LIBRARY
Examples.RaceCarSpatialModel ​
Race car of the dymos example over the arc length: RaceCarModel with every time rate divided by tlap is a state,
This is the formulation dymos integrates: the collocation nodes are fixed in space, so the horizon of a controller built on it is a fixed distance ahead of the car whatever its speed, and the lap time is the state tlap at the end of the horizon rather than a nonlinear function of the state to be pulled towards a reference. The arc length s stays a state with the unit derivative so that the curvature table is evaluated as in the time-domain model. The controls remain the time rates of the steering angle, the throttle and the brake; Ts and time_steps of a controller using this model are distances. ACADOSMPCRaceCarSpatialExample uses it as the plant and as the prediction model, with the simulation running over the arc length:
This component extends from RaceCarModel
Usage ​
MPCComponents.Examples.RaceCarSpatialModel(M=800.0, a=1.8, b=1.6, tw=0.73, Iz=450.0, h=0.3, chi=0.5, CoP=1.6, beta=0.62, k_lambda=44.0, mu0_x=1.68, mu0_y=1.68, k_mu=0.0, tau_x=0.2, tau_y=0.2, rho=1.2, CdA=2.0, ClA=4.0, g=9.8, P_max=960000.0, s0=0.0, n0=0.0, alpha0=0.0, V0=20.0, delta0=0.0, throttle0=0.0, brake0=0.0, blend=racetrack_blend(), seg_curvature=racetrack_curvatures(), seg_length=racetrack_lengths(), N0_f=M * g / 2 * b / (a + b), N0_r=M * g / 2 * a / (a + b), facet_cos=friction_polygon_cos(), facet_sin=friction_polygon_sin(), adh_max=friction_polygon_bound())
Parameters: ​
| Name | Description | Units | Default value |
|---|---|---|---|
nseg | Number of segments of the track (derived) | – | racetrack_segment_count() |
nbound | Number of segment boundaries of the track (derived) | – | nseg + 1 |
nfacet | Number of facets of the polygon inscribed in the friction ellipse (derived) | – | friction_polygon_facets() |
M | Mass | – | 800.0 |
a | Distance from the centre of gravity to the front axle | – | 1.8 |
b | Distance from the centre of gravity to the rear axle | – | 1.6 |
tw | Half track width | – | 0.73 |
Iz | Yaw inertia | – | 450.0 |
h | Height of the centre of gravity | – | 0.3 |
chi | Roll stiffness distribution (share of the lateral load transfer taken by the front axle) | – | 0.5 |
CoP | Distance from the centre of pressure to the front axle | – | 1.6 |
beta | Brake bias (share of the braking force on the front axle) | – | 0.62 |
k_lambda | Lateral tyre stiffness per unit normal load | – | 44.0 |
mu0_x | Longitudinal friction coefficient at the static load | – | 1.68 |
mu0_y | Lateral friction coefficient at the static load | – | 1.68 |
k_mu | Load sensitivity of the friction coefficient | – | 0.0 |
tau_x | Time constant of the longitudinal load transfer | – | 0.2 |
tau_y | Time constant of the lateral load transfer | – | 0.2 |
rho | Air density | – | 1.2 |
CdA | Drag coefficient times frontal area | – | 2.0 |
ClA | Downforce coefficient times wing area | – | 4.0 |
g | Gravitational acceleration | – | 9.8 |
P_max | Engine power limit, the scale of the normalized tractive power power_norm | – | 960000.0 |
s0 | Arc length at the start | – | 0.0 |
n0 | Lateral offset at the start | – | 0.0 |
alpha0 | Relative heading at the start | – | 0.0 |
V0 | Speed at the start (positive: the speed divides the slip angle dynamics) | – | 20.0 |
delta0 | Steering angle at the start | – | 0.0 |
throttle0 | Throttle at the start | – | 0.0 |
brake0 | Brake at the start | – | 0.0 |
blend | Arc length over which the curvature blends from one segment to the next | – | racetrack_blend() |
seg_curvature | Curvature of each segment of the track; an online parameter of the controller | – | racetrack_curvatures() |
seg_length | Length of each segment of the track; an online parameter of the controller | – | racetrack_lengths() |
Connectors ​
delta_rate- This connector represents a real signal as an input to a component (RealInput)throttle_rate- This connector represents a real signal as an input to a component (RealInput)brake_rate- This connector represents a real signal as an input to a component (RealInput)s_out- This connector represents a real signal as an output from a component (RealOutput)n_out- This connector represents a real signal as an output from a component (RealOutput)alpha_out- This connector represents a real signal as an output from a component (RealOutput)V_out- This connector represents a real signal as an output from a component (RealOutput)lambda_out- This connector represents a real signal as an output from a component (RealOutput)omega_out- This connector represents a real signal as an output from a component (RealOutput)ax_out- This connector represents a real signal as an output from a component (RealOutput)ay_out- This connector represents a real signal as an output from a component (RealOutput)delta_out- This connector represents a real signal as an output from a component (RealOutput)throttle_out- This connector represents a real signal as an output from a component (RealOutput)brake_out- This connector represents a real signal as an output from a component (RealOutput)seg_curvature_out- This connector represents a real signal as an output from a component (RealOutput)seg_length_out- This connector represents a real signal as an output from a component (RealOutput)
Variables ​
| Name | Description | Units |
|---|---|---|
s | Arc length along the centerline | – |
s_lap | Arc length within the lap, at which the track tables are evaluated | – |
n | Lateral offset from the centerline | – |
alpha | Heading relative to the centerline | – |
V | Speed | – |
lambda | Body slip angle | – |
omega | Yaw rate | – |
ax | Longitudinal acceleration (filtered) | – |
ay | Lateral acceleration (filtered) | – |
delta | Steering angle | – |
throttle | Throttle: the rear longitudinal tyre force as a fraction of the weight (nonnegative) | – |
brake | Brake: the braking force as a fraction of the weight (nonnegative) | – |
seg_start | Arc length at which each segment starts (derived from the lengths) | – |
t_in | Fraction of the blend into each segment covered, clipped to [0, 1] | – |
t_out | Fraction of the blend out of each segment covered, clipped to [0, 1] | – |
kappa_k | Curvature contributed by the first k segments | – |
kappa | Curvature of the centerline at the arc length s | – |
sdot | Rate of progress along the centerline | – |
downforce | Downforce | – |
drag | Aerodynamic drag | – |
N_fl | Normal load of the front-left tyre | – |
N_fr | Normal load of the front-right tyre | – |
N_rl | Normal load of the rear-left tyre | – |
N_rr | Normal load of the rear-right tyre | – |
S_f | Longitudinal force of one front tyre | – |
S_r | Longitudinal force of one rear tyre | – |
F_fl | Lateral force of the front-left tyre | – |
F_fr | Lateral force of the front-right tyre | – |
F_rl | Lateral force of the rear-left tyre | – |
F_rr | Lateral force of the rear-right tyre | – |
ndot | Rate of the lateral offset | – |
alphadot | Rate of the relative heading | – |
Vdot | Acceleration along the body axis | – |
lambdadot | Rate of the body slip angle | – |
omegadot | Yaw acceleration | – |
axdot | Rate of the filtered longitudinal acceleration | – |
aydot | Rate of the filtered lateral acceleration | – |
power | Tractive power | – |
power_norm | Tractive power relative to the engine power limit (one at the limit) | – |
thrust | Throttle minus brake, the signed thrust of dymos | – |
c_fl | Friction ellipse of the front-left tyre (one at the limit of adherence) | – |
c_fr | Friction ellipse of the front-right tyre | – |
c_rl | Friction ellipse of the rear-left tyre | – |
c_rr | Friction ellipse of the rear-right tyre | – |
sx_fl | Longitudinal tyre force of the front-left tyre relative to its limit of adherence | – |
sx_fr | Longitudinal tyre force of the front-right tyre relative to its limit of adherence | – |
sx_rl | Longitudinal tyre force of the rear-left tyre relative to its limit of adherence | – |
sx_rr | Longitudinal tyre force of the rear-right tyre relative to its limit of adherence | – |
sy_fl | Lateral tyre force of the front-left tyre relative to its limit of adherence, linear in the slip angle | – |
sy_fr | Lateral tyre force of the front-right tyre relative to its limit of adherence | – |
sy_rl | Lateral tyre force of the rear-left tyre relative to its limit of adherence | – |
sy_rr | Lateral tyre force of the rear-right tyre relative to its limit of adherence | – |
adh_fl | Facets of the octagon inscribed in the friction ellipse of the front-left tyre, each at most adh_max: the form the controller constrains | – |
adh_fr | Facets of the octagon inscribed in the friction ellipse of the front-right tyre | – |
adh_rl | Facets of the octagon inscribed in the friction ellipse of the rear-left tyre | – |
adh_rr | Facets of the octagon inscribed in the friction ellipse of the rear-right tyre | – |
tlap | Time elapsed from the start of the horizon | – |
Behavior ​
Behavior of this component cannot be rendered because it includes path variables.
Source ​
dyad
"""
Race car of the dymos example over the arc length: `RaceCarModel` with every time rate divided by
``\\dot s``, so that the independent variable of the model is the distance along the centerline and
the elapsed time `tlap` is a state,
```math
\\frac{dx}{ds} = \\frac{\\dot x}{\\dot s}, \\qquad \\frac{d\\, t_\\mathrm{lap}}{ds} = \\frac{1}{\\dot s}.
```
This is the formulation dymos integrates: the collocation nodes are fixed in space, so the horizon of
a controller built on it is a fixed distance ahead of the car whatever its speed, and the lap time is
the state `tlap` at the end of the horizon rather than a nonlinear function of the state to be pulled
towards a reference. The arc length `s` stays a state with the unit derivative so that the curvature
table is evaluated as in the time-domain model. The controls remain the time rates of the steering
angle, the throttle and the brake; `Ts` and `time_steps` of a controller using this model are
distances. `ACADOSMPCRaceCarSpatialExample` uses it as the plant and as the prediction model, with the
simulation running over the arc length: ``\\dot s`` must stay positive, so the car must be moving.
"""
component RaceCarSpatialModel
extends RaceCarModel()
"Time elapsed from the start of the horizon"
variable tlap::Real
relations
initial tlap = 0
der(s) = 1
der(tlap) = 1 / sdot
der(n) = ndot / sdot
der(alpha) = alphadot / sdot
der(V) = Vdot / sdot
der(lambda) = lambdadot / sdot
der(omega) = omegadot / sdot
der(ax) = axdot / sdot
der(ay) = aydot / sdot
der(delta) = delta_rate / sdot
der(throttle) = throttle_rate / sdot
der(brake) = brake_rate / sdot
metadata {
"Dyad": {
"icons": {"default": "dyad://MPCComponents/RaceCarPlant.svg"},
"doc": {"behavior": false}
}
}
endFlattened Source
dyad
"""
Race car of the dymos example over the arc length: `RaceCarModel` with every time rate divided by
``\\dot s``, so that the independent variable of the model is the distance along the centerline and
the elapsed time `tlap` is a state,
```math
\\frac{dx}{ds} = \\frac{\\dot x}{\\dot s}, \\qquad \\frac{d\\, t_\\mathrm{lap}}{ds} = \\frac{1}{\\dot s}.
```
This is the formulation dymos integrates: the collocation nodes are fixed in space, so the horizon of
a controller built on it is a fixed distance ahead of the car whatever its speed, and the lap time is
the state `tlap` at the end of the horizon rather than a nonlinear function of the state to be pulled
towards a reference. The arc length `s` stays a state with the unit derivative so that the curvature
table is evaluated as in the time-domain model. The controls remain the time rates of the steering
angle, the throttle and the brake; `Ts` and `time_steps` of a controller using this model are
distances. `ACADOSMPCRaceCarSpatialExample` uses it as the plant and as the prediction model, with the
simulation running over the arc length: ``\\dot s`` must stay positive, so the car must be moving.
"""
component RaceCarSpatialModel
"Rate of the steering angle"
delta_rate = RealInput() {}
"Rate of the throttle"
throttle_rate = RealInput() {}
"Rate of the brake"
brake_rate = RealInput() {}
"Arc length along the centerline"
s_out = RealOutput() {}
"Lateral offset from the centerline, positive to the left"
n_out = RealOutput() {}
"Heading relative to the centerline"
alpha_out = RealOutput() {}
"Speed"
V_out = RealOutput() {}
"Body slip angle"
lambda_out = RealOutput() {}
"Yaw rate"
omega_out = RealOutput() {}
"Longitudinal acceleration (filtered)"
ax_out = RealOutput() {}
"Lateral acceleration (filtered)"
ay_out = RealOutput() {}
"Steering angle"
delta_out = RealOutput() {}
"Throttle"
throttle_out = RealOutput() {}
"Brake"
brake_out = RealOutput() {}
"Curvature of each track segment, the table `seg_curvature` as a signal (for a controller that takes the track as online parameters)"
seg_curvature_out = [RealOutput() for k in 1:nseg] {}
"Length of each track segment, the table `seg_length` as a signal"
seg_length_out = [RealOutput() for k in 1:nseg] {}
"Mass"
parameter M::Real = 800.0
"Distance from the centre of gravity to the front axle"
parameter a::Real = 1.8
"Distance from the centre of gravity to the rear axle"
parameter b::Real = 1.6
"Half track width"
parameter tw::Real = 0.73
"Yaw inertia"
parameter Iz::Real = 450.0
"Height of the centre of gravity"
parameter h::Real = 0.3
"Roll stiffness distribution (share of the lateral load transfer taken by the front axle)"
parameter chi::Real = 0.5
"Distance from the centre of pressure to the front axle"
parameter CoP::Real = 1.6
"Brake bias (share of the braking force on the front axle)"
parameter beta::Real = 0.62
"Lateral tyre stiffness per unit normal load"
parameter k_lambda::Real = 44.0
"Longitudinal friction coefficient at the static load"
parameter mu0_x::Real = 1.68
"Lateral friction coefficient at the static load"
parameter mu0_y::Real = 1.68
"Load sensitivity of the friction coefficient"
parameter k_mu::Real = 0.0
"Time constant of the longitudinal load transfer"
parameter tau_x::Real = 0.2
"Time constant of the lateral load transfer"
parameter tau_y::Real = 0.2
"Air density"
parameter rho::Real = 1.2
"Drag coefficient times frontal area"
parameter CdA::Real = 2.0
"Downforce coefficient times wing area"
parameter ClA::Real = 4.0
"Gravitational acceleration"
parameter g::Real = 9.8
"Engine power limit, the scale of the normalized tractive power `power_norm`"
parameter P_max::Real = 960000.0
"Arc length at the start"
parameter s0::Real = 0.0
"Lateral offset at the start"
parameter n0::Real = 0.0
"Relative heading at the start"
parameter alpha0::Real = 0.0
"Speed at the start (positive: the speed divides the slip angle dynamics)"
parameter V0::Real = 20.0
"Steering angle at the start"
parameter delta0::Real = 0.0
"Throttle at the start"
parameter throttle0::Real = 0.0
"Brake at the start"
parameter brake0::Real = 0.0
"Arc length over which the curvature blends from one segment to the next"
parameter blend::Real = racetrack_blend()
"Number of segments of the track (derived)"
final structural parameter nseg::Integer = racetrack_segment_count()
"Number of segment boundaries of the track (derived)"
final structural parameter nbound::Integer = nseg + 1
"Curvature of each segment of the track; an online parameter of the controller"
parameter seg_curvature::Real[nseg] = racetrack_curvatures()
"Length of each segment of the track; an online parameter of the controller"
parameter seg_length::Real[nseg] = racetrack_lengths()
"Static normal load of one front tyre (derived)"
final parameter N0_f::Real = M * g / 2 * b / (a + b)
"Static normal load of one rear tyre (derived)"
final parameter N0_r::Real = M * g / 2 * a / (a + b)
"Number of facets of the polygon inscribed in the friction ellipse (derived)"
final structural parameter nfacet::Integer = friction_polygon_facets()
"Cosine of the direction of each facet normal in the (longitudinal, lateral) plane of the tyre forces (derived)"
final parameter facet_cos::Real[nfacet] = friction_polygon_cos()
"Sine of the direction of each facet normal (derived)"
final parameter facet_sin::Real[nfacet] = friction_polygon_sin()
"Distance of the facets from the origin, the bound of `adh_fl` and its siblings (derived)"
final parameter adh_max::Real = friction_polygon_bound()
"Arc length along the centerline"
variable s::Real
"Arc length within the lap, at which the track tables are evaluated"
variable s_lap::Real
"Lateral offset from the centerline"
variable n::Real
"Heading relative to the centerline"
variable alpha::Real
"Speed"
variable V::Real
"Body slip angle"
variable lambda::Real
"Yaw rate"
variable omega::Real
"Longitudinal acceleration (filtered)"
variable ax::Real
"Lateral acceleration (filtered)"
variable ay::Real
"Steering angle"
variable delta::Real
"Throttle: the rear longitudinal tyre force as a fraction of the weight (nonnegative)"
variable throttle::Real
"Brake: the braking force as a fraction of the weight (nonnegative)"
variable brake::Real
"Arc length at which each segment starts (derived from the lengths)"
variable seg_start::Real[nbound]
"Fraction of the blend into each segment covered, clipped to [0, 1]"
variable t_in::Real[nseg]
"Fraction of the blend out of each segment covered, clipped to [0, 1]"
variable t_out::Real[nseg]
"Curvature contributed by the first k segments"
variable kappa_k::Real[nbound]
"Curvature of the centerline at the arc length s"
variable kappa::Real
"Rate of progress along the centerline"
variable sdot::Real
"Downforce"
variable downforce::Real
"Aerodynamic drag"
variable drag::Real
"Normal load of the front-left tyre"
variable N_fl::Real
"Normal load of the front-right tyre"
variable N_fr::Real
"Normal load of the rear-left tyre"
variable N_rl::Real
"Normal load of the rear-right tyre"
variable N_rr::Real
"Longitudinal force of one front tyre"
variable S_f::Real
"Longitudinal force of one rear tyre"
variable S_r::Real
"Lateral force of the front-left tyre"
variable F_fl::Real
"Lateral force of the front-right tyre"
variable F_fr::Real
"Lateral force of the rear-left tyre"
variable F_rl::Real
"Lateral force of the rear-right tyre"
variable F_rr::Real
"Rate of the lateral offset"
variable ndot::Real
"Rate of the relative heading"
variable alphadot::Real
"Acceleration along the body axis"
variable Vdot::Real
"Rate of the body slip angle"
variable lambdadot::Real
"Yaw acceleration"
variable omegadot::Real
"Rate of the filtered longitudinal acceleration"
variable axdot::Real
"Rate of the filtered lateral acceleration"
variable aydot::Real
"Tractive power"
variable power::Real
"Tractive power relative to the engine power limit (one at the limit)"
variable power_norm::Real
"Throttle minus brake, the signed thrust of dymos"
variable thrust::Real
"Friction ellipse of the front-left tyre (one at the limit of adherence)"
variable c_fl::Real
"Friction ellipse of the front-right tyre"
variable c_fr::Real
"Friction ellipse of the rear-left tyre"
variable c_rl::Real
"Friction ellipse of the rear-right tyre"
variable c_rr::Real
"Longitudinal tyre force of the front-left tyre relative to its limit of adherence"
variable sx_fl::Real
"Longitudinal tyre force of the front-right tyre relative to its limit of adherence"
variable sx_fr::Real
"Longitudinal tyre force of the rear-left tyre relative to its limit of adherence"
variable sx_rl::Real
"Longitudinal tyre force of the rear-right tyre relative to its limit of adherence"
variable sx_rr::Real
"Lateral tyre force of the front-left tyre relative to its limit of adherence, linear in the slip angle"
variable sy_fl::Real
"Lateral tyre force of the front-right tyre relative to its limit of adherence"
variable sy_fr::Real
"Lateral tyre force of the rear-left tyre relative to its limit of adherence"
variable sy_rl::Real
"Lateral tyre force of the rear-right tyre relative to its limit of adherence"
variable sy_rr::Real
"Facets of the octagon inscribed in the friction ellipse of the front-left tyre, each at most `adh_max`: the form the controller constrains"
variable adh_fl::Real[nfacet]
"Facets of the octagon inscribed in the friction ellipse of the front-right tyre"
variable adh_fr::Real[nfacet]
"Facets of the octagon inscribed in the friction ellipse of the rear-left tyre"
variable adh_rl::Real[nfacet]
"Facets of the octagon inscribed in the friction ellipse of the rear-right tyre"
variable adh_rr::Real[nfacet]
"Time elapsed from the start of the horizon"
variable tlap::Real
relations
initial s = s0
initial n = n0
initial alpha = alpha0
initial V = V0
initial lambda = 0
initial omega = 0
initial ax = 0
initial ay = 0
initial delta = delta0
initial throttle = throttle0
initial brake = brake0
# Curvature of the centerline at s: each segment contributes between its start and its end, with a
# cubic blend (3t^2 - 2t^3, continuous with a continuous derivative) of length `blend` at either end.
# The tables describe one lap, so the arc length is reduced to the lap before the lookup; the
# derivative of floor is zero, so the Jacobian of the lookup is that of the tables themselves
seg_start[1] = 0
s_lap = s - seg_start[nbound] * floor(s / seg_start[nbound])
kappa_k[1] = 0
for k in 1:nseg
seg_start[k + 1] = seg_start[k] + seg_length[k]
t_in[k] = max(min((s_lap - seg_start[k]) / blend, 1), 0)
t_out[k] = max(min((s_lap - seg_start[k + 1]) / blend, 1), 0)
kappa_k[k + 1] = kappa_k[k] + seg_curvature[k] * (t_in[k] ^ 2 * (3 - 2 * t_in[k]) - t_out[k] ^ 2 * (3 - 2 * t_out[k]))
end
kappa = kappa_k[nbound]
# Aerodynamics and normal loads
downforce = 0.5 * rho * ClA * V ^ 2
drag = 0.5 * rho * CdA * V ^ 2
N_fl = N0_f + M / 4 * (-(ax * h) / (a + b) + ay * chi * h / tw) + downforce * (1 - CoP / (a + b)) / 2
N_fr = N0_f + M / 4 * (-(ax * h) / (a + b) - ay * chi * h / tw) + downforce * (1 - CoP / (a + b)) / 2
N_rl = N0_r + M / 4 * ((ax * h) / (a + b) + ay * (1 - chi) * h / tw) + downforce * (CoP / (a + b)) / 2
N_rr = N0_r + M / 4 * ((ax * h) / (a + b) - ay * (1 - chi) * h / tw) + downforce * (CoP / (a + b)) / 2
# Tyre forces
S_f = -(M * g / 2) * brake * beta
S_r = (M * g / 2) * (throttle - brake * (1 - beta))
F_rr = N_rr * k_lambda * (lambda + omega * (b + lambda * tw) / V)
F_rl = N_rl * k_lambda * (lambda + omega * (b - lambda * tw) / V)
F_fr = N_fr * k_lambda * (lambda + delta - omega * (a - lambda * tw) / V)
F_fl = N_fl * k_lambda * (lambda + delta - omega * (a + lambda * tw) / V)
# Motion in the track frame: the time rates of the states, which the plant integrates over time and
# the spatial model divides by sdot to integrate over the arc length
sdot = V * cos(alpha - lambda) / (1 - n * kappa)
ndot = V * sin(alpha - lambda)
alphadot = omega - kappa * sdot
Vdot = (2 * S_f + 2 * S_r) / M - delta * (F_fl + F_fr) / M - drag / M - omega * V * lambda
lambdadot = omega - Vdot * lambda / V - delta * 2 * S_f / (M * V) - (F_fl + F_fr + F_rl + F_rr) / (M * V)
omegadot = (a * (F_fr + F_fl) - b * (F_rr + F_rl)) / Iz
axdot = (Vdot + omega * V * lambda - ax) / tau_x
aydot = (omega * V - (V * lambdadot + Vdot * lambda) - ay) / tau_y
# Constrained quantities
power = V * (2 * S_f + 2 * S_r)
power_norm = power / P_max
thrust = throttle - brake
c_fl = (S_f / (N_fl * (mu0_x + k_mu * N_fl / N0_f))) ^ 2 + (F_fl / (N_fl * (mu0_y + k_mu * N_fl / N0_f))) ^ 2
c_fr = (S_f / (N_fr * (mu0_x + k_mu * N_fr / N0_f))) ^ 2 + (F_fr / (N_fr * (mu0_y + k_mu * N_fr / N0_f))) ^ 2
c_rl = (S_r / (N_rl * (mu0_x + k_mu * N_rl / N0_r))) ^ 2 + (F_rl / (N_rl * (mu0_y + k_mu * N_rl / N0_r))) ^ 2
c_rr = (S_r / (N_rr * (mu0_x + k_mu * N_rr / N0_r))) ^ 2 + (F_rr / (N_rr * (mu0_y + k_mu * N_rr / N0_r))) ^ 2
# The same limits as a polygon inscribed in the ellipse of each tyre, in the coordinates (S / mu_x N,
# F / mu_y N). The lateral coordinate is k_lambda times the slip angle over mu_y, linear in the slip
# angle, which is what makes the facets well conditioned for the sequential quadratic programming of
# the controller: the ellipse itself spans a slip angle of only mu_y / k_lambda and is too curved
sx_fl = S_f / ((mu0_x + k_mu * N_fl / N0_f) * N_fl)
sx_fr = S_f / ((mu0_x + k_mu * N_fr / N0_f) * N_fr)
sx_rl = S_r / ((mu0_x + k_mu * N_rl / N0_r) * N_rl)
sx_rr = S_r / ((mu0_x + k_mu * N_rr / N0_r) * N_rr)
sy_fl = k_lambda * (lambda + delta - omega * (a + lambda * tw) / V) / (mu0_y + k_mu * N_fl / N0_f)
sy_fr = k_lambda * (lambda + delta - omega * (a - lambda * tw) / V) / (mu0_y + k_mu * N_fr / N0_f)
sy_rl = k_lambda * (lambda + omega * (b - lambda * tw) / V) / (mu0_y + k_mu * N_rl / N0_r)
sy_rr = k_lambda * (lambda + omega * (b + lambda * tw) / V) / (mu0_y + k_mu * N_rr / N0_r)
for j in 1:nfacet
adh_fl[j] = facet_cos[j] * sx_fl + facet_sin[j] * sy_fl
adh_fr[j] = facet_cos[j] * sx_fr + facet_sin[j] * sy_fr
adh_rl[j] = facet_cos[j] * sx_rl + facet_sin[j] * sy_rl
adh_rr[j] = facet_cos[j] * sx_rr + facet_sin[j] * sy_rr
end
s_out = s
n_out = n
alpha_out = alpha
V_out = V
lambda_out = lambda
omega_out = omega
ax_out = ax
ay_out = ay
delta_out = delta
throttle_out = throttle
brake_out = brake
for k in 1:nseg
seg_curvature_out[k] = seg_curvature[k]
seg_length_out[k] = seg_length[k]
end
initial tlap = 0
der(s) = 1
der(tlap) = 1 / sdot
der(n) = ndot / sdot
der(alpha) = alphadot / sdot
der(V) = Vdot / sdot
der(lambda) = lambdadot / sdot
der(omega) = omegadot / sdot
der(ax) = axdot / sdot
der(ay) = aydot / sdot
der(delta) = delta_rate / sdot
der(throttle) = throttle_rate / sdot
der(brake) = brake_rate / sdot
metadata {
"Dyad": {
"icons": {"default": "dyad://MPCComponents/RaceCarPlant.svg"},
"doc": {"behavior": false}
}
}
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