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1 % This is like robot1, except we only use a Kalman filter.
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2 % The goal is to study how the precision matrix changes.
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3
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4 seed = 0;
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5 rand('state', seed);
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6 randn('state', seed);
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7
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8 if 0
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9 T = 20;
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10 ctrl_signal = [repmat([1 0]', 1, T/4) repmat([0 1]', 1, T/4) ...
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11 repmat([-1 0]', 1, T/4) repmat([0 -1]', 1, T/4)];
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12 else
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13 T = 12;
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14 ctrl_signal = repmat([1 0]', 1, T);
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15 end
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16
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17 nlandmarks = 6;
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18 if 0
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19 true_landmark_pos = [1 1;
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20 4 1;
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21 4 4;
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22 1 4]';
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23 else
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24 true_landmark_pos = 10*rand(2,nlandmarks);
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25 end
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26 figure(1); clf
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27 hold on
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28 for i=1:nlandmarks
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29 %text(true_landmark_pos(1,i), true_landmark_pos(2,i), sprintf('L%d',i));
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30 plot(true_landmark_pos(1,i), true_landmark_pos(2,i), '*')
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31 end
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32 hold off
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33
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34 init_robot_pos = [0 0]';
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35
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36 true_robot_pos = zeros(2, T);
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37 true_data_assoc = zeros(1, T);
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38 true_rel_dist = zeros(2, T);
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39 for t=1:T
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40 if t>1
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41 true_robot_pos(:,t) = true_robot_pos(:,t-1) + ctrl_signal(:,t);
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42 else
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43 true_robot_pos(:,t) = init_robot_pos + ctrl_signal(:,t);
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44 end
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45 %nn = argmin(dist2(true_robot_pos(:,t)', true_landmark_pos'));
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46 nn = wrap(t, nlandmarks); % observe 1, 2, 3, 4, 1, 2, ...
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47 true_data_assoc(t) = nn;
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48 true_rel_dist(:,t) = true_landmark_pos(:, nn) - true_robot_pos(:,t);
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49 end
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50
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51 R = 1e-3*eye(2); % noise added to observation
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52 Q = 1e-3*eye(2); % noise added to robot motion
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53
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54 % Create data set
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55 obs_noise_seq = sample_gaussian([0 0]', R, T)';
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56 obs_rel_pos = true_rel_dist + obs_noise_seq;
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57 %obs_rel_pos = true_rel_dist;
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58
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59
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60 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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61 % Create params for inference
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62
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63 % X(t) = A X(t-1) + B U(t) + noise(Q)
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64
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65 % [L1] = [1 ] * [L1] + [0] * Ut + [0 ]
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66 % [L2] [ 1 ] [L2] [0] [ 0 ]
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67 % [R ]t [ 1] [R ]t-1 [1] [ Q]
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68
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69 % Y(t)|S(t)=s = C(s) X(t) + noise(R)
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70 % Yt|St=1 = [1 0 -1] * [L1] + R
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71 % [L2]
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72 % [R ]
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73
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74 % Create indices into block structure
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75 bs = 2*ones(1, nlandmarks+1); % sizes of blocks in state space
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76 robot_block = block(nlandmarks+1, bs);
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77 for i=1:nlandmarks
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78 landmark_block(:,i) = block(i, bs)';
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79 end
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80 Xsz = 2*(nlandmarks+1); % 2 values for each landmark plus robot
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81 Ysz = 2; % observe relative location
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82 Usz = 2; % input is (dx, dy)
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83
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84
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85 % create block-diagonal trans matrix for each switch
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86 A = zeros(Xsz, Xsz);
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87 for i=1:nlandmarks
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88 bi = landmark_block(:,i);
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89 A(bi, bi) = eye(2);
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90 end
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91 bi = robot_block;
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92 A(bi, bi) = eye(2);
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93 A = repmat(A, [1 1 nlandmarks]); % same for all switch values
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94
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95 % create block-diagonal system cov
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96
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97
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98 Qbig = zeros(Xsz, Xsz);
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99 bi = robot_block;
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100 Qbig(bi,bi) = Q; % only add noise to robot motion
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101 Qbig = repmat(Qbig, [1 1 nlandmarks]);
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102
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103 % create input matrix
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104 B = zeros(Xsz, Usz);
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105 B(robot_block,:) = eye(2); % only add input to robot position
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106 B = repmat(B, [1 1 nlandmarks]);
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107
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108 % create observation matrix for each value of the switch node
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109 % C(:,:,i) = (0 ... I ... -I) where the I is in the i'th posn.
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110 % This computes L(i) - R
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111 C = zeros(Ysz, Xsz, nlandmarks);
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112 for i=1:nlandmarks
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113 C(:, landmark_block(:,i), i) = eye(2);
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114 C(:, robot_block, i) = -eye(2);
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115 end
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116
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117 % create observation cov for each value of the switch node
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118 Rbig = repmat(R, [1 1 nlandmarks]);
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119
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120 % initial conditions
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121 init_x = zeros(Xsz, 1);
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122 init_v = zeros(Xsz, Xsz);
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123 bi = robot_block;
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124 init_x(bi) = init_robot_pos;
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125 init_V(bi, bi) = 1e-5*eye(2); % very sure of robot posn
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126 for i=1:nlandmarks
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127 bi = landmark_block(:,i);
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128 init_V(bi,bi)= 1e5*eye(2); % very uncertain of landmark psosns
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129 %init_x(bi) = true_landmark_pos(:,i);
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130 %init_V(bi,bi)= 1e-5*eye(2); % very sure of landmark psosns
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131 end
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132
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133 [xsmooth, Vsmooth] = kalman_filter(obs_rel_pos, A, C, Qbig, Rbig, init_x, init_V, ...
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134 'model', true_data_assoc, 'u', ctrl_signal, 'B', B);
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135
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136 est_robot_pos = xsmooth(robot_block, :);
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137 est_robot_pos_cov = Vsmooth(robot_block, robot_block, :);
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138
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139 for i=1:nlandmarks
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140 bi = landmark_block(:,i);
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141 est_landmark_pos(:,i) = xsmooth(bi, T);
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142 est_landmark_pos_cov(:,:,i) = Vsmooth(bi, bi, T);
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143 end
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144
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145
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146
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147 P = zeros(size(Vsmooth));
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148 for t=1:T
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149 P(:,:,t) = inv(Vsmooth(:,:,t));
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150 end
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151
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152 figure(1)
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153 for t=1:T
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154 subplot(T/2,2,t)
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155 imagesc(P(1:2:end,1:2:end, t))
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156 colorbar
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157 end
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158
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159 figure(2)
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160 for t=1:T
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161 subplot(T/2,2,t)
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162 imagesc(Vsmooth(1:2:end,1:2:end, t))
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163 colorbar
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164 end
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165
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166
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167
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168 % marginalize out robot position and then check structure
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169 bi = landmark_block(:);
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170 V = Vsmooth(bi,bi,T);
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171 P = inv(V);
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172 P(1:2:end,1:2:end)
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