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paper_gu1_4logic.py
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paper_gu1_4logic.py
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import time
import gurobipy as gp
import matplotlib.pyplot as plt
import numpy as np
from gurobipy import GRB
# Parameters
T = 10 # Prediction horizon (seconds)
dt = 0.1 # Time step (seconds)
N = int(T / dt) # Number of time steps
# State and control bounds
x_min, x_max = 0, np.inf
y_min, y_max = 0, 2.
v_x_min, v_x_max = 0, 20
v_y_min, v_y_max = -1, 1
a_x_min, a_x_max = -2, 2
a_y_min, a_y_max = -0.5, 0.5
j_x_min, j_x_max = -2.5, 2.5
j_y_min, j_y_max = -1, 1
# Other parameters from Table 1
theta_min, theta_max = -0.4, 0.4 # in radians
omega_min, omega_max = -0.26, 0.26 # in radians/s
# Speed bump parameters
x_bump_start, x_bump_end = 30, 32
v_max_bump = 5
# Initial state and reference values
x0, y0 = 0, 0.75
v_x0, v_y0 = 10, 0
a_x0, a_y0 = 0, 0
v_r = 10.0 # Reference speed
y_r = y0 # Reference lateral position (center of the lane)
# wheel base
L = 2.7
# Cost function weights (from Table 1)
q1, q2, q3, q4, q5 = 1, 1, 1, 2, 4
r1, r2 = 40, 40
# Create the model
model = gp.Model("SpeedBump_MIQP")
# Create variables
x = model.addVars(N+1, lb=x_min, ub=x_max, name="x")
y = model.addVars(N+1, lb=y_min, ub=y_max, name="y")
v_x = model.addVars(N+1, lb=v_x_min, ub=v_x_max, name="v_x")
v_y = model.addVars(N+1, lb=v_y_min, ub=v_y_max, name="v_y")
a_x = model.addVars(N+1, lb=a_x_min, ub=a_x_max, name="a_x")
a_y = model.addVars(N+1, lb=a_y_min, ub=a_y_max, name="a_y")
j_x = model.addVars(N, lb=j_x_min, ub=j_x_max, name="j_x")
j_y = model.addVars(N, lb=j_y_min, ub=j_y_max, name="j_y")
epsilon = 1e-6 # Small value for numerical stability
# Binary variables for speed bump logical constraints
delta1 = model.addVars(N+1, vtype=GRB.BINARY, name="delta1")
delta2 = model.addVars(N+1, vtype=GRB.BINARY, name="delta2")
delta3 = model.addVars(N+1, vtype=GRB.BINARY, name="delta3")
# binary variables for steer to left or right
turn_left = model.addVars(N+1, vtype=GRB.BINARY, name="delta4")
turn_right = model.addVars(N+1, vtype=GRB.BINARY, name="delta5")
is_turning = model.addVars(N+1, vtype=GRB.BINARY, name="is_turning")
# Set initial conditions
model.addConstr(x[0] == x0)
model.addConstr(y[0] == y0)
model.addConstr(v_x[0] == v_x0)
model.addConstr(v_y[0] == v_y0)
model.addConstr(a_x[0] == a_x0)
model.addConstr(a_y[0] == a_y0)
# Add dynamics constraints
for k in range(N):
model.addConstr(x[k+1] == x[k] + v_x[k]*dt + 0.5*a_x[k]*dt**2 + (1/6)*j_x[k]*dt**3)
model.addConstr(y[k+1] == y[k] + v_y[k]*dt + 0.5*a_y[k]*dt**2 + (1/6)*j_y[k]*dt**3)
model.addConstr(v_x[k+1] == v_x[k] + a_x[k]*dt + 0.5*j_x[k]*dt**2)
model.addConstr(v_y[k+1] == v_y[k] + a_y[k]*dt + 0.5*j_y[k]*dt**2)
model.addConstr(a_x[k+1] == a_x[k] + j_x[k]*dt)
model.addConstr(a_y[k+1] == a_y[k] + j_y[k]*dt)
# Add constraints for theta and omega (equations 5 and 6 in the paper)
for k in range(N+1):
model.addConstr(v_y[k] >= v_x[k] * np.tan(theta_min))
model.addConstr(v_y[k] <= v_x[k] * np.tan(theta_max))
model.addConstr(a_y[k] >= -v_x[k] * omega_max)
model.addConstr(a_y[k] <= v_x[k] * omega_max)
v_turn = 0.1
# Speed bump logical constraints using indicator constraints
for k in range(N+1):
# δ1(k) = 1 ⇔ x(k) ≥ x_bump_start
model.addGenConstrIndicator(delta1[k], True, x[k] >= x_bump_start)
model.addGenConstrIndicator(delta1[k], False, x[k] <= x_bump_start + epsilon)
# δ2(k) = 1 ⇔ x(k) ≤ x_bump_end
model.addGenConstrIndicator(delta2[k], True, x[k] <= x_bump_end + epsilon)
model.addGenConstrIndicator(delta2[k], False, x[k] >= x_bump_end)
# δ3(k) = 1 ⇔ v_x(k) ≤ v_max_bump
model.addGenConstrIndicator(delta3[k], True, v_x[k] <= v_max_bump)
model.addGenConstrIndicator(delta3[k], False, v_x[k] >= v_max_bump - epsilon)
#model.addGenConstrIndicator(delta4[k], True, v_y[k] >= v_turn)
#model.addGenConstrIndicator(delta5[k], True, v_y[k] <= -v_turn)
# Create binary variables for each turning condition
# Add indicator constraints for turning
model.addGenConstrIndicator(turn_left[k], True, v_y[k] >= v_turn)
model.addGenConstrIndicator(turn_right[k], True, v_y[k] <= -v_turn)
# Add the OR constraint for turning
# Add the OR constraint for turning
model.addGenConstrOr(is_turning[k], [turn_left[k], turn_right[k]], name=f"turn_choice_{k}")
# Add the original condition involving delta1 and delta2
model.addConstr(delta1[k] + delta2[k] - is_turning[k] <= 1)
# Logical implications from equation 7
model.addConstr(-delta1[k] + delta3[k] <= 0)
model.addConstr(-delta2[k] + delta3[k] <= 0)
model.addConstr(delta1[k] + delta2[k] - delta3[k] <= 1)
# Logical implications from steering constraints
# delta1[k] + delta2[k] - delta4[k] <= 1 OR delta1[k] + delta2[k] - delta5[k] <= 1
# Objective function
obj = gp.QuadExpr()
for k in range(N+1):
#obj += q1 * (v_x[k] - v_r)**2 + q2 * a_x[k]**2 + q3 * (y[k] - y_r)**2 + q4 * v_y[k]**2 + q5 * a_y[k]**2
obj += q1 * (v_x[k] - v_r)**2 + q3 * (y[k] - y_r)**2 + q4 * v_y[k]**2 + q5 * a_y[k]**2
for k in range(N):
obj += r1 * j_x[k]**2 + r2 * j_y[k]**2
model.setObjective(obj, GRB.MINIMIZE)
# Optimize the model
t0 = time.time()
model.optimize()
t1 = time.time()
# in ms
print("Elapsed time:", (t1 - t0) * 1000)
# Extract results
x_res = [x[k].X for k in range(N+1)]
y_res = [y[k].X for k in range(N+1)]
v_x_res = [v_x[k].X for k in range(N+1)]
v_y_res = [v_y[k].X for k in range(N+1)]
a_x_res = [a_x[k].X for k in range(N+1)]
a_y_res = [a_y[k].X for k in range(N+1)]
j_x_res = [j_x[k].X for k in range(N)]
j_y_res = [j_y[k].X for k in range(N)]
# compute steering angle
# compute sum of squared jerk
jerk_sum = sum(j_x_res[k]**2 + j_y_res[k]**2 for k in range(N))
print("Sum of squared jerk:", jerk_sum)
# Create the plots
fig, axs = plt.subplots(3, 2, figsize=(15, 15))
# Longitudinal speed plot
axs[0, 0].plot(x_res, v_x_res, 'b-', linewidth=2, label='v_x')
axs[0, 0].set_xlabel('x (m)')
axs[0, 0].set_ylabel('v_x (m/s)')
axs[0, 0].set_title('Longitudinal Speed Profile')
axs[0, 0].axvline(x=x_bump_start, color='r', linestyle='--', label='Speed bump start')
axs[0, 0].axvline(x=x_bump_end, color='r', linestyle='--', label='Speed bump end')
axs[0, 0].axhline(y=v_max_bump, color='g', linestyle=':', label='Max speed in bump')
axs[0, 0].fill_between([x_bump_start, x_bump_end], 0, v_x_max, alpha=0.2, color='r')
axs[0, 0].set_xlim(0, max(x_res))
axs[0, 0].set_ylim(0, v_x_max)
axs[0, 0].legend()
axs[0, 0].grid(True, linestyle=':', alpha=0.7)
# Lateral speed plot
axs[0, 1].plot(x_res, v_y_res, 'b-', linewidth=2, label='v_y')
axs[0, 1].set_xlabel('x (m)')
axs[0, 1].set_ylabel('v_y (m/s)')
axs[0, 1].set_title('Lateral Speed Profile')
axs[0, 1].axvline(x=x_bump_start, color='r', linestyle='--', label='Speed bump start')
axs[0, 1].axvline(x=x_bump_end, color='r', linestyle='--', label='Speed bump end')
axs[0, 1].set_xlim(0, max(x_res))
axs[0, 1].set_ylim(v_y_min, v_y_max)
axs[0, 1].legend()
axs[0, 1].grid(True, linestyle=':', alpha=0.7)
# Longitudinal acceleration plot
axs[1, 0].plot(x_res, a_x_res, 'b-', linewidth=2, label='a_x')
axs[1, 0].set_xlabel('x (m)')
axs[1, 0].set_ylabel('a_x (m/s²)')
axs[1, 0].set_title('Longitudinal Acceleration Profile')
axs[1, 0].axvline(x=x_bump_start, color='r', linestyle='--', label='Speed bump start')
axs[1, 0].axvline(x=x_bump_end, color='r', linestyle='--', label='Speed bump end')
axs[1, 0].set_xlim(0, max(x_res))
axs[1, 0].set_ylim(a_x_min, a_x_max)
axs[1, 0].legend()
axs[1, 0].grid(True, linestyle=':', alpha=0.7)
# Lateral acceleration plot
axs[1, 1].plot(x_res, a_y_res, 'b-', linewidth=2, label='a_y')
axs[1, 1].set_xlabel('x (m)')
axs[1, 1].set_ylabel('a_y (m/s²)')
axs[1, 1].set_title('Lateral Acceleration Profile')
axs[1, 1].axvline(x=x_bump_start, color='r', linestyle='--', label='Speed bump start')
axs[1, 1].axvline(x=x_bump_end, color='r', linestyle='--', label='Speed bump end')
axs[1, 1].set_xlim(0, max(x_res))
axs[1, 1].set_ylim(a_y_min, a_y_max)
axs[1, 1].legend()
axs[1, 1].grid(True, linestyle=':', alpha=0.7)
# Vehicle Trajectory plot
axs[2, 0].plot(x_res, y_res, 'b-', linewidth=2, label='Vehicle Path')
axs[2, 0].set_xlabel('x (m)')
axs[2, 0].set_ylabel('y (m)')
axs[2, 0].set_title('Vehicle Trajectory')
axs[2, 0].axvline(x=x_bump_start, color='r', linestyle='--', label='Speed bump start')
axs[2, 0].axvline(x=x_bump_end, color='r', linestyle='--', label='Speed bump end')
axs[2, 0].set_xlim(0, max(x_res))
axs[2, 0].set_ylim(y_min, y_max)
axs[2, 0].legend()
axs[2, 0].grid(True, linestyle=':', alpha=0.7)
# Steering angle plot
plt.tight_layout()
plt.show()
print("Optimization status:", model.status)
print("Objective value:", model.objVal)
print("Optimization status:", model.status)
print("Objective value:", model.objVal)
print("Optimization status:", model.status)
print("Objective value:", model.objVal)