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Fundamentals of Orbit Determination

University of Colorado Boulder via Coursera

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Provider

Coursera
Pricing

Paid Course
Languages

English
Certificate

Certificate Available
Effort

5 weeks, 8 hours a week
Sessions

Self-Paced
Level

Intermediate
Subtitles

English

Found in

Orbital Mechanics Courses

Overview

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Orbit determination is the process of estimating a spacecraft's trajectory from a set of imperfect tracking observations. It is a foundational component of every spaceflight mission from GPS to Earth-observation satellites to planetary missions like OSIRIS-REx and Mars Reconnaissance Orbiter. This course builds the mathematical machinery on which every orbit-determination process sits: dynamical and measurement models and their partial derivatives, and the representation and propagation of uncertainty. The estimation algorithms themselves (batch least squares, Kalman filtering, smoothing) are covered in the follow-on courses of this specialization; here we develop the foundation they all share. Across five modules you will build force models for two-body and perturbed orbits; derive analytical partial derivatives for both the dynamics and the measurements; develop the equations and Jacobians for range, range-rate, angles, and other common tracking observables; map uncertainty through the dynamics using linear covariance mapping and state noise compensation; augment the state with estimated parameters and dynamic model compensation accelerations; and synthesize everything into an end-to-end Monte Carlo orbit-determination project.

Syllabus

Dynamics Models

This module develops the differential equations that govern spacecraft motion and the partial derivatives that drive the state transition matrix. Starting with the two-body Keplerian problem and then adding the perturbations that real missions must account for, such as the spherical-harmonic gravity field, atmospheric drag, solar radiation pressure, and third-body gravity. With the dynamics in hand, the linearization process about a reference trajectory is covered, including the state transition matrix and its augmented-integration computation. This relies on deriving the partial derivatives of the dynamics, the process of which is reviewed and applied to the orbital dynamics of interest in this course.

Measurement Models

This module develops the mathematical models for the observations that orbit determination filters consume. The ideal range and range-rate equations are dervied, and the real-world complications are discussed, including: light-time correction, coordinate-frame transformations (ECI ↔ ECEF, J2000/ICRF), and the various time scales (TAI, UT1, UTC, TT, TDB) needed to keep the geometry self-consistent. The measurement Jacobians for range and range-rate are derived and when the partials with respect to station coordinates, rotation parameters, or measurement biases must be included is discussed. It closes with a survey of other observable types used in practice such as: optical navigation (center-finding and landmark-based), GPS, inter-satellite ranging, radar, LIDAR/altimetry with crossovers and DDOR.

Uncertainty Propagation

In this module the orbit determination problem becomes statistical. The Gaussian probability density function is introduced as the standard representation of state uncertainty in this course, and the method for transforming a Gaussian under linear maps is developed. The spacecraft state covariance is propagated through the dynamics using the state transition matrix. State noise compensation (SNC) is introduced, which adds a process-noise term to the covariance equation to absorb un-modeled accelerations. SNC tuning is discussed for use in real application.

Parameters and DMC

This module extends the filter beyond just position and velocity. The parameters that real missions need to estimate alongside the state, such as drag and SRP scale factors, station coordinates, range and range-rate biases, and gravitational parameters are discussed. The augmented state vector is developed along with the corresponding partials. This framework is used to incorporate dynamic model compensation (DMC) as a first-order Gauss-Markov stochastic acceleration that captures whatever un-modeled forces show up, without committing to a specific physical mechanism. The corresponding covariance contribution is derived and compared against SNC as two complementary approaches to imperfect dynamics models.

Assessment

The culminating assessment synthesizes the course material into an end-to-end orbit-determination exercise. You will build a Monte Carlo analysis that varies force-model fidelity, measurement geometry, process-noise levels, and SNC settings on a representative tracking scenario, then assess each modeling choice in terms of state perturbations, residual statistics, covariance consistency, and computational cost. This assessment sets up the path forward that connects this course's modeling foundation to the estimation algorithms covered in the follow-on courses of the specialization.