The movement of bodies in space (like spacecraft, satellites, and space stations) must be predicted and controlled with precision in order to ensure safety and efficacy. Kinematics is a field that develops descriptions and predictions of the motion of these bodies in 3D space. This course in Kinematics covers four major topic areas: an introduction to particle kinematics, a deep dive into rigid body kinematics in two parts (starting with classic descriptions of motion using the directional cosine matrix and Euler angles, and concluding with a review of modern descriptors like quaternions and Classical and Modified Rodrigues parameters).

The course ends with a look at static attitude determination, using modern algorithms to predict and execute relative orientations of bodies in space.

After this course, you will be able to…

* Differentiate a vector as seen by another rotating frame and derive frame dependent velocity and acceleration vectors

* Apply the Transport Theorem to solve kinematic particle problems and translate between various sets of attitude descriptions

* Add and subtract relative attitude descriptions and integrate those descriptions numerically to predict orientations over time

* Derive the fundamental attitude coordinate properties of rigid bodies and determine attitude from a series of heading measurements

**Who is this class for:** This class is for working engineering professionals looking to add to their skill sets, graduate students in engineering looking to fill gaps in their knowledge base, and enterprising engineering undergraduates looking to expand their horizons.

### Syllabus

**WEEK 1**

Introduction to Kinematics

This module covers particle kinematics. A special emphasis is placed on a frame-independent vectorial notation. The position velocity and acceleration of particles are derived using rotating frames utilizing the transport theorem.

Graded: Concept Check 1 – Particle Kinematics and Vector Frames

Graded: Concept Check 2 – Angular Velocities

Graded: Concept Check 3 – Vector Differentiation and the Transport Theorem

**WEEK 2**

Rigid Body Kinematics I

This module provides an overview of orientation descriptions of rigid bodies. The 3D heading is here described using either the direction cosine matrix (DCM) or the Euler angle sets. For each set the fundamental attitude addition and subtracts are discussed, as well as the differential kinematic equation which relates coordinate rates to the body angular velocity vector.

Graded: Concept Check 1 – Rigid Body Kinematics

Graded: Concept Check 2 – DCM Definitions

Graded: Concept Check 3 – DCM Properties

Graded: Concept Check 4 – DCM Addition and Subtraction

Graded: Concept Check 5 – DCM Differential Kinematic Equations (ODE)

Graded: Concept Check 6 – Euler Angles Definitions

Graded: Concept Check 7 – Euler Angle and DCM Relation

Graded: Concept Check 8 – Euler Angle Addition and Subtraction

Graded: Concept Check 9 – Euler Angle Differential Kinematic Equations

Graded: Concept Check 10 – Symmetric Euler Angle Addition

**WEEK 3**

Rigid Body Kinematics II

This module covers modern attitude coordinate sets including Euler Parameters (quaternions), principal rotation parameters, Classical Rodrigues parameters, modified Rodrigues parameters, as well as stereographic orientation parameters. For each set the concepts of attitude addition and subtraction is developed, as well as mappings to other coordinate sets.

Graded: Concept Check 1 – Principal Rotation Definitions

Graded: Concept Check 2 – Principal Rotation Parameter relation to DCM

Graded: Concept Check 3 – Principal Rotation Addition

Graded: Concept Check 4 – Euler Parameter Definitions

Graded: Concept Check 5, 6 – Euler Parameter Relationship to DCM

Graded: Concept Check 7 – Euler Parameter Addition

Graded: Concept Check 8 – EP Differential Kinematic Equations

Graded: Concept Check 9 – CRP Definitions

Graded: Concept Check 10 – CRPs Stereographic Projection

Graded: Concept Check 11, 12 – CRP Addition

Graded: Concept Check 13 – CRP Differential Kinematic Equations

Graded: Concept Check 15 – MRPs Definitions

Graded: Concept Check 16 – MRP Stereographic Projection

Graded: Concept Check 17 – MRP Shadow Set

Graded: Concept Check 18 – MRP to DCM Relation

Graded: Concept Check 19 – MRP Addition and Subtraction

Graded: Concept Check 20 – MRP Differential Kinematic Equation

**WEEK 4**

Static Attitude Determination

This module covers how to take an instantaneous set of observations (sun heading, magnetic field direction, star direction, etc.) and compute a corresponding 3D attitude measure. The attitude determination methods covered include the TRIAD method, Devenport’s q-method, QUEST as well as OLAE. The benefits and computation challenges are reviewed for each algorithm.

Graded: Concept Check 1 – Attitude Determination

Graded: Concept Check 2 – TRIAD Method

Graded: Concept Check 3, 4 – Devenport’s q-Method

Graded: Concept Check 5 – QUEST Method

Graded: Concept Check 6 – OLAE Method

Graded: Kinematics Final Assignment

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