Flight Controller Design for an Autonomous MAV Dissertation Submitted in partial fulfillment of the requirements for the Master of Technology Program by Gopinadh Sirigineedi 03301012 Under the guidance of Prof. S.P. Bhat Department of Aerospace Engineering Indian Institute of Technology, Bombay JULY 2005
i Dissertation Approval Sheet Dissertation entitled Flight Controller Design for an Autonomous MAV, submitted by Gopinadh Sirigineedi (Roll No. 03301012) is approved for the degree of Master of Technology. Guide Examiners (Internal) (External) Chairman Date: July, 2005.
ii Certificate Certified that this M.Tech Project Report titled as Flight Controller Design for an Autonomous MAV, submitted by Gopinadh Sirigineedi (Roll No. 03301012) is approved by me for submission. Certified further that, to the best of my knowledge, the report represents work carried out by the student. Date: 8-7-2005. Prof. S.P. Bhat (Project Guide)
iii Abstract The report presents a flight controller designed to enable the Mini Air Vehicle (MAV) Kadet MkII to navigate autonomously from an initial way-point to a final way-point. A guidance strategy reported in the literature is adopted. Since the guidance strategy requires the aircraft to fly in one of three trim states, namely, straight level flight, level coordinated right circular turn, level coordinated left circular turn, linearized models are generated for these three trim states. Three controllers, each consisting of a state feedback controller and a reduced-order observer are designed for each of these linearized aircraft models. Each controller stabilizes respective trim state, and also tracks altitude commands. The performance of linear controllers is demonstrated by simulations performed with a nonlinear aircraft model. Closed-loop guidance is performed on the nonlinear aircraft model. The guidance strategy is modified to smoothen the trajectory of the aircraft by eliminating continuous switching between left and right turns. Simulations for three dimensional way-point navigation are performed. Simulations are used to study the effect of closed-loop poles on guidance. The effect of discrete position and heading updates is simulated. Keywords: Autonomous, State feedback, Reduced-order observer, Guidance Strategy.
iv Contents Abstract Nomenclature List of Figures iii viii x 1 Introduction 1 2 Controller Design for Full Aircraft Dynamics 4 2.1 Linearized Aircraft Model......................... 4 2.2 State Feedback Controller......................... 6 2.2.1 Selection of Closed-Loop Poles................... 7 2.3 Observer Design............................... 8 2.4 Control Law Implementation On Nonlinear Aircraft Model....... 9 3 Closed-Loop Guidance 15 3.1 Guidance Strategy............................. 15 3.2 Closed-Loop Guidance........................... 17 3.3 Modified Guidance Strategy........................ 18 3.4 Guidance Under Discrete Position and Heading Updates........ 22 3.5 Effect of Closed-Loop Poles On Guidance................. 25 4 Conclusion 29 Appendices 29 A Nonlinear Aircraft Equations and Aircraft Data 30 A.1 Nonlinear Aircraft Equations........................ 30 A.2 Aircraft Data................................ 33 B Linear Controller Design for Level Straight Flight 37
v C Linear Controller Design for Level Coordinated Right Circular Turn 41 D Linear Controller Design for Level Coordinated Left Circular Turn 45 E Simulink Models 49 References 52
vi Nomenclature English Symbols A, B, C State, control and output matrices of the linearized aircraft model  State matrix of reduced-order observer A aa, A ab, A ba, A bb Sub-matrices of the state matrix B a, B b Sub-matrices of the control matrix ˆB Sub-matrix of control matrix of reduced-order observer Ĉ Output matrix of reduced-order observer ˆD Direct transmission matrix of reduced-order observer ˆF Sub-matrix of control matrix of reduced-order observer g Acceleration due to gravity in metre/sec 2 h Perturbation in altitude from trim value in metres H Altitude at which the aircraft is flying level and trim in metres h d K K e M n p P P T q Q Q T r R T u u c V Desired change in the altitude from trim value in metres State feedback gain matrix Reduced-order observer gain matrix Maximum turn rate of the aircraft in radians/sec Load factor Perturbation in roll rate from trim value in radians/sec Cross range error of the aircraft in metres Roll rate of the aircraft at trim state in radians/sec Perturbation in pitch rate from trim value in radians/sec Down range error of the aircraft in metres Pitch rate of the aircraft at trim state in radians/sec Perturbation in yaw rate from trim value in radians/sec Yaw rate of the aircraft at trim state in radians/sec Control vector for the aircraft model Input command to the closed-loop aircraft model Velocity of the aircraft
vii v V T x x x x a x b x b x d Perturbation in true air-speed from trim value in metre/sec True air-speed of the aircraft at trim state in metre/sec State vector of the linearized aircraft model Error between aircraft state vector and desired state vector Estimated state vector Measured part of the state vector Unmeasured part of the state vector Estimation of unmeasured part of the state vector Vector of desired aircraft states Greek Symbols α α T β β T φ φ T θ θ T δ a δ at δ e δ et δ n δ nt δ r δ rt η ψ γ ν λ Perturbation in angle of attack from trim value in radians Angle of attack of the aircraft at trim state in radians Perturbation in side-slip angle from trim value in radians Side-slip angle of the aircraft at trim state in radians Perturbation in bank angle from trim value in radians Bank angle of the aircraft at trim state in radians Perturbations of pitch angle from trim value in radians Pitch angle of the aircraft at trim state in radians Perturbation in aileron deflection from trim value in radians Deflection of the aileron when the aircraft is flying at trim state in radians Perturbation in elevator deflection from trim value in radians Deflection in the elevator when the aircraft is flying at trim state in radians Perturbation in propeller speed from trim value in rpm Propeller speed when the aircraft is flying at trim state in rpm Perturbation in rudder deflection from trim value in radians Deflection of the rudder when the aircraft is flying at trim state in radians Reduced-order observer state vector Function used in guidance strategy Angle made by the line with x-axes in modified logic Turn rate of the aircraft Heading angle of the aircraft
viii Subscripts c d f T Command Desired Final destination Trim
ix List of Figures 1.1 Kadet MkII................................. 3 2.1 Block diagram of closed-loop aircraft with trim selector......... 10 2.2 Reduced-order observer based controller................. 11 2.3 Response of the aircraft for trim selector commands........... 12 2.4 True air-speed and altitude responses.................... 13 2.5 Angle-of-attack and sideslip responses................... 13 2.6 Response of propeller speed to trim selector commands.......... 14 2.7 Response of elevator deflection to trim selector commands........ 14 2.8 Response of aileron deflection to selector commands........... 14 2.9 Response of rudder deflection to selector commands............ 14 3.1 Phase portrait under guidance strategy.................. 16 3.2 Schematic diagram of guidance strategy implementation........ 17 3.3 Aircraft path under closed-loop guidance................. 18 3.4 Phase portrait under modified guidance strategy............. 19 3.5 Aircraft path under modified guidance strategy............. 20 3.6 Trim selector commands with original guidance strategy......... 21 3.7 Trim selector commands with modified guidance strategy......... 21 3.8 Trajectory of the aircraft for visiting three way-points.......... 22 3.9 Variation of altitude for visiting three way-points............. 22 3.10 Trim selector commands as the aircraft visits the way-points....... 23 3.11 Response of true air-speed of the aircraft.................. 23 3.12 Variation of angle of attack......................... 23 3.13 Variation of sideslip angle.......................... 23 3.14 Variation of propeller speed......................... 24 3.15 Variation of elevator deflection....................... 24 3.16 Variation of aileron deflection........................ 24 3.17 Variation of rudder deflection........................ 24
x 3.18 Guidance strategy implementation with sample and hold block..... 25 3.19 Guidance under discrete updates..................... 26 3.20 Trajectories of aircraft with faster and slower poles........... 27 3.21 Trim selector commands to the closed-loop aircraft with faster poles.. 28 3.22 Trim selector commands to the closed-loop aircraft with slower poles.. 28 A.1 Variation of coefficient of thrust with advance ratio............ 34 A.2 Variation of coefficient of power with advance ratio............ 34 E.1 Simulink model to implement closed-loop guidance........... 50 E.2 Simulink model of the controller...................... 51
1 Chapter 1 Introduction An aircraft flight control system provides the capability to stabilize and control the aircraft. The evolution of modern aircraft created a need for automatic-pilot control systems. In addition, the widening performance envelope created a need to augment the stability of the aircraft dynamics over some parts of the flight envelope. Because of the large changes in aircraft dynamics, the dynamic mode that is stable and adequately damped in one flight condition may become unstable, or at least inadequately damped, in another flight condition. These problems are overcome by using feedback control to modify the aircraft dynamics. The aircraft motion variables are sensed and used to generate signals that can be fed into the aircraft control actuators, thus modifying the dynamic behavior. This feedback must be adjusted according to the flight conditions. The adjustment process is called gain scheduling because, in its simplest form, it involves only changing the amount of feedback according to the predetermined schedule. Sophisticated control configurations are needed to meet the mission requirements for advanced aircraft. The required vehicle performance during low altitude, low speed and high angle-of-attack, all-weather, day and night operations must be achieved. One of the difficulties dealing with flight control are nonlinearities that must be considered. Lyshevski [1] presents aircraft flight control system design under state and control bounds. Yang and Kung [2] present the application of nonlinear H state feedback theory to flight control which solves the aircraft equations without linearization. The aircraft configuration also impacts control response through variations in centre of gravity and moment of inertia. Keating, Parchter and Houpis [3] present Quantitative Feedback Theory (QFT) based robust flight controller for varying flight conditions. Mini Air Vehicles (MAVs) are miniature airplanes designed to be small, light, and highly resilient. The purpose of Mini Air Vehicles (MAVs) is to provide inexpensive
2 and expendable platforms for surveillance and data collection in situations where larger vehicles are not practical. They can be used for battlefield surveillance or mapping the extent of chemical spills or viral outbreaks. Other applications include use in search and rescue operations, traffic coverage and crop or wildlife monitoring [4]. Most of the MAVs are fully human piloted and make use of off-the-shelf radio control systems. These planes are difficult to fly due to their unconventional designs. Another limitation of human piloted MAVs besides the range of the radio control transmitter, is the range of the pilot s sight. Cameras have been used on MAVs to extend their usable range; however mapping three-dimensional control inputs from a two-dimensional video is foreign for most pilots. So, there is an urgent need for an MAV which can fly autonomously, to extend the operational range and to perform diversified tasks. Unmanned Aerial Vehicles(UAVs) are susceptible to battle damages and failures as there is no pilot on-borad. In such a event, the aerodynamics can change rapidly and deviate significantly from the model used for control design. To stabilize the aircraft dynamics and achieve accurate command tracking in presence of significant model errors, Farrel, Sharma and Polycarpou [5] present a on-line approximation based longitudinal control that is based on ideas from feedback linearization and back stepping. Andrievsky and Fradkov [6] present combined adaptive control law with forced sliding motion and parametric approximation for the attitude control of an unmanned aerial vehicle. In this report, we outline the design of flight controller for a rigid MAV to enable it to fly autonomously. The work done is at its rudimentary stage and assumes gust free and obstacle free environment. The aim of the project is to design a flight controller for an MAV to enable the MAV to navigate autonomously from a initial way-point to a final way-point. Figure 1.1 shows Kadet MkII chosen for present work. The feedback guidance strategy proposed by Bhat and Kumar [7] achieves perfect guidance under the assumption that true ground speed and altitude are constant and the turn rate of the MAV is perfectly tracked. So the controller has to be designed such that the velocity, altitude and turn rate follow the guidance command. In the first stage of the project a controller for tracking velocity and altitude commands was designed using the root-locus approach. The controller obtained demanded unrealistically huge propeller speed and elevator deflection. In the second stage a controller was designed for the longitudinal dynamics of the aircraft by combining a pole placement using state feedback with observer. In the
3 Figure 1.1: Kadet MkII final stage we have designed the controller for complete aircraft dynamics combining the pole placement state feedback using reduced-order observer. Closed-loop guidance is performed with guidance in outer-loop. In Chapter-2 the linearized aircraft models are developed for three trim states required for guidance, that is, straight flight, right turn and left turn. We present a controller that combines state feedback with a reduced-order observer such that the aircraft maintains trimmed straight or coordinated turning flight at a specified speed and turn rate and tracks altitude commands. Three controllers are designed for stabilizing each of the three trim states required for guidance. The state feedback gain is obtained using the pole placement technique. A reduced-order observer is used to estimate the unmeasured states. The linear controllers are then used in simulations with the nonlinear aircraft model and the results are presented. In Chapter-3 the guidance strategy proposed by Bhat and Kumar is described. Simulations of closed-loop guidance are performed with guidance block in the outerloop. It was found that, due to the stringent demand for straight flight, the guidance causes the aircraft to continuously switch between right and left turns without flying straight. Modified guidance strategy is developed, to eliminate this switching behavior. The effectiveness of the guidance is demonstrated by simulating a three-dimensional mission involving visiting of three way-points. Guidance under discrete position and heading updates is simulated. Effect of closed-loop poles on closed-loop guidance is studied. Simulations results are presented to show the effect of discrete position and heading updates.
4 Chapter 2 Controller Design for Full Aircraft Dynamics We wish to implement the guidance strategy described in reference [7] on the nonlinear aircraft model. This guidance strategy achieves perfect guidance under the assumption that the aircraft flies at a constant speed and altitude and follows turn rate commands perfectly. Hence, for successful implementation, the guidance strategy requires a controller that keeps the speed and altitude of the aircraft constant and tracks turn rate commands perfectly. In this chapter we design a controller such that the aircraft maintains trimmed straight or coordinated turning flight at a specified speed and turn rate and tracks constant altitude commands. Three controllers are designed for stabilizing each of the three trim states required for guidance, that is, straight flight, right turn and left turn. Linearized aircraft models are developed for the three trim states required for the guidance strategy. The controllers, which consist of a state feedback controller and an observer for estimating unmeasured states, are designed using these linearized aircraft models. The three controllers are then implemented on a Simulink model of the nonlinear aircraft. Simulation results are presented to demonstrate the performance of the controllers. 2.1 Linearized Aircraft Model A nonlinear aircraft model is developed in MATLAB, using the AeroSim aeronautical simulation block set. Variations in the density of air with altitude are neglected. The value of air density is fixed at 1.1117 kg/m 3, which is the density of air at an altitude of 1000 metres [8]. Consequently, the dynamics of the aircraft do not depend on the
5 altitude of the aircraft. The moment generated by the propeller is also neglected. The dynamics of engine are not considered. Instead, the propulsion system is assumed to consist only of a propeller. The control input to the propulsion system is propeller speed instead of traditionally used throttle setting. Static propeller data is used to find thrust for a given advance ratio and propeller speed. Thus, the propeller is assumed to react instantaneously for given input commands. Mass of the aircraft remains constant throughout the flight. The aerodynamic data and propeller data used in the model is given in Appendix A. The dynamics of the actuators deflecting the control surfaces are not considered. So, the control surfaces deflect instantaneously when input command is given. The nonlinear model is linearized around a trim condition. Three trim conditions are considered a straight level flight at a true air-speed of 17.44 m/s, coordinated level turning flight at a true air-speed of 17.44 m/s and a bank angle of 10 degrees, and coordinated level turning flight at a true air-speed of 17.44 m/s and a bank angle of 10 degrees. A trim routine available in the AeroSim aeronautical simulation blockset was modified such that it runs on an aircraft model without engine block. This routine was used to find the control inputs and aircraft states in each of the three trimmed flight conditions. The values of trim aircraft states and trim control inputs of the aircraft for the three trim states are given in appendices B, C and D. The equations of motion for an aircraft when linearized about a trim condition can be written in linear state variable form as [9, Chap-2] ẋ = Ax + Bu, (2.1) where the state vector [ x = v β α φ θ p q r h ] T (2.2) consists of perturbations v in true air-speed, β in sideslip, α in angle of attack, φ in bank angle, θ in pitch angle, p in roll rate, q in pitch rate, r in yaw rate and h in altitude from the trim values V T, β T, α T, φ T, θ T, P T, Q T, R T, H of the respective quantities. The control vector u = [δ n δ e δ a δ r ] T in equation (2.1) consists of perturbations δ n in the propeller speed, δ e in the elevator deflection, δ a in the aileron deflection and δ r in the rudder deflection from the trim propeller speed δ nt, trim elevator deflection δ et, trim aileron deflection δ at and trim rudder deflection δ rt, respectively. Linearized equations about a general trim condition were derived. These were used along with the trim states and trim inputs generated by the trim routine to generate
6 linearized aircraft models for each trim state. The equations used to derive linearized equations are given in Appendix A. The nonlinear aircraft is linearized about each of the three trim conditions discussed, and three linearized aircraft models are generated. The A and B matrices of the linearized aircraft models are given in appendices B, C and D. 2.2 State Feedback Controller We first design a state feedback controller to stabilize the trim state and track altitude commands. Since, density variations have been neglected, the inputs required to trim the aircraft at a given true air-speed and bank angle as well as the corresponding trim values of the state variables do not depend on the altitude. Specifically, every vector of the form x d =[0 0 0 0 0 0 0 0 h d ] T is an equilibrium solution of the linearized equation (2.1). Hence, the stabilization of the trim state as well as the tracking of the desired constant altitude h d can be simultaneously achieved by stabilizing the desired equilibrium x d, so that the closed-loop solutions satisfy lim t x(t)=x d. Let u, the control vector be given by u = Kx + u c, (2.3) where K R 4 9 is the state feedback gain matrix and u c is the input command. On substituting (2.3) in (2.1) we get ẋ = (A BK)x + Bu c. (2.4) Denoting, x=x x d and noting that x d is a constant vector, (2.4) yields ẋ = (A BK) x + (A BK)x d + Bu c. (2.5) Since Ax d =0, (2.5) yields ẋ = (A BK) x + B(u c Kx d ). (2.6) The error x converges to zero if all the eigenvalues of the matrix A BK have negative real parts and u c Kx d = 0. This gives u c = Kx d, so that u = K(x x d ). (2.7) We next use the pole placement technique to find the gain matrix K such that the matrix A BK has its eigenvalues at specified locations.
7 2.2.1 Selection of Closed-Loop Poles The open-loop eigenvalues of the aircraft model linearized about trimmed level flight are located at 4.22±j7.0651, 0.034±j0.7507, 0, 15.4836, 0.0092 and 0.1377± j2.007. For the aircraft model linearized about level coordinated turn of bank angle 10 degrees the eigenvalues are located at 4.2194 ± j7.0648, 0.0336 ± j0.7574, 0, 15.4806, 0.0113 and 0.1394 ± j2.0062, and for the aircraft model linearized about level coordinated turn of bank angle 10 degrees the eigenvalues are located at 4.2194 ± j7.0648, 0.0336 ± j0.7574, 0, 15.4806, 0.0113 and 0.1394 ± j2.0062. As all the eigenvalues have negative real parts each of the three trim states are stable. The first two eigenvalues correspond to short-period and phugoid modes, respectively. The eigenvalue at zero arises because altitude is taken as a state while developing the aircraft model. The last three eigenvalues correspond to roll, spiral and dutch-roll modes, respectively. Flying qualities for level-1 and category-a flight are selected, as flying qualities for MAVs are not available. The damping ratio of the poles corresponding to the shortperiod mode is selected to be between 0.35 and 1.30. The undamped natural frequency is selected to be between 2.5 rad/sec and 9 rad/sec [10, Chap-3]. Unlike a manned aircraft, for an MAV the phugoid mode should also be well damped with a damping ratio around 0.7 [11]. The time constant of the roll mode eigenvalue should be less than 1 sec. For dutch roll eigenvalue pair, the minimum values of ζ, ζω n and ω n are given as 0.19, 0.35 and 1 rad/s, respectively [10, Chap-3]. The closed-loop poles, that is, the eigenvalues of the matrix A BK are placed at 4 ± j5, 0.3 ± j0.2, 0.3, 5, 0.5 ± j0.5 and 0.3 for all the three trim conditions. The matrix K which places the closed-loop poles at these points is found by using the command place in MATLAB. Choice of poles is made taking into account the settling time of the closed-loop system and the demands on the control inputs. If the eigenvalues of A BK are placed further left, the system response becomes fast, but the demands on the propeller speed and elevator deflection will be higher than in the case of slower system. So a system with faster response will make the control inputs reach the saturation limits even for small commanded increase in the altitude and may cause problems of instability when used in nonlinear aircraft model. The state feedback controller is designed for each of the three linearized aircraft models developed. Three state feedback controllers were designed one for maintaining the aircraft operate around the straight level flight, one for maintaining the aircraft operate around the coordinated right turning flight and the other for maintaining the
8 aircraft operate around the coordinated left turning flight. The controller gain matrix K for aircraft models trimmed for level, 10 degree banked coordinated turn and 10 degree banked coordinated turn, which places the closed-loop poles at the above mentioned locations are given in appendices B, C and D, respectively. 2.3 Observer Design The feedback controller described in the previous section uses all the state variables for feedback. However, we assume that only true air-speed, pitch rate, roll rate, yaw rate and altitude are available for feedback. Hence we next consider the design of reducedorder observer for estimating the unmeasured state variables based on the theory given in reference [12, Chap-12]. The reduced-order observer generates the estimates of the perturbations β, α, φ, θ based on the measurements of the perturbations v, p, q and r and the control vector u. A reduced-order observer is designed for each of the three linearized models. The dynamics of the aircraft are independent of the altitude, and are hence completely captured in the evolution of the eight state variables v, β, α, φ, θ, p, q, r. The state variable h is included only to achieve tracking of the commanded altitude. So the A matrix of the system used for observer design is the 8 8 matrix corresponding to the state variables v, β, α, φ, θ, p, q, r. Consider the linear aircraft model with 8 state variables ẋ = Ax + Bu, y = Cx, (2.8) where the state vector x=[v β α φ θ p q r] T can be partitioned into two parts, the measured variables x a =[v p q r] T and the unmeasured variables x b =[β α φ θ] T. The [ ] output matrix C is given by I 4 4 0 4 4 so that the state variable x a is equal to the output y. Then the partitioned state and output equations become ẋa ẋ b = y = A aa A ba [ I 0 A ab A bb ] x a x a x b x b + B a B b u, (2.9). (2.10)
9 The equations which define the reduced-order observer are [12, Chap-12] η =  η + [ x = ˆB Ĉ η + ˆDy, ˆF ] y u, (2.11) where  = A bb K e A ab, ˆB = ÂK e + A ba K e A aa, ˆF = B b K e B a, Ĉ = 0 4 4 I 4 4, (2.12) ˆD = I 4 4 K e. The state estimate x and the observer state η are given by x = x a x b, (2.13) η = x b K e y. (2.14) The matrix K e is found by placing the eigenvalues of  at desired locations. As a general rule the observer poles must be two to five times faster than the controller poles to make sure the estimation error converges to zero quickly. Such faster decay of the observer error compared with the desired dynamics makes the controller poles dominate the system response [12, Chap-12]. The eigenvalues of  are placed at 16, 15 ± j1 and 17. The observer along with the state feedback controller stabilizes the aircraft around the trim state. The matrix K e for aircraft models trimmed for level, level coordinated right turn and level coordinated left turn are given in appendices B, C and D, respectively. 2.4 Control Law Implementation On Nonlinear Aircraft Model The linear controllers described in Section 2.2 and 2.3 for the three trim states were used in simulations with the nonlinear aircraft model. Each controller consists of a
10 Commanded trim Trim selector U Trim Y Trim x d Controller Saturation Block U Nonlinear Aircraft Model Y Figure 2.1: Block diagram of closed-loop aircraft with trim selector state feedback controller and a reduced-order observer. Each controller stabilizes the aircraft around the respective trim state, for which it is designed. At any given time only one of the three controllers is used. With the help of the three controllers we can operate the aircraft in either of the three trim states. Moreover, the aircraft can be made to switch between any two of the three trim states by switching the corresponding controller. This is necessary because the aircraft should be able to switch from one trim state to other trim state for the guidance strategy to be implemented. The block diagram depicting the selection of controller by the trim selector is shown in Figure 2.1. The inputs to the controller are perturbations from the trim operating point, as the linear controller operates on the perturbations in the states. The block diagram in Figure 2.2 shows the reduced-order observer based controller employed for the nonlinear aircraft model. Inputs to the observer are perturbations from the trim values. As shown in Figure 2.2 the inputs to the reduced-order observer are u and y which are the perturbations in inputs and outputs, respectively, from the trim values U Trim and Y Trim. The nonlinear aircraft model works on the true inputs. So the controller output u is added to the trim input vector U Trim and passed to the nonlinear aircraft model. To prevent the control inputs from going out of operating range, a saturation block has been placed before the aircraft model. The operating speed of propeller is maintained between 3,000 rpm and 10,000 rpm. The deflections of elevator, aileron and rudder are kept between 10 degrees and 10 degrees.
11 x d x State feedback Gain Matrix K u U Trim Saturation Block Nonlinear Aircraft Model Y Y Trim U Trim y Estimates of Side slip angle AOA, bank angle & Pitch angle Reduced Order Observer True Air speed, angular rates Altitude Figure 2.2: Reduced-order observer based controller The load factor for 10 degrees bank angle turn is 1.0154. The expression for turn rate is given by M = g n 2 1, (2.15) V where n is load factor and V is the velocity of the aircraft. For a circular turn at a bank angle of 10 degrees and a true air-speed of 17.44 m/s, the turn rate is 0.0992 rad/s and it takes 63.33 seconds to complete one full circular turn. Figure 2.3 shows the response of the aircraft for commands from the trim selector. The turn rate that has to be tracked by the aircraft is shown in dashed lines. At t = 0 the aircraft is commanded to do a level coordinated turn at a bank angle 10 degrees. At t = 100 sec it is commanded to fly level and straight. At t = 200 sec is commanded to do a level coordinated turn at a bank angle 10 degrees. Figure 2.4 shows the responses of true air-speed and altitude for commands of the trim selector. The maximum variation of true air-speed from trim value of 17.44 m/s is 0.017 m/s. The maximum variation of altitude from the trim altitude of 1000 m is 0.11 m. Figure 2.5 shows the responses of angle of attack and sideslip angle to trim selector commands. Both the angle of attack and sideslip angle vary rapidly when the aircraft switches from one trim state to the other. Figure 2.6 shows the response of propeller speed. The propeller speed falls sharply when the aircraft switches from one trim state to the other and then
12 0.15 0.1 0.05 Turn rate (rad/s) 0 0.05 0.1 0.15 0.2 0 50 100 150 200 250 300 t (s) Figure 2.3: Response of the aircraft for trim selector commands. settles to a constant value. Figure 2.7 shows the response of elevator deflections. The elevator deflects sharply when the aircraft switches from one trim state to the other. The deflections are well within the operating range. Figure 2.8 shows the response of aileron deflections. Initially the aileron deflects away from its final steady value. Figure 2.9 shows the response of rudder deflection to trim selector commands.
13 17.458 1000.02 17.456 17.454 1000 17.452 999.98 True air speed (m/s) 17.45 17.448 17.446 Altitude (m) 999.96 999.94 17.444 999.92 17.442 17.44 999.9 17.438 0 50 100 150 200 250 300 t (s) 999.88 0 50 100 150 200 250 300 t (s) Figure 2.4: True air-speed and altitude responses to trim selector commands. 1 2.45 0.8 2.4 0.6 2.35 0.4 Angle of attack (deg) 2.3 2.25 Side slip angle (deg) 0.2 0 0.2 2.2 0.4 0.6 2.15 0.8 2.1 0 50 100 150 200 250 300 t (s) 1 0 50 100 150 200 250 300 t (s) Figure 2.5: Angle of attack and sideslip responses to trim selector commands.
14 6060 5.6 6040 5.4 6020 5.2 Propeller speed (rpm) 6000 5980 5960 Elevator deflection (deg) 5 4.8 4.6 5940 4.4 5920 4.2 5900 0 50 100 150 200 250 300 t (s) 4 0 50 100 150 200 250 300 t (s) Figure 2.6: Response of propeller speed to trim selector commands. Figure 2.7: Response of elevator deflection to trim selector commands. 0.15 0.3 0.1 0.2 0.05 0.1 Aileron deflection (deg) 0 0.05 Rudder deflection (deg) 0 0.1 0.2 0.1 0.3 0.15 0.4 0.2 0 50 100 150 200 250 300 t (s) 0.5 0 50 100 150 200 250 300 t (s) Figure 2.8: Response of aileron deflection to selector commands. Figure 2.9: Response of rudder deflection to selector commands.
15 Chapter 3 Closed-Loop Guidance 3.1 Guidance Strategy Bhat and Kumar [7] present a guidance strategy to steer a MAV from a given initial position and heading to a specified destination way-point in an obstacle-free environment. The strategy achieves perfect guidance at constant altitude and speed under continuous and perfect position and heading updates, and perfect tracking of turn rate commands for small as well as large inter way-point distances. Reference [7] considers a MAV flying at a constant altitude and speed in a twodimensional plane. A kinematic model for such an aircraft is given by ẋ = V cos λ, ẏ = V sin λ, λ = ν, (3.1) where x and y are the position co-ordinates of the aircraft, λ is the heading angle of the aircraft, ν is the turn rate and V is the speed of the aircraft. If the maximum permissible load factor of the MAV is n, then the maximum permissible rate at which the aircraft can turn is given by equation (2.15). The turn rate constraint ν M leads to a lower bound V/M on the turn radius. Under the assumption of constant velocity, time-optimal trajectories of (3.1) between specified initial and final positions and initial headings under the turn rate constraint ν M consist of arcs of circles of minimum turn radius V/M and straight lines. In other words, a time-optimal turn rate time history takes the values ±M and 0. Let r and r f be the position vectors of the instantaneous location and destination way-point respectively. The cross range error P and the down range error Q to the
16 Down range L R R L Cross range Figure 3.1: Phase portrait under guidance strategy destination are defined as P = 1 V ( V D).ˆk, (3.2) Q = 1 V ( D V ), (3.3) where D = r f r is the relative displacement between the instantaneous location and the destination way-point, V is the instantaneous velocity vector of the MAV, and k is a unit vector orthogonal to the plane of motion of the MAV. The magnitudes of D and V are P 2 + Q 2 and V, respectively. Letting ψ(p, Q) = ( P V/M) 2 + Q 2, the feedback strategy that steers the MAV from a initial position and heading to a specified destination way-point along the shortest path is given by where ν(t) = λ(p (t), Q(t)), (3.4) λ(p, Q) = Msign(P ), ψ(p, Q) V 2 /M 2, P 0, = Msign(P ), ψ(p, Q) < V 2 /M 2, = 0, P = 0, Q > 0, = M, P = 0, Q < 0. (3.5) Figure 3.1 shows the phase portrait of the closed-loop system obtained by applying the guidance strategy (3.5) to the kinematic equations (3.1). It demonstrates that the guidance strategy (3.4) - (3.5) steers the vehicle to the destination way-point by driving cross range and down range to zero in a finite time.
17 Guidance Commanded trim Trim selector x d U Trim Controller Y Trim Saturation Block U Nonlinear Aircraft Model Latitude Longitude Heading Y Figure 3.2: Schematic diagram of guidance strategy implementation 3.2 Closed-Loop Guidance The guidance strategy described in Section 3.1 is used for guiding the aircraft from the initial way-point to the final way-point. A schematic diagram of the nonlinear aircraft model with guidance in the outer-loop is shown in the Figure 3.2. The guidance strategy given in Section 3.1 is implemented in the guidance block. The guidance block takes position and heading updates of the aircraft and gives commands to the trim selector in order to select one of the three trim states described in Section 2.1. The guidance block then commands the aircraft to either fly straight, turn right, or turn left and selects the correct inner-loop controller to ensure that the aircraft operation converges to that trim state. As the distances between the way-points considered by us are small compared to the radius of earth, x and y co-ordinates of the position of the aircraft are obtained by multiplying longitude and latitude by the radius of earth, respectively. The radius of earth is taken as 6371.3 km [13]. Results of simulations performed on closed-loop aircraft with guidance in the outerloop are presented. Figure 3.3 shows the four paths followed by the aircraft to reach each of the way-points (0,200,1000), (0,800,1000) (0,-800,1000) and (1000,0,1000) starting from (0,0,1000) with a initial heading along the x-axes. For inter way-point distances greater than minimum turn radius given by V T /M=175.84 m, the aircraft first takes a turn until the heading is aligned with destination way-point. For inter waypoint distances less than the minimum turn radius, the aircraft first moves away from destination way-point and then turns towards it. The simulation is stopped as soon as the trajectories reach within 20 m of the destination way-point. The aircraft can not come very close to the destination way-point because the aircraft takes time to settle
18 800 600 400 200 Y (m) 0 200 400 600 800 400 200 0 200 400 600 800 1000 1200 X (m) Figure 3.3: Aircraft path under closed-loop guidance down in the trim state while switching between the trim states. The guidance strategy assumes that the guidance commands are obeyed instantaneously, but the aircraft takes a finite time to switch from one trim state to another. This sluggishness of the aircraft to implement guidance commands results in the aircraft missing the target by 20 metres. As shown in Figure 3.3, instead of flying straight to reach the way-point (1000,0,1000), which is aligned with the initial heading of the aircraft, the aircraft reaches the waypoint by performing a series of right and left turns. This is because of the stringent demand of P = 0 by the guidance strategy for straight flight. When the aircraft heading is aligned with the destination way-point the guidance block should command the aircraft to fly straight, but by the time command is issued the aircraft over-shoots and the heading of the aircraft is no longer aligned with the destination way-point. As the condition P = 0 is satisfied only at discrete instants, the guidance block commands the aircraft either to turn right or left. So, the aircraft reaches the destination way-point by performing a sequence of right or left turns and the path of the aircraft is wavy, instead of smooth circular turns and straight lines. 3.3 Modified Guidance Strategy The guidance strategy described in Section 3.1 is modified so that the aircraft reaches the destination way-point smoothly without continuously switching between right and
19 Down range S L R R L Cross range Figure 3.4: Phase portrait under modified guidance strategy left turns. Conditions for straight flight are relaxed such that the aircraft flies straight even if the destination way-point is not aligned with the heading of the aircraft. The guidance strategy is modified such that the aircraft flies straight when the destination way-point is within a cone of angle 180-2γ containing the current heading. The strategy is thus given by ν(t) = λ(p (t), Q(t)), (3.6) where λ(p, Q) = Msign(P ), ψ(p, Q) V 2 /M 2, P 0, Q P tan γ < 0, = Msign(P ), ψ(p, Q) < V 2 /M 2, Q P tan γ < 0, = 0, Q P tan γ 0, Q > 0, = M, P = 0, Q < 0. (3.7) Figure 3.4 shows the phase portrait of the closed-loop system obtained by applying the strategy (3.6) to the kinematic equations (3.1). The phase portrait in the figure demonstrates that the guidance strategy (3.6) - (3.7) steers the vehicle to the destination way-point. Simulations are done with different values of γ, and γ = 80 degrees is found to give better guidance in terms of the termination proximity to the destination way-point. With γ = 80 the aircraft reaches within 20 m to the destination way-point. The path followed by the aircraft with this modified guidance strategy is shown in the Figure 3.5 for the same destination way-points considered in Section 3.2. The aircraft flies straight for a considerable amount of time. The aircraft reaches the way-point (1000,0,0) flying
20 800 600 400 200 Y (m) 0 200 400 600 800 400 200 0 200 400 600 800 1000 1200 X (m) Figure 3.5: Aircraft path under modified guidance strategy straight. Even for other way-points the path followed by aircraft is smooth. Figures 3.6 and 3.7 compare the sequence of trim states commanded by the original guidance strategy to reach the destination way-point (0,800,1000) with that commanded by the modified guidance strategy. Trim selector command of 1 represents right turn, 2 represents straight flight and 3 represents left turn. The aircraft with the original guidance strategy switches between left and right turns and never flies straight, whereas the aircraft with the modified guidance strategy flies straight switching only occasionally to right and left turning trim states. Three dimensional way-point navigation is performed, as the controllers are capable of tracking altitude commands. Figure 3.8 shows the trajectory followed by the aircraft for visiting the three way-points (1500,1000,1005), (4000,0,1030) and (1000,-3000,995) in sequence starting from (0,0,1000) with a initial heading along the x-axes under the modified guidance strategy. The co-ordinates of the next way-point are loaded as soon as the aircraft is within 20 m from the current destination way-point. The first waypoint (1500,1000,1005) is reached at t = 104.2 secs, second way-point at t = 262.9 secs and the final way-point at t = 522.4 secs. The simulation is stopped when the aircraft reaches within 20 m from the third way-point. To reach the way-point the aircraft first climbs to the altitude of the way-point, and then moves at constant altitude. Figure 3.9 shows the altitude variation of the aircraft. As the three way-points are not in the same plane, the altitude of the aircraft changes while visiting the way-points.
21 3 3 Commands to trim selector 2 Commands to trim selector 2 1 1 0 10 20 30 40 50 60 t (s) 0 10 20 30 40 50 60 t (s) Figure 3.6: Trim selector commands with original guidance strategy. Figure 3.7: Trim selector commands with modified guidance strategy. Figure 3.10 shows the trim selector commands as the aircraft visits the three waypoints. The commands change rapidly as the aircraft comes close to the way-point. Figure 3.11 gives true air-speed response of the aircraft. There is sudden variation in the true air-speed just after each way-point is visited. As soon as the aircraft reaches a way-point, the aircraft is commanded to change altitude to match the altitude of the next way-point. So the true air-speed of the aircraft either increases or decreases depending on whether the aircraft is descending or ascending. Figure 3.12 shows the variation of angle of attack. The angle of attack momentarily reaches 10.3 degrees when the aircraft is climbing from an altitude of 1005 metres to 1030 metres. Figure 3.13 shows the variation of sideslip angle. The sideslip angle reaches a maximum of 11.5 degrees. Figure 3.14 shows the variation of engine speed. The engine speed is pushed to the limits of its operating range. The propeller speed reaches the upper limit of 10,000 rpm when the aircraft is climbing and reaches the lower limit of 3,000 rpm when the aircraft is descending. Figure 3.15 shows the variation of elevator deflection. The elevator deflection reaches its upper limit of 10 degrees when the aircraft is descending and reaches the lower limit of 10 degrees when the aircraft is climbing. Figure 3.16 shows the variation of aileron deflection. The variation in aileron deflection is well within its operating range. The aileron deflection reaches a maximum of 0.86 degrees and a minimum of 0.77 degrees. The variation in aileron deflection is so small that it may be difficult for the actuator to deflect the aileron by such small angle. The
22 1035 1030 z (m) 1030 1025 1020 1015 1010 1005 1000 995 1025 1020 Altitude (m) 1015 1010 1005 1000 1000 0 1000 2000 3000 y (m) 0 1000 2000 x (m) 3000 4000 995 990 0 100 200 300 400 500 600 t (s) Figure 3.8: Trajectory of the aircraft for visiting three way-points. Figure 3.9: Variation of altitude for visiting three way-points. resolution of the actuator should be high and the mechanical couplings should not have dead zones for the aileron to be deflected to such small angle. Figure 3.17 shows the variation in rudder deflection as the aircraft visits the way-points. The rudder deflection reaches a maximum of 4.45 degrees and a minimum of 3.85 degrees. 3.4 Guidance Under Discrete Position and Heading Updates One of the main assumptions for the guidance strategy is that the position and heading updates are continuously available. This assumption is not valid in practice. In order to sense the position and heading, we anticipate use of the Global Positioning System (GPS) which receives updates typically at an update frequency of 1Hz. In this case, only discrete position and heading updates are available. In this section we present the effect of discrete updates on the guidance strategy. In between the updates, no data is available to the guidance. To simplify the situation, we assume that the guidance command in between the updates is based only on the last updates of position and heading. To study the effect of discrete position and heading updates on the guidance strategy the GPS is simulated by a sample and hold block with a sampling period of 1 sec. The closed-loop guidance with sample and hold block is shown in Figure 3.18.
23 25 24 3 23 Trim selector commands 2 True air speed (m/s) 22 21 20 19 1 18 17 16 0 100 200 300 400 500 600 t (s) 15 0 100 200 300 400 500 600 t (s) Figure 3.10: Trim selector commands as the aircraft visits the way-points. Figure 3.11: Response of true air-speed of the aircraft. 12 15 10 10 8 Angle of attack (deg) 6 4 Side slip angle (deg) 5 0 2 5 0 2 0 100 200 300 400 500 600 t (s) 10 0 100 200 300 400 500 600 t (s) Figure 3.12: Variation of angle of attack. Figure 3.13: Variation of sideslip angle.
24 11000 10 10000 8 9000 6 Propeller speed (rpm) 8000 7000 6000 5000 4000 Elevator deflection (deg) 4 2 0 2 4 6 3000 8 2000 0 100 200 300 400 500 600 t (s) 10 0 100 200 300 400 500 600 t (s) Figure 3.14: Variation of propeller speed. Figure 3.15: Variation of elevator deflection. 1 5 0.8 4 0.6 3 Aileron deflection (deg) 0.4 0.2 0 0.2 Rudder deflection (deg) 2 1 0 1 0.4 2 0.6 3 0.8 0 100 200 300 400 500 600 t (s) 4 0 100 200 300 400 500 600 t (s) Figure 3.16: Variation of aileron deflection. Figure 3.17: Variation of rudder deflection.
25 Sample & Hold Guidance Commanded trim Trim selector Latitude Longitude Heading x d U Trim Controller Y Trim Saturation Block U Nonlinear Aircraft Model Y Figure 3.18: Schematic diagram of guidance strategy implementation with sample and hold block Figure 3.19 presents two trajectories the one in red generated under discrete position and heading updates, and the one in blue generated under continuous position and heading updates, the modified guidance strategy being used in both the cases. The aircraft starts at (0,0,1000) with a initial heading along the x-axes and reaches the destination way-point (600,500,1000). The simulation is stopped when the aircraft is within 20 m from the destination way-point. The crosses on the trajectory represent the locations where the updates are made available from the GPS block. While the closed-loop aircraft follows straight path when position and heading updates are continuously available, the aircraft deviates from the straight path under discrete position and heading updates. This is expected, because the guidance strategy demands continuous position and heading updates for perfect guidance. As the position and heading are updated at a interval of 1 second, the aircraft deviates from the path under discrete updates. 3.5 Effect of Closed-Loop Poles On Guidance The guidance strategy assumes that the guidance commands are implemented instantaneously, but the closed-loop aircraft takes a nonzero amount of time to settle in a trim state after switching from either of the other two trim states. Clearly faster settling yields better guidance. To study the improvement provided by faster settling,
26 600 500 400 300 y (m) 200 100 0 100 100 0 100 200 300 400 500 600 700 x (m) Figure 3.19: Guidance under discrete updates three controllers are designed to place the closed-loop poles for each of the three linear models at 4 ± j5, 1 ± j1, 1, 5, 1.5 ± j1 and 1 by selecting a suitable K matrix. Now, the closed-loop aircraft will be able to react quickly to the guidance commands, as the poles are placed further left in the complex plane than those given in subsection 2.2.1. The effect of closed-loop poles on guidance is studied by comparing the trajectory and trim selector commands for the closed-loop aircraft with faster poles with those of the aircraft with slower poles. Figure 3.20 shows the trajectory of the closed-loop aircraft with faster poles in red, and that with slower poles in blue. The original guidance strategy is implemented and the aircraft starts at (0,0,1000) with a initial heading along the x-axes in both the cases. The closed-loop aircraft with faster poles goes as close as 2 m to the destination way-point (-300,800,1000) where as the closed-loop aircraft with slower poles can go only up to 20 m to the destination way-point. Because of the small settling times of its states, the closed-loop aircraft with faster poles is quick to settle to one of the three trim states and achieves better guidance. This is evident from red plot which shows that the the closed-loop aircraft with faster poles reaches the way-point along a straight line path. The trajectory is smooth even if the original guidance strategy is used, because the aircraft is quick enough to respond to guidance commands. So, better guidance can be achieved with original guidance strategy, if the aircraft settles faster in the trim state commanded. Figure 3.21 shows the trim selector commands of the closed-loop aircraft with faster