# factorGraph

Bipartite graph of factors and nodes

## Description

A factorGraph object stores a bipartite graph consisting of factors connected to variable nodes. The nodes represent the unknown random variables in an estimation problem, and the factors represent probabilistic constraints on those variables, derived from measurements or prior knowledge. Add factors and nodes to the factor graph by using the addFactor function.

## Creation

### Description

example

G = factorGraph creates an empty factorGraph object.

## Properties

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Number of nodes in the factor graph, specified as a positive integer.

Number of factors in the factor graph, specified as a positive integer.

## Object Functions

 addFactor Add factor to factor graph fixNode Fix or free node in factor graph hasNode Check if node ID exists in factor graph isConnected Check if factor graph is connected isNodeFixed Check if node is fixed nodeIDs Get all node IDs in factor graph nodeState Get or set node state in factor graph nodeType Get node type of node in factor graph optimize Optimize factor graph

## Examples

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Create and optimize a factor graph with custom solver options.

Create Factor Graph and Solver Settings

Create a factor graph and solver options with custom settings. Set the maximum number of iterations to 1000 and set the verbosity of the optimize output to 2.

G = factorGraph;
optns = factorGraphSolverOptions(MaxIterations=1000,VerbosityLevel=2)
optns =
factorGraphSolverOptions with properties:

MaxIterations: 1000
FunctionTolerance: 1.0000e-06
StepTolerance: 1.0000e-08
VerbosityLevel: 2
TrustRegionStrategyType: 1

Create a GPS factor with node identification number of 1 with NED ReferenceFrame and add it to the factor graph.

fgps = factorGPS(1,ReferenceFrame="NED");

Optimize Factor Graph

Optimize the factor graph with the custom settings. The results of the optimization are displayed with the level of detail depending on the VerbosityLevel.

optimize(G,optns);
0  0.000000e+00    0.00e+00    0.00e+00   0.00e+00   0.00e+00  1.00e+04        0    4.20e-05    2.45e-04

Solver Summary (v 2.0.0-eigen-(3.3.4)-no_lapack-eigensparse-no_openmp-no_custom_blas)

Original                  Reduced
Parameter blocks                            1                        1
Parameters                                  7                        7
Effective parameters                        6                        6
Residual blocks                             1                        1
Residuals                                   3                        3

Minimizer                        TRUST_REGION

Sparse linear algebra library    EIGEN_SPARSE

Given                     Used
Linear solver          SPARSE_NORMAL_CHOLESKY   SPARSE_NORMAL_CHOLESKY
Linear solver ordering              AUTOMATIC                        1

Cost:
Initial                          0.000000e+00
Final                            0.000000e+00
Change                           0.000000e+00

Minimizer iterations                        1
Successful steps                            1
Unsuccessful steps                          0

Time (in seconds):
Preprocessor                         0.000203

Residual only evaluation           0.000000 (0)
Jacobian & residual evaluation     0.000014 (1)
Linear solver                      0.000000 (0)
Minimizer                            0.003924

Postprocessor                        0.000009
Total                                0.004136

Create a matrix of positions of the landmarks to use for localization, and the real positions of the robot to compare your factor graph estimate against. Use the exampleHelperPlotPositionsAndLandmarks helper function to visualize the landmark points and the real path of the robot..

landmarks = [0 -3  0;
3  4  0;
7  1  0];
realpos = [0  0  0;
2 -2  0;
5  3  0;
10 2  0];
exampleHelperPlotPositionsAndLandmarks(realpos,landmarks)

Use Landmarks and Other Data as Factors

Create a factor graph, and add a prior factor to loosely fix the start pose of the robot by providing an estimate pose.

fg = factorGraph;
rng(2)
pf = factorPoseSE3Prior(1,Measurement=[0 0 0 1 0 0 0]);

Create factorPoseSE3AndXYZ landmark factor objects that relate the first and second pose nodes to the first landmark point, and then add the landmark factors to the factor graph. The landmark factors used here are for SE(3) state space but the process is identical for landmark factors for SE(2) state space. Add some random number to the relative position between the landmark and the ground truth position to simulate real sensor measurements.

% Landmark 1 Factors
measurementlmf1 = landmarks(1,:) - realpos(1,:) + 0.1*rand(1,3);
measurementlmf2 = landmarks(1,:) - realpos(2,:) + 0.1*rand(1,3);
lmf1 = factorPoseSE3AndPointXYZ([1 5],Measurement=measurementlmf1);
lmf2 = factorPoseSE3AndPointXYZ([2 5],Measurement=measurementlmf2);

Create landmark factors for the second and third landmark points, as well, relating them to the second and third pose nodes and third and fourth pose nodes, respectively. Use the exampleHelperAddNoiseAndAddToFactorGraph helper function to add noise to the measurement for each landmark factor and add the factors to the factor graph. Once you have added all landmark factors to the factor graph, the IDs of the pose nodes are 1, 2, 3, and 4, and the IDs of the landmark nodes are 5, 6, and 7.

% Landmark 2 Factors
lmf3 = factorPoseSE3AndPointXYZ([2 6],Measurement=landmarks(2,:)-realpos(2,:));
lmf4 = factorPoseSE3AndPointXYZ([3 6],Measurement=landmarks(2,:)-realpos(3,:));

% Landmark 3 Factors
lmf5 = factorPoseSE3AndPointXYZ([3 7],Measurement=landmarks(3,:)-realpos(3,:));
lmf6 = factorPoseSE3AndPointXYZ([4 7],Measurement=landmarks(3,:)-realpos(4,:));

Use relative pose factors to relate consecutive poses, and add the factors to the factor graph. To simulate sensor readings for the measurements of each factor, take the difference between a consecutive pair of ground truth positions, append a quaternion of zero, and add noise.

zeroQuat = [1 0 0 0];
rp1 = factorTwoPoseSE3([1 2],Measurement=[realpos(2,:)-realpos(1,:) zeroQuat]);
rp2 = factorTwoPoseSE3([2 3],Measurement=[realpos(3,:)-realpos(2,:) zeroQuat]);
rp3 = factorTwoPoseSE3([3 4],Measurement=[realpos(4,:)-realpos(3,:) zeroQuat]);

Optimize Factor Graph

Optimize the factor graph with the default solver options. The optimization updates the states of all nodes in the factor graph, so the positions of vehicle and the landmarks update.

fgso = factorGraphSolverOptions;
optimize(fg,fgso)
ans = struct with fields:
InitialCost: 71.6462
FinalCost: 0.0140
NumSuccessfulSteps: 5
NumUnsuccessfulSteps: 0
TotalTime: 0.0328
TerminationType: 0
IsSolutionUsable: 1

Visualize and Compare Results

Get and store the updated node states for the vehicle and landmarks and plot the results, comparing the factor graph estimate of the robot path to the known ground truth of the robot.

fgposopt = [fg.nodeState(1); fg.nodeState(2); fg.nodeState(3); fg.nodeState(4)]
fgposopt = 4×7

-0.0000    0.0000   -0.0000    1.0000   -0.0000    0.0000    0.0000
2.0529   -2.0006    0.0528    0.9991    0.0415    0.0115    0.0053
5.0501    3.0537    0.3775    0.9980    0.0569    0.0210    0.0197
10.0939    2.3060    0.0984    0.9883    0.1423    0.0465    0.0274

fglmopt = [fg.nodeState(5); fg.nodeState(6); fg.nodeState(7)]
fglmopt = 3×3

0.0753   -2.9889    0.0527
3.0294    4.0345    0.5962
7.1825    1.2102    0.1122

exampleHelperPlotPositionsAndLandmarks(realpos,landmarks,fgposopt,fglmopt)

## References

[1] Dellaert, Frank. Factor graphs and GTSAM: A Hands-On Introduction. Georgia: Georgia Tech, September, 2012.

## Version History

Introduced in R2022a