UAV Search: Maximising Target AcquisitionContents1 Introduction 12 Quadrotor Helicopter Dynamics 23 Optimal Control 54 Search and Rescue Scenario 75 Maximising Target Acquisition 86 Analysis of Controller 117 Conclusion and Future Work 13Department of Electrical and Computer Engineering 1 INTRODUCTIONUAV Search : Maximising Target AcquisitionAbstractThis paper is an academic experience report describing analysis of optimal control techniquesfor simulated quad-rotor unmanned aerial vehicles (UAVs) performing search and rescuemissions. Analysis of the controller and guidance laws governing the UAV are described indetail culminating in a closed form expression describing the probability of detection over acertain field.1 IntroductionThe introduction of autonomous unmanned vehicles has posed an interesting challenge toengineers. After removing the human navigator from the loop, more robust dynamic automa-tion systems must be implemented to ensure safe and efficient control. Simulated scenarioshave been developed investigating the use of quadrotor helicopters for search and rescuemissions. According to the authors of Sliding mode control of a quadrotor helicopter [6] aquadrotor helicopter is a four rotor helicopter that was first built in 1907 by the BreguetBrothers. The simulated UAV was based on the STARMAC II (Fig. 1) rotorcraft developedby researchers at Stanford University as mentioned in [4].Figure 1: STARMAC quadrotor helicopter developed at Stanford University [3]The quadrotor helicopter is a nonlinear system, hence the dynamics will be altered using1 of 15 IEEE Member #90594047Department of Electrical and Computer Engineering 2 QUADROTOR HELICOPTER DYNAMICSsliding mode control to improve motion control. Sliding mode control implements high-frequency switching control; control switches from one smooth condition to another areinherent in its variable structure control. This allows for the system to ‘slide’ between theboundaries of the controller thus the trajectory of the system is aptly named sliding mode.The current system contains a single quadrotor helicopter at a height of 150m, underthe command from a central ground station. A spiral search pattern is initiated which thequadrotor follows until the target is acquired within the quadrotor’s cameras field of view.Once the quadrotor ‘sees’ the target, a target tracking controller is implemented which allowsthe quadrotor helicopter to follow the target at a safe distance.For the current system, target acquisition is not guaranteed. Many simulations provideevidence that target acquisition can be a slow procedure. Utilising a fixed height for searchingalso increases the chances of the target evading the quadrotor helicopter. The aim of thispaper is to introduce a closed form solution to determining a search height which maximisesthe probability of detection of a target over a certain field.Once a closed form expression has been achieved, a scenario will be introduced to analysethe performance of the proposed solution. The search height of a single quadrotor helicopterwill be calculated to maximise the probability of detection of a ground vehicle starting 2.2kmaway. The area describing possible locations of the ground vehicle will be introduced andabsolute certainty of target acquisition will be defined as the field of view of the cameratotally encompassing the area where the ground vehicle could possibly exist.2 Quadrotor Helicopter DynamicsAccurate simulation requires well-defined dynamics. By studying the free body diagram(Fig. 2a) of a typical quadrotor helicopter, it is possible to derive well-defined dynamics forthe system. According to Jason Hansen, [2], a proposed derivation of the nonlinear dynamicscan be performed in North-East-Down (NED) inertial and body fixed coordinates as detailed2 of 15 IEEE Member #90594047Department of Electrical and Computer Engineering 2 QUADROTOR HELICOPTER DYNAMICSin [3]. Moments generated about the centre of mass can be described as follows(a) (b)Figure 2: (a) Free body diagram of a typical quadrotor [2]. (b) Alternative free body diagram ofquadrotor helicopter [3].My= Iyy¨θ = (T3r) − (T1r) (1)Assuming a linear thrust-to-torque relationship and identical force generation across eachaxis control of the attitude can be related to the followingT1= T − δTT3= T + δT(2)As T corresponds to nominal thrust then δT describes the deviation from nominal thrust.With θ = 0, nominal thrust T relates to the required force output by each rotor to satisfygravitational equilibrium. This is represented mathematically asT =mg4(3)By equation (2), it follows thatTtotal= T1+ T3= (T − δT) + (T + δT) = 2T(4)Note the resulting force is equal as previously mentioned. Combining (4) with (1) allows3 of 15 IEEE Member #90594047Department of Electrical and Computer Engineering 2 QUADROTOR HELICOPTER DYNAMICSforMy= Iyy¨θ = T3r − T1rMy= Iyy¨θ = (T + δT)r − (T − δT)rMy= Iyy¨θ = 2δTr(5)By Laplace transform of (5) we achieveIyys2θ(s) = 2δT (s)r (6)Transposition of formulae of (6) results in an expression relating the quadrotor’s attitudeand δTθ(s)T (s)=2rIyys2(7)Combining the forces corresponding to the x and z components of the quadrotor allowsfor a derivation of the translational acceleration as followsFz= m¨z = T1cos θ + T2cos θ + T3cos θ + T4cos θ − mgFx= m¨x = T1sin θ + T2sin θ + T3sin θ + T4sin θ(8)Combining (2) and (8) allows forFz= m¨z = (T − δT ) cos θ + T cos θ + (T + δT ) cos θ + T cos θ − mgFx= m¨x = (T − δT ) sin θ + T sin θ + (T + δT ) sin θ + T sin θFz= m¨z = 4T cos θ −mgFx= m¨x = 4T sin θ(9)Which can then be linearised about θ as follows4 of 15 IEEE Member #90594047Department of Electrical and Computer Engineering 3 OPTIMAL CONTROLFz= m¨z = 4T − mgFx= m¨x = 4T θ(10)If one assumes mg corresponds to a disturbance in the system then a Laplace transformwill yield the followingFz= ms2Z(s) = 4T (s)Fx= ms2X(s) = 4T θ(s)(11)By transposition of formulae of (11) one yields a relationship describing the quadrotor’sangle and resulting position. The transfer function is as followsZ(s)T (s)=4ms2X(s)θ(s)=T (s)ms2(12)Deriving the aforementioned transfer functions allow for a description of the translationalstates as a function of the delta thrust generated by the rotors according to the author of[2].3 Optimal ControlAs mentioned in [1], for a control system to be considered optimal the system parametersmust be adjusted such that the