328 lines
16 KiB
C++
328 lines
16 KiB
C++
/* ----------------------------------------------------------------------------
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* GTSAM Copyright 2010, Georgia Tech Research Corporation,
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* Atlanta, Georgia 30332-0415
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* All Rights Reserved
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* Authors: Frank Dellaert, et al. (see THANKS for the full author list)
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* See LICENSE for the license information
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* -------------------------------------------------------------------------- */
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/*
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* Point2KalmanFilter.cpp
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*
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* simple linear Kalman filter on a moving 2D point, but done using factor graphs
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*
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* Created on: Aug 19, 2011
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* @Author: Frank Dellaert
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* @Author: Stephen Williams
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*/
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#include <gtsam/slam/PriorFactor.h>
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#include <gtsam/slam/BetweenFactor.h>
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#include <gtsam/nonlinear/LinearizedFactor.h>
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//#include <gtsam/nonlinear/NonlinearOptimization-inl.h>
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#include <gtsam/nonlinear/NonlinearFactorGraph-inl.h>
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#include <gtsam/nonlinear/LieValues-inl.h>
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#include <gtsam/nonlinear/Ordering.h>
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#include <gtsam/nonlinear/Key.h>
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#include <gtsam/linear/GaussianSequentialSolver.h>
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#include <gtsam/linear/GaussianBayesNet.h>
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#include <gtsam/linear/GaussianFactorGraph.h>
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#include <gtsam/linear/NoiseModel.h>
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#include <gtsam/geometry/Point2.h>
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using namespace std;
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using namespace gtsam;
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typedef TypedSymbol<Point2, 'x'> Key; /// Variable labels for a specific type
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typedef LieValues<Key> Values; /// Class to store values - acts as a state for the system
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int main() {
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// [code below basically does SRIF with LDL]
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// Create a factor graph to perform the inference
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NonlinearFactorGraph<Values>::shared_ptr nonlinearFactorGraph(new NonlinearFactorGraph<Values>);
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// Ground truth example
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// Start at origin, move to the right (x-axis): 0,0 0,1 0,2
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// Motion model is just moving to the right (x'-x)^2
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// Measurements are GPS like, (x-z)^2, where z is a 2D measurement
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// i.e., we should get 0,0 0,1 0,2 if there is no noise
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// Create new state variable, x0
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Key x0(0);
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// Initialize state x0 (2D point) at origin by adding a prior factor, i.e., Bayes net P(x0)
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// This is equivalent to x_0 and P_0
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Point2 x_initial(0,0);
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SharedDiagonal P_initial = noiseModel::Diagonal::Sigmas(Vector_(2, 0.1, 0.1));
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PriorFactor<Values, Key> factor1(x0, x_initial, P_initial);
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nonlinearFactorGraph->add(factor1);
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// Now predict the state at t=1, i.e. argmax_{x1} P(x1) = P(x1|x0) P(x0)
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// In Kalman Filter notation, this is x_{t+1|t} and P_{t+1|t}
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// For the Kalman Filter, this requires a motion model, f(x_{t}) = x_{t+1|t)
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// Assuming the system is linear, this will be of the form f(x_{t}) = F*x_{t} + B*u_{t} + w
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// where F is the state transition model/matrix, B is the control input model,
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// and w is zero-mean, Gaussian white noise with covariance Q
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// Note, in some models, Q is actually derived as G*w*G^T where w models uncertainty of some
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// physical property, such as velocity or acceleration, and G is derived from physics
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//
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// For the purposes of this example, let us assume we are using a constant-position model and
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// the controls are driving the point to the right at 1 m/s. Then, F = [1 0 ; 0 1], B = [1 0 ; 0 1]
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// and u = [1 ; 0]. Let us also assume that the process noise Q = [0.1 0 ; 0 0.1];
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//
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// In the case of factor graphs, the factor related to the motion model would be defined as
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// f2 = (f(x_{t}) - x_{t+1}) * Q^-1 * (f(x_{t}) - x_{t+1})^T
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// Conveniently, there is a factor type, called a BetweenFactor, that can generate this factor
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// given the expected difference, f(x_{t}) - x_{t+1}, and Q.
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// so, difference = x_{t+1} - x_{t} = F*x_{t} + B*u_{t} - I*x_{t}
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// = (F - I)*x_{t} + B*u_{t}
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// = B*u_{t} (for our example)
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Key x1(1);
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Point2 difference(1,0);
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SharedDiagonal Q = noiseModel::Diagonal::Sigmas(Vector_(2, 0.1, 0.1));
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BetweenFactor<Values, Key> factor2(x0, x1, difference, Q);
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nonlinearFactorGraph->add(factor2);
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// We have now made the small factor graph f1-(x0)-f2-(x1)
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// where factor f1 is just the prior from time t0, P(x0)
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// and factor f2 is from the motion model
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// Eliminate this in order x0, x1, to get Bayes net P(x0|x1)P(x1)
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// As this is a filter, all we need is the posterior P(x1), so we just keep the root of the Bayes net
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//
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// Because of the way GTSAM works internally, we have used nonlinear class even though this example
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// system is linear. We first convert the nonlinear factor graph into a linear one, using the specified
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// ordering. Linear factors are simply numbered, and are not accessible via named key like the nonlinear
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// variables. Also, the nonlinear factors are linearized around an initial estimate. For a true linear
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// system, the initial estimate is not important.
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// Create the desired ordering
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Ordering::shared_ptr ordering(new Ordering);
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ordering->insert(x0, 0);
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ordering->insert(x1, 1);
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// Create a set of linearization points at (0,0). Since this is a linear system, the actual linearization point doesn't matter
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Values linearizationPoints;
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linearizationPoints.insert(x0, Point2(0,0));
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linearizationPoints.insert(x1, Point2(0,0));
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// Convert the nonlinear factor graph into an "ordered" linear factor graph
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GaussianFactorGraph::shared_ptr linearFactorGraph = nonlinearFactorGraph->linearize(linearizationPoints, *ordering)->dynamicCastFactors<GaussianFactorGraph>();
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//->template dynamicCastFactors<GaussianFactorGraph>()
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// Solve the linear factor graph, converting it into a linear Bayes Network ( P(x0,x1) = P(x0|x1)*P(x1) )
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GaussianSequentialSolver solver0(*linearFactorGraph);
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GaussianBayesNet::shared_ptr linearBayesNet = solver0.eliminate();
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// If needed, the current estimate of x1 may be extracted from the Bayes Network
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VectorValues result = optimize(*linearBayesNet);
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Point2 x1_predict = linearizationPoints[x1].expmap(result[ordering->at(x1)]);
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x1_predict.print("X1 Predict");
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// Convert the root conditional, P(x1) in this case, into a Prior for the next step
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LinearizedFactor<Values,Key>::KeyLookup lookup;
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lookup[0] = x0; lookup[1] = x1;
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LinearizedFactor<Values,Key> factor3(linearBayesNet->back()->toFactor(), lookup, linearizationPoints);
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// Create a new, empty graph and add the prior from the previous step
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nonlinearFactorGraph = NonlinearFactorGraph<Values>::shared_ptr(new NonlinearFactorGraph<Values>);
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nonlinearFactorGraph->add(factor3);
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// Now, a measurement, z1, has been received, and the Kalman Filter should be "Updated"/"Corrected"
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// This is equivalent to saying P(x1|z1) ~ P(z1|x1)*P(x1) ~ f3(x1)*f4(x1;z1)
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// So, we need to create the measurement factor, f4
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// For the Kalman Filter, this the the measurement function, h(x_{t}) = z_{t}
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// Assuming the system is linear, this will be of the form h(x_{t}) = H*x_{t} + v
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// where H is the observation model/matrix, and v is zero-mean, Gaussian white noise with covariance R
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//
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// For the purposes of this example, let us assume we have something like a GPS that returns
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// the current position of the robot. For this simple example, we can use a PriorFactor to model the
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// observation as it depends on only a single state variable, x1. To model real sensor observations
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// generally requires the creation of a new factor type. For example, factors for range sensors, bearing
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// sensors, and camera projections have already been added to GTSAM.
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//
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// In the case of factor graphs, the factor related to the measurements would be defined as
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// f4 = (h(x_{t}) - z_{t}) * R^-1 * (h(x_{t}) - z_{t})^T
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// = (x_{t} - z_{t}) * R^-1 * (x_{t} - z_{t})^T
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// This can again be modeled using the PriorFactor, where the mean is z_{t} and the covariance is R.
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Point2 z1(1.0, 0.0);
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SharedDiagonal R1 = noiseModel::Diagonal::Sigmas(Vector_(2, 0.25, 0.25));
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PriorFactor<Values, Key> factor4(x1, z1, R1);
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nonlinearFactorGraph->add(factor4);
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// We have now made the small factor graph f3-(x1)-f2
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// where factor f3 is the prior from previous time ( P(x1) )
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// and factor f4 is from the measurement, z1 ( P(x1|z1) )
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// Eliminate this in order x1, to get Bayes net P(x1)
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// As this is a filter, all we need is the posterior P(x1), so we just keep the root of the Bayes net
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// We solve as before...
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// Create the desired ordering
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ordering = Ordering::shared_ptr(new Ordering);
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ordering->insert(x1, 0);
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// Create a set of linearization points at (0,0). Since this is a linear system, the actual linearization point doesn't matter
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linearizationPoints.insert(x1, Point2(0,0));
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// Convert the nonlinear factor graph into an "ordered" linear factor graph
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linearFactorGraph = nonlinearFactorGraph->linearize(linearizationPoints, *ordering)->dynamicCastFactors<gtsam::GaussianFactorGraph>();
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// Solve the linear factor graph, converting it into a linear Bayes Network ( P(x0,x1) = P(x0|x1)*P(x1) )
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GaussianSequentialSolver solver1(*linearFactorGraph);
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linearBayesNet = solver1.eliminate();
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// If needed, the current estimate of x1 may be extracted from the Bayes Network
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result = optimize(*linearBayesNet);
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Point2 x1_update = linearizationPoints[x1].expmap(result[ordering->at(x1)]);
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x1_update.print("X1 Update");
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// Convert the root conditional, P(x1) in this case, into a Prior for the next step
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lookup[0] = x1;
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LinearizedFactor<Values,Key> factor5(linearBayesNet->back()->toFactor(), lookup, linearizationPoints);
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// Create a new, empty graph and add the prior from the previous step
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nonlinearFactorGraph = NonlinearFactorGraph<Values>::shared_ptr(new NonlinearFactorGraph<Values>);
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nonlinearFactorGraph->add(factor5);
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// Wash, rinse, repeat for another time step
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Key x2(2);
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difference = Point2(1,0);
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Q = noiseModel::Diagonal::Sigmas(Vector_(2, 0.1, 0.1));
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BetweenFactor<Values, Key> factor6(x1, x2, difference, Q);
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nonlinearFactorGraph->add(factor6);
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// We have now made the small factor graph f5-(x1)-f6-(x2)
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// Eliminate this in order x1, x2, to get Bayes net P(x1|x2)P(x2)
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// Create the desired ordering
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ordering = Ordering::shared_ptr(new Ordering);
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ordering->insert(x1, 0);
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ordering->insert(x2, 1);
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// Create a set of linearization points at (0,0). Since this is a linear system, the actual linearization point doesn't matter
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linearizationPoints.insert(x1, Point2(0,0));
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linearizationPoints.insert(x2, Point2(0,0));
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// Convert the nonlinear factor graph into an "ordered" linear factor graph
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linearFactorGraph = nonlinearFactorGraph->linearize(linearizationPoints, *ordering)->dynamicCastFactors<gtsam::GaussianFactorGraph>();
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// Solve the linear factor graph, converting it into a linear Bayes Network ( P(x1,x2) = P(x1|x2)*P(x2) )
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GaussianSequentialSolver solver2(*linearFactorGraph);
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linearBayesNet = solver2.eliminate();
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// If needed, the current estimate of x2 may be extracted from the Bayes Network
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result = optimize(*linearBayesNet);
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Point2 x2_predict = linearizationPoints[x2].expmap(result[ordering->at(x2)]);
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x2_predict.print("X2 Predict");
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// Convert the root conditional, P(x2) in this case, into a Prior for the next step
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lookup[0] = x1; lookup[1] = x2;
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LinearizedFactor<Values,Key> factor7(linearBayesNet->back()->toFactor(), lookup, linearizationPoints);
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// Create a new, empty graph and add the prior from the previous step
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nonlinearFactorGraph = NonlinearFactorGraph<Values>::shared_ptr(new NonlinearFactorGraph<Values>);
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nonlinearFactorGraph->add(factor7);
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// And update using z2 ...
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Point2 z2(2.0, 0.0);
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SharedDiagonal R2 = noiseModel::Diagonal::Sigmas(Vector_(2, 0.25, 0.25));
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PriorFactor<Values, Key> factor8(x2, z2, R2);
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nonlinearFactorGraph->add(factor8);
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// We have now made the small factor graph f7-(x2)-f8
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// where factor f7 is the prior from previous time ( P(x2) )
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// and factor f8 is from the measurement, z2 ( P(x2|z2) )
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// Eliminate this in order x2, to get Bayes net P(x2)
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// As this is a filter, all we need is the posterior P(x2), so we just keep the root of the Bayes net
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// We solve as before...
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// Create the desired ordering
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ordering = Ordering::shared_ptr(new Ordering);
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ordering->insert(x2, 0);
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// Create a set of linearization points at (0,0). Since this is a linear system, the actual linearization point doesn't matter
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linearizationPoints.insert(x2, Point2(0,0));
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// Convert the nonlinear factor graph into an "ordered" linear factor graph
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linearFactorGraph = nonlinearFactorGraph->linearize(linearizationPoints, *ordering)->dynamicCastFactors<gtsam::GaussianFactorGraph>();
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// Solve the linear factor graph, converting it into a linear Bayes Network ( P(x0,x1) = P(x0|x1)*P(x1) )
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GaussianSequentialSolver solver3(*linearFactorGraph);
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linearBayesNet = solver3.eliminate();
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// If needed, the current estimate of x2 may be extracted from the Bayes Network
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result = optimize(*linearBayesNet);
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Point2 x2_update = linearizationPoints[x2].expmap(result[ordering->at(x2)]);
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x2_update.print("X2 Update");
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// Convert the root conditional, P(x1) in this case, into a Prior for the next step
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lookup[0] = x2;
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LinearizedFactor<Values,Key> factor9(linearBayesNet->back()->toFactor(), lookup, linearizationPoints);
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// Create a new, empty graph and add the prior from the previous step
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nonlinearFactorGraph = NonlinearFactorGraph<Values>::shared_ptr(new NonlinearFactorGraph<Values>);
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nonlinearFactorGraph->add(factor9);
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// Wash, rinse, repeat for another time step
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Key x3(3);
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difference = Point2(1,0);
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Q = noiseModel::Diagonal::Sigmas(Vector_(2, 0.1, 0.1));
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BetweenFactor<Values, Key> factor10(x2, x3, difference, Q);
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nonlinearFactorGraph->add(factor10);
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// We have now made the small factor graph f9-(x2)-f10-(x3)
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// Eliminate this in order x2, x3, to get Bayes net P(x2|x3)P(x3)
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// Create the desired ordering
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ordering = Ordering::shared_ptr(new Ordering);
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ordering->insert(x2, 0);
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ordering->insert(x3, 1);
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// Create a set of linearization points at (0,0). Since this is a linear system, the actual linearization point doesn't matter
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linearizationPoints.insert(x2, Point2(0,0));
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linearizationPoints.insert(x3, Point2(0,0));
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// Convert the nonlinear factor graph into an "ordered" linear factor graph
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linearFactorGraph = nonlinearFactorGraph->linearize(linearizationPoints, *ordering)->dynamicCastFactors<gtsam::GaussianFactorGraph>();
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// Solve the linear factor graph, converting it into a linear Bayes Network ( P(x2,x3) = P(x2|x3)*P(x3) )
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GaussianSequentialSolver solver4(*linearFactorGraph);
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linearBayesNet = solver4.eliminate();
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// If needed, the current estimate of x3 may be extracted from the Bayes Network
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result = optimize(*linearBayesNet);
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Point2 x3_predict = linearizationPoints[x3].expmap(result[ordering->at(x3)]);
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x3_predict.print("X3 Predict");
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// Convert the root conditional, P(x3) in this case, into a Prior for the next step
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lookup[0] = x2; lookup[1] = x3;
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LinearizedFactor<Values,Key> factor11(linearBayesNet->back()->toFactor(), lookup, linearizationPoints);
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// Create a new, empty graph and add the prior from the previous step
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nonlinearFactorGraph = NonlinearFactorGraph<Values>::shared_ptr(new NonlinearFactorGraph<Values>);
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nonlinearFactorGraph->add(factor11);
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// And update using z3 ...
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Point2 z3(3.0, 0.0);
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SharedDiagonal R3 = noiseModel::Diagonal::Sigmas(Vector_(2, 0.25, 0.25));
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PriorFactor<Values, Key> factor12(x3, z3, R3);
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nonlinearFactorGraph->add(factor12);
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// We have now made the small factor graph f11-(x3)-f12
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// where factor f11 is the prior from previous time ( P(x3) )
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// and factor f12 is from the measurement, z3 ( P(x3|z3) )
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// Eliminate this in order x3, to get Bayes net P(x3)
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// As this is a filter, all we need is the posterior P(x3), so we just keep the root of the Bayes net
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// We solve as before...
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// Create the desired ordering
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ordering = Ordering::shared_ptr(new Ordering);
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ordering->insert(x3, 0);
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// Create a set of linearization points at (0,0). Since this is a linear system, the actual linearization point doesn't matter
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linearizationPoints.insert(x3, Point2(0,0));
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// Convert the nonlinear factor graph into an "ordered" linear factor graph
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linearFactorGraph = nonlinearFactorGraph->linearize(linearizationPoints, *ordering)->dynamicCastFactors<gtsam::GaussianFactorGraph>();
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// Solve the linear factor graph, converting it into a linear Bayes Network ( P(x0,x1) = P(x0|x1)*P(x1) )
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GaussianSequentialSolver solver5(*linearFactorGraph);
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linearBayesNet = solver5.eliminate();
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// If needed, the current estimate of x1 may be extracted from the Bayes Network
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result = optimize(*linearBayesNet);
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Point2 x3_update = linearizationPoints[x3].expmap(result[ordering->at(x3)]);
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x3_update.print("X3 Update");
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return 0;
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}
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