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Abstract

Arm mounted unmanned aerial vehicles provide more feasible and attractive solution to manipulate objects in remote areas where access to arm mounted ground vehicles is not possible. In this research, an under-actuated quadrotor unmanned aerial vehicle model equipped with gripper is utilized to grab objects from inaccessible locations. A dual control structure is proposed for controlling and stabilization of the moving unmanned aerial vehicle along with the motions of the gripper. The control structure consists of model reference adaptive control augmented with an optimal baseline controller. Although model reference adaptive control deals with the uncertainties as well as attitude controlling of unmanned aerial vehicle, baseline controller is utilized to control the gripper, remove unwanted constant errors and disturbances during arm movement. The proposed control structure is applied in 6-degree-of-freedom nonlinear model of a quadrotor unmanned aerial vehicle equipped with gripper having (2 degrees of freedom) robotic limb; it is applicable for the simulations to desired path of unmanned aerial vehicle and to grasp object. Moreover, the efficiency of the presented control structure is compared with optimal baseline controller. It is observed that the proposed control algorithm has good transient behavior, better robustness in the presence of continuous uncertainties and gripper movement involved in the model of unmanned aerial vehicle.

Keywords

Adaptive control baseline controller gripper unmanned aerial vehicle and model reference adaptive control with integrator

Introduction

In modern era, unmanned aerial vehicles (UAVs) are hired in a wide range of military and civilian applications. It includes long duration of aerographic sampling, localization, detection of pollution, landscape, path-planning and finding objects in remote areas. Modern UAVs require more precise and accurate tracking control to perform the desired task.^1–3

To reduce its cost, weight factor, energy conservation and improved system performance, it is commonly designed to be an under-actuated UAV, such that it can achieve the same control properties as a fully actuated UAV with less input control laws by Madani et al. (2006).⁴ In addition, a fully actuated UAV along with failed actuators is considered as under-actuated system, which is controlled by an under-actuated control algorithm.

Aerospace robots are concerned; there are two wide categories that robots fall in it, one is remotely operated UAV and other is autonomous operated UAV. Remotely operated vehicles are very common in the aerospace domain. The underwater robots that are able to swim in water are also very common for naval and underwater photography application, which is explained by Nicholson and Healey⁵ and Wynn et al.⁶ An emerging area of research is to apply robotics in the aerographic domain as well. Aerospace robotics has become very popular for many reasons, one among these is to access remote locations and grab objects.

This research proposes a model, controlling of rotorcraft, which is equipped with a gripper and we commonly called it as gripper. The main purpose of the attachment of gripper is that the aerospace robot should be able to grasp and carry payload from remote locations to desired destination.⁷ UAVs equipped with a gripper is very common nowadays, and they have high degree of freedom (DOF) more than six that could prompt to transmuting applications, like fire frightening, infrastructure development and carry payload. We require a perfect precise, accurate controller for continuously moving UAV along with moving gripper end effector with respect to a protest of intrigue definitely enough for handling payload. The movement of the gripper affects the dynamic stability of UAV, additionally muddling the locations from previous works.^8–11

Previously, many researchers have worked on UAV equipped with gripper through 1-DOF to 9-DOF.¹² The host UAV is responsible to offer the extension for the manipulation of DOF. And their results are valuable; some of them use simple grasping instead of ambidextrous handling.¹³

The research work presents an innovative gripper-based UAV model which is controlled by using adaptive hybrid control structure, and its stability is also investigated. Our proposed UAV contains 8 DOF along with three manipulators: 6-DOF for UAV and 2-DOF for the gripper. Previously, an adaptive sliding mode controller (ASMC) was designed for the quad rotor aerial manipulation with 2-DOF gripper by Kim et al.¹⁴ A two-layer-based novel controller was proposed for controlling the inverse kinematic of quadrotor UAV equipped with a manipulator, which is designed by Arleo et al.¹⁵ The design and control of a hyper-redundant manipulator of UAV have 6 DOF, in which 6 DOF is for UAV system and 3 DOF is for the gripper, which was proposed by Danko and Oh.¹⁶

If there are accumulation uncertainties in the dynamics of system, it will deteriorate the preferred close-loop performance of the system and then recovers the preferred performance by expanding the optimal baseline controller with model reference adaptive control (MRAC).

The main innovations of this research are as follows: (1) a dual control structure MRAC augmented with an optimal baseline controller which consists of adaptive control and linear quadratic regulator (LQR) with proportional integral (PI) feedback connection; (2) the baseline controller is able to control under minimal conditions—it means that it does not able to operate under uncertainties in the system. But it would be able to handle the constant disturbances in the system and perform the desired task; (3) the fine tuning of the system is done by using adaptive control laws and its stability is dealt by Lyapunov equation; (4) to minimize the Riccati equation for the calculation of state feedback gain by using optimal matrices is a major design issue in optimal baseline controller.

The content of this manuscript is structured as follows. Section “System model” discusses the complete model of quadrotor UAV equipped with gripper, which is followed by section “Designing of control architecture” in which designing of control architecture is presented, and the “Dual control approach” and dynamic effects of gripper are also discussed in this section. In section “Simulation results and discussions,” the validity of control structure simulations and experiments are defined. Finally, section “Conclusion” concludes the whole manuscript.

System model

The movements of quadrotor UAV along with 6 DOF are defined as in Figure 1.

Figure 1.

Movements of quadrotor aerial vehicle.¹⁶

Assumption 1

The free body diagram and movements of aerial vehicle along $(X, Y, Z)$ are global axes and their Euler angles $(φ, θ, ψ)$ are roll, pitch and yaw, which are defined in Figure 2 and have been taken from Nicol et al.¹⁷

Figure 2.

Supposition of rotor movements.

The fixed body Earth frame, roll, pitch, yaw and rotor movements are defined in Figure 2.

Remark 1

The drag and thrust coefficients must be taken as constant, as well as drag and rolling moments are neglected

{\begin{matrix} τ_{1} = b δ_{i}^{2} \\ q_{1} = d δ_{i}^{2} \end{matrix}

(1)

where d is the drag factor and b is the thrust factor.

Assumption 2

If the propeller (rotors) velocities are varying slowly, its acceleration is insignificant for the dynamics of the system.

Following are the control terminologies of the UAV in which “ $δ$ ” is the rotor speed. $u_{1}$ is the vertical force input, $u_{2}$ is the roll actuator input, $u_{3}$ is the pitch actuator input and $u_{4}$ is the yaw moment input. The state-space model of the nonlinear equations of 6-DOF quadrotor is explained by Tanveer et al.¹⁸

{\begin{matrix} u_{1} = b (δ_{1}^{2} + δ_{2}^{2} + δ_{3}^{2} + δ_{4}^{2}) \\ u_{2} = b (δ_{4}^{2} - δ_{2}^{2}) \\ u_{3} = b (δ_{3}^{4} - δ_{1}^{2}) \\ u_{4} = b (δ_{2}^{2} + δ_{4}^{2} - δ_{1}^{2} - δ_{3}^{2}) \end{matrix}

(2)

The translational movements are defined in Equation (3), which is used to control the translational velocities of quadrotor aerial robot.¹⁹ The real-time constraints of embedded control loop are as follows

f {(x, u)}_{tran .} = {\begin{matrix} \overset{\cdot}{x} \\ \frac{\cos φ \sin θ \cos ψ + \sin φ \sin ψ}{m} * u_{1} \\ \overset{\cdot}{y} \\ \frac{\cos φ \sin θ \sin ψ + \sin φ \cos ψ}{m} * u_{1} \\ \overset{\cdot}{z} \\ - g + \frac{\cos φ \cos θ}{m} * u_{1} \end{matrix}

(3)

Rotational movements are defined in Equation (4) for the controlling of rotational velocities of quadrotor and can be obtained as follows

f {(x, u)}_{rot .} = {\begin{matrix} \overset{\cdot}{φ} \\ \overset{\cdot}{θ} \overset{\cdot}{ψ} (\frac{I_{y} - I_{z}}{I_{x}}) + \frac{1}{I_{x}} * u_{2} \\ \overset{\cdot}{θ} \\ \overset{\cdot}{φ} \overset{\cdot}{ψ} (\frac{I_{z} - I_{x}}{I_{y}}) + \frac{1}{I_{y}} * u_{3} \\ \overset{\cdot}{ψ} \\ \overset{\cdot}{φ} \overset{\cdot}{θ} (\frac{I_{x} - I_{y}}{I_{z}}) + \frac{1}{I_{z}} * u_{4} \end{matrix}

(4)

Newton’s equation of motion states that the rotational dynamics of any rigid body can be obtained as, $τ = I \overset{\cdot}{δ} + δ * (I ω)$ , where $I$ is the total inertia and $δ$ is the angular moment. The sum of all turning effect of the rotor craft is $τ$ which is equal to the sum of input torque, external torque and it is obtained as $τ = τ_{u} + τ_{ext}$ . The linear dynamics of the system are $\overset{\cdot\cdot}{T} = F / m$ , where $m$ is the total weight or mass of the UAV along with gripper and $F$ is the control and external forces that act in a frame of rotor craft. The free body structure of quadrotor UAV having 6 DOF along with gripper having 2 DOF is shown in Figure 3. Driving the model of the UAV input of all rotors and its rotational subsystem was previously used by Papachristos and colleagues.^20–22 The state-space vector, its rotational subsystem and input of all the rotors are used by Bouabdallah et al.,²³ such that the state-space vector is “x” and the I/O vector of UAV is obtained as “ $u_{S} (t)$ ” and “ $y_{S} (t)$ ”

{\begin{matrix} {\overset{\cdot}{x}}_{S} (t) = f (x_{S} (t), u_{S} (t)) \\ y_{S} (t) = h (x_{S} (t)) \end{matrix}

(5)

Figure 3.

(a) Free body diagram of quadrotor aerial vehicle equipped with gripper²⁴ and (b) actual UAV model.

Transformation matrix between the angular velocities $(p, q, r)$ of the system and the Euler angles $(\overset{\cdot}{φ}, \overset{\cdot}{θ}, \overset{\cdot}{ψ})$ is considered as unity matrix.

Assumption 3

Simulated results show that the supposition $[(p, q, r) ≅ (\overset{\cdot}{φ}, \overset{\cdot}{θ}, \overset{\cdot}{ψ})]$ is effective and the states of state-space model can be obtained as follows

{\begin{matrix} x_{S} (t) = [φ \overset{\cdot}{φ} θ \overset{\cdot}{θ} ψ \overset{\cdot}{ψ}] \\ u_{S} (t) = [u_{1}, u_{2}, u_{3}, u_{4}] \\ y_{S} (t) = [p, q, r] \end{matrix}

(6)

Rotation rates including motors input are used to define linearization of physical model of UAV. In the initial stage, steady-state solution must be taken in order to linearize the nonlinear behavior of the system by converging it to zero. In the simplest way, the UAV must linearize to the possible equilibrium points which is as follows

\begin{matrix} {x'}_{So} (t), {u'}_{So} (t) \\ f ({x'}_{So} (t), {u'}_{So} (t)) \equiv 0 \end{matrix}

(7)

Remark 2

The trim point conditions are suitable to linearize the system. However, the operating points of the UAV is are by the input torque on the rotors

{\begin{matrix} A_{m, n} = \frac{δ f_{m} ({x'}_{So} (t), {u'}_{So} (t))}{δ x_{n}} \\ B_{m, n} = \frac{δ f_{m} ({x'}_{So} (t), {u'}_{So} (t))}{δ U_{n}} \\ C_{m, n} = \frac{δ h_{m} ({x'}_{So} (t))}{δ x_{n}} \end{matrix}

(8)

Nonlinear equations of the system are linearized by using Equation (8). The outcome of the final state-space model is eliminated due to non-controllable input or output. State-space model is structurally minimized by the elimination of non-minimal state dynamics, left with the rotational subsystem

{\begin{matrix} A_{s} X + B_{s} u (t) = [\begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \overset{\cdot}{ψ} (\frac{I_{y} - I_{z}}{I_{x}}) & 0 & \overset{\cdot}{θ} (\frac{I_{y} - I_{z}}{I_{x}}) \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & \overset{\cdot}{ψ} (\frac{I_{z} - I_{x}}{I_{y}}) & 0 & 0 & 0 & \overset{\cdot}{φ} (\frac{I_{z} - I_{x}}{I_{y}}) \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & \overset{\cdot}{θ} (\frac{I_{x} - I_{y}}{I_{z}}) & 0 & \overset{\cdot}{φ} (\frac{I_{x} - I_{y}}{I_{z}}) & 0 & 0 \end{matrix}] [\begin{matrix} φ \\ \overset{\cdot}{φ} \\ θ \\ \overset{\cdot}{θ} \\ ψ \\ \overset{\cdot}{ψ} \end{matrix}] + [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & \frac{1}{I_{x}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{I_{y}} & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{I_{z}} \end{matrix}] [\begin{matrix} u_{1} \\ u_{2} \\ \begin{matrix} u_{3} \\ u_{4} \end{matrix} \end{matrix}] \\ Y = C = [\begin{matrix} \begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \end{matrix} \end{matrix}] [\begin{matrix} p \\ q \\ r \end{matrix}] \end{matrix}

(9)

Remark 3

The direct transition matrix D converges to zero due to no direct coupling and output of the system. The output matrix C consists of the measurement of three variables. The introduction of the UAV is edified by utilizing Euler angles that are used to variate the altitude and attitude rotating at global axis.²⁵

Designing of control architecture

The designing of the dual controller approach is to fulfill the desired manipulation job precisely and accurately, by designing the controller which must be able to satisfy some requirements: first, to handle external disturbances and give better robustness, because at the time of flying, the gripper can suddenly carry payload or no payload and it will deteriorate the stability of UAV. Hereafter, controller must be robust to control these sudden effects in the system. Furthermore, another main feature to design the controller is adaptability. If uncertainties lie in the system, it will destroy the desired performance as well as the stability of the system. At the time of gripping or discharging, the desired object can modify the physical quantities of the link of gripper.^26,27 Then, adaptability with respect to these effects is essential to fulfill the desired task successfully. Figure 4 shows the block diagram of adaptive-based dual controller approach.²⁸

Figure 4.

Block diagram of model-based adaptive hybrid controller.

Dual controller approach

Mostly, autonomous systems include autonomous underwater vehicles (AUVs) and UAVs are dealt by the MRAC. The dual controller which is based on adaptive controller is augmented with linear PI feedback controller.²⁸ The augmented architecture opposes adaptive stems from the fact that it is most realistic applications, a system that already has a model-based controller, which is designed to contain proportional as well as feedback integral connection. Such proposed model-based controller has been able to operate under no uncertainty conditions. It will be able to reject the unknown disturbances and would be able to track the desired or referred trajectory with minimum errors. By adding the uncertainty constant in the system, it will interrupt the desired performance of the system. The desired performance of the system is recovered by augmenting the model controller with an adaptive-based dual controller approach.²⁹

With respect to the nonlinear Multiple-Input-Multiple-Output (MIMO) system model of the UAV along with gripper, the system dynamics are already linearized. The uncertainties in the system dynamics which satisfy the matching conditions and the system dynamics are as follows

{\begin{matrix} {\overset{\cdot}{x}}_{s} (t) = A_{s} x_{s} (t) + B_{s} β (u_{s} (t) + F (x_{s} (t)) + τ_{u} \\ y_{s} (t) = C_{s} x_{s} (t) \end{matrix}

(10)

where x_s (t) is the time-dependent state variable of UAV; y_s (t) is the time-dependent output of UAV; A_s, B_s, C_s are matrices of UAV; u_s(t) is the time-dependent controller input; β is the uncertainity constant; $F (x_{s} (t))$ is the nonlinear function of x_s(t); and τ_ext is the robotic arm (external torque).

F (x_{s} (t)) = μ^{T} η (x_{s} (t))

(11)

The $(A_{s}, (B_{s} β))$ dynamics of UAV is assumed to be controllable, $μ^{T}$ is the anonymous parametric matrix and $η$ is the Lipchitz-continuous regressor vector. Thus, supposition of the basic kind of MIMO system is obtained as follows

{\begin{matrix} {\overset{\cdot}{x}}_{s} (t) = A_{s} x_{s} (t) + B_{s} β (u_{s} (t) + μ^{T} η (x_{s} (t))) \\ y_{s} (t) = C_{s} x_{s} (t) + D_{s} β (u_{s} (t) + μ^{T} η (x_{s} (t))) \end{matrix}

(12)

The tracking error related to the output of the system is given by

e_{y_{s}} (t) = y_{s} (t) + y_{r} (t)

(13)

Now, the integral can be obtained as follows

{\overset{\cdot}{e}}_{y_{I}} (t) = e_{y} (t) = y_{s} (t) + y_{r} (t)

(14)

Open loop dynamics of UAV are as follows

\begin{matrix} \underset{\overset{\cdot}{x}}{\underset{︸}{(\begin{matrix} {\overset{\cdot}{e}}_{yI} \\ {\overset{\cdot}{x}}_{s} (t) \end{matrix})}} = \underset{A}{\underset{︸}{(\begin{matrix} O_{n * n} & C_{s} \\ O_{m_{s} * n} & A_{s} \end{matrix})}} \underset{x}{\underset{︸}{(\begin{matrix} e_{yI} (t) \\ x_{s} (t) \end{matrix})}} \\ + \underset{B}{\underset{︸}{(\begin{matrix} D_{s} \\ B_{s} \end{matrix})}} β (u_{s} (t) + F (x_{s} (t))) + \underset{B_{ref}}{\underset{︸}{(\begin{matrix} - I_{n * n} \\ O_{m_{s} * n} \end{matrix})}} y_{r} (t) \end{matrix}

(15)

It is equivalent to, from Equation (17), the measured output in Equation (12)

{\begin{matrix} \begin{matrix} \overset{\cdot}{x} (t) = A_{x} (t) + B β (u_{s} (t) + μ^{T} η (x_{s} (t))) + B_{r} (t) y_{r} (t) \\ y_{s} (t) = \underset{C}{\underset{︸}{(\begin{matrix} O & C_{s} (t) \end{matrix})}} \underset{x}{\underset{︸}{(\begin{matrix} e_{yI} (t) \\ x_{s} (t) \end{matrix})}} \\ + \underset{D}{\underset{︸}{D_{s} (t)}} β (u_{s} (t) + μ^{T} η (x_{s} (t))) u_{s} (t) \\ = C_{x} (t) + D β (u_{s} (t) + μ^{T} η (x_{s} (t))) u_{s} (t) \end{matrix} \end{matrix}

(16)

The linear baseline open-loop dynamics of UAV is obtained as follows

{\begin{matrix} \overset{\cdot}{x} (t) = A_{x} (t) + B u_{s} (t) + B_{r} y_{r} (t) \\ y_{s} (t) = C_{s} (t) + D_{s} (t) u_{s} (t) \end{matrix}

(17)

Now, stabilization of the optimal controller is calculated

\overset{\cdot}{z} (t) = Az (t) + Bv (t)

(18)

where

z (t) = \overset{\cdot}{x} (t) = (\begin{matrix} {\overset{\cdot}{e}}_{yI} \\ {\overset{\cdot}{x}}_{s} \end{matrix}), ν (t) = \overset{\cdot}{u} (t)

(19)

The linear quadratic cost index is designed to control the input “ $ν$ ”

J (ν (t)) = \int_{0}^{\infty} (z^{T} (t) Q z (t) + ν^{T} (t) R ν (t)) d t

(20)

where $Q$ and $R$ are positive definite symmetric matrices. It is optimal form of LQR solution in the feedback form

ν (t) = {\overset{\cdot}{u}}_{s} (t) = \underset{K_{x}^{T}}{\underset{︸}{- R^{- 1} B^{T} P z (t)}} = (K_{I} K_{s}) (\begin{matrix} {\overset{\cdot}{e}}_{yI} (t) \\ {\overset{\cdot}{x}}_{s} (t) \end{matrix})

(21)

From Equation (21), the Riccati-based equation is obtained as follows

A^{T} P + PA + Q - PB R^{- 1} B^{T} P = 0

(22)

By integrating Equation (21), the baseline LQR-PI controller is obtained as follows

\begin{matrix} u_{bl} = - K_{x}^{T} x (t) = - K e_{yI} (t) - K_{s} x (t) \\ = K_{I} \frac{(y_{r} (t) - y_{s} (t))}{s} - K_{s} x_{s} (t) \end{matrix}

(23)

The optimal gain matrix is obtained as follows

K_{x}^{T} = (K_{I} K_{s})

(24)

The resulting model-based reference dynamics of system become

{\overset{\cdot}{x}}_{r} = A_{r} x_{r} (t) + B_{r} y_{r} (t), y_{r} (t) = C_{r} (t) - x_{r} (t)

(25)

where

A_{r} = A_{s} - B_{s} K_{x}^{T}, C_{r} = C_{s} - D_{s} K_{x}^{T}

(26)

where $A_{r} (t)$ is Hurwitz, by designing

u_{s} (t) = \underset{u_{bl}}{\underset{︸}{- K_{x}^{T} x}} + u_{ad} = u_{bl} + u_{ad}

(27)

{\begin{matrix} x = A_{r} (t) x + B β \\ + (u_{ad} + \underset{{\bar{μ}}^{T} \bar{η} (u_{bl}, x_{s} (t))}{\underset{︸}{\overset{K_{u}^{T}}{\overset{︷}{(I_{n * n} - β^{- 1})}} + u_{bl} + μ^{T} η (x_{s} (t))}}) B_{r} y_{r} (t) \\ y = C_{r} (t) x (t) + D β (u_{ad} + {\bar{μ}}^{T} \bar{η} (u_{bl}, x_{s} (t))) \end{matrix}

(28)

which is equivalent to

{\begin{matrix} \overset{\cdot}{x} = A_{r} (t) x (t) + B β (u_{ad} + {\bar{μ}}^{T} \bar{η} (u_{bl}, x_{s} (t))) + B_{r} y_{r} \\ y (t) = C_{r} (t) x (t) + D β (u_{ad} + {\bar{μ}}^{T} \bar{η} (u_{bl}, x_{s} (t))) \end{matrix}

(29)

The regressor vector is as follows

η (u_{bl}, x_{s}) = {(u_{bl}^{T} η^{T} (x_{s}))}^{T}

(30)

The extended matrix for ideal parameters is obtained as follows

\bar{μ} = (K_{u}^{T} μ^{T})^{T}

(31)

The adaptive component is utilized under uncertainties in the condition

u_{ad} = - \hat{{\bar{μ}}^{T}} \bar{η} (u_{bl}, x_{s} (t))

(32)

{\begin{matrix} \overset{\cdot}{x} (t) = A_{r} (t) x (t) - B β \underset{Δ \bar{μ}}{\underset{︸}{{(μ - \bar{μ})}^{T}}} \\ y (t) = C_{r} (t) x (t) + D β Δ {\bar{μ}}^{T} \bar{η} \end{matrix} η + B_{r} (t) y_{r} (t)

(33)

where

Δ \bar{μ} = (μ - \bar{μ})

(34)

Theorem

The uncertainty in the dynamics of our proposed system in Equation (10) is considered, operating under the nominal conditions of adaptive controller in Equation (32) and the updated adaptive law in Equation (42). Now, the reference model of system (25) is assumed and conditions are matched by using Equation (26), which could be driven by the unbounded external disturbance of the gripper. Riccati-based equation is used to define the positive definite symmetric matrix of all the gestures of that gripper, which is bounded with respect to time. The algebraic solution of Lyapunov-based positive definite symmetric matrix is provided in Equation (38).

The state tracking error is as follows

e (t) = x (t) - x_{r} (t)

(35)

The tracking error dynamics is calculated as follows

\overset{\cdot}{e} (t) = A_{r} (t) e (t) - B β Δ {\bar{μ}}^{T} \bar{η}

(36)

By considering the unbounded quadratic Lyapunov function, we obtain

V (e, Δ \bar{μ}) = e^{T} P_{r} e + trace (Δ {\bar{μ}}^{T} Γ_{\bar{μ}}^{- 1} Δ \bar{μ} B)

(37)

The algebraic form of Lyapunov equation is as follows

A_{r}^{T} (t) P_{r} (t) + P_{r} (t) A_{r} (t) = - Q_{r} (t)

(38)

Time differentiating V is written as follows

\begin{matrix} \overset{\cdot}{V} (e, Δ \bar{μ}) = - e^{T} Q_{r} (t) e - 2 e^{T} P_{r} (t) B β Δ {\bar{μ}}^{T} \bar{η} \\ + 2 trace (Δ {\bar{μ}}^{T} Γ_{\bar{μ}}^{- 1} \overset{\cdot}{μ} β) \end{matrix}

(39)

Proof

We have proven that for any bounded command $y_{r}$ , the close loop system output from Equation (33) globally asymptotically tracks the reference model output from Equation (25) as time-cycle goes to infinity. At the same time, the reference model dynamics of Equation (25) are chosen such that it will easily track any bounded command $y_{r}$ , with bounded errors. Therefore, the output will easily track the desired output of the system by Lavretsky and Wise.²⁸

By applying vector trace identity, we obtain

a^{T} b = trace (b, a^{T})

(40)

It yields

\begin{matrix} V (e, Δ μ) = \\ - e^{T} Q_{r} e + 2 trace (Δ μ^{- T} {Γ_{\bar{μ}}^{- 1} \dot{\overset{β}{\bar{μ}}} - \bar{η} e^{T} P_{ref} B} β) \end{matrix}

(41)

The updated adaptive law is as follows

\overset{\cdot}{μ} = Γ_{\bar{μ}} \bar{η} (u_{bl}, x_{s} (t)) e^{T} P_{r} (t) B

(42)

Then

\overset{\cdot}{V} (e, Δ \bar{μ}) = - e^{T} Q_{r} (t) e 0

(43)

By using error dynamics of Equation (36), we obtain

\begin{matrix} y (t) = C_{x} (t) - D β \to \underset{\to 0}{\underset{︸}{Δ {\bar{μ}}^{T} \bar{η}}} C_{r} (t) x (t) \to C_{r} (t) x_{r} (t) \\ = y_{r} (t) \end{matrix}

(44)

The adaptive laws (42) is written as follows

Γ_{\bar{μ}} = (\begin{matrix} Γ_{u} & O_{n * m} \\ O_{N * m} & Γ_{μ} \end{matrix})

(45)

Now the adaptive laws (42) becomes

\dot{\hat{K_{u}}} = Γ_{u} u_{bl} e^{T} P_{r} (t) B; \overset{\cdot}{\hat{μ}} = Γ_{μ} η (x_{s} (t)) e^{T} P_{r} (t) B

(46)

The complete form of model-based adaptive hybrid controller is written as follows

u = u_{bl} + u_{ad} = - \underset{u_{bl} = Baseline}{K_{x}^{T} x (t)} + \underset{u_{ad} = Adaptive Augmentation}{[{\hat{K}}_{u}^{T} u_{bl} - {\hat{μ}}^{T} η (x_{s} (t))]}

(47)

which is also equivalent to

\begin{matrix} u = (I_{n * n} - {\hat{K}}_{u}^{T}) u_{bl} - {\hat{μ}}^{T} η (x_{s} (t)) \\ = - (I_{n * n} - {\hat{K}}_{u}^{T}) K_{x}^{T} x (t) - {\hat{μ}}^{T} η (x_{s} (t)) \\ = (I_{n * n} - {\hat{K}}_{u}^{T}) (K_{I} \frac{(y_{r} (t) - y (t))}{s} - K_{s} x_{s} (t)) \\ - {\hat{μ}}^{T} η (x_{s} (t)) \end{matrix}

(48)

When movements of gripper adaptation is incorporated, such that the effects of gripper is measured as an external torque, rotational control law becomes

τ_{u} = I (k_{r} + k_{v} δ_{v}) - τ_{ext}

(49)

Dynamic effects of gripper: The attached gripper contains two joint links. The sum of external turning effect which is caused by the limb can be described by the total torque produced by individual arm effect

τ_{ext} = τ_{b} + τ_{c}

(50)

where $τ_{b}$ , $τ_{c}$ are the mobile link and link of constant torque and the mass of robotic link is synchronized by the inertial components²¹ (Table 1) shows the simulation parameters of UAV.

Table 1.

Simulation parameters.

Parameters	Definition	Values
b	Thrust factor	3.17 e⁻⁶ N s²
d	Drag Factor	7.42 e⁻⁵ N ms²
g	Gravity	9.8 m/s
m	Mass	4 kg
$I_{Z}$	Inertia at Z	1.121 e⁻³ kg m²
$I_{X}$	Inertia at X	6.002 e⁻⁶ N s² m²
$I_{y}$	Inertia at Y	6.278 e⁻⁶ N s² m²

Simulation results and discussion

The proposed model of quadrotor aerial vehicle equipped with gripper and the proposed control algorithm are presented in this section. To validate the robustness and feasibility of the designed controller in this research, we simulated the model of an under-actuated quad rotor UAV, in two scenarios one is to carrying payload and other is without carrying payload. To evaluate the effectiveness of the designed controller, the dynamic simulations of the proposed UAV model along with gripper are performed in MATLAB 2015/Simulink. The simulation results are shown in Figures 5 –13.

Figure 5.

Control input forces of UAV.

Figure 6.

Control input torques of UAV.

Figure 7.

Euler angles without payload.

Figure 8.

Euler angles with payload.

Figure 9.

Altitude of UAV with and without payload.

Figure 10.

Payload effects due to the mass.

Figure 11.

Payload effects on Euler angle.

Figure 12.

External torque effects of robotic arm without payload.

Figure 13.

Orientation of arm without payload.

Figure 5 shows the inertial control input forces of UAV along $(X, Y, Z)$ global axis, which is followed by the control input torques of UAV that are shown in Figure 6. Figure 7 shows the Euler angle $(φ, θ, ψ)$ rates, at that time when UAV does not carry any payload; it clearly shows the efficiency of the designed controller and the angular rates converge to zero with no steady-state error. Moreover, in Figure 8, the effectiveness and feasibility of our proposed controller are shown when UAV carrying the payload; at that interval of time, certain fluctuations lie in the Euler angles which will converge to zero after it moves to the particular altitude.

In Figure 9, the altitude response of the system is shown where altitude without load shows flat response while with payload, uneven variations can be seen at different time slots. Figure 10 shows the payload effects on Euler angles, in which it is mentioned that the maximum capacity to carry the payload is about 2000 g. In Figure 11, it is notified in the graph that pitch swims 4° around 8 s spot. However, when controlling the pitch “θ,” the coupled dynamics also produce effects on roll $φ$ and yaw $ψ$ as it is not possible to roll the UAV with payload. The equilibrium points for roll “φ” and yaw “ψ” have been achieved at 15 and 17 s, respectively. In yaw, the actuation is less than roll and pitch due to controlling torque of rotor speed. The values of “φ” in the graph are minimal and appear at approximately zero value due to payload friction.

The external torque effects lie along $(X, Y, Z)$ axis of the gripper and are shown in Figure 12. Finally, Figure 13 shows the orientation of gripper and its manipulators’ desired position of the end effector of the arm, and it is observed that it converges to the desired state asymptotically.

Conclusion

This research presented an aerial manipulator combining a quadrotor and a 2-DOF gripper. To achieve the grabbing successful, the dynamic model of the combined system is analyzed and model-based adaptive optimal-based line controller was designed. Moreover, an adaptive law was proposed to accommodate the non-holonomic quadrotor and its velocities to support the three joints of the manipulator and rotorcraft. The proposed control algorithm is verified with and without including manipulation (payload) scenario simulation. When variation occurs in the input or external disturbance appears during flight operation due to the movement of gripper, the designed dual controller shows rapid rate of convergence and better robustness over others; therefore, the desired task of UAV is achieved with less margin of errors. The designed controller can be applied for the stabilization of the dynamics of UAV when carrying a payload. Furthermore, the simulated results show that the aerial gripper is a prominent solution for different scenarios of remote locations such as manipulation, inspection and transportation.

Future work

In this research, a 2-DOF gripper was used to grab the objects by using three manipulators, and its movements are fixed due to the less manipulators in the body frame. If we need more movements, we need to add more manipulators in the body frame of gripper. The modeling would be expanded by the inclusion of more manipulator links to increase the equations of the end effector. The designed dual controller approach is capable to handle multiple manipulations and we will use this same scheme for our future work to increase the manipulators. In future, we will consider vision-based tracking and grabbing of objects.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

This work was supported in part by the National Natural Science Foundation of China under Grant 61573097 and 91748106, in part by Key Laboratory of Integrated Automation of Process Industry (PAL-N201704), in part by the Fundamental Research Funds for the Central Universities (3208008401), in part by the Qing Lan Project and Six Major Top-talent Plan, and in part by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

ORCID iD

Zain Anwar Ali

References

Chao

Cao

Chen

Autopilots for small unmanned aerial vehicles: a survey. Int J Control Autom Syst 2010; 8(1): 36–44.

D’Oleire-Oltmanns

Marzolff

Peter

et al . Unmanned aerial vehicle (UAV) for monitoring soil erosion in Morocco. Remote Sens 2012; 4(11): 3390–3416.

Miller

Parasuraman

Designing for flexible interaction between humans and automation: delegation interfaces for supervisory control. Hum Fact 2007; 49(1): 57–75.

Madani

Benallegue

Backstepping sliding mode control applied to a miniature quadrotor flying robot. Proceedings of IEEE Industrial Electronics the 32nd Annual Conference, pp. 700–705, 2006.

Nicholson

Healey

AJ.

The present state of autonomous underwater vehicle (AUV) applications and technologies. Marine Technol Soc J 2008; 42(1): 44–51.

Wynn

Huvenne

Le Bas

et al . Autonomous underwater vehicles (AUVs): their past, present and future contributions to the advancement of marine geoscience. Marine Geol 2014; 352: 451–468.

Sintov

Avramovich

Shapiro

Design and motion planning of an autonomous climbing robot with claws. Robot Auton Syst 2011; 59(11): 1008–1019.

Bernard

Securitising the desert: an analysis of counterterrorism operations impacts in new theatres, gaps, shortcomings and recommendations for change. Birkbeck L Rev 2016; 4: 29.

Rule

TA.

Airspace in an age of drones. BU L Rev 2015; 95: 155.

10.

Osinga

Roorda

MP.

From Douhet to drones, air warfare, and the evolution of targeting. In: Ducheine

Schmitt

Osinga

(eds) Targeting: the challenges of modern warfare. The Hague: T.M.C. Asser Press, 2016, pp. 27–76.

11.

Pannir

MPO

Aparna

Jency

MVP

. Design and development of an autonomous UAV using image processing algorithm for precision agriculture. IJAICT 2018; 4: 1358–1360.

12.

Santana

ER.

Humanoid robots and artificial intelligence in aircraft assembly: a case study and state-of-the-art review. 2018, http://www.diva-portal.org/smash/record.jsf?pid=diva2%3A1216727&dswid=9152

13.

Narayanan

MS.

Quantitative evaluation of user performance in minimally invasive surgical procedures. Buffalo, NY: State University of New York, 2014.

14.

Kim

Choi

Kim

HJ.

Aerial manipulation using a quadrotor with a two DOF robotic arm. In: Proceedings of the IEEE/RSJ international conference on intelligent robots and systems, Tokyo, Japan, 3–7 November 2013, pp. 4990–4995. New York: IEEE.

15.

Arleo

Caccavale

Muscio

et al . Control of quadrotor aerial vehicles equipped with a robotic arm. In: Proceedings of the 21st Mediterranean conference on control and automation, Chania, 25–28 June 2013, p. I3. New York: IEEE.

16.

Danko

PY.

Design and control of a hyper-redundant manipulator for mobile manipulating unmanned aerial vehicles. J Intell Robot Syst 2014; 73(1–4): 709–723.

17.

Nicol

Macnab

CJB

Ramirez-Serrano

Robust adaptive control of a quadrotor helicopter. Mechatronics 2011; 21(6): 927–938.

18.

Tanveer

Ahmed

Hazry

et al . Stabilized controller design for attitude and altitude controlling of quad-rotor under disturbance and noisy conditions. Am J Appl Sci 2013; 10(8): 819–831.

19.

Raffo

Ortega

Rubio

FR.

An integral predictive/nonlinear H∞ control structure for a quadrotor helicopter. Automatica 201046(1): 29–39.

20.

Papachristos

Alexis

Tzes

Model predictive hovering-translation control of an unmanned Tri-TiltRotor. In: Proceedings of the IEEE international conference on robotics and automation, Karlsruhe, 6–10 May 2013, pp. 5425–5432. New York: IEEE.

21.

Zheng

Xiong

Luo

JL.

Second order sliding mode control for a quadrotor UAV. ISA Trans 2014; 53(4): 1350–1356.

22.

Huang

Xian

Diao

et al . Adaptive tracking control of underactuated quadrotor unmanned aerial vehicles via backstepping. In: Proceedings of the American control conference, Baltimore, MD, 30 June–2 July 2010, pp. 2076–2081. New York: IEEE.

23.

Bouabdallah

Murrieri

Siegwart

Design and control of an indoor micro quadrotor. In: Proceedings of the IEEE international conference on robotics and automation, New Orleans, LA, 26 April–1 May 2004, vol. 5, pp. 4393–4398. New York: IEEE.

24.

Abaunza

Castillo

Victorino

et al . Dual quaternion modeling and control of a quad-rotor aerial manipulator. J Intell Robot Syst 2017; 88(2–4): 267–283.

25.

Warner

DC.

Design, simulation, and experimental verification of a computer model and enhanced position estimator for the NPS AUV II. Doctoral Dissertation, Naval Postgraduate School, Monterey, CA, 1991.

26.

Dahiya

Valle

Robotic tactile sensing: technologies and system. Berlin: Springer Science+Business Media, 2012.

27.

Chaturvedi

DK.

Modeling and simulation of systems using MATLAB and Simulink. Boca Raton, FL: CRC Press, 2009.

28.

Lavretsky

Wise

KA.

Model reference adaptive control with integral feedback connections. In: Robust and adaptive control. London: Springer, 2013, pp. 293–315.

29.

Rahideh

Shaheed

MH.

Constrained output feedback model predictive control for nonlinear systems. Control Eng Pract 2012; 20(4): 431–443.

Modeling and controlling of quadrotor aerial vehicle equipped with a gripper

Abstract

Keywords

Introduction

System model

Assumption 1

Remark 1

Assumption 2

Assumption 3

Remark 2

Remark 3

Designing of control architecture

Dual controller approach

Theorem

Proof

Simulation results and discussion

Conclusion

Future work

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References