UAV Coordinate Frames and Rigid Body Dynamics

Size: px
Start display at page:

Download "UAV Coordinate Frames and Rigid Body Dynamics"

Transcription

1 Brigham Young University BYU ScholarsArchive All Faculty Publications 24-- UAV oordinate Frames and Rigid Body Dynamics Randal Beard Follow this and additional works at: Part of the Electrical and omputer Engineering ommons Original Publication itation Randal W. Beard, Timothy W. McLain, Small Unmanned Aircraft: Theory and Practice, Princeton University Press, 22, ISBN: BYU ScholarsArchive itation Beard, Randal, "UAV oordinate Frames and Rigid Body Dynamics" (24). All Faculty Publications This Report is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in All Faculty Publications by an authorized administrator of BYU ScholarsArchive. For more information, please contact

2 hapter 2 UAV oordinate Frames and Rigid Body Dynamics 2.. Rotation Matrices This section describes the various coordinate systems that are used to describe the position of orientation of aircraft, and the transformation between these coordinate systems. It is necessary to use several different coordinate systems for the following reasons: Newton s equations of motion are given the coordinate frame attached to the UAV. Some of the on-board sensors take measurements in the body frame, e.g., rate gyros, while some of the sensors take measurements in the inertial frame, e.g., GPS. Aerodynamics forces and torques are exerted in the body frame. Most system requirements, e.g., flight trajectories, are specified in the inertial frame. We begin in two dimensions by considering the two coordinate frames shown in Figure 2.. The vector p can be expressed in both the frame (specified by (i, j )) and in the frame (specified by (i, j )). In the frame we have p = p xi + p yj. Alternatively in the frame we have p = p xi + p yj. Setting these two expressions equal to each other gives p xi + p yj = p xi + p yj. 3

3 4 Figure 2.: Rotation in 2D Taking the dot product of both sides with i and j respectively, and stacking the result into matrix form gives Noting that gives where p = ( p x p y ) ( ) ( i i i j = j i j j i i = cos(θ) i j = sin(θ) j i = sin(θ) j j = cos(θ) p = R p, ( cos(θ) R = sin(θ) p x p y ) sin(θ). cos(θ) One interpretation is that R transforms vectors expressed in frame to an equivalent expression in. where Inverting R gives R = ( cos(θ) sin(θ) ) sin(θ) = cos(θ) p = R p = R p, ( cos 2 (θ) + sin 2 (θ) ). cos(θ) sin(θ) ) sin(θ) = R. T cos(θ)

4 UAV oordinate Frames and Rigid Body Dynamics 5 In general, the inverse of a rotation matrix will be its transpose. An alternate interpretation of R is that the coordinate axis has been rotated into the coordinate axis by an angle of θ. In three dimensions, a rotation of θ about the z-axis is given by cos(θ) sin(θ) R = sin(θ) cos(θ). Alternatively, a rotation of θ about the x-axis is given by and a rotation about the y-axis is given by R = cos(θ) sin(θ), sin(θ) cos(θ) cos(θ) sin(θ) R =. sin(θ) cos(θ) 2..2 UAV oordinate Frames For UAVs there are several coordinate systems that are of interest. The inertial frame I. The inertial coordinate system is an earth fixed coordinate system with origin at the defined home location. The x-axis of I points North, the y-axis points East, and the z axis points toward the center of the earth. The vehicle frame v. The origin of the vehicle frame is at the center of mass of the UAV. However, the axes of v are aligned with the axis of the inertial frame I. In other words, the x-axis points North, the y-axis points East, and the z-axis points toward the center of the earth. The body frame b. The origin of the body frame is also at the center of mass of the UAV. The x-axis points out the nose of the UAV, the y-axis points out the right wing, and the z-axis point out the belly. As the attitude of the UAV moves, the body frame remains fixed with respect to the airframe.

5 6 The wind frame w. To maintain lift, the UAV is required to maintain a positive pitch angle with respect to the velocity vector of the UAV. It is often convenient to align the body fixed reference frame with the velocity vector. This is the purpose of the wind frame. The origin of the wind frame is the center of mass of the UAV. The x-axis is aligned with the projection of the velocity vector on the x z plane of the body axis. The y-axis points out the right wing of the UAV and the z-axis is constructed to complete a right handed orthogonal coordinate systems. Figure 2.2 shows the body and the wind axes. The velocity vectory of the UAV points in the direction of the x w axis which makes a angle α with the x b axis. Figure 2.2: The body and the wind axes of the UAV. The x-axis of the body frame b is aligned with the body of the UAV, whereas the x-axis of the wind frame w is aligned with the velocity vector of the UAV. The angle between x b and x w is the angle of attack α. The orientation of an aircraft is specified by three angles, roll (φ), pitch (θ), and yaw (ψ), which are called the Euler angles. The yaw angle is defined as the rotation ψ about the z v -axis in the vehicle frame v. The resulting coordinate frame is denoted and is shown in Figure 2.3. It can be seen from the figure that the transformation from v to is given by cos(ψ) sin(ψ) R yaw (ψ) = R v = sin(ψ) cos(ψ).

6 UAV oordinate Frames and Rigid Body Dynamics 7 v x v y ψ x ψ y v z z v Figure 2.3: Yaw Angle. The pitch angle is defined as the rotation θ about the y axis in. The resulting coordinate frame is denoted by 2 and is shown in Figure 2.4. It can be seen from the figure that the transformation x 2 θ x y 2 y θ 2 z 2 z Figure 2.4: Pitch Angle. from to 2 is given by cos(θ) sin(θ) R pitch (θ) = R 2 =. sin(θ) cos(θ) The roll angle is defined as the rotation φ about the x 2 axis in 2. The resulting coordinate frame is the body frame, denoted by b, and shown in Figure 2.5. It can be seen from the figure that the

7 8 x 2 x b y 2 φ y b b 2 z 2 φ z b Figure 2.5: Roll Angle. transformation from 2 to is given by R roll (φ) = R 2 b = cos(φ) sin(φ). sin(φ) cos(φ) Therefore, the complete transformation from the vehicle frame v to the body frame b is given by p b = R v b p v, where R v b = R roll (φ)r pitch (θ)r yaw (ψ) cθcψ cθsψ sθ = sφsθcψ cφsψ sφsθsψ + cφcψ sφcθ, cφsθcψ + sφsψ cφsθsψ sφcψ cφcθ where cϕ = cos(ϕ) and sϕ = sin(ϕ) UAV State Variables The state variables of the UAV are the following twelve quantities

8 UAV oordinate Frames and Rigid Body Dynamics 9 x = the inertial position of the UAV along x I in I, y = the inertial position of the UAV along y I in I, h = the altitude of the aircraft measured along z v in v, u = the body frame velocity measured along x b in b, v = the body frame velocity measured along y b in b, w = the body frame velocity measured along z b in b, φ = the roll angle defined with respect to 2, θ = the pitch angle defined with respect to, ψ = the yaw angle defined with respect to v, p = the roll rate measured along x b in b, q = the pitch rate measured along y b in b, r = the yaw rate measured along z b in b. The state variables are shown schematically in Figure 2.6. (x, u) (θ, q) (φ, p) Roll Axis (y, v) Pitch Axis (z, w) (ψ, r) Yaw Axis Figure 2.6: Definition of Axes

9 2..4 UAV Kinematics The state variables x, y, and h are inertial frame quantities, whereas the velocities u, v, and w are body frame quantities. Therefore the relationship between position and velocities is given by x u d y = R b v v dt h w u = Rv b T v w cθcψ sφsθcψ cφsψ cφsθcψ + sφsψ u = cθsψ sφsθsψ + cφcψ cφsθsψ sφcψ v. sθ sφcθ cφcθ w The relationship between absolute angles φ, θ, and ψ, and the angular rates p, q, and r is also complicated by the fact that these quantities are defined in different coordinate frames. The angular rates are defined in the body frame b, whereas the roll angle φ is defined in b as shown in Figure 2.5, the pitch angle θ is defined in 2 as shown in Figure 2.4, and the yaw angle ψ is defined in the vehicle frame as shown in Figure 2.3. Therefore we have p φ q = R b b + R 2 b θ + R b r ψ φ = I 3 + R roll (φ) θ + R roll (φ)r pitch (θ) ψ sθ φ = cφ sφcθ θ. sφ cφcθ ψ Inverting we get φ sin(φ) tan(θ) cos(φ) tan(θ) p θ = cos(φ) sin(φ) q. ψ sin(φ) sec(θ) cos(φ) sec(θ) r

10 UAV oordinate Frames and Rigid Body Dynamics 2..5 Rigid Body Dynamics In general, Newton s equations of motion are given by m dv dt = F I J dω dt = M, I where v cg is the inertial velocity of the center of mass, F is the force exerted on the body in the inertial frame, ω is the angular velocity, M is the inertial frame torque, m is the mass, J x J xy J xz J = J xy J y J yz J xz J yz J z is the inertia tensor expressed in the body frame, and da I is the time derivative of a in the inertial frame. It is straightforward to argue [5] that if the body is rotating at an angular velocity of ω, then dt d dt a = d b I dt a + ω a, where d/dt b is the time derivate in the body frame. Therefore the general six degrees-of-freedom model for a rigid body is given by [2, p. ] m dv dt = ω mv + F b J dω dt = ω Jω + M b where v = (u, v, w) T is the body frame velocity of the center of gravity expressed in the body frame, ω = (p, q, r) T is the angular velocity about the center of gravity expressed in the body frame, F = (F x, F y, F z ) T is the external force placed on the center of gravity, expressed in the body frame, M = (L, M, N) T are the torques about the center of gravity, expressed in the body frame.

11 2 2. Exercises Summarizing the equations of motion are given by ẋ cθcψ sφsθcψ cφsψ cφsθcψ + sφsψ u ẏ = cθsψ sφsθsψ + cφcψ cφsθsψ sφcψ v (2.) ḣ sθ sφcθ cφcθ w v = ω v + m F (2.2) φ sin(φ) tan(θ) cos(φ) tan(θ) p θ = cos(φ) sin(φ) q (2.3) ψ sin(φ) sec(θ) cos(φ) sec(θ) r ω = J ( ω Jω + M) (2.4) 2. Exercises 2. Develop a Simulink simulation of the equations of motion given in Equations (2.) (2.4). Place various forces an torques on the UAV and convince yourself that your simulation gives reasonable results. Unzip the file homework.zip. hange the name of the file uaveom_empty.m to uaveom.m. Find the location in the file where it says UAV dynamics go here. Delete the zeros and add the appropriate equations.

16.333: Lecture #3. Frame Rotations. Euler Angles. Quaternions

16.333: Lecture #3. Frame Rotations. Euler Angles. Quaternions 16.333: Lecture #3 Frame Rotations Euler Angles Quaternions Fall 2004 16.333 3 1 Euler Angles For general applications in 3D, often need to perform 3 separate rotations to relate our inertial frame to

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

Autonomous Fixed Wing Air Vehicles

Autonomous Fixed Wing Air Vehicles Guidance and Control of Autonomous Fixed Wing Air Vehicles Randal W. Beard Timothy W. McLain Brigham Young University ii Contents Preface vii 1 Introduction 1 1.1 System Architecture..................................

More information

Quadrotor Dynamics and Control Rev 0.1

Quadrotor Dynamics and Control Rev 0.1 Brigham Young University BYU ScholarsArchive All Faculty Publications 28-5-5 Quadrotor Dynamics and Control Rev.1 Randal Beard beard@byu.edu Follow this and additional works at: https://scholarsarchive.byu.edu/facpub

More information

Chapter 1. Introduction. 1.1 System Architecture

Chapter 1. Introduction. 1.1 System Architecture Chapter 1 Introduction 1.1 System Architecture The objective of this book is to prepare the reader to do research in the exciting and rapidly developing field of autonomous navigation, guidance, and control

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

Modeling and Sliding Mode Control of a Quadrotor Unmanned Aerial Vehicle

Modeling and Sliding Mode Control of a Quadrotor Unmanned Aerial Vehicle Modeling and Sliding Mode Control of a Quadrotor Unmanned Aerial Vehicle Nour BEN AMMAR, Soufiene BOUALLÈGUE and Joseph HAGGÈGE Research Laboratory in Automatic Control LA.R.A), National Engineering School

More information

c 2016 Jan Willem Vervoorst

c 2016 Jan Willem Vervoorst c 216 Jan Willem Vervoorst A MODULAR SIMULATION ENVIRONMENT FOR THE IMPROVED DYNAMIC SIMULATION OF MULTIROTOR UNMANNED AERIAL VEHICLES BY JAN WILLEM VERVOORST THESIS Submitted in partial fulfillment of

More information

Research Article Shorter Path Design and Control for an Underactuated Satellite

Research Article Shorter Path Design and Control for an Underactuated Satellite Hindawi International Journal of Aerospace Engineering Volume 27 Article ID 8536732 9 pages https://doi.org/.55/27/8536732 Research Article Shorter Path Design and Control for an Underactuated Satellite

More information

NONLINEAR CONTROL OF A HELICOPTER BASED UNMANNED AERIAL VEHICLE MODEL

NONLINEAR CONTROL OF A HELICOPTER BASED UNMANNED AERIAL VEHICLE MODEL NONLINEAR CONTROL OF A HELICOPTER BASED UNMANNED AERIAL VEHICLE MODEL T JOHN KOO, YI MA, AND S SHANKAR SASTRY Abstract In this paper, output tracking control of a helicopter based unmanned aerial vehicle

More information

Classical Mechanics. Luis Anchordoqui

Classical Mechanics. Luis Anchordoqui 1 Rigid Body Motion Inertia Tensor Rotational Kinetic Energy Principal Axes of Rotation Steiner s Theorem Euler s Equations for a Rigid Body Eulerian Angles Review of Fundamental Equations 2 Rigid body

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

Spacecraft and Aircraft Dynamics

Spacecraft and Aircraft Dynamics Spacecraft and Aircraft Dynamics Matthew M. Peet Illinois Institute of Technology Lecture 4: Contributions to Longitudinal Stability Aircraft Dynamics Lecture 4 In this lecture, we will discuss Airfoils:

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

1/30. Rigid Body Rotations. Dave Frank

1/30. Rigid Body Rotations. Dave Frank . 1/3 Rigid Body Rotations Dave Frank A Point Particle and Fundamental Quantities z 2/3 m v ω r y x Angular Velocity v = dr dt = ω r Kinetic Energy K = 1 2 mv2 Momentum p = mv Rigid Bodies We treat a rigid

More information

Aerobatic Maneuvering of Miniature Air Vehicles Using Attitude Trajectories

Aerobatic Maneuvering of Miniature Air Vehicles Using Attitude Trajectories Brigham Young University BYU ScholarsArchive All Faculty Publications 28-8 Aerobatic Maneuvering of Miniature Air Vehicles Using Attitude Trajectories James K. Hall Brigham Young University - Provo, hallatjk@gmail.com

More information

2 Rotation Kinematics

2 Rotation Kinematics X P x P r P 2 Rotation Kinematics Consider a rigid body with a fixed point. Rotation about the fixed point is the only possible motion of the body. We represent the rigid body by a body coordinate frame

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

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

Lecture 37: Principal Axes, Translations, and Eulerian Angles

Lecture 37: Principal Axes, Translations, and Eulerian Angles Lecture 37: Principal Axes, Translations, and Eulerian Angles When Can We Find Principal Axes? We can always write down the cubic equation that one must solve to determine the principal moments But if

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

Phys 7221 Homework # 8

Phys 7221 Homework # 8 Phys 71 Homework # 8 Gabriela González November 15, 6 Derivation 5-6: Torque free symmetric top In a torque free, symmetric top, with I x = I y = I, the angular velocity vector ω in body coordinates with

More information

Problem 1: Ship Path-Following Control System (35%)

Problem 1: Ship Path-Following Control System (35%) Problem 1: Ship Path-Following Control System (35%) Consider the kinematic equations: Figure 1: NTNU s research vessel, R/V Gunnerus, and Nomoto model: T ṙ + r = Kδ (1) with T = 22.0 s and K = 0.1 s 1.

More information

Simulation of Kinematic and Dynamic Models of an Underwater Remotely Operated Vehicle

Simulation of Kinematic and Dynamic Models of an Underwater Remotely Operated Vehicle Simulation of Kinematic and Dynamic Models of an Underwater Remotely Operated Vehicle Viviana Martínez, Daniel Sierra and Rodolfo Villamizar Escuela de Ingenierías Eléctrica, Electrónica y de Telecomunicaciones

More information

Rigid Body Rotation. Speaker: Xiaolei Chen Advisor: Prof. Xiaolin Li. Department of Applied Mathematics and Statistics Stony Brook University (SUNY)

Rigid Body Rotation. Speaker: Xiaolei Chen Advisor: Prof. Xiaolin Li. Department of Applied Mathematics and Statistics Stony Brook University (SUNY) Rigid Body Rotation Speaker: Xiaolei Chen Advisor: Prof. Xiaolin Li Department of Applied Mathematics and Statistics Stony Brook University (SUNY) Content Introduction Angular Velocity Angular Momentum

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

Lecture 38: Equations of Rigid-Body Motion

Lecture 38: Equations of Rigid-Body Motion Lecture 38: Equations of Rigid-Body Motion It s going to be easiest to find the equations of motion for the object in the body frame i.e., the frame where the axes are principal axes In general, we can

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

AA 242B/ ME 242B: Mechanical Vibrations (Spring 2016)

AA 242B/ ME 242B: Mechanical Vibrations (Spring 2016) AA 242B/ ME 242B: Mechanical Vibrations (Spring 2016) Homework #2 Due April 17, 2016 This homework focuses on developing a simplified analytical model of the longitudinal dynamics of an aircraft during

More information

Chapter 9. Nonlinear Design Models. Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 1

Chapter 9. Nonlinear Design Models. Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 1 Chapter 9 Nonlinear Design Models Beard & McLain, Small Unmanned Aircraft, Princeton University Press, 2012, Chapter 9, Slide 1 Architecture Destination, obstacles Waypoints Path Definition Airspeed, Altitude,

More information

The Generalized Missile Equations of Motion

The Generalized Missile Equations of Motion 2 The Generalized Missile Equations of Motion 2.1 Coordinate Systems 2.1.1 Transformation Properties of Vectors In a rectangular system of coordinates, a vector can be completely specified by its components.

More information

Dynamic Modeling of Fixed-Wing UAVs

Dynamic Modeling of Fixed-Wing UAVs Autonomous Systems Laboratory Dynamic Modeling of Fixed-Wing UAVs (Fixed-Wing Unmanned Aerial Vehicles) A. Noth, S. Bouabdallah and R. Siegwart Version.0 1/006 1 Introduction Dynamic modeling is an important

More information

DYNAMICS & CONTROL OF UNDERWATER GLIDERS I: STEADY MOTIONS. N. Mahmoudian, J. Geisbert, & C. Woolsey. acas

DYNAMICS & CONTROL OF UNDERWATER GLIDERS I: STEADY MOTIONS. N. Mahmoudian, J. Geisbert, & C. Woolsey. acas DYNAMICS & CONTROL OF UNDERWATER GLIDERS I: STEADY MOTIONS N. Mahmoudian, J. Geisbert, & C. Woolsey acas Virginia Center for Autonomous Systems Virginia Polytechnic Institute & State University Blacksburg,

More information

Lecture 38: Equations of Rigid-Body Motion

Lecture 38: Equations of Rigid-Body Motion Lecture 38: Equations of Rigid-Body Motion It s going to be easiest to find the equations of motion for the object in the body frame i.e., the frame where the axes are principal axes In general, we can

More information

Developing a Framework for Control of Agile Aircraft Platforms in Autonomous Hover

Developing a Framework for Control of Agile Aircraft Platforms in Autonomous Hover Developing a Framework for Control of Agile Aircraft Platforms in Autonomous Hover Kyle J. Krogh A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics

More information

Incremental control and guidance of hybrid aircraft applied to the Cyclone tailsitter UAV

Incremental control and guidance of hybrid aircraft applied to the Cyclone tailsitter UAV Incremental control and guidance of hybrid aircraft applied to the Cyclone tailsitter UAV E.J.J. Smeur, M. Bronz, G.C.H.E. de Croon arxiv:182.714v1 [cs.ro] 2 Feb 218 ABSTRACT Hybrid unmanned aircraft,

More information

GUIDANCE AND CONTROL OF UNMANNED AIRSHIPS FOR WAYPOINT NAVIGATION IN THE PRESENCE OF WIND GHASSAN M.ATMEH

GUIDANCE AND CONTROL OF UNMANNED AIRSHIPS FOR WAYPOINT NAVIGATION IN THE PRESENCE OF WIND GHASSAN M.ATMEH GUIDANCE AND CONTROL OF UNMANNED AIRSHIPS FOR WAYPOINT NAVIGATION IN THE PRESENCE OF WIND by GHASSAN M.ATMEH Presented to the Faculty of the Graduate School of The University of Texas at Arlington in Partial

More information

Inertial Odometry using AR Drone s IMU and calculating measurement s covariance

Inertial Odometry using AR Drone s IMU and calculating measurement s covariance Inertial Odometry using AR Drone s IMU and calculating measurement s covariance Welcome Lab 6 Dr. Ahmad Kamal Nasir 25.02.2015 Dr. Ahmad Kamal Nasir 1 Today s Objectives Introduction to AR-Drone On-board

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

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

Sensor Fault Diagnosis in Quadrotors Using Nonlinear Adaptive Estimators

Sensor Fault Diagnosis in Quadrotors Using Nonlinear Adaptive Estimators 204 Sensor Fault Diagnosis in Quadrotors Using Nonlinear Adaptive Estimators Remus C Avram, Xiaodong Zhang 2 and acob Campbell 3,2 Wright State University, Dayton, Ohio, 45404, USA avram.3@wright.edu xioadong.zhang@wright.edu

More information

Chapter 1 Lecture 2. Introduction 2. Topics. Chapter-1

Chapter 1 Lecture 2. Introduction 2. Topics. Chapter-1 Chapter 1 Lecture 2 Introduction 2 Topics 1.4 Equilibrium of airplane 1.5 Number of equations of motion for airplane in flight 1.5.1 Degrees of freedom 1.5.2 Degrees of freedom for a rigid airplane 1.6

More information

Rotational Kinematics. Description of attitude kinematics using reference frames, rotation matrices, Euler parameters, Euler angles, and quaternions

Rotational Kinematics. Description of attitude kinematics using reference frames, rotation matrices, Euler parameters, Euler angles, and quaternions Rotational Kinematics Description of attitude kinematics using reference frames, rotation matrices, Euler parameters, Euler angles, and quaternions Recall the fundamental dynamics equations ~ f = d dt

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

Optimal Control, Guidance and Estimation. Lecture 16. Overview of Flight Dynamics II. Prof. Radhakant Padhi. Prof. Radhakant Padhi

Optimal Control, Guidance and Estimation. Lecture 16. Overview of Flight Dynamics II. Prof. Radhakant Padhi. Prof. Radhakant Padhi Optimal Control, Guidance and Estimation Lecture 16 Overview of Flight Dynamics II Prof. Radhakant Padhi Dept. of erospace Engineering Indian Institute of Science - Bangalore Point Mass Dynamics Prof.

More information

6. 3D Kinematics DE2-EA 2.1: M4DE. Dr Connor Myant

6. 3D Kinematics DE2-EA 2.1: M4DE. Dr Connor Myant DE2-EA 2.1: M4DE Dr Connor Myant 6. 3D Kinematics Comments and corrections to connor.myant@imperial.ac.uk Lecture resources may be found on Blackboard and at http://connormyant.com Contents Three-Dimensional

More information

Institutionen för systemteknik

Institutionen för systemteknik Institutionen för systemteknik Department of Electrical Engineering Examensarbete Modelling and control of a hexarotor UAV Examensarbete utfört i Reglerteknik vid Tekniska högskolan vid Linköpings universitet

More information

Unmanned Airship Design with Sliding Ballast: Modeling and Experimental Validation

Unmanned Airship Design with Sliding Ballast: Modeling and Experimental Validation Unmanned Airship Design with Sliding Ballast: Modeling and Experimental Validation Eric Lanteigne Department of Mechanical Engineering University of Ottawa Ontario, Canada, K1N6N5 Email: Eric.Lanteigne@uottawa.ca

More information

Flight Control Simulators for Unmanned Fixed-Wing and VTOL Aircraft

Flight Control Simulators for Unmanned Fixed-Wing and VTOL Aircraft Flight Control Simulators for Unmanned Fixed-Wing and VTOL Aircraft Naoharu Yoshitani 1, Shin-ichi Hashimoto 2, Takehiro Kimura 3, Kazuki Motohashi 2 and Shoh Ueno 4 1 Dept. of Aerospace Engineering, Teikyo

More information

SMALL UNMANNED AIRCRAFT

SMALL UNMANNED AIRCRAFT SMALL UNMANNED AIRCRAFT This page intentionally left blank SMALL UNMANNED AIRCRAFT Theory and Practice RANDAL W BEARD TIMOTHY W McLAIN PRINCETON UNIVERSITY PRESS Princeton and Oxford Copyright 2012 by

More information

Lecture Notes - Modeling of Mechanical Systems

Lecture Notes - Modeling of Mechanical Systems Thomas Bak Lecture Notes - Modeling of Mechanical Systems February 19, 2002 Aalborg University Department of Control Engineering Fredrik Bajers Vej 7C DK-9220 Aalborg Denmark 2 Table of Contents Table

More information

Chapter 2 Coordinate Systems and Transformations

Chapter 2 Coordinate Systems and Transformations Chapter 2 Coordinate Systems and Transformations 2.1 Coordinate Systems This chapter describes the coordinate systems used in depicting the position and orientation (pose) of the aerial robot and its manipulator

More information

Physics 312, Winter 2007, Practice Final

Physics 312, Winter 2007, Practice Final Physics 312, Winter 2007, Practice Final Time: Two hours Answer one of Question 1 or Question 2 plus one of Question 3 or Question 4 plus one of Question 5 or Question 6. Each question carries equal weight.

More information

Fault Diagnosis and Fault-Tolerant Control of Quadrotor UAVs

Fault Diagnosis and Fault-Tolerant Control of Quadrotor UAVs Wright State University CORE Scholar Browse all Theses and Dissertations Theses and Dissertations 2016 Fault Diagnosis and Fault-Tolerant Control of Quadrotor UAVs Remus C. Avram Wright State University

More information

Manipulator Dynamics 2. Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

Manipulator Dynamics 2. Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA Manipulator Dynamics 2 Forward Dynamics Problem Given: Joint torques and links geometry, mass, inertia, friction Compute: Angular acceleration of the links (solve differential equations) Solution Dynamic

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

MAE 142 Homework #5 Due Friday, March 13, 2009

MAE 142 Homework #5 Due Friday, March 13, 2009 MAE 142 Homework #5 Due Friday, March 13, 2009 Please read through the entire homework set before beginning. Also, please label clearly your answers and summarize your findings as concisely as possible.

More information

MCE/EEC 647/747: Robot Dynamics and Control. Lecture 2: Rigid Motions and Homogeneous Transformations

MCE/EEC 647/747: Robot Dynamics and Control. Lecture 2: Rigid Motions and Homogeneous Transformations MCE/EEC 647/747: Robot Dynamics and Control Lecture 2: Rigid Motions and Homogeneous Transformations Reading: SHV Chapter 2 Mechanical Engineering Hanz Richter, PhD MCE503 p.1/22 Representing Points, Vectors

More information

Artificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik. Robot Dynamics. Dr.-Ing. John Nassour J.

Artificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik. Robot Dynamics. Dr.-Ing. John Nassour J. Artificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik Robot Dynamics Dr.-Ing. John Nassour 25.1.218 J.Nassour 1 Introduction Dynamics concerns the motion of bodies Includes Kinematics

More information

arxiv: v1 [math.ds] 18 Nov 2008

arxiv: v1 [math.ds] 18 Nov 2008 arxiv:0811.2889v1 [math.ds] 18 Nov 2008 Abstract Quaternions And Dynamics Basile Graf basile.graf@epfl.ch February, 2007 We give a simple and self contained introduction to quaternions and their practical

More information

HYBRID SYSTEM BASED DESIGN FOR THE COORDINATION AND CONTROL OF MULTIPLE AUTONOMOUS VEHICLES. Charles A. Clifton. Thesis

HYBRID SYSTEM BASED DESIGN FOR THE COORDINATION AND CONTROL OF MULTIPLE AUTONOMOUS VEHICLES. Charles A. Clifton. Thesis HYBRID SYSTEM BASED DESIGN FOR THE COORDINATION AND CONTROL OF MULTIPLE AUTONOMOUS VEHICLES By Charles A. Clifton Thesis Submitted to the Faculty of the Graduate School of Vanderbilt University in partial

More information

Modeling and Control of mini UAV

Modeling and Control of mini UAV Modeling and Control of mini UAV Gerardo Ramon Flores Colunga, A. Guerrero, Juan Antonio Escareño, Rogelio Lozano To cite this version: Gerardo Ramon Flores Colunga, A. Guerrero, Juan Antonio Escareño,

More information

Aerial Rendezvous Between an Unmanned Air Vehicle and an Orbiting Target Vehicle

Aerial Rendezvous Between an Unmanned Air Vehicle and an Orbiting Target Vehicle Brigham Young University BYU ScholarsArchive All Theses and Dissertations 2011-10-18 Aerial Rendezvous Between an Unmanned Air Vehicle and an Orbiting Target Vehicle Mark Andrew Owen Brigham Young University

More information

(3.1) a 2nd-order vector differential equation, as the two 1st-order vector differential equations (3.3)

(3.1) a 2nd-order vector differential equation, as the two 1st-order vector differential equations (3.3) Chapter 3 Kinematics As noted in the Introduction, the study of dynamics can be decomposed into the study of kinematics and kinetics. For the translational motion of a particle of mass m, this decomposition

More information

In this section of notes, we look at the calculation of forces and torques for a manipulator in two settings:

In this section of notes, we look at the calculation of forces and torques for a manipulator in two settings: Introduction Up to this point we have considered only the kinematics of a manipulator. That is, only the specification of motion without regard to the forces and torques required to cause motion In this

More information

AN INTEGRATOR BACKSTEPPING CONTROLLER FOR A STANDARD HELICOPTER YITAO LIU THESIS

AN INTEGRATOR BACKSTEPPING CONTROLLER FOR A STANDARD HELICOPTER YITAO LIU THESIS AN INEGRAOR BACKSEPPING CONROLLER FOR A SANDARD HELICOPER BY YIAO LIU HESIS Submitted in partial fulfillment of the requirements for the degree of Master of Science in Electrical and Computer Engineering

More information

Minimal representations of orientation

Minimal representations of orientation Robotics 1 Minimal representations of orientation (Euler and roll-pitch-yaw angles) Homogeneous transformations Prof. lessandro De Luca Robotics 1 1 Minimal representations rotation matrices: 9 elements

More information

Quadrotor Control in the Presence of Unknown Mass Properties. Rikky Ricardo Petrus Rufino Duivenvoorden

Quadrotor Control in the Presence of Unknown Mass Properties. Rikky Ricardo Petrus Rufino Duivenvoorden Quadrotor Control in the Presence of Unknown Mass Properties by Rikky Ricardo Petrus Rufino Duivenvoorden A thesis submitted in conformity with the requirements for the degree of Master of Applied Science

More information

Rotational Motion. Chapter 4. P. J. Grandinetti. Sep. 1, Chem P. J. Grandinetti (Chem. 4300) Rotational Motion Sep.

Rotational Motion. Chapter 4. P. J. Grandinetti. Sep. 1, Chem P. J. Grandinetti (Chem. 4300) Rotational Motion Sep. Rotational Motion Chapter 4 P. J. Grandinetti Chem. 4300 Sep. 1, 2017 P. J. Grandinetti (Chem. 4300) Rotational Motion Sep. 1, 2017 1 / 76 Angular Momentum The angular momentum of a particle with respect

More information

8 Velocity Kinematics

8 Velocity Kinematics 8 Velocity Kinematics Velocity analysis of a robot is divided into forward and inverse velocity kinematics. Having the time rate of joint variables and determination of the Cartesian velocity of end-effector

More information

Small lightweight aircraft navigation in the presence of wind

Small lightweight aircraft navigation in the presence of wind Small lightweight aircraft navigation in the presence of wind Cornel-Alexandru Brezoescu To cite this version: Cornel-Alexandru Brezoescu. Small lightweight aircraft navigation in the presence of wind.

More information

UNCLASSIFIED. Control of Spinning Symmetric Airframes. Curtis P. Mracek Max Stafford and Mike Unger Raytheon Systems Company Tucson, AZ 85734

UNCLASSIFIED. Control of Spinning Symmetric Airframes. Curtis P. Mracek Max Stafford and Mike Unger Raytheon Systems Company Tucson, AZ 85734 Control of Spinning Symmetric Airframes Curtis P. Mracek Max Stafford and Mike Unger Raytheon Systems Company Tucson, AZ 85734 Several missiles and projectiles have periods during their flight where the

More information

State Estimation for Autopilot Control of Small Unmanned Aerial Vehicles in Windy Conditions

State Estimation for Autopilot Control of Small Unmanned Aerial Vehicles in Windy Conditions University of Colorado, Boulder CU Scholar Aerospace Engineering Sciences Graduate Theses & Dissertations Aerospace Engineering Sciences Summer 7-23-2014 State Estimation for Autopilot Control of Small

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

Chapter 6: Momentum Analysis

Chapter 6: Momentum Analysis 6-1 Introduction 6-2Newton s Law and Conservation of Momentum 6-3 Choosing a Control Volume 6-4 Forces Acting on a Control Volume 6-5Linear Momentum Equation 6-6 Angular Momentum 6-7 The Second Law of

More information

NONLINEAR SIMULATION OF A MICRO AIR VEHICLE

NONLINEAR SIMULATION OF A MICRO AIR VEHICLE NONLINEAR SIMULATION OF A MICRO AIR VEHICLE By JASON JOSEPH JACKOWSKI A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

More information

Quadrotor Modeling and Control

Quadrotor Modeling and Control 16-311 Introduction to Robotics Guest Lecture on Aerial Robotics Quadrotor Modeling and Control Nathan Michael February 05, 2014 Lecture Outline Modeling: Dynamic model from first principles Propeller

More information

Modelling of Opposed Lateral and Longitudinal Tilting Dual-Fan Unmanned Aerial Vehicle

Modelling of Opposed Lateral and Longitudinal Tilting Dual-Fan Unmanned Aerial Vehicle Modelling of Opposed Lateral and Longitudinal Tilting Dual-Fan Unmanned Aerial Vehicle N. Amiri A. Ramirez-Serrano R. Davies Electrical Engineering Department, University of Calgary, Canada (e-mail: namiri@ucalgary.ca).

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

Generalization of the Euler Angles

Generalization of the Euler Angles The Journal of the Astronautical Sciences, Vol. 51, No. 2, AprilJune 2003, pp. 123123 Generalization of the Euler Angles Malcolm D. Shuster 1 and F. Landis Markley 2 Abstract It is shown that the Euler

More information

Quadrotors Flight Formation Control Using a Leader-Follower Approach*

Quadrotors Flight Formation Control Using a Leader-Follower Approach* 23 European Conference (ECC) July 7-9, 23, Zürich, Switzerland. Quadrotors Flight Formation Using a Leader-Follower Approach* D. A. Mercado, R. Castro and R. Lozano 2 Abstract In this paper it is presented

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

Two dimensional rate gyro bias estimation for precise pitch and roll attitude determination utilizing a dual arc accelerometer array

Two dimensional rate gyro bias estimation for precise pitch and roll attitude determination utilizing a dual arc accelerometer array Rochester Institute of Technology RIT Scholar Works Theses Thesis/Dissertation Collections -- Two dimensional rate gyro bias estimation for precise pitch and roll attitude determination utilizing a dual

More information

Towed-body Trajectory Tracking in Aerial Recovery of Micro Air Vehicle in the presence of Wind

Towed-body Trajectory Tracking in Aerial Recovery of Micro Air Vehicle in the presence of Wind American Control Conference on O'Farrell Street, San Francisco, CA, USA June 9 - July, Towed-body Tracking in Aerial Recovery of Micro Air Vehicle in the presence of Wind Liang Sun and Randal W. Beard

More information

Circumnavigation with a group of quadrotor helicopters

Circumnavigation with a group of quadrotor helicopters Circumnavigation with a group of quadrotor helicopters JOHANNA ORIHUELA SWARTLING Master s Degree Project Stockholm, Sweden March 2014 XR-EE-RT 2014:007 Abstract The primary goal of this thesis has been

More information

Mechanics Physics 151

Mechanics Physics 151 Mechanics Phsics 151 Lecture 8 Rigid Bod Motion (Chapter 4) What We Did Last Time! Discussed scattering problem! Foundation for all experimental phsics! Defined and calculated cross sections! Differential

More information

Control Allocation for Fault Tolerant Control of a VTOL Octorotor

Control Allocation for Fault Tolerant Control of a VTOL Octorotor UKACC International Conference on Control 212 Cardiff, UK, 3-5 September 212 Control Allocation for Fault Tolerant Control of a VTOL Octorotor Aryeh Marks Department of Aerospace Engineering Cranfield

More information

Kinematics. Basilio Bona. Semester 1, DAUIN Politecnico di Torino. B. Bona (DAUIN) Kinematics Semester 1, / 15

Kinematics. Basilio Bona. Semester 1, DAUIN Politecnico di Torino. B. Bona (DAUIN) Kinematics Semester 1, / 15 Kinematics Basilio Bona DAUIN Politecnico di Torino Semester 1, 2016-17 B. Bona (DAUIN) Kinematics Semester 1, 2016-17 1 / 15 Introduction The kinematic quantities used to represent a body frame are: position

More information

Some history. F p. 1/??

Some history. F p. 1/?? Some history F 12 10 18 p. 1/?? F 12 10 18 p. 1/?? Some history 1600: Galileo Galilei 1564 1642 cf. section 7.0 Johannes Kepler 1571 1630 cf. section 3.7 1700: Isaac Newton 1643 1727 cf. section 1.1 1750

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

CS491/691: Introduction to Aerial Robotics

CS491/691: Introduction to Aerial Robotics CS491/691: Introduction to Aerial Robotics Topic: Midterm Preparation Dr. Kostas Alexis (CSE) Areas of Focus Coordinate system transformations (CST) MAV Dynamics (MAVD) Navigation Sensors (NS) State Estimation

More information

AE Stability and Control of Aerospace Vehicles

AE Stability and Control of Aerospace Vehicles AE 430 - Stability and Control of Aerospace Vehicles Aircraft Equations of Motion Dynamic Stability Degree of dynamic stability: time it takes the motion to damp to half or to double the amplitude of its

More information

Dynamic Feedback Control for a Quadrotor Unmanned Aerial Vehicle

Dynamic Feedback Control for a Quadrotor Unmanned Aerial Vehicle Dynamic Feedback Control for a Quadrotor Unmanned Aerial Vehicle N. K. M Sirdi, Abdellah Mokhtari LSIS Laboratoire de Sciences de l Information et des Systèmes, CNRS UMR 6168. Dom. Univ. St- Jérôme, Av.

More information

2.004: MODELING, DYNAMICS, & CONTROL II Spring Term 2003

2.004: MODELING, DYNAMICS, & CONTROL II Spring Term 2003 .4: MODELING, DYNAMICS, & CONTROL II Spring Term 3 PLEASE ALSO NOTE THAT ALL PRELAB EXERCISE ARE DUE AT THE START (WITHIN 1 MINUTES) OF THE LAB SESSION, NO LATE WORK IS ACCEPTED. Pre-Lab Exercise for Experiment

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

Mathematical Modelling and Dynamics Analysis of Flat Multirotor Configurations

Mathematical Modelling and Dynamics Analysis of Flat Multirotor Configurations Mathematical Modelling and Dynamics Analysis of Flat Multirotor Configurations DENIS KOTARSKI, Department of Mechanical Engineering, Karlovac University of Applied Sciences, J.J. Strossmayera 9, Karlovac,

More information

The Jacobian. Jesse van den Kieboom

The Jacobian. Jesse van den Kieboom The Jacobian Jesse van den Kieboom jesse.vandenkieboom@epfl.ch 1 Introduction 1 1 Introduction The Jacobian is an important concept in robotics. Although the general concept of the Jacobian in robotics

More information

Chapter 6: Momentum Analysis of Flow Systems

Chapter 6: Momentum Analysis of Flow Systems Chapter 6: Momentum Analysis of Flow Systems Introduction Fluid flow problems can be analyzed using one of three basic approaches: differential, experimental, and integral (or control volume). In Chap.

More information

Motion in Three Dimensions

Motion in Three Dimensions Motion in Three Dimensions We ve learned about the relationship between position, velocity and acceleration in one dimension Now we need to extend those ideas to the three-dimensional world In the 1-D

More information

Classical Mechanics III (8.09) Fall 2014 Assignment 3

Classical Mechanics III (8.09) Fall 2014 Assignment 3 Classical Mechanics III (8.09) Fall 2014 Assignment 3 Massachusetts Institute of Technology Physics Department Due September 29, 2014 September 22, 2014 6:00pm Announcements This week we continue our discussion

More information