linear Kalman prototyping
Frank Dellaert
parent
94f4f2959e
commit
16cf864892
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@ -31,6 +31,9 @@ sources += GaussianISAM.cpp
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check_PROGRAMS += tests/testHessianFactor tests/testGaussianFactor tests/testGaussianConditional
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check_PROGRAMS += tests/testGaussianFactorGraph tests/testGaussianJunctionTree
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# Kalman Filter
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check_PROGRAMS += tests/testKalmanFilter
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# Iterative Methods
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headers += iterative-inl.h
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sources += iterative.cpp SubgraphPreconditioner.cpp
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@ -0,0 +1,451 @@
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/* ----------------------------------------------------------------------------
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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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* testKalmanFilter.cpp
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*
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* Test simple linear Kalman filter on a moving 2D point
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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/linear/GaussianSequentialSolver.h>
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#include <gtsam/linear/HessianFactor.h>
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#include <CppUnitLite/TestHarness.h>
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using namespace std;
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using namespace gtsam;
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class KalmanFilter {
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private:
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size_t n_; /** dimensionality of state */
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/**
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* The Kalman filter posterior density is a Gaussian Conditional with no parents
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*/
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GaussianConditional::shared_ptr density_;
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/**
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* solve a factor graph fragment and store result as density_
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*/
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void solve(GaussianFactorGraph& factorGraph) {
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// Solve the factor graph
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GaussianSequentialSolver solver(factorGraph);
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GaussianBayesNet::shared_ptr bayesNet = solver.eliminate();
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// As this is a filter, all we need is the posterior P(x_t),
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// so we just keep the root of the Bayes net
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density_ = bayesNet->back();
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}
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public:
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/**
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* Constructor from prior density at time k=0
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* In Kalman Filter notation, these are is x_{0|0} and P_{0|0}
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* @param x estimate at time 0
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* @param P covariance at time 0
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*
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*/
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KalmanFilter(const Vector& x, const Matrix& P) :
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n_(x.size()) {
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// Create a Hessian Factor from (x,P)
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HessianFactor::shared_ptr factor(new HessianFactor(0, x, P));
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#ifdef LOWLEVEL
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// Eliminate it directly using LDL
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size_t nrFrontals = 1;
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Eigen::LDLT<Matrix>::TranspositionType pi = factor->partialLDL(nrFrontals);
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vector<Index> keys;
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keys.push_back(0);
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density_ = factor->splitEliminatedFactor(nrFrontals, keys, pi);
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#else
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// Create a factor graph f(x), eliminate it into P(x)
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GaussianFactorGraph factorGraph;
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factorGraph.push_back(factor);
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solve(factorGraph);
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#endif
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}
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/**
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* Return mean of posterior P(x|Z) at given all measurements Z
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*/
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Vector mean() const {
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Index nVars = 1;
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VectorValues x(nVars, n_);
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density_->rhs(x);
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density_->solveInPlace(x);
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return x[0];
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}
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/**
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* Return information matrix of posterior P(x|Z) at given all measurements Z
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*/
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Matrix information() const {
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return density_->computeInformation();
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}
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/**
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* Return covariance of posterior P(x|Z) at given all measurements Z
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*/
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Matrix covariance() const {
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return inverse(information());
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}
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/**
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* Predict the state P(x_{t+1}|Z^t)
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* In Kalman Filter notation, this is x_{t+1|t} and P_{t+1|t}
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* After the call, that is the density that can be queried.
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* Details and parameters:
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* In a linear Kalman Filter, the motion model is 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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void predict(const Matrix& F, const Matrix& B, const Vector& u,
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const Matrix& Q) {
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// We will create a 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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GaussianFactorGraph factorGraph;
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// push back f1
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factorGraph.push_back(density_->toFactor());
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// The factor related to the motion model is defined as
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// f2(x_{t},x_{t+1}) = (F*x_{t} - x_{t+1}) * Q^-1 * (F*x_{t} - x_{t+1})^T
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// Eliminate graph in order x0, x1, to get Bayes net P(x0|x1)P(x1)
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solve(factorGraph);
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}
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};
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// KalmanFilter
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/* ************************************************************************* */
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TEST(GaussianConditional, constructor) {
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// [code below basically does SRIF with LDL]
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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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// Initialize state x0 (2D point) at origin
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Vector x00 = Vector_(2,0.0,0.0);
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Matrix P00 = 0.1*eye(2,2);
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// Initialize a Kalman filter
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KalmanFilter kalmanFilter(x00,P00);
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EXPECT(assert_equal(x00, kalmanFilter.mean()));
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EXPECT(assert_equal(P00, kalmanFilter.covariance()));
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// Now predict the state at t=1, i.e. P(x_1)
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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.
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// Then, F = [1 0 ; 0 1], B = [1 0 ; 0 1] and u = [1 ; 0].
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// Let us also assume that the process noise Q = [0.1 0 ; 0 0.1];
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Matrix F = eye(2,2);
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Matrix B = eye(2,2);
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Vector u = Vector_(2,1.0,0.0);
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Matrix Q = 0.2*eye(2,2);
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kalmanFilter.predict(F, B, u, Q);
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Vector x10 = Vector_(2,0.0,0.0);
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Matrix P10 = 0.1*eye(2,2);
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EXPECT(assert_equal(x10, kalmanFilter.mean()));
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EXPECT(assert_equal(P10, kalmanFilter.covariance()));
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/*
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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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Vector x1_predict = linearizationPoints[x1].expmap(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->get_R(), cg0->get_d() - cg0->get_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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Vector 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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// 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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Vector x1_update = linearizationPoints[x1].expmap(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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Key x2(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 = Vector(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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// 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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Vector x2_predict = linearizationPoints[x2].expmap(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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Vector 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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// 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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Vector x2_update = linearizationPoints[x2].expmap(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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Key x3(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 = Vector(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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// 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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Vector x3_predict = linearizationPoints[x3].expmap(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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|
||||
|
||||
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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
|
||||
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());
|
||||
|
||||
// Create the desired ordering
|
||||
ordering = Ordering::shared_ptr(new Ordering);
|
||||
ordering->insert(x3, 0);
|
||||
|
||||
// And update using z3 ...
|
||||
Vector z3(3.0, 0.0);
|
||||
SharedDiagonal R3 = noiseModel::Diagonal::Sigmas(Vector_(2, 0.25, 0.25));
|
||||
PriorFactor<Values, Key> factor12(x3, z3, R3);
|
||||
|
||||
// Linearize the factor and add it to the linear factor graph
|
||||
linearFactorGraph->push_back(factor12.linearize(linearizationPoints, *ordering));
|
||||
|
||||
// We have now made the small factor graph f11-(x3)-f12
|
||||
// where factor f11 is the prior from previous time ( P(x3) )
|
||||
// and factor f12 is from the measurement, z3 ( P(x3|z3) )
|
||||
// Eliminate this in order x3, to get Bayes net P(x3)
|
||||
// As this is a filter, all we need is the posterior P(x3), so we just keep the root of the Bayes net
|
||||
// We solve as before...
|
||||
|
||||
// Solve the linear factor graph, converting it into a linear Bayes Network ( P(x0,x1) = P(x0|x1)*P(x1) )
|
||||
GaussianSequentialSolver solver5(*linearFactorGraph);
|
||||
linearBayesNet = solver5.eliminate();
|
||||
|
||||
// Extract the current estimate of x2 from the Bayes Network
|
||||
result = optimize(*linearBayesNet);
|
||||
Vector x3_update = linearizationPoints[x3].expmap(result[ordering->at(x3)]);
|
||||
x3_update.print("X3 Update");
|
||||
|
||||
// Update the linearization point to the new estimate
|
||||
linearizationPoints.update(x3, x3_update);
|
||||
*/
|
||||
}
|
||||
|
||||
/* ************************************************************************* */
|
||||
int main() { TestResult tr; return TestRegistry::runAllTests(tr);}
|
||||
/* ************************************************************************* */
|
||||
|
||||
Loading…
Reference in New Issue