fast_lio/include/IKFoM_toolkit/esekfom/esekfom.hpp

/*
 *  Copyright (c) 2019--2023, The University of Hong Kong
 *  All rights reserved.
 *
 *  Author: Dongjiao HE <hdj65822@connect.hku.hk>
 *
 *  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 the Universitaet Bremen nor the names of its
 *     contributors may be used to endorse or promote products derived
 *     from this software without specific prior written permission.
 *
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 */

#ifndef ESEKFOM_EKF_HPP
#define ESEKFOM_EKF_HPP


#include <vector>
#include <cstdlib>

#include <boost/bind.hpp>
#include <Eigen/Core>
#include <Eigen/Geometry>
#include <Eigen/Dense>
#include <Eigen/Sparse>

#include "../mtk/types/vect.hpp"
#include "../mtk/types/SOn.hpp"
#include "../mtk/types/S2.hpp"
#include "../mtk/startIdx.hpp"
#include "../mtk/build_manifold.hpp"
#include "util.hpp"

//#define USE_sparse


namespace esekfom {

using namespace Eigen;

//used for iterated error state EKF update
//for the aim to calculate  measurement (z), estimate measurement (h), partial differention matrices (h_x, h_v) and the noise covariance (R) at the same time, by only one function.
//applied for measurement as a manifold.
template<typename S, typename M, int measurement_noise_dof = M::DOF>
struct share_datastruct
{
	bool valid;
	bool converge;
	M z;
	Eigen::Matrix<typename S::scalar, M::DOF, measurement_noise_dof> h_v;
	Eigen::Matrix<typename S::scalar, M::DOF, S::DOF> h_x;
	Eigen::Matrix<typename S::scalar, measurement_noise_dof, measurement_noise_dof> R;
};

//used for iterated error state EKF update
//for the aim to calculate  measurement (z), estimate measurement (h), partial differention matrices (h_x, h_v) and the noise covariance (R) at the same time, by only one function.
//applied for measurement as an Eigen matrix whose dimension is changing
template<typename T>
struct dyn_share_datastruct
{
	bool valid;
	bool converge;
	Eigen::Matrix<T, Eigen::Dynamic, 1> z;
	Eigen::Matrix<T, Eigen::Dynamic, 1> h;
	Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> h_v;
	Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> h_x;
	Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> R;
};

//used for iterated error state EKF update
//for the aim to calculate  measurement (z), estimate measurement (h), partial differention matrices (h_x, h_v) and the noise covariance (R) at the same time, by only one function.
//applied for measurement as a dynamic manifold whose dimension or type is changing
template<typename T>
struct dyn_runtime_share_datastruct
{
	bool valid;
	bool converge;
	//Z z;
	Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> h_v;
	Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> h_x;
	Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> R;
};

template<typename state, int process_noise_dof, typename input = state, typename measurement=state, int measurement_noise_dof=0>
class esekf{

	typedef esekf self;
	enum{
		n = state::DOF, m = state::DIM, l = measurement::DOF
	};

public:
	
	typedef typename state::scalar scalar_type;
	typedef Matrix<scalar_type, n, n> cov;
	typedef Matrix<scalar_type, m, n> cov_;
	typedef SparseMatrix<scalar_type> spMt;
	typedef Matrix<scalar_type, n, 1> vectorized_state;
	typedef Matrix<scalar_type, m, 1> flatted_state;
	typedef flatted_state processModel(state &, const input &);
	typedef Eigen::Matrix<scalar_type, m, n> processMatrix1(state &, const input &);
	typedef Eigen::Matrix<scalar_type, m, process_noise_dof> processMatrix2(state &, const input &);
	typedef Eigen::Matrix<scalar_type, process_noise_dof, process_noise_dof> processnoisecovariance;
	typedef measurement measurementModel(state &, bool &);
	typedef measurement measurementModel_share(state &, share_datastruct<state, measurement, measurement_noise_dof> &);
	typedef Eigen::Matrix<scalar_type, Eigen::Dynamic, 1> measurementModel_dyn(state &, bool &);
	//typedef Eigen::Matrix<scalar_type, Eigen::Dynamic, 1> measurementModel_dyn_share(state &,  dyn_share_datastruct<scalar_type> &);
	typedef void measurementModel_dyn_share(state &,  dyn_share_datastruct<scalar_type> &);
	typedef Eigen::Matrix<scalar_type ,l, n> measurementMatrix1(state &, bool&);
	typedef Eigen::Matrix<scalar_type , Eigen::Dynamic, n> measurementMatrix1_dyn(state &, bool&);
	typedef Eigen::Matrix<scalar_type ,l, measurement_noise_dof> measurementMatrix2(state &, bool&);
	typedef Eigen::Matrix<scalar_type ,Eigen::Dynamic, Eigen::Dynamic> measurementMatrix2_dyn(state &, bool&);
	typedef Eigen::Matrix<scalar_type, measurement_noise_dof, measurement_noise_dof> measurementnoisecovariance;
	typedef Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> measurementnoisecovariance_dyn;

	esekf(const state &x = state(),
		const cov  &P = cov::Identity()): x_(x), P_(P){
	#ifdef USE_sparse
		SparseMatrix<scalar_type> ref(n, n);
		ref.setIdentity();
		l_ = ref;
		f_x_2 = ref;
		f_x_1 = ref;
	#endif
	};

	//receive system-specific models and their differentions.
	//for measurement as a manifold.
	void init(processModel f_in, processMatrix1 f_x_in, processMatrix2 f_w_in, measurementModel h_in, measurementMatrix1 h_x_in, measurementMatrix2 h_v_in, int maximum_iteration, scalar_type limit_vector[n])
	{
		f = f_in;
		f_x = f_x_in;
		f_w = f_w_in;
		h = h_in;
		h_x = h_x_in;
		h_v = h_v_in;

		maximum_iter = maximum_iteration;
		for(int i=0; i<n; i++)
		{
			limit[i] = limit_vector[i];
		}

		x_.build_S2_state();
		x_.build_SO3_state();
		x_.build_vect_state();
	}

	//receive system-specific models and their differentions.
	//for measurement as an Eigen matrix whose dimention is chaing.
	void init_dyn(processModel f_in, processMatrix1 f_x_in, processMatrix2 f_w_in, measurementModel_dyn h_in, measurementMatrix1_dyn h_x_in, measurementMatrix2_dyn h_v_in, int maximum_iteration, scalar_type limit_vector[n])
	{
		f = f_in;
		f_x = f_x_in;
		f_w = f_w_in;
		h_dyn = h_in;
		h_x_dyn = h_x_in;
		h_v_dyn = h_v_in;


		maximum_iter = maximum_iteration;
		for(int i=0; i<n; i++)
		{
			limit[i] = limit_vector[i];
		}
		
		x_.build_S2_state();
		x_.build_SO3_state();
		x_.build_vect_state();
	}

	//receive system-specific models and their differentions.
	//for measurement as a dynamic manifold whose dimension or type is changing.
	void init_dyn_runtime(processModel f_in, processMatrix1 f_x_in, processMatrix2 f_w_in, measurementMatrix1_dyn h_x_in, measurementMatrix2_dyn h_v_in, int maximum_iteration, scalar_type limit_vector[n])
	{
		f = f_in;
		f_x = f_x_in;
		f_w = f_w_in;
		h_x_dyn = h_x_in;
		h_v_dyn = h_v_in;

		maximum_iter = maximum_iteration;
		for(int i=0; i<n; i++)
		{
			limit[i] = limit_vector[i];
		}

		x_.build_S2_state();
		x_.build_SO3_state();
		x_.build_vect_state();
	}

	//receive system-specific models and their differentions
	//for measurement as a manifold.
	//calculate  measurement (z), estimate measurement (h), partial differention matrices (h_x, h_v) and the noise covariance (R) at the same time, by only one function (h_share_in).
	void init_share(processModel f_in, processMatrix1 f_x_in, processMatrix2 f_w_in, measurementModel_share h_share_in, int maximum_iteration, scalar_type limit_vector[n])
	{
		f = f_in;
		f_x = f_x_in;
		f_w = f_w_in;
		h_share = h_share_in;

		maximum_iter = maximum_iteration;
		for(int i=0; i<n; i++)
		{
			limit[i] = limit_vector[i];
		}

		x_.build_S2_state();
		x_.build_SO3_state();
		x_.build_vect_state();
	}

	//receive system-specific models and their differentions
	//for measurement as an Eigen matrix whose dimension is changing.
	//calculate  measurement (z), estimate measurement (h), partial differention matrices (h_x, h_v) and the noise covariance (R) at the same time, by only one function (h_dyn_share_in).
	void init_dyn_share(processModel f_in, processMatrix1 f_x_in, processMatrix2 f_w_in, measurementModel_dyn_share h_dyn_share_in, int maximum_iteration, scalar_type limit_vector[n])
	{
		f = f_in;
		f_x = f_x_in;
		f_w = f_w_in;
		h_dyn_share = h_dyn_share_in;

		maximum_iter = maximum_iteration;
		for(int i=0; i<n; i++)
		{
			limit[i] = limit_vector[i];
		}

		x_.build_S2_state();
		x_.build_SO3_state();
		x_.build_vect_state();
	}


	//receive system-specific models and their differentions
	//for measurement as a dynamic manifold whose dimension  or type is changing.
	//calculate  measurement (z), estimate measurement (h), partial differention matrices (h_x, h_v) and the noise covariance (R) at the same time, by only one function (h_dyn_share_in).
	//for any scenarios where it is needed
	void init_dyn_runtime_share(processModel f_in, processMatrix1 f_x_in, processMatrix2 f_w_in, int maximum_iteration, scalar_type limit_vector[n])
	{
		f = f_in;
		f_x = f_x_in;
		f_w = f_w_in;

		maximum_iter = maximum_iteration;
		for(int i=0; i<n; i++)
		{
			limit[i] = limit_vector[i];
		}

		x_.build_S2_state();
		x_.build_SO3_state();
		x_.build_vect_state();
	}

	// iterated error state EKF propogation
	void predict(double &dt, processnoisecovariance &Q, const input &i_in){
		flatted_state f_ = f(x_, i_in);
		cov_ f_x_ = f_x(x_, i_in);
		cov f_x_final;

		Matrix<scalar_type, m, process_noise_dof> f_w_ = f_w(x_, i_in);
		Matrix<scalar_type, n, process_noise_dof> f_w_final;
		state x_before = x_;
		x_.oplus(f_, dt);

		F_x1 = cov::Identity();
		for (std::vector<std::pair<std::pair<int, int>, int> >::iterator it = x_.vect_state.begin(); it != x_.vect_state.end(); it++) {
			int idx = (*it).first.first;
			int dim = (*it).first.second;
			int dof = (*it).second;
			for(int i = 0; i < n; i++){
				for(int j=0; j<dof; j++)
				{f_x_final(idx+j, i) = f_x_(dim+j, i);}	
			}
			for(int i = 0; i < process_noise_dof; i++){
				for(int j=0; j<dof; j++)
				{f_w_final(idx+j, i) = f_w_(dim+j, i);}
			}
		}
		Matrix<scalar_type, 3, 3> res_temp_SO3;
		MTK::vect<3, scalar_type> seg_SO3;
		for (std::vector<std::pair<int, int> >::iterator it = x_.SO3_state.begin(); it != x_.SO3_state.end(); it++) {
			int idx = (*it).first;
			int dim = (*it).second;
			for(int i = 0; i < 3; i++){
				seg_SO3(i) = -1 * f_(dim + i) * dt;
			}
			MTK::SO3<scalar_type> res;
			res.w() = MTK::exp<scalar_type, 3>(res.vec(), seg_SO3, scalar_type(1/2));
		#ifdef USE_sparse
			res_temp_SO3 = res.toRotationMatrix();
			for(int i = 0; i < 3; i++){
				for(int j = 0; j < 3; j++){
					f_x_1.coeffRef(idx + i, idx + j) = res_temp_SO3(i, j);
				}
			}
		#else
			F_x1.template block<3, 3>(idx, idx) = res.toRotationMatrix();
		#endif			
			res_temp_SO3 = MTK::A_matrix(seg_SO3);
			for(int i = 0; i < n; i++){
				f_x_final. template block<3, 1>(idx, i) = res_temp_SO3 * (f_x_. template block<3, 1>(dim, i));	
			}
			for(int i = 0; i < process_noise_dof; i++){
				f_w_final. template block<3, 1>(idx, i) = res_temp_SO3 * (f_w_. template block<3, 1>(dim, i));
			}
		}
		
		
		Matrix<scalar_type, 2, 3> res_temp_S2;
		Matrix<scalar_type, 2, 2> res_temp_S2_;
		MTK::vect<3, scalar_type> seg_S2;
		for (std::vector<std::pair<int, int> >::iterator it = x_.S2_state.begin(); it != x_.S2_state.end(); it++) {
			int idx = (*it).first;
			int dim = (*it).second;
			for(int i = 0; i < 3; i++){
				seg_S2(i) = f_(dim + i) * dt;
			}
			MTK::vect<2, scalar_type> vec = MTK::vect<2, scalar_type>::Zero();
			MTK::SO3<scalar_type> res;
			res.w() = MTK::exp<scalar_type, 3>(res.vec(), seg_S2, scalar_type(1/2));
			Eigen::Matrix<scalar_type, 2, 3> Nx;
			Eigen::Matrix<scalar_type, 3, 2> Mx;
			x_.S2_Nx_yy(Nx, idx);
			x_before.S2_Mx(Mx, vec, idx);
		#ifdef USE_sparse
			res_temp_S2_ = Nx * res.toRotationMatrix() * Mx;
			for(int i = 0; i < 2; i++){
				for(int j = 0; j < 2; j++){
					f_x_1.coeffRef(idx + i, idx + j) = res_temp_S2_(i, j);
				}
			}
		#else
			F_x1.template block<2, 2>(idx, idx) = Nx * res.toRotationMatrix() * Mx;
		#endif

			Eigen::Matrix<scalar_type, 3, 3> x_before_hat;
			x_before.S2_hat(x_before_hat, idx);
			res_temp_S2 = -Nx * res.toRotationMatrix() * x_before_hat*MTK::A_matrix(seg_S2).transpose();
			
			for(int i = 0; i < n; i++){
				f_x_final. template block<2, 1>(idx, i) = res_temp_S2 * (f_x_. template block<3, 1>(dim, i));
				
			}
			for(int i = 0; i < process_noise_dof; i++){
				f_w_final. template block<2, 1>(idx, i) = res_temp_S2 * (f_w_. template block<3, 1>(dim, i));
			}
		}
	
	#ifdef USE_sparse
		f_x_1.makeCompressed();
		spMt f_x2 = f_x_final.sparseView();
		spMt f_w1 = f_w_final.sparseView();
		spMt xp = f_x_1 + f_x2 * dt;
		P_ = xp * P_ * xp.transpose() + (f_w1 * dt) * Q * (f_w1 * dt).transpose();
	#else
		F_x1 += f_x_final * dt;
		P_ = (F_x1) * P_ * (F_x1).transpose() + (dt * f_w_final) * Q * (dt * f_w_final).transpose();
	#endif
	}

	//iterated error state EKF update for measurement as a manifold.
	void update_iterated(measurement& z, measurementnoisecovariance &R) {
		
		if(!(is_same<typename measurement::scalar, scalar_type>())){
			std::cerr << "the scalar type of measurment must be the same as the state" << std::endl;
			std::exit(100);
		}
		int t = 0;
		bool converg = true;   
		bool valid = true;    
		state x_propagated = x_;
		cov P_propagated = P_;
		
		for(int i=-1; i<maximum_iter; i++)
		{
			vectorized_state dx, dx_new;
			x_.boxminus(dx, x_propagated);
			dx_new = dx;
		#ifdef USE_sparse
			spMt h_x_ = h_x(x_, valid).sparseView();
			spMt h_v_ = h_v(x_, valid).sparseView();
			spMt R_ = R.sparseView();
		#else
			Matrix<scalar_type, l, n> h_x_ = h_x(x_, valid);
			Matrix<scalar_type, l, Eigen::Dynamic> h_v_ = h_v(x_, valid);
		#endif	
			if(! valid)
			{
				continue; 
			}

			P_ = P_propagated;
			
			Matrix<scalar_type, 3, 3> res_temp_SO3;
			MTK::vect<3, scalar_type> seg_SO3;
			for (std::vector<std::pair<int, int> >::iterator it = x_.SO3_state.begin(); it != x_.SO3_state.end(); it++) {
				int idx = (*it).first;
				int dim = (*it).second;
				for(int i = 0; i < 3; i++){
					seg_SO3(i) = dx(idx+i);
				}

				res_temp_SO3 = A_matrix(seg_SO3).transpose();
				dx_new.template block<3, 1>(idx, 0) = res_temp_SO3 * dx.template block<3, 1>(idx, 0);
				for(int i = 0; i < n; i++){
					P_. template block<3, 1>(idx, i) = res_temp_SO3 * (P_. template block<3, 1>(idx, i));	
				}
				for(int i = 0; i < n; i++){
					P_. template block<1, 3>(i, idx) =(P_. template block<1, 3>(i, idx)) *  res_temp_SO3.transpose();	
				}
			}
		
		
			Matrix<scalar_type, 2, 2> res_temp_S2;
			MTK::vect<2, scalar_type> seg_S2;
			for (std::vector<std::pair<int, int> >::iterator it = x_.S2_state.begin(); it != x_.S2_state.end(); it++) {
				int idx = (*it).first;
				int dim = (*it).second;
				for(int i = 0; i < 2; i++){
					seg_S2(i) = dx(idx + i);
				}
				
				Eigen::Matrix<scalar_type, 2, 3> Nx;
				Eigen::Matrix<scalar_type, 3, 2> Mx;
				x_.S2_Nx_yy(Nx, idx);
				x_propagated.S2_Mx(Mx, seg_S2, idx);
				res_temp_S2 = Nx * Mx; 
				dx_new.template block<2, 1>(idx, 0) = res_temp_S2 * dx.template block<2, 1>(idx, 0);
				for(int i = 0; i < n; i++){
					P_. template block<2, 1>(idx, i) = res_temp_S2 * (P_. template block<2, 1>(idx, i));	
				}
				for(int i = 0; i < n; i++){
					P_. template block<1, 2>(i, idx) = (P_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
				}
			}

			Matrix<scalar_type, n, l> K_;
			if(n > l)
			{
			#ifdef USE_sparse
				Matrix<scalar_type, l, l> K_temp = h_x_ * P_ * h_x_.transpose();
				spMt R_temp = h_v_ * R_ * h_v_.transpose();
				K_temp += R_temp;
				K_ = P_ * h_x_.transpose() * K_temp.inverse();
			#else
				K_= P_ * h_x_.transpose() * (h_x_ * P_ * h_x_.transpose() + h_v_ * R * h_v_.transpose()).inverse();
			#endif
			}
			else
			{
			#ifdef USE_sparse
				measurementnoisecovariance b = measurementnoisecovariance::Identity();
				Eigen::SparseQR<Eigen::SparseMatrix<scalar_type>, Eigen::COLAMDOrdering<int>> solver; 
				solver.compute(R_);
				measurementnoisecovariance R_in_temp = solver.solve(b);
				spMt R_in = R_in_temp.sparseView();
				spMt K_temp = h_x_.transpose() * R_in * h_x_;
				cov P_temp = P_.inverse();
				P_temp += K_temp;
				K_ = P_temp.inverse() * h_x_.transpose() * R_in;
			#else
				measurementnoisecovariance R_in = (h_v_*R*h_v_.transpose()).inverse();
				K_ = (h_x_.transpose() * R_in * h_x_ + P_.inverse()).inverse() * h_x_.transpose() * R_in;
			#endif 
			}
			Matrix<scalar_type, l, 1> innovation; 
			z.boxminus(innovation, h(x_, valid));
			cov K_x = K_ * h_x_;
			Matrix<scalar_type, n, 1> dx_ = K_ * innovation + (K_x  - Matrix<scalar_type, n, n>::Identity()) * dx_new;
        	state x_before = x_;
			x_.boxplus(dx_);

			converg = true;
			for(int i = 0; i < n ; i++)
			{
				if(std::fabs(dx_[i]) > limit[i])
				{
					converg = false;
					break;
				}
			}

			if(converg) t++;
	        
			if(t > 1 || i == maximum_iter - 1)
			{
				L_ = P_;
		
				Matrix<scalar_type, 3, 3> res_temp_SO3;
				MTK::vect<3, scalar_type> seg_SO3;
				for(typename std::vector<std::pair<int, int> >::iterator it = x_.SO3_state.begin(); it != x_.SO3_state.end(); it++) {
					int idx = (*it).first;
					for(int i = 0; i < 3; i++){
						seg_SO3(i) = dx_(i + idx);
					}
					res_temp_SO3 = A_matrix(seg_SO3).transpose();
					for(int i = 0; i < n; i++){
						L_. template block<3, 1>(idx, i) = res_temp_SO3 * (P_. template block<3, 1>(idx, i)); 
					}
					if(n > l)
					{
						for(int i = 0; i < l; i++){
							K_. template block<3, 1>(idx, i) = res_temp_SO3 * (K_. template block<3, 1>(idx, i));
						}
					}
					else
					{
						for(int i = 0; i < n; i++){
							K_x. template block<3, 1>(idx, i) = res_temp_SO3 * (K_x. template block<3, 1>(idx, i));
						}
					}
					for(int i = 0; i < n; i++){
						L_. template block<1, 3>(i, idx) = (L_. template block<1, 3>(i, idx)) * res_temp_SO3.transpose();
						P_. template block<1, 3>(i, idx) = (P_. template block<1, 3>(i, idx)) * res_temp_SO3.transpose();
					}
				}

				Matrix<scalar_type, 2, 2> res_temp_S2;
				MTK::vect<2, scalar_type> seg_S2;
				for(typename std::vector<std::pair<int, int> >::iterator it = x_.S2_state.begin(); it != x_.S2_state.end(); it++) {
					int idx = (*it).first;
			
					for(int i = 0; i < 2; i++){
						seg_S2(i) = dx_(i + idx);
					}
			
					Eigen::Matrix<scalar_type, 2, 3> Nx;
					Eigen::Matrix<scalar_type, 3, 2> Mx;
					x_.S2_Nx_yy(Nx, idx);
					x_propagated.S2_Mx(Mx, seg_S2, idx);
					res_temp_S2 = Nx * Mx; 
		
					for(int i = 0; i < n; i++){
						L_. template block<2, 1>(idx, i) = res_temp_S2 * (P_. template block<2, 1>(idx, i)); 
					}
					if(n > l)
					{
						for(int i = 0; i < l; i++){
							K_. template block<2, 1>(idx, i) = res_temp_S2 * (K_. template block<2, 1>(idx, i));
						}
					}
					else
					{
						for(int i = 0; i < n; i++){
							K_x. template block<2, 1>(idx, i) = res_temp_S2 * (K_x. template block<2, 1>(idx, i));
						}
					}
					for(int i = 0; i < n; i++){
						L_. template block<1, 2>(i, idx) = (L_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
						P_. template block<1, 2>(i, idx) = (P_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
					}
				}
				if(n > l)
				{
					P_ = L_ - K_ * h_x_ * P_;
				}
				else
				{
					P_ = L_ - K_x * P_;
				}
				return;
			}
		}
	}

	//iterated error state EKF update for measurement as a manifold.
	//calculate measurement (z), estimate measurement (h), partial differention matrices (h_x, h_v) and the noise covariance (R) at the same time, by only one function.
	void update_iterated_share() {
		
		if(!(is_same<typename measurement::scalar, scalar_type>())){
			std::cerr << "the scalar type of measurment must be the same as the state" << std::endl;
			std::exit(100);
		}

		int t = 0;
		share_datastruct<state, measurement, measurement_noise_dof> _share;
		_share.valid = true;
		_share.converge = true;
		state x_propagated = x_;
		cov P_propagated = P_;
		
		for(int i=-1; i<maximum_iter; i++)
		{
			vectorized_state dx, dx_new;
			x_.boxminus(dx, x_propagated);
			dx_new = dx;
			measurement h = h_share(x_, _share);
			measurement z = _share.z;
			measurementnoisecovariance R = _share.R;
		#ifdef USE_sparse
			spMt h_x_ = _share.h_x.sparseView();
			spMt h_v_ = _share.h_v.sparseView();
			spMt R_ = _share.R.sparseView();
		#else
			Matrix<scalar_type, l, n> h_x_ = _share.h_x;
			Matrix<scalar_type, l, Eigen::Dynamic> h_v_ = _share.h_v;
		#endif	
			if(! _share.valid)
			{
				continue; 
			}

			P_ = P_propagated;
			
			Matrix<scalar_type, 3, 3> res_temp_SO3;
			MTK::vect<3, scalar_type> seg_SO3;
			for (std::vector<std::pair<int, int> >::iterator it = x_.SO3_state.begin(); it != x_.SO3_state.end(); it++) {
				int idx = (*it).first;
				int dim = (*it).second;
				for(int i = 0; i < 3; i++){
					seg_SO3(i) = dx(idx+i);
				}

				res_temp_SO3 = A_matrix(seg_SO3).transpose();
				dx_new.template block<3, 1>(idx, 0) = res_temp_SO3 * dx.template block<3, 1>(idx, 0);
				for(int i = 0; i < n; i++){
					P_. template block<3, 1>(idx, i) = res_temp_SO3 * (P_. template block<3, 1>(idx, i));	
				}
				for(int i = 0; i < n; i++){
					P_. template block<1, 3>(i, idx) =(P_. template block<1, 3>(i, idx)) *  res_temp_SO3.transpose();	
				}
			}
		
		
			Matrix<scalar_type, 2, 2> res_temp_S2;
			MTK::vect<2, scalar_type> seg_S2;
			for (std::vector<std::pair<int, int> >::iterator it = x_.S2_state.begin(); it != x_.S2_state.end(); it++) {
				int idx = (*it).first;
				int dim = (*it).second;
				for(int i = 0; i < 2; i++){
					seg_S2(i) = dx(idx + i);
				}
				
				Eigen::Matrix<scalar_type, 2, 3> Nx;
				Eigen::Matrix<scalar_type, 3, 2> Mx;
				x_.S2_Nx_yy(Nx, idx);
				x_propagated.S2_Mx(Mx, seg_S2, idx);
				res_temp_S2 = Nx * Mx; 
				dx_new.template block<2, 1>(idx, 0) = res_temp_S2 * dx.template block<2, 1>(idx, 0);
				for(int i = 0; i < n; i++){
					P_. template block<2, 1>(idx, i) = res_temp_S2 * (P_. template block<2, 1>(idx, i));	
				}
				for(int i = 0; i < n; i++){
					P_. template block<1, 2>(i, idx) = (P_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
				}
			}

			Matrix<scalar_type, n, l> K_;
			if(n > l)
			{
			#ifdef USE_sparse
				Matrix<scalar_type, l, l> K_temp = h_x_ * P_ * h_x_.transpose();
				spMt R_temp = h_v_ * R_ * h_v_.transpose();
				K_temp += R_temp;
				K_ = P_ * h_x_.transpose() * K_temp.inverse();
			#else
				K_= P_ * h_x_.transpose() * (h_x_ * P_ * h_x_.transpose() + h_v_ * R * h_v_.transpose()).inverse();
			#endif
			}
			else
			{
			#ifdef USE_sparse
				measurementnoisecovariance b = measurementnoisecovariance::Identity();
				Eigen::SparseQR<Eigen::SparseMatrix<scalar_type>, Eigen::COLAMDOrdering<int>> solver; 
				solver.compute(R_);
				measurementnoisecovariance R_in_temp = solver.solve(b);
				spMt R_in = R_in_temp.sparseView();
				spMt K_temp = h_x_.transpose() * R_in * h_x_;
				cov P_temp = P_.inverse();
				P_temp += K_temp;
				K_ = P_temp.inverse() * h_x_.transpose() * R_in;
			#else
				measurementnoisecovariance R_in = (h_v_*R*h_v_.transpose()).inverse();
				K_ = (h_x_.transpose() * R_in * h_x_ + P_.inverse()).inverse() * h_x_.transpose() * R_in;
			#endif 
			}
			Matrix<scalar_type, l, 1> innovation; 
			z.boxminus(innovation, h);
			cov K_x = K_ * h_x_;
			Matrix<scalar_type, n, 1> dx_ = K_ * innovation + (K_x  - Matrix<scalar_type, n, n>::Identity()) * dx_new;
        	state x_before = x_;
			x_.boxplus(dx_);

			_share.converge = true;
			for(int i = 0; i < n ; i++)
			{
				if(std::fabs(dx_[i]) > limit[i])
				{
					_share.converge = false;
					break;
				}
			}

			if(_share.converge) t++;
	        
			if(t > 1 || i == maximum_iter - 1)
			{
				L_ = P_;
		
				Matrix<scalar_type, 3, 3> res_temp_SO3;
				MTK::vect<3, scalar_type> seg_SO3;
				for(typename std::vector<std::pair<int, int> >::iterator it = x_.SO3_state.begin(); it != x_.SO3_state.end(); it++) {
					int idx = (*it).first;
					for(int i = 0; i < 3; i++){
						seg_SO3(i) = dx_(i + idx);
					}
					res_temp_SO3 = A_matrix(seg_SO3).transpose();
					for(int i = 0; i < n; i++){
						L_. template block<3, 1>(idx, i) = res_temp_SO3 * (P_. template block<3, 1>(idx, i)); 
					}
					if(n > l)
					{
						for(int i = 0; i < l; i++){
							K_. template block<3, 1>(idx, i) = res_temp_SO3 * (K_. template block<3, 1>(idx, i));
						}
					}
					else
					{
						for(int i = 0; i < n; i++){
							K_x. template block<3, 1>(idx, i) = res_temp_SO3 * (K_x. template block<3, 1>(idx, i));
						}
					}
					for(int i = 0; i < n; i++){
						L_. template block<1, 3>(i, idx) = (L_. template block<1, 3>(i, idx)) * res_temp_SO3.transpose();
						P_. template block<1, 3>(i, idx) = (P_. template block<1, 3>(i, idx)) * res_temp_SO3.transpose();
					}
				}

				Matrix<scalar_type, 2, 2> res_temp_S2;
				MTK::vect<2, scalar_type> seg_S2;
				for(typename std::vector<std::pair<int, int> >::iterator it = x_.S2_state.begin(); it != x_.S2_state.end(); it++) {
					int idx = (*it).first;
			
					for(int i = 0; i < 2; i++){
						seg_S2(i) = dx_(i + idx);
					}
			
					Eigen::Matrix<scalar_type, 2, 3> Nx;
					Eigen::Matrix<scalar_type, 3, 2> Mx;
					x_.S2_Nx_yy(Nx, idx);
					x_propagated.S2_Mx(Mx, seg_S2, idx);
					res_temp_S2 = Nx * Mx; 
		
					for(int i = 0; i < n; i++){
						L_. template block<2, 1>(idx, i) = res_temp_S2 * (P_. template block<2, 1>(idx, i)); 
					}
					if(n > l)
					{
						for(int i = 0; i < l; i++){
							K_. template block<2, 1>(idx, i) = res_temp_S2 * (K_. template block<2, 1>(idx, i));
						}
					}
					else
					{
						for(int i = 0; i < n; i++){
							K_x. template block<2, 1>(idx, i) = res_temp_S2 * (K_x. template block<2, 1>(idx, i));
						}
					}
					for(int i = 0; i < n; i++){
						L_. template block<1, 2>(i, idx) = (L_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
						P_. template block<1, 2>(i, idx) = (P_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
					}
				}
				if(n > l)
				{
					P_ = L_ - K_ * h_x_ * P_;
				}
				else
				{
					P_ = L_ - K_x * P_;
				}
				return;
			}
		}
	}

	//iterated error state EKF update for measurement as an Eigen matrix whose dimension is changing.
	void update_iterated_dyn(Eigen::Matrix<scalar_type, Eigen::Dynamic, 1> z, measurementnoisecovariance_dyn R) {
	
		int t = 0;
		bool valid = true;
		bool converg = true;
		state x_propagated = x_;
		cov P_propagated = P_;
		int dof_Measurement;
		int dof_Measurement_noise = R.rows();
		for(int i=-1; i<maximum_iter; i++)
		{
			valid = true;
		#ifdef USE_sparse
			spMt h_x_ = h_x_dyn(x_, valid).sparseView();
			spMt h_v_ = h_v_dyn(x_, valid).sparseView();
			spMt R_ = R.sparseView();
		#else
			Matrix<scalar_type, Eigen::Dynamic, n> h_x_ = h_x_dyn(x_, valid);
			Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> h_v_ = h_v_dyn(x_, valid);
		#endif	
			Matrix<scalar_type, Eigen::Dynamic, 1> h_ = h_dyn(x_, valid);
			dof_Measurement = h_.rows();
			vectorized_state dx, dx_new;
			x_.boxminus(dx, x_propagated);
			dx_new = dx;
			if(! valid)
			{
				continue; 
			}

			P_ = P_propagated;
			Matrix<scalar_type, 3, 3> res_temp_SO3;
			MTK::vect<3, scalar_type> seg_SO3;
			for (std::vector<std::pair<int, int> >::iterator it = x_.SO3_state.begin(); it != x_.SO3_state.end(); it++) {
				int idx = (*it).first;
				int dim = (*it).second;
				for(int i = 0; i < 3; i++){
					seg_SO3(i) = dx(idx+i);
				}

				res_temp_SO3 = MTK::A_matrix(seg_SO3).transpose();
				dx_new.template block<3, 1>(idx, 0) = res_temp_SO3 * dx_new.template block<3, 1>(idx, 0);
				for(int i = 0; i < n; i++){
					P_. template block<3, 1>(idx, i) = res_temp_SO3 * (P_. template block<3, 1>(idx, i));	
				}
				for(int i = 0; i < n; i++){
					P_. template block<1, 3>(i, idx) =(P_. template block<1, 3>(i, idx)) *  res_temp_SO3.transpose();	
				}
			}
		
			Matrix<scalar_type, 2, 2> res_temp_S2;
			MTK::vect<2, scalar_type> seg_S2;
			for (std::vector<std::pair<int, int> >::iterator it = x_.S2_state.begin(); it != x_.S2_state.end(); it++) {
				int idx = (*it).first;
				int dim = (*it).second;
				for(int i = 0; i < 2; i++){
					seg_S2(i) = dx(idx + i);
				}

				Eigen::Matrix<scalar_type, 2, 3> Nx;
				Eigen::Matrix<scalar_type, 3, 2> Mx;
				x_.S2_Nx_yy(Nx, idx);
				x_propagated.S2_Mx(Mx, seg_S2, idx);
				res_temp_S2 = Nx * Mx; 
				dx_new.template block<2, 1>(idx, 0) = res_temp_S2 * dx_new.template block<2, 1>(idx, 0);
				for(int i = 0; i < n; i++){
					P_. template block<2, 1>(idx, i) = res_temp_S2 * (P_. template block<2, 1>(idx, i));	
				}
				for(int i = 0; i < n; i++){
					P_. template block<1, 2>(i, idx) = (P_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
				}
			}

			Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> K_;
			if(n > dof_Measurement)
			{
				#ifdef USE_sparse
				Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> K_temp = h_x_ * P_ * h_x_.transpose();
				spMt R_temp = h_v_ * R_ * h_v_.transpose();
				K_temp += R_temp;
				K_ = P_ * h_x_.transpose() * K_temp.inverse();
			#else
				K_= P_ * h_x_.transpose() * (h_x_ * P_ * h_x_.transpose() + h_v_ * R * h_v_.transpose()).inverse();
			#endif
			}
			else
			{
			#ifdef USE_sparse
				Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> b = Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic>::Identity(dof_Measurement_noise, dof_Measurement_noise);
				Eigen::SparseQR<Eigen::SparseMatrix<scalar_type>, Eigen::COLAMDOrdering<int>> solver; 
				solver.compute(R_);
				Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> R_in_temp = solver.solve(b);
				spMt R_in = R_in_temp.sparseView();
				spMt K_temp = h_x_.transpose() * R_in * h_x_;
				cov P_temp = P_.inverse();
				P_temp += K_temp;
				K_ = P_temp.inverse() * h_x_.transpose() * R_in;
			#else
				Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> R_in = (h_v_*R*h_v_.transpose()).inverse();
				K_ = (h_x_.transpose() * R_in * h_x_ + P_.inverse()).inverse() * h_x_.transpose() * R_in;
			#endif 
			}
			cov K_x = K_ * h_x_;
			Matrix<scalar_type, n, 1> dx_ = K_ * (z - h_) + (K_x - Matrix<scalar_type, n, n>::Identity()) * dx_new; 
			state x_before = x_;
			x_.boxplus(dx_);
			converg = true;
			for(int i = 0; i < n ; i++)
			{
				if(std::fabs(dx_[i]) > limit[i])
				{
					converg = false;
					break;
				}
			}
			if(converg) t++;
			if(t > 1 || i == maximum_iter - 1)
			{
				L_ = P_;
				std::cout << "iteration time:" << t << "," << i << std::endl;
		
				Matrix<scalar_type, 3, 3> res_temp_SO3;
				MTK::vect<3, scalar_type> seg_SO3;
				for(typename std::vector<std::pair<int, int> >::iterator it = x_.SO3_state.begin(); it != x_.SO3_state.end(); it++) {
					int idx = (*it).first;
					for(int i = 0; i < 3; i++){
						seg_SO3(i) = dx_(i + idx);
					}
					res_temp_SO3 = MTK::A_matrix(seg_SO3).transpose();
					for(int i = 0; i < n; i++){
						L_. template block<3, 1>(idx, i) = res_temp_SO3 * (P_. template block<3, 1>(idx, i)); 
					}
					if(n > dof_Measurement)
					{
						for(int i = 0; i < dof_Measurement; i++){
							K_. template block<3, 1>(idx, i) = res_temp_SO3 * (K_. template block<3, 1>(idx, i));
						}
					}
					else
					{
						for(int i = 0; i < n; i++){
							K_x. template block<3, 1>(idx, i) = res_temp_SO3 * (K_x. template block<3, 1>(idx, i));
						}
					}
					for(int i = 0; i < n; i++){
						L_. template block<1, 3>(i, idx) = (L_. template block<1, 3>(i, idx)) * res_temp_SO3.transpose();
						P_. template block<1, 3>(i, idx) = (P_. template block<1, 3>(i, idx)) * res_temp_SO3.transpose();
					}
				}

				Matrix<scalar_type, 2, 2> res_temp_S2;
				MTK::vect<2, scalar_type> seg_S2;
				for(typename std::vector<std::pair<int, int> >::iterator it = x_.S2_state.begin(); it != x_.S2_state.end(); it++) {
					int idx = (*it).first;
			
					for(int i = 0; i < 2; i++){
						seg_S2(i) = dx_(i + idx);
					}
			
					Eigen::Matrix<scalar_type, 2, 3> Nx;
					Eigen::Matrix<scalar_type, 3, 2> Mx;
					x_.S2_Nx_yy(Nx, idx);
					x_propagated.S2_Mx(Mx, seg_S2, idx);
					res_temp_S2 = Nx * Mx; 
		
					for(int i = 0; i < n; i++){
						L_. template block<2, 1>(idx, i) = res_temp_S2 * (P_. template block<2, 1>(idx, i)); 
					}
					if(n > dof_Measurement)
					{
						for(int i = 0; i < dof_Measurement; i++){
							K_. template block<2, 1>(idx, i) = res_temp_S2 * (K_. template block<2, 1>(idx, i));
						}
					}
					else
					{
						for(int i = 0; i < n; i++){
							K_x. template block<2, 1>(idx, i) = res_temp_S2 * (K_x. template block<2, 1>(idx, i));
						}
					}
					for(int i = 0; i < n; i++){
						L_. template block<1, 2>(i, idx) = (L_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
						P_. template block<1, 2>(i, idx) = (P_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
					}
				}
				if(n > dof_Measurement)
				{
					P_ = L_ - K_*h_x_*P_;
				}
				else
				{
					P_ = L_ - K_x * P_;
				}
				return;
			}
		}
	}
	//iterated error state EKF update for measurement as an Eigen matrix whose dimension is changing.
	//calculate measurement (z), estimate measurement (h), partial differention matrices (h_x, h_v) and the noise covariance (R) at the same time, by only one function.
	void update_iterated_dyn_share() {
		
		int t = 0;
		dyn_share_datastruct<scalar_type> dyn_share;
		dyn_share.valid = true;
		dyn_share.converge = true;
		state x_propagated = x_;
		cov P_propagated = P_;
		int dof_Measurement;
		int dof_Measurement_noise;
		for(int i=-1; i<maximum_iter; i++)
		{
			dyn_share.valid = true;
			h_dyn_share (x_,  dyn_share);
			//Matrix<scalar_type, Eigen::Dynamic, 1> h = h_dyn_share (x_,  dyn_share);
			Matrix<scalar_type, Eigen::Dynamic, 1> z = dyn_share.z;
			Matrix<scalar_type, Eigen::Dynamic, 1> h = dyn_share.h;
		#ifdef USE_sparse
			spMt h_x = dyn_share.h_x.sparseView();
			spMt h_v = dyn_share.h_v.sparseView();
			spMt R_ = dyn_share.R.sparseView();
		#else
			Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> R = dyn_share.R; 
			Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> h_x = dyn_share.h_x;
			Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> h_v = dyn_share.h_v;
		#endif	
			dof_Measurement = h_x.rows();
			dof_Measurement_noise = dyn_share.R.rows();
			vectorized_state dx, dx_new;
			x_.boxminus(dx, x_propagated);
			dx_new = dx;
			if(! (dyn_share.valid))
			{
				continue;
			}

			P_ = P_propagated;
			Matrix<scalar_type, 3, 3> res_temp_SO3;
			MTK::vect<3, scalar_type> seg_SO3;
			for (std::vector<std::pair<int, int> >::iterator it = x_.SO3_state.begin(); it != x_.SO3_state.end(); it++) {
				int idx = (*it).first;
				int dim = (*it).second;
				for(int i = 0; i < 3; i++){
					seg_SO3(i) = dx(idx+i);
				}

				res_temp_SO3 = MTK::A_matrix(seg_SO3).transpose();
				dx_new.template block<3, 1>(idx, 0) = res_temp_SO3 * dx_new.template block<3, 1>(idx, 0);
				for(int i = 0; i < n; i++){
					P_. template block<3, 1>(idx, i) = res_temp_SO3 * (P_. template block<3, 1>(idx, i));	
				}
				for(int i = 0; i < n; i++){
					P_. template block<1, 3>(i, idx) =(P_. template block<1, 3>(i, idx)) *  res_temp_SO3.transpose();	
				}
			}
		
			Matrix<scalar_type, 2, 2> res_temp_S2;
			MTK::vect<2, scalar_type> seg_S2;
			for (std::vector<std::pair<int, int> >::iterator it = x_.S2_state.begin(); it != x_.S2_state.end(); it++) {
				int idx = (*it).first;
				int dim = (*it).second;
				for(int i = 0; i < 2; i++){
					seg_S2(i) = dx(idx + i);
				}

				Eigen::Matrix<scalar_type, 2, 3> Nx;
				Eigen::Matrix<scalar_type, 3, 2> Mx;
				x_.S2_Nx_yy(Nx, idx);
				x_propagated.S2_Mx(Mx, seg_S2, idx);
				res_temp_S2 = Nx * Mx; 
				dx_new.template block<2, 1>(idx, 0) = res_temp_S2 * dx_new.template block<2, 1>(idx, 0);
				for(int i = 0; i < n; i++){
					P_. template block<2, 1>(idx, i) = res_temp_S2 * (P_. template block<2, 1>(idx, i));	
				}
				for(int i = 0; i < n; i++){
					P_. template block<1, 2>(i, idx) = (P_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
				}
			}

			Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> K_;
			if(n > dof_Measurement)
			{
			#ifdef USE_sparse
				Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> K_temp = h_x * P_ * h_x.transpose();
				spMt R_temp = h_v * R_ * h_v.transpose();
				K_temp += R_temp;
				K_ = P_ * h_x.transpose() * K_temp.inverse();
			#else
				K_= P_ * h_x.transpose() * (h_x * P_ * h_x.transpose() + h_v * R * h_v.transpose()).inverse();
			#endif
			}
			else
			{
			#ifdef USE_sparse
				Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> b = Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic>::Identity(dof_Measurement_noise, dof_Measurement_noise);
				Eigen::SparseQR<Eigen::SparseMatrix<scalar_type>, Eigen::COLAMDOrdering<int>> solver; 
				solver.compute(R_);
				Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> R_in_temp = solver.solve(b);
				spMt R_in = R_in_temp.sparseView();
				spMt K_temp = h_x.transpose() * R_in * h_x;
				cov P_temp = P_.inverse();
				P_temp += K_temp;
				K_ = P_temp.inverse() * h_x.transpose() * R_in;
			#else
				Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> R_in = (h_v*R*h_v.transpose()).inverse();
				K_ = (h_x.transpose() * R_in * h_x + P_.inverse()).inverse() * h_x.transpose() * R_in;
			#endif 
			}

			cov K_x = K_ * h_x;
			Matrix<scalar_type, n, 1> dx_ = K_ * (z - h) + (K_x - Matrix<scalar_type, n, n>::Identity()) * dx_new; 
			state x_before = x_;
			x_.boxplus(dx_);
			dyn_share.converge = true;
			for(int i = 0; i < n ; i++)
			{
				if(std::fabs(dx_[i]) > limit[i])
				{
					dyn_share.converge = false;
					break;
				}
			}
			if(dyn_share.converge) t++;
			if(t > 1 || i == maximum_iter - 1)
			{
				L_ = P_;
				std::cout << "iteration time:" << t << "," << i << std::endl;
		
				Matrix<scalar_type, 3, 3> res_temp_SO3;
				MTK::vect<3, scalar_type> seg_SO3;
				for(typename std::vector<std::pair<int, int> >::iterator it = x_.SO3_state.begin(); it != x_.SO3_state.end(); it++) {
					int idx = (*it).first;
					for(int i = 0; i < 3; i++){
						seg_SO3(i) = dx_(i + idx);
					}
					res_temp_SO3 = MTK::A_matrix(seg_SO3).transpose();
					for(int i = 0; i < int(n); i++){
						L_. template block<3, 1>(idx, i) = res_temp_SO3 * (P_. template block<3, 1>(idx, i)); 
					}
					if(n > dof_Measurement)
					{
						for(int i = 0; i < dof_Measurement; i++){
							K_. template block<3, 1>(idx, i) = res_temp_SO3 * (K_. template block<3, 1>(idx, i));
						}
					}
					else
					{
						for(int i = 0; i < n; i++){
							K_x. template block<3, 1>(idx, i) = res_temp_SO3 * (K_x. template block<3, 1>(idx, i));
						}
					}
					for(int i = 0; i < n; i++){
						L_. template block<1, 3>(i, idx) = (L_. template block<1, 3>(i, idx)) * res_temp_SO3.transpose();
						P_. template block<1, 3>(i, idx) = (P_. template block<1, 3>(i, idx)) * res_temp_SO3.transpose();
					}
				}

				Matrix<scalar_type, 2, 2> res_temp_S2;
				MTK::vect<2, scalar_type> seg_S2;
				for(typename std::vector<std::pair<int, int> >::iterator it = x_.S2_state.begin(); it != x_.S2_state.end(); it++) {
					int idx = (*it).first;
			
					for(int i = 0; i < 2; i++){
						seg_S2(i) = dx_(i + idx);
					}
			
					Eigen::Matrix<scalar_type, 2, 3> Nx;
					Eigen::Matrix<scalar_type, 3, 2> Mx;
					x_.S2_Nx_yy(Nx, idx);
					x_propagated.S2_Mx(Mx, seg_S2, idx);
					res_temp_S2 = Nx * Mx; 
		
					for(int i = 0; i < n; i++){
						L_. template block<2, 1>(idx, i) = res_temp_S2 * (P_. template block<2, 1>(idx, i)); 
					}
					if(n > dof_Measurement)
					{
						for(int i = 0; i < dof_Measurement; i++){
							K_. template block<2, 1>(idx, i) = res_temp_S2 * (K_. template block<2, 1>(idx, i));
						}
					}
					else
					{
						for(int i = 0; i < n; i++){
							K_x. template block<2, 1>(idx, i) = res_temp_S2 * (K_x. template block<2, 1>(idx, i));
						}
					}
					for(int i = 0; i < n; i++){
						L_. template block<1, 2>(i, idx) = (L_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
						P_. template block<1, 2>(i, idx) = (P_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
					}
				}
				if(n > dof_Measurement)
				{
					P_ = L_ - K_*h_x*P_;
				}
				else
				{
					P_ = L_ - K_x * P_;
				}
				return;
			}
		}
	}

	//iterated error state EKF update for measurement as a dynamic manifold, whose dimension or type is changing.
	//the measurement and the measurement model are received in a dynamic manner.
	template<typename measurement_runtime, typename measurementModel_runtime>
	void update_iterated_dyn_runtime(measurement_runtime z, measurementnoisecovariance_dyn R, measurementModel_runtime h_runtime) {
	
		int t = 0;
		bool valid = true;
		bool converg = true;
		state x_propagated = x_;
		cov P_propagated = P_;
		int dof_Measurement;
		int dof_Measurement_noise;
		for(int i=-1; i<maximum_iter; i++)
		{
			valid = true;
		#ifdef USE_sparse
			spMt h_x_ = h_x_dyn(x_, valid).sparseView();
			spMt h_v_ = h_v_dyn(x_, valid).sparseView();
			spMt R_ = R.sparseView();
		#else
			Matrix<scalar_type, Eigen::Dynamic, n> h_x_ = h_x_dyn(x_, valid);
			Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> h_v_ = h_v_dyn(x_, valid);
		#endif	
			measurement_runtime h_ = h_runtime(x_, valid);
			dof_Measurement = measurement_runtime::DOF;
			dof_Measurement_noise = R.rows();
			vectorized_state dx, dx_new;
			x_.boxminus(dx, x_propagated);
			dx_new = dx;
			if(! valid)
			{
				continue; 
			}

			P_ = P_propagated;
			Matrix<scalar_type, 3, 3> res_temp_SO3;
			MTK::vect<3, scalar_type> seg_SO3;
			for (std::vector<std::pair<int, int> >::iterator it = x_.SO3_state.begin(); it != x_.SO3_state.end(); it++) {
				int idx = (*it).first;
				int dim = (*it).second;
				for(int i = 0; i < 3; i++){
					seg_SO3(i) = dx(idx+i);
				}

				res_temp_SO3 = MTK::A_matrix(seg_SO3).transpose();
				dx_new.template block<3, 1>(idx, 0) = res_temp_SO3 * dx_new.template block<3, 1>(idx, 0);
				for(int i = 0; i < n; i++){
					P_. template block<3, 1>(idx, i) = res_temp_SO3 * (P_. template block<3, 1>(idx, i));	
				}
				for(int i = 0; i < n; i++){
					P_. template block<1, 3>(i, idx) =(P_. template block<1, 3>(i, idx)) *  res_temp_SO3.transpose();	
				}
			}
		
			Matrix<scalar_type, 2, 2> res_temp_S2;
			MTK::vect<2, scalar_type> seg_S2;
			for (std::vector<std::pair<int, int> >::iterator it = x_.S2_state.begin(); it != x_.S2_state.end(); it++) {
				int idx = (*it).first;
				int dim = (*it).second;
				for(int i = 0; i < 2; i++){
					seg_S2(i) = dx(idx + i);
				}

				Eigen::Matrix<scalar_type, 2, 3> Nx;
				Eigen::Matrix<scalar_type, 3, 2> Mx;
				x_.S2_Nx_yy(Nx, idx);
				x_propagated.S2_Mx(Mx, seg_S2, idx);
				res_temp_S2 = Nx * Mx; 
				dx_new.template block<2, 1>(idx, 0) = res_temp_S2 * dx_new.template block<2, 1>(idx, 0);
				for(int i = 0; i < n; i++){
					P_. template block<2, 1>(idx, i) = res_temp_S2 * (P_. template block<2, 1>(idx, i));	
				}
				for(int i = 0; i < n; i++){
					P_. template block<1, 2>(i, idx) = (P_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
				}
			}

			Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> K_;
			if(n > dof_Measurement)
			{
			#ifdef USE_sparse
				Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> K_temp = h_x_ * P_ * h_x_.transpose();
				spMt R_temp = h_v_ * R_ * h_v_.transpose();
				K_temp += R_temp;
				K_ = P_ * h_x_.transpose() * K_temp.inverse();
			#else
				K_= P_ * h_x_.transpose() * (h_x_ * P_ * h_x_.transpose() + h_v_ * R * h_v_.transpose()).inverse();
			#endif
			}
			else
			{
			#ifdef USE_sparse
				Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> b = Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic>::Identity(dof_Measurement_noise, dof_Measurement_noise);
				Eigen::SparseQR<Eigen::SparseMatrix<scalar_type>, Eigen::COLAMDOrdering<int>> solver; 
				solver.compute(R_);
				Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> R_in_temp = solver.solve(b);
				spMt R_in = R_in_temp.sparseView();
				spMt K_temp = h_x_.transpose() * R_in * h_x_;
				cov P_temp = P_.inverse();
				P_temp += K_temp;
				K_ = P_temp.inverse() * h_x_.transpose() * R_in;
			#else
				Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> R_in = (h_v_*R*h_v_.transpose()).inverse();
				K_ = (h_x_.transpose() * R_in * h_x_ + P_.inverse()).inverse() * h_x_.transpose() * R_in;
			#endif 
			}
			cov K_x = K_ * h_x_;
			Eigen::Matrix<scalar_type, measurement_runtime::DOF, 1> innovation;
			z.boxminus(innovation, h_);
			Matrix<scalar_type, n, 1> dx_ = K_ * innovation + (K_x - Matrix<scalar_type, n, n>::Identity()) * dx_new; 
			state x_before = x_;
			x_.boxplus(dx_);
			converg = true;
			for(int i = 0; i < n ; i++)
			{
				if(std::fabs(dx_[i]) > limit[i])
				{
					converg = false;
					break;
				}
			}
			if(converg) t++;
			if(t > 1 || i == maximum_iter - 1)
			{
				L_ = P_;
				std::cout << "iteration time:" << t << "," << i << std::endl;
		
				Matrix<scalar_type, 3, 3> res_temp_SO3;
				MTK::vect<3, scalar_type> seg_SO3;
				for(typename std::vector<std::pair<int, int> >::iterator it = x_.SO3_state.begin(); it != x_.SO3_state.end(); it++) {
					int idx = (*it).first;
					for(int i = 0; i < 3; i++){
						seg_SO3(i) = dx_(i + idx);
					}
					res_temp_SO3 = MTK::A_matrix(seg_SO3).transpose();
					for(int i = 0; i < n; i++){
						L_. template block<3, 1>(idx, i) = res_temp_SO3 * (P_. template block<3, 1>(idx, i)); 
					}
					if(n > dof_Measurement)
					{
						for(int i = 0; i < dof_Measurement; i++){
							K_. template block<3, 1>(idx, i) = res_temp_SO3 * (K_. template block<3, 1>(idx, i));
						}
					}
					else
					{
						for(int i = 0; i < n; i++){
							K_x. template block<3, 1>(idx, i) = res_temp_SO3 * (K_x. template block<3, 1>(idx, i));
						}
					}
					for(int i = 0; i < n; i++){
						L_. template block<1, 3>(i, idx) = (L_. template block<1, 3>(i, idx)) * res_temp_SO3.transpose();
						P_. template block<1, 3>(i, idx) = (P_. template block<1, 3>(i, idx)) * res_temp_SO3.transpose();
					}
				}

				Matrix<scalar_type, 2, 2> res_temp_S2;
				MTK::vect<2, scalar_type> seg_S2;
				for(typename std::vector<std::pair<int, int> >::iterator it = x_.S2_state.begin(); it != x_.S2_state.end(); it++) {
					int idx = (*it).first;
			
					for(int i = 0; i < 2; i++){
						seg_S2(i) = dx_(i + idx);
					}
			
					Eigen::Matrix<scalar_type, 2, 3> Nx;
					Eigen::Matrix<scalar_type, 3, 2> Mx;
					x_.S2_Nx_yy(Nx, idx);
					x_propagated.S2_Mx(Mx, seg_S2, idx);
					res_temp_S2 = Nx * Mx; 
		
					for(int i = 0; i < n; i++){
						L_. template block<2, 1>(idx, i) = res_temp_S2 * (P_. template block<2, 1>(idx, i)); 
					}
					if(n > dof_Measurement)
					{
						for(int i = 0; i < dof_Measurement; i++){
							K_. template block<2, 1>(idx, i) = res_temp_S2 * (K_. template block<2, 1>(idx, i));
						}
					}
					else
					{
						for(int i = 0; i < n; i++){
							K_x. template block<2, 1>(idx, i) = res_temp_S2 * (K_x. template block<2, 1>(idx, i));
						}
					}
					for(int i = 0; i < n; i++){
						L_. template block<1, 2>(i, idx) = (L_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
						P_. template block<1, 2>(i, idx) = (P_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
					}
				}
				if(n > dof_Measurement)
				{
					P_ = L_ - K_*h_x_*P_;
				}
				else
				{
					P_ = L_ - K_x * P_;
				}
				return;
			}
		}
	}

	//iterated error state EKF update for measurement as a dynamic manifold, whose dimension or type is changing.
	//the measurement and the measurement model are received in a dynamic manner.
	//calculate measurement (z), estimate measurement (h), partial differention matrices (h_x, h_v) and the noise covariance (R) at the same time, by only one function.
	template<typename measurement_runtime, typename measurementModel_dyn_runtime_share>
	void update_iterated_dyn_runtime_share(measurement_runtime z, measurementModel_dyn_runtime_share h) {
		
		int t = 0;
		dyn_runtime_share_datastruct<scalar_type> dyn_share;
		dyn_share.valid = true;
		dyn_share.converge = true;
		state x_propagated = x_;
		cov P_propagated = P_;
		int dof_Measurement;
		int dof_Measurement_noise;
		for(int i=-1; i<maximum_iter; i++)
		{
			dyn_share.valid = true;
			measurement_runtime h_ = h(x_,  dyn_share); 
			//measurement_runtime z = dyn_share.z;
		#ifdef USE_sparse
			spMt h_x = dyn_share.h_x.sparseView();
			spMt h_v = dyn_share.h_v.sparseView();
			spMt R_ = dyn_share.R.sparseView();
		#else
			Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> R = dyn_share.R; 
			Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> h_x = dyn_share.h_x;
			Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> h_v = dyn_share.h_v;
		#endif	
			dof_Measurement = measurement_runtime::DOF;
			dof_Measurement_noise = dyn_share.R.rows();
			vectorized_state dx, dx_new;
			x_.boxminus(dx, x_propagated);
			dx_new = dx;
			if(! (dyn_share.valid))
			{
				continue;
			}

			P_ = P_propagated;
			Matrix<scalar_type, 3, 3> res_temp_SO3;
			MTK::vect<3, scalar_type> seg_SO3;
			for (std::vector<std::pair<int, int> >::iterator it = x_.SO3_state.begin(); it != x_.SO3_state.end(); it++) {
				int idx = (*it).first;
				int dim = (*it).second;
				for(int i = 0; i < 3; i++){
					seg_SO3(i) = dx(idx+i);
				}

				res_temp_SO3 = MTK::A_matrix(seg_SO3).transpose();
				dx_new.template block<3, 1>(idx, 0) = res_temp_SO3 * dx_new.template block<3, 1>(idx, 0);
				for(int i = 0; i < n; i++){
					P_. template block<3, 1>(idx, i) = res_temp_SO3 * (P_. template block<3, 1>(idx, i));	
				}
				for(int i = 0; i < n; i++){
					P_. template block<1, 3>(i, idx) =(P_. template block<1, 3>(i, idx)) *  res_temp_SO3.transpose();	
				}
			}
		
			Matrix<scalar_type, 2, 2> res_temp_S2;
			MTK::vect<2, scalar_type> seg_S2;
			for (std::vector<std::pair<int, int> >::iterator it = x_.S2_state.begin(); it != x_.S2_state.end(); it++) {
				int idx = (*it).first;
				int dim = (*it).second;
				for(int i = 0; i < 2; i++){
					seg_S2(i) = dx(idx + i);
				}

				Eigen::Matrix<scalar_type, 2, 3> Nx;
				Eigen::Matrix<scalar_type, 3, 2> Mx;
				x_.S2_Nx_yy(Nx, idx);
				x_propagated.S2_Mx(Mx, seg_S2, idx);
				res_temp_S2 = Nx * Mx; 
				dx_new.template block<2, 1>(idx, 0) = res_temp_S2 * dx_new.template block<2, 1>(idx, 0);
				for(int i = 0; i < n; i++){
					P_. template block<2, 1>(idx, i) = res_temp_S2 * (P_. template block<2, 1>(idx, i));	
				}
				for(int i = 0; i < n; i++){
					P_. template block<1, 2>(i, idx) = (P_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
				}
			}

			Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> K_;
			if(n > dof_Measurement)
			{
			#ifdef USE_sparse
				Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> K_temp = h_x * P_ * h_x.transpose();
				spMt R_temp = h_v * R_ * h_v.transpose();
				K_temp += R_temp;
				K_ = P_ * h_x.transpose() * K_temp.inverse();
			#else
				K_= P_ * h_x.transpose() * (h_x * P_ * h_x.transpose() + h_v * R * h_v.transpose()).inverse();
			#endif
			}
			else
			{
			#ifdef USE_sparse
				Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> b = Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic>::Identity(dof_Measurement_noise, dof_Measurement_noise);
				Eigen::SparseQR<Eigen::SparseMatrix<scalar_type>, Eigen::COLAMDOrdering<int>> solver; 
				solver.compute(R_);
				Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> R_in_temp = solver.solve(b);
				spMt R_in =R_in_temp.sparseView();
				spMt K_temp = h_x.transpose() * R_in * h_x;
				cov P_temp = P_.inverse();
				P_temp += K_temp;
				K_ = P_temp.inverse() * h_x.transpose() * R_in;
			#else
				Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> R_in = (h_v*R*h_v.transpose()).inverse();
				K_ = (h_x.transpose() * R_in * h_x + P_.inverse()).inverse() * h_x.transpose() * R_in;
			#endif 
			}
			cov K_x = K_ * h_x;
			Eigen::Matrix<scalar_type, measurement_runtime::DOF, 1> innovation;
			z.boxminus(innovation, h_);
			Matrix<scalar_type, n, 1> dx_ = K_ * innovation + (K_x - Matrix<scalar_type, n, n>::Identity()) * dx_new; 
			state x_before = x_;
			x_.boxplus(dx_);
			dyn_share.converge = true;
			for(int i = 0; i < n ; i++)
			{
				if(std::fabs(dx_[i]) > limit[i])
				{
					dyn_share.converge = false;
					break;
				}
			}
			if(dyn_share.converge) t++;
			if(t > 1 || i == maximum_iter - 1)
			{
				L_ = P_;
				std::cout << "iteration time:" << t << "," << i << std::endl;
		
				Matrix<scalar_type, 3, 3> res_temp_SO3;
				MTK::vect<3, scalar_type> seg_SO3;
				for(typename std::vector<std::pair<int, int> >::iterator it = x_.SO3_state.begin(); it != x_.SO3_state.end(); it++) {
					int idx = (*it).first;
					for(int i = 0; i < 3; i++){
						seg_SO3(i) = dx_(i + idx);
					}
					res_temp_SO3 = MTK::A_matrix(seg_SO3).transpose();
					for(int i = 0; i < int(n); i++){
						L_. template block<3, 1>(idx, i) = res_temp_SO3 * (P_. template block<3, 1>(idx, i)); 
					}
					if(n > dof_Measurement)
					{
						for(int i = 0; i < dof_Measurement; i++){
							K_. template block<3, 1>(idx, i) = res_temp_SO3 * (K_. template block<3, 1>(idx, i));
						}
					}
					else
					{
						for(int i = 0; i < n; i++){
							K_x. template block<3, 1>(idx, i) = res_temp_SO3 * (K_x. template block<3, 1>(idx, i));
						}
					}
					for(int i = 0; i < n; i++){
						L_. template block<1, 3>(i, idx) = (L_. template block<1, 3>(i, idx)) * res_temp_SO3.transpose();
						P_. template block<1, 3>(i, idx) = (P_. template block<1, 3>(i, idx)) * res_temp_SO3.transpose();
					}
				}

				Matrix<scalar_type, 2, 2> res_temp_S2;
				MTK::vect<2, scalar_type> seg_S2;
				for(typename std::vector<std::pair<int, int> >::iterator it = x_.S2_state.begin(); it != x_.S2_state.end(); it++) {
					int idx = (*it).first;
			
					for(int i = 0; i < 2; i++){
						seg_S2(i) = dx_(i + idx);
					}
			
					Eigen::Matrix<scalar_type, 2, 3> Nx;
					Eigen::Matrix<scalar_type, 3, 2> Mx;
					x_.S2_Nx_yy(Nx, idx);
					x_propagated.S2_Mx(Mx, seg_S2, idx);
					res_temp_S2 = Nx * Mx; 
		
					for(int i = 0; i < n; i++){
						L_. template block<2, 1>(idx, i) = res_temp_S2 * (P_. template block<2, 1>(idx, i)); 
					}
					if(n > dof_Measurement)
					{
						for(int i = 0; i < dof_Measurement; i++){
							K_. template block<2, 1>(idx, i) = res_temp_S2 * (K_. template block<2, 1>(idx, i));
						}
					}
					else
					{
						for(int i = 0; i < n; i++){
							K_x. template block<2, 1>(idx, i) = res_temp_S2 * (K_x. template block<2, 1>(idx, i));
						}
					}
					for(int i = 0; i < n; i++){
						L_. template block<1, 2>(i, idx) = (L_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
						P_. template block<1, 2>(i, idx) = (P_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
					}
				}
				if(n > dof_Measurement)
				{
					P_ = L_ - K_*h_x * P_;
				}
				else
				{
					P_ = L_ - K_x * P_;
				}
				return;
			}
		}
	}
	
	//iterated error state EKF update modified for one specific system.
	void update_iterated_dyn_share_modified(double R, double &solve_time) {
		
		dyn_share_datastruct<scalar_type> dyn_share;
		dyn_share.valid = true;
		dyn_share.converge = true;
		int t = 0;
		state x_propagated = x_;
		cov P_propagated = P_;
		int dof_Measurement; 
		
		Matrix<scalar_type, n, 1> K_h;
		Matrix<scalar_type, n, n> K_x; 
		
		vectorized_state dx_new = vectorized_state::Zero();
		for(int i=-1; i<maximum_iter; i++)
		{
			dyn_share.valid = true;	
			h_dyn_share(x_, dyn_share);

			if(! dyn_share.valid)
			{
				continue; 
			}

			//Matrix<scalar_type, Eigen::Dynamic, 1> h = h_dyn_share(x_, dyn_share);
			#ifdef USE_sparse
				spMt h_x_ = dyn_share.h_x.sparseView();
			#else
				Eigen::Matrix<scalar_type, Eigen::Dynamic, 12> h_x_ = dyn_share.h_x;
			#endif
			double solve_start = omp_get_wtime();
			dof_Measurement = h_x_.rows();
			vectorized_state dx;
			x_.boxminus(dx, x_propagated);
			dx_new = dx;
			
			
			
			P_ = P_propagated;
			
			Matrix<scalar_type, 3, 3> res_temp_SO3;
			MTK::vect<3, scalar_type> seg_SO3;
			for (std::vector<std::pair<int, int> >::iterator it = x_.SO3_state.begin(); it != x_.SO3_state.end(); it++) {
				int idx = (*it).first;
				int dim = (*it).second;
				for(int i = 0; i < 3; i++){
					seg_SO3(i) = dx(idx+i);
				}

				res_temp_SO3 = MTK::A_matrix(seg_SO3).transpose();
				dx_new.template block<3, 1>(idx, 0) = res_temp_SO3 * dx_new.template block<3, 1>(idx, 0);
				for(int i = 0; i < n; i++){
					P_. template block<3, 1>(idx, i) = res_temp_SO3 * (P_. template block<3, 1>(idx, i));	
				}
				for(int i = 0; i < n; i++){
					P_. template block<1, 3>(i, idx) =(P_. template block<1, 3>(i, idx)) *  res_temp_SO3.transpose();	
				}
			}

			Matrix<scalar_type, 2, 2> res_temp_S2;
			MTK::vect<2, scalar_type> seg_S2;
			for (std::vector<std::pair<int, int> >::iterator it = x_.S2_state.begin(); it != x_.S2_state.end(); it++) {
				int idx = (*it).first;
				int dim = (*it).second;
				for(int i = 0; i < 2; i++){
					seg_S2(i) = dx(idx + i);
				}

				Eigen::Matrix<scalar_type, 2, 3> Nx;
				Eigen::Matrix<scalar_type, 3, 2> Mx;
				x_.S2_Nx_yy(Nx, idx);
				x_propagated.S2_Mx(Mx, seg_S2, idx);
				res_temp_S2 = Nx * Mx; 
				dx_new.template block<2, 1>(idx, 0) = res_temp_S2 * dx_new.template block<2, 1>(idx, 0);
				for(int i = 0; i < n; i++){
					P_. template block<2, 1>(idx, i) = res_temp_S2 * (P_. template block<2, 1>(idx, i));	
				}
				for(int i = 0; i < n; i++){
					P_. template block<1, 2>(i, idx) = (P_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
				}
			}
			//Matrix<scalar_type, n, Eigen::Dynamic> K_;
			//Matrix<scalar_type, n, 1> K_h;
			//Matrix<scalar_type, n, n> K_x; 

			/*
			if(n > dof_Measurement)
			{
				K_= P_ * h_x_.transpose() * (h_x_ * P_ * h_x_.transpose()/R + Eigen::Matrix<double, Dynamic, Dynamic>::Identity(dof_Measurement, dof_Measurement)).inverse()/R;
			}
			else
			{
				K_= (h_x_.transpose() * h_x_ + (P_/R).inverse()).inverse()*h_x_.transpose();
			}
			*/

			if(n > dof_Measurement)
			{
			//#ifdef USE_sparse
				//Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> K_temp = h_x * P_ * h_x.transpose();
				//spMt R_temp = h_v * R_ * h_v.transpose();
				//K_temp += R_temp;
				Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> h_x_cur = Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic>::Zero(dof_Measurement, n);
				h_x_cur.topLeftCorner(dof_Measurement, 12) = h_x_;
				/*
				h_x_cur.col(0) = h_x_.col(0);
				h_x_cur.col(1) = h_x_.col(1);
				h_x_cur.col(2) = h_x_.col(2);
				h_x_cur.col(3) = h_x_.col(3);
				h_x_cur.col(4) = h_x_.col(4);
				h_x_cur.col(5) = h_x_.col(5);
				h_x_cur.col(6) = h_x_.col(6);
				h_x_cur.col(7) = h_x_.col(7);
				h_x_cur.col(8) = h_x_.col(8);
				h_x_cur.col(9) = h_x_.col(9);
				h_x_cur.col(10) = h_x_.col(10);
				h_x_cur.col(11) = h_x_.col(11);
				*/
				
				Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> K_ = P_ * h_x_cur.transpose() * (h_x_cur * P_ * h_x_cur.transpose()/R + Eigen::Matrix<double, Dynamic, Dynamic>::Identity(dof_Measurement, dof_Measurement)).inverse()/R;
				K_h = K_ * dyn_share.h;
				K_x = K_ * h_x_cur;
			//#else
			//	K_= P_ * h_x.transpose() * (h_x * P_ * h_x.transpose() + h_v * R * h_v.transpose()).inverse();
			//#endif
			}
			else
			{
			#ifdef USE_sparse
				//Eigen::Matrix<scalar_type, n, n> b = Eigen::Matrix<scalar_type, n, n>::Identity();
				//Eigen::SparseQR<Eigen::SparseMatrix<scalar_type>, Eigen::COLAMDOrdering<int>> solver; 
				spMt A = h_x_.transpose() * h_x_;
				cov P_temp = (P_/R).inverse();
				P_temp. template block<12, 12>(0, 0) += A;
				P_temp = P_temp.inverse();
				/*
				Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> h_x_cur = Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic>::Zero(dof_Measurement, n);
				h_x_cur.col(0) = h_x_.col(0);
				h_x_cur.col(1) = h_x_.col(1);
				h_x_cur.col(2) = h_x_.col(2);
				h_x_cur.col(3) = h_x_.col(3);
				h_x_cur.col(4) = h_x_.col(4);
				h_x_cur.col(5) = h_x_.col(5);
				h_x_cur.col(6) = h_x_.col(6);
				h_x_cur.col(7) = h_x_.col(7);
				h_x_cur.col(8) = h_x_.col(8);
				h_x_cur.col(9) = h_x_.col(9);
				h_x_cur.col(10) = h_x_.col(10);
				h_x_cur.col(11) = h_x_.col(11);
				*/
				K_ = P_temp. template block<n, 12>(0, 0) * h_x_.transpose();
				K_x = cov::Zero();
				K_x. template block<n, 12>(0, 0) = P_inv. template block<n, 12>(0, 0) * HTH;
				/*
				solver.compute(R_);
				Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> R_in_temp = solver.solve(b);
				spMt R_in =R_in_temp.sparseView();
				spMt K_temp = h_x.transpose() * R_in * h_x;
				cov P_temp = P_.inverse();
				P_temp += K_temp;
				K_ = P_temp.inverse() * h_x.transpose() * R_in;
				*/
			#else
				cov P_temp = (P_/R).inverse();
				//Eigen::Matrix<scalar_type, 12, Eigen::Dynamic> h_T = h_x_.transpose();
				Eigen::Matrix<scalar_type, 12, 12> HTH = h_x_.transpose() * h_x_; 
				P_temp. template block<12, 12>(0, 0) += HTH;
				/*
				Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> h_x_cur = Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic>::Zero(dof_Measurement, n);
				//std::cout << "line 1767" << std::endl;
				h_x_cur.col(0) = h_x_.col(0);
				h_x_cur.col(1) = h_x_.col(1);
				h_x_cur.col(2) = h_x_.col(2);
				h_x_cur.col(3) = h_x_.col(3);
				h_x_cur.col(4) = h_x_.col(4);
				h_x_cur.col(5) = h_x_.col(5);
				h_x_cur.col(6) = h_x_.col(6);
				h_x_cur.col(7) = h_x_.col(7);
				h_x_cur.col(8) = h_x_.col(8);
				h_x_cur.col(9) = h_x_.col(9);
				h_x_cur.col(10) = h_x_.col(10);
				h_x_cur.col(11) = h_x_.col(11);
				*/
				cov P_inv = P_temp.inverse();
				//std::cout << "line 1781" << std::endl;
				K_h = P_inv. template block<n, 12>(0, 0) * h_x_.transpose() * dyn_share.h;
				//std::cout << "line 1780" << std::endl;
				//cov HTH_cur = cov::Zero();
				//HTH_cur. template block<12, 12>(0, 0) = HTH;
				K_x.setZero(); // = cov::Zero();
				K_x. template block<n, 12>(0, 0) = P_inv. template block<n, 12>(0, 0) * HTH;
				//K_= (h_x_.transpose() * h_x_ + (P_/R).inverse()).inverse()*h_x_.transpose();
			#endif 
			}

			//K_x = K_ * h_x_;
			Matrix<scalar_type, n, 1> dx_ = K_h + (K_x - Matrix<scalar_type, n, n>::Identity()) * dx_new; 
			state x_before = x_;
			x_.boxplus(dx_);
			dyn_share.converge = true;
			for(int i = 0; i < n ; i++)
			{
				if(std::fabs(dx_[i]) > limit[i])
				{
					dyn_share.converge = false;
					break;
				}
			}
			if(dyn_share.converge) t++;
			
			if(!t && i == maximum_iter - 2)
			{
				dyn_share.converge = true;
			}

			if(t > 1 || i == maximum_iter - 1)
			{
				L_ = P_;
				//std::cout << "iteration time" << t << "," << i << std::endl; 
				Matrix<scalar_type, 3, 3> res_temp_SO3;
				MTK::vect<3, scalar_type> seg_SO3;
				for(typename std::vector<std::pair<int, int> >::iterator it = x_.SO3_state.begin(); it != x_.SO3_state.end(); it++) {
					int idx = (*it).first;
					for(int i = 0; i < 3; i++){
						seg_SO3(i) = dx_(i + idx);
					}
					res_temp_SO3 = MTK::A_matrix(seg_SO3).transpose();
					for(int i = 0; i < n; i++){
						L_. template block<3, 1>(idx, i) = res_temp_SO3 * (P_. template block<3, 1>(idx, i)); 
					}
					// if(n > dof_Measurement)
					// {
					// 	for(int i = 0; i < dof_Measurement; i++){
					// 		K_.template block<3, 1>(idx, i) = res_temp_SO3 * (K_. template block<3, 1>(idx, i));
					// 	}
					// }
					// else
					// {
						for(int i = 0; i < 12; i++){
							K_x. template block<3, 1>(idx, i) = res_temp_SO3 * (K_x. template block<3, 1>(idx, i));
						}
					//}
					for(int i = 0; i < n; i++){
						L_. template block<1, 3>(i, idx) = (L_. template block<1, 3>(i, idx)) * res_temp_SO3.transpose();
						P_. template block<1, 3>(i, idx) = (P_. template block<1, 3>(i, idx)) * res_temp_SO3.transpose();
					}
				}

				Matrix<scalar_type, 2, 2> res_temp_S2;
				MTK::vect<2, scalar_type> seg_S2;
				for(typename std::vector<std::pair<int, int> >::iterator it = x_.S2_state.begin(); it != x_.S2_state.end(); it++) {
					int idx = (*it).first;

					for(int i = 0; i < 2; i++){
						seg_S2(i) = dx_(i + idx);
					}

					Eigen::Matrix<scalar_type, 2, 3> Nx;
					Eigen::Matrix<scalar_type, 3, 2> Mx;
					x_.S2_Nx_yy(Nx, idx);
					x_propagated.S2_Mx(Mx, seg_S2, idx);
					res_temp_S2 = Nx * Mx; 
					for(int i = 0; i < n; i++){
						L_. template block<2, 1>(idx, i) = res_temp_S2 * (P_. template block<2, 1>(idx, i)); 
					}
					// if(n > dof_Measurement)
					// {
					// 	for(int i = 0; i < dof_Measurement; i++){
					// 		K_. template block<2, 1>(idx, i) = res_temp_S2 * (K_. template block<2, 1>(idx, i));
					// 	}
					// }
					// else
					// {
						for(int i = 0; i < 12; i++){
							K_x. template block<2, 1>(idx, i) = res_temp_S2 * (K_x. template block<2, 1>(idx, i));
						}
					//}
					for(int i = 0; i < n; i++){
						L_. template block<1, 2>(i, idx) = (L_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
						P_. template block<1, 2>(i, idx) = (P_. template block<1, 2>(i, idx)) * res_temp_S2.transpose();
					}
				}

				// if(n > dof_Measurement)
				// {
				// 	Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic> h_x_cur = Eigen::Matrix<scalar_type, Eigen::Dynamic, Eigen::Dynamic>::Zero(dof_Measurement, n);
				// 	h_x_cur.topLeftCorner(dof_Measurement, 12) = h_x_;
				// 	/*
				// 	h_x_cur.col(0) = h_x_.col(0);
				// 	h_x_cur.col(1) = h_x_.col(1);
				// 	h_x_cur.col(2) = h_x_.col(2);
				// 	h_x_cur.col(3) = h_x_.col(3);
				// 	h_x_cur.col(4) = h_x_.col(4);
				// 	h_x_cur.col(5) = h_x_.col(5);
				// 	h_x_cur.col(6) = h_x_.col(6);
				// 	h_x_cur.col(7) = h_x_.col(7);
				// 	h_x_cur.col(8) = h_x_.col(8);
				// 	h_x_cur.col(9) = h_x_.col(9);
				// 	h_x_cur.col(10) = h_x_.col(10);
				// 	h_x_cur.col(11) = h_x_.col(11);
				// 	*/
				// 	P_ = L_ - K_*h_x_cur * P_;
				// }
				// else
				//{
					P_ = L_ - K_x.template block<n, 12>(0, 0) * P_.template block<12, n>(0, 0);
				//}
				solve_time += omp_get_wtime() - solve_start;
				return;
			}
			solve_time += omp_get_wtime() - solve_start;
		}
	}

	void change_x(state &input_state)
	{
		x_ = input_state;
		if((!x_.vect_state.size())&&(!x_.SO3_state.size())&&(!x_.S2_state.size()))
		{
			x_.build_S2_state();
			x_.build_SO3_state();
			x_.build_vect_state();
		}
	}

	void change_P(cov &input_cov)
	{
		P_ = input_cov;
	}

	const state& get_x() const {
		return x_;
	}
	const cov& get_P() const {
		return P_;
	}
private:
	state x_;
	measurement m_;
	cov P_;
	spMt l_;
	spMt f_x_1;
	spMt f_x_2;
	cov F_x1 = cov::Identity();
	cov F_x2 = cov::Identity();
	cov L_ = cov::Identity();

	processModel *f;
	processMatrix1 *f_x;
	processMatrix2 *f_w;

	measurementModel *h;
	measurementMatrix1 *h_x;
	measurementMatrix2 *h_v;

	measurementModel_dyn *h_dyn;
	measurementMatrix1_dyn *h_x_dyn;
	measurementMatrix2_dyn *h_v_dyn;

	measurementModel_share *h_share;
	measurementModel_dyn_share *h_dyn_share;

	int maximum_iter = 0;
	scalar_type limit[n];
	
	template <typename T>
    T check_safe_update( T _temp_vec )
    {
        T temp_vec = _temp_vec;
        if ( std::isnan( temp_vec(0, 0) ) )
        {
            temp_vec.setZero();
            return temp_vec;
        }
        double angular_dis = temp_vec.block( 0, 0, 3, 1 ).norm() * 57.3;
        double pos_dis = temp_vec.block( 3, 0, 3, 1 ).norm();
        if ( angular_dis >= 20 || pos_dis > 1 )
        {
            printf( "Angular dis = %.2f, pos dis = %.2f\r\n", angular_dis, pos_dis );
            temp_vec.setZero();
        }
        return temp_vec;
    }
public:
	EIGEN_MAKE_ALIGNED_OPERATOR_NEW
};

} // namespace esekfom

#endif //  ESEKFOM_EKF_HPP