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- U(s): desired lateral distance from the path which is always 0.
- C(s) = c / s: constant compensation term of the controller computed by using the model steer input-curvature radius.
- M(s) = m / s: constant term that accounts for the error of the model used to determine the compensation.
- D(s) = d / s^2: disturbance term that models the curve in the path.
- w: the wheelbase.
- alpha: the steer factor.
- delta: steer input in (-100, 100).
- X(s): controlled variable, the lateral distance from the path.
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If we set c to cancel the effect of d, then the error is caused by m. By applying the final value theorem and cancel the effect of d with c, one obtains
lim x(t) = m / k2 for t -> inf
I computed Thus the final error depends only on k2 as reasonable to expect. Recalling that m must be a steer input, it is easy to compute the maximum value of m by . By looking at the maximum error in the validation data for the #STEER INPUT - CURVATURE RADIUS RELATIONSHIP I obtained m = 13.3045. For the derivation of the model check formulae.pdf. The model does not consider take into account the measurements disturbances (including leaps included) and it considers negligible disturbances associated with the actuator, projection on the path and slip angles.
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We want the final error ef, the settling time ts and the frequency of oscillation wd to be low or, in other words, we want to keep low the quantities ts = -4.6 / Re(p) and wd = Im(p). With k1 = k2 = 0.3 and v0 = 800 we obtain ef = 44.3434mm, ts = 3.37s and wd = 0.9314.
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RESULTS
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Figure distgamma.png shows the measured distance from the path and the difference between the car and the path direction.
DOCUMENTATION EXPANSION
These are likely to be expanded and moved on the main wiki page at some point.
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