""" Graph based SLAM example author: Atsushi Sakai (@Atsushi_twi) Ref [A Tutorial on Graph-Based SLAM](http://www2.informatik.uni-freiburg.de/~stachnis/pdf/grisetti10titsmag.pdf) """ import numpy as np import math import copy import itertools import matplotlib.pyplot as plt # Simulation parameter Qsim = np.diag([0.2, np.deg2rad(1.0)])**2 Rsim = np.diag([0.1, np.deg2rad(10.0)])**2 DT = 2.0 # time tick [s] SIM_TIME = 100.0 # simulation time [s] MAX_RANGE = 30.0 # maximum observation range STATE_SIZE = 3 # State size [x,y,yaw] # Covariance parameter of Graph Based SLAM C_SIGMA1 = 0.1 C_SIGMA2 = 0.1 C_SIGMA3 = np.deg2rad(1.0) MAX_ITR = 20 # Maximum iteration show_graph_dtime = 20.0 # [s] show_animation = True class Edge(): def __init__(self): self.e = np.zeros((3, 1)) self.omega = np.zeros((3, 3)) # information matrix self.d1 = 0.0 self.d2 = 0.0 self.yaw1 = 0.0 self.yaw2 = 0.0 self.angle1 = 0.0 self.angle2 = 0.0 self.id1 = 0 self.id2 = 0 def cal_observation_sigma(d): sigma = np.zeros((3, 3)) sigma[0, 0] = C_SIGMA1**2 sigma[1, 1] = C_SIGMA2**2 sigma[2, 2] = C_SIGMA3**2 return sigma def calc_rotational_matrix(angle): Rt = np.matrix([[math.cos(angle), -math.sin(angle), 0], [math.sin(angle), math.cos(angle), 0], [0, 0, 1.0]]) return Rt def calc_edge(x1, y1, yaw1, x2, y2, yaw2, d1, angle1, phi1, d2, angle2, phi2, t1, t2): edge = Edge() tangle1 = pi_2_pi(yaw1 + angle1) tangle2 = pi_2_pi(yaw2 + angle2) tmp1 = d1 * math.cos(tangle1) tmp2 = d2 * math.cos(tangle2) tmp3 = d1 * math.sin(tangle1) tmp4 = d2 * math.sin(tangle2) edge.e[0, 0] = x2 - x1 - tmp1 + tmp2 edge.e[1, 0] = y2 - y1 - tmp3 + tmp4 hyaw = phi1 - phi2 + angle1 - angle2 edge.e[2, 0] = pi_2_pi(yaw2 - yaw1 - hyaw) Rt1 = calc_rotational_matrix(tangle1) Rt2 = calc_rotational_matrix(tangle2) sig1 = cal_observation_sigma(d1) sig2 = cal_observation_sigma(d2) edge.omega = np.linalg.inv(Rt1 * sig1 * Rt1.T + Rt2 * sig2 * Rt2.T) edge.d1, edge.d2 = d1, d2 edge.yaw1, edge.yaw2 = yaw1, yaw2 edge.angle1, edge.angle2 = angle1, angle2 edge.id1, edge.id2 = t1, t2 return edge def calc_edges(xlist, zlist): edges = [] cost = 0.0 zids = list(itertools.combinations(range(len(zlist)), 2)) for (t1, t2) in zids: x1, y1, yaw1 = xlist[0, t1], xlist[1, t1], xlist[2, t1] x2, y2, yaw2 = xlist[0, t2], xlist[1, t2], xlist[2, t2] if zlist[t1] is None or zlist[t2] is None: continue # No observation for iz1 in range(len(zlist[t1][:, 0])): for iz2 in range(len(zlist[t2][:, 0])): if zlist[t1][iz1, 3] == zlist[t2][iz2, 3]: d1 = zlist[t1][iz1, 0] angle1, phi1 = zlist[t1][iz1, 1], zlist[t1][iz1, 2] d2 = zlist[t2][iz2, 0] angle2, phi2 = zlist[t2][iz2, 1], zlist[t2][iz2, 2] edge = calc_edge(x1, y1, yaw1, x2, y2, yaw2, d1, angle1, phi1, d2, angle2, phi2, t1, t2) edges.append(edge) cost += (edge.e.T * edge.omega * edge.e)[0, 0] print("cost:", cost, ",nedge:", len(edges)) return edges def calc_jacobian(edge): t1 = edge.yaw1 + edge.angle1 A = np.matrix([[-1.0, 0, edge.d1 * math.sin(t1)], [0, -1.0, -edge.d1 * math.cos(t1)], [0, 0, -1.0]]) t2 = edge.yaw2 + edge.angle2 B = np.matrix([[1.0, 0, -edge.d2 * math.sin(t2)], [0, 1.0, edge.d2 * math.cos(t2)], [0, 0, 1.0]]) return A, B def fill_H_and_b(H, b, edge): A, B = calc_jacobian(edge) id1 = edge.id1 * STATE_SIZE id2 = edge.id2 * STATE_SIZE H[id1:id1 + STATE_SIZE, id1:id1 + STATE_SIZE] += A.T * edge.omega * A H[id1:id1 + STATE_SIZE, id2:id2 + STATE_SIZE] += A.T * edge.omega * B H[id2:id2 + STATE_SIZE, id1:id1 + STATE_SIZE] += B.T * edge.omega * A H[id2:id2 + STATE_SIZE, id2:id2 + STATE_SIZE] += B.T * edge.omega * B b[id1:id1 + STATE_SIZE, 0] += (A.T * edge.omega * edge.e) b[id2:id2 + STATE_SIZE, 0] += (B.T * edge.omega * edge.e) return H, b def graph_based_slam(x_init, hz): print("start graph based slam") zlist = copy.deepcopy(hz) zlist.insert(1, zlist[0]) x_opt = copy.deepcopy(x_init) nt = x_opt.shape[1] n = nt * STATE_SIZE for itr in range(MAX_ITR): edges = calc_edges(x_opt, zlist) H = np.matrix(np.zeros((n, n))) b = np.matrix(np.zeros((n, 1))) for edge in edges: H, b = fill_H_and_b(H, b, edge) # to fix origin H[0:STATE_SIZE, 0:STATE_SIZE] += np.identity(STATE_SIZE) dx = - np.linalg.inv(H).dot(b) for i in range(nt): x_opt[0:3, i] += dx[i * 3:i * 3 + 3, 0] diff = dx.T.dot(dx) print("iteration: %d, diff: %f" % (itr + 1, diff)) if diff < 1.0e-5: break return x_opt def calc_input(): v = 1.0 # [m/s] yawrate = 0.1 # [rad/s] u = np.matrix([v, yawrate]).T return u def observation(xTrue, xd, u, RFID): xTrue = motion_model(xTrue, u) # add noise to gps x-y z = np.matrix(np.zeros((0, 4))) for i in range(len(RFID[:, 0])): dx = RFID[i, 0] - xTrue[0, 0] dy = RFID[i, 1] - xTrue[1, 0] d = math.sqrt(dx**2 + dy**2) angle = pi_2_pi(math.atan2(dy, dx)) - xTrue[2, 0] phi = pi_2_pi(math.atan2(dy, dx)) if d <= MAX_RANGE: dn = d + np.random.randn() * Qsim[0, 0] # add noise anglen = angle + np.random.randn() * Qsim[1, 1] # add noise zi = np.matrix([dn, anglen, phi, i]) z = np.vstack((z, zi)) # add noise to input ud1 = u[0, 0] + np.random.randn() * Rsim[0, 0] ud2 = u[1, 0] + np.random.randn() * Rsim[1, 1] ud = np.matrix([ud1, ud2]).T xd = motion_model(xd, ud) return xTrue, z, xd, ud def motion_model(x, u): F = np.matrix([[1.0, 0, 0], [0, 1.0, 0], [0, 0, 1.0]]) B = np.matrix([[DT * math.cos(x[2, 0]), 0], [DT * math.sin(x[2, 0]), 0], [0.0, DT]]) x = F * x + B * u return x def pi_2_pi(angle): return (angle + math.pi) % (2 * math.pi) - math.pi def main(): print(__file__ + " start!!") time = 0.0 # RFID positions [x, y, yaw] RFID = np.array([[10.0, -2.0, 0.0], [15.0, 10.0, 0.0], [3.0, 15.0, 0.0], [-5.0, 20.0, 0.0], [-5.0, 5.0, 0.0] ]) # State Vector [x y yaw v]' xTrue = np.matrix(np.zeros((STATE_SIZE, 1))) xDR = np.matrix(np.zeros((STATE_SIZE, 1))) # Dead reckoning # history hxTrue = xTrue hxDR = xTrue hz = [] dtime = 0.0 while SIM_TIME >= time: time += DT dtime += DT u = calc_input() xTrue, z, xDR, ud = observation(xTrue, xDR, u, RFID) hxDR = np.hstack((hxDR, xDR)) hxTrue = np.hstack((hxTrue, xTrue)) hz.append(z) if dtime >= show_graph_dtime: x_opt = graph_based_slam(hxDR, hz) dtime = 0.0 if show_animation: plt.cla() plt.plot(RFID[:, 0], RFID[:, 1], "*k") plt.plot(np.array(hxTrue[0, :]).flatten(), np.array(hxTrue[1, :]).flatten(), "-b") plt.plot(np.array(hxDR[0, :]).flatten(), np.array(hxDR[1, :]).flatten(), "-k") plt.plot(np.array(x_opt[0, :]).flatten(), np.array(x_opt[1, :]).flatten(), "-r") plt.axis("equal") plt.grid(True) plt.title("Time" + str(time)[0:5]) plt.pause(1.0) if __name__ == '__main__': main()