Gait Analysis for a Tiltrotor: The Dynamic Invertible Gait

Abstract

A conventional feedback-linearization-based controller, when applied to a tiltrotor (eight inputs), results in extensive changes in tilting angles, which are not expected in practice. To solve this problem, we introduce the novel concept of “UAV gait” to restrict the tilting angles. The gait plan was initially used to solve the control problems in quadruped (four-legged) robots. Applying this approach, accompanied by feedback linearization, to a tiltrotor may give rise to the well-known non-invertible problem in the decoupling matrix. In this study, we explored invertible gait in a tiltrotor, and applied feedback linearization to stabilize the attitude and the altitude. The conditions necessary to achieve a full-rank decoupling matrix were deduced and simplified to near-zero roll and zero pitch. This paper proposes several invertible gaits to conduct an attitude–altitude control test. The accepted gaits within the region of interest were visualized. The simulation was conducted in Simulink, MATLAB. The results show promising responses in stabilizing attitude and altitude.

1. Introduction

In the past decade, tiltrotors have attracted great interest. Tiltrotors are a novel type of quadrotor [1,2,3,4,5,6,7,8], wherein the axes of the propellers tilt, imparting the ability to change the direction of each thrust. Typical control methods to stabilize a tiltrotor include LQR and PID [9,10,11], backstepping and sliding mode [12,13,14,15,16], feedback linearization [17,18,19,20,21,22,23], optimal control [24,25,26], adaptive control [27,28], etc. Feedback linearization [29,30,31] explicitly decouples the nonlinear parts, and enables one to utilize the over-actuated properties. This approach is not only effective for a standard tiltrotor, but also for a tiltrotor with predetermined tilting angles [32].

Given the benefits of feedback linearization, we have achieved capabilities such as tracking with a rapid response for sophisticated references [17,18,22]. However, several potential risks may hinder the applications of this technique. One of these is saturation restriction. This limit refers to the upper bound of a given motor and the non-negative bound of the thrust; negative thrust is usually not acceptable in applications. Some research has focused on avoiding either of the above bounds [33,34,35,36]. Although hitting a boundary does not necessarily result in instability, the corresponding stability criteria for these cases can be hard to trace or generalize [37,38,39].

Another typical issue of feedback linearization when applied in over-actuated systems is the so-called “State Drift” phenomenon, as defined in [40]; the state or input may drift as the simulation continues. In [1,22], the authors defined the optimal control conditions to avert this problem. The extra requirements restrict the freedom of the inputs to some extent. Unfortunately, relevant stability proof has not been provided. An alternative method to avoid state drift is to reduce the number of inputs by predefining some of them [40]. This method is inspired by gait plan, which is widely adopted in quadruped (four-legged) robots [41,42,43,44,45].

The number of inputs (the magnitude of each thrust and the direction of each thrust) of the tiltrotor in this research was eight at most. The number of degrees of freedom was six (attitude and position). The current feedback-linearization-based control strategies used to stabilize this tiltrotor are categorized into two groups.

One seeks to manipulate all eight inputs so that the tiltrotor becomes an over-actuated system with the potential to simultaneously maneuver in all degrees of freedom. The other sacrifices the number of inputs to make it equal to the number of degrees of freedom; a typical means of achieving this is to make the tilting angles of the thrusts at the opposite arms equal.

However, both strategies lead to unrealistic changes in tilting angles. The tilting angles undergo large changes during flight [1,22]; some of these are unexpectedly fast. These requirements are hardly practical; a reasonable tilting angle should be within a small range and not extend beyond a certain period ($2\pi$).

In this research, we prohibited changes in tilting angles during our flight and analyzed each combination of four tilting angles to find those wherein feedback linearization was applicable (i.e., the decoupling matrix was invertible). This means that the number of inputs was four in this research. Furthermore, the accepted region for designing the gait was explored in an attitude–altitude stabilization test.

We compared the control result by completing the same task but with the other controller, which fully utilized all the inputs (over-actuated) of the same tiltrotor with identical parameters.

The rest of this paper is structured as follows. Section 2 introduces the dynamics of the tiltrotor. Then, the developed feedback-linearization-based controller is presented in Section 3. The necessary conditions for an applicable gait are analyzed in Section 4. In the simulation presented in Section 5, the attitude and the altitude of the tiltrotor were stabilized by the controller proposed in Section 3. Section 6 shows the results and the sufficient conditions for an applicable gait in the region of interest. The conclusions and discussions are addressed in Section 7.

2. Dynamics of the Tiltrotor

A tiltrotor is a type of modified quadrotor whose directions of thrust are able to change [1,2,3,4,5,6,7]. Figure 1 illustrates a tiltrotor. Here, the direction of each thrust is shown in the corresponding yellow plane. For details of the kinematics, [1,2,4,5,7] are recommended.

Among the quadrotor dynamics discussed in previous research, the widely accepted model from [1,2,7,22] is constructed by separately analyzing each part (body frame, tilting motor frames, and propeller frames) using Newton’s Law. The controller was developed based on simplified dynamics via Equations (1)–(5).

The position $P={\left[\begin{array}{ccc}X& Y& Z\end{array}\right]}^T$ is given by:

$$\begin{array}{l}\ddot{P}=\left[\begin{array}{c}0\\ 0\\ -g\end{array}\right]+\frac{1}{m}·{}_{}{}^{W}R·F\left(\alpha \right)·\left[\begin{array}{c}{\varpi }_1·\left|{\varpi }_1\right|\\ {\varpi }_2·\left|{\varpi }_2\right|\\ {\varpi }_3·\left|{\varpi }_3\right|\\ {\varpi }_4·\left|{\varpi }_4\right|\end{array}\right]\\ \triangleq \left[\begin{array}{c}0\\ 0\\ -g\end{array}\right]+\frac{1}{m}·{}_{}{}^{W}R·F\left(\alpha \right)·w\end{array}$$

(1)

where $m$ is the total mass, $g$ is the gravitational acceleration, and ${\varpi }_i$, ($i=1,2,3,4$) is the angular velocity of the propeller (${\varpi }_{1,3}<0$, ${\varpi }_{2,4}>0$) with respect to the propeller-fixed frame, $w={\left[w_1 w_2 w_3 w_4\right]}^T$. $w_i={\varpi }_i·\left|{\varpi }_i\right|$, $m$ is the mass of the tilt-rotor. Rotor 1 and Rotor 3 were assumed to rotate clockwise. Rotor 2 and Rotor 4 were assumed to rotate counter-clockwise. ${}_{}{}^{W}R$ is the rotational matrix [46] between the inertial frame and the body-fixed frame (Equation (2)):

$${}_{}{}^{W}R=\left[\begin{array}{ccc}c\theta ·c\psi & s\varphi ·s\theta ·c\psi -c\varphi ·s\psi & c\varphi ·s\theta ·c\psi +s\varphi ·s\psi \\ c\theta ·s\psi & s\varphi ·s\theta ·s\psi +c\varphi ·c\psi & c\varphi ·s\theta ·s\psi -s\varphi ·c\psi \\ -s\theta & s\varphi ·c\theta & c\varphi ·c\theta \end{array}\right],$$

(2)

where $s\Lambda =\sin \left(\Lambda \right)$ and $c\Lambda =\cos \left(\Lambda \right)$. $\varphi$, $\theta$, and $\psi$ are the roll angle, pitch angle, and yaw angle, respectively. The tilting angles $\alpha =\left[\begin{array}{cccc}{\alpha }_1& {\alpha }_2& {\alpha }_3& {\alpha }_4\end{array}\right]$. The positive directions of the tilting angles are defined in Figure 1. $F\left(\alpha \right)$ is defined as:

$$F\left(\alpha \right)=\left[\begin{array}{cccc}0& K_f·s2& 0& -K_f·s4\\ K_f·s1& 0& -K_f·s3& 0\\ -K_f·c1& K_f·c2& -K_f·c3& K_f·c4\end{array}\right]$$

(3)

where $si=\sin \left({\alpha }_i\right)$, $ci=\cos \left({\alpha }_i\right)$, and ($i=1,2,3,4$). $K_f$ ($8.048\times {10}^{-6}\mathrm{N}·{\mathrm{s}}^2/{\mathrm{rad}}^2$) is the coefficient of the thrust.

The angular velocity of the body with respect to its own frame, ${\omega }_B={\left[\begin{array}{ccc}p& q& r\end{array}\right]}^T$, is governed by Newton–Euler Formula as:

$${\dot{\omega }}_B=I_B{}^{-1}·\tau \left(\alpha \right)·w$$

(4)

where $I_B$ is the matrix of moments of inertia, $K_m$ ($2.423\times {10}^{-7}\mathrm{N}·\mathrm{m}·{\mathrm{s}}^2/{\mathrm{rad}}^2$) is the coefficient of the drag, and $L$ is the length of the arm:

$$\tau \left(\alpha \right)=\left[\begin{array}{cccc}0& L·K_f·c2-K_m·s2& 0& -L·K_f·c4+K_m·s4\\ L·K_f·c1+K_m·s1& 0& -L·K_f·c3-K_m·s3& 0\\ L·K_f·s1-K_m·c1& -L·K_f·s2-K_m·c2& L·K_f·s3-K_m·c3& -L·K_f·s4-K_m·c4\end{array}\right].$$

(5)

Thus far, we have determined the dynamics of a tiltrotor. There are several remarks we can make.

Firstly, the tilting angles (${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$) are predetermined before controlling. This indicates that they are constant during flight, i.e.,:

$${\dot{\alpha }}_i\equiv 0,i=1, 2, 3, 4.$$

(6)

In this research, we will determine the proper tilting angles required to support our flight.

Secondly, the relationship [47,48,49] between the angular velocity of the body, ${\omega }_B$, and the attitude rotation matrix (${}_{}{}^{W}R$) is given by:

$${}_{}{}^W\dot{R}={}_{}{}^{W}R·{\widehat{\omega }}_B$$

(7)

where “$\widehat{}$” is the hat operation used to produce the skew matrix, and ${}_{}{}^W\dot{R}$ represents the derivative of rotation matrix.

Our simulator was built based on Equations (1)–(7).

In the controller design process, we approximated the relationship between the angular velocity of the body, ${\omega }_B$, and the attitude angle, ($\varphi ,\theta ,\psi$), by:

$$\left[\begin{array}{c}\dot{\varphi }\\ \dot{\theta }\\ \dot{\psi }\end{array}\right]={\omega }_B.$$

(8)

Instead of further exploiting Equation (7), the controller was designed based on Equations (1)–(6) and (8).

The parameters of this tiltrotor are as follows: $m=0.429 \mathrm{kg}$, $L=0.1785 \mathrm{m}$, $g=9.8 \mathrm{N}/\mathrm{kg}$, and $I_B=\mathrm{diag} \left(\left[2.24\times {10}^{-3}, 2.99\times {10}^{-3}, 4.80\times {10}^{-3}\right]\right) \mathrm{kg}·{\mathrm{m}}^2$.

3. Feedback Linearization and Control

The control scenario consisted of two sections. Firstly, the nonlinear dynamics were dynamically inverted by feedback linearization. Secondly, the linearized system was stabilized based on a third-order PD controller. The rest of this section introduces these strategies.

3.1. Feedback Linearization

The first step in feedback linearization is to select the output. In this research, we chose attitude–altitude, described in Equation (9), as our output, because the choice of position–yaw may introduce a further singular zone [50,51]:

$$\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\\ y_4\end{array}\right]=\left[\begin{array}{c}\varphi \\ \theta \\ \psi \\ Z\end{array}\right].$$

(9)

Calculating the second derivative of Equation (9) yields:

$$\left[\begin{array}{c}{\ddot{y}}_1\\ {\ddot{y}}_2\\ {\ddot{y}}_3\\ {\ddot{y}}_4\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\\ -g\end{array}\right]+{\left[\begin{array}{c}{I}_B{}^{-1}·\tau \left(\alpha \right)\\ \left[\begin{array}{ccc}0& 0& 1\end{array}\right]·\frac{K_f}{m}·{}_{}{}^{W}R·F\left(\alpha \right)\end{array}\right]}^{4\times 4}·w.$$

(10)

Notably, we derive Equation (11) if ${\varpi }_{1,3}<0$, ${\varpi }_{2,4}>0$:

$${\left({\varpi }_i·\left|{\varpi }_i\right|\right)}^{\prime }=2·{\dot{\varpi }}_i·\left|{\varpi }_i\right|.$$

(11)

Differentiating Equation (10) yields:

$$\begin{array}{ll}\hfill \left[\begin{array}{c}{\stackrel{⃛}{y}}_1\\ {\stackrel{⃛}{y}}_2\\ {\stackrel{⃛}{y}}_3\\ {\stackrel{⃛}{y}}_4\end{array}\right]& ={\left[\begin{array}{c}{I}_B{}^{-1}·\tau \left(\alpha \right)\\ \left[\begin{array}{ccc}0& 0& 1\end{array}\right]·\frac{K_f}{m}·{}_{}{}^{W}R·F\left(\alpha \right)·2·\left[\begin{array}{cccc}\left|{\varpi }_1\right|& & & \\ & \left|{\varpi }_2\right|& & \\ & & \left|{\varpi }_3\right|& \\ & & & \left|{\varpi }_4\right|\end{array}\right]\end{array}\right]}^{4\times 4}·\left[\begin{array}{c}{\dot{\varpi }}_1\\ {\dot{\varpi }}_2\\ {\dot{\varpi }}_3\\ {\dot{\varpi }}_4\end{array}\right]\\ & +\left[\begin{array}{ccc}0& 0& 1\end{array}\right]·\frac{K_f}{m}·{}_{}{}^{W}R·{\widehat{\omega }}_B·F\left(\alpha \right)·w·\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]\\ & \triangleq \overline{\Delta }·\left[\begin{array}{c}{\dot{\varpi }}_1\\ {\dot{\varpi }}_2\\ {\dot{\varpi }}_3\\ {\dot{\varpi }}_4\end{array}\right]+Ma\end{array}$$

(12)

where $\overline{\Delta }$ is called the decoupling matrix [32], and ${\left[\begin{array}{cccc}{\dot{\varpi }}_1& {\dot{\varpi }}_2& {\dot{\varpi }}_3& {\dot{\varpi }}_4\end{array}\right]}^T\triangleq U$ is the new input vector.

From Equation (12), we may derive the decoupled relationship in Equation (13), which is compatible with the controller design process:

$$\left[\begin{array}{c}{\dot{\varpi }}_1\\ {\dot{\varpi }}_2\\ {\dot{\varpi }}_3\\ {\dot{\varpi }}_4\end{array}\right]={\overline{\Delta }}^{-1}·\left(\left[\begin{array}{c}{\stackrel{⃛}{y}}_{1d}\\ {\stackrel{⃛}{y}}_{2d}\\ {\stackrel{⃛}{y}}_{3d}\\ {\stackrel{⃛}{y}}_{4d}\end{array}\right]-Ma\right).$$

(13)

Obviously, the necessary condition for receiving Equation (13) is that the decoupling matrix ($\overline{\Delta }$) is invertible. Section 4 deepens this discussion.

Once solving Equation (13), the controller may be applied to this linearized system. In this research, we deployed third-order PD controllers.

3.2. Third-Order PD Controllers

We designed the third-order PD controllers as:

$$\left[\begin{array}{c}{\stackrel{⃛}{y}}_{1d}\\ {\stackrel{⃛}{y}}_{2d}\\ {\stackrel{⃛}{y}}_{3d}\end{array}\right]=\left[\begin{array}{c}{\stackrel{⃛}{y}}_{1r}\\ {\stackrel{⃛}{y}}_{2r}\\ {\stackrel{⃛}{y}}_{3r}\end{array}\right]+K_{P1}{}^{3\times 3}·\left(\left[\begin{array}{c}{\ddot{y}}_{1r}\\ {\ddot{y}}_{2r}\\ {\ddot{y}}_{3r}\end{array}\right]-\left[\begin{array}{c}{\ddot{y}}_1\\ {\ddot{y}}_2\\ {\ddot{y}}_3\end{array}\right]\right)+K_{P2}{}^{3\times 3}·\left(\left[\begin{array}{c}{\dot{y}}_{1r}\\ {\dot{y}}_{2r}\\ {\dot{y}}_{3r}\end{array}\right]-\left[\begin{array}{c}{\dot{y}}_1\\ {\dot{y}}_2\\ {\dot{y}}_3\end{array}\right]\right)+K_{P3}{}^{3\times 3}·\left(\left[\begin{array}{c}{y}_{1r}\\ y_{2r}\\ y_{3r}\end{array}\right]-\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\end{array}\right]\right),$$

(14)

$${\stackrel{⃛}{y}}_{4d}={\stackrel{⃛}{y}}_{4r}+K_{P{Z}_1}·\left({\ddot{y}}_{4r}-{\ddot{y}}_4\right)+K_{P{Z}_2}·\left({\dot{y}}_{4r}-{\dot{y}}_4\right)+K_{P{Z}_3}·\left(y_{4r}-y_4\right),$$

(15)

where $K_{Pi}$ ($i=1,2,3$) is the three-by-three diagonal control coefficient matrix, $K_{P{Z}_i}$ ($i=1,2,3$) is the control coefficient (scalar), $y_j$ ($j=1, 2, 3, 4$) is the state, and $y_{jr}$ ($j=1,2,3,4$) is the reference.

The control parameters in this section are specified as follows: $K_{P1}=K_{P2}=K_{P3}=\mathrm{diag}\left(\left[1,1,1\right]\right)$, $K_{P{Z}_1}=10$, $K_{P{Z}_2}=5$, and $K_{P{Z}_3}=10$.

4. Applicable Gait (Necessary Conditions)

As discussed, the necessary condition for applying this control is that the decoupling matrix ($\overline{\Delta }$) is invertible. In this section, we find the equivalent conditions to derive an invertible decoupling matrix.

Notice the relationship:

$$\overline{\Delta }~\left[\begin{array}{c}\tau \left(\alpha \right)\\ \left[\begin{array}{ccc}0& 0& 1\end{array}\right]·{}_{}{}^{W}R·F\left(\alpha \right)\end{array}\right]$$

(16)

where “$A~B$” indicates that Matrix $A$ is equivalent to Matrix $B$. Two matrices are called equivalent if, and only if, there exist invertible matrices $P$ and $Q$, so that $A=P·B·Q$.

The following propositions and proofs are specifically applicable only to the tiltrotor with our chosen control parameters and coefficients.

Proposition 1.

The decoupling matrix is invertible if, and only if:

$$\begin{array}{l}1.000·c1·c2·c3·s4·s\theta -1.000·c1·s2·c3·c4·s\theta \\ -2.880·c1·c2·s3·s4·s\theta +2.880·c1·s2·s3·c4·s\theta \\ -2.880·s1·c2·c3·s4·s\theta +2.880·s1·s2·c3·c4·s\theta \\ -1.000·s1·c2·s3·s4·s\theta +1.000·s1·s2·s3·c4·s\theta \\ +4.000·c1·c2·c3·c4·c\varphi ·c\theta +5.592·c1·c2·c3·s4\\ ·c\varphi ·c\theta -5.592·c1·c2·s3·c4·c\varphi ·c\theta +5.592·c1·s2\\ ·c3·c4·c\varphi ·c\theta -5.592·s1·c2·c3·c4·c\varphi ·c\theta +1.000\\ ·c1·c2·s3·c4·s\varphi ·c\theta +0.9716·c1·c2·s3·s4·c\varphi ·c\theta \\ -2.000·c1·s2·c3·s4·c\varphi ·c\theta +0.9716·c1·s2·s3·c4\\ ·c\varphi ·c\theta -1.000·s1·c2·c3·c4·s\varphi ·c\theta +0.9716·s1·c2\\ ·c3·s4·c\varphi ·c\theta -2.000·s1·c2·s3·c4·c\varphi ·c\theta +0.9716\\ ·s1·s2·c3·c4·c\varphi ·c\theta +2.880·c1·c2·s3·s4·s\varphi ·c\theta \\ +2.880·c1·s2·s3·c4·s\varphi ·c\theta -0.1687·c1·s2·s3·s4·\\ c\varphi ·c\theta -2.880·s1·c2·c3·s4·s\varphi ·c\theta +0.1687·s1·c2\\ ·s3·s4·c\varphi ·c\theta -2.880·s1·s2·c3·c4·s\varphi ·c\theta -0.1687\\ ·s1·s2·c3·s4·c\varphi ·c\theta +0.1687·s1·s2·s3·c4·c\varphi ·c\theta \\ -1.000·c1·s2·s3·s4·s\varphi ·c\theta +1.000·s1·s2·c3·s4\\ ·s\varphi ·c\theta \\ \ne 0\end{array}$$

(17)

Proof of Proposition 1.

Expanding the second matrix in Equation (16) yields:

$$\overline{\Delta }~\left[\begin{array}{cccc}0& L·K_f·c2-K_m·s2& 0& -L·K_f·c4+K_m·s4\\ L·K_f·c1+K_m·s1& 0& -L·K_f·c3-K_m·s3& 0\\ L·K_f·s1-K_m·c1& -L·K_f·s2-K_m·c2& L·K_f·s3-K_m·c3& -L·K_f·s4-K_m·c4\\ c\theta ·s\varphi ·s1-c\theta ·c\varphi ·c1& -s\theta ·s2+c\theta ·c\varphi ·c2& -c\theta ·s\varphi ·s3-c\theta ·c\varphi ·c3& s\theta ·s4+c\theta ·c\varphi ·c4\end{array}\right].$$

(18)

Calculating the determinant of the second matrix in Equation (18) yields Condition (17). □

Proposition 2.

When the roll angle and pitch angle of the tiltrotor are close to zero, the decoupling matrix is invertible if, and only if:

$$\begin{array}{l}4.000·c1·c2·c3·c4+5.592\\ ·(+c1·c2·c3·s4-c1·c2·s3·c4+c1·s2·c3·c4-s1·c2·c3·c4)\\ +0.9716\\ ·(+c1·c2·s3·s4+c1·s2·s3·c4+s1·c2·c3·s4+s1·s2·c3·c4)\\ +2.000·(-c1·s2·c3·s4-s1·c2·s3·c4)+0.1687\\ ·(-c1·s2·s3·s4+s1·c2·s3·s4-s1·s2·c3·s4+s1·s2·s3·c4)\\ \ne 0.\end{array}$$

(19)

Proof of Proposition 2.

Make the assumptions in Equations (20) and (21):

$$\theta =0.$$

(20)

$$\varphi =0.$$

(21)

Substituting Equations (20) and (21) into Condition (17) yields Condition (19). □

Remark 1.

One may believe that Equation (18) would be an applicable gait-selecting zone, where any combination of $\left({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4\right)$satisfying Proposition 1 would lead to a stable result. Unfortunately, this is not always true, because the proven Proposition only guarantees the invertibility of the decoupling matrix. Hitting the non-negative angular velocity bound for${\varpi }_i$can result in instability. Obviously, neither Proposition 1 nor Proposition 2 rules out this situation. This is why we call the Propositions the “necessary” conditions for an applicable gait. The sufficient condition within the range of interest is presented in the simulation inSection 5 and Section 6.

Remark 2.

Notably, Condition (17) not only restricts the gait (${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$), but also rules out some attitudes ($\varphi ,\theta$); a specific gait can be driven to violate Condition (17) while steering to some specific attitudes. In actuality, a similar attitude-based condition means the failure makes the decoupling matrix invertible in [50], hindering further applications in feedback linearization (position–yaw output) before modification. This is because roll and pitch are not directly controlled via the position–yaw output choice. However, the adverse effect of this property is weakened in our research. The output choice (attitude–altitude) enables the tiltrotor to directly steer the attitude. Thus, roll angle and pitch angle can be relatively arbitrarily assigned. Steering the attitude away from the region violating Condition (17) is consequently made possible by this controller.

5. Attitude–Altitude Stabilization Test

The conditions in Section 4 are necessary in achieving the applicable gait. The sufficient conditions are explored empirically in this section.

To reduce the complexity, the preliminary step was to restrict the gait to some extent and find the gait region of interest. The next step was to determine the exploration direction of the candidate gaits. The final step was to conduct the simulation for each gait along the exploring direction.

5.1. Restricted Gait Region

The simulation was conducted at near-zero roll angle and pitch angle ($\varphi =0,\theta =0$). Thus, the following analysis is based on Condition (19) rather than on Condition (17).

Although Condition (19) includes no information about the roll angle and pitch angle, there are still four parameters in the gait (${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$) to be determined, which is a complicated issue. Here, we simplify the gait first.

Instead of exploring the entire space of (${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$), we explore four gaits with one of the following restrictions in each gait: ${\alpha }_1={\alpha }_3$, ${\alpha }_1=\frac{1}{2}·{\alpha }_3$, ${\alpha }_1=-{\alpha }_3$, and ${\alpha }_1=-\frac{1}{2}·{\alpha }_3$.

5.1.1. Case 1 (Equal)

This section demonstrates the result for some cases satisfying ${\alpha }_1={\alpha }_3$. Specifically, ${\alpha }_1={\alpha }_3=-0.15,-0.075, 0, 0.075, 0.15$. Figure 2 plots the left side of Condition (19). The result is five surfaces about $\left({\alpha }_2,{\alpha }_4\right)$. Intercepting the surfaces in Figure 2 by the zero plane ($\mathrm{Determinant} =0$) yields Figure 3. Figure 3 plots $\left({\alpha }_2,{\alpha }_4\right)$, violating Condition (19).

5.1.2. Case 2 (Half)

This section presents the results for some cases satisfying ${\alpha }_1=\frac{1}{2}·{\alpha }_3$. Specifically, ${\alpha }_1=\frac{1}{2}·{\alpha }_3=-0.2,-0.1, 0, 0.1, 0.2$. Figure 4 plots the left side of Condition (19). The result is five surfaces about $\left({\alpha }_2, {\alpha }_4\right)$. Intercepting the surfaces in Figure 4 by the zero plane ($\mathrm{Determinant} =0$) yields Figure 5. Figure 5 plots the $\left({\alpha }_2,{\alpha }_4\right)$ violating Condition (19).

5.1.3. Case 3 (Negative)

This section presents the results for some cases satisfying ${\alpha }_1=-{\alpha }_3$. Specifically, ${\alpha }_1=-{\alpha }_3=-1.4,-0.7, 0, 0.7, 1.4$. Figure 6 plots the left side of Condition (19). The result is five surfaces about $\left({\alpha }_2,{\alpha }_4\right)$. Intercepting the surfaces in Figure 6 by the zero plane ($\mathrm{Determinant} =0$) yields Figure 7. Figure 7 plots $\left({\alpha }_2,{\alpha }_4\right)$, violating Condition (19).

5.1.4. Case 4 (Negative Half)

This section presents the result for some cases satisfying ${\alpha }_1=-\frac{1}{2}·{\alpha }_3$. Specifically, ${\alpha }_1=-\frac{1}{2}·{\alpha }_3=-0.3,-0.15, 0, 0.15, 0.3$. Figure 8 plots the left side of Condition (19). The result is five surfaces about $\left({\alpha }_2,{\alpha }_4\right)$. Intercepting the surfaces in Figure 8 by the zero plane ($\mathrm{Determinant} =0$) yields Figure 9. Figure 9 plots $\left({\alpha }_2,{\alpha }_4\right)$, violating Condition (19).

5.2. Interested Gait Region and Direction of Exploration

Notably, choosing the gait of $\left({\alpha }_2,{\alpha }_4\right)$ on the relevant curve in Figure 3, Figure 5, Figure 7 and Figure 9 is strictly prohibited. The decline in the relevant curve indicates the violation of Condition (19). It is also worth mentioning that the tiltrotor becomes the conventional quadrotor in terms of gait, satisfying $\left({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4\right)=\left(0,0,0,0\right)$. Our gait analysis includes this special case.

Another point worth mentioning is the continuous requirement of gait switching. For example, switching from $\left({\alpha }_2,{\alpha }_4\right)=\left(1,1\right)$ to $\left({\alpha }_2,{\alpha }_4\right)=\left(-1,-1\right)$ is determined to violate Condition (19) for any of the cases shown in Figure 3, Figure 5, Figure 7 and Figure 9. This is because the relevant switching process will cross the curve, which should be prohibited.

Given these concerns, we only choose part of the region containing $\left({\alpha }_2,{\alpha }_4\right)=\left(0,0\right)$ as the gait region of interest.

5.2.1. Interested Gait Region

It can be concluded from the green region in Figure 10, Figure 11, Figure 12 and Figure 13 that any $\left({\alpha }_2,{\alpha }_4\right)$ located within or on the edge of the triangular zone defined in (22) for each case in Section 5.1 will not violate Condition (19).

The rest of the paper focuses on the gait $\left({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4\right)$, satisfying:

$$\mathrm{Region} \mathrm{within} \mathrm{or} \mathrm{on} \Delta MUV \mathrm{governed} \mathrm{by} U\left(-1.5,1.5\right), V\left(1.5,-1.5\right), M\left(1,1\right).$$

(22)

These gaits naturally satisfy the necessary condition given in Proposition 2. Section 5.2.2 outlines the direction of exploration for the simulation identifying the sufficient conditions of stability within the zone in (22).

5.2.2. Direction of Exploration

Exploring the entire space defined in (22) is not necessary or achievable. Thus, we explore parts of this region based on the definition of the directions of exploration in (23)–(25):

$${\alpha }_4=-{\alpha }_2,{\alpha }_2\in \left[-\frac{3}{2} , 0\right].$$

(23)

$${\alpha }_4=-{\alpha }_2,{\alpha }_2\in \left[0 , \frac{3}{2}\right].$$

(24)

$${\alpha }_4={\alpha }_2,{\alpha }_2\in \left[0 , 1\right].$$

(25)

The exploration in each direction given in (23)–(25) starts from $\left|{\alpha }_2\right|=0$. With the predetermined ${\alpha }_1 \mathrm{and} {\alpha }_3$, the exploration ends at a critical ${\alpha }_{2M}$, defined in (26):

$$\begin{gathered} \forall {\alpha }_2 satisfying \left|{\alpha }_2\right|⩽\left|{\alpha }_{2C}\right|,\left({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4\right)\Rightarrow stable, \\ \exists {\alpha }_2 satisfying \left|{\alpha }_2\right|>\left|{\alpha }_{2C}\right|,\left({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4\right)\Rightarrow unstable. \\ {\alpha }_{2M}=\mathrm{sign}\left({\alpha }_2\right)·\max \left(\left|{\alpha }_{2C}\right|\right). \end{gathered}$$

(26)

The expected output is the three gaits $\left({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4\right)$ corresponding to the three ${\alpha }_{2M}$ along the directions given in (23)–(25).

5.3. Attitude–Altitude Control

Feedback linearization relies on the choice of output. In this research, the selected output was the attitude–altitude vector; we focus on only controlling the attitude and altitude. Notably, this control scheme can also be used to reach a desired position with a conventional quadrotor [52], which can be regarded as the special case in our tiltrotor, where $\left({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4\right)=\left(0,0,0,0\right)$. However, position control in a tiltrotor was not the primary interest of this study.

In this simulation, the tiltrotor was expected to achieve attitude–altitude self-adjustment.

The initial attitude angles of the tiltrotor were assigned as ${\varphi }_i=0, {\theta }_i=0, \mathrm{and} {\varphi }_i=0$. The initial angular velocity vector of the tiltrotor with respect to the body-fixed frame was ${\omega }_B={\left[0,0,0\right]}^T$. The initial position vector was ${\left[0,0,0\right]}^T$. The initial velocity vector was ${\left[0,0,0\right]}^T$.

The input vector is the derivative of each value of angular velocity of the propeller ${\left[{\dot{\varpi }}_1,{\dot{\varpi }}_2,{\dot{\varpi }}_3,{\dot{\varpi }}_4\right]}^T$; therefore, assigning an initial velocity for each propeller was necessary. The absolute value of each initial angular velocity was 300 (rad). Notably, these angular velocities are not sufficient to compensate for the effect of gravity, even for the case $\left({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4\right)=\left(0,0,0,0\right)$.

The reference was a four-dimensional attitude–altitude vector, ${\left[{\varphi }_r,{\theta }_r,{\psi }_r,Z_r\right]}^T$. To maintain zero attitude and zero height, the designed reference was ${\left[0,0,0,0\right]}^T$. Based on the control parameters set in Section 3 and each gait designed in Section 5, we recorded the gaits giving stable results.

6. Results

This section presents the results of the tests in the previous section. Section 6.1 shows the histories of the attitude, altitude, and angular velocities during flights of two typical gaits. Section 6.2 displays the admissible gaits. These are the results of the work described in Section 5.2.2. Section 6.3 presents the results for the same task completed with another control strategy, which utilized all the inputs (over-actuated) on the same tiltrotor with identical parameters.

6.1. Flight History

Section 6.1 presents the results from two gaits. Gait 1: $\left({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4\right)=\left(-0.1, 0.1,-0.2, 0.1\right)$. Gait 2: $\left({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4\right)=\left(-0.15,-0.1, 0.3,-0.1\right)$.

6.1.1. Gait 1

For the gait $\left({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4\right)=\left(-0.1, 0.1,-0.2, 0.1\right)$, the attitude history, altitude history and angular velocities history of the propellers are plotted in Figure 14, Figure 15 and Figure 16, respectively.

Notably, none of the angular velocities touched 0, which guarantees the invertibility of the decoupling matrix in (12).

6.1.2. Gait 2

For the gait $\left({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4\right)=\left(-0.15,-0.1,0.3,-0.1\right)$, the attitude history, altitude history and angular velocities history of the propellers are plotted in Figure 17, Figure 18 and Figure 19, respectively.

Similarly, none of the angular velocities touched 0.

6.2. Applicable Gait (Sufficient Conditions)

This section displays the applicability of each gait from Section 5.1.1, Section 5.1.2, Section 5.1.3 and Section 5.1.4 (with five different $\left({\alpha }_1,{\alpha }_3\right)$ pairs in each section), determined via the method in (26).

Surprisingly, the resulting applicable values of $\left({\alpha }_2,{\alpha }_4\right)$ leading to stable results in each $\left({\alpha }_1,{\alpha }_3\right)$ in 20 ($5\times 4$) restrictions were highly similar. Except for one gait ($\left({\alpha }_1,{\alpha }_3\right)=\left(0.2,0.4\right)$), which led to instability, the regions of applicable $\left({\alpha }_2,{\alpha }_4\right)$ for the remaining 19 cases contained the same triangular region governed by $\left(-1.3,1.3\right)$, $\left(1.3,-1.3\right)$, and $\left(1,1\right)$.

The specific gaits leading to stable control results for different $\left({\alpha }_1,{\alpha }_3\right)$ are detailed in Figure 20, Figure 21, Figure 22 and Figure 23.

6.3. Over-Actuated Control

This section presents the control result for an over-actuated controller with eight control inputs, used to complete the same task with identical parameters and initial conditions. The attitude history, the altitude history, the angular velocities history of the propellers, and the tilting angles history are plotted in Figure 24, Figure 25, Figure 26 and Figure 27, respectively.

Although the response was faster than in the control method proposed in this research, the tilting angles changed over-rapidly and intensively, which is commonly believed to be unrealistic.

7. Conclusions and Discussions

The necessary conditions to develop an invertible decoupling matrix for the attitude–altitude-based feedback linearization method have been determined and visualized for a tiltrotor.

For the gait restricted by ${\alpha }_{1 }={\alpha }_3,{\alpha }_1=\frac{1}{2}·{\alpha }_3,{\alpha }_{1 }=-{\alpha }_3,{\alpha }_1=-\frac{1}{2}·{\alpha }_3$ within the specific region of interest given in (22), the feedback linearization had great success in terms of stabilization if, and only if, $\left({\alpha }_2,{\alpha }_4\right)$ lay inside the relevant region defined in Figure 20, Figure 21, Figure 22 and Figure 23.

The angular velocities in each simulation that manifested a stable result avoided touching the non-negative constraints.

When the over-actuated controller was used for completing the same task, the tilting angle was maneuvered over-rapidly and intensively. In return, the tiltrotor received a faster response and less overshoot.

For the defined $\left({\alpha }_1,{\alpha }_3\right)$ of interest, the control result is very likely to be unstable if $\left({\alpha }_2,{\alpha }_4\right)$ lies outside the relevant colored areas in Figure 20, Figure 21, Figure 22 and Figure 23.

These conclusions can offer effective guidance for designing the gait in a safe (invertible) region. For example, a different kind of feedback-linearization-based gait may allow $\left({\alpha }_2,{\alpha }_4\right)$ to lie within the triangular region governed by $\left(-1.3,1.3\right)$, $\left(1.3,-1.3\right)$, and $\left(1,1\right)$, because this region is most closely related to achieving a stable result.

The ${\alpha }_2-{\alpha }_4$ planes (Figure 3, Figure 5, Figure 7 and Figure 9) in Section 5.2 were divided into several regions by the invertibility-violating curves. We have argued that the applicable gait should lie in the same region. Furthermore, we asserted that crossing these curves is inevitable when switching from the gait in one region to the gait in another region. However, this is not always true.

It can be proven that the decoupling matrix for the gait $\left({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4\right)=\left(0,0,0,0\right)$ is always invertible, introducing no invertibility-violating curves. Thus, one may avoid crossing the invertibility-violating curves while switching the gait by adjusting $\left({\alpha }_1,{\alpha }_3\right)$ at the same time, e.g., the tiltrotor may return to the safe gait $\left({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4\right)=\left(0,0,0,0\right)$ as the middle step before switching to the desired gait. Further discussions are beyond the scope of this research.

Our next step is to develop a periodical gait for the tiltrotor in order to track a more complicated reference.