leads to straightforward tuning and transparent implementation (Ren et al., 14 Oct 2025).

6. Implications, Applications, and Framework Scope

The UC-FAS methodology, as realized in the described 6-DOF quadrotor context, unifies various transformation approaches within the FAS paradigm. It enables exact feedback linearization and decoupling for practical nonlinear robotic and aerial platforms originally deemed underactuated by conventional means. The systematic elimination of high-order derivative estimation and observer requirements distinguishes the approach, providing a standardized pathway for advanced nonlinear control synthesis, rapid tuning, and guaranteed stability under broad actuator and configuration limits (Ren et al., 14 Oct 2025).