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An Algorithm to Design Prescribed Length Codes for Single-Tracked - - PowerPoint PPT Presentation

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Problem Solution Comments An Algorithm to Design Prescribed Length Codes for Single-Tracked Shaft Encoders IEEE International Conference on Mechatronics 2009 B. Balle , E. Ventura, J. M. Fuertes Universitat Politcnica de Catalunya April 17,

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Problem Solution Comments

An Algorithm to Design Prescribed Length Codes for Single-Tracked Shaft Encoders

IEEE International Conference on Mechatronics 2009

  • B. Balle, E. Ventura, J. M. Fuertes

Universitat Politècnica de Catalunya

April 17, 2009

  • B. Balle, E. Ventura, J. M. Fuertes

Prescribed Length Shaft Encoders

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Problem Solution Comments

Outline

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Problem

2

Solution

3

Comments

  • B. Balle, E. Ventura, J. M. Fuertes

Prescribed Length Shaft Encoders

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SLIDE 3

Problem Solution Comments

Outline

1

Problem

2

Solution

3

Comments

  • B. Balle, E. Ventura, J. M. Fuertes

Prescribed Length Shaft Encoders

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Problem Solution Comments

Problem statement

Problem Design shaft encoders with any desired resolution shaft encoder ≡ digital absolute shaft encoder Applications: aerospace, aviation, computer-aided machinery, semiconductor manufacturing, robotics, medical imaging, telescopes...

  • B. Balle, E. Ventura, J. M. Fuertes

Prescribed Length Shaft Encoders

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Problem Solution Comments

Encoder diagrams

Conceptual shaft encoder

  • B. Balle, E. Ventura, J. M. Fuertes

Prescribed Length Shaft Encoders

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Problem Solution Comments

Encoder diagrams

Multi-tracked shaft encoder

  • B. Balle, E. Ventura, J. M. Fuertes

Prescribed Length Shaft Encoders

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Problem Solution Comments

Encoder diagrams

Single-tracked shaft encoder

  • B. Balle, E. Ventura, J. M. Fuertes

Prescribed Length Shaft Encoders

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Problem Solution Comments

Encoder diagrams

Single-tracked shaft encoder Gain: reduction of moving mass

  • B. Balle, E. Ventura, J. M. Fuertes

Prescribed Length Shaft Encoders

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Problem Solution Comments

Construction of single-tracked encoders

Parameters: q: detector’s arity (q = 2) e: desired resolution (e = 8) n: number of detectors (n = 3) Problem Construct a (q, n, e)-closed sequence (with n ≥ ⌈logq e⌉) Fact Maximal LFSRs can generate such sequences when e = qn − 1 and q = pm for some prime p and integer m > 0

  • B. Balle, E. Ventura, J. M. Fuertes

Prescribed Length Shaft Encoders

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Problem Solution Comments

Outline

1

Problem

2

Solution

3

Comments

  • B. Balle, E. Ventura, J. M. Fuertes

Prescribed Length Shaft Encoders

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Problem Solution Comments

Idea behind the solution

a3 u3 u2 u1 u0 a0 a1 a2

Use sequences generated by non-maximal LFSR Connection and seed polynomials: a(x) = x4 − (a3x3 + a2x2 + a1x + a0) ∈ Fq [X] u(x) = u3x3 + u2x2 + u1x + u0 ∈ Fq [X] Parameters: q: detector’s arity – size of the field e: desired resolution – length of the sequence n: number of detectors – degree of a(x)

  • B. Balle, E. Ventura, J. M. Fuertes

Prescribed Length Shaft Encoders

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Problem Solution Comments

Main result

Problem Given q and e, find a polynomial a(x) ∈ Fq [X] of order e and minimal degree n. Theorem This problem can be solved algorithmically Fact Using the solution a(x) as connection polynomial and u(x) = 1 as seed polynomial, the resulting LFSR generates a (q, n, e)-closed sequence

  • B. Balle, E. Ventura, J. M. Fuertes

Prescribed Length Shaft Encoders

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Problem Solution Comments

Outline

1

Problem

2

Solution

3

Comments

  • B. Balle, E. Ventura, J. M. Fuertes

Prescribed Length Shaft Encoders

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Problem Solution Comments

About our solution

Benefits: Works “out of the box” with q-ary detectors and any resolution Minimizes the number of detectors required among all sequences generated by a LFSRs Algorithmically efficient in practice Avoids extra circuitry used by previously proposed solutions Drawbacks: May use more detectors than strictly necessary Generated codes do not satisfy Gray property

  • B. Balle, E. Ventura, J. M. Fuertes

Prescribed Length Shaft Encoders

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Problem Solution Comments

Future work

Build a real implementation Estimate the number of extra detectors required Use the redundancy in the code for error correction purposes Generalize the theory to non-linear feedback logics

  • B. Balle, E. Ventura, J. M. Fuertes

Prescribed Length Shaft Encoders

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Problem Solution Comments

Questions?

  • B. Balle, E. Ventura, J. M. Fuertes

Prescribed Length Shaft Encoders