This course is part of the Advanced Spacecraft Dynamics and Control Specialization

Hanspeter Schaub

Offered By

Advanced Spacecraft Dynamics and Control SpecializationUniversity of Colorado Boulder

About this Course

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This course is part 2 of the specialization Advanced Spacecraft Dynamics and Control. It assumes you have a strong foundation in spacecraft dynamics and control, including particle dynamics, rotating frame, rigid body kinematics and kinetics.  The focus of the course is to understand key analytical mechanics methodologies to develop equations of motion in an algebraically efficient manner.  The course starts by first developing D’Alembert’s principle and how the associated virtual work and virtual displacement concepts allows us to ignore non-working force terms.  Unconstrained systems and holonomic constrains are investigated.  Next Kane's equations and the virtual power form of D'Alembert's equations are briefly reviewed for particles.
Next Lagrange’s equations are developed which still assume a finite set of generalized coordinates, but can be applied to multiple rigid bodies as well.  Lagrange multipliers are employed to apply Pfaffian constraints.  

Finally, Hamilton’s extended principle is developed to allow us to consider a dynamical system with flexible components.  Here there are an infinite number of degrees of freedom.  The course focuses on how to develop spacecraft related partial differential equations, but does not study numerically solving them.  The course ends comparing the presented assumed mode methods to classical final element solutions.

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Course 2 of 3 in the

Advanced Spacecraft Dynamics and Control Specialization

Advanced Level

Graduate course on Spacecraft Dynamics and Control, background in vector calculus, linear algebra, basic differential equations.

Approx. 32 hours to complete

English

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What you will learn

Use virtual work methods to develop equations of motion of mechanical systems.
Understand how to use Lagrange multipliers to study constrained dynamical systems.
Be able to derive the equations of motion of a spacecraft with flexible sub-components.

Skills you will gain

Lagrangian Dynamics
holonomic constraints
D'Alembert's Principle
Hamilton's Extended Principle
multi-body dynamics

Flexible deadlines

Reset deadlines in accordance to your schedule.

Shareable Certificate

Earn a Certificate upon completion

100% online

Start instantly and learn at your own schedule.

Course 2 of 3 in the

Advanced Spacecraft Dynamics and Control Specialization

Advanced Level

Graduate course on Spacecraft Dynamics and Control, background in vector calculus, linear algebra, basic differential equations.

Approx. 32 hours to complete

English

Could your company benefit from training employees on in-demand skills?

Try Coursera for Business

Instructor

Hanspeter Schaub

Glenn L. Murphy Chair in Engineering, Professor

Department of Aerospace Engineering Sciences

28,773 Learners

7 Courses

Offered by

University of Colorado Boulder

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Syllabus - What you will learn from this course

Week1

Week 1

11 hours to complete

Generalized Methods of Analytical Mechanics

11 hours to complete

24 videos (Total 288 min)

24 videos

Welcome to the Course!3mMotivation for Analytical Mechanics21mIntroduction1mVirtual Displacements16mTaking First Order Variations16mVirtual Work11mExample: Circularly Orbiting Particle7mExample: Planar Spinning Body5mClassical Form of D'Alembert's Principle19mExample: Falling Rod Revisited15mExample: Generalized Forces on Particle14mVirtual Power Form of D'Alembert's Equations16mExample: Cart-Pendulum System22mExample: Planar Orbital Motion13mTorques Acting on a Rigid Body8mExample: Generalized Force on 2-Link System12mHolonomic Constraints8mExample: Spherical Pendulum6mExample: Constrained 3D Particle Motion16mMultiple Constraints7mPfaffian Constraints10mGeneral Constrained Optimization7mExample: Extremum on Circles4mDiscussion on Constrainted Optimization18m

10 practice exercises

Virtual Displacements15mTaking First Order Variations30mVirtual Work1hClassical Form of D'Alembert's Principle1h 30mVirtual Power Form of D'Alembert's Equations1hTorques Acting on a Rigid Body30mHolonomic Constraints1hMultiple Constraints10mPfaffian Constraints15mConstrained Optimization30m

Week2

Week 2

11 hours to complete

Energy Based Equations of Motion

11 hours to complete

19 videos (Total 185 min)

19 videos

Derivation of Basic Lagrange's Equations12mReview: Lagrangian Dynamics7mExample: Particle in a Plane10mLagrange's Equations with Conservative Forces7mExample: Cart-Pendulum revisited with Lagrange's equationsrev10mConstrained Lagrange's Equations13mExample: Particle in Rotating Tube10mExample: Rolling Wheel26mExample: Falling Ring5mCompact Matrix Form of Lagrange's Equations10mCyclic Coordinates4mExample: Falling Planar Particle3mExample: Planar Particle on a Spring3mRouthian Reduction11mExample: Falling Planar Particle With Routhian4mMotivation for Boltzmann Hamel Equations13mQuasi Velocity Coordinates2mBoltzmann Hamel Equation Development11mExample: Rigid Body Motion in Free Space12m

6 practice exercises

Basic Lagrange's Equations30mLagrange's Equations with Conservative Forces1h 30mConstrained Lagrange's Equations2h 30mCompact Matrix Form of Lagrange's Equations1hCyclic Coordinates1hBoltzmann Hamel Equations1h

Week3

Week 3

10 hours to complete

Variational Methods in Analytical Dynamics

10 hours to complete

21 videos (Total 256 min)

21 videos

Motivation for Variational Methods5mVariational Calculus20mHamilton's Principle Function3mHamilton's Variational Principles13mExample: Spring-Mass-Damper System9mExtremun of Hamilton's Principle Function7mHamilton's Law of Varying Action3mExample: Particle In Gravity Field10mExample: Linear Oscillator System6mReview of Hamilton's Extended Principle7mNon-Uniform Axially Elastic Rod29mExample: Elastic Rod with External Force37mMotivation for Hybrid Systems3mHybrid Coordinate Definitions4mHybrid Lagrangian Formulation10mExample: Axial Rod and Spring-Mass System12mExample: Hub with Euler-Bernoulli Beam9mMotivation for Reduction to a Finite Set of Coordinates1mAssumed Modes Method16mExample27mInput Shaped Attitude Control16m

7 practice exercises

Variational Calculus30mHamilton's Principles1h 30mHamilton's Law of Varying Action45mNon-Uniform Axially Elastic Rod1h 30mHybrid Dynamical Systems1hFinite Dimensional Modeling15mInput Shaped Attitude Control15m

About the Advanced Spacecraft Dynamics and Control Specialization

This Specialization on advanced spacecraft dynamcis and control is intended for experienced spacecraft dynamics and GNC engineers and researchers. It is assumed the viewer has completed the prior spacecraft dynamics specialization already. Through 3 courses we cover the topics of momentum-based attitude dynamics and control, we derive analytical methods to model complex spacecraft systems, and finally conclude with a captstone project course. After this course you will be prepared to model the dynamics of spacecraft systems with time varying components (reacton wheels, CMS, deployable panels, etc.).

Frequently Asked Questions

When will I have access to the lectures and assignments?
Access to lectures and assignments depends on your type of enrollment. If you take a course in audit mode, you will be able to see most course materials for free. To access graded assignments and to earn a Certificate, you will need to purchase the Certificate experience, during or after your audit. If you don't see the audit option:
The course may not offer an audit option. You can try a Free Trial instead, or apply for Financial Aid.
The course may offer 'Full Course, No Certificate' instead. This option lets you see all course materials, submit required assessments, and get a final grade. This also means that you will not be able to purchase a Certificate experience.
What will I get if I subscribe to this Specialization?
When you enroll in the course, you get access to all of the courses in the Specialization, and you earn a certificate when you complete the work. Your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. If you only want to read and view the course content, you can audit the course for free.
Is financial aid available?
Yes. In select learning programs, you can apply for financial aid or a scholarship if you can’t afford the enrollment fee. If fin aid or scholarship is available for your learning program selection, you’ll find a link to apply on the description page.
I have not taken the earlier classes on Spacecraft Dynamics and Control, can I jump right into this class?
This course does stand on its on. It is still recommended that you have a strong foundation in particle dynamics, rotating frames, rigid body kinematics, etc.

More questions? Visit the Learner Help Center.

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Analytical Mechanics for Spacecraft Dynamics

About this Course

What you will learn

Skills you will gain