378 lines
17 KiB
C++
378 lines
17 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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* @file elaboratePoint2KalmanFilter.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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* This example manually creates all of the needed data structures
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*
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* @date 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/nonlinear/PriorFactor.h>
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#include <gtsam/slam/BetweenFactor.h>
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//#include <gtsam/nonlinear/Ordering.h>
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#include <gtsam/inference/Symbol.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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int main() {
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// [code below basically does SRIF with Cholesky]
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// Create a factor graph to perform the inference
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GaussianFactorGraph::shared_ptr linearFactorGraph(new GaussianFactorGraph);
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// Create the desired ordering
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Ordering::shared_ptr ordering(new Ordering);
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// Create a structure to hold the linearization points
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Values linearizationPoints;
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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
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Symbol x0('x',0);
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ordering->insert(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((Vec(2) << 0.1, 0.1));
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PriorFactor<Point2> factor1(x0, x_initial, P_initial);
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// Linearize the factor and add it to the linear factor graph
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linearizationPoints.insert(x0, x_initial);
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linearFactorGraph->push_back(factor1.linearize(linearizationPoints, *ordering));
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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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Symbol x1('x',1);
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ordering->insert(x1, 1);
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Point2 difference(1,0);
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SharedDiagonal Q = noiseModel::Diagonal::Sigmas((Vec(2) << 0.1, 0.1));
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BetweenFactor<Point2> factor2(x0, x1, difference, Q);
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// Linearize the factor and add it to the linear factor graph
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linearizationPoints.insert(x1, x_initial);
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linearFactorGraph->push_back(factor2.linearize(linearizationPoints, *ordering));
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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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// 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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// Extract the current estimate of x1,P1 from the Bayes Network
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VectorValues result = optimize(*linearBayesNet);
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Point2 x1_predict = linearizationPoints.at<Point2>(x1).retract(result[ordering->at(x1)]);
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x1_predict.print("X1 Predict");
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// Update the new linearization point to the new estimate
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linearizationPoints.update(x1, x1_predict);
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// Create a new, empty graph and add the prior from the previous step
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linearFactorGraph = GaussianFactorGraph::shared_ptr(new GaussianFactorGraph);
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// Convert the root conditional, P(x1) in this case, into a Prior for the next step
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// Some care must be done here, as the linearization point in future steps will be different
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// than what was used when the factor was created.
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// f = || F*dx1' - (F*x0 - x1) ||^2, originally linearized at x1 = x0
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// After this step, the factor needs to be linearized around x1 = x1_predict
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// This changes the factor to f = || F*dx1'' - b'' ||^2
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// = || F*(dx1' - (dx1' - dx1'')) - b'' ||^2
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// = || F*dx1' - F*(dx1' - dx1'') - b'' ||^2
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// = || F*dx1' - (b'' + F(dx1' - dx1'')) ||^2
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// -> b' = b'' + F(dx1' - dx1'')
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// -> b'' = b' - F(dx1' - dx1'')
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// = || F*dx1'' - (b' - F(dx1' - dx1'')) ||^2
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// = || F*dx1'' - (b' - F(x_predict - x_inital)) ||^2
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const GaussianConditional::shared_ptr& cg0 = linearBayesNet->back();
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assert(cg0->nrFrontals() == 1);
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assert(cg0->nrParents() == 0);
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linearFactorGraph->add(0, cg0->R(), cg0->d() - cg0->R()*result[ordering->at(x1)], noiseModel::Diagonal::Sigmas(cg0->get_sigmas(), true));
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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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// 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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// where f3 is the prior from the previous step, and
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// where f4 is a measurement factor
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//
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// So, now we need to create the measurement factor, f4
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// For the Kalman Filter, this is 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 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((Vec(2) << 0.25, 0.25));
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PriorFactor<Point2> factor4(x1, z1, R1);
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// Linearize the factor and add it to the linear factor graph
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linearFactorGraph->push_back(factor4.linearize(linearizationPoints, *ordering));
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// We have now made the small factor graph f3-(x1)-f4
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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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// 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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// Extract the current estimate of x1 from the Bayes Network
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result = optimize(*linearBayesNet);
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Point2 x1_update = linearizationPoints.at<Point2>(x1).retract(result[ordering->at(x1)]);
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x1_update.print("X1 Update");
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// Update the linearization point to the new estimate
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linearizationPoints.update(x1, x1_update);
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// Wash, rinse, repeat for another time step
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// Create a new, empty graph and add the prior from the previous step
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linearFactorGraph = GaussianFactorGraph::shared_ptr(new GaussianFactorGraph);
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// Convert the root conditional, P(x1) in this case, into a Prior for the next step
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// The linearization point of this prior must be moved to the new estimate of x, and the key/index needs to be reset to 0,
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// the first key in the next iteration
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const GaussianConditional::shared_ptr& cg1 = linearBayesNet->back();
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assert(cg1->nrFrontals() == 1);
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assert(cg1->nrParents() == 0);
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JacobianFactor tmpPrior1 = JacobianFactor(*cg1);
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linearFactorGraph->add(0, tmpPrior1.getA(tmpPrior1.begin()), tmpPrior1.getb() - tmpPrior1.getA(tmpPrior1.begin()) * result[ordering->at(x1)], tmpPrior1.get_model());
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// Create a key for the new state
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Symbol x2('x',2);
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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 nonlinear factor describing the motion model
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difference = Point2(1,0);
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Q = noiseModel::Diagonal::Sigmas((Vec(2) <, 0.1, 0.1));
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BetweenFactor<Point2> factor6(x1, x2, difference, Q);
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// Linearize the factor and add it to the linear factor graph
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linearizationPoints.insert(x2, x1_update);
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linearFactorGraph->push_back(factor6.linearize(linearizationPoints, *ordering));
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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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// Extract the current estimate of x2 from the Bayes Network
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result = optimize(*linearBayesNet);
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Point2 x2_predict = linearizationPoints.at<Point2>(x2).retract(result[ordering->at(x2)]);
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x2_predict.print("X2 Predict");
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// Update the linearization point to the new estimate
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linearizationPoints.update(x2, x2_predict);
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// Now add the next measurement
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// Create a new, empty graph and add the prior from the previous step
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linearFactorGraph = GaussianFactorGraph::shared_ptr(new GaussianFactorGraph);
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// Convert the root conditional, P(x1) in this case, into a Prior for the next step
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const GaussianConditional::shared_ptr& cg2 = linearBayesNet->back();
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assert(cg2->nrFrontals() == 1);
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assert(cg2->nrParents() == 0);
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JacobianFactor tmpPrior2 = JacobianFactor(*cg2);
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linearFactorGraph->add(0, tmpPrior2.getA(tmpPrior2.begin()), tmpPrior2.getb() - tmpPrior2.getA(tmpPrior2.begin()) * result[ordering->at(x2)], tmpPrior2.get_model());
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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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// And update using z2 ...
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Point2 z2(2.0, 0.0);
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SharedDiagonal R2 = noiseModel::Diagonal::Sigmas((Vec(2) << 0.25, 0.25));
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PriorFactor<Point2> factor8(x2, z2, R2);
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// Linearize the factor and add it to the linear factor graph
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linearFactorGraph->push_back(factor8.linearize(linearizationPoints, *ordering));
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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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// 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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// Extract the current estimate of x2 from the Bayes Network
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result = optimize(*linearBayesNet);
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Point2 x2_update = linearizationPoints.at<Point2>(x2).retract(result[ordering->at(x2)]);
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x2_update.print("X2 Update");
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// Update the linearization point to the new estimate
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linearizationPoints.update(x2, x2_update);
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// Wash, rinse, repeat for a third time step
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// Create a new, empty graph and add the prior from the previous step
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linearFactorGraph = GaussianFactorGraph::shared_ptr(new GaussianFactorGraph);
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// Convert the root conditional, P(x1) in this case, into a Prior for the next step
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const GaussianConditional::shared_ptr& cg3 = linearBayesNet->back();
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assert(cg3->nrFrontals() == 1);
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assert(cg3->nrParents() == 0);
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JacobianFactor tmpPrior3 = JacobianFactor(*cg3);
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linearFactorGraph->add(0, tmpPrior3.getA(tmpPrior3.begin()), tmpPrior3.getb() - tmpPrior3.getA(tmpPrior3.begin()) * result[ordering->at(x2)], tmpPrior3.get_model());
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// Create a key for the new state
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Symbol x3('x',3);
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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 nonlinear factor describing the motion model
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difference = Point2(1,0);
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Q = noiseModel::Diagonal::Sigmas((Vec(2) << 0.1, 0.1));
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BetweenFactor<Point2> factor10(x2, x3, difference, Q);
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// Linearize the factor and add it to the linear factor graph
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linearizationPoints.insert(x3, x2_update);
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linearFactorGraph->push_back(factor10.linearize(linearizationPoints, *ordering));
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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 solver4(*linearFactorGraph);
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linearBayesNet = solver4.eliminate();
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// Extract the current estimate of x3 from the Bayes Network
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result = optimize(*linearBayesNet);
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Point2 x3_predict = linearizationPoints.at<Point2>(x3).retract(result[ordering->at(x3)]);
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x3_predict.print("X3 Predict");
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// Update the linearization point to the new estimate
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linearizationPoints.update(x3, x3_predict);
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// Now add the next measurement
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// Create a new, empty graph and add the prior from the previous step
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linearFactorGraph = GaussianFactorGraph::shared_ptr(new GaussianFactorGraph);
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// Convert the root conditional, P(x1) in this case, into a Prior for the next step
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const GaussianConditional::shared_ptr& cg4 = linearBayesNet->back();
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assert(cg4->nrFrontals() == 1);
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assert(cg4->nrParents() == 0);
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JacobianFactor tmpPrior4 = JacobianFactor(*cg4);
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linearFactorGraph->add(0, tmpPrior4.getA(tmpPrior4.begin()), tmpPrior4.getb() - tmpPrior4.getA(tmpPrior4.begin()) * result[ordering->at(x3)], tmpPrior4.get_model());
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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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// And update using z3 ...
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Point2 z3(3.0, 0.0);
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SharedDiagonal R3 = noiseModel::Diagonal::Sigmas((Vec(2) << 0.25, 0.25));
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PriorFactor<Point2> factor12(x3, z3, R3);
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// Linearize the factor and add it to the linear factor graph
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linearFactorGraph->push_back(factor12.linearize(linearizationPoints, *ordering));
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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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// 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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// Extract the current estimate of x2 from the Bayes Network
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result = optimize(*linearBayesNet);
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Point2 x3_update = linearizationPoints.at<Point2>(x3).retract(result[ordering->at(x3)]);
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x3_update.print("X3 Update");
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// Update the linearization point to the new estimate
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linearizationPoints.update(x3, x3_update);
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return 0;
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}
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