Stability and nonlinear controllability analysis of a quadrotor-like autonomous underwater vehicle considering variety of cases

Abstract

A quadrotor-like autonomous underwater vehicle that is similar to, yet different from quadrotor unmanned aerial vehicles, has been reported recently. This article investigates the stability and nonlinear controllability properties of the vehicle. First, the 12-degree-of-freedom model of the vehicle deploying an X shape actuation system is developed. Then, a stability property is investigated showing that the vehicle cannot be stabilized by a time invariant smooth state feedback law. After that, by adopting a nonlinear controllability analysis tool in geometric control theory, the small-time local controllability of the vehicle is analyzed for a variety of cases, including the vertical plane motion, the horizontal plane motion, and the three-dimensional space motion. Finally, different small-time local controllability conditions for different cases are developed. The result shows that the small-time local controllability holds for vertical plane motion and horizontal plane motion. However, the full degree of freedom kinodynamics model (i.e. 12 states) of the vehicle does not satisfy the small-time local controllability from zero-velocity states.

Introduction

Autonomous underwater vehicles (AUVs) have been under development since the early 1970s. Due to the dramatic advancements in computing and sensor techniques of present technologies, AUVs have been used to perform a variety of different tasks, including scientific, military research, and commercial applications, ^{1 –6} to name but a few. The challenging problems involved in these applications are accurate localization and navigation. In general, taking weight, complexity, reliability, and efficiency into account, AUVs are commonly designed without lateral actuators to be a typical underactuated control system. For such a system, localization and navigation are difficult, partly because it exhibits nonholonomic constraints and is not fully feedback linearizable. ^7,8 In fact, a nonholonomic system cannot be stabilized by a smooth static state feedback. ⁹ It is significant to study the controllability of AUVs to find its limitation of localization and navigation.

Intuitively, controllability is a problem about the ability to drive a system from one state to any other state in finite time with an admissible control input. In the past few decades, much research have been carried out to analyze the controllability for both linear and nonlinear systems. For the controllability of a linear system, Kalman et al. ¹⁰ presented a necessary and sufficient condition, known as Kalman Rank Condition over half a century ago. Gao and Zhao ¹¹ discussed the linear controllability of a two-dimensional translational oscillators with rotating actuator (2-DTORA) when the 2-DTORA system is on the horizontal plane and on a slope, respectively.

Generally speaking, it is much challenging for studying nonlinear controllability due to various types of nonlinear systems. Among different approaches, the local linearization and geometric control theory are two main tools that have been widely used. However, the controllability condition about a given point using the local linearization approach is a sufficient but not necessary one, that is, there exists a class of nonlinear systems that are linearly uncontrollable but nonlinear controllable. ¹² Alternately, geometric control theory is one of the most suitable frameworks for analyzing the controllability of nonlinear systems, with which various notations of controllability have been defined in the literature. ¹³ Here, we focus on the small-time local controllability (STLC). Sussmann ¹⁴ proposed a general sufficient condition for STLC of a nonlinear system at a given point. The authors show that many results proposed previously were particular cases. Goodwine and Burdick ¹⁵ proposed a method for assessing the nonlinear controllability of systems with unilateral control inputs based on the STLC condition proposed by Sussmann. ¹⁴

There are many studies related to STLC issues in various fields. Gui et al. ¹⁶ proved a spacecraft-control moment gyro system is STLC. Muralidharan and Mahindrakar ¹⁷ proved that a spherical robot is STLC at the equilibrium using the Bianchini and Stefani’s condition, ¹⁸ which is a particular case of the result proposed by Sussmann. ¹⁴ Liljeback et al. ¹⁹ showed that a snake robot does not satisfy sufficient conditions for STLC. Du et al. ²⁰ considered the controllability analysis and fault-tolerant control problems for a class of hexacopters. For a multirotor unmanned aerial vehicle (UAV), Saied et al. ²¹ investigated the attitude controllability issue using geometry control theory and considering different rotor failures and explored the STLC of the vehicle attitude dynamics subjected to unilateral control inputs. Hassan et al. ^22,23 investigated the airplane flight dynamics using the linear controllability analysis and geometric control formulation. For the underwater applications, Pettersen and Egeland ²⁴ proved that the STLC of the surface vessel is satisfied. Compared to the surface vessel, ²⁴ the quadrotor-like AUV (QLAUV) introduced in the article possesses more complex dynamic constraints. The result is not applicable to judge the STLC of the QLAUV. Moreover, the work of Saied et al. ²¹ explores only the attitude controllability issue but not expands the result to full-dimension states for a UAV.

This article investigates the stability and nonlinear controllability of the QLAUV. Similar to quadrotor UAVs, the QLAUV is equipped with only four identical thrusters for motion control. ²⁵ The unique characteristic of the vehicle is that the configuration shape of thrusters is like a character X. Thus, the thrust configuration is named by the X shape actuation system. Compared to traditional AUVs, it can achieve more kinds of motions, for example, yaw control at the zero surge speed state. A stability analysis is investigated, proving that there exists no smooth static state feedback law that makes the QLAUV asymptotically stabilize to equilibrium. Furthermore, the STLC of the QLAUV is analyzed for different cases including the vertical plane motion, the horizontal plane motion, and the three-dimensional (3-D) space motion.

The rest of the article is organized as follows. QLAUV kinematic and dynamic models are presented in the second section. In the third section, we investigate the stability of the QLAUV considering 12-degree-of-freedom (DOF) states. The STLC analysis based on geometric control theory is introduced in the fourth section. In the fifth section, we analyze the STLC of the QLAUV with regard to different cases, which is followed by some concluding remarks.

QLAUV nonlinear dynamic modeling

This section introduces primarily the kinematic and dynamic motion equations for the QLAUV. Subsection “Vehicle description” describes the vehicle as a whole. Subsection “Vehicle kinematics and dynamics” gives the kinematic and dynamic equations of the system; complex hydrodynamic effects are also taken into account. Finally, subsection “The ‘X shape’ Actuation system” analyzes the X shape actuation system of the QLAUV, related to forces and torques equations.

Vehicle description

The QLAUV was firstly introduced by Bian et al. ²⁵ and has been reported recently by Bian and Xiang ²⁶ to implement various motions using a sliding mode controller. It is developed under the project of the State Key Laboratory of Industrial Control Technology. This QLAUV is about 120 cm in length and 28 cm in diameter. To reduce the negative impact of the hydrodynamic forces, the hull of the vehicle is designed to a cylindrical shape, and the hand and the tail are elliptical. Figures 1 and 2 present the actual shape and the mechanical drawing of the QLAUV, respectively. As shown in Figures 1 and 2, the QLAUV has four identical thrusters, which are symmetrically configured apart at two sides of the hull. Furthermore, there exists a deflection angle between the body of the thrusters and the longitudinal axis of the vehicle. Besides, the direction of the heads of the thrusters is opposite: On one side, the heads of the two thrusters are close; on the other side, the heads keep away from each other.

Figure 1.

Quadrotor-like autonomous underwater vehicle.

Figure 2.

QLAUV profile. (a) Left lateral view, (b) right lateral view, and (c) top view. QLAUV: quadrotor-like autonomous underwater vehicle.

Vehicle kinematics and dynamics

Define an inertial coordinate frame $\left\{N\right\}=(x_n,y_n,z_n)$ with origin o_n and a body-fixed coordinate frame $\left\{B\right\}=(x_b,y_b,z_b)$ with origin o_b, as presented in Figure 3, where x_b, y_b, and z_b are chosen to coincide with the principal axes of inertia. Assume that the center of gravity (CG) of the vehicle is coincident with o_b and the underwater vehicle is naturally buoyant. With these assumptions and neglecting hydrostatics produced by gravitational/buoyancy forces and torques due to the existence of the metacentric height, the kinematics and dynamics modeling of the QLAUV moving in 12-DOF is ²⁷

$\begin{array}{l}\dot{\eta }=J(\eta )\nu \\ \dot{\nu }=-{(M_{RB}+M_A)}^{-1}((C_{RB}(\nu )+C_A(\nu ))\nu \hspace{0.17em}+\hspace{0.17em}D(\nu )\nu -\Gamma )\end{array}$

1

where $\eta =[x,y,z,\varphi ,\theta ,\psi {]}^T$ and $\nu =[u,v,w,p,q,r{]}^T$ . Notations x, y, and z denote the displacements in the surge, sway, and heave directions, respectively, and u, v, and w represent the surge, sway, and heave velocities, respectively. Furthermore, notations ϕ, θ, ψ, p, q, and r denote the roll, pitch, and yaw angles and angular velocities, respectively. The terms $J(\eta )$ , M_RB, M_A, $C_{RB}(\nu )$ , $C_A(\nu )$ , $D(\nu )$ , and Γ denote the kinematic transformation, rigid-body inertia mass, hydrodynamic added mass, rigid-body Coriolis, hydrodynamic Coriolis, damping, and actuation system, respectively, and will be defined in the sequel.

Figure 3.

Inertial frame and body-fixed frame for QLAUV. QLAUV: quadrotor-like autonomous underwater vehicle.

The kinematic transformation $J(\eta )$ is given by

$\begin{array}{l}J(\eta )=\left[\begin{array}{cc}{J}_1(\eta )& 0_{3\times 3}\\ 0_{3\times 3}& J_2(\eta )\end{array}\right]\\ J_1(\eta )=\left[\begin{array}{ccc}{\text{c}}_{\psi }{\text{c}}_{\theta }& {\text{c}}_{\psi }{\text{s}}_{\theta }{\text{s}}_{\varphi }-{\text{s}}_{\psi }{\text{c}}_{\theta }& {\text{c}}_{\psi }{\text{s}}_{\theta }{\text{c}}_{\varphi }+{\text{s}}_{\psi }{\text{s}}_{\theta }\\ {\text{s}}_{\psi }{\text{c}}_{\theta }& {\text{s}}_{\psi }{\text{s}}_{\theta }{\text{c}}_{\varphi }+{\text{c}}_{\psi }{\text{c}}_{\theta }& {\text{s}}_{\psi }{\text{s}}_{\theta }{\text{c}}_{\varphi }-{\text{c}}_{\psi }{\text{s}}_{\theta }\\ -{\text{s}}_{\theta }& {\text{c}}_{\theta }{\text{s}}_{\varphi }& {\text{c}}_{\theta }{\text{c}}_{\varphi }\end{array}\right]\\ J_2(\eta )=\left[\begin{array}{ccc}1& {\text{s}}_{\varphi }{t}_{\theta }& {\text{c}}_{\varphi }{t}_{\theta }\\ 0& {\text{c}}_{\varphi }& -{\text{s}}_{\varphi }\\ 0& {\text{s}}_{\varphi }/{\text{c}}_{\theta }& {\text{c}}_{\varphi }/{\text{c}}_{\theta }\end{array}\right]\end{array}$

2

where the abbreviations ${\text{c}}_j=\text{cos}(j)$ , ${\text{s}}_j=\text{sin}(j)$ , and $t_j=\text{tan}(j)$ are used for the sake of brevity. Assume that the vehicle has three planes of symmetry so that the inertia, added mass, and damping matrices are diagonal. In this setting, the inertia mass M_RB and added mass M_A are expressed as

$\begin{array}{c}{M}_{RB}=\text{diag}(m,m,m,I_x,I_y,I_z)\\ M_A=-\text{diag}(X_{\dot{u}},Y_{\dot{v}},Z_{\dot{w}},K_{\dot{p}},M_{\dot{q}},N_{\dot{r}})\end{array}$

3

One can use Kirchhoff’s equation ²⁷ to derive the rigid-body Coriolis $C_{RB}(\nu )$ and hydrodynamic Coriolis $C_A(\nu )$ , of which detailed forms are given by

$\begin{array}{l}{C}_{RB}(\nu )=\\ \left[\begin{array}{cccccc}0& 0& 0& 0& mw& -mv\\ 0& 0& 0& -mw& 0& mu\\ 0& 0& 0& mv& -mu& 0\\ 0& mw& -mv& 0& I_{z}r& -I_{y}q\\ -mw& 0& mu& -I_{z}r& 0& I_{x}p\\ mv& -mu& 0& I_{y}q& -I_{x}p& 0\end{array}\right]\\ C_A(\nu )=\\ \left[\begin{array}{cccccc}0& 0& 0& 0& -Z_{\dot{w}}w& Y_{\dot{v}}v\\ 0& 0& 0& Z_{\dot{w}}w& 0& -X_{\dot{u}}u\\ 0& 0& 0& -Y_{\dot{v}}v& X_{\dot{u}}u& 0\\ 0& -Z_{\dot{w}}w& Y_{\dot{v}}v& 0& -N_{\dot{r}}r& M_{\dot{q}}q\\ Z_{\dot{w}}w& 0& -X_{\dot{u}}u& N_{\dot{r}}r& 0& -K_{\dot{p}}p\\ -Y_{\dot{v}}v& X_{\dot{u}}u& 0& -M_{\dot{q}}q& K_{\dot{p}}q& 0\end{array}\right]\end{array}$

4

Also, the damping matrix D is expressed as

$D=\text{diag}(X_u,Y_v,Z_w,K_p,M_q,N_r)$

5

For simplicity, the higher-order nonlinear terms of D are neglected.

The “X shape” actuation system Γ

The actuation system should be described in the body-fixed reference frame due to the fact that the dynamic model of the QLAUV is derived in the frame. More specifically, both the thruster’s geometric center and the vehicle’s CG are designed to be in the x_b y_b-plane. The configuration of the thrusters is described in Figure 4, in which thruster i presents the i th thruster, $i=1,2,3,4$ ; the index i is limited to the same set for the remainder of the article if not be stated explicitly. Figure 4 shows that thrusters 1 and 2 signed blue are located on the right side of the QLAUV, and thrusters 3 and 4 signed red are located on the left. As the shape of the thruster configuration in a left view shown in Figure 4 looks like a character X, the actuation system of the QLAUV is called the X shape actuation system.

Figure 4.

The configuration of the thrusters in a left view.

As described in subsection “Vehicle description,” the centers of the four thrusters are located symmetrically in the $x_b{y}_b$ -plane with respect to the x_b and y_b axes. Let a vector q = $(x_0,y_0,\alpha )$ denote the positions of the thrusters with respect to the origin o_b, where $x_0,y_0,\text{\hspace{0.17em}and\hspace{0.17em}}\alpha$ represent the longitudinal position, transversal position, and deflection angle with respect to the longitudinal axis, respectively. Then, the positions of the four thrusters are $(x_0,y_0,\alpha )$ , $(-x_0,y_0,\alpha )$ , $(-x_0,-y_0,\alpha )$ , and $(x_0,-y_0,\alpha )$ , respectively, because the deflection angles of four thrusters with respect to the longitudinal axis are equal.

According to the geometry of the QLAUV, the X shape actuation system, that is, the mapping between the thrusters’ lifts and the total forces and torques is given by

$\Gamma =B(q)F$

6

where $\Gamma =[X,Y,Z,K,M,N{]}^T$ denotes the total forces and torques, in which X, Y, and Z are the forces on the x_b-, y_b-, and z_b-axes, respectively, and K, M, and N describe the torques about the x_b-, y_b-, and z_b-axes, respectively. $F=[F_1,F_2,F_3,F_4{]}^T$ is the thrusters’ lifts, with F_i being the lift produced by the thruster i; the force-distributed matrix $B(q)$ denoting the mapping between Γ and F is related to the vector q. For simplicity, the matrix $B(q)$ is abbreviated as B and its parameterized form is

$B=\left[\begin{array}{cccc}\text{cos}\alpha & -\text{cos}\alpha & \text{cos}\alpha & -\text{cos}\alpha \\ 0& 0& 0& 0\\ \text{sin}\alpha & \text{sin}\alpha & \text{sin}\alpha & \text{sin}\alpha \\ y_0\text{sin}\alpha & y_0\text{sin}\alpha & -y_0\text{sin}\alpha & -y_0\text{sin}\alpha \\ -x_0\text{sin}\alpha & x_0\text{sin}\alpha & x_0\text{sin}\alpha & -x_0\text{sin}\alpha \\ -y_0\text{cos}\alpha & y_0\text{cos}\alpha & y_0\text{cos}\alpha & -y_0\text{cos}\alpha \end{array}\right]$

7

The elements of B are derived by the geometric analysis of the forces imposed on the QLAUV. For example, as shown in Figure 5, $f_{i{x}_b}$ and $f_{i{z}_b}$ describe the components of F_i with regard to the x_b and z_b axes, respectively, and o_i denotes the application point of F_i. The amplitude of $f_{i{x}_b}$ can be presented by $F_i\text{cos}\alpha$ . Thus, the total force X for surge motion can be calculated by

Figure 5.

The decomposition of the thrusters’ lifts in the plane of projection of $x_b{z}_b$ -plane.

$\begin{array}{c}X={\mathit{f}}_{1x_b}+{\mathit{f}}_{2x_b}+{\mathit{f}}_{3x_b}+{\mathit{f}}_{4x_b}\\ =F_1\text{cos}\alpha -F_2\text{cos}\alpha +F_3\text{cos}\alpha -F_4\text{cos}\alpha \\ =\left[\begin{array}{cccc}\text{cos}\alpha & -\text{cos}\alpha & \text{cos}\alpha & -\text{cos}\alpha \end{array}\right]F\end{array}$

8

which forms the first row of B. Similar processes can be followed in the other directions of motion.

The columns of B imply that each thruster can provide forces or torques in the directions of surge, heave, roll, pitch, and yaw. Some simple motions such as the surge, heave, or yaw motion for the QLAUV should be implemented through the counteraction and superposition of the thrusters so that the actuation system is strongly coupled. A transformation matrix T satisfying

$\frac{1}{4\text{sin}\alpha }T=\left[\begin{array}{cccc}\text{tan}\alpha & 1& 1/y_0& -1/x_0\\ -\text{tan}\alpha & 1& 1/y_0& 1/x_0\\ \text{tan}\alpha & 1& -1/y_0& 1/x_0\\ -\text{tan}\alpha & 1& -1/y_0& -1/x_0\end{array}\right]$

9

is introduced here to decouple the actuation system. Define $F^{\prime }=[{F^{\prime }}_1,{F^{\prime }}_2,{F^{\prime }}_3,{F^{\prime }}_4{]}^T$ satisfying

$F=T{F}^{\prime }$

10

by which the mapping of the actuation system becomes

$\Gamma ={\left[\begin{array}{cccccc}{F^{\prime }}_1& 0& {F^{\prime }}_2& {F^{\prime }}_3& {F^{\prime }}_4& \beta {F^{\prime }}_4\end{array}\right]}^T$

11

Now $\beta =(y_0\text{cos}\alpha )/(x_0\text{sin}\alpha )$ is the only coupling term in the actuation system. To ensure the column full rank of the matrix B and nonsingularity of the matrix T, the parameter α should satisfy $\alpha \ne k\pi /2$ , $k\in N$ , which means that $\beta \ne 0$ . Compared to traditional AUVs deploying the thruster for speed control and the rudder for angle control, the QLAUV can implement more independent motions in different directions. For instance, since yaw angle control is independent of surge control, the QLAUV can implement yaw control at the zero surge speed state.

Applying the above notations, the system (1) can be rewritten into the following affine nonlinear model

$x=\mathit{f}(x)+{\displaystyle \sum _{i=1}^4}{\mathit{g}}_i{{\mathit{F}}^{\prime }}_i$

12

where $\mathit{x}\in R^{12}$ , $\mathit{f}(\mathit{x})$ , and ${\mathit{g}}_i$ are smooth vector fields, and ${\mathit{F}}_i^{\text{'}}$ denotes the decoupled thruster force. The detailed model is expressed in Appendix 1.

Stabilizability of QLAUV

As shown in the actuation system (11), the QLAUV cannot achieve lateral movements directly since the second element of the system (11) is zero, which is a typical underactuated system. The following result shows that there exists no smooth static state feedback law that ensures asymptotic stability regarding the equilibrium.

Theorem 1

Consider the kinematic and dynamic models (1) of the QLAUV and the actuation system (11), and suppose $(\eta ,\nu {)=(0}_{6\times 1}{\mathrm{,0}}_{6\times 1})$ is an equilibrium of the system. Then, there exists no smooth static feedback law that ensures asymptotic stability regarding the equilibrium.

Proof

Define the following mapping $\Pi (\eta ,\nu ,\Gamma ):R^6\times R^6\times R^4\to R^{12}$ as

$\Pi (\eta ,\nu ,\Gamma )=\left[\begin{array}{c}J(\eta )\nu \\ {(M_{RB}+M_A)}^{-1}{\Gamma }_a\end{array}\right]$

13

where

${\Gamma }_a=\Gamma -(C_{RB}(\nu )+C_A(\nu )+D(\nu ))\nu$

14

Consider the points of the form

$\epsilon =\left[\begin{array}{c}{\mathit{0}}_{6\times 1}\\ {(M_{RB}+M_A)}^{-1}∊\end{array}\right]$

15

where $∊={\left[{∊}_1,{∊}_2,{∊}_3,{∊}_4,{∊}_5,{∊}_6\right]}^T\in R^6$ , and ∊₂ is an arbitrary nonzero number. Note that ε is a point near the equilibrium. As $J(\eta )$ is of full rank, the equation $\Pi (\eta ,\nu ,\Gamma )=\epsilon$ implies $\mathit{\nu }={\mathit{0}}_{6\times 1}$ , which means that $\Gamma =∊$ . However, it is not solvable since the second term of Γ is equal to zero. Thus, no points of the form ε are in the image of the mapping $\Pi (\eta ,\nu ,\Gamma )$ . As the fact above violates the third necessary condition, ⁸ the system cannot be asymptotically stabilized by smooth static state feedback.

Small-time local controllability

Consider an affine nonlinear system as follows

$\dot{\mathit{x}}=\mathit{f}(\mathit{x})+{\displaystyle \mathbf{\sum }_{i=1}^m}{\mathit{g}}_i(\mathit{x})u_i$

16

where $\mathit{x}\in R^n$ is the system state. $\mathit{f}(\mathit{x})$ is the drift vector field and ${\mathit{g}}_i(\mathit{x})$ is the control input field related to the control input u_i, assume that x₀ is an equilibrium point (i.e. $\mathit{f}({\mathit{x}}_0)=0$ ). The sufficient but not necessary condition for the local controllability of the system at ${\mathit{x}}_0$ is that the linearized system at ${\mathit{x}}_0$

$\Delta \dot{\mathit{x}}=\left[\frac{\partial \mathit{f}}{dt}\right]{|}_{{\mathit{x}}_0}\Delta \mathit{x}+{\displaystyle \mathbf{\sum }_{i=1}^m}{\mathit{g}}_i({\mathit{x}}_0)u_i$

17

to be controllable. That is, the linearized version of Kalman rank condition

$\begin{array}{l}\mathit{M}=[{\mathit{g}}_1\mathrm{,...,}{\mathit{g}}_m\mathrm{,}\left[\frac{\partial \mathit{f}}{\partial \mathit{x}}\right]{|}_{{\mathit{x}}_0}{\mathit{g}}_1\mathrm{,...,}{\left[\frac{\partial \mathit{f}}{\partial \mathit{x}}\right]|}_{{\mathit{x}}_0}{\mathit{g}}_m\mathrm{,}\\ \mathrm{.}\mathrm{..}{\mathrm{,}{\left[\frac{\partial \mathit{f}}{\partial \mathit{x}}\right]}^{n-1}|}_{{\mathit{x}}_0}{\mathit{g}}_1\mathrm{,...,}{{\left[\frac{\partial \mathit{f}}{\partial \mathit{x}}\right]}^{n-1}|}_{{\mathit{x}}_0}{\mathit{g}}_m]\end{array}$

18

to be of full row rank. ¹² Alternatively, geometry control theory has been widely used to study the STLC. The reader can refer to the study of Kou et al. ²⁸ for some definitions, such as the reachable set $R_T(x_0)$ , accessibility algebra ζ, and accessibility distribution C. Some necessary theorems used to judge the STLC of the system (16) are recalled as follows.

Theorem 2

Consider the system $(16)$ , if dim $C(x_0)=n$ (i.e. the accessibility algebra spans the tangent space to M at x₀), then for any $T>0$ , the set $R_T(x_0)$ has a nonempty interior, that is, the system has the accessibility property from x₀. ²¹

When the hypotheses of this theorem hold, one can say that the system satisfies Lie Algebra Rank Condition (LARC) at x₀. ²⁹

For a driftless system where the vector filed f equals a zero vector, the system is STLC if it verifies the LARC. ²¹ However, the condition is not sufficient with a drift system, as some of the Lie brackets involved the drift term may have directional constraints, which obstructs the controllability of the system. ³⁰ For the systems with drift, Sussman ¹⁴ proposed a general theorem which states that some additional conditions about good and bad Lie brackets should hold besides the LARC in order to guarantee the STLC.

Theorem 3

A system that satisfies the LARC by good Lie brackets terms up to degree i is STLC if all bad Lie brackets of degree $j\le i$ are neutralized. ¹⁴

It is worth mentioning that if a Lie bracket can be presented by a linear combination of good Lie brackets of lower degree, it is neutralized and such a bracket does not obstruct STLC. Particularly, a bad Lie bracket is neutralized if it is equal to zero at the equilibrium. Algorithm 1 shows a procedure to study the STLC of any given affine system.

Algorithm 1

The procedure to study the STLC of a given system.

Require: An affine nonlinear control system
1: if dim (C) = n then
2:  if The system is driftless then
3:   The system is STLC.
4:  else
5:   if The system satisfies Theorem 3 then
6:    The system is STLC.
7:   else
8:    The STLC of the system cannot be judged.
9:   end if
10:  end if
11: else
12:  The system is not STLC.
13: end if

Sussman ¹⁴ gives a simple approach to judging whether a Lie bracket is good or not. Consider any Lie bracket X consisting of the drift vector f and control vectors ${\mathit{g}}_1\mathrm{,...},{\mathit{g}}_m$ , a θ-degree of X denoted by ${\delta }_{\theta }(X)$ is defined as follows

${\delta }_{\theta }(X)=\frac{1}{\theta }{\delta }^f(X)+{\displaystyle \sum _{i=1}^m}{\delta }^{{\mathit{g}}_i}(X)$

where θ is an arbitrary term satisfying $\theta \in [1,\infty )$ ; here, θ is set to be $1$ . ${\delta }^f(X)$ and ${\delta }^{{\mathit{g}}_i}(X)$ denote the number of times f and g _i , respectively, appearing in the bracket X. If ${\delta }^f(X)$ is odd and ${\delta }^{{\mathit{g}}_i}(X)$ is even (including 0), the bracket X is called a bad bracket. The good brackets are those that are not bad. Some good and bad Lie brackets are presented in Table 1.

Table 1.

Some Lie brackets.

Lie brackets (X)	${\delta }^f(X)+{\displaystyle {\sum }_{i=1}^2}{\delta }^{{\mathit{g}}_i}(X)$	Type
f	1	Bad
$\text{[}f,{\mathit{g}}_1]$	2	Good
$\text{[}{\mathit{g}}_1,[f,{\mathit{g}}_1]$	3	Bad
$\text{[}{\mathit{g}}_1,[f,{\mathit{g}}_2]$	3	Good
$[{\mathit{g}}_2,[f,{\mathit{g}}_2]$	3	Bad

Geometric control formulation for QLAUV kinematics and dynamics

Simple computations show that the local controllability cannot be judged by the method of linearizing the system of the QLAUV at its equilibrium. In this part, we investigate the STLC of the QLAUV considering different cases such as the dynamic and kinodynamic models using geometric control theory.

Controllability of dynamics

This part mainly studies the controllability for QLAUV dynamics, which focuses on velocities and angular velocities of the vehicle, leaving displacements and attitudes alone.

Three-DOF dynamics

We firstly study the three-DOF dynamics of the QLAUV in an affine form as in equation (16), where the state vector $\mathit{x}=[u,v,r{]}^T$ focuses on the surge, sway, and yaw angular velocities in the horizontal plane. The state equation is expressed as follows

$\begin{array}{l}\left[\begin{array}{c}\dot{u}\\ \dot{v}\\ \dot{r}\end{array}\right]=\left[\begin{array}{c}{f}_1\\ f_2\\ f_6\end{array}\right]+\left[\begin{array}{c}\frac{1}{m_1}\\ 0\\ 0\end{array}\right]{F^{\prime }}_1+\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]{F^{\prime }}_2+\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]{F^{\prime }}_3+\left[\begin{array}{c}0\\ 0\\ \frac{\beta }{m_6}\end{array}\right]{F^{\prime }}_4\end{array}$

19

The drift field $\mathit{f}$ is written as $\mathit{f}(x)=[f_{1-2},f_6{]}^T$ , and the explanation of the simplified form $f_{1-2}$ can be found in Appendix 1. Moreover, ${\mathit{g}}_i$ is the corresponding control input vector with proper orders relative to the input ${F^{\prime }}_i$ . Similar expressions are used with a slight abuse at the rest of the section. Here, both ${\mathit{g}}_2$ and ${\mathit{g}}_3$ are equal to ${\mathit{0}}_{3\times 1}$ and they are left out for Lie bracket calculations. According to Lie brackets and Lie Algebra, the accessibility distribution is given by

$C=\text{span}\{{\mathit{g}}_1,{\mathit{g}}_4,[{\mathit{g}}_1,[\mathit{f},{\mathit{g}}_4]]\}$

20

with det $(C)=(-{\beta }^2)/(m_1{m}_2{m}_6^2)\ne 0$ when $\beta \ne 0$ , so its dimension is equal to that of the state $\mathit{x}$ . Let ρ denote the maximum degree of the Lie brackets in the accessibility distribution C. Then, the system satisfies LARC by good brackets with $\rho =3$ .

Following the approach in the previous section, all bad brackets where the degree is less than or equal to 3 include

$\mathit{f},[{\mathit{g}}_1,[\mathit{f},{\mathit{g}}_1]],[{\mathit{g}}_2,[\mathit{f},{\mathit{g}}_2]],[{\mathit{g}}_3,[\mathit{f},{\mathit{g}}_3]],[{\mathit{g}}_4,[\mathit{f},{\mathit{g}}_4]]$

21

All the above brackets are equal to ${\mathit{0}}_{3\times 1}$ , which indicates that they are neutralized in zero-velocity states. Therefore, considering $u,v,r$ degrees only, the system is STLC from zero-velocity states.

Remark 1

It is worth to be highlighted that even if the system drift vector $\mathit{f}$ equals zero at the equilibrium, the system cannot be regarded as a driftless one, because the drift vector $\mathit{f}$ shows the complex internal dynamics of the system (19), and that indicates that the system possesses more possibilities in the directions of motion.

For example, consider the system (19) as a driftless one at the equilibrium to yields

$\begin{array}{l}\left[\begin{array}{c}\dot{u}\\ \dot{v}\\ \dot{r}\end{array}\right]=\left[\begin{array}{c}\frac{1}{m_1}\\ 0\\ 0\end{array}\right]{F^{\prime }}_1+\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]{F\text{'}}_2+\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]{F^{\prime }}_3+\left[\begin{array}{c}0\\ 0\\ \frac{\beta }{m_6}\end{array}\right]{F^{\prime }}_4\end{array}$

22

Note that any control input vector ${\mathit{g}}_i$ is a constant one, which means that all the Lie brackets comprising any number of ${\mathit{g}}_i$ are equal to ${\mathit{0}}_{3\times 1}$ . Accordingly, $\text{det}(C)=2$ . The result means that the system does not satisfy the LARC. However, the conclusion mentioned above implies that the system satisfies LARC by good brackets with $\rho =3$ , which is conflicted with the result where the system (19) is regarded as a driftless one.

Four-DOF dynamics

This part considers one more variable, the heave velocity w, so that the state vector is $\mathit{x}=[u,v,w,r{]}^T$ . The state equation is expressed as follows

$\left[\begin{array}{c}\dot{u}\\ \dot{v}\\ \dot{w}\\ \dot{r}\end{array}\right]=\left[\begin{array}{c}{f}_1\\ f_2\\ f_3\\ f_6\end{array}\right]+\left[\begin{array}{c}\frac{1}{m_1}\\ 0\\ 0\\ 0\end{array}\right]{F^{\prime }}_1+\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right]{F^{\prime }}_2+\left[\begin{array}{c}0\\ 0\\ \frac{1}{m_3}\\ 0\end{array}\right]{F^{\prime }}_3+\left[\begin{array}{c}0\\ 0\\ 0\\ \frac{\beta }{m_6}\end{array}\right]{F^{\prime }}_4$

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The drift field $\mathit{f}$ is written as $\mathit{f}(x)=[{\mathit{f}}_{1-3},{\mathit{f}}_6{]}^T$ , and the control input vector ${\mathit{g}}_2$ is left out for Lie bracket calculations as it is a zero vector. Consider $\mathit{f}$ , ${\mathit{g}}_1$ , ${\mathit{g}}_3$ , and ${\mathit{g}}_4$ to calculate the accessibility distribution as follows

$C=\text{span}\{{\mathit{g}}_1,{\mathit{g}}_3,{\mathit{g}}_4,[{\mathit{g}}_1,[\mathit{f},{\mathit{g}}_4]]\}$

24

with det $(C)=({\beta }^2)/(m_1{m}_2{m}_3{m}_6^2)\ne 0$ . It satisfies the LARC with $\rho =3$ , and the corresponding bad brackets are same as equation (21). In zero-velocity states, they are equal to ${\mathit{0}}_{4\times 1}$ ; thus, considering $u,v,w,r$ degrees, the system satisfies STLC from zero-velocity states.

Six-DOF dynamics

Considering all linear and angular velocity vectors, where $\mathit{x}=[u,v,w,p,q,r{]}^T$ is the state. The drift field f is given by $\mathit{f}(x)=[f_{1-6}{]}^T$ , and the control input fields ${\mathit{g}}_i$ are

$\begin{array}{l}{\mathit{g}}_1={[1/m_1,\mathrm{0,0,0,0,0}]}^T\\ {\mathit{g}}_2={[0,0,1/m_3,\mathrm{0,0,0}]}^T\\ {\mathit{g}}_3={[0,0,0,1/m_4,\mathrm{0,0}]}^T\\ {\mathit{g}}_4=[0,0,0,0,1/m_5,\beta /m_6{]}^T\end{array}$

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The accessibility distribution is given by

$C=\text{span}\{{\mathit{g}}_1,{\mathit{g}}_2,{\mathit{g}}_3,{\mathit{g}}_4,[{\mathit{g}}_1,[\mathit{f},{\mathit{g}}_2]],[{\mathit{g}}_1,[\mathit{f},{\mathit{g}}_4]]\}$

26

with det $(C)=({\beta }^2(m_1-m_3))/(m_1^2{m}_2{m}_3^2{m}_4{m}_5{m}_6^2)\ne 0$ when $m_1\ne m_3$ . This system also satisfies the LARC with $\rho =3$ , and the corresponding bad brackets are same as equation (21). In zero-velocity states, the first four of those brackets are equal to ${\mathit{0}}_{6\times 1}$ , and the last bad bracket is expressed as

$\begin{array}{c}[{\mathit{g}}_4,[\mathit{f},{\mathit{g}}_4]]={\left[\begin{array}{cccccc}0& 0& 0& \frac{-2\beta (m_5-m_6)}{m_4{m}_5{m}_6}& 0& 0\end{array}\right]}^T\\ ={\mathit{g}}_3\times \frac{-2\beta (m_5-m_6)}{m_5{m}_6}\end{array}$

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It can be neutralized by a lower degree good bracket ${\mathit{g}}_3$ . Thus, the system considering all linear and angular velocity satisfies STLC from zero-velocity states if $m_1\ne m_3$ .

Controllability of kinodynamics

The above subsection analyzes the controllability of the underwater vehicle without considering the displaced position and orientation, which is vital to the path following, trajectory tracking, path planning, and so on. This subsection further analyzes the controllability of the vehicle considering both kinematics and dynamics.

Vertical plane kinodynamics

This part considers only the surge and heave displacements and velocities, where $\mathit{x}=[u,w,x,z{]}^T$ is the state. The drift field $\mathit{f}$ is expressed as $\mathit{f}(x)=[f_1,f_3,f_7,f_9{]}^T$ . The detail state equation is

$\left[\begin{array}{c}\dot{u}\\ \dot{w}\\ \dot{x}\\ \dot{z}\end{array}\right]=\left[\begin{array}{c}{f}_1\\ f_3\\ f_5\\ f_7\end{array}\right]+\left[\begin{array}{c}\frac{1}{m_1}\\ 0\\ 0\\ 0\end{array}\right]{F^{\prime }}_1+\left[\begin{array}{c}0\\ \frac{1}{m_3}\\ 0\\ 0\end{array}\right]{F^{\prime }}_2+\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right]{F^{\prime }}_3+\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right]{F^{\prime }}_4$

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As shown in the actuation system (11), there exists no coupled terms between the direction of surge and heave movements, and the surge and heave motion can be directly controllable. However, this does not imply that the system is STLC. The main results are presented as follows.

Theorem 4

The system (1) considering vertical plane kinodynamics only is STLC in zero-velocity states, satisfying the yaw angle $\psi \ne (k+0.5)\pi ,$ $k\in N$ .

Proof

The accessibility distribution of the system (28) is given by

$C=\text{span}\{{\mathit{g}}_1,{\mathit{g}}_2,[\mathit{f},{\mathit{g}}_1],[\mathit{f},{\mathit{g}}_2]\}$

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with det $(C)=(c_{\varphi }{c}_{\psi }+s_{\theta }{s}_{\psi }{s}_{\varphi })/(m_1^2{m}_3^2)$ . Owing to the existence of the metacentric height, $\theta \approx 0$ and $\varphi \approx 0$ , then, det $(C)\approx (c_{\psi })/(m_1^2{m}_3^2)$ . Thus, the system satisfies LARC with $\rho =2$ provided the following condition

$\psi \ne (k+0.5)\pi ,k\in N$

30

Note that $\mathit{f}$ is the only bad bracket where the degree is less than or equal to 2. In zero-velocity states, $\mathit{f}$ is a zero vector, and thereby the system is STLC under the condition that equation (30) holds.

A straightforward explanation about condition (30) is: when yaw angle ψ is approximate to $0.5\pi$ , the x-axis with respect to the inertial coordinate frame is parallel to the y_b-axis with respect to the body-fixed coordinate system. Since the vehicle cannot provide the sway velocity in this direction, and the angles and the angular speeds are not controlled, it is not possible to satisfy the STLC.

Horizon plane kinodynamics

This part investigates the kinematic and dynamic models of the QLAUV in the horizontal plane. The system state vector $\mathit{x}=[u,v,r,x,y,\psi {]}^T$ , the drift field $\mathit{f}=[f_{1-2},f_{6-8},f_{12}{]}^T$ , and the control input fields ${\mathit{g}}_i$ are

$\begin{array}{l}{\mathit{g}}_1={[1/m_1,\mathrm{0,0,0,0,0}]}^T\\ {\mathit{g}}_2={\mathit{g}}_3={\mathit{\text{0}}}_{6\times 1}\\ {\mathit{g}}_4={[0,0,\beta /m_6,\mathrm{0,0,0}]}^T\end{array}$

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The accessibility distribution is given by

$\begin{array}{l}C=\text{span}\{{\mathit{g}}_1,{\mathit{g}}_4,[\mathit{f},{\mathit{g}}_1],[\mathit{f},{\mathit{g}}_4],[{\mathit{g}}_1,[\mathit{f},{\mathit{g}}_1]],\\ [{\mathit{g}}_1,[\mathit{f},[\mathit{f},{\mathit{g}}_4]]]\}\end{array}$

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with det $(C)=(-{\beta }^4{(c_{\varphi })}^2(m_1-m_2))/(m_1^3{m}_2^2{m}_6^4)$ . The roll angle $\varphi \approx 0$ , then one have

$\text{det}(C)\approx \frac{{\beta }^4(m_2-m_1)}{(m_1^3{m}_2^2{m}_6^4)}\ne 0$

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when $m_1\ne m_2$ . Thus, the system satisfies LARC with $\rho =4$ . As the degree of the bad brackets is odd, the corresponding bad brackets where the degree is less than or equal to $4$ are identical to equation (21). Simple computation indicates that above bad brackets are equal to ${\mathit{0}}_{6\times 1}$ in zero-velocity states, so the system satisfies STLC if $m_1\ne m_2$ .

3-D space kinodynamics

This part considers the 3-D full-degree kinematic and dynamic models of the QLAUV. Here, the linear velocities transformation is omitted as it does not affect controllability (i.e. ignorable coordinates). ²³ The system state vector $\mathit{x}=[u,v,w,p,q,r,\varphi ,\theta ,\psi {]}^T$ , the drift field $\mathit{f}=[f_{1-6},f_{10-12}{]}^T$ , and the control input fields ${\mathit{g}}_i$ are

$\begin{array}{l}{\mathit{g}}_1={[1/m_1,\mathrm{0,0,0,0,0,0,0,0}]}^T\\ {\mathit{g}}_2={[0,0,1/m_3,\mathrm{0,0,0,0,0,0}]}^T\\ {\mathit{g}}_3={[0,0,0,1/m_4,\mathrm{0,0,0,0,0}]}^T\\ {\mathit{g}}_4={[0,0,0,0,1/m_5,\beta /m_6,\mathrm{0,0,0}]}^T\end{array}$

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The accessibility distribution is given by

$\begin{array}{l}C=\text{span}\{{\mathit{g}}_1,{\mathit{g}}_2,{\mathit{g}}_3,{\mathit{g}}_4,[\mathit{f},{\mathit{g}}_1],[\mathit{f},{\mathit{g}}_2],[\mathit{f},{\mathit{g}}_3],[\mathit{f},{\mathit{g}}_4],\\ [{\mathit{g}}_1,[\mathit{f},[\mathit{f},{\mathit{g}}_2]]]\}\end{array}$

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Tedious computation indicates that in zero-velocity states, rank $(C)=7<9$ , the LARC is not verified at the equilibrium states, and the system is not STLC from zero-velocity states. That implies that with the quadrotor-like thruster configuration, it is impossible to drive the vehicle to any speed in any 3-D space.

Remark 2

Note that the controllability of some cases abovementioned such as subsection “Four-DOF dynamics” and subsection “Vertical plane kinodynamics” can be analyzed by the linearized version of Kalman rank condition (18). For example, the linearized form of four-DOF dynamic system at the equilibrium is described as

$\mathit{\Delta }\dot{\mathit{x}}=\mathit{A}\mathit{\Delta }\mathit{x}+{\displaystyle \mathbf{\sum }_{i=1}^4}{\mathit{g}}_i{F^{\prime }}_i$

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where $\Delta \mathit{x}=[\Delta u,\Delta v,\Delta w,\Delta r{]}^T$ . The expression of g _i is same as one in subsection “Four-DOF dynamics” because ${\mathit{g}}_i$ is independent of $\mathit{x}$ . System matrix $\mathit{A}$ is equal to the Jacobian matrix of $\mathit{f}$ with respect to $\mathit{x}$ at the equilibrium ${\mathit{x}}_0{=[0,0,0,0]}^T$ , and its detailed form is

$A={\frac{\partial f}{\partial x}|}_{x_0}=\left[\begin{array}{cccc}-\frac{X_u}{m_1}& 0& \frac{m_{1}q}{m_3}& 0\\ 0& -\frac{Y_v}{m_2}& -\frac{m_{2}p}{m_3}& 0\\ -\frac{m_{3}q}{m_1}& \frac{m_{3}p}{m_2}& -\frac{Z_w}{m_3}& 0\\ 0& 0& 0& -\frac{N_r}{m_6}\end{array}\right]$

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The rank of the controllability matrix

$\mathit{M}=\left[\begin{array}{c}{\mathit{g}}_1\mathrm{,...},{\mathit{g}}_4,\mathit{A}{\mathit{g}}_1\mathrm{,...},\mathit{A}{\mathit{g}}_4\mathrm{,...},{\mathit{A}}^3{\mathit{g}}_1\mathrm{,...},{\mathit{A}}^3{\mathit{g}}_4\end{array}\right]$

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is found to be 4 if $p\ne 0$ which means that the four-DOF dynamic system is locally controllable at the equilibrium.

Conclusion

In the article, the stability property and controllability of the QLAUV have been investigated. First, it is proved that the QLAUV is not stabilizable by time invariant smooth state feedback. Then, by use of geometric control theory, we analyzed the STLC of the QLAUV. Furthermore, we investigated different STLC conditions with regard to the dynamic and kinodynamic models. Although the STLC holds in the case of vertical plane motion and horizontal plane motion, the full-DOF kinodynamics model (i.e. 12 states) of the QLAUV system does not satisfy the LARC and is not STLC from zero-velocity states.