mirror of
https://github.com/AtsushiSakai/PythonRobotics.git
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318 lines
7.9 KiB
Python
318 lines
7.9 KiB
Python
"""
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Graph based SLAM example
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author: Atsushi Sakai (@Atsushi_twi)
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"""
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import numpy as np
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import math
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import copy
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import itertools
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import matplotlib.pyplot as plt
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# Simulation parameter
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Qsim = np.diag([0.2, math.radians(1.0)])**2
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Rsim = np.diag([0.1, math.radians(10.0)])**2
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DT = 2.0 # time tick [s]
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SIM_TIME = 100.0 # simulation time [s]
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MAX_RANGE = 30.0 # maximum observation range
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STATE_SIZE = 3 # State size [x,y,yaw]
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# Covariance parameter of Graph Based SLAM
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C_SIGMA1 = 0.1
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C_SIGMA2 = 0.1
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C_SIGMA3 = math.radians(1.0)
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MAX_ITR = 20 # Maximum iteration
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show_graph_dtime = 20.0 # [s]
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show_animation = True
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class Edge():
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def __init__(self):
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self.e = np.zeros((3, 1))
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self.omega = np.zeros((3, 3)) # information matrix
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self.d1 = 0.0
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self.d2 = 0.0
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self.yaw1 = 0.0
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self.yaw2 = 0.0
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self.angle1 = 0.0
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self.angle2 = 0.0
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self.id1 = 0
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self.id2 = 0
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def cal_observation_sigma(d):
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sigma = np.zeros((3, 3))
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sigma[0, 0] = C_SIGMA1**2
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sigma[1, 1] = C_SIGMA2**2
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sigma[2, 2] = C_SIGMA3**2
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return sigma
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def calc_rotational_matrix(angle):
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Rt = np.matrix([[math.cos(angle), -math.sin(angle), 0],
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[math.sin(angle), math.cos(angle), 0],
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[0, 0, 1.0]])
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return Rt
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def calc_edge(x1, y1, yaw1, x2, y2, yaw2, d1,
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angle1, phi1, d2, angle2, phi2, t1, t2):
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edge = Edge()
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tangle1 = pi_2_pi(yaw1 + angle1)
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tangle2 = pi_2_pi(yaw2 + angle2)
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tmp1 = d1 * math.cos(tangle1)
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tmp2 = d2 * math.cos(tangle2)
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tmp3 = d1 * math.sin(tangle1)
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tmp4 = d2 * math.sin(tangle2)
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edge.e[0, 0] = x2 - x1 - tmp1 + tmp2
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edge.e[1, 0] = y2 - y1 - tmp3 + tmp4
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hyaw = phi1 - phi2 + angle1 - angle2
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edge.e[2, 0] = pi_2_pi(yaw2 - yaw1 - hyaw)
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Rt1 = calc_rotational_matrix(tangle1)
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Rt2 = calc_rotational_matrix(tangle2)
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sig1 = cal_observation_sigma(d1)
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sig2 = cal_observation_sigma(d2)
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edge.omega = np.linalg.inv(Rt1 * sig1 * Rt1.T + Rt2 * sig2 * Rt2.T)
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edge.d1, edge.d2 = d1, d2
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edge.yaw1, edge.yaw2 = yaw1, yaw2
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edge.angle1, edge.angle2 = angle1, angle2
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edge.id1, edge.id2 = t1, t2
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return edge
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def calc_edges(xlist, zlist):
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edges = []
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cost = 0.0
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zids = list(itertools.combinations(range(len(zlist)), 2))
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for (t1, t2) in zids:
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x1, y1, yaw1 = xlist[0, t1], xlist[1, t1], xlist[2, t1]
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x2, y2, yaw2 = xlist[0, t2], xlist[1, t2], xlist[2, t2]
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if zlist[t1] is None or zlist[t2] is None:
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continue # No observation
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for iz1 in range(len(zlist[t1][:, 0])):
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for iz2 in range(len(zlist[t2][:, 0])):
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if zlist[t1][iz1, 3] == zlist[t2][iz2, 3]:
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d1 = zlist[t1][iz1, 0]
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angle1, phi1 = zlist[t1][iz1, 1], zlist[t1][iz1, 2]
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d2 = zlist[t2][iz2, 0]
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angle2, phi2 = zlist[t2][iz2, 1], zlist[t2][iz2, 2]
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edge = calc_edge(x1, y1, yaw1, x2, y2, yaw2, d1,
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angle1, phi1, d2, angle2, phi2, t1, t2)
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edges.append(edge)
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cost += (edge.e.T * edge.omega * edge.e)[0, 0]
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print("cost:", cost, ",nedge:", len(edges))
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return edges
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def calc_jacobian(edge):
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t1 = edge.yaw1 + edge.angle1
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A = np.matrix([[-1.0, 0, edge.d1 * math.sin(t1)],
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[0, -1.0, -edge.d1 * math.cos(t1)],
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[0, 0, -1.0]])
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t2 = edge.yaw2 + edge.angle2
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B = np.matrix([[1.0, 0, -edge.d2 * math.sin(t2)],
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[0, 1.0, edge.d2 * math.cos(t2)],
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[0, 0, 1.0]])
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return A, B
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def fill_H_and_b(H, b, edge):
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A, B = calc_jacobian(edge)
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id1 = edge.id1 * STATE_SIZE
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id2 = edge.id2 * STATE_SIZE
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H[id1:id1 + STATE_SIZE, id1:id1 + STATE_SIZE] += A.T * edge.omega * A
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H[id1:id1 + STATE_SIZE, id2:id2 + STATE_SIZE] += A.T * edge.omega * B
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H[id2:id2 + STATE_SIZE, id1:id1 + STATE_SIZE] += B.T * edge.omega * A
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H[id2:id2 + STATE_SIZE, id2:id2 + STATE_SIZE] += B.T * edge.omega * B
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b[id1:id1 + STATE_SIZE, 0] += (A.T * edge.omega * edge.e)
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b[id2:id2 + STATE_SIZE, 0] += (B.T * edge.omega * edge.e)
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return H, b
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def graph_based_slam(x_init, hz):
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print("start graph based slam")
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zlist = copy.deepcopy(hz)
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zlist.insert(1, zlist[0])
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x_opt = copy.deepcopy(x_init)
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nt = x_opt.shape[1]
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n = nt * STATE_SIZE
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for itr in range(MAX_ITR):
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edges = calc_edges(x_opt, zlist)
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H = np.matrix(np.zeros((n, n)))
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b = np.matrix(np.zeros((n, 1)))
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for edge in edges:
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H, b = fill_H_and_b(H, b, edge)
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# to fix origin
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H[0:STATE_SIZE, 0:STATE_SIZE] += np.identity(STATE_SIZE)
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dx = - np.linalg.inv(H).dot(b)
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for i in range(nt):
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x_opt[0:3, i] += dx[i * 3:i * 3 + 3, 0]
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diff = dx.T.dot(dx)
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print("iteration: %d, diff: %f" % (itr + 1, diff))
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if diff < 1.0e-5:
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break
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return x_opt
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def calc_input():
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v = 1.0 # [m/s]
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yawrate = 0.1 # [rad/s]
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u = np.matrix([v, yawrate]).T
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return u
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def observation(xTrue, xd, u, RFID):
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xTrue = motion_model(xTrue, u)
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# add noise to gps x-y
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z = np.matrix(np.zeros((0, 4)))
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for i in range(len(RFID[:, 0])):
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dx = RFID[i, 0] - xTrue[0, 0]
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dy = RFID[i, 1] - xTrue[1, 0]
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d = math.sqrt(dx**2 + dy**2)
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angle = pi_2_pi(math.atan2(dy, dx)) - xTrue[2, 0]
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phi = pi_2_pi(math.atan2(dy, dx))
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if d <= MAX_RANGE:
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dn = d + np.random.randn() * Qsim[0, 0] # add noise
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anglen = angle + np.random.randn() * Qsim[1, 1] # add noise
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zi = np.matrix([dn, anglen, phi, i])
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z = np.vstack((z, zi))
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# add noise to input
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ud1 = u[0, 0] + np.random.randn() * Rsim[0, 0]
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ud2 = u[1, 0] + np.random.randn() * Rsim[1, 1]
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ud = np.matrix([ud1, ud2]).T
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xd = motion_model(xd, ud)
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return xTrue, z, xd, ud
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def motion_model(x, u):
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F = np.matrix([[1.0, 0, 0],
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[0, 1.0, 0],
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[0, 0, 1.0]])
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B = np.matrix([[DT * math.cos(x[2, 0]), 0],
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[DT * math.sin(x[2, 0]), 0],
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[0.0, DT]])
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x = F * x + B * u
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return x
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def pi_2_pi(angle):
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while(angle > math.pi):
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angle = angle - 2.0 * math.pi
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while(angle < -math.pi):
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angle = angle + 2.0 * math.pi
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return angle
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def main():
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print(__file__ + " start!!")
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time = 0.0
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# RFID positions [x, y, yaw]
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RFID = np.array([[10.0, -2.0, 0.0],
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[15.0, 10.0, 0.0],
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[3.0, 15.0, 0.0],
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[-5.0, 20.0, 0.0],
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[-5.0, 5.0, 0.0]
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])
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# State Vector [x y yaw v]'
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xTrue = np.matrix(np.zeros((STATE_SIZE, 1)))
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xDR = np.matrix(np.zeros((STATE_SIZE, 1))) # Dead reckoning
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# history
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hxTrue = xTrue
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hxDR = xTrue
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hz = []
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dtime = 0.0
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while SIM_TIME >= time:
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time += DT
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dtime += DT
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u = calc_input()
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xTrue, z, xDR, ud = observation(xTrue, xDR, u, RFID)
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hxDR = np.hstack((hxDR, xDR))
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hxTrue = np.hstack((hxTrue, xTrue))
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hz.append(z)
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if dtime >= show_graph_dtime:
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x_opt = graph_based_slam(hxDR, hz)
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dtime = 0.0
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if show_animation:
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plt.cla()
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plt.plot(RFID[:, 0], RFID[:, 1], "*k")
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plt.plot(np.array(hxTrue[0, :]).flatten(),
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np.array(hxTrue[1, :]).flatten(), "-b")
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plt.plot(np.array(hxDR[0, :]).flatten(),
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np.array(hxDR[1, :]).flatten(), "-k")
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plt.plot(np.array(x_opt[0, :]).flatten(),
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np.array(x_opt[1, :]).flatten(), "-r")
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plt.axis("equal")
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plt.grid(True)
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plt.title("Time" + str(time)[0:5])
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plt.pause(1.0)
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if __name__ == '__main__':
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main()
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