import numpy as np
import osqp
from scipy import sparse
import matplotlib.pyplot as plt
# Colors
PREDICTION = '#BA4A00'
##################
# MPC Controller #
##################
class MPC:
def __init__(self, model, N, Q, R, QN, StateConstraints, InputConstraints,
ay_max):
"""
Constructor for the Model Predictive Controller.
:param model: bicycle model object to be controlled
:param N: time horizon | int
:param Q: state cost matrix
:param R: input cost matrix
:param QN: final state cost matrix
:param StateConstraints: dictionary of state constraints
:param InputConstraints: dictionary of input constraints
:param ay_max: maximum allowed lateral acceleration in curves
"""
# Parameters
self.N = N # horizon
self.Q = Q # weight matrix state vector
self.R = R # weight matrix input vector
self.QN = QN # weight matrix terminal
# Model
self.model = model
# Dimensions
self.nx = self.model.n_states
self.nu = 2
# Constraints
self.state_constraints = StateConstraints
self.input_constraints = InputConstraints
# Maximum lateral acceleration
self.ay_max = ay_max
# Current control and prediction
self.current_prediction = None
# Counter for old control signals in case of infeasible problem
self.infeasibility_counter = 0
# Current control signals
self.current_control = np.zeros((self.nu*self.N))
# Initialize Optimization Problem
self.optimizer = osqp.OSQP()
def _init_problem(self):
"""
Initialize optimization problem for current time step.
"""
# Constraints
umin = self.input_constraints['umin']
umax = self.input_constraints['umax']
xmin = self.state_constraints['xmin']
xmax = self.state_constraints['xmax']
# LTV System Matrices
A = np.zeros((self.nx * (self.N + 1), self.nx * (self.N + 1)))
B = np.zeros((self.nx * (self.N + 1), self.nu * (self.N)))
# Reference vector for state and input variables
ur = np.zeros(self.nu*self.N)
xr = np.zeros(self.nx*(self.N+1))
# Offset for equality constraint (due to B * (u - ur))
uq = np.zeros(self.N * self.nx)
# Dynamic state constraints
xmin_dyn = np.kron(np.ones(self.N + 1), xmin)
xmax_dyn = np.kron(np.ones(self.N + 1), xmax)
# Dynamic input constraints
umax_dyn = np.kron(np.ones(self.N), umax)
# Get curvature predictions from previous control signals
kappa_pred = np.tan(np.array(self.current_control[3::] +
self.current_control[-1:])) / self.model.length
# Iterate over horizon
for n in range(self.N):
# Get information about current waypoint
current_waypoint = self.model.reference_path.get_waypoint(self.model.wp_id + n)
next_waypoint = self.model.reference_path.get_waypoint(self.model.wp_id + n + 1)
delta_s = next_waypoint - current_waypoint
kappa_ref = current_waypoint.kappa
v_ref = current_waypoint.v_ref
# Compute LTV matrices
f, A_lin, B_lin = self.model.linearize(v_ref, kappa_ref, delta_s)
A[(n+1) * self.nx: (n+2)*self.nx, n * self.nx:(n+1)*self.nx] = A_lin
B[(n+1) * self.nx: (n+2)*self.nx, n * self.nu:(n+1)*self.nu] = B_lin
# Set reference for input signal
ur[n*self.nu:(n+1)*self.nu] = np.array([v_ref, kappa_ref])
# Compute equality constraint offset (B*ur)
uq[n * self.nx:(n+1)*self.nx] = B_lin.dot(np.array
([v_ref, kappa_ref])) - f
# Constrain maximum speed based on predicted car curvature
vmax_dyn = np.sqrt(self.ay_max / (np.abs(kappa_pred[n]) + 1e-12))
if vmax_dyn < umax_dyn[self.nu*n]:
umax_dyn[self.nu*n] = vmax_dyn
# Compute dynamic constraints on e_y
ub, lb, _ = self.model.reference_path.update_path_constraints(
self.model.wp_id+1, self.N, 2*self.model.safety_margin,
self.model.safety_margin)
xmin_dyn[0] = self.model.spatial_state.e_y
xmax_dyn[0] = self.model.spatial_state.e_y
xmin_dyn[self.nx::self.nx] = lb
xmax_dyn[self.nx::self.nx] = ub
# Set reference for state as center-line of drivable area
xr[self.nx::self.nx] = (lb + ub) / 2
# Get equality matrix
Ax = sparse.kron(sparse.eye(self.N + 1),
-sparse.eye(self.nx)) + sparse.csc_matrix(A)
Bu = sparse.csc_matrix(B)
Aeq = sparse.hstack([Ax, Bu])
# Get inequality matrix
Aineq = sparse.eye((self.N + 1) * self.nx + self.N * self.nu)
# Combine constraint matrices
A = sparse.vstack([Aeq, Aineq], format='csc')
# Get upper and lower bound vectors for equality constraints
lineq = np.hstack([xmin_dyn,
np.kron(np.ones(self.N), umin)])
uineq = np.hstack([xmax_dyn, umax_dyn])
# Get upper and lower bound vectors for inequality constraints
x0 = np.array(self.model.spatial_state[:])
leq = np.hstack([-x0, uq])
ueq = leq
# Combine upper and lower bound vectors
l = np.hstack([leq, lineq])
u = np.hstack([ueq, uineq])
# Set cost matrices
P = sparse.block_diag([sparse.kron(sparse.eye(self.N), self.Q), self.QN,
sparse.kron(sparse.eye(self.N), self.R)], format='csc')
q = np.hstack(
[-np.tile(np.diag(self.Q.A), self.N) * xr[:-self.nx],
-self.QN.dot(xr[-self.nx:]),
-np.tile(np.diag(self.R.A), self.N) * ur])
# Initialize optimizer
self.optimizer = osqp.OSQP()
self.optimizer.setup(P=P, q=q, A=A, l=l, u=u, verbose=False)
def get_control(self):
"""
Get control signal given the current position of the car. Solves a
finite time optimization problem based on the linearized car model.
"""
# Number of state variables
nx = self.model.n_states
nu = 2
# Update current waypoint
self.model.get_current_waypoint()
# Update spatial state
self.model.spatial_state = self.model.t2s(reference_state=
self.model.temporal_state, reference_waypoint=
self.model.current_waypoint)
# Initialize optimization problem
self._init_problem()
# Solve optimization problem
dec = self.optimizer.solve()
try:
# Get control signals
control_signals = np.array(dec.x[-self.N*nu:])
control_signals[1::2] = np.arctan(control_signals[1::2] *
self.model.length)
v = control_signals[0]
delta = control_signals[1]
# Update control signals
self.current_control = control_signals
# Get predicted spatial states
x = np.reshape(dec.x[:(self.N+1)*nx], (self.N+1, nx))
# Update predicted temporal states
self.current_prediction = self.update_prediction(x)
# Get current control signal
u = np.array([v, delta])
# if problem solved, reset infeasibility counter
self.infeasibility_counter = 0
except:
print('Infeasible problem. Previously predicted'
' control signal used!')
id = nu * (self.infeasibility_counter + 1)
u = np.array(self.current_control[id:id+2])
# increase infeasibility counter
self.infeasibility_counter += 1
if self.infeasibility_counter == (self.N - 1):
print('No control signal computed!')
exit(1)
return u
def update_prediction(self, spatial_state_prediction):
"""
Transform the predicted states to predicted x and y coordinates.
Mainly for visualization purposes.
:param spatial_state_prediction: list of predicted state variables
:return: lists of predicted x and y coordinates
"""
# Containers for x and y coordinates of predicted states
x_pred, y_pred = [], []
# Iterate over prediction horizon
for n in range(2, self.N):
# Get associated waypoint
associated_waypoint = self.model.reference_path.\
get_waypoint(self.model.wp_id+n)
# Transform predicted spatial state to temporal state
predicted_temporal_state = self.model.s2t(associated_waypoint,
spatial_state_prediction[n, :])
# Save predicted coordinates in world coordinate frame
x_pred.append(predicted_temporal_state.x)
y_pred.append(predicted_temporal_state.y)
return x_pred, y_pred
def show_prediction(self):
"""
Display predicted car trajectory in current axis.
"""
if self.current_prediction is not None:
plt.scatter(self.current_prediction[0], self.current_prediction[1],
c=PREDICTION, s=30)