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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.
- 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.
The transfer function Laplace transform of the system is
X(s) = (-v0 * d * w + alpha * v0^2 * c + alpha * v0^2 * m) / (s * (w * s^2 + alpha * v0^2 * k2 * s + alpha * v0^2 * k1))
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 the maximum value of m by looking at the validation data for the #STEER INPUT - CURVATURE RADIUS RELATIONSHIP. For the derivation of the model check formulae.pdf. The model does not consider the measurements disturbances (including leaps) and considers negligible disturbances associated with the actuator, projection on the path and slip angles.
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where beta and gamma are respectively the direction of the car and of the path in radians. The system has a pole in 0 and a pair of complex poles for
|k1| < +- sqrt(4 * w * k2 / (alpha ) = sqrt(4914 * k2v0^2)
which is true for any reasonable calibration of the controller. The pair of poles are
p = - alpha * v0^2 * k1 / 2w +- i * sqrt(4w * alpha * v0^2 * k_2 - alpha^2 * v_0^4 * k1^2) / 2w
We want to 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 = 0.23 and v0 = 800 we obtain ef = 44.34, ts = 53.059s37s and wd = 0.99599314.
DOCUMENTATION EXPANSION
These are likely to be expanded and moved on the main wiki page at some point.
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