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+// 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 - odometry_mean) / 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 + range_reading - corridor_length) /
+        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;
+}