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For a simulation for testing on automatic cruise control, I came across the equation:

$$ v_{n+1} = (1 - k_1 / m) v_n + (1 - k_b) \begin{pmatrix} T_n \\ θ_n \\ \end{pmatrix} $$

where:

  • $T$ = throttle position
  • $k_1$ = viscous friction
  • $k_b = k_2 / m$
  • $k_2 = m g \sin(θ)$
  • $v$ = velocity

$k_b$ doesn't make sense. The matrix part doesn't make sense to me. Can anyone expound the equation?

Why isn't the angle put in sin or cos first?

Also, why is there viscous friction in solid physics? Isn't the angular component in $k_2$ enough?

SOURCE:

A Fuzzy Logic Book: "scribd.com/doc/105335356/124/INDUSTRIAL-APPLICATIONS"; Page 508. Number 13.2.

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Hi James, and welcome to Physics Stack Exchange! Thanks for taking the consideration to put the homework tag on your question; in this case I don't think it's actually necessary though because you're asking a purely conceptual question, not an educational one. Also, could you link or cite the source where you found this equation? It does look kind of odd and it may help interpret it if we had access to the original source. – David Zaslavsky♦ Sep 18 '12 at 17:38
@DavidZaslavsky, Here's the source. A Fuzzy Logic Book: "scribd.com/doc/105335356/124/INDUSTRIAL-APPLICATIONS"; Page 508. Number 13.2. Thank you for the welcome. – James Sep 18 '12 at 20:40
1  
I meant edit the question so that it includes the source. – David Zaslavsky♦ Sep 18 '12 at 23:17

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