examples/robot_pose_mle.cc - ceres-solver - Git at Google

 // Ceres Solver - A fast non-linear least squares minimizer
 // Copyright 2015 Google Inc. All rights reserved.
 // http://ceres-solver.org/
 //
 // Redistribution and use in source and binary forms, with or without
 // modification, are permitted provided that the following conditions are met:
 //
 // * Redistributions of source code must retain the above copyright notice,
 //   this list of conditions and the following disclaimer.
 // * Redistributions in binary form must reproduce the above copyright notice,
 //   this list of conditions and the following disclaimer in the documentation
 //   and/or other materials provided with the distribution.
 // * Neither the name of Google Inc. nor the names of its contributors may be
 //   used to endorse or promote products derived from this software without
 //   specific prior written permission.
 //
 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
 // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
 // ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
 // LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
 // CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
 // SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
 // INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
 // CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
 // ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
 // POSSIBILITY OF SUCH DAMAGE.
 //
 // Author: joydeepb@ri.cmu.edu (Joydeep Biswas)
 //
 // This example demonstrates how to use the DynamicAutoDiffCostFunction
 // variant of CostFunction. The DynamicAutoDiffCostFunction is meant to
 // be used in cases where the number of parameter blocks or the sizes are not
 // known at compile time.
 //
 // This example simulates a robot traversing down a 1-dimension hallway with
 // noise odometry readings and noisy range readings of the end of the hallway.
 // By fusing the noisy odometry and sensor readings this example demonstrates
 // how to compute the maximum likelihood estimate (MLE) of the robot's pose at
 // each timestep.
 //
 // The robot starts at the origin, and it is travels to the end of a corridor of
 // fixed length specified by the "--corridor_length" flag. It executes a series
 // of motion commands to move forward a fixed length, specified by the
 // "--pose_separation" flag, at which pose it receives relative odometry
 // measurements as well as a range reading of the distance to the end of the
 // hallway. The odometry readings are drawn with Gaussian noise and standard
 // deviation specified by the "--odometry_stddev" flag, and the range readings
 // similarly with standard deviation specified by the "--range-stddev" flag.
 //
 // There are two types of residuals in this problem:
 // 1) The OdometryConstraint residual, that accounts for the odometry readings
 //    between successive pose estimatess of the robot.
 // 2) The RangeConstraint residual, that accounts for the errors in the observed
 //    range readings from each pose.
 //
 // The OdometryConstraint residual is modeled as an AutoDiffCostFunction with
 // a fixed parameter block size of 1, which is the relative odometry being
 // solved for, between a pair of successive poses of the robot. Differences
 // between observed and computed relative odometry values are penalized weighted
 // by the known standard deviation of the odometry readings.
 //
 // The RangeConstraint residual is modeled as a DynamicAutoDiffCostFunction
 // which sums up the relative odometry estimates to compute the estimated
 // global pose of the robot, and then computes the expected range reading.
 // Differences between the observed and expected range readings are then
 // penalized weighted by the standard deviation of readings of the sensor.
 // Since the number of poses of the robot is not known at compile time, this
 // cost function is implemented as a DynamicAutoDiffCostFunction.
 //
 // The outputs of the example are the initial values of the odometry and range
 // readings, and the range and odometry errors for every pose of the robot.
 // After computing the MLE, the computed poses and corrected odometry values
 // are printed out, along with the corresponding range and odometry errors. Note
 // that as an MLE of a noisy system the errors will not be reduced to zero, but
 // the odometry estimates will be updated to maximize the joint likelihood of
 // all odometry and range readings of the robot.
 //
 // Mathematical Formulation
 // ======================================================
 //
 // Let p_0, .., p_N be (N+1) robot poses, where the robot moves down the
 // corridor starting from p_0 and ending at p_N. We assume that p_0 is the
 // origin of the coordinate system.
 // Odometry u_i is the observed relative odometry between pose p_(i-1) and p_i,
 // and range reading y_i is the range reading of the end of the corridor from
 // pose p_i. Both odometry as well as range readings are noisy, but we wish to
 // compute the maximum likelihood estimate (MLE) of corrected odometry values
 // u*_0 to u*_(N-1), such that the Belief is optimized:
 //
 // Belief(u*_(0:N-1) | u_(0:N-1), y_(0:N-1))                                  1.
 //   =        P(u*_(0:N-1) | u_(0:N-1), y_(0:N-1))                            2.
 //   \propto  P(y_(0:N-1) | u*_(0:N-1), u_(0:N-1)) P(u*_(0:N-1) | u_(0:N-1))  3.
 //   =       \prod_i{ P(y_i | u*_(0:i)) P(u*_i | u_i) }                       4.
 //
 // Here, the subscript "(0:i)" is used as shorthand to indicate entries from all
 // timesteps 0 to i for that variable, both inclusive.
 //
 // Bayes' rule is used to derive eq. 3 from 2, and the independence of
 // odometry observations and range readings is expolited to derive 4 from 3.
 //
 // Thus, the Belief, up to scale, is factored as a product of a number of
 // terms, two for each pose, where for each pose term there is one term for the
 // range reading, P(y_i | u*_(0:i) and one term for the odometry reading,
 // P(u*_i | u_i) . Note that the term for the range reading is dependent on all
 // odometry values u*_(0:i), while the odometry term, P(u*_i | u_i) depends only
 // on a single value, u_i. Both the range reading as well as odoemtry
 // probability terms are modeled as the Normal distribution, and have the form:
 //
 // p(x) \propto \exp{-((x - x_mean) / x_stddev)^2}
 //
 // where x refers to either the MLE odometry u* or range reading y, and x_mean
 // is the corresponding mean value, u for the odometry terms, and y_expected,
 // the expected range reading based on all the previous odometry terms.
 // The MLE is thus found by finding those values x* which minimize:
 //
 // x* = \arg\min{((x - x_mean) / x_stddev)^2}
 //
 // which is in the nonlinear least-square form, suited to being solved by Ceres.
 // The non-linear component arise from the computation of x_mean. The residuals
 // ((x - x_mean) / x_stddev) for the residuals that Ceres will optimize. As
 // mentioned earlier, the odometry term for each pose depends only on one
 // variable, and will be computed by an AutoDiffCostFunction, while the term
 // for the range reading will depend on all previous odometry observations, and
 // will be computed by a DynamicAutoDiffCostFunction since the number of
 // odoemtry observations will only be known at run time.

 #include <cstdio>
 #include <math.h>
 #include <vector>

 #include "ceres/ceres.h"
 #include "ceres/dynamic_autodiff_cost_function.h"
 #include "gflags/gflags.h"
 #include "glog/logging.h"
 #include "random.h"

 using ceres::AutoDiffCostFunction;
 using ceres::DynamicAutoDiffCostFunction;
 using ceres::CauchyLoss;
 using ceres::CostFunction;
 using ceres::LossFunction;
 using ceres::Problem;
 using ceres::Solve;
 using ceres::Solver;
 using ceres::examples::RandNormal;
 using std::min;
 using std::vector;

 DEFINE_double(corridor_length, 30.0, "Length of the corridor that the robot is "
               "travelling down.");

 DEFINE_double(pose_separation, 0.5, "The distance that the robot traverses "
               "between successive odometry updates.");

 DEFINE_double(odometry_stddev, 0.1, "The standard deviation of "
               "odometry error of the robot.");

 DEFINE_double(range_stddev, 0.01, "The standard deviation of range readings of "
               "the robot.");

 // The stride length of the dynamic_autodiff_cost_function evaluator.
 static const int kStride = 10;

 struct OdometryConstraint {
   typedef AutoDiffCostFunction<OdometryConstraint, 1, 1> OdometryCostFunction;

   OdometryConstraint(double odometry_mean, double odometry_stddev) :
       odometry_mean(odometry_mean), odometry_stddev(odometry_stddev) {}

   template <typename T>
   bool operator()(const T* const odometry, T* residual) const {
     *residual = (*odometry - T(odometry_mean)) / T(odometry_stddev);
     return true;
   }

   static OdometryCostFunction* Create(const double odometry_value) {
     return new OdometryCostFunction(
         new OdometryConstraint(odometry_value, FLAGS_odometry_stddev));
   }

   const double odometry_mean;
   const double odometry_stddev;
 };

 struct RangeConstraint {
   typedef DynamicAutoDiffCostFunction<RangeConstraint, kStride>
       RangeCostFunction;

   RangeConstraint(
       int pose_index,
       double range_reading,
       double range_stddev,
       double corridor_length) :
       pose_index(pose_index), range_reading(range_reading),
       range_stddev(range_stddev), corridor_length(corridor_length) {}

   template <typename T>
   bool operator()(T const* const* relative_poses, T* residuals) const {
     T global_pose(0);
     for (int i = 0; i <= pose_index; ++i) {
       global_pose += relative_poses[i][0];
     }
     residuals[0] = (global_pose + T(range_reading) - T(corridor_length)) /
         T(range_stddev);
     return true;
   }

   // Factory method to create a CostFunction from a RangeConstraint to
   // conveniently add to a ceres problem.
   static RangeCostFunction* Create(const int pose_index,
                                    const double range_reading,
                                    vector<double>* odometry_values,
                                    vector<double*>* parameter_blocks) {
     RangeConstraint* constraint = new RangeConstraint(
         pose_index, range_reading, FLAGS_range_stddev, FLAGS_corridor_length);
     RangeCostFunction* cost_function = new RangeCostFunction(constraint);
     // Add all the parameter blocks that affect this constraint.
     parameter_blocks->clear();
     for (int i = 0; i <= pose_index; ++i) {
       parameter_blocks->push_back(&((*odometry_values)[i]));
       cost_function->AddParameterBlock(1);
     }
     cost_function->SetNumResiduals(1);
     return (cost_function);
   }

   const int pose_index;
   const double range_reading;
   const double range_stddev;
   const double corridor_length;
 };

 void SimulateRobot(vector<double>* odometry_values,
                    vector<double>* range_readings) {
   const int num_steps = static_cast<int>(
       ceil(FLAGS_corridor_length / FLAGS_pose_separation));

   // The robot starts out at the origin.
   double robot_location = 0.0;
   for (int i = 0; i < num_steps; ++i) {
     const double actual_odometry_value = min(
         FLAGS_pose_separation, FLAGS_corridor_length - robot_location);
     robot_location += actual_odometry_value;
     const double actual_range = FLAGS_corridor_length - robot_location;
     const double observed_odometry =
         RandNormal() * FLAGS_odometry_stddev + actual_odometry_value;
     const double observed_range =
         RandNormal() * FLAGS_range_stddev + actual_range;
     odometry_values->push_back(observed_odometry);
     range_readings->push_back(observed_range);
   }
 }

 void PrintState(const vector<double>& odometry_readings,
                 const vector<double>& range_readings) {
   CHECK_EQ(odometry_readings.size(), range_readings.size());
   double robot_location = 0.0;
   printf("pose: location     odom    range  r.error  o.error\n");
   for (int i = 0; i < odometry_readings.size(); ++i) {
     robot_location += odometry_readings[i];
     const double range_error =
         robot_location + range_readings[i] - FLAGS_corridor_length;
     const double odometry_error =
         FLAGS_pose_separation - odometry_readings[i];
     printf("%4d: %8.3f %8.3f %8.3f %8.3f %8.3f\n",
            static_cast<int>(i), robot_location, odometry_readings[i],
            range_readings[i], range_error, odometry_error);
   }
 }

 int main(int argc, char** argv) {
   google::InitGoogleLogging(argv[0]);
   CERES_GFLAGS_NAMESPACE::ParseCommandLineFlags(&argc, &argv, true);
   // Make sure that the arguments parsed are all positive.
   CHECK_GT(FLAGS_corridor_length, 0.0);
   CHECK_GT(FLAGS_pose_separation, 0.0);
   CHECK_GT(FLAGS_odometry_stddev, 0.0);
   CHECK_GT(FLAGS_range_stddev, 0.0);

   vector<double> odometry_values;
   vector<double> range_readings;
   SimulateRobot(&odometry_values, &range_readings);

   printf("Initial values:\n");
   PrintState(odometry_values, range_readings);
   ceres::Problem problem;

   for (int i = 0; i < odometry_values.size(); ++i) {
     // Create and add a DynamicAutoDiffCostFunction for the RangeConstraint from
     // pose i.
     vector<double*> parameter_blocks;
     RangeConstraint::RangeCostFunction* range_cost_function =
         RangeConstraint::Create(
             i, range_readings[i], &odometry_values, &parameter_blocks);
     problem.AddResidualBlock(range_cost_function, NULL, parameter_blocks);

     // Create and add an AutoDiffCostFunction for the OdometryConstraint for
     // pose i.
     problem.AddResidualBlock(OdometryConstraint::Create(odometry_values[i]),
                              NULL,
                              &(odometry_values[i]));
   }

   ceres::Solver::Options solver_options;
   solver_options.minimizer_progress_to_stdout = true;

   Solver::Summary summary;
   printf("Solving...\n");
   Solve(solver_options, &problem, &summary);
   printf("Done.\n");
   std::cout << summary.FullReport() << "\n";
   printf("Final values:\n");
   PrintState(odometry_values, range_readings);
   return 0;
 }
	// Ceres Solver - A fast non-linear least squares minimizer
	// Copyright 2015 Google Inc. All rights reserved.
	// http://ceres-solver.org/
	//
	// Redistribution and use in source and binary forms, with or without
	// modification, are permitted provided that the following conditions are met:
	//
	// * Redistributions of source code must retain the above copyright notice,
	// this list of conditions and the following disclaimer.
	// * Redistributions in binary form must reproduce the above copyright notice,
	// this list of conditions and the following disclaimer in the documentation
	// and/or other materials provided with the distribution.
	// * Neither the name of Google Inc. nor the names of its contributors may be
	// used to endorse or promote products derived from this software without
	// specific prior written permission.
	//
	// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
	// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
	// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
	// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
	// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
	// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
	// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
	// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
	// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
	// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
	// POSSIBILITY OF SUCH DAMAGE.
	//
	// Author: joydeepb@ri.cmu.edu (Joydeep Biswas)
	//
	// This example demonstrates how to use the DynamicAutoDiffCostFunction
	// variant of CostFunction. The DynamicAutoDiffCostFunction is meant to
	// be used in cases where the number of parameter blocks or the sizes are not
	// known at compile time.
	//
	// This example simulates a robot traversing down a 1-dimension hallway with
	// noise odometry readings and noisy range readings of the end of the hallway.
	// By fusing the noisy odometry and sensor readings this example demonstrates
	// how to compute the maximum likelihood estimate (MLE) of the robot's pose at
	// each timestep.
	//
	// The robot starts at the origin, and it is travels to the end of a corridor of
	// fixed length specified by the "--corridor_length" flag. It executes a series
	// of motion commands to move forward a fixed length, specified by the
	// "--pose_separation" flag, at which pose it receives relative odometry
	// measurements as well as a range reading of the distance to the end of the
	// hallway. The odometry readings are drawn with Gaussian noise and standard
	// deviation specified by the "--odometry_stddev" flag, and the range readings
	// similarly with standard deviation specified by the "--range-stddev" flag.
	//
	// There are two types of residuals in this problem:
	// 1) The OdometryConstraint residual, that accounts for the odometry readings
	// between successive pose estimatess of the robot.
	// 2) The RangeConstraint residual, that accounts for the errors in the observed
	// range readings from each pose.
	//
	// The OdometryConstraint residual is modeled as an AutoDiffCostFunction with
	// a fixed parameter block size of 1, which is the relative odometry being
	// solved for, between a pair of successive poses of the robot. Differences
	// between observed and computed relative odometry values are penalized weighted
	// by the known standard deviation of the odometry readings.
	//
	// The RangeConstraint residual is modeled as a DynamicAutoDiffCostFunction
	// which sums up the relative odometry estimates to compute the estimated
	// global pose of the robot, and then computes the expected range reading.
	// Differences between the observed and expected range readings are then
	// penalized weighted by the standard deviation of readings of the sensor.
	// Since the number of poses of the robot is not known at compile time, this
	// cost function is implemented as a DynamicAutoDiffCostFunction.
	//
	// The outputs of the example are the initial values of the odometry and range
	// readings, and the range and odometry errors for every pose of the robot.
	// After computing the MLE, the computed poses and corrected odometry values
	// are printed out, along with the corresponding range and odometry errors. Note
	// that as an MLE of a noisy system the errors will not be reduced to zero, but
	// the odometry estimates will be updated to maximize the joint likelihood of
	// all odometry and range readings of the robot.
	//
	// Mathematical Formulation
	// ======================================================
	//
	// Let p_0, .., p_N be (N+1) robot poses, where the robot moves down the
	// corridor starting from p_0 and ending at p_N. We assume that p_0 is the
	// origin of the coordinate system.
	// Odometry u_i is the observed relative odometry between pose p_(i-1) and p_i,
	// and range reading y_i is the range reading of the end of the corridor from
	// pose p_i. Both odometry as well as range readings are noisy, but we wish to
	// compute the maximum likelihood estimate (MLE) of corrected odometry values
	// u_0 to u_(N-1), such that the Belief is optimized:
	//
	// Belief(u*_(0:N-1) \| u_(0:N-1), y_(0:N-1)) 1.
	// = P(u*_(0:N-1) \| u_(0:N-1), y_(0:N-1)) 2.
	// \propto P(y_(0:N-1) \| u_(0:N-1), u_(0:N-1)) P(u_(0:N-1) \| u_(0:N-1)) 3.
	// = \prod_i{ P(y_i \| u_(0:i)) P(u_i \| u_i) } 4.
	//
	// Here, the subscript "(0:i)" is used as shorthand to indicate entries from all
	// timesteps 0 to i for that variable, both inclusive.
	//
	// Bayes' rule is used to derive eq. 3 from 2, and the independence of
	// odometry observations and range readings is expolited to derive 4 from 3.
	//
	// Thus, the Belief, up to scale, is factored as a product of a number of
	// terms, two for each pose, where for each pose term there is one term for the
	// range reading, P(y_i \| u*_(0:i) and one term for the odometry reading,
	// P(u*_i \| u_i) . Note that the term for the range reading is dependent on all
	// odometry values u_(0:i), while the odometry term, P(u_i \| u_i) depends only
	// on a single value, u_i. Both the range reading as well as odoemtry
	// probability terms are modeled as the Normal distribution, and have the form:
	//
	// p(x) \propto \exp{-((x - x_mean) / x_stddev)^2}
	//
	// where x refers to either the MLE odometry u* or range reading y, and x_mean
	// is the corresponding mean value, u for the odometry terms, and y_expected,
	// the expected range reading based on all the previous odometry terms.
	// The MLE is thus found by finding those values x* which minimize:
	//
	// x* = \arg\min{((x - x_mean) / x_stddev)^2}
	//
	// which is in the nonlinear least-square form, suited to being solved by Ceres.
	// The non-linear component arise from the computation of x_mean. The residuals
	// ((x - x_mean) / x_stddev) for the residuals that Ceres will optimize. As
	// mentioned earlier, the odometry term for each pose depends only on one
	// variable, and will be computed by an AutoDiffCostFunction, while the term
	// for the range reading will depend on all previous odometry observations, and
	// will be computed by a DynamicAutoDiffCostFunction since the number of
	// odoemtry observations will only be known at run time.

	#include <cstdio>
	#include <math.h>
	#include <vector>

	#include "ceres/ceres.h"
	#include "ceres/dynamic_autodiff_cost_function.h"
	#include "gflags/gflags.h"
	#include "glog/logging.h"
	#include "random.h"

	using ceres::AutoDiffCostFunction;
	using ceres::DynamicAutoDiffCostFunction;
	using ceres::CauchyLoss;
	using ceres::CostFunction;
	using ceres::LossFunction;
	using ceres::Problem;
	using ceres::Solve;
	using ceres::Solver;
	using ceres::examples::RandNormal;
	using std::min;
	using std::vector;

	DEFINE_double(corridor_length, 30.0, "Length of the corridor that the robot is "
	"travelling down.");

	DEFINE_double(pose_separation, 0.5, "The distance that the robot traverses "
	"between successive odometry updates.");

	DEFINE_double(odometry_stddev, 0.1, "The standard deviation of "
	"odometry error of the robot.");

	DEFINE_double(range_stddev, 0.01, "The standard deviation of range readings of "
	"the robot.");

	// The stride length of the dynamic_autodiff_cost_function evaluator.
	static const int kStride = 10;

	struct OdometryConstraint {
	typedef AutoDiffCostFunction<OdometryConstraint, 1, 1> OdometryCostFunction;

	OdometryConstraint(double odometry_mean, double odometry_stddev) :
	odometry_mean(odometry_mean), odometry_stddev(odometry_stddev) {}

	template <typename T>
	bool operator()(const T* const odometry, T* residual) const {
	residual = (odometry - T(odometry_mean)) / T(odometry_stddev);
	return true;
	}

	static OdometryCostFunction* Create(const double odometry_value) {
	return new OdometryCostFunction(
	new OdometryConstraint(odometry_value, FLAGS_odometry_stddev));
	}

	const double odometry_mean;
	const double odometry_stddev;
	};

	struct RangeConstraint {
	typedef DynamicAutoDiffCostFunction<RangeConstraint, kStride>
	RangeCostFunction;

	RangeConstraint(
	int pose_index,
	double range_reading,
	double range_stddev,
	double corridor_length) :
	pose_index(pose_index), range_reading(range_reading),
	range_stddev(range_stddev), corridor_length(corridor_length) {}

	template <typename T>
	bool operator()(T const* const* relative_poses, T* residuals) const {
	T global_pose(0);
	for (int i = 0; i <= pose_index; ++i) {
	global_pose += relative_poses[i][0];
	}
	residuals[0] = (global_pose + T(range_reading) - T(corridor_length)) /
	T(range_stddev);
	return true;
	}

	// Factory method to create a CostFunction from a RangeConstraint to
	// conveniently add to a ceres problem.
	static RangeCostFunction* Create(const int pose_index,
	const double range_reading,
	vector<double>* odometry_values,
	vector<double> parameter_blocks) {
	RangeConstraint* constraint = new RangeConstraint(
	pose_index, range_reading, FLAGS_range_stddev, FLAGS_corridor_length);
	RangeCostFunction* cost_function = new RangeCostFunction(constraint);
	// Add all the parameter blocks that affect this constraint.
	parameter_blocks->clear();
	for (int i = 0; i <= pose_index; ++i) {
	parameter_blocks->push_back(&((*odometry_values)[i]));
	cost_function->AddParameterBlock(1);
	}
	cost_function->SetNumResiduals(1);
	return (cost_function);
	}

	const int pose_index;
	const double range_reading;
	const double range_stddev;
	const double corridor_length;
	};

	void SimulateRobot(vector<double>* odometry_values,
	vector<double>* range_readings) {
	const int num_steps = static_cast<int>(
	ceil(FLAGS_corridor_length / FLAGS_pose_separation));

	// The robot starts out at the origin.
	double robot_location = 0.0;
	for (int i = 0; i < num_steps; ++i) {
	const double actual_odometry_value = min(
	FLAGS_pose_separation, FLAGS_corridor_length - robot_location);
	robot_location += actual_odometry_value;
	const double actual_range = FLAGS_corridor_length - robot_location;
	const double observed_odometry =
	RandNormal() * FLAGS_odometry_stddev + actual_odometry_value;
	const double observed_range =
	RandNormal() * FLAGS_range_stddev + actual_range;
	odometry_values->push_back(observed_odometry);
	range_readings->push_back(observed_range);
	}
	}

	void PrintState(const vector<double>& odometry_readings,
	const vector<double>& range_readings) {
	CHECK_EQ(odometry_readings.size(), range_readings.size());
	double robot_location = 0.0;
	printf("pose: location odom range r.error o.error\n");
	for (int i = 0; i < odometry_readings.size(); ++i) {
	robot_location += odometry_readings[i];
	const double range_error =
	robot_location + range_readings[i] - FLAGS_corridor_length;
	const double odometry_error =
	FLAGS_pose_separation - odometry_readings[i];
	printf("%4d: %8.3f %8.3f %8.3f %8.3f %8.3f\n",
	static_cast<int>(i), robot_location, odometry_readings[i],
	range_readings[i], range_error, odometry_error);
	}
	}

	int main(int argc, char** argv) {
	google::InitGoogleLogging(argv[0]);
	CERES_GFLAGS_NAMESPACE::ParseCommandLineFlags(&argc, &argv, true);
	// Make sure that the arguments parsed are all positive.
	CHECK_GT(FLAGS_corridor_length, 0.0);
	CHECK_GT(FLAGS_pose_separation, 0.0);
	CHECK_GT(FLAGS_odometry_stddev, 0.0);
	CHECK_GT(FLAGS_range_stddev, 0.0);

	vector<double> odometry_values;
	vector<double> range_readings;
	SimulateRobot(&odometry_values, &range_readings);

	printf("Initial values:\n");
	PrintState(odometry_values, range_readings);
	ceres::Problem problem;

	for (int i = 0; i < odometry_values.size(); ++i) {
	// Create and add a DynamicAutoDiffCostFunction for the RangeConstraint from
	// pose i.
	vector<double*> parameter_blocks;
	RangeConstraint::RangeCostFunction* range_cost_function =
	RangeConstraint::Create(
	i, range_readings[i], &odometry_values, &parameter_blocks);
	problem.AddResidualBlock(range_cost_function, NULL, parameter_blocks);

	// Create and add an AutoDiffCostFunction for the OdometryConstraint for
	// pose i.
	problem.AddResidualBlock(OdometryConstraint::Create(odometry_values[i]),
	NULL,
	&(odometry_values[i]));
	}

	ceres::Solver::Options solver_options;
	solver_options.minimizer_progress_to_stdout = true;

	Solver::Summary summary;
	printf("Solving...\n");
	Solve(solver_options, &problem, &summary);
	printf("Done.\n");
	std::cout << summary.FullReport() << "\n";
	printf("Final values:\n");
	PrintState(odometry_values, range_readings);
	return 0;
	}