Linearized Equations of Motion!

Size: px
Start display at page:

Download "Linearized Equations of Motion!"

Transcription

1 Linearized Equations of Motion Robert Stengel, Aircraft Flight Dynamics MAE 331, 216 Learning Objectives Develop linear equations to describe small perturbational motions Apply to aircraft dynamic equations Reading: Flight Dynamics , , , , Copyright 216 by Robert Stengel. All rights reserved. For educational use only Assignment 6 due: End of day, Nov. 22, 216 Code the aerodynamic, thrust, and inertial properties of the HondaJet in AeroModel.m MATLAB function for FLIGHTver2.m 2

2 Review Questions Describe the functions of airplane control systems: Elevator/Stabilator Ailerons/Elevons Rudder Spoilers How can you control the yaw dynamics of a flying wing airplane? What are compensating ailerons? Why aren t all control surfaces all moving? How are the mechanical dynamics of control systems modeled? What is a horn balance, and how does it work? What are control tabs, and how do they work? 3 How Is System Response Calculated? Linear and nonlinear, time-varying and timeinvariant dynamic models Numerical integration time domain Linear, time-invariant LTI dynamic models Numerical integration time domain State transition time domain Transfer functions frequency domain 4

3 Integration Algorithms Exact x T = x + f x t Rectangular Euler Integration xt k = xt k1 + xt k1,t k xt k1 + f [ xt k1,ut k1,wt k1 ]t t = t k t k1 T,u t,w t dt Trapezoidal modified Euler Integration ~MATLAB s ode23 xt k xt k1 + 1 [ 2 x + x 1 2 ] where x 1 = f [ xt k1,ut k1,wt k1 ]t = f [ xt k1 + x 1,ut k,wt k ]t See MATLAB manual for descriptions of ode45 and ode15s 5 Numerical Integration: MATLAB Ordinary Differential Equation Solvers * Explicit Runge-Kutta Algorithm Adams-Bashforth-Moulton Algorithm Numerical Differentiation Formula Modified Rosenbrock Method Trapezoidal Rule Trapezoidal Rule w/back Differentiation * Shampine, L. F. and M. W. Reichelt, The MATLAB ODE Suite, SIAM Journal on Scientific Computing, Vol. 18, 1997, pp

4 Nominal and Actual Trajectories Nominal or reference trajectory and control history { x N t, u N t,w N t} for t in [t o,t f ] x : dynamic state u : control input w : disturbance input Actual trajectory perturbed by Small initial condition variation, x o t o Small control variation, ut { xt, ut,wt } for t in [t o,t f ] = x N t + xt, u N t + ut,w N t + wt { } 7 Both Paths Satisfy the Dynamic Equations Dynamic models for the actual and the nominal problems are the same x N t = f[x N t,u N t,w N t], x N t o xt = f[xt,ut,wt], given given x t o Differences in initial condition and forcing... xt o = xt o x N t o ut = ut u N t wt = wt w N t in * + t o,t f, -... perturb rate of change and the state xt = x N t + xt xt = x N t + xt in * t o,t f +, 8

5 Approximate Neighboring Trajectory as a Linear Perturbation to the Nominal Trajectory x N t = f[x N t,u N t,w N t,t] xt = x N t + xt = f[x N t + xt,u N t + ut,w N t + wt,t] Approximate the new trajectory as the sum of the nominal path plus a linear perturbation xt = x N t + xt f[x N t,u N t,w N t,t]+ f x xt + f u ut + f w wt 9 Linearized Equation Approximates Perturbation Dynamics Solve for the nominal and perturbation trajectories separately Nominal Equation x N t = f[x N t,u N t,w N t,t], x N t o given Perturbation Equation dimx = n 1 dimu = m 1 dimw = s 1 xt f xt x x=x N t + f ut u=u N t u w=w N t x=x N t + f wt u=u N t w w=w N t x=x N t u=u N t w=w N t Ftxt+ Gtut+ Ltwt, x t o given dimx = n 1 dimu = m 1 dimw = s 1 1

6 Jacobian Matrices Express Solution Sensitivity to Small Perturbations Stability matrix, F, is square Ft = f x x=x N t u=u N t w=w N t dimf = n n = f 1 f 1 x 1 x n f n f n x 1 x n x=x N t u=u N t w=w N t 11 Sensitivity to Control Perturbations, G Gt = f = u x=x N t u=u N t w=w N t dimg = n m f 1 f 1 u 1 u m f n f n u 1 u m x=x N t u=u N t w=w N t 12

7 Sensitivity to Disturbance Perturbations, G Lt = f = w x=x N t u=u N t w=w N t diml = n s f 1 f 1 w 1 w s f n f n w 1 w s x=x N t u=u N t w=w N t 13 Scalar Example of Nominal and Perturbation Equations Actual System x t = x t + 2 t + 3u t + 4w 3 t Nominal System x N t = x N t + 2 N t + 3u N t + 4w 3 N t Perturbation System 2 x t = x t + 4x N tx t + 3u t +12w N tw t 14

8 Comparison of Damped Linear and Nonlinear Systems Linear Spring x 1 t = t t = 1x 1 t t Displacement Rate of Change Linear Spring Force vs. Displacement Spring Damper Linear plus Stiffening Cubic Spring x 1 t = t t = 1x 1 t 1x 3 1 t t Linear plus Weakening Cubic Spring x 1 t = t t = 1x 1 t +.8x 3 1 t t Cubic Spring Force vs. Displacement Cubic Spring Force vs. Displacement 15 MATLAB Simulation of Linear and Nonlinear Dynamic Systems MATLAB Main Script Nonlinear and Linear Examples clear tspan = [ 1]; xo = [, 1]; [t1,x1 = ode23nonlin,tspan,xo; xo = [, 1]; [t2,x2] = ode23nonlin,tspan,xo; xo = [, 1]; [t3,x3] = ode23lin,tspan,xo; xo = [, 1]; [t4,x4] = ode23lin,tspan,xo; subplot2,1,1 plott1,x1:,1,k,t2,x2:,1,b,t3,x3:,1,r,t4,x4:,1,g ylabelposition, grid subplot2,1,2 plott1,x1:,2,k,t2,x2:,2,b,t3,x3:,2,r,t4,x4:,2,g xlabeltime, ylabelrate, grid Linear System x 1 t = t x 2 t = 1x 1 t t function xdot = Lint,x Linear Ordinary Differential Equation x1 = Position x2 = Rate xdot = [x2-1*x1 - x2]; Nonlinear System x 1 t = t x 2 t = 1x 1 t +.8x 3 1 t t function xdot = NonLint,x Nonlinear Ordinary Differential Equation x1 = Position x2 = Rate xdot = [x2-1*x1 +.8*x1^3 - x2]; 16

9 Linear and Stiffening Cubic Springs: Small and Large Initial Conditions Cubic term Linear and nonlinear responses are indistinguishable with small initial condition 17 Linear and Weakening Cubic Springs: Small and Large Initial Conditions Cubic term 18

10 Linear, Time-Varying LTV Approximation of Perturbation Dynamics 19 Stiffening Linear-Cubic Spring Example x 1N N Nonlinear, time-invariant NTI equation x 1 t = f 1 = t t = f 2 = 1x 1 t 1x 1 3 t t Integrate equations to produce nominal path t f f 1N f 2N dt x 1N t N t Analytical evaluation of partial derivatives in,t f f 1 x 1 = ; f 1 = 1 f 2 = 1 3 1N t; x 1 f 2 = 1 f 1 u = ; f 1 w = f 2 u = ; f 2 w = 2

11 Nominal NTI and Perturbation LTV Dynamic Equations Nonlinear, time-invariant NTI nominal equation x N t = f[x N t], x N given x 1N t = N t N t = 1x 1N t 1x 1N 3 t N t x 1N N Example = 9 Perturbations approximated by linear, time-varying LTV equation x 1 t t xt = Ftxt, x given = N t 1 x 1 t t Example x 1 = 1 21 Comparison of Approximate and Exact Solutions Initial Conditions N = 9 =1 N t+ t =1 t =1 x N t xt x N t+ xt xt x N t xt x N t + xt xt 22

12 Suppose Nominal Initial Condition is Zero Nominal solution remains at equilibrium x N t = f[x N t], x N =, x N t = in [, ] Perturbation equation is linear and time-invariant LTI x 1 t t = x 1 t t 23 Separation of the Equations of Motion into Longitudinal and Lateral- Directional Sets 24

13 Grumman F9F Rate of change of Translational Velocity Rate of change of Translational Position Rate of change of Angular Velocity I xy = I yz = Rate of change of Angular Position Rigid-Body Equations of Motion Scalar Notation u = X / m gsin + rv qw v = Y / m + gsin cos ru + pw w = Z / m + g cos cos + qu pv x I = cos cos u + cos sin + sin sin cos v + sin sin + cos sin cos w y I = cos sin u + cos cos + sin sin sin v + sin cos + cos sin sin w z I = sin u + sin cos v + cos cos w 2 I xx I zz I xz p = I zz L + I xz N { I xz I yy I xx I zz p + I 2 xz + I zz I zz I yy r }q pr I xz p 2 r 2 q = M I xx I zz I yy r = I xz L + I xx N { I xz I yy I xx I zz r + I 2 xz + I xx I xx I yy p }q = p + qsin + r cos tan = qcos r sin = qsin + r cos sec 2 I xx I zz I xz State Vector x 1 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 1 x 11 x 12 u v w x y z = p q r 25 Reorder the State Vector x 1 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 1 x 11 x 12 u v w x y z = p q r First six elements of the state are longitudinal variables Second six elements of the state are lateraldirectional variables x 1 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 1 x 11 x 12 new = x Lon x LatDir = u w x z q v y p r * 26

14 Longitudinal Equations of Motion Dynamics of velocity, position, angular rate, and angle primarily in the vertical plane u = X / m gsin + rv qw x 1 = f 1 w = Z / m + gcos cos + qu pv = f 2 x I = cos cos u + cos sin + sin sin cos v + sin sin + cos sin cos w x 3 = f 3 z I = sin u + sin cos v + cos cos w x 4 = f 4 pr I xz p 2 r 2 q = M I xx I zz I yy x 5 = f 5 = qcos rsin x 6 = f 6 x Lon t = f[x Lon t,u Lon t,w Lon t] 27 Lateral-Directional Equations of Motion Dynamics of velocity, position, angular rate, and angle primarily out of the vertical plane v = Y / m + gsin cos ru + pw x 7 = f 7 y I = cos sin u + cos cos + sin sin sin v + sin cos + cos sin sin w x 8 = f 8 I 2 xxi zz I xz I I I 2 xx zz xz p = I zz L + I xz N { I xz I yy I xx I zz p + I 2 xz + I zz I zz I yy r }q r = I xz L + I xx N { I xz I yy I xx I zz r + I 2 xz + I xx I xx I yy p }q = p + qsin + r cos tan x 11 = f 11 = qsin + r cos sec x 12 = f 12 x 9 = f 9 x 1 = f 1 x LD t = f[x LD t,u LD t,w LD t] 28

15 Sensitivity to Small Motions 12 x 12 stability matrix for the entire system Ft = f 1 f... 1 x 1 x f 12 f x 1 x 12 Four 6 x 6 blocks distinguish longitudinal and lateral-directional effects Effects of longitudinal perturbations on longitudinal motion Effects of lateral-directional perturbations on longitudinal motion F = Effects of longitudinal perturbations on lateral-directional motion F Lon LatDir F Lon Lon F LatDir F LatDir Effects of lateral-directional perturbations on lateral-directional motion 29 Sensitivity to Small Control Inputs Gt = 12 x 6 control matrix for the entire system f 1 E f 1 T f 1 F f 1 A f 1 R f 1 SF f 2 E f 2 T f 2 F f 2 A f 2 R f 2 SF f 12 E f 12 T f 12 F f 12 A f 12 R f 12 SF Four 6 x 3 blocks distinguish longitudinal and lateraldirectional control effects Effects of longitudinal controls on longitudinal motion G = Effects of longitudinal controls on lateral-directional motion G Lon LatDir G Lon Effects of lateral-directional controls on longitudinal motion Lon G LatDir G LatDir Effects of lateral-directional controls on lateral-directional motion 3

16 Sensitivity to Small Disturbance Inputs Disturbance input vector and perturbation wt = u w t w w t q w t v w t p w t r w t Axial wind, m / s Normal wind, m / s Pitching wind shear, deg / s or rad / s Lateral wind, m / s Rolling wind shear, deg / s or rad / s Yawing wind shear, deg / s or rad / s wt = u w t w w t q w t v w t p w t r w t Four 6 x 3 blocks distinguish longitudinal and lateral-directional effects Effects of longitudinal disturbances on longitudinal motion L = L Lon LatDir L Lon Effects of longitudinal disturbances on lateral-directional motion Effects of lateral-directional disturbances on longitudinal motion Lon L LatDir L LatDir Effects of lateral-directional disturbances on lateral-directional motion 31 Decoupling Approximation for Small Perturbations from Steady, Level Flight 32

17 Restrict the Nominal Flight Path to the Vertical Plane Nominal lateral-directional motions are zero x Lat DirN = x Lat DirN = Nominal longitudinal equations reduce to u N = X / m gsin N q N w N w N = Z / m + gcos N + q N u N x IN = cos N u N + sin N w N z IN = sin N u N + cos N w N q N = M I yy N = q N x 1 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 1 x 11 x 12 Nominal State Vector N = x Lon x LatDir = N 33 u N w N x N z N q N N Restrict the Nominal Flight Path to Steady, Level Flight Specify nominal airspeed V N and altitude h N = z N Calculate conditions for trimmed equilibrium flight See Flight Dynamics and FLIGHT program for a solution method Trimmed State Vector is constant = X / m gsin N q N w N = Z / m + gcos N + q N u N V N = cos N u N + sin N w N = sin N u N + cos N w N = M I yy = q N u w x z q Trim = u Trim w Trim V N t t z N Trim 34

18 Small Longitudinal and Lateral- Directional Perturbation Effects Assume the airplane is symmetric and its nominal path is steady, level flight Small longitudinal and lateral-directional perturbations are approximately uncoupled from each other 12 x 12 system is block diagonal constant, i.e., linear, time-invariant LTI decoupled into two separate 6 x 6 systems F = F Lon F LatDir G = G Lon G LatDir L = L Lon L LatDir 35 6 x 6 LTI Longitudinal Perturbation Model Dynamic Equation x Lon t = F Lon x Lon t + G Lon u Lon t + L Lon w Lon t x Lon = State Vector x 1 x 3 x 4 x 5 x 6 Lon = u w x z q Control Vector u Lon = T E F Disturbance Vector w Lon = u wind w wind q wind 36

19 LTI Longitudinal Response to Initial Pitch Rate 37 6 x 6 LTI Lateral-Directional Perturbation Model Dynamic Equation x Lat Dir t = F Lat Dir x Lat Dir t + G Lat Dir u Lat Dir t + L Lat Dir w Lat Dir t x Lat Dir = State Vector x 1 x 3 x 4 x 5 x 6 Lat Dir v y p = r * Control Vector u Lat Dir = A R SF Disturbance Vector w Lon = v wind p wind r wind 38

20 LTI Lateral-Directional Response to Initial Yaw Rate 39 Next Time: Longitudinal Dynamics Reading: Flight Dynamics , Airplane Stability and Control Chapter 7 Learning Objectives 6 th -order -> 4 th -order -> hybrid equations Dynamic stability derivatives Long-period phugoid mode Short-period mode 4

21 Supplemental Material 41 How Do We Calculate the Partial Derivatives? Ft = f x x=x N t u=u N t w=w N t Gt = f u x=x N t u=u N t w=w N t Lt = f w x=x N t u=u N t w=w N t Numerically First differences in fx,u,w Analytically Symbolic evaluation of analytical models of F, G, and L 42

22 Numerical Estimation of the Jacobian Matrix, Ft f 1 t x 1 f 2 t x 1 f 1 f 2 x 1 + x 1 x n x 1 + x 1 x n * f 1 x 1 * x 1 x n x=xn t u=u N t w=w N t 2x 1 ; * f 2 x 1 * x 1 x n x=xn t u=u N t w=w N t 2x 1 ; f 1 t f 2 t f 1 f 2 x 1 + x n x 1 + x n * f 1 2 * f 2 2 Continue for all n x n elements of Ft x 1 * x n x 1 * x n x=xn t u=u N t w=w N t x=xn t u=u N t w=w N t 43 Numerical Estimation of the Jacobian Matrix, Gt f 1 t u 1 f 2 t u 1 f 1 f 2 u 1 + u 1 u 2 u m u 1 + u 1 u 2 u m * f 1 u 1 * u 1 u 2 u m x=xn t u=u N t w=w N t 2u 1 ; * f 2 u 1 * u 1 u 2 u m x=xn t u=u N t w=w N t 2u 1 ; f 1 t u 2 f 2 t u 2 f 1 f 2 u 1 u 2 + u 2 u m u 1 u 2 + u 2 u m * f 1 2u 2 * f 2 2u 2 Continue for all n x m elements of Gt u 1 u 2 * u 2 u m u 1 u 2 * u 2 u m x=xn t u=u N t w=w N t x=xn t u=u N t w=w N t 44

23 f 1 t w 1 f 2 t w 1 f 1 f 2 Numerical Estimation of the Jacobian Matrix, Lt w 1 + w 1 w 2 w s w 1 + w 1 w 2 w s * f 1 w 1 * w 1 w 2 w s x=xn t u=u N t w=w N t 2w 1 ; * f 2 w 1 * w 1 w 2 w s x=xn t u=u N t w=w N t 2w 1 ; f 1 t w 2 f 2 t w 2 f 1 f 2 w 1 w 2 + w 2 w s w 1 w 2 + w 2 w s * f 1 2w 2 * f 2 Continue for all n x s elements of Lt 2w 2 w 1 w 2 * w 2 w s w 1 w 2 * w 2 w s 45 x=xn t u=u N t w=w N t x=xn t u=u N t w=w N t

Time Response of Dynamic Systems! Multi-Dimensional Trajectories Position, velocity, and acceleration are vectors

Time Response of Dynamic Systems! Multi-Dimensional Trajectories Position, velocity, and acceleration are vectors Time Response of Dynamic Systems Robert Stengel Robotics and Intelligent Systems MAE 345, Princeton University, 217 Multi-dimensional trajectories Numerical integration Linear and nonlinear systems Linearization

More information

Both Paths Satisfy the Dynamic Equations

Both Paths Satisfy the Dynamic Equations Liearized Equatios ad Modes of Motio Robert Stegel, Aircraft Flight Dyamics MAE 331, 008 Liearizatio of oliear dyamic models Nomial flight path Perturbatios about the omial flight path Modes of motio Nomial

More information

Alternative Expressions for the Velocity Vector Velocity restricted to the vertical plane. Longitudinal Equations of Motion

Alternative Expressions for the Velocity Vector Velocity restricted to the vertical plane. Longitudinal Equations of Motion Linearized Longitudinal Equations of Motion Robert Stengel, Aircraft Flig Dynamics MAE 33, 008 Separate solutions for nominal and perturbation flig paths Assume that nominal path is steady and in the vertical

More information

Lecture AC-1. Aircraft Dynamics. Copy right 2003 by Jon at h an H ow

Lecture AC-1. Aircraft Dynamics. Copy right 2003 by Jon at h an H ow Lecture AC-1 Aircraft Dynamics Copy right 23 by Jon at h an H ow 1 Spring 23 16.61 AC 1 2 Aircraft Dynamics First note that it is possible to develop a very good approximation of a key motion of an aircraft

More information

Translational and Rotational Dynamics!

Translational and Rotational Dynamics! Translational and Rotational Dynamics Robert Stengel Robotics and Intelligent Systems MAE 345, Princeton University, 217 Copyright 217 by Robert Stengel. All rights reserved. For educational use only.

More information

Chapter 4 The Equations of Motion

Chapter 4 The Equations of Motion Chapter 4 The Equations of Motion Flight Mechanics and Control AEM 4303 Bérénice Mettler University of Minnesota Feb. 20-27, 2013 (v. 2/26/13) Bérénice Mettler (University of Minnesota) Chapter 4 The Equations

More information

Linearized Equations of Motion Robert Stengel, Aircraft Flight Dynamics MAE 331, 2018

Linearized Equations of Motion Robert Stengel, Aircraft Flight Dynamics MAE 331, 2018 Linearized Equaions of Moion Rober Sengel, Aircraf Fligh Dynamics MAE 331, 2018 Learning Objecives Develop linear equaions o describe small perurbaional moions Apply o aircraf dynamic equaions Reading:

More information

Control Systems! Copyright 2017 by Robert Stengel. All rights reserved. For educational use only.

Control Systems! Copyright 2017 by Robert Stengel. All rights reserved. For educational use only. Control Systems Robert Stengel Robotics and Intelligent Systems MAE 345, Princeton University, 2017 Analog vs. digital systems Continuous- and Discretetime Dynamic Models Frequency Response Transfer Functions

More information

Aircraft Equations of Motion: Translation and Rotation Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018

Aircraft Equations of Motion: Translation and Rotation Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018 Aircraft Equations of Motion: Translation and Rotation Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018 Learning Objectives What use are the equations of motion? How is the angular orientation of

More information

Stochastic and Adaptive Optimal Control

Stochastic and Adaptive Optimal Control Stochastic and Adaptive Optimal Control Robert Stengel Optimal Control and Estimation, MAE 546 Princeton University, 2018! Nonlinear systems with random inputs and perfect measurements! Stochastic neighboring-optimal

More information

Flight Dynamics and Control

Flight Dynamics and Control Flight Dynamics and Control Lecture 1: Introduction G. Dimitriadis University of Liege Reference material Lecture Notes Flight Dynamics Principles, M.V. Cook, Arnold, 1997 Fundamentals of Airplane Flight

More information

Dynamics and Control of Rotorcraft

Dynamics and Control of Rotorcraft Dynamics and Control of Rotorcraft Helicopter Aerodynamics and Dynamics Abhishek Department of Aerospace Engineering Indian Institute of Technology, Kanpur February 3, 2018 Overview Flight Dynamics Model

More information

Linearized Longitudinal Equations of Motion Robert Stengel, Aircraft Flight Dynamics MAE 331, 2018

Linearized Longitudinal Equations of Motion Robert Stengel, Aircraft Flight Dynamics MAE 331, 2018 Linearized Longitudinal Equations of Motion Robert Stengel, Aircraft Flight Dynamics MAE 331, 018 Learning Objectives 6 th -order -> 4 th -order -> hybrid equations Dynamic stability derivatives Long-period

More information

Nonlinear State Estimation! Extended Kalman Filters!

Nonlinear State Estimation! Extended Kalman Filters! Nonlinear State Estimation! Extended Kalman Filters! Robert Stengel! Optimal Control and Estimation, MAE 546! Princeton University, 2017!! Deformation of the probability distribution!! Neighboring-optimal

More information

Optimal Control and Estimation MAE 546, Princeton University Robert Stengel, Preliminaries!

Optimal Control and Estimation MAE 546, Princeton University Robert Stengel, Preliminaries! Optimal Control and Estimation MAE 546, Princeton University Robert Stengel, 2018 Copyright 2018 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/mae546.html

More information

Minimization with Equality Constraints

Minimization with Equality Constraints Path Constraints and Numerical Optimization Robert Stengel Optimal Control and Estimation, MAE 546, Princeton University, 2015 State and Control Equality Constraints Pseudoinverse State and Control Inequality

More information

Topic # Feedback Control

Topic # Feedback Control Topic #7 16.31 Feedback Control State-Space Systems What are state-space models? Why should we use them? How are they related to the transfer functions used in classical control design and how do we develop

More information

Dynamic Optimal Control!

Dynamic Optimal Control! Dynamic Optimal Control! Robert Stengel! Robotics and Intelligent Systems MAE 345, Princeton University, 2017 Learning Objectives Examples of cost functions Necessary conditions for optimality Calculation

More information

Adaptive Trim and Trajectory Following for a Tilt-Rotor Tricopter Ahmad Ansari, Anna Prach, and Dennis S. Bernstein

Adaptive Trim and Trajectory Following for a Tilt-Rotor Tricopter Ahmad Ansari, Anna Prach, and Dennis S. Bernstein 7 American Control Conference Sheraton Seattle Hotel May 4 6, 7, Seattle, USA Adaptive Trim and Trajectory Following for a Tilt-Rotor Tricopter Ahmad Ansari, Anna Prach, and Dennis S. Bernstein Abstract

More information

Introduction to Flight Dynamics

Introduction to Flight Dynamics Chapter 1 Introduction to Flight Dynamics Flight dynamics deals principally with the response of aerospace vehicles to perturbations in their flight environments and to control inputs. In order to understand

More information

Time Response of Linear, Time-Invariant (LTI) Systems Robert Stengel, Aircraft Flight Dynamics MAE 331, 2018

Time Response of Linear, Time-Invariant (LTI) Systems Robert Stengel, Aircraft Flight Dynamics MAE 331, 2018 Time Response of Linear, Time-Invariant LTI Systems Robert Stengel, Aircraft Flight Dynamics MAE 331, 2018 Learning Objectives Methods of time-domain analysis Continuous- and discrete-time models Transient

More information

Minimization of Static! Cost Functions!

Minimization of Static! Cost Functions! Minimization of Static Cost Functions Robert Stengel Optimal Control and Estimation, MAE 546, Princeton University, 2017 J = Static cost function with constant control parameter vector, u Conditions for

More information

Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor

Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor Robot Dynamics Rotary Wing AS: Control of a Quadrotor 5-85- V Marco Hutter, Roland Siegwart and Thomas Stastny Robot Dynamics - Rotary Wing AS: Control of a Quadrotor 7..6 Contents Rotary Wing AS. Introduction

More information

FLIGHT DYNAMICS. Robert F. Stengel. Princeton University Press Princeton and Oxford

FLIGHT DYNAMICS. Robert F. Stengel. Princeton University Press Princeton and Oxford FLIGHT DYNAMICS Robert F. Stengel Princeton University Press Princeton and Oxford Preface XV Chapter One Introduction 1 1.1 ELEMENTS OF THE AIRPLANE 1 Airframe Components 1 Propulsion Systems 4 1.2 REPRESENTATIVE

More information

AB-267 DYNAMICS & CONTROL OF FLEXIBLE AIRCRAFT

AB-267 DYNAMICS & CONTROL OF FLEXIBLE AIRCRAFT FLÁIO SILESTRE DYNAMICS & CONTROL OF FLEXIBLE AIRCRAFT LECTURE NOTES LAGRANGIAN MECHANICS APPLIED TO RIGID-BODY DYNAMICS IMAGE CREDITS: BOEING FLÁIO SILESTRE Introduction Lagrangian Mechanics shall be

More information

Optimal Control and Estimation MAE 546, Princeton University Robert Stengel, Preliminaries!

Optimal Control and Estimation MAE 546, Princeton University Robert Stengel, Preliminaries! Optimal Control and Estimation MAE 546, Princeton University Robert Stengel, 2017 Copyright 2017 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/mae546.html

More information

Time-Invariant Linear Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 2015

Time-Invariant Linear Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 2015 Time-Invariant Linear Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 15 Asymptotic approach from time-varying to constant gains Elimination of cross weighting

More information

Aircraft Flight Dynamics!

Aircraft Flight Dynamics! Aircraft Flight Dynamics Robert Stengel MAE 331, Princeton University, 2016 Course Overview Introduction to Flight Dynamics Math Preliminaries Copyright 2016 by Robert Stengel. All rights reserved. For

More information

Atmospheric Hazards to Flight! Robert Stengel,! Aircraft Flight Dynamics, MAE 331, Frames of Reference

Atmospheric Hazards to Flight! Robert Stengel,! Aircraft Flight Dynamics, MAE 331, Frames of Reference Atmospheric Hazards to Flight! Robert Stengel,! Aircraft Flight Dynamics, MAE 331, 2016!! Microbursts!! Wind Rotors!! Wake Vortices!! Clear Air Turbulence Copyright 2016 by Robert Stengel. All rights reserved.

More information

Lecture #AC 3. Aircraft Lateral Dynamics. Spiral, Roll, and Dutch Roll Modes

Lecture #AC 3. Aircraft Lateral Dynamics. Spiral, Roll, and Dutch Roll Modes Lecture #AC 3 Aircraft Lateral Dynamics Spiral, Roll, and Dutch Roll Modes Copy right 2003 by Jon at h an H ow 1 Spring 2003 16.61 AC 3 2 Aircraft Lateral Dynamics Using a procedure similar to the longitudinal

More information

Minimization with Equality Constraints!

Minimization with Equality Constraints! Path Constraints and Numerical Optimization Robert Stengel Optimal Control and Estimation, MAE 546, Princeton University, 2017 State and Control Equality Constraints Pseudoinverse State and Control Inequality

More information

Time-Invariant Linear Quadratic Regulators!

Time-Invariant Linear Quadratic Regulators! Time-Invariant Linear Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 17 Asymptotic approach from time-varying to constant gains Elimination of cross weighting

More information

Frequency Domain System Identification for a Small, Low-Cost, Fixed-Wing UAV

Frequency Domain System Identification for a Small, Low-Cost, Fixed-Wing UAV Frequency Domain System Identification for a Small, Low-Cost, Fixed-Wing UAV Andrei Dorobantu, Austin M. Murch, Bernie Mettler, and Gary J. Balas, Department of Aerospace Engineering & Mechanics University

More information

April 15, 2011 Sample Quiz and Exam Questions D. A. Caughey Page 1 of 9

April 15, 2011 Sample Quiz and Exam Questions D. A. Caughey Page 1 of 9 April 15, 2011 Sample Quiz Exam Questions D. A. Caughey Page 1 of 9 These pages include virtually all Quiz, Midterm, Final Examination questions I have used in M&AE 5070 over the years. Note that some

More information

Stochastic Optimal Control!

Stochastic Optimal Control! Stochastic Control! Robert Stengel! Robotics and Intelligent Systems, MAE 345, Princeton University, 2015 Learning Objectives Overview of the Linear-Quadratic-Gaussian (LQG) Regulator Introduction to Stochastic

More information

Lecture 8: Calculus and Differential Equations

Lecture 8: Calculus and Differential Equations Lecture 8: Calculus and Differential Equations Dr. Mohammed Hawa Electrical Engineering Department University of Jordan EE201: Computer Applications. See Textbook Chapter 9. Numerical Methods MATLAB provides

More information

Lecture 8: Calculus and Differential Equations

Lecture 8: Calculus and Differential Equations Lecture 8: Calculus and Differential Equations Dr. Mohammed Hawa Electrical Engineering Department University of Jordan EE21: Computer Applications. See Textbook Chapter 9. Numerical Methods MATLAB provides

More information

Mech 6091 Flight Control System Course Project. Team Member: Bai, Jing Cui, Yi Wang, Xiaoli

Mech 6091 Flight Control System Course Project. Team Member: Bai, Jing Cui, Yi Wang, Xiaoli Mech 6091 Flight Control System Course Project Team Member: Bai, Jing Cui, Yi Wang, Xiaoli Outline 1. Linearization of Nonlinear F-16 Model 2. Longitudinal SAS and Autopilot Design 3. Lateral SAS and Autopilot

More information

Manifesto on Numerical Integration of Equations of Motion Using Matlab

Manifesto on Numerical Integration of Equations of Motion Using Matlab Manifesto on Numerical Integration of Equations of Motion Using Matlab C. Hall April 11, 2002 This handout is intended to help you understand numerical integration and to put it into practice using Matlab

More information

Aircraft Flight Dynamics Robert Stengel MAE 331, Princeton University, 2018

Aircraft Flight Dynamics Robert Stengel MAE 331, Princeton University, 2018 Aircraft Flight Dynamics Robert Stengel MAE 331, Princeton University, 2018 Course Overview Introduction to Flight Dynamics Math Preliminaries Copyright 2018 by Robert Stengel. All rights reserved. For

More information

Chapter 9b: Numerical Methods for Calculus and Differential Equations. Initial-Value Problems Euler Method Time-Step Independence MATLAB ODE Solvers

Chapter 9b: Numerical Methods for Calculus and Differential Equations. Initial-Value Problems Euler Method Time-Step Independence MATLAB ODE Solvers Chapter 9b: Numerical Methods for Calculus and Differential Equations Initial-Value Problems Euler Method Time-Step Independence MATLAB ODE Solvers Acceleration Initial-Value Problems Consider a skydiver

More information

Robot Dynamics Fixed-wing UAVs: Dynamic Modeling and Control

Robot Dynamics Fixed-wing UAVs: Dynamic Modeling and Control Robot Dynamics Fixed-wing UAVs: Dynamic Modeling and Control 151-0851-00 V Marco Hutter, Roland Siegwart, and Thomas Stastny 05.12.2017 1 Contents Fixed-wing UAVs 1. ntroduction 2. Aerodynamic Basics 3.

More information

Fundamentals of Airplane Flight Mechanics

Fundamentals of Airplane Flight Mechanics David G. Hull Fundamentals of Airplane Flight Mechanics With 125 Figures and 25 Tables y Springer Introduction to Airplane Flight Mechanics 1 1.1 Airframe Anatomy 2 1.2 Engine Anatomy 5 1.3 Equations of

More information

Equations of Motion for Micro Air Vehicles

Equations of Motion for Micro Air Vehicles Equations of Motion for Micro Air Vehicles September, 005 Randal W. Beard, Associate Professor Department of Electrical and Computer Engineering Brigham Young University Provo, Utah 84604 USA voice: (80

More information

Gliding, Climbing, and Turning Flight Performance! Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2016

Gliding, Climbing, and Turning Flight Performance! Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2016 Gliding, Climbing, and Turning Flight Performance! Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2016 Learning Objectives Conditions for gliding flight Parameters for maximizing climb angle and rate

More information

Lecture 11 Overview of Flight Dynamics I. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

Lecture 11 Overview of Flight Dynamics I. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Lecture 11 Overview of Flight Dynamics I Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Point Mass Dynamics Dr. Radhakant Padhi Asst. Professor

More information

/ m U) β - r dr/dt=(n β / C) β+ (N r /C) r [8+8] (c) Effective angle of attack. [4+6+6]

/ m U) β - r dr/dt=(n β / C) β+ (N r /C) r [8+8] (c) Effective angle of attack. [4+6+6] Code No: R05322101 Set No. 1 1. (a) Explain the following terms with examples i. Stability ii. Equilibrium. (b) Comment upon the requirements of stability of a i. Military fighter aircraft ii. Commercial

More information

Flight and Orbital Mechanics

Flight and Orbital Mechanics Flight and Orbital Mechanics Lecture slides Challenge the future 1 Flight and Orbital Mechanics Lecture 7 Equations of motion Mark Voskuijl Semester 1-2012 Delft University of Technology Challenge the

More information

Direct spatial motion simulation of aircraft subjected to engine failure

Direct spatial motion simulation of aircraft subjected to engine failure Direct spatial motion simulation of aircraft subjected to engine failure Yassir ABBAS*,1, Mohammed MADBOULI 2, Gamal EL-BAYOUMI 2 *Corresponding author *,1 Aeronautical Engineering Department, Engineering

More information

Applications Linear Control Design Techniques in Aircraft Control I

Applications Linear Control Design Techniques in Aircraft Control I Lecture 29 Applications Linear Control Design Techniques in Aircraft Control I Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Topics Brief Review

More information

Modeling of a Small Unmanned Aerial Vehicle

Modeling of a Small Unmanned Aerial Vehicle Modeling of a Small Unmanned Aerial Vehicle A. Elsayed Ahmed, A. Hafez, A. N. Ouda, H. Eldin Hussein Ahmed, H. Mohamed Abd-Elkader Abstract Unmanned aircraft systems (UAS) are playing increasingly prominent

More information

Dynamics and Control Preliminary Examination Topics

Dynamics and Control Preliminary Examination Topics Dynamics and Control Preliminary Examination Topics 1. Particle and Rigid Body Dynamics Meirovitch, Leonard; Methods of Analytical Dynamics, McGraw-Hill, Inc New York, NY, 1970 Chapters 1-5 2. Atmospheric

More information

Robot Control Basics CS 685

Robot Control Basics CS 685 Robot Control Basics CS 685 Control basics Use some concepts from control theory to understand and learn how to control robots Control Theory general field studies control and understanding of behavior

More information

TTK4190 Guidance and Control Exam Suggested Solution Spring 2011

TTK4190 Guidance and Control Exam Suggested Solution Spring 2011 TTK4190 Guidance and Control Exam Suggested Solution Spring 011 Problem 1 A) The weight and buoyancy of the vehicle can be found as follows: W = mg = 15 9.81 = 16.3 N (1) B = 106 4 ( ) 0.6 3 3 π 9.81 =

More information

First-Order Low-Pass Filter!

First-Order Low-Pass Filter! Filters, Cost Functions, and Controller Structures! Robert Stengel! Optimal Control and Estimation MAE 546! Princeton University, 217!! Dynamic systems as low-pass filters!! Frequency response of dynamic

More information

Performance. 7. Aircraft Performance -Basics

Performance. 7. Aircraft Performance -Basics Performance 7. Aircraft Performance -Basics In general we are interested in finding out certain performance characteristics of a vehicle. These typically include: how fast and how slow an aircraft can

More information

A SIMPLIFIED ANALYSIS OF NONLINEAR LONGITUDINAL DYNAMICS AND CONCEPTUAL CONTROL SYSTEM DESIGN

A SIMPLIFIED ANALYSIS OF NONLINEAR LONGITUDINAL DYNAMICS AND CONCEPTUAL CONTROL SYSTEM DESIGN A SIMPLIFIED ANALYSIS OF NONLINEAR LONGITUDINAL DYNAMICS AND CONCEPTUAL CONTROL SYSTEM DESIGN ROBBIE BUNGE 1. Introduction The longitudinal dynamics of fixed-wing aircraft are a case in which classical

More information

MODELING OF A SMALL UNMANNED AERIAL VEHICLE

MODELING OF A SMALL UNMANNED AERIAL VEHICLE MODELING OF A SMALL UNMANNED AERIAL VEHICLE AHMED ELSAYED AHMED Electrical Engineering Dept., Shoubra Faculty of Engineering, Benha University, Qaliuobia, Egypt (Telephone: +201007124097), Email: eng_medoelbanna

More information

Numerical Methods for the Solution of Differential Equations

Numerical Methods for the Solution of Differential Equations Numerical Methods for the Solution of Differential Equations Markus Grasmair Vienna, winter term 2011 2012 Analytical Solutions of Ordinary Differential Equations 1. Find the general solution of the differential

More information

Gliding, Climbing, and Turning Flight Performance Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018

Gliding, Climbing, and Turning Flight Performance Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018 Gliding, Climbing, and Turning Flight Performance Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018 Learning Objectives Conditions for gliding flight Parameters for maximizing climb angle and rate

More information

STATE VARIABLE (SV) SYSTEMS

STATE VARIABLE (SV) SYSTEMS Copyright F.L. Lewis 999 All rights reserved Updated:Tuesday, August 05, 008 STATE VARIABLE (SV) SYSTEMS A natural description for dynamical systems is the nonlinear state-space or state variable (SV)

More information

18. Linearization: the phugoid equation as example

18. Linearization: the phugoid equation as example 79 18. Linearization: the phugoid equation as example Linearization is one of the most important and widely used mathematical terms in applications to Science and Engineering. In the context of Differential

More information

Investigating the Performance of Adaptive Methods in application to Autopilot of General aviation Aircraft

Investigating the Performance of Adaptive Methods in application to Autopilot of General aviation Aircraft I J C T A, 8(5), 2015, pp 2423-2431 International Science Press Investigating the Performance of Adaptive Methods in application to Autopilot of General aviation Aircraft V Rajesari 1 and L Padma Suresh

More information

Linear-Quadratic-Gaussian Controllers!

Linear-Quadratic-Gaussian Controllers! Linear-Quadratic-Gaussian Controllers! Robert Stengel! Optimal Control and Estimation MAE 546! Princeton University, 2017!! LTI dynamic system!! Certainty Equivalence and the Separation Theorem!! Asymptotic

More information

Design, Analysis and Research Corporation (DARcorporation) ERRATA: Airplane Flight Dynamics and Automatic Flight Controls Part I

Design, Analysis and Research Corporation (DARcorporation) ERRATA: Airplane Flight Dynamics and Automatic Flight Controls Part I Design, Analysis and Research orporation (DARcorporation) ERRATA: Airplane Flight Dynamics and Automatic Flight ontrols Part I opyright 995 by Dr. Jan Roskam Year of Print, 995 (Errata Revised August 7,

More information

Aircraft Flight Dynamics & Vortex Lattice Codes

Aircraft Flight Dynamics & Vortex Lattice Codes Aircraft Flight Dynamics Vortex Lattice Codes AA241X April 14 2014 Stanford University Overview 1. Equations of motion 2. Non-dimensional EOM Aerodynamics 3. Trim Analysis Longitudinal Lateral 4. Linearized

More information

Contribution of Airplane design parameters on Roll Coupling اي داءالبارامترات التصميميه للطائره على ازدواج الحركي

Contribution of Airplane design parameters on Roll Coupling اي داءالبارامترات التصميميه للطائره على ازدواج الحركي International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:13 No:06 7 Contribution of Airplane design parameters on Roll Coupling اي داءالبارامترات التصميميه للطائره على ازدواج الحركي

More information

Pitch Control of Flight System using Dynamic Inversion and PID Controller

Pitch Control of Flight System using Dynamic Inversion and PID Controller Pitch Control of Flight System using Dynamic Inversion and PID Controller Jisha Shaji Dept. of Electrical &Electronics Engineering Mar Baselios College of Engineering & Technology Thiruvananthapuram, India

More information

Flying and Swimming Robots!

Flying and Swimming Robots! Flying and Swimming Robots! Robert Stengel! Robotics and Intelligent Systems MAE 345, Princeton University, 2017! Aircraft! Aquatic robots! Space robots! Quaternions! Simulink/Simscape/ SimMechanics Copyright

More information

FAULT-TOLERANT NONLINEAR FLIGHT CONTROL

FAULT-TOLERANT NONLINEAR FLIGHT CONTROL Proceedings of COBEM Copyright by ABCM st Brazilian Congress of Mechanical Engineering October 4-8,, Natal, RN, Brazil FAULT-TOLERANT NONLINEAR FLIGHT CONTROL Filipe Alves Pereira da Silva, filipe.alves@gmail.com

More information

Department of Aerospace Engineering and Mechanics University of Minnesota Written Preliminary Examination: Control Systems Friday, April 9, 2010

Department of Aerospace Engineering and Mechanics University of Minnesota Written Preliminary Examination: Control Systems Friday, April 9, 2010 Department of Aerospace Engineering and Mechanics University of Minnesota Written Preliminary Examination: Control Systems Friday, April 9, 2010 Problem 1: Control of Short Period Dynamics Consider the

More information

Stability and Control

Stability and Control Stability and Control Introduction An important concept that must be considered when designing an aircraft, missile, or other type of vehicle, is that of stability and control. The study of stability is

More information

Flight Dynamics & Control Equations of Motion of 6 dof Rigid Aircraft-Kinematics

Flight Dynamics & Control Equations of Motion of 6 dof Rigid Aircraft-Kinematics Flight Dynamic & Control Equation of Motion of 6 dof Rigid Aircraft-Kinematic Harry G. Kwatny Department of Mechanical Engineering & Mechanic Drexel Univerity Outline Rotation Matrix Angular Velocity Euler

More information

Mechanics of Flight. Warren F. Phillips. John Wiley & Sons, Inc. Professor Mechanical and Aerospace Engineering Utah State University WILEY

Mechanics of Flight. Warren F. Phillips. John Wiley & Sons, Inc. Professor Mechanical and Aerospace Engineering Utah State University WILEY Mechanics of Flight Warren F. Phillips Professor Mechanical and Aerospace Engineering Utah State University WILEY John Wiley & Sons, Inc. CONTENTS Preface Acknowledgments xi xiii 1. Overview of Aerodynamics

More information

Aircraft Dynamics First order and Second order system

Aircraft Dynamics First order and Second order system Aircraft Dynamics First order and Second order system Prepared by A.Kaviyarasu Assistant Professor Department of Aerospace Engineering Madras Institute Of Technology Chromepet, Chennai Aircraft dynamic

More information

Design, Analysis and Research Corporation (DARcorporation) ERRATA: Airplane Flight Dynamics and Automatic Flight Controls Part I

Design, Analysis and Research Corporation (DARcorporation) ERRATA: Airplane Flight Dynamics and Automatic Flight Controls Part I Design, Analysis and Research Corporation (DARcorporation) ERRATA: Airplane Flight Dynamics and Automatic Flight Controls Part I Copyright 00 by Dr. Jan Roskam Year of Print, 00 (Errata Revised August

More information

Chapter 1: Introduction

Chapter 1: Introduction Chapter 1: Introduction Definition: A differential equation is an equation involving the derivative of a function. If the function depends on a single variable, then only ordinary derivatives appear and

More information

Development of a novel method for autonomous navigation and landing of unmanned aerial vehicles

Development of a novel method for autonomous navigation and landing of unmanned aerial vehicles Rochester Institute of Technology RIT Scholar Works Theses Thesis/Dissertation Collections 8-1-2009 Development of a novel method for autonomous navigation and landing of unmanned aerial vehicles David

More information

Dynamic modeling and control system design for tri-rotor UAV

Dynamic modeling and control system design for tri-rotor UAV Loughborough University Institutional Repository Dynamic modeling and control system design for tri-rotor UAV This item was submitted to Loughborough University's Institutional Repository by the/an author.

More information

Sailing Performance and Maneuverability of a Traditional Ryukyuan Tribute Ship

Sailing Performance and Maneuverability of a Traditional Ryukyuan Tribute Ship sia Navigation Conference 009 ailing Performance and Maneuverability of a Traditional Ryukyuan Tribute hip by Yutaka MUYM (Kanazawa Institute of Technology) Hikaru YGI (Tokai University ) Yutaka TERO (Tokai

More information

AEROSPACE ENGINEERING

AEROSPACE ENGINEERING AEROSPACE ENGINEERING Subject Code: AE Course Structure Sections/Units Topics Section A Engineering Mathematics Topics (Core) 1 Linear Algebra 2 Calculus 3 Differential Equations 1 Fourier Series Topics

More information

Adaptive State Estimation Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 2018

Adaptive State Estimation Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 2018 Adaptive State Estimation Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 218! Nonlinearity of adaptation! Parameter-adaptive filtering! Test for whiteness of the residual!

More information

Experimental Aircraft Parameter Estimation

Experimental Aircraft Parameter Estimation Experimental Aircraft Parameter Estimation AA241X May 14 2014 Stanford University Overview 1. System & Parameter Identification 2. Energy Performance Estimation Propulsion OFF Propulsion ON 3. Stability

More information

MECH 6091 Flight Control Systems Final Course Project

MECH 6091 Flight Control Systems Final Course Project MECH 6091 Flight Control Systems Final Course Project F-16 Autopilot Design Lizeth Buendia Rodrigo Lezama Daniel Delgado December 16, 2011 1 AGENDA Theoretical Background F-16 Model and Linearization Controller

More information

Giovanni Tarantino, Dipartimento di Fisica e Tecnologie Relative, Università di Palermo (Italia)

Giovanni Tarantino, Dipartimento di Fisica e Tecnologie Relative, Università di Palermo (Italia) THE INTERACTIVE PHYSICS FLIGHT SIMULATOR Giovanni Tarantino, Dipartimento di Fisica e Tecnologie Relative, Università di Palermo (Italia) Abstract This paper describes a modelling approach to the dynamics

More information

Aim. Unit abstract. Learning outcomes. QCF level: 6 Credit value: 15

Aim. Unit abstract. Learning outcomes. QCF level: 6 Credit value: 15 Unit T23: Flight Dynamics Unit code: J/504/0132 QCF level: 6 Credit value: 15 Aim The aim of this unit is to develop learners understanding of aircraft flight dynamic principles by considering and analysing

More information

Optimal Control, Guidance and Estimation. Lecture 17. Overview of Flight Dynamics III. Prof. Radhakant Padhi. Prof.

Optimal Control, Guidance and Estimation. Lecture 17. Overview of Flight Dynamics III. Prof. Radhakant Padhi. Prof. Optimal Control, Guidance and Estimation Lecture 17 Overview of Flight Dynamics III Prof. adhakant Padhi Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Six DOF Model Prof. adhakant

More information

CHAPTER 1. Introduction

CHAPTER 1. Introduction CHAPTER 1 Introduction Linear geometric control theory was initiated in the beginning of the 1970 s, see for example, [1, 7]. A good summary of the subject is the book by Wonham [17]. The term geometric

More information

Six Degree of Freedom Dynamical Model of a. Morphing Aircraft

Six Degree of Freedom Dynamical Model of a. Morphing Aircraft AIAA Atmospheric Flight Mechanics Conference 10-13 August 2009, Chicago, Illinois AIAA 2009-5849 Six Degree of Freedom Dynamical Model of a Morphing Aircraft Adam Niksch, John Valasek, Thomas W. Strganac,

More information

What is flight dynamics? AE540: Flight Dynamics and Control I. What is flight control? Is the study of aircraft motion and its characteristics.

What is flight dynamics? AE540: Flight Dynamics and Control I. What is flight control? Is the study of aircraft motion and its characteristics. KING FAHD UNIVERSITY Department of Aerospace Engineering AE540: Flight Dynamics and Control I Instructor Dr. Ayman Hamdy Kassem What is flight dynamics? Is the study of aircraft motion and its characteristics.

More information

Orbital Mechanics! Space System Design, MAE 342, Princeton University! Robert Stengel

Orbital Mechanics! Space System Design, MAE 342, Princeton University! Robert Stengel Orbital Mechanics Space System Design, MAE 342, Princeton University Robert Stengel Conic section orbits Equations of motion Momentum and energy Kepler s Equation Position and velocity in orbit Copyright

More information

1 h 9 e $ s i n t h e o r y, a p p l i c a t i a n

1 h 9 e $ s i n t h e o r y, a p p l i c a t i a n T : 99 9 \ E \ : \ 4 7 8 \ \ \ \ - \ \ T \ \ \ : \ 99 9 T : 99-9 9 E : 4 7 8 / T V 9 \ E \ \ : 4 \ 7 8 / T \ V \ 9 T - w - - V w w - T w w \ T \ \ \ w \ w \ - \ w \ \ w \ \ \ T \ w \ w \ w \ w \ \ w \

More information

Fourth Order RK-Method

Fourth Order RK-Method Fourth Order RK-Method The most commonly used method is Runge-Kutta fourth order method. The fourth order RK-method is y i+1 = y i + 1 6 (k 1 + 2k 2 + 2k 3 + k 4 ), Ordinary Differential Equations (ODE)

More information

EVOLVING DOCUMENT ME 5070 Flight Dynamics

EVOLVING DOCUMENT ME 5070 Flight Dynamics EVOLVING DOCUMENT ME 5070 Flight Dynamics Homework Date of this version: March 20, 2015 Hyperlinks look like this Dates in headings below are the dates of the associated lecture Due January 27, 2015 1

More information

Modelling the dynamics of an unmanned aircraft using a graphical simulation tool

Modelling the dynamics of an unmanned aircraft using a graphical simulation tool Modelling the dynamics of an unmanned aircraft using a graphical simulation tool Sara Jansson Ahmed Department of Aeronautical and Vehicle Engineering KTH - Royal Institute of Technology SE 100 44 Stockholm,

More information

GUST RESPONSE ANALYSIS IN THE FLIGHT FORMATION OF UNMANNED AIR VEHICLES

GUST RESPONSE ANALYSIS IN THE FLIGHT FORMATION OF UNMANNED AIR VEHICLES 28 TH INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES GUST RESPONSE ANALYSIS IN THE FLIGHT FORMATION OF UNMANNED AIR VEHICLES Yoshinobu Inada*, Kyohei Ohta*, and Hideaki Takanobu** *Tokai University,

More information

Stability and Control Analysis in Twin-Boom Vertical Stabilizer Unmanned Aerial Vehicle (UAV)

Stability and Control Analysis in Twin-Boom Vertical Stabilizer Unmanned Aerial Vehicle (UAV) International Journal of Scientific and Research Publications, Volume 4, Issue 2, February 2014 1 Stability and Control Analysis in Twin-Boom Vertical Stabilizer Unmanned Aerial Vehicle UAV Lasantha Kurukularachchi*;

More information

PRINCIPLES OF FLIGHT

PRINCIPLES OF FLIGHT 1 Considering a positive cambered aerofoil, the pitching moment when Cl=0 is: A infinite B positive (nose-up). C negative (nose-down). D equal to zero. 2 The angle between the aeroplane longitudinal axis

More information

Linear Flight Control Techniques for Unmanned Aerial Vehicles

Linear Flight Control Techniques for Unmanned Aerial Vehicles Chapter 1 Linear Flight Control Techniques for Unmanned Aerial Vehicles Jonathan P. How, Emilio Frazzoli, and Girish Chowdhary August 2, 2012 1 Abstract This chapter presents an overview of linear flight

More information

Susceptibility of F/A-18 Flight Control Laws to the Falling Leaf Mode Part I: Linear Analysis

Susceptibility of F/A-18 Flight Control Laws to the Falling Leaf Mode Part I: Linear Analysis Susceptibility of F/A-18 Flight Control Laws to the Falling Leaf Mode Part I: Linear Analysis Abhijit Chakraborty, Peter Seiler and Gary J. Balas Department of Aerospace Engineering & Mechanics University

More information