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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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The "viscous friction" component is a loss of speed that is proportional to velocity. This proportionality is the same that occurs through viscous losses such as viscous dampers or the losses associated with bearings.

$$k_b= \frac{m\,g\,sin(\theta)}{m}$$ This corresponds to the forward or rearward force on the car due to gravity on a hill, divided by the mass, which would correspond to the acceleration due to gravity on a hill.

This acceleration should not be multiplied by the throttle position in a physically accurate simulation. Nor should an angle be multiplied by an acceleration, so I think there must be a transcription error somewhere but I was unable to look at the source book.

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