Flight Dynamics with Spatial Vector Algebra

Author: Onur Tuncer, PhD
Istanbul Technical University, Dept. of Aeronautical Engineering
Maslak, Istanbul, Türkiye
onur.tuncer@itu.edu.tr
Version: 0.8.7 (May 13th, 2026)

Contents

• Introduction and Overview
  • Abstract
  • Nomenclature
  • Abbreviations
• Spatial Vector Algebra Basics
  • Spatial Motion Vectors (Twists)
  • Spatial Force Vectors (Wrenches)
  • Spatial Cross Product for Motion Vectors
  • Spatial Cross Product for Forces
  • Spatial Transforms
  • Spatial Inertia
  • Newton Euler Equations in Spatial Form
  • Benefits of SVA for GNC and EKF Design
  • GNC and EKF Pipeline Diagram
• Coordinate Frames and Basic Notation
  • Coordinate Frames
  • Frame Relationship Figure
  • Notation Summary
• Kinematics, Forces, and Spatial Dynamics
  • Kinematics
  • Forces
  • Spatial Dynamics
  • Mass Dynamics
  • Final ODE System
• Launch-Site Initialization and Frame Chain
  • Frame Chain Overview
  • WGS–84 Geodetic Coordinates to ECEF
  • Local NEU Frame at the Launch Site
  • ECEF to ECI Transformation
  • Position and Velocity in the ECI Frame
  • Rail Direction from Azimuth and Elevation
  • Construction of the Rail Frame
  • Body Orientation on the Launch Rail
  • Final Quaternion Mapping (ECI → Body)
• Software Implementation
• Product Manifolds
  • Product manifolds
• Group Exponential and Logarithm
  • Introduction
  • Number line example: \((\mathbb{R}, +)\)
  • Euclidean vector space: \((\mathbb{R}^n, +)\)
  • A second number-line example: \((\mathbb{R}_{>0}, \times)\)
  • Euclidean “counterpart” of \(\SO(3)\): small-angle approximation
  • Concrete paired examples: Euclidean vs group
  • Summary: the dictionary
• RKMK Numerical Integration on Product Manifolds
  • Lie Group × Euclidean Factors
  • Why product manifolds show up in GNC?
  • Product manifolds and tangent spaces
  • Dynamics on \(G\times \R^n\)
  • Retractions and the exponential map
  • From classical RK to RKMK
  • RKMK on \(G\times \R^n\): coupled stage equations
  • Concrete special case: \(\SO(3)\times \R^n\)
  • Note \(\R^n\) is also a Lie group
  • Explicit example: RK4-RKMK on \(G\times \R^n\)
  • Implicit RKMK on product manifolds (stiff dynamics)
  • Quaternion normalization vs. Lie-group integration
  • Summary for \(G\times\R^n\) implementations
  • Appendix
• Unified Notation
  • Goal
  • State as a product manifold
  • Spatial motion vectors as coordinates of \(\Lie{se}(3)\)
  • Kinematics on \(\SE(3)\) using spatial velocity
  • Featherstone cross operators as \(\ad\) and \(\ad^{\*}\)
  • Rigid-body dynamics in unified form
  • ODE on the product manifold \(\SE(3)\times\R^m\)
  • RKMK integration in the same notation
  • Practical Ramifications
  • Note: \(\R^m\) is also a Lie group
  • Appendix
• Aerodynamic and Wind Policies
  • Atmosphere-Relative Velocity
  • Drag-Only Policy (DragOnlyAeroPolicy)
  • Drag and Rotary-Damping Policy (BrickDampingAeroPolicy)
  • Wind Models
  • Wind-Aware Drag Policy (WindAwareDragPolicy<Wind>)
  • Snapshot Air-Data Corrections
  • Policy Composition Summary
• DAVE-ML Models
  • Overview
  • C++ Architecture
  • F-16 DAVE-ML Data Files
  • Validation and Test Suite
  • References
• Appendices
  • Algorithmic Differentiation
  • Functional Mock-up Units (FMUs)
• Aetherion C++ API Reference
  • Spatial Algebra
  • Lie-Group ODE Solvers
  • Flight Dynamics
  • Rigid Body
  • Environment
  • Coordinate Transforms
  • Serialization
  • Simulation Framework
• Examples
  • Dragless Sphere (NASA TM-2015-218675 Atmospheric Scenario 1)
  • Tumbling Brick, No Damping (NASA TM-2015-218675 Atmospheric Scenario 2)
  • Tumbling Brick, With Aerodynamic Damping (NASA TM-2015-218675 Atmospheric Scenario 3)
  • Sphere with Atmospheric Drag (NASA TM-2015-218675 Atmospheric Scenario 6)
  • Dropped Sphere, Steady Wind (NASA TM-2015-218675 Atmospheric Scenario 7)
  • Dropped Sphere, 2D Wind Shear (NASA TM-2015-218675 Atmospheric Scenario 8)
  • Eastward Cannonball (NASA TM-2015-218675 Atmospheric Scenario 9)
  • Northward Cannonball (NASA TM-2015-218675 Atmospheric Scenario 10)
  • F-16 Steady Straight-and-Level Flight (NASA TM-2015-218675 Atmospheric Scenario 11)
  • F-16 Supersonic Trim Check (NASA TM-2015-218675 Atmospheric Check-Case 12)
  • F-16 Subsonic Altitude Change (NASA TM-2015-218675 Atmospheric Scenario 13.1)
  • F-16 Subsonic Airspeed Change (NASA TM-2015-218675 Atmospheric Scenario 13.2)
  • F-16 Subsonic Heading Change (NASA TM-2015-218675 Atmospheric Scenario 13.3)
  • F-16 Subsonic Lateral Side Step (NASA TM-2015-218675 Atmospheric Scenario 13.4)
  • F-16 North Pole Circumnavigation (NASA TM-2015-218675 Atmospheric Scenario 15)
  • F-16 Equator/Date-Line Circumnavigation (NASA TM-2015-218675 Atmospheric Scenario 16)
  • Two-Stage Rocket to Orbit (NASA TM-2015-218675 Atmospheric Scenario 17)

Index