本研究主要分為兩個部分：四旋翼之編隊飛行與防撞設計。第一部分主要討論基於共識控制與模型預測控制之四旋翼編隊策略。首先，四旋翼之編隊問題可以分為水平方向與垂直方向之運動。在水平方向運動當中，長機以模型預測控制執行航點之追隨，並在自主飛行的同時根據自身的預測軌跡、速度方向與指定編隊隊形計算各個僚機的編隊飛行參考軌跡。另一方面，僚機以共識控制搭配模型預測控制來執行參考軌跡的追隨。在垂直方向運動當中，長機以模型預測控制執行高度與爬升率的追隨並計算各僚機之參考軌跡。僚機在垂直方向運動則單以模型預測控制來執行參考軌跡的追隨。最終在Matlab/Simulink的環境下進行非線性的模擬以驗證本研究所提出之編隊飛行策略的可行性。
  本研究的第二部分主要在探討四旋翼彼此間的防撞問題。本研究依據幾何與相對運動關係提出兩架四旋翼間在水平上之防撞策略。防撞策略主要分為判斷與計算兩個部分。首先，防撞策略的第一部分根據幾何與相對運動關係判斷兩架四旋翼間是否有碰撞危險。在有碰撞危險的情況下，防撞策略的第二部分將介入並計算防撞控制。完整的防撞控制主要包含方向以及大小。防撞控制的方向之決定是依據當下的幾何與相對運動的情況，本研究根據其中一種幾何情況做推導，並引入一般化防撞控制將防撞控制拓展到所有幾何情況；防撞控制的大小是以一個從相對速度與視線向量(Line of sight, LOS)的內積所推導出的方程式計算之，此方程式包含四旋翼之狀態變數與四旋翼水平上之幾何夾角。最終根據不同飛行情況在Matlab/Simulink的環境下進行非線性的模擬以驗證本研究所提出之防撞策略的可行性。

This study includes two parts. In the first part, the distributed consensus control and model predictive control (MPC)-based formation strategies for quadrotors are proposed. First, the formation-control problem is decoupled into horizontal and vertical motions. The distributed consensus control and MPC-based formation strategy are implemented in the follower's horizontal formation control. In the horizontal motion, the leader tracks the given waypoints by simply using the MPC, and generates the desired formation trajectory for each follower based on its flight information, predicted trajectory, and the given formation pattern. On the other hand, the followers carry out the formation flight based on the proposed horizontal formation strategy and the desired formation trajectories generated by the leader. In the vertical motion, formation control is carried out using only the MPC for both the leader and the follower. Likewise, the leader tracks the desired altitude/climb rate and generates the desired formation trajectories for the followers, and the followers track the desired formation trajectories generated by the leader using the MPC. The optimization problem considered in the MPC differs for the horizontal and vertical motions. The problem is formulated as a quadratic programming (QP) problem for the horizontal motion, and as a linear quadratic tracker (LQT) for the vertical motion. Simulation of a comprehensive maneuver was carried out under a Matlab/Simulink environment to examine the performance of the proposed formation strategies.
  In the second part, a collision avoidance strategy based on the relative motions and horizontal geometric relationship between two quadrotors is proposed. The avoidance strategy includes two parts. The first part of the avoidance strategy determines the existence of the collision warning condition by using the line of sight (LOS), relative motions, and horizontal geometric relationship between two quadrotors. The second part of the avoidance strategy generates the avoidance control if the collision warning condition exists. A complete avoidance control includes two parts, the direction and the magnitude. The direction of the avoidance control for the quadrotor under consideration is determined based on the relative position between two quadrotors with respect to the translated frame of the quadrotor under consideration. The magnitude of the avoidance control for the quadrotor under consideration is derived based on the relative velocity, line of sight (LOS), and the quadrotor's dynamics model. Simulations were carried out under Matlab/Simulink environment to examine the performance of the proposed avoidance strategy.

Contents
List of Figures iii
List of Tables vi
I Quadrotors Formation Strategies Based on Consensus and Model
Predictive Control 1
1 Introduction 2
2 Background 3
3 Dynamic Model of a Quadrotor 5
3.1 Quadrotor’s Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 Linearization of the EOMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Control Law Preliminaries 10
4.1 Consensus Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2 Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2.1 Linear Quadratic Tracker . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2.2 Unconstrained Quadratic Programming . . . . . . . . . . . . . . . . . . . 12
5 Formation Control Strategies 15
5.1 Formation Control Strategy: Leader . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.1.1 Generation of Desired Formation Trajectory in Horizontal Motion . . . . 15
5.1.2 Generation of Desired Formation Trajectory in Vertical Motion . . . . . 17
5.2 Formation Control Strategy: Follower . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2.1 Formation Control Strategy in Horizontal Motion . . . . . . . . . . . . . 19
5.2.2 Formation Control Strategy in Vertical Motion . . . . . . . . . . . . . . . 21
6 Preparation for Formation Simulation 22
6.1 Transformation of Variables of Discrete State-Space model . . . . . . . . . . . . 22
6.2 Simulink Environment Setup for the Formation Simulation . . . . . . . . . . . . 23
7 Simulation Results 28
7.1 Quadrotor Parameters, Control Gains, and Weighting Matrices . . . . . . . . . . 28
7.2 Performance of Level Flight with Different Flight Path and Formation Patterns . 30
7.3 Fixed-point Altitude tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.4 Comprehensive Maneuver Simulation . . . . . . . . . . . . . . . . . . . . . . . . 39
8 Discussion 44
i
9 Conclusion 48
II Quadrotor Horizontal Collision Avoidance Strategy based on
Geometric Approach 49
10 Introduction 50
11 Background 51
12 Horizontal Collision Avoidance Strategy 52
12.1 Collision Avoidance Strategy: Part 1 . . . . . . . . . . . . . . . . . . . . . . . . 53
12.2 Collision Avoidance Strategy: Part 2 . . . . . . . . . . . . . . . . . . . . . . . . 54
12.2.1 Determination of the Direction of the Avoidance Control . . . . . . . . . 55
12.2.2 Determination of the Magnitude of the Avoidance Control . . . . . . . . 58
12.2.3 Functions of the Angle Between the Relative Velocity and the LOS . . . 61
13 Preparation for Simulation 62
13.1 Modification to the Pitch/roll Module of Followers #1 and #2 . . . . . . . . . . 62
14 Simulations 65
14.1 Gain Scheduling for Consensus Control and Parameters for collision avoidance
strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
14.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
14.2.1 Simulation Results, Case I . . . . . . . . . . . . . . . . . . . . . . . . . . 67
14.2.2 Simulation Results, Case II . . . . . . . . . . . . . . . . . . . . . . . . . 70
14.2.3 Simulation Results, Case III . . . . . . . . . . . . . . . . . . . . . . . . . 74
15 Discussion 79
16 Conclusion 81
ii
List of Figures
Figure 1 Network Structure Comparison. . . . . . . . . . . . . . . . . . . . . . . . . 3
Figure 2 Thrusts, torques, quadrotor’s parameter, and reference frames. . . . . . . . 5
Figure 3 Formation control strategy of the leader. . . . . . . . . . . . . . . . . . . . 15
Figure 4 (a) Relative position of the leader and the follower (left); (b) tangential
coordinate and leader’s velocity V L in a horizontal plane (right). . . . . . . 16
Figure 5 Follower’s control strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Figure 6 Simulated quadrotor system: leader. . . . . . . . . . . . . . . . . . . . . . . 24
Figure 7 Simulink/Aerospace Blockset 6DOF block. . . . . . . . . . . . . . . . . . . 24
Figure 8 Configuration of the control system. . . . . . . . . . . . . . . . . . . . . . . 25
Figure 9 Configuration of altitude control of leader (top) and follower (bottom). . . 26
Figure 10 Configuration of leader’s horizontal motion control. . . . . . . . . . . . . . 27
Figure 11 Configuration of follower’s horizontal motion control. . . . . . . . . . . . . 27
Figure 12 Flight Senarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Figure 13 Top view of quadrotors’ trajectories in square(left) and triangle(right) formation.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Figure 14 Quadrotors’ position x and y in square(left) and triangle(right) formation. 32
Figure 15 Quadrotors’ altitude in square(top) and triangle(bottom) formation. . . . 33
Figure 16 Quadrotors’ attitude in square(left) and tirangle(right) formation. . . . . . 33
Figure 17 Top view of quadrotors’ trajectories in square(left) and triangle(right) formation.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Figure 18 Quadrotors’ position x and y in square(left) and triangle(right) formation. 34
Figure 19 Quadrotors’ altitude in square(top) and triangle(bottom) formation. . . . 35
Figure 20 Quadrotors’ attitude in square(left) and tirangle(right) formation. . . . . . 35
Figure 21 Top view of quadrotors’ trajectories in square(left) and triangle(right) formation.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Figure 22 Quadrotors’ position x and y in square(left) and triangle(right) formation. 36
Figure 23 Quadrotors’ altitude in square(top) and triangle(bottom) formation. . . . 37
Figure 24 Quadrotors’ attitude in square(left) and tirangle(right) formation. . . . . . 37
Figure 25 Quadrotors Altitude and Climb Rate v.s. Time. . . . . . . . . . . . . . . . 39
iii
Figure 26 Comprehensive maneuver mission trajectory. . . . . . . . . . . . . . . . . . 40
Figure 27 Quadrotor trajectories of comprehensive maneuvers. . . . . . . . . . . . . . 41
Figure 28 Top view of quadrotor trajectories throughout the comprehensive maneuver. 41
Figure 29 Quadrotor altitude and climb rate throughout the comprehensive maneuver. 42
Figure 30 Quadrotor roll angle throughout the comprehensive maneuver. . . . . . . . 42
Figure 31 Quadrotor pitch angle throughout the comprehensive maneuver. . . . . . . 43
Figure 32 Quadrotor yaw angle throughout the comprehensive maneuver. . . . . . . . 43
Figure 33 Zoom-in of the responses of altitude and climb rate during 190 to 210 seconds. 45
Figure 34 Responses of pitch and roll angles from start to 55 seconds. . . . . . . . . . 45
Figure 35 Responses of pitch and roll angles from 255 to 280 seconds. . . . . . . . . . 46
Figure 36 Zoom-in of altitude vs. time during upward circling motion . . . . . . . . . 47
Figure 37 Block Diagram of Quadrotor’s Control Architecture. . . . . . . . . . . . . . 52
Figure 38 Block Diagram of Quadrotor’s Pitch/Roll Control. . . . . . . . . . . . . . . 53
Figure 39 Horizontal Geometric Relationship of two Quadrotors . . . . . . . . . . . . 54
Figure 40 Avoidance Geometric Relationship of two Quadrotors . . . . . . . . . . . . 55
Figure 41 Configuration of the Modified Pitch/roll Control Module for Follower #1. . 63
Figure 42 Configuration of Formation Strategy of Follower #1. . . . . . . . . . . . . . 63
Figure 43 Second Part of the Avoidance Strategy: Avoidance Control Generator. . . . 64
Figure 44 Inside of the Avoidance Control Generator. . . . . . . . . . . . . . . . . . . 64
Figure 45 Trajectories of Followers #1 and #2 with and without the collision avoidance
strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Figure 46 Attitude of Follower #1 with and without the collision avoidance strategy. . 68
Figure 47 Attitude of Follower #2 with and without the collision avoidance strategy. . 69
Figure 48 Distance between Followers #1 and #2 throughout the flight. . . . . . . . . 70
Figure 49 Angles when quadrotors within caution radius. . . . . . . . . . . . . . . . . 70
Figure 50 Trajectories of Followers #1 and #2 with and without the collision avoidance
strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Figure 51 Attitude of Follower #1 with and without the collision avoidance strategy. . 72
Figure 52 Attitude of Follower #2 with and without the collision avoidance strategy. . 73
Figure 53 Distance between Followers #1 and #2 throughout the flight. . . . . . . . . 74
Figure 54 Angles when quadrotors are within caution radius. . . . . . . . . . . . . . . 74
Figure 55 Trajectories of Followers #1 and #2 with and without the collision avoidance
strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Figure 56 Attitude of Follower #1 with and without the collision avoidance strategy. . 76
Figure 57 Attitude of Follower #2 with and without the collision avoidance strategy. . 77
Figure 58 Distance between Followers #1 and #2 throughout the flight. . . . . . . . . 78
iv
Figure 59 Angles when quadrotors are within caution radius. . . . . . . . . . . . . . . 78

List of Tables
Table 1 Equilibrium pt. and small disturbance expression of each state and controls
in horizontal motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Table 2 Equilibrium pt. and small disturbance expression of each state and controls
in vertical and directional motions. . . . . . . . . . . . . . . . . . . . . . . 8
Table 3 Cases of leader’s vertical motion with desired altitude and climb rate . . . . 17
Table 4 Parameters of the quadrotor . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Table 5 Initial Position of the Quadrotors . . . . . . . . . . . . . . . . . . . . . . . 31
Table 6 Coordinate of the Followers’ Desired Formation Position of the Formation
Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Table 7 Altitude Maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Table 8 Determination of Ic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Table 9 Determination of Kf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Table 10 Parameters for collision avoidance strategy. . . . . . . . . . . . . . . . . . . 66
Table 11 Initial Position of the Quadrotors. . . . . . . . . . . . . . . . . . . . . . . . 66
Table 12 Followers’ Desired Formation Position Before and After Exchanging. . . . . 67

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