science

Legendre pseudospectral method

Legendre pseudospectral method is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Legendre pseudospectral method rather than just read about it. In short: The Legendre pseudospectral method for optimal control problems is based on Legendre polynomials. It is part of the larger theory of pseudospectral optimal control, a term coined by Ross.

Key takeaways

Legendre pseudospectral method belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Legendre pseudospectral method to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Legendre pseudospectral method from memory before moving on to harder problems.

Reference excerpt

The Legendre pseudospectral method for optimal control problems is based on Legendre polynomials. It is part of the larger theory of pseudospectral optimal control, a term coined by Ross. A basic version of the Legendre pseudospectral was originally proposed by Elnagar and his coworkers in 1995. Since then, Ross, Fahroo and their coworkers have extended, generalized and applied the method for a large range of problems. An application that has received wide publicity is the use of their method for generating real time trajectories for the International Space Station.

Fundamentals There are three basic types of Legendre pseudospectral methods:

One based on Gauss-Lobatto points First proposed by Elnagar et al and subsequently extended by Fahroo and Ross to incorporate the covector mapping theorem. Forms the basis for solving general nonlinear finite-horizon optimal control problems. Incorporated in several software products DIDO, OTIS, PSOPT Archived 2016-10-24 at the Wayback Machine One based on Gauss-Radau points First proposed by Fahroo and Ross and subsequently extended (by Fahroo and Ross) to incorporate a covector mapping theorem. Forms the basis for solving general nonlinear infinite-horizon optimal control problems. Forms the basis for solving general nonlinear finite-horizon problems with one free endpoint. One based on Gauss points First proposed by Reddien Forms the basis for solving finite-horizon problems with free endpoints Incorporated in several software products GPOPS, PROPT

Software The first software to implement the Legendre pseudospectral method was DIDO in 2001. Subsequently, the method was incorporated in the NASA code OTIS. Years later, many other software products emerged at an increasing pace, such as PSOPT, PROPT and GPOPS.

Flight implementations The Legendre pseudospectral method (based on Gauss-Lobatto points) has been implemented in flight by NASA on several spacecraft through the use of the software, DIDO. The first flight implementation was on November 5, 2006, when NASA used DIDO to maneuver the International Space Station to perform the Zero Propellant Maneuver. The Zero Propellant Maneuver was discovered by Nazareth Bedrossian using DIDO. Watch a video of this historic maneuver.

See also DIDO Chebyshev pseudospectral method Ross–Fahroo pseudospectral methods Ross–Fahroo lemma Covector mapping principle

References

Worked examples

Example 1 — a first encounter with Legendre pseudospectral method

Start with the simplest possible case. Write down what Legendre pseudospectral method claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Legendre pseudospectral method before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Legendre pseudospectral method ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Legendre pseudospectral method

In research: Legendre pseudospectral method appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Legendre pseudospectral method in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Legendre pseudospectral method is common in secondary-school and first-year university syllabi. It links to neighbouring topics Control theory, Numerical analysis, Optimal control, so understanding it makes those chapters shorter.
In everyday life: Look for Legendre pseudospectral method outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Legendre pseudospectral method in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Legendre pseudospectral method means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Legendre pseudospectral method out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Legendre pseudospectral method in simple terms?

The Legendre pseudospectral method for optimal control problems is based on Legendre polynomials. It is part of the larger theory of pseudospectral optimal control, a term coined by Ross.

Why does Legendre pseudospectral method matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Legendre pseudospectral method?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Legendre pseudospectral method.