Linearized Equations of Motion!
|
|
- Megan Farmer
- 6 years ago
- Views:
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 Robert Stengel Robotics and Intelligent Systems MAE 345, Princeton University, 217 Multi-dimensional trajectories Numerical integration Linear and nonlinear systems Linearization
More informationBoth 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 informationAlternative 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 informationLecture 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 informationTranslational 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 informationChapter 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 informationLinearized 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 informationControl 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 informationAircraft 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 informationStochastic 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 informationFlight 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 informationDynamics 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 informationLinearized 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 informationNonlinear 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 informationOptimal 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 informationMinimization 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 informationTopic # 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 informationDynamic 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 informationAdaptive 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 informationIntroduction 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 informationTime 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 informationMinimization 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 informationRobot 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 informationFLIGHT 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 informationAB-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 informationOptimal 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 informationTime-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 informationAircraft 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 informationAtmospheric 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 informationLecture #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 informationMinimization 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 informationTime-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 informationFrequency 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 informationApril 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 informationStochastic 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 informationLecture 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 informationLecture 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 informationMech 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 informationManifesto 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 informationAircraft 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 informationChapter 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 informationRobot 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 informationFundamentals 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 informationEquations 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 informationGliding, 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 informationLecture 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]
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 informationFlight 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 informationDirect 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 informationApplications 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 informationModeling 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 informationDynamics 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 informationRobot 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 informationTTK4190 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 informationFirst-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 informationPerformance. 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 informationA 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 informationMODELING 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 informationNumerical 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 informationGliding, 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 informationSTATE 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 information18. 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 informationInvestigating 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 informationLinear-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 informationDesign, 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 informationAircraft 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 informationContribution 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 informationPitch 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 informationFlying 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 informationFAULT-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 informationDepartment 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 informationStability 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 informationFlight 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 informationMechanics 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 informationAircraft 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 informationDesign, 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 informationChapter 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 informationDevelopment 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 informationDynamic 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 informationSailing 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 informationAEROSPACE 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 informationAdaptive 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 informationExperimental 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 informationMECH 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 informationGiovanni 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 informationAim. 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 informationOptimal 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 informationCHAPTER 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 informationSix 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 informationWhat 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 informationOrbital 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 information1 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 informationFourth 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 informationEVOLVING 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 informationModelling 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 informationGUST 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 informationStability 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 informationPRINCIPLES 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 informationLinear 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 informationSusceptibility 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