Robust Adaptive Output Tracking for Quadrotor Helicopters

Abstract

Quadrotor helicopters are drawing considerable attention both for their mobility and their potential to perform multiple tasks in complete autonomy. Moreover, the numerous limitations characterizing these aircraft, such as their underactuation, make quadrotors ideal testbeds for innovative theoretical approaches to the problem of controlling autonomous mechanical systems. In this chapter, we propose a robust model reference adaptive control architecture and design an autopilot for quadrotors, which guarantees satisfactory output tracking despite uncertainties in the vehicle’s mass, matrix of inertia, and location of the center of mass. The feasibility of our results is supported by a detailed analysis of the quadrotor’s equations of motion. Specifically, considering the vehicle’s equations of motion as a time-varying nonlinear dynamical system and avoiding the common assumption that the vehicle’s Euler angles are small at all times, we prove that the proposed autopilot guarantees satisfactory output tracking and verifies sufficient conditions for a weak form of controllability of the closed-loop system known as strong accessibility. A numerical example illustrates the applicability of the theoretical results presented and clearly shows how the proposed autopilot outperforms in strong wind conditions autopilots designed using a commonly employed proportional-derivative control law and a conventional model reference adaptive control law.

1. Introduction

Quadrotor helicopters, also known as “quadrotors,” are currently employed in diverse scenarios, which range from search and rescue missions to infrastructure inspection, precision agriculture, and wildlife monitoring ([1, Ch. 1], [2, 3]). Employing quadrotors in enclosed industrial environments or in proximity of untrained personnel is still considered as a challenge for the high reliability required to these aircraft. Additional complexity in the use of quadrotors for commercial applications, such as parcel delivery, is that users demand satisfactory trajectory following capabilities without tuning the controller’s gains prior to each mission, whenever the payload is changed.

Autopilots for commercial-off-the-shelf quadrotors are currently designed assuming that the vehicle’s and the payload’s inertial properties are known and constant in time. Moreover, it is assumed that the propulsion system is able to deliver maximum thrust whenever needed. These assumptions considerably simplify the design of control algorithms for quadrotor helicopters, but also undermine these vehicles’ reliability in challenging work conditions, such as in case the propulsion system is partly damaged or the payload is not rigidly attached to the vehicle. For instance, the authors in [4] show that if the payload’s mass and matrix of inertia vary in time, then autopilots for quadrotors designed using classical control techniques, such as the proportional-derivative control, are inadequate to guarantee satisfactory trajectory tracking.

In recent years, numerous authors, such as Bouadi et al. [5]; Dydek et al. [6]; Jafarnejadsani et al. [7]; Loukianov [8]; Mohammadi & Shahri [9]; Zheng et al. [10], to name a few, employed nonlinear robust control techniques, such as sliding mode control, model reference adaptive control (MRAC), adaptive sliding mode control, and ${\mathcal{L}}_1$ adaptive control, to design autopilots for quadrotors that are able to account for inaccurate modeling assumptions and compensate failures in the propulsion system. These autopilots are generally designed assuming perfect knowledge of the location of the quadrotor’s center of mass, supposing that the vehicle’s Euler angles are small at all times, and neglecting the inertial counter-torque. Furthermore, in several cases also the aerodynamic force and the corresponding moment are omitted. Because of these simplifying assumptions, these autopilots are inadequate for aircraft performing aggressive maneuvers, flying in adverse weather conditions, and transporting payloads not rigidly connected to the vehicle’s frame [11]. The vehicle’s guidance system is usually delegated to avoiding obstacles detected by proximity sensors and cameras installed aboard. For details, see the recent works by Faust et al. [12]; Gao & Shen [13]; Lin & Saripalli [14].

In the first part of this chapter, we present the equations of motion of quadrotors and analyze those properties needed to design effective nonlinear robust controls that enable output tracking. Specifically, we present the equations of motion of quadrotors without assuming a priori that the Euler angles are small and without neglecting the inertial counter-torque and the gyroscopic effect. Since the inertial counter-torque cannot be expressed as an algebraic function of the quadrotor’s state and control vectors, we account for this effect as an unmatched time-varying disturbance on the vehicle’s dynamics and hence, we consider the equations of motion of a quadrotor as a nonlinear time-varying dynamical system. Successively, we verify for the first time sufficient conditions for the strong accessibility of quadrotors’ altitude and rotational dynamics; strong accessibility [15] is a weak form of controllability for nonlinear time-varying dynamical systems. As a result of this analysis, we show that a conservative control law for quadrotors must prevent rotations of a ±π/2 angle about either of the two horizontal axes of the body reference frame; otherwise, the vehicle may be uncontrollable.

In the second part of this chapter, we present a robust autopilot for quadrotors, which is based on a version of the e-modification of the MRAC architecture [16]. This autopilot is characterized by numerous unique features. For instance, we assume that the quadrotor’s inertial properties, such as mass, moment of inertia, and location of the center of mass are unknown. Moreover, we assume that the quadrotor’s reference frame is centered at an arbitrary point, which does not necessarily coincide with the vehicle’s center of mass. In addition, we suppose that the coefficients characterizing the aerodynamic force and moment are unknown. In order to reduce the rotational kinematic and dynamic equations to a form that is suitable for MRAC, we employ an output-feedback linearization approach so that the controlled rotational dynamics is captured by a linear dynamical system, whose virtual input is designed using the proposed robust MRAC architecture.

We design the autopilot’s outer loop so that it regulates the vehicle’s position in the inertial frame and the inner loop so that it regulates the vehicle’s attitude. In conventional autopilots for quadrotors, the outer loop regulates the vehicle’s position in the horizontal plane and the inner loop controls the quadrotor’s altitude and orientation. As in conventional architectures, our outer loop defines the reference pitch and roll angles for the inner loop to track. However, our control architecture allows us to verify a priori that the reference pitch and roll angles meet the sufficient conditions for strong accessibility of the quadrotor’s altitude and rotational dynamics. Conventional autopilots’ outer loop may generate large reference pitch and roll angles that are not guaranteed to lay in the vehicle’s reachable set.

The conventional MRAC architecture ([17], Ch. 9) is designed to regulate time-invariant dynamical systems and, for this reason, autopilots for quadrotors based on the classical MRAC are unable to account for time-varying terms in the vehicle’s dynamics, such as the inertial counter-torque. Moreover, autopilots for quadrotors based on the classical MRAC architecture are robust to both matched and parametric uncertainties, but not unmatched uncertainties, such as aerodynamic forces. The autopilots presented in this chapter, instead, are robust to unmatched uncertainties as well and account for the fact that quadrotors are inherently time-varying dynamical systems.

A numerical example illustrates our theoretical framework by designing a control law that allows a quadrotor to follow a circular trajectory, although the vehicle’s inertial properties are unknown, one of the motors is suddenly turned off, the payload is dropped over the course of the mission, and the wind blows at 16 m/s, which is usually considered as a prohibitive velocity for conventional quadrotors. In this example, we clearly show how the proposed robust control algorithm outperforms both the classical proportional-derivative (PD) control and the conventional MRAC. Indeed, it is shown that quadrotors implementing autopilots based on the PD framework crash as soon as one motor is turned off. Moreover, we verify that quadrotors implementing autopilots based on the classical MRAC framework [6] are unable to fly in the presence of wind gusts faster than 6 m/s. Lastly, we show that, to fly in a wind blowing at 16 m/s, our autopilot requires a control effort that is smaller than the one required by a conventional MRAC-based autopilot to fly in a 6 m/s wind.

2. Notation and definitions

In this section, we establish the notation and the definitions used in this chapter. Let $\mathbb{R}$ denote the set of real numbers, $\mathbb{R}$ⁿ the set of n × 1 real column vectors, and $\mathbb{R}$^n × m the set of n × m real matrices. We write 1_n for the n × n identity matrix, 0_n × m for the zero n × m matrix, and A^T for the transpose of the matrix A ∈ $\mathbb{R}$^n × m. Given a = [a₁, a₂, a₃]^T ∈ $\mathbb{R}$³, $a^{\times }\triangleq \left[\begin{array}{lll}0& -a_3& a_2\\ a_3& 0& -a_1\\ -a_2& a_1& 0\end{array}\right]$ denotes the cross product operator. We write ∥ · ∥ for the Euclidean vector norm and ∥ · ∥_F for the Frobenius matrix norm, that is, given B ∈ $\mathbb{R}$^n × m, $\parallel B{\parallel }_{\mathrm{F}}\triangleq {\left[\mathrm{t}r\left(B{B}^{\mathrm{T}}\right)\right]}^{\frac{1}{2}}$. The Fréchet derivative of the continuously differentiable function V : $\mathbb{R}$ⁿ → $\mathbb{R}$ at x is denoted by V^′(x) ≜ ∂V(x)/∂x.

Definition 2.1 ([18], Def. 6.8). The Lie derivative of the continuously differentiable function V : $\mathbb{R}$ⁿ → $\mathbb{R}$ along the vector field f : $\mathbb{R}$ⁿ → $\mathbb{R}$ⁿ is defined as

$L_{f}V\left(x\right)\triangleq V^{\prime }\left(x\right)f\left(x\right),x\in {\mathbb{R}}^n.$E1

The zeroth-order and the higher-order Lie derivatives are, respectively, defined as

$L_f^{0}V\left(x\right)\triangleq V\left(x\right),L_f^{k}V\left(x\right)\triangleq L_f\left(L_f^{k-1}V\left(x\right)\right),x\in {\mathbb{R}}^n,k\ge 1.$E2

Given the continuously differentiable functions f, g : $\mathbb{R}$ⁿ → $\mathbb{R}$ⁿ, the Lie bracket of f(·) and g(·) is defined as

$\mathrm{a}{\mathrm{d}}_{f}g\left(x\right)\triangleq \frac{\partial g\left(x\right)}{\partial x}f\left(x\right)-\frac{\partial f\left(x\right)}{\partial x}g\left(x\right),x\in {\mathbb{R}}^n.$E3

To recall the definition of uniform ultimate boundedness, consider the nonlinear time-varying dynamical system

$\dot{x}\left(t\right)=f\left(t,x\left(t\right)\right),x\left(t_0\right)=x_0,t\ge t_0,$E4

where x(t) ∈ $\mathbb{R}$ⁿ, t ≥ t₀, f : [t₀, ∞) × $\mathbb{R}$ⁿ → $\mathbb{R}$ⁿ is jointly continuous in its arguments, f(t, ·) is locally Lipschitz continuous in x uniformly in t for all t in compact subsets of t ∈ [t₀, ∞), and 0 = f (t, 0), t ≥ t₀.

Definition 2.2 ([19], Def. 4.6). The nonlinear time-varying dynamical system (4) is uniformly ultimately bounded with ultimate bound b > 0 if there exists c > 0 independent of t₀ and for every a ∈ (0, c), there exists T = T (a, c) ≥ 0, independent of t₀, such that if ∥x₀ ∥  ≤ a, then ∥x(t) ∥  ≤ b, t ≥ t₀ + T.

3. Robust MRAC for output tracking

In order to enable robust output tracking, in this section we present a robust nonlinear control law that is based on the e-modification of the conventional model reference adaptive control [16]. This control law guarantees that after a finite-time transient, the plant’s measured output tracks a given reference signal within some bounded error despite model uncertainties and external disturbances. In practice, the proposed controller guarantees uniform ultimate boundedness of the output tracking error.

Consider the nonlinear time-varying plant and the plant sensors’ dynamics

${\dot{x}}_{\mathrm{p}}\left(t\right)=A_{\mathrm{p}}{x}_{\mathrm{p}}\left(t\right)+B_{\mathrm{p}}\mathrm{\Lambda }\left[u\left(t\right)+{\mathrm{\Theta }}^{\mathrm{T}}\mathrm{\Phi }\left(x_{\mathrm{p}}\left(t\right)\right)\right]+\widehat{\xi }\left(t\right),x_{\mathrm{p}}\left(t_0\right)=x_{\mathrm{p},0},t\ge t_0,$E5

$\dot{y}\left(t\right)=\epsilon C_{\mathrm{p}}{x}_{\mathrm{p}}\left(t\right)-\mathit{\epsilon y}\left(t\right),y\left(t_0\right)=C_{\mathrm{p}}{x}_{\mathrm{p},0},$E6

where $x_{\mathrm{p}}\left(t\right)\in {\mathcal{D}}_{\mathrm{p}}\subseteq {\mathbb{R}}^{n_{\mathrm{p}}}$, t ≥ t₀, denotes the plant’s trajectory, $0_{n_{\mathrm{p}}\times 1}\in {\mathcal{D}}_{\mathrm{p}}$, u(t) ∈ $\mathbb{R}$^m denotes the control input, y(t) ∈ $\mathbb{R}$^m denotes the measured output, ε > 0, A_p ∈ $\mathbb{R}$^{n_p × n_p} is unknown, B_p ∈ $\mathbb{R}$^n_p × m, C_p ∈ $\mathbb{R}$^m × n_p, Λ ∈ $\mathbb{R}$^m × m is diagonal, positive-definite, and unknown, Θ ∈ $\mathbb{R}$^N × m is unknown, the regressor vector Φ : $\mathbb{R}$^n_P → $\mathbb{R}$^N is Lipschitz continuous in its argument, and $\widehat{\xi }:\left[t_0,\infty \right)\to {\mathbb{R}}^{n_{\mathrm{p}}}$ is continuous in its argument and unknown. We assume that $\parallel \widehat{\xi }\left(t\right)\parallel \le {\widehat{\xi }}_{\mathrm{m}\mathit{ax}}$, t ≥ t₀, and Λ is such that the pair (A_p, B_pΛ) is controllable and Λ_min1_m ≤ Λ, for some Λ_min > 0. Both Λ and Θ^TΦ(x_p), $x_{\mathrm{p}}\in {\mathcal{D}}_{\mathrm{p}}$, capture the plant’s matched and parametric uncertainties, such as malfunctions in the control system; the term $\widehat{\xi }\left(·\right)$ captures the plant’s unmatched uncertainties, such as external disturbances.

Eq. (6) models the plant sensors as linear dynamical systems, whose uncontrolled dynamics is exponentially stable and characterized by the parameter ε > 0 ([20], Ch. 2). Given the reference signal y_cmd : [t₀, ∞) → $\mathbb{R}$^m, which is continuous with its first derivative, define $y_{\mathrm{cmd},2}\left(t\right)\triangleq {\dot{y}}_{\mathrm{cmd}}\left(t\right)$, t ≥ t₀, and assume that both y_cmd(·) and y_cmd, 2(·) are bounded, that is, ‖y_cmd(t)‖ ≤ y_max, 1, t ≥ t₀, and ‖y_cmd, 2(t)‖ ≤ y_max, 2, for some y_max, 1, y_max, 2 > 0.

The following theorem provides a robust MRAC for the nonlinear time-varying dynamical system (5) and (6) such that the measured output y(·) is able to eventually track the reference signal y_cmd(·) with bounded error, that is, there exist b > 0 and c > 0 independent of t₀, and for every a ∈ (0, c), there exists a finite-time T = T(a, c) ≥ 0, independent of t₀, such that if ∥y(t₀) − y_cmd(t₀) ∥  ≤ a, then

$\parallel y\left(t\right)-y_{\mathrm{cmd}}\left(t\right)\parallel \le b,t\ge t_0+T.$E7

For the statement of this result, let n ≜ n_p + m and $x\left(t\right)\triangleq {\left[x_{\mathrm{p}}^{\mathrm{T}}\left(t\right),{\left[y\left(t\right)-y_{\mathrm{cmd}}\left(t\right)\right]}^{\mathrm{T}}\right]}^{\mathrm{T}}\in {\mathbb{R}}^n$, t ≥ t₀, note that (5) and (6) are equivalent to

$\dot{x}\left(t\right)=\mathit{Ax}\left(t\right)+B\mathrm{\Lambda }\left[u\left(t\right)+{\mathrm{\Theta }}^{\mathrm{T}}\mathrm{\Phi }\left(x_{\mathrm{p}}\left(t\right)\right)\right]+\xi \left(t\right),x\left(t_0\right)=\left[\begin{array}{l}{x}_{\mathrm{p},0}\\ C_{\mathrm{p}}{x}_{\mathrm{p},0}-y_{\mathrm{cmd}}\left(t_0\right)\end{array}\right],t\ge t_0,$E8

where $x\left(t\right)\in \mathcal{D}\hspace{0.17em}\subseteq \hspace{0.17em}{\mathbb{R}}^n$, $\mathcal{D}\triangleq {\mathcal{D}}_{\mathrm{p}}\times {\mathbb{R}}^m$, $A\triangleq \left[\begin{array}{ll}{A}_{\mathrm{p}}& 0_{n_{\mathrm{p}}\times m}\\ \epsilon C_{\mathrm{p}}& -\epsilon {\mathbf{1}}_m\end{array}\right]$, $B\triangleq \left[\begin{array}{l}{B}_{\mathrm{p}}\\ 0_{m\times m}\end{array}\right]$, $B_1\triangleq \left[\begin{array}{l}{0}_{n_{\mathrm{p}}\times m}\\ -{\mathbf{1}}_m\end{array}\right]$, and $\xi \left(t\right)\triangleq B_1\left[y_{\mathrm{cmd},2}\left(t\right)+\epsilon y_{\mathrm{cmd}}\left(t\right)\right]+\left[\begin{array}{l}{\mathbf{1}}_{n_{\mathrm{p}}}\\ 0_{m\times n_{\mathrm{p}}}\end{array}\right]\widehat{\xi }\left(t\right)$, and consider the reference dynamical model

${\dot{x}}_{\mathrm{r}\mathit{ef}}\left(t\right)=A_{\mathrm{r}\mathit{ef}}{x}_{\mathrm{r}\mathit{ef}}\left(t\right)+B_{\mathrm{r}\mathit{ef}}{y}_{\mathrm{cmd}}\left(t\right),x_{\mathrm{r}\mathit{ef}}\left(t_0\right)=\left[\begin{array}{l}{x}_{\mathrm{p},0}\\ C_{\mathrm{p}}{x}_{\mathrm{p},0}-y_{\mathrm{cmd}}\left(t_0\right)\end{array}\right],t\ge t_0,$E9

where $A_{\mathrm{ref}}=\left[\begin{array}{ll}{A}_{\mathrm{r}\mathit{ef},1}& 0_{n_{\mathrm{p}}\times m}\\ 0_{m\times n_{\mathrm{p}}}& A_{\mathrm{ref},2}\end{array}\right]$, A_ref, 1 ∈ ${\mathbb{R}}^{n_p\times n_p}$ is Hurwitz, A_ref, 2 ∈ $\mathbb{R}$^m × m is Hurwitz, and B_ref ∈ $\mathbb{R}$^n × m is such that the pair (A_ref, B_ref) is controllable.

Theorem 3.1 Consider the nonlinear time-varying dynamical system given by Eqs. (5) and (6), the augmented dynamical system (8), and the linear time-invariant reference dynamical model (9). Define e(t) ≜ x(t) − x_ref(t), t ≥ t₀, and let

$\gamma \left(t,x_{\mathrm{p}},x\right)={\widehat{K}}_x^{\mathrm{T}}\left(t\right)x+{\widehat{\mathrm{K}}}_{\mathrm{cmd}}^{\mathrm{T}}\left(t\right){\mathrm{y}}_{\mathrm{cmd}}\left(t\right)-{\widehat{\mathrm{\Theta }}}^{\mathrm{T}}\left(t\right)\mathrm{\Phi }\left(x_{\mathrm{p}}\right),\left(t,x_{\mathrm{p}},x\right)\in \left[t_0,\infty \right)\times {\mathcal{D}}_{\mathrm{p}}\times \mathcal{D},$E10

where

${\dot{\widehat{K}}}_x\left(t\right)=-{\mathrm{\Gamma }}_x\left[x\left(t\right)e^{\mathrm{T}}\left(t\right)\mathit{PB}+{\sigma }_1‖B^{\mathrm{T}}\mathit{Pe}\left(t\right)‖{\widehat{K}}_x\left(t\right)\right],{\widehat{K}}_x\left(t_0\right)=0_{n\times m},t\ge t_0,$E11

${\dot{\widehat{K}}}_{\mathrm{cmd}}\left(t\right)=-{\mathrm{\Gamma }}_{\mathrm{cmd}}\left[y_{\mathrm{cmd}}\left(t\right)e^{\mathrm{T}}\left(t\right)\mathit{PB}+{\sigma }_2‖B^{\mathrm{T}}\mathit{Pe}\left(t\right)‖{\widehat{K}}_{\mathrm{cmd}}\left(t\right)\right],{\widehat{K}}_{\mathrm{cmd}}\left(t_0\right)=0_{m\times m},$E12

$\dot{\widehat{\mathrm{\Theta }}}\left(t\right)={\mathrm{\Gamma }}_{\mathrm{\Theta }}\left[\mathrm{\Phi }\left(x_{\mathrm{p}}\left(t\right)\right)e^{\mathrm{T}}\left(t\right)\mathit{PB}-{\sigma }_3‖B^{\mathrm{T}}\mathit{Pe}\left(t\right)‖\widehat{\mathrm{\Theta }}\left(t\right)\right],\widehat{\mathrm{\Theta }}\left(t_0\right)=0_{N\times m},$E13

the learning gain matrices Γ_x ∈ $\mathbb{R}$^n × n, Γ_cmd ∈ $\mathbb{R}$^m × m, and Γ_Θ ∈ $\mathbb{R}$^N × N are symmetric positive-definite, P ∈ $\mathbb{R}$^n × n is the symmetric positive-definite solution of the Lyapunov equation

$0=A_{\mathrm{ref}}^{\mathrm{T}}P+P{A}_{\mathrm{ref}}+Q,$E14

Q ∈ $\mathbb{R}$^n × n is symmetric positive-definite, and σ₁, σ₂, σ₃ > 0. If there exists K_x ∈ $\mathbb{R}$^n × m and K_cmd ∈ $\mathbb{R}$^m × m such that

$A_{\mathrm{r}\mathit{ef}}=A+B\mathrm{\Lambda }{K}_x^{\mathrm{T}},$E15

$B_{\mathrm{r}\mathit{ef}}=B\mathrm{\Lambda }{K}_{\mathrm{cmd}}^{\mathrm{T}},$E16

then the nonlinear time-varying dynamical system (8) with u(t) = γ(t, x_p(t), x(t)), t ≥ t₀, is uniformly ultimately bounded and (7) is verified.

Proof: Let $\mathrm{\Delta }{K}_x\triangleq {\widehat{K}}_x-K_x$, $\mathrm{\Delta }{K}_{\mathrm{cmd}}\triangleq {\widehat{K}}_{\mathrm{cmd}}-{\mathrm{K}}_{\mathrm{cmd}}$, and $\mathrm{\Delta \Theta }\triangleq \widehat{\mathrm{\Theta }}-\mathrm{\Theta }$ and consider the Lyapunov function candidate

$V\left(e,\mathrm{\Delta }{K}_x,\mathrm{\Delta }{K}_{\mathrm{cmd}},\mathrm{\Delta \Theta }\right)=e^{\mathrm{T}}\mathit{Pe}+\mathrm{t}r\left(\left[\mathrm{\Delta }{K}_x^{\mathrm{T}}{\mathrm{\Gamma }}_x^{\mathrm{T}}\mathrm{\Delta }{K}_x+\mathrm{\Delta }{K}_{\mathrm{cmd}}^{\mathrm{T}}{\mathrm{\Gamma }}_{\mathrm{cmd}}^{\mathrm{T}}\mathrm{\Delta }{K}_{\mathrm{c}\mathit{md}}+\mathrm{\Delta }{\mathrm{\Theta }}^{\mathrm{T}}{\mathrm{\Gamma }}_{\mathrm{\Theta }}^{\mathrm{T}}\mathrm{\Delta \Theta }\right]\mathrm{\Lambda }\right),\left(e,\mathrm{\Delta }{K}_x,\mathrm{\Delta }{K}_{\mathrm{cmd}},\mathrm{\Delta \Theta }\right)\in {\mathbb{R}}^n\times {\mathbb{R}}^{n\times m}\times {\mathbb{R}}^{m\times m}\times {\mathbb{R}}^{N\times m},$E17

where tr(·) denotes the trace operator. The error dynamics is given by

$\dot{e}\left(t\right)=A_{\mathrm{ref}}e\left(t\right)+B\mathrm{\Lambda }\left[\mathrm{\Delta }{K}_x^{\mathrm{T}}x\left(t\right)+\mathrm{\Delta }{K}_{\mathrm{cmd}}^{\mathrm{T}}{y}_{\mathrm{cmd}}\left(t\right)-\mathrm{\Delta }{\mathrm{\Theta }}^{\mathrm{T}}\mathrm{\Phi }\left(x_{\mathrm{p}}\left(t\right)\right)\right]+\xi \left(t\right),e\left(t_0\right)=0,t\ge t_0,$E18

and using the same arguments as in ([17], pp. 324-325), one can prove that

$\dot{V}\left(e,\mathrm{\Delta }{K}_x,\mathrm{\Delta }{K}_{\mathrm{cmd}},\mathrm{\Delta \Theta }\right)<0\left(e,\mathrm{\Delta }{K}_x,\mathrm{\Delta }{K}_{\mathrm{cmd}},\mathrm{\Delta \Theta }\right)\in \mathrm{\Omega },$E19

for some compact set Ω ⊂ $\mathbb{R}$ⁿ × $\mathbb{R}$^n × m × $\mathbb{R}$^m × m × $\mathbb{R}$^N × m. Thus, it follows from Theorem 4.18 of Khalil [19] that the nonlinear dynamical system given by (18) and (11)–(13) is uniformly ultimately bounded.

Next, let $x_{\mathrm{ref}}\left(t\right)={\left[x_{\mathrm{ref},1}^{\mathrm{T}}\left(t\right),x_{\mathrm{ref},2}^{\mathrm{T}}\left(t\right)\right]}^{\mathrm{T}}$, t ≥ t₀, verify (9), where x_ref, 1(t) ∈ $\mathbb{R}$^n_p and x_ref, 2(t) ∈ $\mathbb{R}$^m. It follows from the uniform ultimate boundedness of (18) that

$‖y\left(t\right)-y_{\mathrm{c}\mathit{md}}\left(t\right)-x_{\mathrm{ref},2}\left(t\right)‖\le \widehat{b},t\ge T+t_0,$E20

for some $\widehat{b}>0$ and T ≥ 0, which are independent of t₀. Moreover, since A_ref is block-diagonal and Hurwitz, B₁ = [0_m × np, −1_m]^T, and y_cmd(·) is bounded, it follows from (9) that x_ref, 2(·) is uniformly bounded ([18], 245), that is, ‖x_ref, 2(t)‖ ≤ b₂, t ≥ t₀, for some b₂ ≥ 0 independent of t₀. Thus, it follows from (20) that (7) is verified with $b=\widehat{b}+b_2$.□

It is important to notice that although the matrix A_p, which characterizes the plant’s uncontrolled linearized dynamics, is unknown and hence, the augmented matrix A is unknown, the structure of the matrix A_p is usually known. Thus, in problems of practical interest it is generally possible to verify the matching conditions (15) and (16), although the matrix A is unknown ([17], Ch. 9).

Remark 3.1 If the adaptive gains ${\widehat{K}}_x\left(·\right)$, ${\widehat{K}}_{\mathrm{cmd}}\left(·\right)$, and $\widehat{\Theta }\left(·\right)$ verify (11)–(13), respectively, with σ₁ = σ₂ = σ₃ = 0, then the control law (10) reduces to the conventional model reference adaptive control law for the augmented dynamical system (8) ([17], p. 298). However, conventional MRAC does not guarantee uniform ultimate boundedness of the closed-loop system in the presence of the matched uncertainty $\widehat{\xi }\left(·\right)$ ([17], pp. 317-319).

4. Modeling assumptions on quadrotors’ dynamics

Let $\mathbb{I}=\left\{O;X,Y,Z\right\}$ denote an orthonormal reference frame fixed with the Earth and centered at some point O, and let $\mathbb{J}=\left\{A;x\left(t\right),y\left(t\right),z\left(t\right)\right\}$, t ≥ t₀, denote an orthonormal reference frame fixed with the quadrotor and centered at some point A, which is arbitrarily chosen. The axes of the reference frames $\mathbb{I}$ and $\mathbb{J}$ form two orthonormal bases of $\mathbb{R}$³ and if a vector a ∈ $\mathbb{R}$³ is expressed in $\mathbb{I}$, then this vector is denoted by $a^{\mathbb{I}}$. Alternatively, if a ∈ $\mathbb{R}$³ is expressed in $\mathbb{J}$, then no superscript is used. In this chapter, we consider the reference frame $\mathbb{I}$ as an inertial reference frame; quadrotors move at subsonic velocities and are usually operated at altitudes considerably lower than 10 kilometers and hence, the error induced by this modeling assumption is negligible ([21], Ch. 5). The Z axis is chosen so that the quadrotor’s weight is given by $F_{\mathrm{g}}^{\mathbb{I}}=m_{\mathrm{Q}}\mathit{gZ}$, where m_Q > 0 denotes the vehicle’s mass and g denotes the gravitational acceleration; the X and Y axes are chosen arbitrarily. The axis z(·) points down and the axis x(·) is aligned to one of the quadrotor’s arms; see Figure 1.

Figure 1.

Schematic representation of a quadrotor helicopter.

The attitude of the reference frame $\mathbb{J}$ with respect to the reference frame $\mathbb{I}$ is captured by the roll, pitch, and yaw angles using a 3-2-1 rotation sequence ([22], Ch. 1). In particular, we denote by ψ : [t₀, ∞) → [0, 2π) the yaw angle and $\phi ,\theta :\left[t_0,\infty \right)\to \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ the roll and pitch angles, respectively. The angular velocity of $\mathbb{J}$ with respect to $\mathbb{I}$ is denoted by ω : [t₀, ∞) → $\mathbb{R}$³ ([22], Def. 1.9). The position of the point A with respect to the origin O of the inertial reference frame $\mathbb{I}$ is denoted by r_A : [t₀, ∞) → $\mathbb{R}$³ and the velocity of A with respect to $\mathbb{I}$ is denoted by v_A : [t₀, ∞) → $\mathbb{R}$³.

The position of the quadrotor’s center of mass C with respect to the reference point A is denoted by r_C ∈ $\mathbb{R}$³. The matrix of inertia of the quadrotor, excluding its propellers, with respect to A is denoted by I ∈ $\mathbb{R}$^3 × 3 and the matrix of inertia of each propeller with respect to A is denoted by I_P ∈ $\mathbb{R}$^3 × 3. The spin rate of the ith propeller is denoted by Ω_P, i : [t₀, ∞) → $\mathbb{R}$, i = 1, …, 4. In this chapter, we model the quadrotor’s frame as a rigid body and propellers as thin disks. Moreover, we assume that the vehicle’s inertial properties, such as the mass m_Q, the inertia matrix I, and the location of the center of mass r_C, are constant, but unknown. The quadrotor’s estimated mass is denoted by ${\widehat{m}}_{\mathrm{Q}}>0$ and the quadrotor’s estimated matrix of inertia with respect to A is given by the symmetric, positive-definite matrix $\widehat{I}\in {\mathbb{R}}^{3\times 3}$.

5. Quadrotors’ equations of motion

In this section, we present the equations of motion of quadrotors. Specifically, a quadrotor’s translational kinematic equation is given by ([22], Ex. 1.12)

${\dot{r}}_A^{\mathbb{I}}\left(t\right)=R\left(\phi \left(t\right),\theta \left(t\right),\psi \left(t\right)\right)v_A\left(t\right),r_A^{\mathbb{I}}\left(t_0\right)=r_{A,0}^{\mathbb{I}},t\ge t_0,$E21

where

$R\left(\phi ,\theta ,\psi \right)\triangleq \left[\begin{array}{lll}cos\psi & -sin\psi & 0\\ sin\psi & cos\psi & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{lll}cos\theta & 0& sin\theta \\ 0& 1& 0\\ -sin\theta & 0& cos\theta \end{array}\right]\left[\begin{array}{lll}1& 0& 0\\ 0& cos\phi & -sin\phi \\ 0& sin\phi & cos\phi \end{array}\right],\left(\phi ,\theta ,\psi \right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times \left[0,2\pi \right),$

and the rotational kinematic equation is given by ([22], Th. 1.7)

$\left[\begin{array}{l}\dot{\phi }\left(t\right)\\ \dot{\theta }\left(t\right)\\ \dot{\psi }\left(t\right)\end{array}\right]=\mathrm{\Gamma }\left(\phi \left(t\right),\theta \left(t\right)\right)\omega \left(t\right),\left[\begin{array}{l}\phi \left(t_0\right)\\ \theta \left(t_0\right)\\ \psi \left(t_0\right)\end{array}\right]=\left[\begin{array}{l}{\phi }_0\\ {\theta }_0\\ {\psi }_0\end{array}\right],$E22

where

$\mathrm{\Gamma }\left(\phi ,\theta \right)\triangleq \left[\begin{array}{lll}1& sin\phi tan\theta & cos\phi tan\theta \\ 0& cos\phi & -sin\phi \\ 0& sin\phi sec\theta & cos\phi sec\theta \end{array}\right],\left(\phi ,\theta \right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times \left(-\frac{\pi }{2},\frac{\pi }{2}\right).$

Under the modeling assumptions outlined in Section 4, a quadrotor’s translational dynamic equation is given by [4]

$F_{\mathrm{T}}\left(t\right)+F_{\mathrm{g}}\left(\phi \left(t\right),\theta \left(t\right)\right)+F\left(v_{\mathrm{A}}\left(t\right)\right)=m_{\mathrm{Q}}\left[{\dot{v}}_A\left(t\right)+{\omega }^{\times }\left(t\right)v_A\left(t\right)+{\dot{\omega }}^{\times }\left(t\right)r_C\right.\left.+{\omega }^{\times }\left(t\right){\omega }^{\times }\left(t\right)r_C\right],v_A\left(t_0\right)=v_{A,0},t\ge t_0,$E23

where F_T(t) = [0, 0, u₁(t)]^T denotes the thrust force, that is, the force produced by the propellers that allows a quadrotor to hover,

$F_{\mathrm{g}}\left(\phi ,\theta \right)=m_{\mathrm{Q}}g{\left[-sin\theta ,cos\theta sin\phi ,cos\theta cos\phi \right]}^{\mathrm{T}},\left(\phi ,\theta \right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times \left(-\frac{\pi }{2},\frac{\pi }{2}\right),$E24

denotes the quadrotor’s weight, and F : $\mathbb{R}$³ → $\mathbb{R}$³ denotes the aerodynamic force acting on the quadrotor [23]. The rotational dynamic equation of a quadrotor is given by [4]

$M_{\mathrm{T}}\left(t\right)+M_{\mathrm{g}}\left(\phi \left(t\right),\theta \left(t\right)\right)+M\left(\omega \left(t\right)\right)=m_{\mathrm{Q}}{r}_C^{\times }\left[{\dot{v}}_A\left(t\right)+{\omega }^{\times }\left(t\right)v_A\left(t\right)\right]+I\dot{\omega }\left(t\right)+{\omega }^{\times }\left(t\right)\mathit{I\omega }\left(t\right)+I_{\mathrm{P}}\sum _{i=1}^4‍{\left[0,0,{\dot{\mathrm{\Omega }}}_{\mathrm{P},i}\left(t\right)\right]}^{\mathrm{T}}+{\omega }^{\times }\left(t\right)I_{\mathrm{P}}\sum _{i=1}^4‍{\left[0,0,{\mathrm{\Omega }}_{\mathrm{P},i}\left(t\right)\right]}^{\mathrm{T}},\omega \left(t_0\right)={\omega }_0,t\ge t_0,$E25

where M_T(t) = [u₂(t), u₃(t), u₄(t)]^T denotes the moment of the forces induced by the propellers, $M_{\mathrm{g}}\left(\phi ,\theta \right)\triangleq r_C^{\times }{F}_{\mathrm{g}}\left(\phi ,\theta \right)$, $\left(\phi ,\theta \right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$, denotes the moment of the quadrotor’s weight with respect to A, and M : $\mathbb{R}$³ → $\mathbb{R}$³ denotes the moment of the aerodynamic force with respect to A. The terms $I_{\mathrm{P}}{\sum }_{i=1}^4‍{\left[0,0,{\dot{\mathrm{\Omega }}}_{P_i}\left(t\right)\right]}^{\mathrm{T}}$, t ≥ t₀, and ${\omega }^{\times }\left(t\right)I_{\mathrm{P}}{\sum }_{i=1}^4‍{\left[0,0,{\mathrm{\Omega }}_{P_i}\left(t\right)\right]}^{\mathrm{T}}$ in (25) are known as inertial counter-torque and gyroscopic effect, respectively. In this chapter, we refer to (21)–(23) and (25) as the equations of motion of a quadrotor helicopter.

We model the aerodynamic force and the moment of the aerodynamic force as

$F\left(v_A\right)=-\parallel v_A\parallel K_F{v}_A,v_A\in {\mathbb{R}}^3,$E26

$M\left(\omega \right)=-\parallel \omega \parallel K_M\omega ,\omega \in {\mathbb{R}}^3,$E27

where K_F, K_M ∈ $\mathbb{R}$^3 × 3 are diagonal, positive-definite, and unknown; for details, refer to [23]. The aerodynamic force (26) is expressed in the reference frame $\mathbb{J}$. The next result allows expressing F(·) in the reference frame $\mathbb{I}$.

Proposition 5.1 Consider the translational kinematic equation (21) and let (26) capture the aerodynamic forces acting on a quadrotor. It holds that

$F^{\mathbb{I}}\left(v_A\right)=-‖{\dot{r}}_A^{\mathbb{I}}‖R\left(\phi ,\theta ,\psi \right)K_F{R}^{\mathrm{T}}\left(\phi ,\theta ,\psi \right){\dot{r}}_A^{\mathbb{I}},\left({\dot{r}}_A,\phi ,\theta ,\psi \right)\in {\mathbb{R}}^3\times \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times \left[0,2\pi \right).$E28

Proof: It follows from (26) that

$F^{\mathbb{I}}\left(v_A\right)=-\parallel v_A\parallel {\left[K_F{v}_A\right]}^{\mathbb{I}}=-\parallel v_A\parallel R\left(\phi ,\theta ,\psi \right)K_F{v}_A$E29

for all $\left(v_A,\phi ,\theta ,\psi \right)\in {\mathbb{R}}^3\times \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times \left[0,2\pi \right)$, and it follows from (21) that

$F^{\mathbb{I}}\left(v_A\right)=-\parallel v_A\parallel R\left(\phi ,\theta ,\psi \right)K_F{R}^{-1}\left(\phi ,\theta ,\psi \right){\dot{r}}_A^{\mathbb{I}}.$E30

Eq. (28) now follows from (30), since R(·,  · , ·) is an orthogonal matrix and hence, per definition, R⁻¹(φ, θ, ψ) = R^T(φ, θ, ψ), $\left(\phi ,\theta ,\psi \right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times \left[0,2\pi \right)$, ([22], Def. A.13) and $\parallel v_A\parallel =‖R^{\mathrm{T}}\left(\phi ,\theta ,\psi \right){\dot{r}}_A‖=\parallel {\dot{r}}_A\parallel$ ([24], p. 132).□

Eq. (26) captures the aerodynamic drag acting on a quadrotor in absence of wind. If the wind velocity $v_W^{\mathbb{I}}:\left[t_0,\infty \right)\to {\mathbb{R}}^3$ is not identically equal to zero, then it follows from (28) that the aerodynamic force is given by

$F^{\mathbb{I}}\left(t,v_A\right)=-‖v_{\mathrm{w}}^{\mathbb{I}}\left(t\right)-{\dot{r}}_A^{\mathbb{I}}‖R\left(\phi ,\theta ,\psi \right)K_F{R}^{\mathrm{T}}\left(\phi ,\theta ,\psi \right)\left[v_{\mathrm{w}}^{\mathbb{I}}\left(t\right)-{\dot{r}}_A^{\mathbb{I}}\right],\left(t,{\dot{r}}_A,\phi ,\theta ,\psi \right)\in \left[t_0,\infty \right)\times {\mathbb{R}}^3\times \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times \left[0,2\pi \right).$E31

It is reasonable to assume that the wind velocity does not affect the moment of the aerodynamic force (27).

A quadrotor’s control vector is given by u(t) = [u₁(t), u₂(t), u₃(t), u₄(t)]^T, t ≥ t₀, that is, the third component of the thrust force F_T(·) and the moment of the forces induced by the propellers M_T(·). One can verify that ([1], Ch. 2)

$\left[\begin{array}{l}{u}_1\left(t\right)\\ u_2\left(t\right)\\ u_3\left(t\right)\\ u_4\left(t\right)\end{array}\right]=\left[\begin{array}{llll}1& 1& 1& 1\\ 0& l& 0& -l\\ l& 0& -l& 0\\ -c_{\mathrm{T}}& c_{\mathrm{T}}& -c_{\mathrm{T}}& c_{\mathrm{T}}\end{array}\right]\left[\begin{array}{l}{T}_1\left(t\right)\\ T_2\left(t\right)\\ T_3\left(t\right)\\ T_4\left(t\right)\end{array}\right],t\ge t_0,$E32

where T_i : [t₀, ∞) → $\mathbb{R}$, i = 1, …, 4, denotes the component of the force produced by the ith propeller along the −z(·) axis of the reference frame $\mathbb{J}$, l > 0 denotes the length of each propeller’s arm, and c_T > 0 denotes each propeller’s drag coefficient.

Remark 5.1 The state vector for the equations of motion of a quadrotor (21)–(23) and (25) is defined as ${\left[r_A^T,v_A^T,\phi ,\theta ,\psi ,{\omega }^T\right]}^T\in {\mathbb{R}}^{12}$ and the inertial counter-torque $I_P{\sum }_{i=1}^4‍{\left[0,0,{\dot{\mathrm{\Omega }}}_{P_i}\left(t\right)\right]}^T$, t ≥ t₀, can be explicitly related through algebraic expressions to neither the state vector x nor the control input u. Thus, the inertial counter-torque must be considered as a time-varying term in a quadrotor’s rotational dynamic equations. Furthermore, it is common practice not to relate the gyroscopic effect ${\omega }^{\times }\left(t\right)I_P{\sum }_{i=1}^4‍{\left[0,0,{\Omega }_{P_i}\left(t\right)\right]}^T$, t ≥ t₀, with the control input u through (32) ([1], Ch. 2). Hence, also the gyroscopic effect must be accounted for as a time-varying term in a quadrotor’s equations of motion. For these reasons, (21)–(23) and (25) are a nonlinear time-varying dynamical system.

6. Proposed control system for quadrotors

In this section, we outline a control strategy for quadrotors and verify that this strategy does not defy the vehicle’s limits given by its controllability and underactuation.

6.1. Proposed control strategy

The configuration of a quadrotor, whose frame is modeled as a rigid body, is uniquely identified by the position in the inertial space of the reference point A, that is, $r_A^{\mathbb{I}}\left(t\right)=\left[r_X\left(t\right),r_Y\left(t\right),\right.$ r_Z(t)]^T, t ≥ t₀, and the Euler angles φ(t), θ(t), and ψ(t). Observing the equations of motion of a quadrotor (21)–(23) and (25), one can show that the four control inputs u₁(·), …, u₄(·) are unable to instantaneously and simultaneously accelerate the six independent generalized coordinates r_X(·), r_Y(·), r_Z(·), φ(·), θ(·), and ψ(·), and hence quadrotors are underactuated mechanical systems ([25], Def. 2.9). However, it follows from (21)–(23) and (25) that the control inputs u₁(·), …, u₄(·) are able to instantaneously and simultaneously accelerate the independent generalized coordinates r_Z(·), φ(·), θ(·), and ψ(·), which uniquely capture the vehicle’s altitude and orientation dynamics.

In practical applications, quadrotors are employed to transport detection devices, such as antennas or cameras, that must be taken to some specific location and pointed in some given direction. For this reason, one usually needs to regulate a quadrotor’s position $r_A^{\mathbb{I}}\left(·\right)$ and yaw angle ψ(·). To meet this goal despite quadrotors’ underactuation, we apply the following control strategy. Let [r_X, ref (t), r_Y, ref (t), r_Z, ref (t)]^T ∈ $\mathbb{R}$³, t ≥ t₀, denote the quadrotor’s reference trajectory, let ψ_ref (t) ∈ [0, 2π) denote the quadrotor’s reference yaw angle, and assume that r_X, ref (·), r_Y, ref (·), r_Z, ref (·), and ψ_ref (·), are continuous with their first two derivatives and bounded with their first derivatives. It follows from Example 1.4 of [22] that (21) and (23) are equivalent to

${\ddot{r}}_A^{\mathbb{I}}\left(t\right)=\left[\begin{array}{l}{u}_X\left(t\right)\\ u_Y\left(t\right)\\ u_Z\left(t\right)\end{array}\right]+m_{\mathrm{Q}}^{-1}{F}^{\mathbb{I}}\left(v_{\mathrm{A}}\left(t\right),\omega \left(t\right)\right)-{\ddot{r}}_C^{\mathbb{I}}\left(t\right)+{\mu }^{\mathbb{I}}\left(t\right),\left[\begin{array}{l}{r}_A^{\mathbb{I}}\left(t_0\right)\\ v_A^{\mathbb{I}}\left(t_0\right)\end{array}\right]=\left[\begin{array}{l}{r}_{A,0}^{\mathbb{I}}\\ v_{A,0}^{\mathbb{I}}\end{array}\right],t\ge t_0,$E33

where [26]

${\mu }^{\mathbb{I}}\left(t\right)\triangleq m_Q^{-1}{u}_1\left(t\right)\left[R(\phi \left(t\right),\theta \left(t\right),\psi \left(t\right))-R({\phi }_{\mathrm{r}\mathit{ef}}\left(t\right),{\theta }_{\mathrm{r}\mathit{ef}}\left(t\right),{\psi }_{\mathrm{r}\mathit{ef}}\left(t\right))\right]Z,$E34

${\phi }_{\mathrm{r}\mathit{ef}}\left(t\right)\triangleq {sin}^{-1}\frac{u_X\left(t\right)sin{\psi }_{\mathrm{r}\mathit{ef}}\left(t\right)-u_Y\left(t\right)cos{\psi }_{\mathrm{r}\mathit{ef}}\left(t\right)}{\sqrt{u_X^2\left(t\right)+u_Y^2\left(t\right)+u_Z^2\left(t\right)}},$E35

${\theta }_{\mathrm{r}\mathit{ef}}\left(t\right)\triangleq {tan}^{-1}\frac{u_X\left(t\right)cos{\psi }_{\mathrm{r}\mathit{ef}}\left(t\right)+u_Y\left(t\right)sin{\psi }_{\mathrm{r}\mathit{ef}}\left(t\right)}{u_Z\left(t\right)}.$E36

Thus, a feedback control law for the virtual control input [u_X(·), u_Y(·), u_Z(·)]^T is designed so that, after a finite-time transient, r_A(·) tracks [r_X, ref(·), r_Y, ref(·), r_Z, ref(·)]^T with bounded error. Furthermore, a feedback control law for the control input [u₂(·), u₃(·), u₄(·)]^T is designed so that, after a finite-time transient, [φ(·), θ(·), ψ(·)]^T tracks [φ_ref(·), θ_ref(·), ψ_ref(·)]^T with bounded error. Since the quadrotor’s mass m_Q is unknown, we compute the component of the quadrotor’s thrust along the z(·) axis of the reference frame $\mathbb{J}$ as

$u_1\left(t\right)={\widehat{m}}_{\mathrm{Q}}\sqrt{u_X^2\left(t\right)+u_Y^2\left(t\right)+u_Z^2\left(t\right)},t\ge t_0.$E37

Figure 2 provides a schematic representation of the proposed control strategy.

Figure 2.

Proposed control scheme for a quadrotor.

Eqs. (35) and (36) constrain the nonlinear dynamical system given by (33), (22), and (25) and enforce its underactuation. Note that (36) is well-defined, since u_Z(t) ≠ 0, t ≥ t₀, is a necessary condition for a quadrotor to fly, and (35) is well-defined since u₁(t) ≠ 0, t ≥ t₀, is a necessary condition to fly and

$\frac{u_X\left(t\right)sin{\psi }_{\mathrm{r}\mathit{ef}}\left(t\right)-u_Y\left(t\right)cos{\psi }_{\mathrm{r}\mathit{ef}}\left(t\right)}{\sqrt{u_X^2\left(t\right)+u_Y^2\left(t\right)+u_Z^2\left(t\right)}}=\sqrt{\frac{u_X^2\left(t\right)+u_Y^2\left(t\right)}{u_X^2\left(t\right)+u_Y^2\left(t\right)+u_Z^2\left(t\right)}}cos\left({\psi }_{\mathrm{r}\mathit{ef}}\left(t\right)+\alpha \left(t\right)\right)\le 1,$

where α(t) ≜ tan⁻¹(u_Y(t)/u_X(t)). Alternative control strategies, which rely on the assumption that the roll and pitch angles are small, are provided by Islam et al. [27]; Kotarski et al. [28]; Liu & Hedrick [29].

6.2. Strong accessibility of quadrotors

In this section, we prove that quadrotors are not controllable whenever the z(·) axis is parallel to the horizontal plane. Specifically, it is well-known that $\theta \left(t\right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$, t ≥ t₀, is a necessary condition for a 3-2-1 rotation sequence to uniquely identify a quadrotor’s orientation in space and guarantee finiteness of the yaw rate for finite angular velocities ([22], p. 19). In the following, we verify for the first time the conditions for quadrotors’ strong accessibility [15], which is a weaker form of controllability for nonlinear dynamical systems [30–32], and prove that if $\phi \left(t^{\ast }\right)=\left|\frac{\pi }{2}\right|$ for some t^∗ ≥ t₀, then the quadrotor may not be controllable, that is, there may not exist a continuous control input that is able to regulate the vehicle’s altitude and orientation dynamics at t = t^∗.

To the authors’ best knowledge, the controllability of quadrotors’ altitude and orientation dynamics has been studied considering simplified models, which assume that the vehicle’s pitch and roll angles are small at all times [5, 33]. Moreover, existing results on the controllability of quadrotors neglect the fact that, as discussed in Remark 5.1, these vehicles are time-varying dynamical systems and rely on sufficient conditions for the controllability of time-invariant dynamical systems [30, 34].

In the following, we recall the notions of reachable set and strong accessibility for the nonlinear time-varying dynamical system

$\dot{x}\left(t\right)=f\left(t,x\left(t\right)\right)+G\left(x\left(t\right)\right)u\left(t\right),x\left(t_0\right)=x_0,t\ge t_0,$E38

where x(t) ∈ $\mathbb{R}$ⁿ, t ≥ t₀, u(t) ∈ $\mathbb{R}$^m is continuous, and both f : [t₀, ∞) × $\mathbb{R}$ⁿ → $\mathbb{R}$ⁿ and G : $\mathbb{R}$ⁿ → $\mathbb{R}$^n × m are continuously differentiable.

Definition 6.1 ([15]). Consider the nonlinear time-varying dynamical system (38), let $\mathcal{M}$ be a real analytic manifold of dimension n, and let y ∈ $\mathcal{M}$ and t₁, t₂ ≥ t₀. The reachable set $\mathcal{R}$(y, t₁, t₂) of (38) from (y, t₁) at t₂ is the set of all states that can be reached at time t₂ by following the solution of (38) with initial condition y, initial time t₁, and some continuous control input u(·). The nonlinear time-varying dynamical system (38) is strongly accessible at y ∈ $\mathcal{M}$ at time t₁ if $\mathcal{R}$(y, t₁, t₂) has a non-empty interior in $\mathcal{M}$ for every t₂ > t₁. The nonlinear time-varying dynamical system (38) is strongly accessible on $\mathcal{M}$ if it is strongly accessible at every y ∈ $\mathcal{M}$ and every t₁ ≥ t₀.

In practice, Definition 6.1 states that if the nonlinear time-varying dynamical system (38) is strongly accessible on $\mathcal{M}$, then for every point in the reachable set of (38), there exists a continuous control input such that the system’s trajectory is contained both in the reachable set and the manifold $\mathcal{M}$ at all times. The next theorem provides sufficient conditions for the strong accessibility of the nonlinear dynamical system (38). For the statement of this result, consider the augmented time-invariant dynamical system

$\dot{\stackrel{~}{x}}\left(t\right)=\tilde{f}\left(\tilde{x}\left(t\right)\right)+\tilde{G}\left(\tilde{x}\left(t\right)\right)u\left(t\right),\tilde{x}\left(0\right)={\left[x_0^{\mathrm{T}},t_0\right]}^{\mathrm{T}},t\ge 0,$E39

where $\tilde{x}\triangleq {\left[x^{\mathrm{T}},t\right]}^{\mathrm{T}}$, $\tilde{f}\left(\tilde{x}\right)\triangleq {\left[f^{\mathrm{T}}\left(x\right),1\right]}^{\mathrm{T}}$, $\tilde{G}\left(\tilde{x}\right)\triangleq {\left[G^{\mathrm{T}}\left(x\right),0_{n\times 1}\right]}^{\mathrm{T}}$, and recall that the controllability matrix of the augmented time-invariant dynamical system (39) is defined as [15]

$\mathcal{C}\left(\tilde{x}\right)\triangleq \left[{\tilde{g}}_1\left(\tilde{x}\right),\dots ,{\tilde{g}}_m\left(\tilde{x}\right),\mathrm{a}{d}_{\tilde{f}}{\tilde{g}}_1\left(\tilde{x}\right),\dots ,\mathrm{a}{d}_{\tilde{f}}{\tilde{g}}_m\left(\tilde{x}\right)\right],\tilde{x}\in {\mathbb{R}}^n\times \left[t_0,\infty \right),$E40

where $\tilde{G}\left(\tilde{x}\right)=\left[{\tilde{g}}_1\left(\tilde{x}\right),\dots {\tilde{g}}_m\left(\tilde{x}\right)\right]$.

Theorem 6.1 ([15]). Consider the nonlinear dynamical system (38). If $\mathit{rank}\mathcal{C}\left(\tilde{x}\right)=n$ for all $\tilde{x}\in \mathcal{M}\times \left[t_0,\infty \right)$, then (38) is strongly accessible.

It follows from (21)–(23) and (25) that a quadrotor’s altitude and orientation are captured by (38) with n = 8, m = 4, $x={\left[r_Z,\phi ,\theta ,\psi ,{\dot{r}}_Z,{\omega }^{\mathrm{T}}\right]}^{\mathrm{T}}$, $f:\left[t_0,\infty \right)\times \mathcal{D}\times {\mathbb{R}}^4\to {\mathbb{R}}^4$, $\mathcal{D}=\left[0,\infty \right)\times \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times$ $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times \left[0,2\pi \right)\times \mathbb{R}\times {\mathbb{R}}^3$, and

$G\left(x\right)=m_{\mathrm{Q}}^{-1}\left[\begin{array}{ll}{0}_{4\times 1}& 0_{4\times 3}\\ cos\phi cos\theta & 0_{1\times 3}\\ 0_{3\times 1}& m_{\mathrm{Q}}\mathrm{\Gamma }\left(\phi ,\theta \right)I^{-1}\end{array}\right];$E41

the explicit expression for f(·, ·) is omitted for brevity. In this case, the controllability matrix $\mathcal{C}\left(·\right)$ of the fully actuated, augmented time-invariant dynamical system (39) is such that

$\mathrm{d}\mathit{et}\mathcal{C}\left(\tilde{x}\right)=\frac{cos\phi }{m_{\mathrm{Q}}^2\mathrm{d}e{t}^{2}I},\tilde{x}\in {\mathbb{R}}^8\times \left[t_0,\infty \right),$E42

and it follows from Theorem 6.1 that $\phi \left(t\right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$, t ≥ t₀, is a sufficient condition to guarantee that a quadrotor’s altitude r_Z(t) and orientation [φ(t), θ(t), ψ(t)]^T can be regulated by some continuous control input u(t), while ${\left[{\dot{r}}_Z\left(t\right),\dot{\phi }\left(t\right),\dot{\theta }\left(t\right),\dot{\psi }\left(t\right)\right]}^{\mathrm{T}}$ remain bounded at all times; in practice, to preserve controllability, a conservative control law for quadrotors must prevent rotations of an angle of ±π/2 about the x(·) axis of the reference frame $\mathbb{J}$. Note that it follows from (35) that ${\phi }_{\mathrm{ref}}\left(t\right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$, t ≥ t₀, and hence the reference roll angle verifies sufficient conditions for strong accessibility of quadrotors’ altitude and orientation dynamics.

7. Nonlinear robust control of quadrotors

In this section, we apply the results presented in Sections 3–6 to design control laws so that a quadrotor can follow a given trajectory with bounded error. Specifically, we design a control law for u(·) so that a quadrotor can track both the given reference trajectory [r_X, ref(t), r_Y, ref(t), r_Z, ref(t)]^T, t ≥ t₀, and the reference yaw angle ψ_ref(t). In practice, we design control laws both for the virtual control input [u_X(t), u_Y(t), u_Z(t)]^T, t ≥ t₀, and the moment of the propellers’ thrust [u₂(t), u₃(t), u₄(t)]^T, so that a quadrotor tracks [r_X, ref(t), r_Y, ref(t), r_Z, ref(t)]^T, the reference roll angle (35), the reference pitch angle (36), and the reference yaw angle ψ_ref(t).

It follows form (33) that if the aerodynamic force is modeled as in (31), then a quadrotor’s translational kinematic and dynamic equations are given by

${\dot{x}}_{\mathrm{p},\mathrm{P}}\left(t\right)=A_{\mathrm{p},\mathrm{P}}{x}_{\mathrm{p},\mathrm{P}}\left(t\right)+B_{\mathrm{p},P}{\mathrm{\Lambda }}_{\mathrm{P}}\left(\left[\begin{array}{l}{u}_X\left(t\right)\\ u_Y\left(t\right)\\ u_Z\left(t\right)\end{array}\right]+{\mathrm{\Theta }}_{\mathrm{P}}^{\mathrm{T}}\mathrm{\Phi }\left(x_{\mathrm{p},\mathrm{P}}\left(t\right)\right)\right)+{\widehat{\xi }}_{\mathrm{P}}\left(t\right),x_{\mathrm{p},\mathrm{P}}\left(t_0\right)={\left[r_{A,0}^{\mathrm{T}},{\left(v_{A,0}^{\mathbb{I}}\right)}^{\mathrm{T}}\right]}^{\mathrm{T}},t\ge t_0,$E43

${\dot{y}}_{\mathrm{P}}\left(t\right)=\epsilon C_{\mathrm{p},\mathrm{P}}{x}_{\mathrm{p},\mathrm{P}}\left(t\right)-\epsilon y_{\mathrm{P}}\left(t\right),y_{\mathrm{P}}\left(t_0\right)=C_{\mathrm{p},\mathrm{P}}{x}_{\mathrm{p},\mathrm{P}}\left(t_0\right),$E44

where $x_{\mathrm{p},\mathrm{P}}\left(t\right)={\left[{\left(r_A^{\mathbb{I}}\left(t\right)\right)}^{\mathrm{T}}{\left({\dot{r}}_A^{\mathbb{I}}\left(t\right)\right)}^{\mathrm{T}}\right]}^{\mathrm{T}}$, t ≥ t₀, $A_{\mathrm{p},\mathrm{P}}=\left[\begin{array}{ll}{0}_{3\times 3}& {\mathbf{1}}_3\\ 0_{3\times 3}& 0_{3\times 3}\end{array}\right]$, $B_{\mathrm{p},\mathrm{P}}=\left[\begin{array}{l}{0}_{3\times 3}\\ {\mathbf{1}}_3\end{array}\right]$, ${\mathrm{\Lambda }}_{\mathrm{P}}=m_{\mathrm{Q}}^{-1}{\mathbf{1}}_3$, C_p, P = [1₃, 0_3 × 3], ${\widehat{\xi }}_{\mathrm{P}}\left(t\right)\in {\mathbb{R}}^6$, $\left[{\mathbf{1}}_3,0_{3\times 3}\right]{\widehat{\xi }}_{\mathrm{P}}\left(t\right)=0_{3\times 3}$, and

$\left[0_{3\times 3},{\mathbf{1}}_3\right]{\widehat{\xi }}_{\mathrm{P}}\left(t\right)=m_{\mathrm{Q}}^{-1}‖v_{\mathrm{W}}^{\mathbb{I}}\left(t\right)-{\dot{r}}_A^{\mathbb{I}}\left(t\right)‖B_{\mathrm{p},\mathrm{P}}R\left(\phi \left(t\right),\theta \left(t\right),\psi \left(t\right)\right)·K_F{R}^{\mathrm{T}}\left(\phi \left(t\right),\theta \left(t\right),\psi \left(t\right)\right)\left[v_{\mathrm{W}}^{\mathbb{I}}\left(t\right)-{\dot{r}}_A^{\mathbb{I}}\left(t\right)\right]+{\mu }^{\mathbb{I}}\left(t\right)-{\ddot{r}}_C^{\mathbb{I}}\left(t\right).$E45

Although ${\mathrm{\Theta }}_{\mathrm{P}}^{\mathrm{T}}\mathrm{\Phi }\left(x_{\mathrm{p},\mathrm{P}}\left(·\right)\right)$ follows neither from (33) nor (28), this nonlinear term has been introduced to account for failures of the control system; in this section, we assume that

$\mathrm{\Phi }\left(z\right)=tanh\mathrm{z},z\in {\mathbb{R}}^n,$E46

which is globally Lipschitz continuous. Since any quadrotor’s velocity and acceleration are bounded, it follows from (45) that also the unmatched uncertainty ${\widehat{\xi }}_{\mathrm{P}}\left(·\right)$ is bounded. Eq. (44) captures the plant sensor’s dynamics ([20], Ch. 2).

It follows from (22) and (25) that a quadrotor’s rotational dynamics is captured by

$\left[\begin{array}{l}\dot{\phi }\left(t\right)\\ \dot{\theta }\left(t\right)\\ \dot{\psi }\left(t\right)\\ \dot{\omega }\left(t\right)\end{array}\right]=f\left(\phi \left(t\right),\theta \left(t\right),\psi \left(t\right),\omega \left(t\right)\right)+\left[\begin{array}{l}{0}_{3\times 3}\\ {\widehat{I}}^{-1}\end{array}\right]\left[\begin{array}{l}{u}_2\left(t\right)\\ u_3\left(t\right)\\ u_4\left(t\right)\end{array}\right]+{\widehat{\xi }}_{\mathrm{A}}\left(t\right),{\left[\phi \left(t_0\right),\theta \left(t_0\right),\psi \left(t_0\right),{\omega }^{\mathrm{T}}\left(t_0\right)\right]}^{\mathrm{T}}={\left[{\phi }_0,{\theta }_0,{\psi }_0,{\omega }_0^{\mathrm{T}}\right]}^{\mathrm{T}},t\ge t_0,$E47

where $f\left(\phi ,\theta ,\psi ,\omega \right)={\left[{\omega }^{\mathrm{T}}{\mathrm{\Gamma }}^{\mathrm{T}}\left(\phi ,\theta \right),{\left(-{\widehat{I}}^{-1}{\omega }^{\times }\widehat{I}\omega \right)}^{\mathrm{T}}\right]}^{\mathrm{T}}$, ${\widehat{\xi }}_{\mathrm{A}}\left(t\right)\in {\mathbb{R}}^6$, $\left[{\mathbf{1}}_3,0_{3\times 3}\right]{\widehat{\xi }}_{\mathrm{A}}\left(t\right)=0_3$,

$\left[0_{3\times 3},{\mathbf{1}}_3\right]{\widehat{\xi }}_{\mathrm{A}}\left(t\right)=I^{-1}{r}_C^{\times }\left[F_{\mathrm{g}}(\phi \left(t\right),\theta \left(t\right))-m_{\mathrm{Q}}{\ddot{r}}_A^{\mathbb{I}}\left(t\right)\right]+I^{-1}M\left(\omega \left(t\right)\right)+\left[{\widehat{I}}^{-1}{\omega }^{\times }\left(t\right)\widehat{I}-I^{-1}{\omega }^{\times }\left(t\right)I\right]\omega \left(t\right)+\left(I-\widehat{I}\right){\left[u_2\left(t\right),u_3\left(t\right),u_4\left(t\right)\right]}^{\mathrm{T}}-I^{-1}{I}_{\mathrm{P}}\sum _{i=1}^4‍\left[\begin{array}{l}0\\ 0\\ {\dot{\mathrm{\Omega }}}_{\mathrm{P},i}\left(t\right)\end{array}\right]-I^{-1}{\omega }^{\times }\left(t\right)I_{\mathrm{P}}\sum _{i=1}^4‍\left[\begin{array}{l}0\\ 0\\ {\mathrm{\Omega }}_{\mathrm{P},i}\left(t\right)\end{array}\right]\mathrm{,}$E48

F_g(·, ·) is given by (24), r_A(·) verifies (43), and M(·) is given by (27). Let $x_{\mathrm{p},\mathrm{A}}={\left[\phi ,\dot{\phi },\theta ,\dot{\theta },\psi ,\dot{\psi }\right]}^{\mathrm{T}}$, $\eta \left(x_{\mathrm{p},\mathrm{A}}\right)={\left[\dot{\phi },\dot{\theta },\dot{\psi }\right]}^{\mathrm{T}}$, $\beta \left(x_{\mathrm{p},\mathrm{A}}\right)={\left[L_f^2\phi ,L_f^2\theta ,L_f^2\psi ,\right]}^{\mathrm{T}}$, and v ∈ $\mathbb{R}$³; the explicit expression of β(·) is omitted for brevity. By proceeding as in Example 6.3 of [35], one can prove that the nonlinear dynamical system (47) is feedback linearizable ([31], Ch. 5). Specifically, (47) with

${\left[u_2,u_3,u_4\right]}^{\mathrm{T}}=\widehat{I}{\mathrm{\Gamma }}^{-1}\left(\phi ,\theta \right)\left[\eta \left(x_{\mathrm{p},\mathrm{A}}\right)-\beta \left(x_{\mathrm{p},\mathrm{A}}\right)+v\right],\left(x_{\mathrm{p},\mathrm{A}},v\right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times \mathbb{R}\times \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times \mathbb{R}\times \left[0,2\pi \right)\times \mathbb{R}\times {\mathbb{R}}^3,$E49

is equivalent to

${\dot{x}}_{\mathrm{p},\mathrm{A}}\left(t\right)=A_{\mathrm{p},\mathrm{A}}{x}_{\mathrm{p},\mathrm{A}}\left(t\right)+B_{\mathrm{p},\mathrm{A}}{\mathrm{\Lambda }}_{\mathrm{A}}\left[v\left(t\right)+{\mathrm{\Theta }}_{\mathrm{A}}^{\mathrm{T}}\mathrm{\Phi }\left(x_{\mathrm{p},\mathrm{A}}\left(t\right)\right)\right]+{\widehat{\xi }}_{\mathrm{A}}\left(t\right),x_{\mathrm{p},\mathrm{A}}\left(t_0\right)={\left[{\phi }_0,{\dot{\phi }}_0,{\theta }_0,{\dot{\theta }}_0,{\psi }_0,{\dot{\psi }}_0\right]}^{\mathrm{T}},t\ge t_0,$E50

${\dot{y}}_{\mathrm{A}}\left(t\right)=\epsilon C_{\mathrm{p},\mathrm{A}}{x}_{\mathrm{p},\mathrm{A}}\left(t\right)-\epsilon y_{\mathrm{A}}\left(t\right),y_{\mathrm{A}}\left(t_0\right)=C_{\mathrm{p},\mathrm{A}}{x}_{\mathrm{p},\mathrm{A}}\left(t_0\right),$E51

where

$A_{\mathrm{p},\mathrm{A}}=\left[\begin{array}{llllll}0& 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 1\end{array}\right],B_{\mathrm{p},\mathrm{A}}=\left[\begin{array}{lll}0& 0& 0\\ 1& 0& 0\\ 0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right],C_{\mathrm{p},\mathrm{A}}={\left[\begin{array}{lll}1& 0& 0\\ 0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]}^{\mathrm{T}},$E52

${\left[{\dot{\phi }}_0,{\dot{\theta }}_0,{\dot{\psi }}_0\right]}^{\mathrm{T}}=\mathrm{\Gamma }\left({\phi }_0,{\theta }_0\right){\omega }_0$, Λ ∈ $\mathbb{R}$^3 × 3 is diagonal positive-definite, and Φ(·) given by (46). Although Λ_A = 1₃ [35], we assume that Λ_A is unknown and accounts for failures of the propulsion system and erroneous modeling assumptions. Similarly, the term ${\mathrm{\Theta }}_{\mathrm{A}}^{\mathrm{T}}\mathrm{\Phi }\left(x_{\mathrm{p},\mathrm{A}}\left(·\right)\right)$ has been introduced to capture matched uncertainties. Since any quadrotor’s angular velocity, angular acceleration, and propeller’s spin rate are bounded, it follows from (48) that also the unmatched uncertainty ${\widehat{\xi }}_{\mathrm{A}}\left(·\right)$ is bounded. Eq. (51) captures the plant sensor’s dynamics ([20], Ch. 2).

The next theorem provides feedback control laws both for [u_X(·), u_Y(·), u_Z(·)]^T and [u₂(·), u₃(·), u₄(·)]^T so that the measured output signal y_P(·) tracks the reference signal

$y_{\mathrm{cmd},\mathrm{P}}\left(t\right)={\left[r_{X,\mathrm{ref}}\left(t\right),r_{Y,\mathrm{ref}}\left(t\right),r_{Z,\mathrm{ref}}\left(t\right)\right]}^{\mathrm{T}},t\ge t_0,$E53

and the measured output signal y_A(·) tracks the reference signal

$y_{\mathrm{cmd},\mathrm{A}}\left(t\right)={\left[{\phi }_{\mathrm{ref}}\left(t\right),{\theta }_{\mathrm{ref}}\left(t\right),{\psi }_{\mathrm{ref}}\left(t\right)\right]}^{\mathrm{T}},$E54

where φ_ref(·) and θ_ref(·) are given by (35) and (36), respectively, with some bounded error despite model uncertainties, external disturbances, and failures of the propulsion system. For the statement of this result, consider both the nonlinear dynamical system given by (43) and (44) and the nonlinear dynamical system given by (50) and (51), and note that these systems are equivalent to (5) and (6) with $x_{\mathrm{p}}={\left[x_{\mathrm{p},\mathrm{P}}^{\mathrm{T}},x_{\mathrm{p},\mathrm{A}}^{\mathrm{T}}\right]}^{\mathrm{T}}$, ${\mathcal{D}}_{\mathrm{p}}={\mathbb{R}}^3\times {\mathbb{R}}^3\times \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times \mathbb{R}\times \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times \mathbb{R}\times \left[0,2\pi \right)\times \mathbb{R}$, u = [u_X, u_Y, u_Z, v^T]^T, $y_{\mathrm{cmd}}={\left[y_{\mathrm{cmd},\mathrm{P}}^{\mathrm{T}},y_{\mathrm{cmd},\mathrm{A}}^{\mathrm{T}}\right]}^{\mathrm{T}}$, $y_{\mathrm{cmd},2}={\left[y_{\mathrm{cmd},2,\mathrm{P}}^{\mathrm{T}},y_{\mathrm{cmd},2,\mathrm{A}}^{\mathrm{T}}\right]}^{\mathrm{T}}$, n_p = 12, m = 6, $\widehat{\xi }={\left[{\widehat{\xi }}_{\mathrm{P}}^{\mathrm{T}},{\widehat{\xi }}_{\mathrm{A}}^{\mathrm{T}}\right]}^{\mathrm{T}}$, and

$A_{\mathrm{p}}=\left[\begin{array}{ll}{A}_{\mathrm{p},\mathrm{P}}& 0_{6\times 6}\\ 0_{6\times 6}& A_{\mathrm{p},\mathrm{A}}\end{array}\right],B_{\mathrm{p}}=\left[\begin{array}{ll}{B}_{\mathrm{p},\mathrm{P}}& 0_{6\times 3}\\ 0_{6\times 3}& B_{\mathrm{p},\mathrm{A}}\end{array}\right],C_{\mathrm{p}}=\left[\begin{array}{ll}{C}_{\mathrm{p},P}& 0_{3\times 6}\\ 0_{3\times 6}& C_{\mathrm{p},\mathrm{A}}\end{array}\right],$

$\mathrm{\Lambda }=\left[\begin{array}{ll}{\mathrm{\Lambda }}_{\mathrm{P}}& 0_{3\times 3}\\ 0_{3\times 3}& {\mathrm{\Lambda }}_{\mathrm{A}}\end{array}\right],\mathrm{\Theta }=\left[\begin{array}{ll}{\mathrm{\Theta }}_{\mathrm{P}}& 0_{6\times 3}\\ 0_{6\times 3}& {\mathrm{\Theta }}_{\mathrm{A}}\end{array}\right],\mathrm{\Phi }\left(x_{\mathrm{p}}\right)=\left[\begin{array}{l}\mathrm{\Phi }\left(x_{\mathrm{p},\mathrm{P}}\right)\\ \mathrm{\Phi }\left(x_{\mathrm{p},\mathrm{A}}\right)\end{array}\right].$

Theorem 7.1 Consider the nonlinear dynamical system given by (43) and (44), the nonlinear dynamical system given by (50) and (51), the reference signals (53) and (54), the augmented dynamical system (8), the reference dynamical model (9), the feedback control law γ(·,  · , ·) given by (10), and the adaptation laws (11)–(13). If there exist K_x ∈ $\mathbb{R}$^18 × 6 and K_cmd ∈ $\mathbb{R}$^6 × 6 such that (15) and (16) and satisfied, then (8) with u = γ(t, x_p, x) is uniformly ultimately bounded. Furthermore, there exist b > 0 and c > 0 independent of t₀, and for every a ∈ (0, c), there exists a finite-time T = T(a, c) ≥ 0, independent of t₀, such that if ∥y_P(t₀) − y_cmd, P(t₀) ∥  ≤ a and ∥y_A(t₀) − y_cmd, A(t₀) ∥  ≤ a, then

$\parallel y_{\mathrm{P}}\left(t\right)-y_{\mathrm{cmd},\mathrm{P}}\left(t\right)\parallel \le b,t\ge t_0+T,$E55

$\parallel y_{\mathrm{A}}\left(t\right)-y_{\mathrm{cmd},\mathrm{A}}\left(t\right)\parallel \le b.$E56

Lastly, the thrust force generated by the quadrotor’s propellers is such that

$u_1\left(t\right)={\widehat{m}}_{\mathrm{Q}}‖{\gamma }_{\mathrm{P}}(t,x_{\mathrm{p}}\left(t\right),x\left(t\right))‖,t\ge t_0,$E57

and the moment of the thrust force generated by the quadrotor’s propellers is given by

$\left[\begin{array}{l}{u}_2\left(t\right)\\ u_3\left(t\right)\\ u_4\left(t\right)\end{array}\right]=\widehat{I}{\mathrm{\Gamma }}^{-1}\left(\phi \left(t\right),\theta \left(t\right)\right)\left[\eta \left(x_{\mathrm{p},\mathrm{A}}\left(t\right)\right)-\beta \left(x_{\mathrm{p},\mathrm{A}}\left(t\right)\right)+{\gamma }_{\mathrm{A}}(t,x_{\mathrm{p}}\left(t\right),x\left(t\right))\right],$E58

where $\gamma \left(t,x_{\mathrm{p}},x\right)={\left[{\gamma }_{\mathrm{P}}^{\mathrm{T}}\left(t,x_{\mathrm{p}},x\right),{\gamma }_{\mathrm{A}}^{\mathrm{T}}(t,x_{\mathrm{p}},x)\right]}^{\mathrm{T}}$, (t, x_p, x) ∈ [t₀, ∞) × $\mathbb{R}$¹² × $\mathbb{R}$¹⁸, γ_P(t, x_p, x) ∈ $\mathbb{R}$³, and γ_A(t, x_p, x) ∈ $\mathbb{R}$³.

Proof: Uniform ultimate boundedness of (8) with u = γ(t, x_p, x) is a direct consequence of Theorem 3.1. Thus, both the nonlinear dynamical system given by (43) and (44) with [u_X, u_Y, u_Z]^T = γ_P(t, x_p, x), $\left(t,x_{\mathrm{p}},x\right)\in \left[t_0,\infty \right)\times {\mathcal{D}}_{\mathrm{p}}\times \mathcal{D}$, and the nonlinear dynamical system given by (50) and (51) with [u₂, u₃, u₄]^T = γ_A(t, x_p, x) are uniformly ultimately bounded. Consequently, it follows from Definition 2.2 that there exist b > 0 and c > 0 independent of t₀, and for every a ∈ (0, c), there exists a finite-time T = T(a, c) ≥ 0, independent of t₀, such that if ∥y_P(t₀) − y_cmd, P(t₀) ∥  ≤ a and ∥y_A(t₀) − y_cmd, A(t₀) ∥  ≤ a, then (55) and (56) are satisfied. Lastly, (57) directly follows from (37), and (58) directly follows from (49).

8. Illustrative numerical example

In this section, we provide a numerical example to illustrate both the applicability and the advantages of the theoretical results presented in this chapter. Specifically, we design a nonlinear robust control algorithm that allows a quadrotor helicopter to follow a circular trajectory, although the vehicle’s inertial properties are unknown, one of the motors is suddenly turned off, the payload is dropped over the course of the mission, and the wind blows at strong velocity.

Consider a quadrotor of mass m_Q = 1 kg and matrix of inertia I = 1₃ kg m², let the propellers be characterized by the matrix of inertia $I_{\mathrm{p}}=\left[\begin{array}{lll}0.025& 0& 0\\ 0& 0.025& 0\\ 0& 0& 0.05\end{array}\right]\mathrm{kg}{\mathrm{m}}^2$, and let that the sensor’s dynamics be characterized by ε = 10. We assume that the vehicle’s mass and matrix of inertia are unknown and estimated to be ${\widehat{m}}_{\mathrm{Q}}=1.25\mathrm{kg}$ and $\widehat{I}=0.8·{\mathbf{1}}_3\mathrm{kg}{\mathrm{m}}^2$, respectively. Moreover, we assume that the aerodynamic force (31) and the aerodynamic moment (27) are characterized by K_F = K_M = 0.01 · 1₃, which we assume unknown, and the wind velocity is given by $v_{\mathrm{W}}^{\mathbb{I}}\left(t\right)={\left[16,0,0\right]}^{\mathrm{T}}\mathrm{m}/\mathrm{s}$, t ≥ t₀; it is worthwhile to note that this wind speed is considered as excessive for quadrotors equipped with conventional autopilots.

Figure 3 shows the quadrotor’s trajectory obtained applying the control laws (57) and (58) to track a circular path of radius 0.3 m at an altitude of 0.75 m despite the fact that the quadrotor’s payload of 0.5 kg is dropped at t ≥ 40 s and one of the motors is turned off at t = 90 s. These results have been obtained by setting σ₁ = σ₂ = σ₃ = 2, Γ_cmd = 100 · 1₆, and Γ_x and Γ_Θ as block-diagonal matrices, whose non-zero blocks are Γ_{x, (1, 1)} = 1000 · 1₉, Γ_{x, (2, 2)} = 2000 · 1₉, Γ_{Θ, (1, 1)} = 200 · 1₉, abd Γ_{Θ, (2, 2)} = 1600 · 1₉.

Figure 3.

Reference trajectory and trajectory followed by the quadrotor implementing the proposed control algorithm. The vehicle is disturbed by some wind constantly blowing at 16 m/s.

Figure 4 shows both the quadrotor’s altitude as function of time and the altitude of an identical quadrotor implementing an autopilot based on the classical PD framework [36] and flying in absence of wind. It is clear how the quadrotor implementing our control algorithm is able to fly at the desired altitude despite the fact that the payload is dropped at t = 40 s and a motor is turned off at t = 90 s. The quadrotor implementing the PD algorithm is unable to reach the desired altitude because of the large error in the vehicle’s mass’ estimate. Moreover, this quadrotor reaches a considerably higher altitude after the payload is dropped and crashes after one of the propellers is turned off.

Figure 4.

Altitude of a quadrotor implementing the proposed control algorithm and altitude of an identical quadrotor implementing an autopilot based on the classical PD control.

The first plot in Figure 5 shows the control inputs (57) and (58). The second plot in Figure 5 shows the control inputs computed using a conventional MRAC framework [6] for a quadrotor tracking the same circular path despite a wind blowing at 6 m/s; numerical simulations show that quadrotors implementing the conventional MRAC framework are unable to fly in the presence of wind gusts faster than 6 m/s. It is clear that our autopilot requires a control effort that is smaller than the effort required by a conventional MRAC-based autopilot to fly in weaker wind.

Figure 5.

Control input for a quadrotor implementing the proposed control algorithm and control input for an identical quadrotor implementing a conventional MRAC-based autopilot.

9. Conclusion

In this chapter, we presented a robust MRAC architecture, which we employed to design autopilots for quadrotor helicopters. The proposed autopilot is the first to account for the fact that quadrotors are nonlinear time-varying dynamical systems, the exact location of the vehicle’s center of mass is usually unknown, and the aircraft reference frame is centered at some point that does not necessarily coincide with the vehicle’s barycenter. Moreover, our autopilot does not rely on the assumption that the Euler angles are small at all times and accounts both for the inertial counter-torque and the gyroscopic effect.

The applicability of our theoretical results has been illustrated by a numerical example and it is clearly shown how the proposed autopilot is able to track a given reference trajectory despite the fact that the payload is dropped during the mission, one of the motors is turned off, and the wind blows at the prohibitive velocity of 16 m/s. It is also shown that quadrotors implementing autopilots based on the classical PD framework crash if one of the propellers stops functioning. Lastly, it is shown that our autopilot requires a control effort that is smaller than the effort required by conventional MRAC-based autopilots to fly in less strong wind.

Acknowledgments

This work was supported in part by the NOAA/Office of Oceanic and Atmospheric Research under NOAA-University of Oklahoma Cooperative Agreement #NA16OAR4320115, U.S. Department of Commerce, the National Science Foundation under Grant no. 1700640, and the National Science Foundation REU under Grant Oklahoma EPSCoR-2017-3. Lastly, the author wishes to thank Mr. Keyvan Mohammadi for performing the numerical simulation presented in Section 8.