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Aerospace Toolbox User's Guide R2021b

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How to Contact MathWorks Latest news: www.mathworks.com Sales and services: www.mathworks.com/sales_and_services User community: www.mathworks.com/matlabcentral Technical support: www.mathworks.com/support/contact_us Phone: 508-647-7000 The MathWorks, Inc. 1 Apple Hill Drive Natick, MA 01760-2098 Aerospace Toolbox User's Guide © COPYRIGHT 2006–2021 by The MathWorks, Inc. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or reproduced in any form without prior written consent from The MathWorks, Inc. FEDERAL ACQUISITION: This provision applies to all acquisitions of the Program and Documentation by, for, or through the federal government of the United States. By accepting delivery of the Program or Documentation, the government hereby agrees that this software or documentation qualifies as commercial computer software or commercial computer software documentation as such terms are used or defined in FAR 12.212, DFARS Part 227.72, and DFARS 252.227-7014. Accordingly, the terms and conditions of this Agreement and only those rights specified in this Agreement, shall pertain to and govern the use, modification, reproduction, release, performance, display, and disclosure of the Program and Documentation by the federal government (or other entity acquiring for or through the federal government) and shall supersede any conflicting contractual terms or conditions. If this License fails to meet the government's needs or is inconsistent in any respect with federal procurement law, the government agrees to return the Program and Documentation, unused, to The MathWorks, Inc. Trademarks MATLAB and Simulink are registered trademarks of The MathWorks, Inc. See www.mathworks.com/trademarks for a list of additional trademarks. Other product or brand names may be trademarks or registered trademarks of their respective holders. Patents MathWorks products are protected by one or more U.S. patents. Please see www.mathworks.com/patents for more information.

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Revision History September 2006 Online only New for Version 1.0 (Release 2006b) March 2007 Online only Revised for Version 1.1 (Release 2007a) September 2007 First printing Revised for Version 2.0 (Release 2007b) March 2008 Online only Revised for Version 2.1 (Release 2008a) October 2008 Online only Revised for Version 2.2 (Release 2008b) March 2009 Online only Revised for Version 2.3 (Release 2009a) September 2009 Online only Revised for Version 2.4 (Release 2009b) March 2010 Online only Revised for Version 2.5 (Release 2010a) September 2010 Online only Revised for Version 2.6 (Release 2010b) April 2011 Online only Revised for Version 2.7 (Release 2011a) September 2011 Online only Revised for Version 2.8 (Release 2011b) March 2012 Online only Revised for Version 2.9 (Release 2012a) September 2012 Online only Revised for Version 2.10 (Release 2012b) March 2013 Online only Revised for Version 2.11 (Release 2013a) September 2013 Online only Revised for Version 2.12 (Release 2013b) March 2014 Online only Revised for Version 2.13 (Release 2014a) October 2014 Online only Revised for Version 2.14 (Release 2014b) March 2015 Online only Revised for Version 2.15 (Release 2015a) September 2015 Online only Revised for Version 2.16 (Release 2015b) March 2016 Online only Revised for Version 2.17 (Release 2016a) September 2016 Online only Revised for Version 2.18 (Release 2016b) March 2017 Online only Revised for Version 2.19 (Release 2017a) September 2017 Online only Revised for Version 2.20 (Release 2017b) March 2018 Online only Revised for Version 2.21 (Release 2018a) September 2018 Online only Revised for Version 3.0 (Release 2018b) March 2019 Online only Revised for Version 3.1 (Release 2019a) September 2019 Online only Revised for Version 3.2 (Release 2019b) March 2020 Online only Revised for Version 3.3 (Release 2020a) September 2020 Online only Revised for Version 3.4 (Release 2020b) March 2021 Online only Revised for Version 4.0 (Release 2021a) September 2021 Online only Revised for Version 4.1 (Release 2021b)

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Getting Started 1 Aerospace Toolbox Product Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2 Aerospace Toolbox and Aerospace Blockset . . . . . . . . . . . . . . . . . . . . . . . . 1-3 Using Aerospace Toolbox 2 Fundamental Coordinate System Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 2-2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2 Motion with Respect to Other Planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2 Coordinate Systems for Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4 Body Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4 Wind Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5 Coordinate Systems for Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6 Geocentric and Geodetic Latitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6 NED Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6 ECI Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7 ECEF Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-8 Coordinate Systems for Display . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-9 Aerospace Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-10 Digital DATCOM Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-11 Digital DATCOM Data Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-11 USAF Digital DATCOM File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-11 Data from DATCOM Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-11 Imported DATCOM Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12 Missing DATCOM Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13 Aerodynamic Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-15 Aerospace Toolbox Animation Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-18 Aero.Animation Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-19 Running the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-19 v Contents

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Simulated and Actual Flight Data Using Aero.Animation Objects . . . . . 2-20 Creating and Configuring an Animation Object . . . . . . . . . . . . . . . . . . . . 2-20 Loading Recorded Data for Flight Trajectories . . . . . . . . . . . . . . . . . . . . 2-20 Displaying Body Geometries in a Figure Window . . . . . . . . . . . . . . . . . . 2-21 Recording Animation Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-21 Playing Back Flight Trajectories Using the Animation Object . . . . . . . . . 2-21 Viewing Recorded Animation Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-22 Manipulating the Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-22 Moving and Repositioning Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-23 Creating a Transparency in the First Body . . . . . . . . . . . . . . . . . . . . . . . 2-24 Changing the Color of the Second Body . . . . . . . . . . . . . . . . . . . . . . . . . 2-25 Turning Off the Landing Gear of the Second Body . . . . . . . . . . . . . . . . . . 2-26 Aero.VirtualRealityAnimation Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-27 Running the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-27 Example of Visualize Aircraft Takeoff and Chase Helicopter with the Virtual Reality Animation Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-28 Visualize Aircraft Takeoff via Virtual Reality Animation Object . . . . . . . . 2-28 Aero.FlightGearAnimation Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-40 About the FlightGear Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-40 Configuring Your Computer for FlightGear . . . . . . . . . . . . . . . . . . . . . . . 2-40 Install and Start FlightGear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-41 Installing Additional FlightGear Scenery . . . . . . . . . . . . . . . . . . . . . . . . . 2-42 Flight Simulator Interface Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-42 Running the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-44 Flight Trajectory Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-45 Loading Recorded Flight Trajectory Data . . . . . . . . . . . . . . . . . . . . . . . . 2-45 Creating a Time Series Object from Trajectory Data . . . . . . . . . . . . . . . . 2-45 Creating a FlightGearAnimation Object . . . . . . . . . . . . . . . . . . . . . . . . . 2-45 Modifying the FlightGearAnimation Object Properties . . . . . . . . . . . . . . 2-46 Generating the Run Script . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-46 Starting the FlightGear Flight Simulator . . . . . . . . . . . . . . . . . . . . . . . . . 2-47 Playing Back the Flight Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-47 Create and Configure Flight Instrument Component and an Animation Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-49 Load and Visualize Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-49 Create Flight Instrument Components . . . . . . . . . . . . . . . . . . . . . . . . . . 2-50 Flight Instrument Components in App Designer . . . . . . . . . . . . . . . . . . . 2-52 Start App Designer and Create a New App . . . . . . . . . . . . . . . . . . . . . . . 2-52 Drag Aerospace Components into the App . . . . . . . . . . . . . . . . . . . . . . . 2-52 Add Code to Load and Visualize Data for the App . . . . . . . . . . . . . . . . . . 2-53 Add Code to Trigger a Display of the Animation Object . . . . . . . . . . . . . . 2-54 Add Code to Close the Animation Window with UIfigure Window . . . . . . 2-55 Save and Run the App . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-56 Work with Fixed-Wing Aircraft Using Functions . . . . . . . . . . . . . . . . . . . . 2-58 Suggested Workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-58 Static Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-59 Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-59 vi Contents

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Analyze Fixed-Wing Aircraft with Objects . . . . . . . . . . . . . . . . . . . . . . . . . 2-60 Suggested Workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-60 Static Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-61 Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-61 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-61 Satellite Scenario Key Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-63 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-63 Orbital Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-68 Two Line Element (TLE) Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-70 Satellite Scenario Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-72 Flight Control Analysis Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-74 Plot Short-Period Undamped Natural Frequency Results . . . . . . . . . . . . . 2-74 Add-On for Ephemeris and Geoid Data Support 3 Add Ephemeris and Geoid Data for Aerospace Products . . . . . . . . . . . . . . 3-2 Functions 4 Aerospace Toolbox Examples 5 Importing from USAF Digital DATCOM Files . . . . . . . . . . . . . . . . . . . . . . . 5-2 Create a Flight Animation from Trajectory Data . . . . . . . . . . . . . . . . . . . 5-17 Estimating G Forces for Flight Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-20 Calculating Best Glide Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-25 Overlaying Simulated and Actual Flight Data . . . . . . . . . . . . . . . . . . . . . . 5-30 Comparing Zonal Harmonic Gravity Model to Other Gravity Models . . . 5-39 Visualize Aircraft Takeoff via Virtual Reality Animation Object . . . . . . . 5-46 Calculating Compressor Power Required in a Supersonic Wind Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-58 vii

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Analyzing Flow with Friction Through an Insulated Constant Area Duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-64 Determining Heat Transfer and Mass Flow Rate in a Ramjet Combustion Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-69 Solving for the Exit Flow of a Supersonic Nozzle . . . . . . . . . . . . . . . . . . . 5-75 Visualizing World Magnetic Model Contours for 2020 Epoch . . . . . . . . . 5-85 Visualizing Geoid Height for Earth Geopotential Model 1996 . . . . . . . . 5-93 Marine Navigation Using Planetary Ephemerides . . . . . . . . . . . . . . . . . . 5-98 Estimate Sun Analemma Using Planetary Ephemerides and ECI to AER Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-107 Display Flight Trajectory Data Using Flight Instruments and Flight Animation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-110 Aerospace Flight Instruments in App Designer . . . . . . . . . . . . . . . . . . . 5-113 Perform Controls and Static Stability Analysis with Linearized Fixed-Wing Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-114 Customize Fixed-Wing Aircraft with Additional Aircraft States . . . . . . 5-121 Determine Nonlinear Dynamics and Static Stability of Fixed-Wing Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-129 Modeling Satellite Constellations Using Ephemeris Data . . . . . . . . . . . 5-138 Satellite Constellation Access to a Ground Station . . . . . . . . . . . . . . . . 5-148 Comparison of Orbit Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-162 Detect and Track LEO Satellite Constellation with Ground Radars . . . 5-170 Get Started With Fixed-Wing Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-181 Customize Fixed-Wing Aircraft with the Object Interface . . . . . . . . . . . 5-197 AC3D Files and Thumbnails A AC3D Files and Thumbnails Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2 viii Contents

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Getting Started • “Aerospace Toolbox Product Description” on page 1-2 • “Aerospace Toolbox and Aerospace Blockset” on page 1-3 1

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Aerospace Toolbox Product Description Analyze and visualize aerospace vehicle motion using reference standards and models Aerospace Toolbox provides standards-based tools and functions for analyzing the motion, mission, and environment of aerospace vehicles. It includes aerospace math operations, coordinate system and spatial transformations, and validated environment models for interpreting flight data. The toolbox also includes 2D and 3D visualization tools and standard cockpit instruments for observing vehicle motion. For flight vehicles, you can import Data Compendium (Datcom) files directly into MATLAB® to represent vehicle aerodynamics. The aerodynamics can be combined with reference parameters to define your aircraft configuration and dynamics for control design and flying qualities analysis. Aerospace Toolbox lets you design and analyze scenarios consisting of satellites and ground stations. You can propagate satellite trajectories from orbital elements or two-line element sets, load in satellite and constellation ephemerides, perform mission analysis tasks such as line-of-sight access, and visualize the scenario as a ground track or globe. 1 Getting Started 1-2

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Aerospace Toolbox and Aerospace Blockset The Aerospace product family includes the Aerospace Toolbox and Aerospace Blockset products. The toolbox provides static data analysis capabilities, while the blockset provides an environment for dynamic modeling and vehicle component modeling and simulation. The Aerospace Blockset software uses part of the functionality of the toolbox as an engine. Use these products together to model aerospace systems in the MATLAB and Simulink® environments. Aerospace Toolbox and Aerospace Blockset 1-3

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Using Aerospace Toolbox • “Fundamental Coordinate System Concepts” on page 2-2 • “Coordinate Systems for Modeling” on page 2-4 • “Coordinate Systems for Navigation” on page 2-6 • “Coordinate Systems for Display” on page 2-9 • “Aerospace Units” on page 2-10 • “Digital DATCOM Data” on page 2-11 • “Aerospace Toolbox Animation Objects” on page 2-18 • “Aero.Animation Objects” on page 2-19 • “Simulated and Actual Flight Data Using Aero.Animation Objects” on page 2-20 • “Aero.VirtualRealityAnimation Objects” on page 2-27 • “Example of Visualize Aircraft Takeoff and Chase Helicopter with the Virtual Reality Animation Object” on page 2-28 • “Aero.FlightGearAnimation Objects” on page 2-40 • “Flight Trajectory Data” on page 2-45 • “Create and Configure Flight Instrument Component and an Animation Object” on page 2-49 • “Flight Instrument Components in App Designer” on page 2-52 • “Work with Fixed-Wing Aircraft Using Functions” on page 2-58 • “Analyze Fixed-Wing Aircraft with Objects” on page 2-60 • “Satellite Scenario Key Concepts” on page 2-63 • “Satellite Scenario Overview” on page 2-72 • “Flight Control Analysis Tools” on page 2-74 2

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Fundamental Coordinate System Concepts Coordinate systems allow you to track an aircraft or spacecraft position and orientation in space. The Aerospace Toolbox coordinate systems are based on these underlying concepts from geodesy, astronomy, and physics. For more information on geographic information, see “Mapping Toolbox”. Definitions The Aerospace Toolbox software uses right-handed (RH) Cartesian coordinate systems. The rightmost rule establishes the x-y-z sequence of coordinate axes. An inertial frame is a nonaccelerating motion reference frame. Loosely speaking, acceleration is defined with respect to the distant cosmos. In an inertial frame, Newton's second law (force = mass X acceleration) holds. Strictly defined, an inertial frame is a member of the set of all frames not accelerating relative to one another. A noninertial frame is any frame accelerating relative to an inertial frame. Its acceleration, in general, includes both translational and rotational components, resulting in pseudoforces (pseudogravity, as well as Coriolis and centrifugal forces). The toolbox models the Earth shape (the geoid) as an oblate spheroid, a special type of ellipsoid with two longer axes equal (defining the equatorial plane) and a third, slightly shorter (geopolar) axis of symmetry. The equator is the intersection of the equatorial plane and the Earth surface. The geographic poles are the intersection of the Earth surface and the geopolar axis. In general, the Earth geopolar and rotation axes are not identical. Latitudes parallel the equator. Longitudes parallel the geopolar axis. The zero longitude or prime meridian passes through Greenwich, England. Approximations The Aerospace Toolbox software makes three standard approximations in defining coordinate systems relative to the Earth. • The Earth surface or geoid is an oblate spheroid, defined by its longer equatorial and shorter geopolar axes. In reality, the Earth is slightly deformed with respect to the standard geoid. • The Earth rotation axis and equatorial plane are perpendicular, so that the rotation and geopolar axes are identical. In reality, these axes are slightly misaligned, and the equatorial plane wobbles as the Earth rotates. This effect is negligible in most applications. • The only noninertial effect in Earth-fixed coordinates is due to the Earth rotation about its axis. This is a rotating, geocentric system. The toolbox ignores the Earth motion around the Sun, the Sun motion in the Galaxy, and the Galaxy's motion through cosmos. In most applications, only the Earth rotation matters. This approximation must be changed for spacecraft sent into deep space, that is, outside the Earth-Moon system, and a heliocentric system is preferred. Motion with Respect to Other Planets The Aerospace Toolbox software uses the standard WGS-84 geoid to model the Earth. You can change the equatorial axis length, the flattening, and the rotation rate. 2 Using Aerospace Toolbox 2-2

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You can represent the motion of spacecraft with respect to any celestial body that is well approximated by an oblate spheroid by changing the spheroid size, flattening, and rotation rate. If the celestial body is rotating westward (retrogradely), make the rotation rate negative. References [1] Recommended Practice for Atmospheric and Space Flight Vehicle Coordinate Systems, R-004-1992, ANSI/AIAA, February 1992. [2] Rogers, R. M., Applied Mathematics in Integrated Navigation Systems, AIAA, Reston, Virginia, 2000. [3] Stevens, B. L., and F. L. Lewis, Aircraft Control, and Simulation, 2nd ed., Wiley-Interscience, New York, 2003. [4] Thomson, W. T., Introduction to Space Dynamics, John Wiley & Sons, New York, 1961/Dover Publications, Mineola, New York, 1986. See Also Related Examples • “Coordinate Systems for Modeling” on page 2-4 • “Coordinate Systems for Navigation” on page 2-6 • “Coordinate Systems for Display” on page 2-9 External Websites • World Geodetic System 1984 (WGS 84) Fundamental Coordinate System Concepts 2-3

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Coordinate Systems for Modeling Modeling aircraft and spacecraft are simplest if you use a coordinate system fixed in the body itself. In the case of aircraft, the forward direction is modified by the presence of wind, and the craft's motion through the air is not the same as its motion relative to the ground. Body Coordinates The noninertial body coordinate system is fixed in both origin and orientation to the moving craft. The craft is assumed to be rigid. The orientation of the body coordinate axes is fixed in the shape of body. • The x-axis points through the nose of the craft. • The y-axis points to the right of the x-axis (facing in the pilot's direction of view), perpendicular to the x-axis. • The z-axis points down through the bottom of the craft, perpendicular to the x-y plane and satisfying the RH rule. Translational Degrees of Freedom Translations are defined by moving along these axes by distances x, y, and z from the origin. Rotational Degrees of Freedom Rotations are defined by the Euler angles P, Q, R or Φ, Θ, Ψ. They are • P or Φ: Roll about the x-axis • Q or Θ: Pitch about the y-axis • R or Ψ: Yaw about the z-axis Unless otherwise specified, by default the software uses ZYX rotation order for Euler angles. 2 Using Aerospace Toolbox 2-4

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Wind Coordinates The noninertial wind coordinate system has its origin fixed in the rigid aircraft. The coordinate system orientation is defined relative to the craft's velocity V. The orientation of the wind coordinate axes is fixed by the velocity V. • The x-axis points in the direction of V. • The y-axis points to the right of the x-axis (facing in the direction of V), perpendicular to the x- axis. • The z-axis points perpendicular to the x-y plane in whatever way needed to satisfy the RH rule with respect to the x- and y-axes. Translational Degrees of Freedom Translations are defined by moving along these axes by distances x, y, and z from the origin. Rotational Degrees of Freedom Rotations are defined by the Euler angles Φ, γ, χ. They are • Φ: Bank angle about the x-axis • γ: Flight path about the y-axis • χ: Heading angle about the z-axis Unless otherwise specified, by default the software uses ZYX rotation order for Euler angles. See Also Related Examples • “Fundamental Coordinate System Concepts” on page 2-2 • “Coordinate Systems for Navigation” on page 2-6 • “Coordinate Systems for Display” on page 2-9 Coordinate Systems for Modeling 2-5

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Coordinate Systems for Navigation Modeling aerospace trajectories requires positioning and orienting the aircraft or spacecraft with respect to the rotating Earth. Navigation coordinates are defined with respect to the center and surface of the Earth. Geocentric and Geodetic Latitudes The geocentric latitude λ on the Earth surface is defined by the angle subtended by the radius vector from the Earth center to the surface point with the equatorial plane. The geodetic latitude μ on the Earth surface is defined by the angle subtended by the surface normal vector n and the equatorial plane. NED Coordinates The north-east-down (NED) system is a noninertial system with its origin fixed at the aircraft or spacecraft's center of gravity. Its axes are oriented along the geodetic directions defined by the Earth surface. • The x-axis points north parallel to the geoid surface, in the polar direction. • The y-axis points east parallel to the geoid surface, along a latitude curve. • The z-axis points downward, toward the Earth surface, antiparallel to the surface's outward normal n. Flying at a constant altitude means flying at a constant z above the Earth's surface. 2 Using Aerospace Toolbox 2-6

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ECI Coordinates The Earth-centered inertial (ECI) system is non-rotating. For most applications, assume this frame to be inertial, although the equinox and equatorial plane move very slightly over time. The ECI system is considered to be truly inertial for high-precision orbit calculations when the equator and equinox are defined at a particular epoch (e.g. J2000). Aerospace functions and blocks that use a particular realization of the ECI coordinate system provide that information in their documentation. The ECI system origin is fixed at the center of the Earth (see figure). • The x-axis points towards the vernal equinox (First Point of Aries ♈). • The y-axis points 90 degrees to the east of the x-axis in the equatorial plane. • The z-axis points northward along the Earth rotation axis. Coordinate Systems for Navigation 2-7

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Earth-Centered Coordinates ECEF Coordinates The Earth-center, Earth-fixed (ECEF) system is a noninertial system that rotates with the Earth. Its origin is fixed at the center of the Earth. • The z-axis points northward along the Earth's rotation axis. • The x-axis points outward along the intersection of the Earth's equatorial plane and prime meridian. • The y-axis points into the eastward quadrant, perpendicular to the x-z plane so as to satisfy the RH rule. See Also Related Examples • “Fundamental Coordinate System Concepts” on page 2-2 • “Coordinate Systems for Modeling” on page 2-4 • “Coordinate Systems for Display” on page 2-9 2 Using Aerospace Toolbox 2-8

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MATLAB Aerospace Toolbox Users Guide (MathWorks, Inc.) (z-library.sk, 1lib.sk, z-lib.sk)

MATLAB Aerospace Toolbox Users Guide (MathWorks, Inc.) (z-library.sk, 1lib.sk, z-lib.sk)

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