Figure 2.1 Coordinate frame of virtual structure. 22

Figure 2.2 Changes in headings of the UAVs. 23

Figure 2.3 Relative errors of UAV positions. 24

Figure 2.4 Controller for eliminating relative errors. 25

Figure 3.1 Interaction graphs without a leader. 34

Figure 3.2 Interaction graphs with node  as the leader. 36

Figure 3.3 Position error responses for Scenario 1. 53

Figure 3.4 Velocity error responses for Scenario 1. 53

Figure 3.5 Position error responses for Scenario 2. 54

Figure 3.6 Velocity error responses for Scenario 2. 54

Figure 3.7 ‐axis position error responses for Scenario 3. 56

Figure 3.8 ‐axis position error responses for Scenario 3. 56

Figure 3.9 2‐D formation trajectories with respect to time for Scenario 3. 57

Figure 3.10 ‐axis position error responses for Scenario 4. 57

Figure 3.11 ‐axis position error responses for Scenario 4. 58

Figure 3.12 2‐D formation trajectories with respect to time for Scenario 4. 58

Figure 3.13 ‐axis position error responses for Scenario 5. 59

Figure 3.14 ‐axis position error responses for Scenario 5. 59

Figure 3.15 2‐D formation trajectories with respect to time for Scenario 5. 60

Figure 4.1 The interaction graph for Case 1. 73

Figure 4.2 The ‐axis position trajectories for Case 1. 74

Figure 4.3 The ‐axis position trajectories for Case 1. 75

Figure 4.4 The planar position trajectories vs. time for Case 1. 75

Figure 4.5 The interaction graph for Case 2. 76

Figure 4.6 The ‐axis position trajectories for Case 2. 77

Figure 4.7 The ‐axis position trajectories for Case 2. 77

Figure 4.8 The planar position trajectories vs. time for Case 2. 78

Figure 5.1 The proposed robust control configuration for vehicle . 88

Figure 6.1 Application 1 without synchronization strategy. 125

Figure 6.2 Application 1 with synchronization strategy. 126

Figure 6.3 Application 2 with only internal synchronization. 128

Figure 6.4 Application 2 with both internal and external synchronization. 129

Figure 7.1 The schematic diagram of formation flight. 132

Figure 7.2 The vortex‐induced force and moment coefficients. 134

Figure 7.3 Biot–Savart law. 134

Figure 7.4 Induced velocity on follower by the right‐hand vortex filament. 135

Figure 7.5 Rotation of the aerodynamic forces. 136

Figure 7.6 Continuous vortex sheet. 137

Figure 7.7 Comparison of the predictions of SHVM and CVSM. 139

Figure 7.8 The structure of the formation flight controller. 142

Figure 7.9 Robust control configuration for close‐formation flight. 143

Figure 7.10 Uncertainty and disturbance estimator for . 146

Figure 7.11 Formation flight control without synchronization. 148

Figure 7.12 Formation flight control with synchronization. 149

Figure 7.13 Three different simulation cases. 150

Figure 7.14 Results for the SHVM: Part 1. 151

Figure 7.15 Results for the SHVM: Part 2. 152

Figure 7.16 Results for the CVSM: Part 1. 153

Figure 7.17 Results for the CVSM: Part 2. 154

Figure 7.18 UDE performance for the SHVM and the CVSM: Part 1. 155

Figure 7.19 UDE performance for the SHVM and the CVSM: Part 2. 156

Figure 8.1 An omnivision system with a camera. 159

Figure 8.2 Equipment used: (left) Ikea STABIL stainless steel double‐boiler insert; (right) Logitech C510 high definition webcam. 160

Figure 8.3 Overhead view of a follower UGV. 160

Figure 8.4 The leader UGV is outfitted for easy visual recognition 161

Figure 8.5 A follower UGV placed at the origin of the RWC system. 161

Figure 8.6 Image axes and camera axes depicted in a raw image. 162

Figure 8.7 Mappings from RWC to CAPC in pixels: left, ; right, . 162

Figure 8.8 The mapped pixels. 162

Figure 8.9 The leader robot as seen by a follower robot: left, in RGB space; right, in HSV (false color). 164

Figure 8.10 Object and background pixels following HSV thresholding. 164

Figure 8.11 An example of erosion applied to a binary image. 165

Figure 8.12 The centroids of the different color regions. 166

Figure 8.13 The geometry involved: left, and ; right, . 166

Figure 8.14 Schematic of the synchronization control scheme. 167

Figure 8.15 Geometric formation parameters. 168

Figure 8.16 Formation trajectories: left, without synchronization; right, with synchronization. 171

Figure 8.17 Tracking error: left, without synchronization; right, with synchronization. 172

Figure 8.18 Leader–follower distances: left, without synchronization; right, with synchronization. 172

Figure 8.19 Follower angular position: left, without synchronization; right, with synchronization. 173

Time‐varying formation along a linear trajectory. 173

Figure 8.21 Desired position coordinates as functions of time. 174

Figure 8.22 Formation trajectories: left, without synchronization; right, with synchronization. 174

Figure 8.23 Follower tracking error: left, without synchronization; right, with synchronization. 174

Figure 8.24 Leader–follower distances: left, without synchronization; right, with synchronization. 175

Figure 8.25 Follower angular position: left, without synchronization; right, with synchronization. 175

Figure 8.26 A group of agents connected with springs and dampers. 176

Figure 8.27 Passivity formation trajectories: left, (0 300 s); right, (0 25 s). 179

Figure 8.28 Follower resultant force magnitudes. 179

Figure 8.29 Follower angular errors (0 300 s). 180

Figure 8.30 Formation spring energy. 180

Figure 8.31 Passivity controller instability. 180

Figure 8.32 iRobot Create (Roomba) robot. 181

Figure 8.33 Optitrack infrared camera setup. 182

Figure 8.34 Close‐up of an infrared camera. 182

Figure 8.35 Optitrack infrared marker patterns on UGVs. 183

Figure 8.36 Ground truth formation trajectories and shapes. 184

Figure 8.37 Ground truth tracking errors. 184

Figure 8.38 Desired formation coordinates as functions of time. 184

Figure 8.39 Ground truth formation trajectories and shapes. 185

Figure 8.40 Ground truth tracking errors. 186

Figure 9.1 The setup of four 3DOF‐Helis. 194

Figure 9.2 The hardware configuration of the experimental setup. 195

Figure 9.3 The 3DOF‐Heli components. 196

Figure 9.4 3DOF‐Heli body frame. 196

Figure 9.5 Hardware configuration of 3DOF‐Heli body. 197

Figure 9.6 Experimental setup with three 3DOF‐Helis. 202

Figure 9.7 The position trajectory of the ADS. 204

Figure 9.8 The experimental results for Control Strategy I. (a) Trajectories of elevation angle; (b) Trajectories of travel angle; (c) Tracking errors of elevation angle; (d) Synchronization errors of elevation angle; (e) Control voltages for front motors; (f) Control voltages for back motors. 205

Figure 9.9 The experimental results for Control Strategy II. (a) Trajectories of elevation angle; (b) Trajectories of travel angle; (c) Tracking errors of elevation angle; (d) Synchronization errors of elevation angle; (e) Control voltages for front motors; (f) Control voltages for back motors. 206

Figure 9.10 The experimental results for Control Strategy III. (a) Trajectories of elevation angle; (b) Trajectories of travel angle; (c) Tracking errors of elevation angle; (d) Synchronization errors of elevation angle; (e) Control voltages for front motors (f) Control voltages for back motors. 207

Figure 9.11 The communication topology for Experiments 2 and 3. 209

Figure 9.12 Experimental results for Case 1. 211

Figure 9.13 Experimental results for Case 2. 212

Figure 9.14 Experimental results for Case 3. 213

Figure 9.15 Positions of ADSs of H2 (blue) and H4 (magenta). 214

Figure 9.16 Experimental results for Case 4. 215

Figure 9.17 Responses of elevation axis for Case 1: (a) angle; (b) angular velocity; (c) UDE output. 218

Figure 9.18 Responses of pitch axis for Case 1: (a) angle; (b) angular velocity; (c) UDE output. 219

Figure 9.19 Control voltages for Case 1: (a) ; (b) . 219

Figure 9.20 Positions of the ADSs of H2 and H4. 220

Figure 9.21 Responses of elevation axis for Case 2: (a) angle; (b) angular velocity; (c) UDE output. 220

Figure 9.22 Responses of pitch axis for Case 2: (a) angle; (b) angular velocity; (c) UDE output. 221

Figure 9.23 Control voltages for Case 2: (a) ; (b) . 221

Figure 9.24 Responses for Case 3 with ADS off: (a) elevation; (b) pitch. 222

Figure 9.25 Responses for Case 3 with ADS on: (a) elevation; (b) pitch. 222