Orbital Mechanics Modeling

Lesson 1: State Space Representation | Advanced Dynamics

Student:

Date:

The Satellite Problem

Consider a satellite of mass \(m\) orbiting a planet in a simplified 1D radial model. The motion is governed by gravity and a radial thruster providing force \(u(t)\). The governing differential equation for the altitude \(r(t)\) is:

\[ m\ddot{r} + \frac{G M m}{r^2} = u(t) \]

Assume small perturbations \(\delta r\) around a circular orbit of radius \(R_0\), such that we can linearize the system.

1

Vector Formulation

Define your state vector \(\mathbf{x}\) and its time derivative \(\dot{\mathbf{x}}\). Use \(x_1 = r\) and \(x_2 = \dot{r}\).

State Vector \(\mathbf{x}\)

Derivative Vector \(\dot{\mathbf{x}}\)

2

System Dynamics

Express the system in the standard linear form \(\dot{\mathbf{x}} = \mathbf{Ax} + \mathbf{Bu}\). Assume a linearized gravity term around \(R_0\): \(g(r) \approx g(R_0) + \frac{\partial g}{\partial r}\big|_{R_0}(r - R_0)\).

Derive the System Matrix \(\mathbf{A}\) and Input Matrix \(\mathbf{B}\)

3

Phase Space Trajectory

Sketch the expected trajectory in the \(r\)-\(\dot{r}\) phase plane for a scenario where the thruster is off (\(u=0\)) and the satellite starts with an initial altitude higher than its stable equilibrium, with zero initial radial velocity.

Altitude (\(r\))

Radial Velocity (\(\dot{r}\))

Analysis Questions:

1. Does the state vector \(\mathbf{x}\) converge to a point, a limit cycle, or diverge? Justify based on physical intuition.

2. How would the introduction of atmospheric drag (a term proportional to \(-\dot{r}\)) change the "flow" of the vector field?

Advanced Projection

If we measure altitude through a radar system with a delay \(\tau\), such that our output is \(y(t) = r(t-\tau)\), can this be represented in the standard \(\mathbf{y} = \mathbf{Cx} + \mathbf{Du}\) form? Why or why not?

Dynamic Systems and State Space Vectors

State Space Representation

Eigenstructure and Modal Analysis

Vector Fields and Stability

Controllability and Observability Vectors

Lyapunov Functions and Global Stability

State Space Representation Slides

Orbital Mechanics Modeling Worksheet

Eigenstructure Modal Slides

Vibration Modes Modal Analysis Activity

Vector Fields Stability Slides

Jacobian Linearization Case Study

Controllability Observability Slides

Controllability Rank Activity

Lyapunov Functions Stability Slides

Global Stability Proof Project

Similar Sequences

Lab Chemistry Mastery

Olutoyin Makinde

The Digital Launch

brittaink@urbandaleschools.com

The Identity Studio

brittaink@urbandaleschools.com

Innovation Lab: Identity Studio

brittaink@urbandaleschools.com

The Big Payday

Melanie Quick

CS is the Superpower

Elizabeth Whitver

Exposure Mastery

Landry.Stahl

Sticky Matter

Hadlee Richardson

Electron Empire

ssneierson

Square Sage Mastery

mr.sigma

Signal and Story

Tom Ingram

Service Sync

Marlyce Kemp

Efficiency Engine

Matthew.Sprout

Sense Detectives

Michele Magee

Web Interactiva

Norman René Trujillo Zapata

Magnetic Forces Investigation

Olutoyin Makinde

Circuit Mastery

Olutoyin Makinde

Maîtrise ICT Expert

Sagesse Artificielle IA Agri

Bluteau Marie

Plumbing Essentials DWV Sequence

mcapeless

Molecular Frontiers

Tanya Salman

Chemistry Core

Taryn Sigl

Chemical Conversions Mastery

worton