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# BSD 3-Clause License
#
# Copyright (c) 2022 University of Wisconsin - Madison
# All rights reserved.
#
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#
# * Redistributions of source code must retain the above copyright notice, this
# list of conditions and the following disclaimer.
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# * Redistributions in binary form must reproduce the above copyright notice,
# this list of conditions and the following disclaimer in the documentation
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# * Neither the name of the copyright holder nor the names of its
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# this software without specific prior written permission.
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# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
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# FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
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#// =============================================================================
#// Author: Harry Zhang
#// =============================================================================
import osqp
import numpy as np
import scipy as sp
from scipy import sparse
import time
def mpc_wpts_solver(e,u,vel,vel_ref):
"""This function is the mpc solver for error based tracking problem
Use: [throttle, steering] = mpc_wpts_solver(error state, current control input, current velocity of the vehicle, reference velocity)
"""
x0 = np.array(e)
u0 = np.array(u)
# parameters related to the vehicle dynamics
r_wheel = 0.08451952624
i_wheel = 1e-3
gamma = 1/3
tau_0 = 0.09
omega_0 = 161.185
c1 = 1e-4
l_car = 0.5
nsim = 1
delta_t = 0.1
start_time = time.time()
for i in range(nsim):
#time dependent variables in Ad and Bd matrices
alpha = u0[0]
delta = u0[1] * 0.6
e1 = x0[0]
e2 = x0[1]
e3 = x0[2]
#reference/target point
xr = np.array([0, 0, 0.,0])
# Discrete time model of a vehicle
Ad = sparse.csc_matrix([
[1.0, vel*np.tan(delta)/l_car*delta_t, -vel_ref*np.sin(e3)*delta_t, 0.],
[-vel*np.tan(delta)/l_car*delta_t, 1.0, vel_ref*np.cos(e3)*delta_t, 0.],
[0., 0., 1.0, 0],
[0., 0., 0., 1.0-(c1*omega_0+tau_0/(i_wheel*omega_0))*delta_t],
])
#Ad = Ad * delta_t + sparse.eye(4)
Bd = sparse.csc_matrix([
[0., vel*e2/(l_car*np.cos(delta)*np.cos(delta))*delta_t],
[0., -vel*e1/(l_car*np.cos(delta)*np.cos(delta))*delta_t],
[0., -vel/(l_car*np.cos(delta)*np.cos(delta))*delta_t],
[-gamma*r_wheel*tau_0/i_wheel*delta_t, 0]
])
[nx, nu] = Bd.shape
# Constraints
#u0 = 10.5916
umin = np.array([0.0,-0.6])
umax = np.array([1, 0.6])
xmin = np.array([-np.inf,-np.inf,-np.inf,-np.inf])
xmax = np.array([np.inf, np.inf, np.inf,np.inf])
# xmin = np.array([-0.75,-0.75,-np.inf,-np.inf])
# xmax = np.array([0.75, 0.75, np.inf,np.inf])
# Objective function
Q = sparse.diags([0.0, 5500., 3500., 300.])
QN = Q
R = sparse.diags([10., 1000.])
# Prediction horizon
N = 5
# Cast MPC problem to a QP: x = (x(0),x(1),...,x(N),u(0),...,u(N-1))
# - quadratic objective
P = sparse.block_diag([sparse.kron(sparse.eye(N), Q), QN,
sparse.kron(sparse.eye(N), R)], format='csc')
# - linear objective
q = np.hstack([np.kron(np.ones(N), -Q.dot(xr)), -QN.dot(xr),
np.zeros(N*nu)])
# - linear dynamics
Ax = sparse.kron(sparse.eye(N+1),-sparse.eye(nx)) + sparse.kron(sparse.eye(N+1, k=-1), Ad)
Bu = sparse.kron(sparse.vstack([sparse.csc_matrix((1, N)), sparse.eye(N)]), Bd)
Aeq = sparse.hstack([Ax, Bu])
leq = np.hstack([-x0, np.zeros(N*nx)])
ueq = leq
# - input and state constraints
Aineq = sparse.eye((N+1)*nx + N*nu)
lineq = np.hstack([np.kron(np.ones(N+1), xmin), np.kron(np.ones(N), umin)])
uineq = np.hstack([np.kron(np.ones(N+1), xmax), np.kron(np.ones(N), umax)])
# - OSQP constraints
A = sparse.vstack([Aeq, Aineq], format='csc')
l = np.hstack([leq, lineq])
u = np.hstack([ueq, uineq])
# Create an OSQP object
prob = osqp.OSQP()
# Setup workspace
prob.setup(P, q, A, l, u, warm_start=True,max_iter=10000)
# Simulate in closed loop
# Solve
res = prob.solve()
# Check solver status
if res.info.status != 'solved':
raise ValueError('OSQP did not solve the problem!')
# Apply first control input to the plant
ctrl = res.x[-N*nu:-(N-1)*nu]
u0 = ctrl
x0 = Ad.dot(x0) + Bd.dot(ctrl)
# Update initial state
l[:nx] = -x0
u[:nx] = -x0
prob.update(l=l, u=u)
# print('For the error state: e=',e )
# print('Control input:',ctrl)
#print('Next predict state',x0)
#print(res.x)
#print("--- %s seconds ---" % (time.time() - start_time))
return ctrl |