mpc_python_learn/mpc_demo/cvxpy_mpc.py

import numpy as np
np.seterr(divide='ignore', invalid='ignore')

from scipy.integrate import odeint
from scipy.interpolate import interp1d
import cvxpy as cp

from utils import road_curve, f, df

from mpc_config import Params
P=Params()

def get_linear_model(x_bar,u_bar):
    """
    """
    L=0.3

    x = x_bar[0]
    y = x_bar[1]
    v = x_bar[2]
    theta = x_bar[3]

    a = u_bar[0]
    delta = u_bar[1]

    A = np.zeros((P.N,P.N))
    A[0,2]=np.cos(theta)
    A[0,3]=-v*np.sin(theta)
    A[1,2]=np.sin(theta)
    A[1,3]=v*np.cos(theta)
    A[3,2]=v*np.tan(delta)/L
    A_lin=np.eye(P.N)+P.dt*A

    B = np.zeros((P.N,P.M))
    B[2,0]=1
    B[3,1]=v/(L*np.cos(delta)**2)
    B_lin=P.dt*B

    f_xu=np.array([v*np.cos(theta), v*np.sin(theta), a,v*np.tan(delta)/L]).reshape(P.N,1)
    C_lin = P.dt*(f_xu - np.dot(A,x_bar.reshape(P.N,1)) - np.dot(B,u_bar.reshape(P.M,1)))

    return np.round(A_lin,4), np.round(B_lin,4), np.round(C_lin,4)


def optimize(state,u_bar,track,ref_vel=1.):
    '''
    :param state:
    :param u_bar:
    :param track:
    :returns:
    '''

    MAX_SPEED = 1.25
    MAX_STEER = 1.57/2
    MAX_ACC = 1.0

    # compute polynomial coefficients of the track
    K=road_curve(state,track)

    # dynamics starting state w.r.t vehicle frame
    x_bar=np.zeros((P.N,P.T+1))
    x_bar[2,0]=state[2]

    #prediction for linearization of costrains
    for t in range (1,P.T+1):
        xt=x_bar[:,t-1].reshape(P.N,1)
        ut=u_bar[:,t-1].reshape(P.M,1)
        A,B,C=get_linear_model(xt,ut)
        xt_plus_one = np.squeeze(np.dot(A,xt)+np.dot(B,ut)+C)
        x_bar[:,t]= xt_plus_one

    #CVXPY Linear MPC problem statement
    cost = 0
    constr = []
    x = cp.Variable((P.N, P.T+1))
    u = cp.Variable((P.M, P.T))

    for t in range(P.T):

        cost += 30*cp.sum_squares(x[3,t]-np.arctan(df(x_bar[0,t],K))) # psi
        cost += 20*cp.sum_squares(f(x_bar[0,t],K)-x[1,t]) # cte
        cost += 10*cp.sum_squares(ref_vel-x[2,t]) # desired v

        # Actuation rate of change
        if t < (P.T - 1):
            cost += cp.quad_form(u[:, t + 1] - u[:, t], 10*np.eye(P.M))

        # Actuation effort
        cost += cp.quad_form( u[:, t],10*np.eye(P.M))

        # Kinrmatics Constrains (Linearized model)
        A,B,C=get_linear_model(x_bar[:,t],u_bar[:,t])
        constr += [x[:,t+1] == A@x[:,t] + B@u[:,t] + C.flatten()]

    # sums problem objectives and concatenates constraints.
    constr += [x[:,0] == x_bar[:,0]] #<--watch out the start condition
    constr += [x[2, :] <= MAX_SPEED]
    constr += [x[2, :] >= 0.0]
    constr += [cp.abs(u[0, :]) <= MAX_ACC]
    constr += [cp.abs(u[1, :]) <= MAX_STEER]

    # Solve
    prob = cp.Problem(cp.Minimize(cost), constr)
    solution = prob.solve(solver=cp.OSQP, verbose=False)

    #retrieved optimized U and assign to u_bar to linearize in next step
    u_bar=np.vstack((np.array(u.value[0, :]).flatten(),
                    (np.array(u.value[1, :]).flatten())))

    return u_bar