Article Figures & Data
Figures
- FIGURE 1
A factor graph representing the system described in Equation (1)
The xk circles are hidden variable nodes, with the number of these nodes proportional to the number of time steps. The yellow, black, and green rectangles are factor nodes.
- FIGURE 2
An illustration of the three equivalent representations of a factor graph.
The factor graph is described in Section 2.3, the equivalent optimization problem in Section 3.1, and the adjacency matrix for the graph and how it can be used to solve the optimization in Section 3.2.
- FIGURE 3
Computation time required to solve a factor graph estimation problem as a function of size
- FIGURE 4
Example of marginalization to achieve a sliding window of size 2
Note how the graph is reduced in size by “marginalizing” a hidden variable and replacing it with a different factor
- FIGURE 5
Example of marginalization in a more complex graph
In this case, the node being removed (labeled “m” on the left) is replaced by a factor that connects to all nodes to which “m” was originally connected.
- FIGURE 6
Comparative results for EKF vs. factor graph estimation for a 2D satellite problem Note that the factor-graph-based estimator is far more accurate across all time steps, especially near the beginning of the time over which measurements are received. (a) EKF estimation, RMSE = 736 km (b) Factor graph estimation, RMSE = 17.5 km
- FIGURE 7
Examples of different systems that can be solved using factor graphs that are not standard Kalman-filter-type graphs: (a) Graph representing a typical SLAM problem, (b) Graph with relative measurements between different time steps, (c) Graph with different dynamics models between time steps and nonperiodic measurements, (d) Graph with additional hidden variable nodes for selecting the weighting of the dynamics models, (e) Graph for use with highly fine-scaled time-based sensors.
- FIGURE 8
Two figures showing robust cost functions and how selecting an accurate weighting value yields correct derivatives: (a) Huber cost function and scaled Gaussians created to have the same derivative, (b) Geman–McClure cost function and scaled Gaussians created to have the same derivative.
- FIGURE 9
Example of a GNSS factor graph, where six GNSS pseudorange factors (black squares) are used to estimate x1 and x2 and four are used to estimate x3
The estimated state, x, includes receiver clock error and atmospheric delay estimates; carrier-phase factors can also be added provided that x includes estimated phase biases. Factors for dynamics (green squares) may incorporate internal navigation.
Tables
scalars Lower-case, non-bold letters represent scalar quantities, e.g., x. vectors Lower-case, bold letters represent vector quantities, e.g., x. Vectors are assumed to be column vectors. matrices Upper-case, bold letters represent matrix values, e.g., A. sets Calligraphic, upper-case letters represent sets, e.g., 
. In some cases, if a set of values can be easily converted to a vector by stacking all of the values in the set, . represents that stacked vector.A⊤ Implies the transpose of the matrix A. time indices Subscripts denote time indices, e.g., xk is the value of x at time step k. functions Lower-case letters followed by parentheses represent functions, e.g., f(x) is a function of x. To avoid confusion, if multiplication is implied by parentheses, a space will be placed between the letter and the parentheses, e.g., K(z – h(x)) represents the matrix K multiplied by the difference between z and the function h of x. N(0, Σ) Represents a zero-mean normal distribution with covariance Σ. Denotes the 2-norm of the vector v weighted by the covariance matrix A, i.e., .p(x) Represents the probability distribution of the random variable x. - TABLE 1
Effect of Sliding Window Size and Batch vs. Real-Time Estimation
RMSE values should be compared against the EKF, which obtains an RMSE of 736 km over entire estimation problem, and 201.2km over the 2nd half of the results.
Window Size RMSE (delayed) RMSE (real-time) Second Half (real-time) Full 17.4 km 493.8 km 13.7 km 750 86.8 km 495.4 km 13.7 km 500 98.1 km 498.5 km 13.7 km 300 170.3 km 474.3 km 13.7 km 200 377.5 km 498.5 km 13.7 km 100 405.9 km 508.4 km 13.7 km - TABLE 2
Example Unimodal Functions Used for Robust Optimization and Weighting Function Used in the Optimization Routine
Name Function Weighting Huber Geman–McClure
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