Adaptive Backstepping Control of an Unmanned Aerial Manipulator

1. Introduction

Research on unmanned aerial vehicles (UAVs) has recently focussed on tasks involving manipulation of the environment. Normally, this interaction is achieved by mounting an “arm” to the UAV so that various manipulation tasks can be achieved. The combined UAV and arm system is referred to as an unmanned aerial manipulator (UAM). Applications such as non-destructive testing of infrastructure require the UAM to keep a sensor in contact with the surface being inspected. The sensor must follow a desired trajectory on the surface and a predefined force must be applied.

Initial work on UAM motion control tends to focus on single-link arms which are rigidly attached to the UAV. UAM force and motion control dates back to [1], where the “arm” consists of a single rod rigidly attached to the UAV. An additional propeller is used to apply force to a surface. A heuristically designed proportional-integral-derivative (PID) controller structure is proposed, and no stability result is given for the force/motion control problem. Work in [2] provides a rigorous stability result using a nonlinear output feedback. The work assumes the single-fixed link arm does not affect the CoM of the system. The dynamics of the end-effector position is decomposed into tangential and normal directions. Error stability requires the beam to be attached "above" the UAV CoM. This work is extended in [3] where it is shown that the end-effector can generate any contact force if it is strictly located either below or above the UAV CoM. The article [4] studies the problem of UAV interaction with the environment without an arm. A’’ cascade controller allows the UAV to "push" against objects using pads attached to the UAV frame. The stability analysis is based on a planar quasi-static motion assumption. This apparatus is improved in [5] where tilted propellers are used. A heuristic PID control is proposed without a formal stability result.

Feedback control for multi-DoF arm UAMs can be classified into two main approaches: (i) separate controllers for the UAV and the arm, and (ii) full UAM control. In Approach (i), the coupling forces/torques acting on the UAV due to the arm are treated as a disturbance [6,7,8,9,10,11]. Approach (i) generally leads to a simpler subsystem-at-a-time design. Approach (ii) views the entire UAM as a single coupled dynamics [12]. This approach is challenging as the entire dynamics is very complex and complexity dramatically increases with the number of arm DoF [13,14,15,16,17]. Hence, this paper follows Approach (i) and compensates for the effect of the arm by introducing uncertain inertial parameters and force/torque disturbances into a UAV dynamics.

Approach (i) is widely found in the literature e.g., [6,7,8]. The work of [6] considers a UAV with a 4-DoF arm. For the UAV inner-loop, i.e., attitude dynamics, a disturbance observer based on [18] is used to estimate the effect of the arm on the UAV. With these estimates, feedback control is designed for the inner-loop that compensates for the disturbance. The outer-loop is treated separately. The effect of coupling between the loops on closed-loop stability is not considered. To validate their approach on a real UAM, the UAV outer-loop controller is taken from [19]. Work in [20] uses model reference control for the UAV outer-loop only. The UAV mass is taken as the unknown parameter. The effect of coupling forces and torques due to the arm on the UAV is not considered. Unlike [6] and [20], our work rejects the arm disturbance while considering the full UAV dynamics. The article [21] considers a UAV with a 3-DoF arm. It proposes a solution for constant setpoint regulation using a Luenberger observer to estimate the torque disturbance due to the arm. Our work is not limited to regulation and designs a trajectory-tracking feedback. We remark that the literature on robust UAV control, i.e., without an arm, is related to Approach (i). Examples of this include [22,23].

In this paper, our control objective is trajectory tracking of the UAV CoM and yaw. The effect of the arm on the UAV is modelled in two ways: it adds a force and torque disturbance to the UAV and introduces parametric uncertainty e.g., arm motion alters the UAV CoM and inertia matrix. We are inspired by the work of [24,25] on UAV motion control. These contributions only consider the force and torque disturbances, while all the inertial parameters are assumed known. Our work also allows for unknown inertial UAV parameters. We propose an adaptive backstepping control scheme that is capable of compensating force/torque disturbances and parameter uncertainty. This work is based on the first author’s thesis [26].

This paper is organized as follows. Section 2 presents the model of the UAM. Section 3 presents model assumptions, formulates the control problem, proposes a control design, and performs stability analysis. Numerical simulations are presented in Section 4. Finally, Section 5 concludes the paper.

2. Modelling

Consider a UAM consisting of a UAV with attached robotic arm. In this section, we present the UAV model and introduce a force/torque disturbance and parametric uncertainty to model the effect of the arm. The UAV model neglects the effects of motor dynamics, rotor drag, and the rotor gyroscopic effects. The details on modelling these higher-order effects are discussed in [27,28].

Consider Figure 1. Let $\mathcal{N}$ denote an inertial navigation frame with basis vectors $\{n_1,n_2,n_3\}$ pointing north, east, down, respectively. Let $\mathcal{B}$ denote a body-fixed frame whose origin coincides with the UAV CoM and whose basis vectors are $\{b_1,b_2,b_3\}$ and point forward, right, and downward, relative to the UAV. The position of the origin of $\mathcal{B}$ relative to the origin of $\mathcal{N}$ is denoted, $p\in \mathbb{R}$, which is expressed in $\mathcal{N}$. The linear velocity the UAV CoM, $v\in \mathbb{R}$, is given by

$$\dot{p}=v.$$

(1)

The orientation of $\mathcal{B}$ relative to $\mathcal{N}$ is described by the rotation matrix $R\in \mathrm{SO}\left(3\right)$. The angular velocity $\omega \in {\mathbb{R}}^3$ of the UAV relative to $\mathcal{N}$ expressed in $\mathcal{B}$ is related to R by

$$\dot{R}=R\mathrm{S}\left(\omega \right),$$

(2)

where

$$\mathrm{S}\left(x\right)=\left[\begin{array}{ccc}0& -x_3& x_2\\ x_3& 0& -x_1\\ -x_2& x_1& 0\end{array}\right],\begin{array}{c}\hfill x=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]\end{array}\in {\mathbb{R}}^3.$$

The UAV input is total propeller thrust $u\in {\mathbb{R}}_{>0}$ which creates a force in $-b_3$ direction, and torque $\tau \in {\mathbb{R}}^3$ on $\mathcal{B}$. Letting $m\in \mathbb{R}$ denote UAV mass and $J\in {\mathbb{R}}^{3\times 3}$ denote UAV inertia, the UAV dynamics is

$$\begin{array}{cc}\hfill m\dot{v}& =mg{e}_3-F,\hfill \\ \hfill J\dot{\omega }& =-\mathrm{S}\left(\omega \right)J\omega +\tau \hfill \end{array}$$

(3)

where $F=-uR{e}_3$, $e_3={[0,0,1]}^{\top }$, and g is gravitational acceleration. We remark that (3) models the UAV in isolation from the arm. To account for the effect of the arm, we introduce disturbance force $d_f\in {\mathbb{R}}^3$ and torque $d_{\tau }\in {\mathbb{R}}^3$ that can be interpreted as a change in input due to the arm. Further, we consider inertial parameters $a\in \mathbb{R}$ and J as uncertain:

$$\begin{array}{cc}\hfill \dot{p}=& v,\hfill \end{array}$$

(4a)

$$\begin{array}{cc}\hfill \dot{v}=& g{e}_3-auR{e}_3+d_f,\hfill \end{array}$$

(4b)

$$\begin{array}{cc}\hfill \dot{R}=& R\mathrm{S}\left(\omega \right),\hfill \end{array}$$

(4c)

$$\begin{array}{cc}\hfill J\dot{\omega }=& -\mathrm{S}\left(\omega \right)J\omega +\tau +d_{\tau }.\hfill \end{array}$$

(4d)

Further details on the UAM model simplification are in [26] Section 2.2.4. 2.2.5.

Note that J in (Section 2) is a positive definite symmetric matrix with six unique entries. These entries of J define the vector $J_v={[J_{xx},J_{yy},J_{zz},J_{xy},J_{yz},J_{zx}]}^{\top }\in {\mathbb{R}}^6$. Using $J_v$ we can re-write the $\mathrm{S}\left(\omega \right)J\omega$ term in (4d) as

$$\begin{array}{cc}\hfill \mathrm{S}\left(\omega \right)J\omega & =\Phi \left(\omega \right)J_v\hfill \end{array}$$

(5)

where $\Phi \left(\omega \right)\in {\mathbb{R}}^{3\times 6}$. In model (Section 2) we take a, $J_v$, $d_f$ and $d_{\tau }$ as unknown constants and denote $\widehat{a}\in \mathbb{R}>0$, ${\widehat{J}}_v\in {\mathbb{R}}^6$, ${\widehat{d}}_f\in {\mathbb{R}}^3$, and ${\widehat{d}}_{\tau }\in {\mathbb{R}}^3$ as their estimates. Estimation errors are defined as $\tilde{a}=a-\widehat{a}$, ${\tilde{J}}_v=J_v-{\widehat{J}}_v$, ${\tilde{d}}_f=d_f-{\widehat{d}}_f$, and ${\tilde{d}}_{\tau }=d_{\tau }-{\widehat{d}}_{\tau }$.

3. Control Design

This section presents the control design for the UAM model (4). Figure 2 shows the structure of the proposed control, where the arm is controlled separately using a PID control. In this figure, we denote $\alpha$ as the arm’s joint angles and their desired reference as ${\alpha }_d$. The proposed control compensates for the effect of the arm using an adaptive backstepping design method. The control objective is trajectory tracking for UAV CoM position p and UAV heading $\psi$. We first present the design for tracking p and then extend it for tracking $\psi$.

3.1. Position Tracking Control

Let $p_d,v_d\in {\mathbb{R}}^3$ represent the desired UAV CoM position and velocity, respectively. We define error coordinate$z_1=p-p_d$. Taking its time-derivative gives

$${\dot{z}}_1=\dot{p}-{\dot{p}}_d=v-v_d.$$

(6)

For this dynamics, consider the Lyapunov function $V_1={\displaystyle \frac{1}{2}}{z}_1^{\top }{z}_1$. Taking its time-derivative gives

$$\begin{array}{cc}\hfill \begin{array}{c}\hfill {\dot{V}}_1\end{array}& \begin{array}{c}\hfill =z_1^{\top }{\dot{z}}_1\end{array}\hfill \\ \hfill & \begin{array}{c}\hfill =z_1^{\top }(v-v_d).\end{array}\hfill & \hfill \begin{array}{c}\hfill (\mathrm{from}\left(6\right))\end{array}\end{array}$$

Adding and subtracting $k_1{z}_1^{\top }{z}_1$ to ${\dot{V}}_1$, where $k_1>0$ is a controller gain,we obtain

$${\dot{V}}_1=-k_1{\parallel z_1\parallel }^2+z_1^{\top }(v-v_d+k_1{z}_1).$$

(7)

From (7) we define $z_2=v-v_d+k_1{z}_1$ so we can rewrite (6) as

$${\dot{z}}_1=-k_1{z}_1+z_2.$$

(8)

Taking the time-derivative of $z_2$, substituting the translational velocity dynamics from (4b), using $a=\widehat{a}+\tilde{a}$, and $d_f={\widehat{d}}_f+{\tilde{d}}_f$, we arrive at

$${\dot{z}}_2=g{e}_3-\widehat{a}uR{e}_3+{\widehat{d}}_f-{\dot{v}}_d-k_1^2{z}_1+k_1{z}_2-\tilde{a}uR{e}_3+{\tilde{d}}_f.$$

(9)

For the above dynamics, consider the Lyapunov function $V_2=V_1+{\displaystyle \frac{1}{2}}{z}_2^{\top }{z}_2$. Its time derivative is

$${\dot{V}}_2={\dot{V}}_1+z_2^{\top }{\dot{z}}_2$$

In this expression, substituting for ${\dot{V}}_1$ from (7) and for ${\dot{z}}_2$ from (9), and simplifying, we arrive at

$${\dot{V}}_2=-k_1{\parallel z_1\parallel }^2+z_2^{\top }(g{e}_3-\widehat{a}uR{e}_3+{\widehat{d}}_f-{\dot{v}}_d-k_1^2{z}_1+k_1{z}_2-\tilde{a}uR{e}_3+{\tilde{d}}_f).$$

Adding and subtracting $k_2{z}_2^{\top }{z}_2$ in the above equation, where $k_2>0$ is a controller gain we obtain

$$\begin{array}{cc}\hfill {\dot{V}}_2=& -k_1\parallel z_1{\parallel }^2-k_2\parallel z_2{\parallel }^2+z_2^{\top }(g{e}_3-\widehat{a}uR{e}_3+{\widehat{d}}_f-{\dot{v}}_d+(1-k_1^2)z_1\hfill \\ & +(k_1+k_2)z_2)-\tilde{a}{z}_2^{\top }uR{e}_3+{\tilde{d}}_f^{\top }{z}_2.\hfill \end{array}$$

(10)

In the above equation, we define the coefficient of $z_2^{\top }$ as

$$z_3=g{e}_3-\widehat{a}uR{e}_3+{\widehat{d}}_f-{\dot{v}}_d+(1-k_1^2)z_1+(k_1+k_2)z_2,$$

(11)

which can be rearranged as

$$z_3-z_1-k_2{z}_2=g{e}_3-\widehat{a}uR{e}_3+{\widehat{d}}_f-{\dot{v}}_d-k_1^2{z}_1+k_1{z}_2.$$

Substituting this into (9) gives

$${\dot{z}}_2=z_3-z_1-k_2{z}_2-\tilde{a}uR{e}_3+{\tilde{d}}_f.$$

(12)

The time-derivative of (11) is

$${\dot{z}}_3=-\dot{\widehat{a}}uR{e}_3-\widehat{a}\dot{u}R{e}_3-\widehat{a}u\dot{R}{e}_3+{\dot{\widehat{d}}}_f-{\ddot{v}}_d+(1-k_1^2){\dot{z}}_1+(k_1+k_2){\dot{z}}_2.$$

Substituting (8) and (12) into the last equation, we get

$$\begin{array}{cc}\hfill {\dot{z}}_3=& -\dot{\widehat{a}}uR{e}_3-\widehat{a}\dot{u}R{e}_3-\widehat{a}uR\mathrm{S}\left(\omega \right)e_3+{\dot{\widehat{d}}}_f-{\ddot{v}}_d+(1-k_1^2)(-k_1{z}_1+z_2)\hfill \\ \hfill & +(k_1+k_2)(z_3-z_1-k_2{z}_2-\tilde{a}uR{e}_3+{\tilde{d}}_f).\hfill \end{array}$$

Simplifying leads us to

$$\begin{array}{cc}\hfill {\dot{z}}_3=& -\dot{\widehat{a}}uR{e}_3-\widehat{a}\dot{u}R{e}_3-\widehat{a}uR\mathrm{S}\left(\omega \right)e_3+{\dot{\widehat{d}}}_f-{\ddot{v}}_d\hfill \\ & +(k_1(k_1^2-1)-k_2-k_1)z_1+(1-k_2(k_1+k_2)-k_1^2)z_2\hfill \\ & +(k_1+k_2)z_3-(k_1+k_2)\tilde{a}uR{e}_3+(k_1+k_2){\tilde{d}}_f.\hfill \end{array}$$

(13)

For the above dynamics, consider the Lyapunov function $V_3=V_2+{\displaystyle \frac{1}{2}}{z}_3^{\top }{z}_3$ whose time-derivative is

$${\dot{V}}_3=-k_1\parallel z_1{\parallel }^2-k_2{\parallel z_2\parallel }^2+z_2^{\top }{z}_3+z_3^{\top }{\dot{z}}_3-\tilde{a}{z}_2^{\top }uR{e}_3+{\tilde{d}}_f^{\top }{z}_2.$$

Substituting (13) in the last equation leads to

$$\begin{array}{cc}\hfill {\dot{V}}_3=& -k_1\parallel z_1{\parallel }^2-k_2{\parallel z_2\parallel }^2+z_3^{\top }(z_2-\dot{\widehat{a}}uR{e}_3-\widehat{a}\dot{u}R{e}_3-\widehat{a}uR\mathrm{S}\left(\omega \right)e_3+{\dot{\widehat{d}}}_f-{\ddot{v}}_d\hfill \\ & +(k_1(k_1^2-1)-k_2-k_1)z_1+(1-k_2(k_1+k_2)-k_1^2)z_2+(k_1+k_2)z_3\hfill \\ & -(k_1+k_2)\tilde{a}uR{e}_3+(k_1+k_2){\tilde{d}}_f)-\tilde{a}{z}_2^{\top }uR{e}_3+{\tilde{d}}_f^{\top }{z}_2.\hfill \end{array}$$

Adding and subtracting $k_3{z}_3^{\top }{z}_3$ from the last expression, where $k_3>0$ is a controller gain, and simplifying gives

$$\begin{array}{cc}\hfill {\dot{V}}_3=& -k_1\parallel z_1{\parallel }^2-k_2\parallel z_2{\parallel }^2-k_3{\parallel z_3\parallel }^2+z_3^{\top }(-\dot{\widehat{a}}uR{e}_3-\widehat{a}\dot{u}R{e}_3-\widehat{a}uR\mathrm{S}\left(\omega \right)e_3+{\dot{\widehat{d}}}_f\hfill \\ & -{\ddot{v}}_d+(k_1(k_1^2-1)-k_2-k_1)z_1+(2-k_2(k_1+k_2)-k_1^2)z_2+(k_1+k_2+k_3)z_3)\hfill \\ & -\tilde{a}(z_2^{\top }+(k_1+k_2)z_3^{\top })uR{e}_3+{\tilde{d}}_f^{\top }(z_2+(k_1+k_2)z_3).\hfill \end{array}$$

Now, consider a modified Lyapunov function

$$V_{3b}=V_3+{\displaystyle \frac{1}{2\lambda }}{\tilde{a}}^2+{\displaystyle \frac{1}{2k_{d_f}}}{\tilde{d}}_f^{\top }{\tilde{d}}_f,$$

(14)

where $k_{d_f}>0,\lambda >0$are controller gains. The time-derivative of $V_{3b}$ is

$$\begin{array}{cc}\hfill {\dot{V}}_{3b}=& -k_1\parallel z_1{\parallel }^2-k_2\parallel z_2{\parallel }^2-k_3{\parallel z_3\parallel }^2+z_3^{\top }(-\dot{\widehat{a}}uR{e}_3-\widehat{a}\dot{u}R{e}_3-\widehat{a}uR\mathrm{S}\left(\omega \right)e_3+{\dot{\widehat{d}}}_f\hfill \\ & -{\ddot{v}}_d+(k_1(k_1^2-1)-k_2-k_1)z_1+(2-k_2(k_1+k_2)-k_1^2)z_2+(k_1+k_2+k_3)\hfill \\ & \times z_3)-\tilde{a}(z_2^{\top }+(k_1+k_2)z_3^{\top })uR{e}_3+{\tilde{d}}_f^{\top }(z_2+(k_1+k_2)z_3)+{\displaystyle \frac{1}{\lambda }}\tilde{a}\dot{\tilde{a}}+{\displaystyle \frac{1}{k_{d_f}}}{\tilde{d}}_f^{\top }{\dot{\tilde{d}}}_f.\hfill \end{array}$$

With some simplification, we get

$$\begin{array}{cc}\hfill {\dot{V}}_{3b}=& -k_1\parallel z_1{\parallel }^2-k_2\parallel z_2{\parallel }^2-k_3{\parallel z_3\parallel }^2+z_3^{\top }(-\dot{\widehat{a}}uR{e}_3-\widehat{a}\dot{u}R{e}_3-\widehat{a}uR\mathrm{S}\left(\omega \right)e_3\hfill \\ & +{\dot{\widehat{d}}}_f-{\ddot{v}}_d+(k_1(k_1^2-1)-k_2-k_1)z_1+(2-k_2(k_1+k_2)-k_1^2)z_2\hfill \\ \hfill & +(k_1+k_2+k_3)z_3)+\tilde{a}\left(\lambda \dot{\tilde{a}}-(z_2^{\top }+(k_1+k_2)z_3^{\top })uR{e}_3\right)\hfill \\ & +{\tilde{d}}_f^{\top }\left({\displaystyle \frac{1}{k_{d_f}}}{\dot{\tilde{d}}}_f+z_2+(k_1+k_2)z_3\right).\hfill \end{array}$$

(15)

We define ${\sigma }_1,{\sigma }_2\in {\mathbb{R}}^3$ to re-write (15):

$$\begin{array}{cc}{\sigma }_1=\hfill & (k_1(k_1^2-1)-k_2-k_1)z_1+(2-k_2(k_1+k_2)-k_1^2)z_2+(k_1+k_2+k_3)z_3-{\ddot{v}}_d,\hfill \end{array}$$

(16a)

$$\begin{array}{c}{\sigma }_2=z_2+(k_1+k_2)z_3.\hfill \end{array}$$

(16b)

Since $d_f$ is constant, we have ${\dot{\tilde{d}}}_f=-{\dot{\widehat{d}}}_f$. Substituting this equation in (15) and using the definitions of ${\sigma }_1,{\sigma }_2$, we obtain

$$\begin{array}{cc}\hfill {\dot{V}}_{3b}=& -k_1\parallel z_1{\parallel }^2-k_2\parallel z_2{\parallel }^2-k_3{\parallel z_3\parallel }^2+z_3^{\top }(-\dot{\widehat{a}}uR{e}_3-\widehat{a}\dot{u}R{e}_3\hfill \\ & -\widehat{a}uR\mathrm{S}\left(\omega \right)e_3+{\dot{\widehat{d}}}_f+{\sigma }_1)+\tilde{a}({\displaystyle \frac{1}{\lambda }}\dot{\tilde{a}}-{\sigma }_2^{\top }uR{e}_3)+{\tilde{d}}_f^{\top }(-{\displaystyle \frac{1}{k_{d_f}}}{\dot{\widehat{d}}}_f+{\sigma }_2).\hfill \end{array}$$

Adding and subtracting $k_{d_f}{z}_3^{\top }{\sigma }_2$ yields

$$\begin{array}{cc}\hfill {\dot{V}}_{3b}=& -k_1\parallel z_1{\parallel }^2-k_2\parallel z_2{\parallel }^2-k_3{\parallel z_3\parallel }^2+z_3^{\top }(-\dot{\widehat{a}}uR{e}_3-\widehat{a}\dot{u}R{e}_3-\widehat{a}uR\mathrm{S}\left(\omega \right)e_3\hfill \\ & +{\dot{\widehat{d}}}_f-k_{d_f}{\sigma }_2+k_{d_f}{\sigma }_2-{\ddot{v}}_d+{\sigma }_1)+\tilde{a}({\displaystyle \frac{1}{\lambda }}\dot{\tilde{a}}-{\sigma }_2^{\top }uR{e}_3)+{\tilde{d}}_f^{\top }(-{\displaystyle \frac{1}{k_{d_f}}}{\dot{\widehat{d}}}_f+{\sigma }_2),\hfill \end{array}$$

and simplification gives

$$\begin{array}{cc}\hfill {\dot{V}}_{3b}=& -k_1\parallel z_1{\parallel }^2-k_2\parallel z_2{\parallel }^2-k_3\parallel z_3{\parallel }^2+z_3^{\top }(-\dot{\widehat{a}}uR{e}_3-\widehat{a}\dot{u}R{e}_3\hfill \\ & -\widehat{a}uR\mathrm{S}\left(\omega \right)e_3+{\sigma }_1+k_{d_f}{\sigma }_2)-k_{d_f}{z}_3^{\top }(-{\displaystyle \frac{1}{k_{d_f}}}{\dot{\widehat{d}}}_f+{\sigma }_2)\hfill \\ & +\tilde{a}({\displaystyle \frac{1}{\lambda }}\dot{\tilde{a}}-{\sigma }_2^{\top }uR{e}_3)+{\tilde{d}}_f^{\top }(-{\displaystyle \frac{1}{k_{d_f}}}{\dot{\widehat{d}}}_f+{\sigma }_2).\hfill \end{array}$$

(17)

Observe

$$\begin{array}{cc}\hfill {\sigma }_1+k_{d_f}{\sigma }_2=& R{R}^{\top }({\sigma }_1+k_{d_f}{\sigma }_2)=RI{R}^{\top }({\sigma }_1+k_{d_f}{\sigma }_2)\hfill \\ \hfill =& R(I-e_3{e}_3^{\top }+e_3{e}_3^{\top })R^{\top }({\sigma }_1+k_{d_f}{\sigma }_2)\hfill \\ \hfill =& \left(R(I-e_3{e}_3^{\top })R^{\top }+R{e}_3{e}_3^{\top }{R}^{\top }\right)({\sigma }_1+k_{d_f}{\sigma }_2).\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill {\dot{V}}_{3b}=& -k_1\parallel z_1{\parallel }^2-k_2\parallel z_2{\parallel }^2-k_3{\parallel z_3\parallel }^2+z_3^{\top }(-\dot{\widehat{a}}uR{e}_3-\widehat{a}\dot{u}R{e}_3\hfill \\ & +R{e}_3{e}_3^{\top }{R}^{\top }({\sigma }_1+k_{d_f}{\sigma }_2)-\widehat{a}uR\mathrm{S}\left(\omega \right)e_3+R(I-e_3{e}_3^{\top })R^{\top }({\sigma }_1+k_{d_f}{\sigma }_2))\hfill \\ & -k_{d_f}{z}_3^{\top }(-{\displaystyle \frac{1}{k_{d_f}}}{\dot{\widehat{d}}}_f+{\sigma }_2)+\tilde{a}({\displaystyle \frac{1}{\lambda }}\dot{\tilde{a}}-{\sigma }_2^{\top }uR{e}_3)+{\tilde{d}}_f^{\top }(-{\displaystyle \frac{1}{k_{d_f}}}{\dot{\widehat{d}}}_f+{\sigma }_2).\hfill \end{array}$$

Let’s factor out $R{e}_3$ from the first three terms inside the parentheses of the $z_3^{\top }$ coefficient and equate them to zero. In other words, let’s impose $\widehat{a}\dot{u}+\dot{\widehat{a}}u-e_3^{\top }{R}^{\top }({\sigma }_1+k_{d_f}{\sigma }_2)=0$, which gives

$$\widehat{a}\dot{u}+\dot{\widehat{a}}u=e_3^{\top }{R}^{\top }({\sigma }_1+k_{d_f}{\sigma }_2).$$

(18)

Therefore,

$$\begin{array}{cc}\hfill {\dot{V}}_{3b}=& -k_1\parallel z_1{\parallel }^2-k_2\parallel z_2{\parallel }^2-k_3{\parallel z_3\parallel }^2+z_3^{\top }R(-\widehat{a}u\mathrm{S}\left(\omega \right)e_3\hfill \\ & +(I-e_3{e}_3^{\top })R^{\top }({\sigma }_1+k_{d_f}{\sigma }_2))+({\tilde{d}}_f^{\top }-k_{d_f}{z}_3^{\top })(-{\displaystyle \frac{1}{k_{d_f}}}{\dot{\widehat{d}}}_f+{\sigma }_2)\hfill \\ & +\tilde{a}({\displaystyle \frac{1}{\lambda }}\dot{\tilde{a}}-{\sigma }_2^{\top }uR{e}_3).\hfill \end{array}$$

(19)

The term $\widehat{a}u\mathrm{S}\left({\omega }_d\right)e_3$ in (19) can be interpreted as a virtual control, whose desired value is taken as

$$\widehat{a}u\mathrm{S}\left({\omega }_d\right)e_3=(I-e_3{e}_3^{\top })R^{\top }({\sigma }_1+k_{d_f}{\sigma }_2).$$

Since $\mathrm{S}\left({\omega }_d\right)e_3=-\mathrm{S}\left(e_3\right){\omega }_d$, the last equation can be rewritten as

$$\widehat{a}u\mathrm{S}\left(e_3\right){\omega }_d=-(I-e_3{e}_3^{\top })R^{\top }({\sigma }_1+k_{d_f}{\sigma }_2).$$

(20)

Extracting individual components of ${\omega }_d$, we have

$$\begin{array}{cc}\hfill {\omega }_{d1}=& -{\displaystyle \frac{e_2^{\top }}{\widehat{a}u}}(I-e_3{e}_3^{\top })R^{\top }({\sigma }_1+k_{d_f}{\sigma }_2),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {\omega }_{d2}=& {\displaystyle \frac{e_1^{\top }}{\widehat{a}u}}(I-e_3{e}_3^{\top })R^{\top }({\sigma }_1+k_{d_f}{\sigma }_2),\hfill \end{array}$$

(22)

while (19) becomes

$$\begin{array}{cc}\hfill {\dot{V}}_{3b}=& -k_1\parallel z_1{\parallel }^2-k_2\parallel z_2{\parallel }^2-k_3{\parallel z_3\parallel }^2+z_3^{\top }\left(-\widehat{a}uR\mathrm{S}\left(\omega \right)e_3+\widehat{a}uR\mathrm{S}\left({\omega }_d\right)e_3\right)\hfill \\ \hfill & +({\tilde{d}}_f^{\top }-k_{d_f}{z}_3^{\top })(-{\displaystyle \frac{1}{k_{d_f}}}{\dot{\widehat{d}}}_f+{\sigma }_2)+\tilde{a}({\displaystyle \frac{1}{\lambda }}\dot{\tilde{a}}-{\sigma }_2^{\top }uR{e}_3).\hfill \end{array}$$

Simplifying the last expression gives

$$\begin{array}{cc}\hfill {\dot{V}}_{3b}=& -k_1\parallel z_1{\parallel }^2-k_2\parallel z_2{\parallel }^2-k_3{\parallel z_3\parallel }^2+z_3^{\top }\widehat{a}uR\mathrm{S}\left(e_3\right)\left(\omega -{\omega }_d\right)\hfill \\ \hfill & +({\tilde{d}}_f^{\top }-k_{d_f}{z}_3^{\top })(-{\displaystyle \frac{1}{k_{d_f}}}{\dot{\widehat{d}}}_f+{\sigma }_2)+\tilde{a}({\displaystyle \frac{1}{\lambda }}\dot{\tilde{a}}-{\sigma }_2^{\top }uR{e}_3).\hfill \end{array}$$

(23)

Now, consider the following parameter update laws,

$$\begin{array}{cc}\hfill {\dot{\widehat{d}}}_f=& k_{d_f}{\sigma }_2,\hfill \\ \hfill \dot{\widehat{a}}=& -\lambda {\sigma }_2^{\top }uR{e}_3,\hfill \end{array}$$

(24)

Substituting these adaptive laws in (23), we have

$$\begin{array}{cc}\hfill {\dot{V}}_{3b}=& -k_1\parallel z_1{\parallel }^2-k_2\parallel z_2{\parallel }^2-k_3{\parallel z_3\parallel }^2+z_3^{\top }\widehat{a}uR\mathrm{S}\left(e_3\right)\left(\omega -{\omega }_d\right).\hfill \end{array}$$

(25)

Let the error in angular velocity be $z_4=\omega -{\omega }_d$. Using (4d), we can write

$$J{\dot{z}}_4=J\dot{\omega }-J{\dot{\omega }}_d=-\mathrm{S}\left(\omega \right)J\omega +\tau +d_{\tau }-J{\dot{\omega }}_d.$$

(26)

Define

$$\begin{array}{cc}\hfill {\overline{\omega }}_d=& -{\displaystyle \frac{1}{\widehat{a}u}}(\dot{\widehat{a}}u+\widehat{a}\dot{u})(I-e_3{e}_3^T){\omega }_d+{\displaystyle \frac{1}{\widehat{a}u}}{\mathrm{S}}^{\top }\left(e_3\right)\mathrm{S}\left(\omega \right)R^T({\sigma }_1+k_{d_f}{\sigma }_2)\hfill \\ & -{\displaystyle \frac{1}{\widehat{a}u}}{\mathrm{S}}^{\top }\left(e_3\right)R^T({\overline{\sigma }}_1+k_{d_f}{\overline{\sigma }}_2)+{\dot{\omega }}_{d3}{e}_3,\hfill \end{array}$$

Using $J_v$ instead of J, we have

$$\begin{array}{cc}\hfill J{\overline{\omega }}_d& =\Psi \left({\overline{\omega }}_d\right)J_v,\hfill \end{array}$$

(27)

where

$$\Psi \left({\overline{\omega }}_d\right)=\left[\begin{array}{cccccc}{\overline{\omega }}_{d1}& 0& 0& {\overline{\omega }}_{d2}& 0& {\overline{\omega }}_{d3}\\ 0& {\overline{\omega }}_{d2}& 0& {\overline{\omega }}_{d1}& {\overline{\omega }}_{d3}& 0\\ 0& 0& {\overline{\omega }}_{d3}& 0& {\overline{\omega }}_{d2}& {\overline{\omega }}_{d1}\end{array}\right],$$

Therefore, (26) can be rewritten as

$$J{\dot{z}}_4=-(\Phi \left(\omega \right)+\Psi \left({\dot{\omega }}_d\right))J_v+\tau +d_{\tau }.$$

(28)

Substituting $J_v={\widehat{J}}_v+{\tilde{J}}_v$ and $d_{\tau }={\widehat{d}}_{\tau }+{\tilde{d}}_{\tau }$ in (28) gives

$$J{\dot{z}}_4=-(\Phi \left(\omega \right)+\Psi \left({\dot{\omega }}_d\right))({\widehat{J}}_v+{\tilde{J}}_v)+\tau +{\widehat{d}}_{\tau }+{\tilde{d}}_{\tau }.$$

(29)

Now, consider the Lyapunov function

$$V_4=V_{3b}+{\displaystyle \frac{1}{2}}{z}_4^{\top }J{z}_4+{\displaystyle \frac{1}{2k_{d_{\tau }}}}{\tilde{d}}_{\tau }^{\top }{\tilde{d}}_{\tau }+{\displaystyle \frac{1}{2}}{\tilde{J}}_v^{\top }{\Gamma }^{-1}{\tilde{J}}_v,$$

(30)

where $k_{d_{\tau }}>0$, $\Gamma \in {\mathbb{R}}^{6\times 6}>0$ are controller gains. Differentiating the above equation with respect to time gives

$${\dot{V}}_4={\dot{V}}_{3b}+z_4^{\top }J{\dot{z}}_4+{\displaystyle \frac{1}{k_{d_{\tau }}}}{\tilde{d}}_{\tau }^{\top }{\dot{\tilde{d}}}_{\tau }+{\tilde{J}}_v^{\top }{\Gamma }^{-1}{\dot{\tilde{J}}}_v.$$

(31)

Substituting (25) and (29) into (31) gives

$$\begin{array}{cc}\hfill {\dot{V}}_4=& -k_1\parallel z_1{\parallel }^2-k_2\parallel z_2{\parallel }^2-k_3{\parallel z_3\parallel }^2+z_3^{\top }\widehat{a}uR\mathrm{S}\left(e_3\right)z_4\hfill \\ & +z_4^{\top }(-(\Phi \left(\omega \right)+\Psi \left({\dot{\omega }}_d\right))({\widehat{J}}_v+{\tilde{J}}_v)+\tau +{\widehat{d}}_{\tau }+{\tilde{d}}_{\tau })\hfill \\ & +{\displaystyle \frac{1}{k_{d_{\tau }}}}{\tilde{d}}_{\tau }^{\top }{\dot{\tilde{d}}}_{\tau }+{\tilde{J}}_v^{\top }{\Gamma }^{-1}{\dot{\tilde{J}}}_v.\hfill \end{array}$$

(32)

Let $k_4>0$ be a controller gain. Adding and subtracting $k_4{z}_4^{\top }{z}_4$ in the above expression, factoring $z_4^{\top }$, we obtain

$$\begin{array}{cc}\hfill {\dot{V}}_4=& -k_1\parallel z_1{\parallel }^2-k_2\parallel z_2{\parallel }^2-k_3\parallel z_3{\parallel }^2-k_4{\parallel z_4\parallel }^2\hfill \\ & +z_4^{\top }(\widehat{a}u{R}_b^{n^{\top }}{\mathrm{S}}^{\top }\left(e_3\right)z_3+k_4{z}_4-(\Phi \left(\omega \right)+\Psi \left({\dot{\omega }}_d\right)){\widehat{J}}_v+\tau +{\widehat{d}}_{\tau })\hfill \\ & +{\tilde{J}}_v^{\top }({\Gamma }^{-1}{\dot{\tilde{J}}}_v-{(\Phi \left(\omega \right)+\Psi \left({\dot{\omega }}_d\right))}^{\top }{z}_4)+{\tilde{d}}_{\tau }^{\top }({\displaystyle \frac{1}{k_{d_{\tau }}}}{\dot{\tilde{d}}}_{\tau }+z_4).\hfill \end{array}$$

(33)

At this point, we assign the expression for input $\tau$:

$$\tau =-\widehat{a}u{R}_b^{n^{\top }}{\mathrm{S}}^{\top }\left(e_3\right)z_3-k_4{z}_4+(\Phi \left(\omega \right)+\Psi \left({\dot{\omega }}_d\right)){\widehat{J}}_v-{\widehat{d}}_{\tau },$$

(34)

and define the parameter update laws

$$\begin{array}{cc}\hfill {\dot{\widehat{J}}}_v=& -\Gamma {(\Phi \left(\omega \right)+\Psi \left({\dot{\omega }}_d\right))}^{\top }{z}_4,\hfill \\ \hfill {\dot{\widehat{d}}}_{\tau }=& k_{d_{\tau }}{z}_4.\hfill \end{array}$$

(35)

Substituting (34) and (35) in (33), and recalling that $J_v$ and $d_{\tau }$ are constants, we get

$${\dot{V}}_4=-k_1\parallel z_1{\parallel }^2-k_2\parallel z_2{\parallel }^2-k_3\parallel z_3{\parallel }^2-k_4{\parallel z_4\parallel }^2,$$

(36)

which is negative semi-definite.

Theorem 1. 

Consider the UAM dynamics (4) and assume $\dot{a}=0$, ${\dot{d}}_f=0$, ${\dot{d}}_{\tau }=0$, ${\dot{J}}_v=0$, and $u>0$, $\widehat{a}>0$. Given a smooth time-varying trajectory $p_d\left(t\right)$, feedback law (34) with $k_1,k_2,k_3,k_4>0$, and parameter update laws (24), (35), then the tracking error $z^{\top }=[z_1^{\top },z_2^{\top },z_3^{\top },z_4^{\top }]$ is globally asymptotically convergent to 0 and parameter estimates $\widehat{a},{\widehat{d}}_f,{\widehat{d}}_{\tau }$ and ${\widehat{J}}_v$ are bounded.

Proof. 

Consider the quadratic positive definite Lyapunov function $V_4$ in (30). According to [29] Lemma 4.3, there exist two positive definite functions that lower and upper bound $V_4$. The time-derivative of $V_4$ after substitution of (34), (24), (35), is a negative semi-definite function. Using Barbalat’s lemma [29] Theorem 8.4, we can conclude $z\to 0$ as $t\to \infty$, and parameter and disturbance estimates $\widehat{a}$, ${\widehat{d}}_f$, ${\widehat{d}}_{\tau }$ and ${\widehat{J}}_v$ are bounded. Since $V_4$ is radially unbounded, we conclude that the equilibrium point $z=0$ is globally asymptotically stable. □

3.2. Position and Yaw Tracking Control

Thus far, the adaptive backstepping control design achieves trajectory tracking for p. Since the UAV has 4 control inputs, we can extend the control to track UAV heading, which is practically relevant for manipulation tasks. We identify UAV heading with yaw angle $\psi \in (-\pi ,\pi ]$. Let the desired UAV yaw trajectory be ${\psi }_d\in (-\pi ,\pi ]$ and define tracking error $z_{\psi }=\psi -{\psi }_d$. Let

$$V_{\psi }={\displaystyle \frac{1}{2}}{z}_{\psi }^2.$$

(37)

Thus,

$${\dot{V}}_{\psi }=z_{\psi }{\dot{z}}_{\psi }=z_{\psi }(\dot{\psi }-{\dot{\psi }}_d)$$

It can be computed that $\dot{\psi }=e_3^{\top }{W}^{-1}\omega$, where for 3-2-1 Euler angles parameterization, we have

$$W=\left[\begin{array}{ccc}1& 0& -sin\theta \\ 0& cos\varphi & sin\varphi cos\theta \\ 0& -sin\varphi & cos\varphi cos\theta \end{array}\right]$$

and $\theta$, $\varphi$ represent UAV pitch, roll, respectively. Therefore, we can write

$${\dot{V}}_{\psi }=z_{\psi }{\dot{z}}_{\psi }=z_{\psi }(e_3^{\top }{W}^{-1}\omega -{\dot{\psi }}_d)$$

Adding and subtracting $k_{\psi }{z}_{\psi }^2$, where $k_{\psi }>0$ is a control gain, gives

$${\dot{V}}_{\psi }=-k_{\psi }{z}_{\psi }^2+z_{\psi }(k_{\psi }{z}_{\psi }+e_3^{\top }{W}^{-1}\omega -{\dot{\psi }}_d)$$

(38)

Define

$$e_3^{\top }{W}^{-1}{\omega }_d=-k_{\psi }{z}_{\psi }+{\dot{\psi }}_d$$

(39)

which gives

$${\omega }_{d3}={\displaystyle \frac{{\mathrm{c}}_{\theta }}{{\mathrm{c}}_{\varphi }}}(-k_{\psi }{z}_{\psi }+{\dot{\psi }}_d+{\displaystyle \frac{{\mathrm{s}}_{\varphi }}{{\mathrm{c}}_{\theta }}}{\omega }_{d2}),$$

(40)

where ${\omega }_{d2}$ is given in (22). Substituting (39) in (38) results in

$$\begin{array}{cc}\hfill {\dot{V}}_{\psi }& =-k_{\psi }{z}_{\psi }^2+z_{\psi }\left(e_3^{\top }{W}^{-1}(\omega -{\omega }_d)\right)\hfill \\ \hfill & =-k_{\psi }{z}_{\psi }^2+z_4^{\top }{z}_{\psi }{W}^{-1^{\top }}{e}_3.\hfill \end{array}$$

(41)

We are now ready to state the stability result for the position-yaw tracking controller.

Theorem 2. 

Consider the UAM dynamics (4) and assuming $\dot{a}=0$, ${\dot{d}}_f=0$, ${\dot{d}}_{\tau }=0$ and ${\dot{J}}_v=0$ and $u>0$, $\widehat{a}>0$. Given a smooth trajectory $p_d\left(t\right),{\psi }_d\left(t\right)$, control (18) and

$$\tau =-z_{\psi }{W}^{-T}{e}_3-\widehat{a}u{R}_b^{n^{\top }}{\mathrm{S}}^{\top }\left(e_3\right)z_3-k_4{z}_4+(\Phi \left(\omega \right)+\Psi \left({\dot{\omega }}_d\right)){\widehat{J}}_v-{\widehat{d}}_{\tau }$$

(42)

with $k_1>0,k_2>0,k_3>0,k_4>0$, and parameter update laws (24),(35), then tracking error $z^{\top }=[z_1^{\top },z_2^{\top },z_3^{\top },z_4^{\top },z_{\psi }]$ converges asymptotically to 0 while estimates $\widehat{a},{\widehat{d}}_f,{\widehat{d}}_{\tau }$ and ${\widehat{J}}_v$ are bounded.

Proof. 

Consider the Lypunov function $V_{4,\psi }=V_4+V_{\psi }$ with time derivative

$${\dot{V}}_{4,\psi }={\dot{V}}_4+{\dot{V}}_{\psi ,1}.$$

(43)

Substituting for ${\dot{V}}_4$ using (33), using parameter update laws (35), and the expression for ${\dot{V}}_{\psi }$ in (41), gives

$$\begin{array}{cc}\hfill {\dot{V}}_{4,\psi }=& -k_1\parallel z_1{\parallel }^2-k_2\parallel z_2{\parallel }^2-k_3\parallel z_3{\parallel }^2-k_4{\parallel z_4\parallel }^2-k_{\psi }{z}_{\psi }^2+z_4^{\top }{z}_{\psi }{W}^{-1^{\top }}{e}_3\hfill \\ & +z_4^{\top }(\widehat{a}u{R}_b^{n^{\top }}{\mathrm{S}}^{\top }\left(e_3\right)z_3+k_4{z}_4-(\Phi \left(\omega \right)+\Psi \left({\dot{\omega }}_d\right)){\widehat{J}}_v+\tau +{\widehat{d}}_{\tau })\hfill \end{array}$$

(44)

Substituting (42) gives

$$\begin{array}{cc}\hfill {\dot{V}}_{4,\psi }=& -k_1\parallel z_1{\parallel }^2-k_2\parallel z_2{\parallel }^2-k_3\parallel z_3{\parallel }^2-k_4\parallel z_4{\parallel }^2-k_{\psi }{\parallel z_{\psi }\parallel }^2\hfill \end{array}$$

(45)

which is negative semi-definite. As in the proof of Theorem 1, we conclude $z\to 0$ as $t\to \infty$ and parameters remain bounded. □

Remark 1. 

The controllers presented in (34) and (42) depend on ${\dot{\omega }}_d$, however, the above derivation gives expression for ${\omega }_d$. The value of${\dot{\omega }}_d$ can be obtained numerically by low-pass filtered finite difference. The algebraic expression for ${\omega }_d$ depends on ${\tilde{d}}_f$ which is not known. As well, the unknown inertia vector $J_v$ in the angular velocity dynamics makes the stability result difficult to prove. Work in [26] provides further details for alternate solutions to this problem.

4. Simulation Results

This section presents numerical simulations using MATLAB Simscape Multibody to validate our control design. The files to generate these simulations can be found at https://github.com/ANCL/UAM_control. A multibody simulation is necessary to avoid the complexity of simulating the closed-loop in a traditional state-space format. The UAM used in the simulation is shown in Figure 3. The arm is a 3-link 2-DoF manipulator with two revolute joints. Link 1 is attached to the CoM of the UAV and rotates in the $b_1-b_3$ plane about the origin of $\mathcal{B}$. Link 2 is a revolute joint which also rotates in the same plane. Link 3 is fixed to Link 2 and acts as a payload mass. The arm parameters are given in Table 2. The UAV parameters are given in Table 1. We consider three simulations: a figure-8 desired position with fixed and moving arm cases, and a pick and place operation.

Table 1. UAV parameters.

Parameter	Value	Parameter	Value
m	1.6 $\mathrm{k}\mathrm{g}$	$J_{xx}$, $J_{yy}$	0.03 $\mathrm{k}\mathrm{g} {\mathrm{m}}^2$
$J_{zz}$	0.05 $\mathrm{k}\mathrm{g} {\mathrm{m}}^2$	$J_{xy}$, $J_{zx}$, $J_{yz}$	0.0 $\mathrm{k}\mathrm{g} {\mathrm{m}}^2$

Table 1. UAV parameters.

Parameter	Value	Parameter	Value
m	1.6 $\mathrm{k}\mathrm{g}$	$J_{xx}$, $J_{yy}$	0.03 $\mathrm{k}\mathrm{g} {\mathrm{m}}^2$
$J_{zz}$	0.05 $\mathrm{k}\mathrm{g} {\mathrm{m}}^2$	$J_{xy}$, $J_{zx}$, $J_{yz}$	0.0 $\mathrm{k}\mathrm{g} {\mathrm{m}}^2$

Table 2. Arm parameters.

Parameter	Link 1	Link 2	Link 3
Joint Type	Revolute	Revolute	Fixed
Mass ($\mathrm{k}\mathrm{g}$)	0.1	0.1	0.1
Length ($\mathrm{m}$)	0.2	0.3	0.1
Inertia ($\mathrm{k}\mathrm{g}{\mathrm{m}}^2$)	$[7.508,7.533,0.0417]\times {10}^{-4}$	$[7.508,7.533,0.0417]\times {10}^{-4}$	$[1.667,1.667,1.667]\times {10}^{-4}$

Table 2. Arm parameters.

Parameter	Link 1	Link 2	Link 3
Joint Type	Revolute	Revolute	Fixed
Mass ($\mathrm{k}\mathrm{g}$)	0.1	0.1	0.1
Length ($\mathrm{m}$)	0.2	0.3	0.1
Inertia ($\mathrm{k}\mathrm{g}{\mathrm{m}}^2$)	$[7.508,7.533,0.0417]\times {10}^{-4}$	$[7.508,7.533,0.0417]\times {10}^{-4}$	$[1.667,1.667,1.667]\times {10}^{-4}$

4.1. Figure-8 Reference Trajectory

For the figure-8 trajectory, in addition to the coupling forces and torques due to the arm, we consider a constant external disturbance force of ${[0.3,0.3,0.3]}^{\top }\mathrm{N}$} acting at the UAV CoM. The controller gains used for all simulations are in Table 3. Initial conditions are $p\left(0\right)={[-2,2,0]}^{\top }$$\mathrm{m}$, $v\left(0\right)=0$$\mathrm{m}/\mathrm{s}$, $\eta \left(0\right)=0$$\mathrm{rad}$ and $\omega \left(0\right)=0$$\mathrm{rad}/\mathrm{s}$, $\widehat{a}\left(0\right)={\displaystyle \frac{1}{0.7m}}$, and ${\widehat{J}}_v\left(0\right)=0.6J_v$ i.e., we consider a 30% error in mass and a 40% error in inertia initially. The initial conditions for the disturbances are ${\widehat{d}}_f\left(0\right)={\widehat{d}}_{\tau }\left(0\right)=0$.

Figure 4a–f present the simulation results for the fixed arm case ${\alpha }_1={\alpha }_2=0$ where the arm is locked in the $b_3$ direction. The position tracking errors in Figure 4b converge to a small neighbourhood of 0. Estimates $\widehat{a}$, ${\widehat{d}}_f$, ${\widehat{d}}_{\tau }$, ${\widehat{J}}_v$ remain bounded. These estimates are shown in Figure 4c–e. We observe that in steady-state these parameters are time-varying due to the time-varying reference trajectory. That is, the UAV must vary its attitude to track the position trajectory, and this creates a time-varying disturbance. Increasing controller gains leads to faster tracking error convergence and smaller ultimate error bounds caused by time-varying disturbances.Figure 4f shows the control inputs.

The simulation results for the moving arm case are given in Figure 5a–f. The arm motion is taken as ${\alpha }_1\left(t\right)={\displaystyle \frac{\pi }{3}}sin\left(\pi t\right)$ and ${\alpha }_2\left(t\right)={\displaystyle \frac{3\pi }{2}}t$. Figure 5b shows the UAV CoM position errors, which admit small oscillations around the origin. Relative to the fixed arm case, we observe larger error in steady-state. However, from a practical perspective, the error is not significant. The estimated parameters and disturbances are bounded and shown in Figure 5c–e. As in the fixed arm case, these parameters are time-varying in steady state, and large variations are observed. Figure 5f shows the control inputs.

We now present a practical pick and place motion where the UAM takes off to a target position, extends its arm, picks up a payload, flies to a desired location, and places the payload at the desired location. Multiple “table” landmarks are created in MATLAB Simscape Multibody that correspond to the various locations of the task. A small cubic payload of mass $0.1\mathrm{k}\mathrm{g}$ and dimension $0.1\mathrm{m}$ is attached to the end-effector. The UAV CoM start location is $p\left(0\right)={[-2,2,0]}^{\top }\mathrm{m}$. The first and second desired UAV CoM positions and yaw are denoted Location 1 and Location 2, respectively. Location 1 is $p_d^1={[1.4379,2.0000,-1.4121]}^{\top }\mathrm{m},{\psi }_d^1=0$ $\mathrm{rad}$ and Location 2 is $p_d^2={[-2,-1.4379,-0.9121]}^{\top }\mathrm{m}$, ${\psi }_d^2=-\pi /2$ $\mathrm{rad}$. When picking up the payload, the UAV position is lowered using a function $h\left(t_0\right)=[0,0,0.1sin(\pi \left(t_0\right)/5)){]}^{\top }$. Table 4 shows the desired trajectory for the UAV and arm. The Home configuration for the arm is selected to be ${\alpha }_1=-\pi /4,{\alpha }_2=3\pi /4$. The desired values of p and $\psi$ are shown in Figure 6a. The position errors are given in Figure 6b. The figures label each phase of the task. Evidently, the figures show tracking error is asymptotically convergent as expected by the theory. In this task, the assumptions of Theorem 2 hold as desired position and arm configuraiton is eventualy constant. Hence, this simulation case illustrates a practical case where assumptions hold.

5. Conclusion

In this paper, we have presented a control design for a UAM consisting of a UAV and multi-DoF robotic arm. The design is based on a UAV model that accounts for the arm’s effect by including torque and force disturbances and parametric uncertainty. An adaptive backstepping control law is presented that achieves UAV CoM position and yaw tracking when disturbances and parameters are constant. Numerical simulations demonstrate that bounded tracking error results when disturbances and parameters are varying. A practical pick and place task illustrates a case where tracking error converges. Future work involves indoor and output experimental validation of the control with an open source platform as in [30]. An intermediate step for this validation is software-in-the-loop testing that includes various non-idealities such as actuator saturation, sampling effects, state estimation, and rotor drag [31].