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pb_lqr.py
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# Simulation for inverted pendulum
# Author: Yu Okamoto
import math
import numpy as np
import matplotlib as mpl
mpl.use('tkagg')
import matplotlib.pyplot as plt
import matplotlib.animation as animation
import scipy.linalg as spl
# LQR function
def lqr(A, B, Q, R):
# solve Riccati equation
X = spl.solve_continuous_are(A, B, Q, R)
# compute the LQR gain
K = spl.inv(R).dot(B.T).dot(X)
#eigVals, eigVecs = scipy.linalg.eig(A-B*K)
print "lqr gain ", K
return K
# Rotation array (matrix)
def R3x(rad):
return np.array([[1.0, 0.0, 0.0],
[0.0, np.cos(rad)[0], -np.sin(rad)[0]],
[0.0, np.sin(rad)[0], np.cos(rad)[0]]])
def R3y(rad):
return np.array([[np.cos(rad)[0], 0.0, np.sin(rad)[0]],
[0.0, 1.0, 0.0],
[-np.sin(rad)[0], 0.0, np.cos(rad)[0]]])
def R3z(rad):
return np.array([[np.cos(rad)[0], -np.sin(rad)[0], 0.0],
[np.sin(rad)[0], np.cos(rad)[0], 0.0],
[0.0, 0.0, 1.0]])
# parameters for simulation
dt = 0.02
simtime = 8.0
L = 0.77 # distance from base to c.o.m [m]
g = 9.81 # gravitational acceleration [m/s^2]
A = np.array([[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], # r
[g / L, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -g, 0], # r dot
[0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], # s
[0, 0, g / L, 0, 0, 0, 0, 0, 0, 0, g, 0, 0], # s dot
[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], # x
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, g, 0], # x dot
[0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], # y
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -g, 0, 0], # y dot
[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], # z
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], # z dot
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], # gamma
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], # beta
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]) # alpha
B = np.array([[0, 0, 0, 0], # u = [Wx,
[0, 0, 0, 0], # Wy,
[0, 0, 0, 0], # Wz,
[0, 0, 0, 0], # a]
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 1],
[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0]])
C = np.array([[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0]])
# initial state
x = np.array([[30*np.pi/180.0], # r
[0.0], # r dot
[0.0], # s
[0.0], # s dot
[0.0], # x
[0.0], # x dot
[0.0], # y
[0.0], # y dot
[1.0], # z
[0.0], # z dot
[0.0], # gamma
[0.0], # beta
[0.0]]) # alpha
# reference
y_ref = np.array([[0.0], # x [m]
[0.0], # y [m]
[1.0]]) # z [m]
# extend system for servo
As = np.hstack([np.vstack([A, -C]),
np.vstack([np.zeros([A.shape[0], C.shape[0]]), np.zeros([C.shape[0], C.shape[0]])])])
Bs = np.vstack([B, np.zeros([C.shape[0], B.shape[1]])])
Cs = np.hstack([C, np.zeros([y_ref.shape[0], C.shape[0]])])
Is = np.vstack([np.zeros([A.shape[0], y_ref.shape[0]]), np.identity(y_ref.shape[0])])
xs = np.vstack([x, np.zeros([y_ref.shape[0], 1])])
xs_init = xs
dxs = np.zeros([As.shape[0], 1])
# LQR
Q = np.diag([1.0, # r
1.0, # r_dot
1.0, # s
1.0, # s_dot
0.5, # x
0.5, # x_dot
0.5, # y
0.5, # y_dot
0.5, # z
0.5, # z_dot
1.0, # gamma
1.0, # beta
1.0, # alpha
1.0, # delta x
1.0, # delta y
0.1 # delta z
])
R = np.diag([1.0,
1.0,
1.0,
0.001])
K = lqr(As, Bs, Q, R)
# variables for simulation and plot
t = np.arange(0, simtime + dt / 100, dt)
xp = np.zeros([int(simtime / dt) + 1, As.shape[0]])
up = np.zeros([int(simtime / dt) + 1, 4])
dxp = np.zeros([int(simtime / dt) + 1, As.shape[0]])
# zp = np.zeros([int(simtime / dt) + 1, 1])
# Store initial xs and dxs (t = 0)
xp[0, :] = np.transpose(xs)
dxp[0, :] = np.transpose(dxs)
up[0, :] = 0
# zp[0] = math.sqrt(math.pow(L, 2) - math.pow(xs[0], 2) - math.pow(xs[2], 2))
y = Cs.dot(xs)
fig = plt.figure(figsize=(6, 6))
ax = fig.add_subplot(111, autoscale_on=False, xlim=(-4, 4), ylim=(-4.0, 4.0))
ax.grid()
line_quad, = ax.plot([], [], 'b-', lw=1)
line_pole, = ax.plot([], [], 'ro-', lw=3)
time_template = 'time = %.1fs'
time_text = ax.text(0.05, 0.9, '', transform=ax.transAxes)
def init():
line_quad.set_data([], [])
line_pole.set_data([], [])
time_text.set_text('')
global xs
xs = xs_init
return line_quad, line_pole, time_text
plotted = False
# simulation loop
def sim_loop(i):
global xs, dxs, y, xp, dxp, zp, up, plotted
# calc input from lqr gain
xse = xs.copy()
xse[4] -= y_ref[0]
xse[6] -= y_ref[1]
xse[8] -= y_ref[2]
u = -K.dot(xse) # [Wx, Wy, Wz, a]
# creates variables for the xs(t-1) so it is easier to code
r = xs[0]
r_dot = xs[1]
s = xs[2]
s_dot = xs[3]
x = xs[4]
x_dot = xs[5]
y = xs[6]
y_dot = xs[7]
z = xs[8]
z_dot = xs[9]
gamma = xs[10]
beta = xs[11]
alpha = xs[12]
# create variables for dxs
r_dd = dxs[1] # r double dot
s_dd = dxs[3] # s double dot
# create variables for inputs
Wx = u[0]
Wy = u[1]
Wz = u[2]
# gravity copensation
Rov = (R3z(alpha).dot(R3y(beta))).dot(R3x(gamma)) # rotation matrix from o to v
g_acc = np.array([[0],
[0],
[-g]])
# print R3z(alpha)
# print R3z(alpha).dot(R3y(beta))
# print Rov
Rinv = np.linalg.inv(Rov)
# print Rinv.dot(-g_acc)
u[3] += np.linalg.norm(Rinv.dot(-g_acc))
a = u[3]
# update state equation
## x_dd, y_dd, z_dd
acc = np.array([[0],
[0],
[a]])
trans_dd = Rov.dot(acc) + g_acc
# print trans_dd, plotted
# create variables for translational double dot so it is easier to code
x_dd = trans_dd[0] # x double dot
y_dd = trans_dd[1] # y double dot
z_dd = trans_dd[2] # z double dot
omega_vector = np.array([Wx,
Wy,
Wz])
Rot = np.array([[1, np.sin(beta) * np.sin(gamma) / np.cos(beta), np.sin(beta) * np.cos(gamma) / np.cos(beta)],
[0, np.cos(gamma), -np.sin(gamma)],
[0, np.sin(gamma) / np.cos(beta), np.cos(gamma) / np.cos(beta)]])
angle_dot = Rot.dot(omega_vector)
# Create temporary buffer to store the nonlinear A.dot(xs) part
dxs_temp = np.zeros([As.shape[0], 1])
# dxs_temp = np.zeros([As.shape[0],1])
# print math.pow(L, 2) - math.pow(r, 2) - math.pow(s, 2)
zeta = math.sqrt(math.pow(L, 2) - math.pow(r, 2) - math.pow(s, 2))
dxs_temp[0] = r_dot # r_dot
dxs_temp[2] = s_dot # s_dot
dxs_temp[4] = x_dot # x_dot
dxs_temp[6] = y_dot # y_dot
dxs_temp[8] = z_dot # z_dot
dxs_temp[10] = angle_dot[0] # gamma_dot
dxs_temp[11] = angle_dot[1] # beta_dot
dxs_temp[12] = angle_dot[2] # alpha_dot
dxs_temp[5] = x_dd # x_dd
dxs_temp[7] = y_dd # y_dd
dxs_temp[9] = z_dd # z_dd
# r_dd
dxs_temp[1] = 1 / ((L * L - s * s) * zeta * zeta) * (-math.pow(r, 4) * x_dd
- math.pow((L * L - s * s), 2) * x_dd
- 2 * r * r *
(s * r_dot * s_dot + (-L * L + s * s) * x_dd)
+ math.pow(r, 3) * (s_dot * s_dot +
s * s_dd - zeta * (g + z_dd))
+ r * (-L * L * s * s_dd
+ math.pow(s, 3) * s_dd
+ s * s *
(r_dot * r_dot - zeta * (g + z_dd))
+ L * L * (-r_dot * r_dot - s_dot * s_dot + zeta * (g + z_dd))))
# s_dd
dxs_temp[3] = 1 / ((L * L - r * r) * zeta * zeta) * (-math.pow(s, 4) * y_dd
- math.pow((L * L - r * r), 2) * y_dd
- 2 * s * s *
(r * r_dot * s_dot + (-L * L + r * r) * y_dd)
+ math.pow(s, 3) * (r_dot * r_dot +
r * r_dd - zeta * (g + z_dd))
+ s * (-L * L * r * r_dd
+ math.pow(r, 3) * r_dd
+ r * r *
(s_dot * s_dot - zeta * (g + z_dd))
+ L * L * (-r_dot * r_dot - s_dot * s_dot + zeta * (g + z_dd))))
# Create vector for error states
error = - np.vstack([ np.zeros([ A.shape[0],1 ]), Cs.dot(xs) ]) + Is.dot(y_ref)
# print -error
# Update
dxs = dxs_temp + error
xs = xs + dxs * dt
y = Cs.dot(xs)
# record for plot
xp[i] = np.transpose(xs)
dxp[i] = np.transpose(dxs)
up[i] = np.transpose(u)
# zp[i] = zeta
# draw animation
#right
lc = L*np.cos(beta)
ls = L*np.sin(beta)
tx = [ x, x+0.5*lc ]
ty = [ z, z-0.5*ls ]
tx = np.vstack([tx, tx[-1] + 0.25*ls ])
ty = np.vstack([ty, ty[-1] + 0.25*lc ])
tx = np.vstack([tx, tx[-1] + 0.125*lc ])
ty = np.vstack([ty, ty[-1] - 0.125*ls ])
tx = np.vstack([tx, tx[-2] - 0.125*lc ])
ty = np.vstack([ty, ty[-2] + 0.125*ls ])
#back to center
tx = np.vstack([tx, tx[-3] ])
ty = np.vstack([ty, ty[-3] ])
tx = np.vstack([tx, tx[-5] ])
ty = np.vstack([ty, ty[-5] ])
#left
tx = np.vstack([tx, tx[0] - 0.5*lc ])
ty = np.vstack([ty, ty[0] + 0.5*ls ])
tx = np.vstack([tx, tx[-1] + 0.25*ls ])
ty = np.vstack([ty, ty[-1] + 0.25*lc ])
tx = np.vstack([tx, tx[-1] - 0.125*lc ])
ty = np.vstack([ty, ty[-1] + 0.125*ls ])
tx = np.vstack([tx, tx[-2] + 0.125*lc ])
ty = np.vstack([ty, ty[-2] - 0.125*ls ])
#back to center
tx = np.vstack([tx, tx[-3] ])
ty = np.vstack([ty, ty[-3] ])
tx = np.vstack([tx, tx[-5] ])
ty = np.vstack([ty, ty[-5] ])
line_quad.set_data(tx, ty)
tx = [x, x+2.0*L*np.sin(xs[0]) ]
ty = [z, z+2.0*L*np.cos(xs[0]) ]
line_pole.set_data(tx, ty)
time_text.set_text(time_template % (i*dt))
if i==len(xp[:,0])-1 and plotted != True:
plt.figure()
# plt.rcParams["font.family"] = "Times New Roman"
plt.subplot(3, 1, 1)
plt.plot(t, xp[:, 4])
plt.plot(t, xp[:, 6])
plt.plot(t, xp[:, 8])
plt.plot([t[0],t[-1]], [y_ref[0], y_ref[0]], 'r:')
plt.legend([r'$x$',r'$x_{\rm{ref}}$'])
plt.ylabel(r'$x \rm{[m]}$')
plt.subplot(3, 1, 2)
plt.plot(t, xp[:, 0]*180/np.pi)
plt.plot([t[0],t[-1]], [y_ref[1], y_ref[1]], 'r:')
plt.legend([r'$\theta$',r'$\theta_{\rm{ref}}$'], loc='upper right')
plt.ylabel(r'$\theta$ [deg]')
plt.subplot(3, 1, 3)
plt.plot(t, up)
plt.ylabel(r'$f \rm{[N]}$')
plt.xlabel(r'$t \rm{[s]}$')
plotted = True
plt.show(block=False)
return line_quad, line_pole, time_text
ani = animation.FuncAnimation(fig, sim_loop, frames=np.arange(0, len(xp[:,0])),
interval=100, blit=True, init_func=init)
plt.show()