From 8307ef7d9911e6b5dae3d931d4c529ab84f39f28 Mon Sep 17 00:00:00 2001
From: Shamil Mamedov <sh.mamedov@innopolis.ru>
Date: Thu, 26 Dec 2019 12:09:57 +0300
Subject: [PATCH] identification of drive parameters fails even in simluation
 with some noise. We thoughts that weighted least sqaures can help, but for
 now it doesn't. It seems that we don't do it right.

---
 data_ur10/data_pltng.m         |   82 +-
 data_ur10/data_prcsng.m        |  161 +--
 data_ur10/ur-19_12_23_free.csv | 2036 +++++++++++++++++++++++++++++++
 data_ur10/ur-19_12_23_load.csv | 2089 ++++++++++++++++++++++++++++++++
 regressorWithMotorDynamics.m   |    6 +-
 ur10_indntfcn_real.m           |  137 ++-
 ur10_indntfcn_smltn.m          |  230 +++-
 ur10_indntfcn_smltn.m~         |  464 +++++++
 ur_exprmt_dsgn.m               |    6 +-
 9 files changed, 4991 insertions(+), 220 deletions(-)
 create mode 100644 data_ur10/ur-19_12_23_free.csv
 create mode 100644 data_ur10/ur-19_12_23_load.csv
 create mode 100644 ur10_indntfcn_smltn.m~

diff --git a/data_ur10/data_pltng.m b/data_ur10/data_pltng.m
index 342d6b0..f4dadb9 100644
--- a/data_ur10/data_pltng.m
+++ b/data_ur10/data_pltng.m
@@ -1,105 +1,102 @@
-% ------------------------------------------------------------------------
-% Positions
-% ------------------------------------------------------------------------
+% ---------------------------------------------------------------------
+% In this script we plot data processing of real data from UR
+% ---------------------------------------------------------------------
+trajectory = unloadedTrajectory; % choose trajectory
+
+%% Positions
+% Ploting ideal posititions and obtained ones
 % %{ 
 figure
 subplot(2,1,1)
     hold on
     for i = 1:3
-        plot(t_msrd, q_msrd(:,i))
-        plot(traj_par.t, q(i,:), '--')
+        plot(trajectory.t, trajectory.q(:,i))
     end
     xlabel('$t$,\ sec','interpreter','latex')
-    legend('$q_1$, rad','$q_2$, rad','$q_3$, rad','interpreter','latex')
+    legend('$q_1$','$q_2$','$q_3$','interpreter','latex')
     grid on
 subplot(2,1,2)
     hold on
     for i = 4:6
-        plot(t_msrd, qd_msrd(:,i))
-        plot(traj_par.t, qd(i,:), '--')
+        plot(trajectory.t, trajectory.q(:,i))
     end
     xlabel('$t$,\ sec','interpreter','latex')
-    legend('$q_4$, rad','$q_5$, rad','$q_6$, rad','interpreter','latex')
+    legend('$q_4$','$q_5$','$q_6$','interpreter','latex')
     grid on
 %}
-% ------------------------------------------------------------------------
-% Velocities
-% ------------------------------------------------------------------------
-%{
+
+%% Velocities
+% %{
 figure
 subplot(2,1,1)
     hold on
     for i = 1:3
-        plot(t_msrd, qd_msrd(:,i))
-        plot(traj_par.t, qd(i,:), '--')
+        plot(trajectory.t, trajectory.qd(:,i))
+        plot(trajectory.t, trajectory.qd_fltrd(:,i))
     end
     xlabel('$t$,\ sec','interpreter','latex')
-    legend('$\dot{q}_1$, rad/s','$\dot{q}_2$, rad/s','$\dot{q}_3$, rad/s',...
+    legend('$\dot{q}_1$','$\dot{q}_2$','$\dot{q}_3$',...
                 'interpreter','latex')
     grid on
 subplot(2,1,2)
     hold on
     for i = 4:6
-        plot(t_msrd, qd_msrd(:,i))
-        plot(traj_par.t, qd(i,:), '--')
+        plot(trajectory.t, trajectory.qd(:,i))
+        plot(trajectory.t, trajectory.qd_fltrd(:,i))
     end
     xlabel('$t$,\ sec','interpreter','latex')
-    legend('$\dot{q}_4$, rad/s','$\dot{q}_5$, rad/s','$\dot{q}_6$, rad/s',...
+    legend('$\dot{q}_4$','$\dot{q}_5$','$\dot{q}_6$',...
                 'interpreter','latex')
     grid on
 %}
 
-% ------------------------------------------------------------------------
-% Accelerations
-% ------------------------------------------------------------------------
-%{
+%% Accelerations
+% %{
 figure
 subplot(2,1,1)
     hold on
     for i = 1:3
-        plot(t_msrd, q2d_est(:,i))
-        plot(traj_par.t, q2d(i,:), '--')
+        plot(trajectory.t, trajectory.q2d_est(:,i))
     end
     xlabel('$t$,\ sec','interpreter','latex')
     ylabel('$\ddot{q}$,\ rad','interpreter','latex')
+    legend('$\ddot{q}_1$','$\ddot{q}_2$','$\ddot{q}_3$',...
+                'interpreter','latex')
     ylim([-2.5, 2.5])
     grid on
 subplot(2,1,2)
     hold on
     for i = 4:6
-        plot(t_msrd, q2d_est(:,i))
-        plot(traj_par.t, q2d(i,:), '--')
+        plot(trajectory.t, trajectory.q2d_est(:,i))
     end
     xlabel('$t$,\ sec','interpreter','latex')
     ylabel('$\ddot{q}$,\ rad','interpreter','latex')
+    legend('$\ddot{q}_4$','$\ddot{q}_5$','$\ddot{q}_6$',...
+                'interpreter','latex')
     ylim([-2.5, 2.5])
     grid on
 %} 
 
-% ------------------------------------------------------------------------
-% measured torque vs desired torque
-% ------------------------------------------------------------------------
-%{
+%% measured currents vs desired currents
+% %{
 for i = 1:6
     figure
-    plot(t_msrd,i_msrd(:,i))
+    plot(trajectory.t,trajectory.i(:,i))
     hold on
-    plot(t_msrd,i_des(:,i))
+    plot(trajectory.t,trajectory.i_des(:,i))
     grid on
     legend('msrd','dsrd','interpreter','latex')
 end
 %}
 
-% ------------------------------------------------------------------------
-% currents vs Filteres currents
-% ------------------------------------------------------------------------
-%{
+%% currents vs Filteres currents
+% %{
 figure
 subplot(2,1,1)
     hold on
     for i = 1:3
-        plot(t_msrd,i_msrd(:,i))
-        plot(t_msrd,i_fltrd(:,i))
+        plot(trajectory.t,trajectory.i(:,i))
+        plot(trajectory.t,trajectory.i_fltrd(:,i))
     end
     xlabel('$t$,\ sec','interpreter','latex')
     legend('$i_1$, A','$i_2$, A','$i_3$, A','interpreter','latex')
@@ -107,14 +104,11 @@ subplot(2,1,1)
 subplot(2,1,2)
     hold on
     for i = 4:6
-        plot(t_msrd,i_msrd(:,i))
-        plot(t_msrd,i_fltrd(:,i))
+        plot(trajectory.t,trajectory.i(:,i))
+        plot(trajectory.t,trajectory.i_fltrd(:,i))
     end
     xlabel('$t$,\ sec','interpreter','latex')
     legend('$i_4$, A','$i_5$, A','$i_6$, A','interpreter','latex')
     grid on
 
 %}
-
-
-
diff --git a/data_ur10/data_prcsng.m b/data_ur10/data_prcsng.m
index 9356335..cec5589 100644
--- a/data_ur10/data_prcsng.m
+++ b/data_ur10/data_prcsng.m
@@ -3,40 +3,44 @@ clc; clear all; close all;
 % ------------------------------------------------------------------------
 % Load and calculate desired trajectory
 % ------------------------------------------------------------------------
-load('opt_sltn_N5T20.mat');
+load('ptrnSrch_N5T20QR.mat');
 [q,qd,q2d] = mixed_traj(traj_par.t,c_pol,a,b,traj_par.wf,traj_par.N);
 
 
 % ------------------------------------------------------------------------
 % Load the real trajectory of the robot without load
 % ------------------------------------------------------------------------
-traj_data = csvread('ur-19_10_29-20_sec.csv');
+traj_data = csvread('ur-19_12_23_free.csv');
 
-int_idx = 256; % get the point when the trajectory begins
-fnl_idx = 2158;
+int_idx = 1; % get the point when the trajectory begins
+fnl_idx = 2016;
 
-t_msrd = traj_data(int_idx:fnl_idx,1) - traj_data(int_idx,1); %subtract offset
-q_msrd = traj_data(int_idx:fnl_idx,2:7);
-qd_msrd = traj_data(int_idx:fnl_idx,8:13);
-i_msrd = traj_data(int_idx:fnl_idx,14:19);
-i_des = traj_data(int_idx:fnl_idx,20:25);
-tau_des = traj_data(int_idx:fnl_idx,26:31);
+unloadedTrajectory = struct; % structure with trajectory data
+
+unloadedTrajectory.t = traj_data(int_idx:fnl_idx,1) - traj_data(int_idx,1);
+unloadedTrajectory.q = traj_data(int_idx:fnl_idx,2:7);
+unloadedTrajectory.qd = traj_data(int_idx:fnl_idx,8:13);
+unloadedTrajectory.i = traj_data(int_idx:fnl_idx,14:19);
+unloadedTrajectory.i_des = traj_data(int_idx:fnl_idx,20:25);
+unloadedTrajectory.tau_des = traj_data(int_idx:fnl_idx,26:31);
 
 
 % -----------------------------------------------------------------------
 % Load the real trajectory of the robot with load
 % -----------------------------------------------------------------------
-traj_ldd = csvread('ur-19_11_05-20_sec_mass_2.csv');
+traj_ldd = csvread('ur-19_12_23_load.csv');
 
-int_idx_ldd = 8;
-fnl_idx_ldd = 1877;
+int_idx_ldd = 1;
+fnl_idx_ldd = 2040;
 
-t_msrd_ldd = traj_ldd(int_idx_ldd:fnl_idx_ldd,1) - traj_ldd(int_idx_ldd,1); %subtract offset
-q_msrd_ldd = traj_ldd(int_idx_ldd:fnl_idx_ldd,2:7);
-qd_msrd_ldd = traj_ldd(int_idx_ldd:fnl_idx_ldd,8:13);
-i_msrd_ldd = traj_ldd(int_idx_ldd:fnl_idx_ldd,14:19);
-i_des_ldd = traj_ldd(int_idx_ldd:fnl_idx_ldd,20:25);
-tau_des_ldd = traj_ldd(int_idx_ldd:fnl_idx_ldd,26:31);
+loadedTrajectory = struct;
+
+loadedTrajectory.t = traj_ldd(int_idx_ldd:fnl_idx_ldd,1) - traj_ldd(int_idx_ldd,1); 
+loadedTrajectory.q = traj_ldd(int_idx_ldd:fnl_idx_ldd,2:7);
+loadedTrajectory.qd = traj_ldd(int_idx_ldd:fnl_idx_ldd,8:13);
+loadedTrajectory.i = traj_ldd(int_idx_ldd:fnl_idx_ldd,14:19);
+loadedTrajectory.i_des = traj_ldd(int_idx_ldd:fnl_idx_ldd,20:25);
+loadedTrajectory.tau_des = traj_ldd(int_idx_ldd:fnl_idx_ldd,26:31);
 
 
 % ------------------------------------------------------------------------
@@ -46,14 +50,14 @@ tau_des_ldd = traj_ldd(int_idx_ldd:fnl_idx_ldd,26:31);
 vel_filt = designfilt('lowpassiir','FilterOrder',3, ...
         'HalfPowerFrequency',0.2,'DesignMethod','butter');
     
-qd_fltrd = zeros(size(qd_msrd));
+unloadedTrajectory.qd_fltrd = zeros(size(unloadedTrajectory.qd));
 for i = 1:6
-    qd_fltrd(:,i) = filtfilt(vel_filt,qd_msrd(:,i));
+    unloadedTrajectory.qd_fltrd(:,i) = filtfilt(vel_filt,unloadedTrajectory.qd(:,i));
 end
 
-qd_fltrd_ldd = zeros(size(qd_msrd_ldd));
+loadedTrajectory.qd_fltrd = zeros(size(loadedTrajectory.qd));
 for i = 1:6
-    qd_fltrd_ldd(:,i) = filtfilt(vel_filt,qd_msrd_ldd(:,i));
+    loadedTrajectory.qd_fltrd(:,i) = filtfilt(vel_filt,loadedTrajectory.qd(:,i));
 end
 
 
@@ -61,31 +65,34 @@ end
 % Estimating accelerations
 % ------------------------------------------------------------------------
 % Three point central difference
-q2d_est = zeros(size(qd_fltrd));
-for i = 2:length(qd_fltrd)-1
-   dlta_qd_fltrd =  qd_fltrd(i+1,:) -  qd_fltrd(i-1,:);
-   dlta_t_msrd = t_msrd(i+1) - t_msrd(i-1);
-   q2d_est(i,:) = dlta_qd_fltrd/dlta_t_msrd;
+unloadedTrajectory.q2d_est = zeros(size(unloadedTrajectory.qd_fltrd));
+for i = 2:length(unloadedTrajectory.qd_fltrd)-1
+   dlta_qd_fltrd =  unloadedTrajectory.qd_fltrd(i+1,:) -  ...
+                        unloadedTrajectory.qd_fltrd(i-1,:);
+   dlta_t_msrd = unloadedTrajectory.t(i+1) - unloadedTrajectory.t(i-1);
+   unloadedTrajectory.q2d_est(i,:) = dlta_qd_fltrd/dlta_t_msrd;
 end
 
-q2d_est_ldd = zeros(size(qd_fltrd_ldd));
-for i = 2:length(qd_fltrd_ldd)-1
-   dlta_qd_fltrd_ldd =  qd_fltrd_ldd(i+1,:) -  qd_fltrd_ldd(i-1,:);
-   dlta_t_msrd_ldd = t_msrd_ldd(i+1) - t_msrd_ldd(i-1);
-   q2d_est_ldd(i,:) = dlta_qd_fltrd_ldd/dlta_t_msrd_ldd;
+loadedTrajectory.q2d_est = zeros(size(loadedTrajectory.qd_fltrd));
+for i = 2:length(loadedTrajectory.qd_fltrd)-1
+   dlta_qd_fltrd_ldd =  loadedTrajectory.qd_fltrd(i+1,:) - ...
+                            loadedTrajectory.qd_fltrd(i-1,:);
+   dlta_t_msrd_ldd = loadedTrajectory.t(i+1) - loadedTrajectory.t(i-1);
+   loadedTrajectory.q2d_est(i,:) = dlta_qd_fltrd_ldd/dlta_t_msrd_ldd;
 end
 
 % Zeros phase filtering acceleration obtained by finite difference
 accel_filt = designfilt('lowpassiir','FilterOrder',5, ...
         'HalfPowerFrequency',0.05,'DesignMethod','butter');
 for i = 1:6
-    q2d_est(:,i) = filtfilt(accel_filt,q2d_est(:,i));
+    unloadedTrajectory.q2d_est(:,i) = filtfilt(accel_filt,unloadedTrajectory.q2d_est(:,i));
 end
 
 for i = 1:6
-    q2d_est_ldd(:,i) = filtfilt(accel_filt,q2d_est_ldd(:,i));
+    loadedTrajectory.q2d_est(:,i) = filtfilt(accel_filt,loadedTrajectory.q2d_est(:,i));
 end
 
+
 % ------------------------------------------------------------------------
 % Filtering current
 % ------------------------------------------------------------------------
@@ -94,90 +101,10 @@ curr_filt = designfilt('lowpassiir','FilterOrder',5, ...
         'HalfPowerFrequency',0.1,'DesignMethod','butter');
 
 for i = 1:6
-    i_fltrd(:,i) = filtfilt(curr_filt,i_msrd(:,i));
+    unloadedTrajectory.i_fltrd(:,i) = filtfilt(curr_filt,unloadedTrajectory.i(:,i));
 end
 
 for i = 1:6
-    i_fltrd_ldd(:,i) = filtfilt(curr_filt,i_msrd_ldd(:,i));
+    loadedTrajectory.i_fltrd(:,i) = filtfilt(curr_filt,loadedTrajectory.i(:,i));
 end
 
-
-
-
-
-
-return
-% ------------------------------------------------------------------------
-% Construncting regressor and performing least squares
-% ------------------------------------------------------------------------
-
-W1 = []; W2 = []; W3 = []; W4 = []; W5 = []; W6 = [];
-I1 = []; I2 = []; I3 = []; I4 = []; I5 = []; I6 = [];
-
-n1 = 1;
-n2 = 2;
-n3 = 9;
-n4 = 16;
-n5 = 23;
-n6 = 30;
-xi = 5e-4;
-
-for i = 1:length(t_msrd)
-    Yi = base_regressor(q_msrd(i,:)',qd_fltrd(i,:)',q2d_est(i,:)');
-% Complementing regressor with friction terms
-
-%     drvi = ur10_drv_rgsr(qd_fltrd(i,:)');
-    drvi = ur10_drv_rgsr(q2d_est(i,:)');
-    frctni = ur10_frctn_rgsr(qd_fltrd(i,:)');
-    
-% As drive gains are unknown we want to build regressor for each joint in
-% order to identify paramters as seen from this joint point of view
-    Yi_l1 = Yi(:,1);
-    Yi_l2 = Yi(:,2:8);
-    Yi_l3 = Yi(:,9:15);
-    Yi_l4 = Yi(:,16:22);
-    Yi_l5 = Yi(:,23:29);
-    Yi_l6 = Yi(:,30:36);
-           
-%     Yi_j2 = [Yi_l2(2,:), drvi(2), Yi_l3(2,:), Yi_l4(2,:), Yi_l5(2,:), Yi_l6(2,:)];
-    Yi_j2 = [Yi_l2(2,:), Yi_l3(2,:), Yi_l4(2,:), Yi_l5(2,:), Yi_l6(2,:)];
-    Yi_j3 = [Yi_l3(3,:), drvi(3), Yi_l4(3,:), Yi_l5(3,:), Yi_l6(3,:)];
-    Yi_j4 = [Yi_l4(4,:), drvi(4), Yi_l5(4,:), Yi_l6(4,:)];
-    Yi_j5 = [Yi_l5(5,:), drvi(5), Yi_l6(5,:)];
-    Yi_j6 = [Yi_l6(6,:), drvi(6)];
-    
-%     W1 = vertcat(W1,[Yi(1,n1:4),Yi(1,6:11),Yi(1,13:18),Yi(1,20:end), frctni(1,:)]);
-%     W2 = vertcat(W2,[Yi(2,n2:end),frctni(2,:)]);
-%     W3 = vertcat(W3,[Yi(3,n3:end),frctni(3,:)]);
-%     W4 = vertcat(W4,[Yi(4,n4:end),frctni(4,:)]);
-%     W5 = vertcat(W5,[Yi(5,n5:end),frctni(5,:)]);
-%     W6 = vertcat(W6,[Yi(6,n6:end),frctni(6,:)]);
-    
-    W2 = vertcat(W2,[Yi_j2, frctni(2,:)]);
-    W3 = vertcat(W3,[Yi_j3, frctni(3,:)]);
-    W4 = vertcat(W4,[Yi_j4, frctni(4,:)]);
-    W5 = vertcat(W5,[Yi_j5, frctni(5,:)]);
-    W6 = vertcat(W6,[Yi_j6, frctni(6,:)]);
-
-%     I1 = vertcat(I1,i_fltrd(i,1));
-    I2 = vertcat(I2,i_fltrd(i,2));
-    I3 = vertcat(I3,i_fltrd(i,3));
-    I4 = vertcat(I4,i_fltrd(i,4));
-    I5 = vertcat(I5,i_fltrd(i,5));
-    I6 = vertcat(I6,i_fltrd(i,6));
-end
-
-% pi1_hat = (W1'*W1)\(W1'*I1);
-pi2_hat = (W2'*W2)\(W2'*I2);
-pi3_hat = (W3'*W3)\(W3'*I3);
-pi4_hat = (W4'*W4)\(W4'*I4);
-pi5_hat = (W5'*W5)\(W5'*I5);
-pi6_hat = (W6'*W6)\(W6'*I6);
-
-
-
-
-
-
-
-
diff --git a/data_ur10/ur-19_12_23_free.csv b/data_ur10/ur-19_12_23_free.csv
new file mode 100644
index 0000000..c3c8fdd
--- /dev/null
+++ b/data_ur10/ur-19_12_23_free.csv
@@ -0,0 +1,2036 @@
+446.4150,-0.0000,-1.5707,-0.0000,-1.5708,0.0000,0.0000,-0.0000,-0.0000,0.0000,0.0000,0.0000,0.0000,-0.1218,-0.5965,-0.1634,-0.1541,-0.0634,0.1899,0.2638,-0.0549,0.0826,-0.0094,0.0227,0.0313,1.1319,-0.2874,0.2702,-0.0226,0.0527,0.0775
+446.4250,0.0000,-1.5708,0.0000,-1.5708,-0.0000,0.0000,0.0013,-0.0000,0.0012,0.0000,0.0000,0.0015,0.4204,-0.7715,-0.0838,-0.2022,-0.0482,0.2459,0.5138,-0.1010,0.1636,-0.0202,0.0489,0.0668,1.1319,-0.2925,0.2671,-0.0229,0.0527,0.0775
+446.4350,0.0001,-1.5708,0.0000,-1.5709,0.0000,0.0000,0.0039,-0.0008,0.0010,0.0000,0.0000,0.0044,0.8786,-0.7488,0.0796,-0.1969,0.0095,0.2487,0.7637,-0.1465,0.2451,-0.0310,0.0752,0.1023,1.1319,-0.2904,0.2681,-0.0228,0.0527,0.0775
+446.4450,0.0001,-1.5708,0.0001,-1.5708,0.0001,0.0001,0.0071,-0.0009,0.0031,0.0000,0.0031,0.0068,0.9813,-0.8976,0.1840,-0.2085,0.0217,0.2413,1.0137,-0.1918,0.3266,-0.0418,0.1014,0.1378,1.1319,-0.2859,0.2697,-0.0226,0.0527,0.0775
+446.4570,0.0002,-1.5708,0.0001,-1.5708,0.0001,0.0002,0.0097,-0.0018,0.0061,-0.0015,0.0054,0.0078,0.9089,-0.8313,0.1776,-0.2080,0.0549,0.2641,1.3137,-0.2467,0.4241,-0.0548,0.1330,0.1804,1.1319,-0.2873,0.2682,-0.0226,0.0527,0.0775
+446.4670,0.0003,-1.5708,0.0002,-1.5708,0.0001,0.0003,0.0108,-0.0018,0.0067,-0.0025,0.0054,0.0087,0.8515,-0.8159,0.2393,-0.2377,0.0797,0.2404,1.2868,-0.2351,0.4184,-0.0556,0.1352,0.1829,0.0000,0.0003,-0.0043,-0.0001,0.0000,0.0000
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diff --git a/data_ur10/ur-19_12_23_load.csv b/data_ur10/ur-19_12_23_load.csv
new file mode 100644
index 0000000..0273faa
--- /dev/null
+++ b/data_ur10/ur-19_12_23_load.csv
@@ -0,0 +1,2089 @@
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+781.7330,0.0063,-1.5576,0.0212,-1.5646,0.0114,-0.0070,0.0001,-0.0004,-0.0006,0.0000,0.0000,-0.0000,0.3159,-0.6677,-0.2585,-0.1245,-0.1984,-0.1046,-0.0000,-0.2141,-0.1503,-0.0115,0.0034,0.0000,-0.0000,-2.5640,-1.4785,-0.1235,0.0365,0.0000
+781.7430,0.0062,-1.5576,0.0212,-1.5646,0.0114,-0.0070,0.0001,-0.0004,-0.0006,0.0000,0.0000,-0.0000,0.2916,-0.6676,-0.1960,-0.1555,-0.1679,-0.1108,-0.0000,-0.2135,-0.1499,-0.0115,0.0034,0.0000,-0.0000,-2.5570,-1.4745,-0.1233,0.0364,0.0000
+781.7550,0.0063,-1.5576,0.0212,-1.5646,0.0114,-0.0070,0.0000,-0.0004,-0.0004,0.0000,0.0000,-0.0000,0.3047,-0.7504,-0.2174,-0.1351,-0.1860,-0.1111,-0.0000,-0.2133,-0.1496,-0.0114,0.0034,0.0000,-0.0000,-2.5548,-1.4723,-0.1230,0.0363,0.0000
+781.7650,0.0062,-1.5576,0.0211,-1.5646,0.0113,-0.0070,0.0000,-0.0004,-0.0003,0.0000,0.0000,-0.0000,0.3158,-0.6641,-0.1874,-0.1192,-0.1782,-0.1226,0.0000,-0.2136,-0.1499,-0.0114,0.0034,0.0000,0.0000,-2.5579,-1.4744,-0.1232,0.0364,0.0000
+781.7750,0.0063,-1.5576,0.0211,-1.5646,0.0114,-0.0071,0.0000,-0.0002,-0.0003,0.0000,0.0000,-0.0000,0.2151,-0.6652,-0.2504,-0.1149,-0.2019,-0.1221,0.0000,-0.2136,-0.1499,-0.0115,0.0034,0.0000,0.0000,-2.5586,-1.4751,-0.1234,0.0364,0.0000
+781.7870,0.0062,-1.5576,0.0212,-1.5646,0.0114,-0.0070,0.0000,-0.0001,-0.0001,0.0000,0.0000,-0.0001,0.3408,-0.6216,-0.2664,-0.1287,-0.1847,-0.1231,0.0000,-0.2136,-0.1499,-0.0115,0.0034,0.0000,0.0000,-2.5575,-1.4750,-0.1233,0.0364,0.0000
+781.7970,0.0063,-1.5576,0.0212,-1.5646,0.0113,-0.0070,0.0000,0.0001,-0.0001,0.0000,0.0000,-0.0000,0.2253,-0.7598,-0.2626,-0.1361,-0.1761,-0.1122,0.0000,-0.2136,-0.1500,-0.0115,0.0034,0.0000,0.0000,-2.5581,-1.4756,-0.1234,0.0364,0.0000
+781.8090,0.0063,-1.5576,0.0212,-1.5646,0.0113,-0.0070,0.0000,0.0001,-0.0000,0.0000,0.0000,-0.0001,0.2070,-0.7734,-0.2971,-0.1168,-0.2218,-0.1275,-0.0000,-0.2139,-0.1501,-0.0115,0.0034,0.0000,-0.0000,-2.5620,-1.4765,-0.1234,0.0364,0.0000
+781.8190,0.0063,-1.5575,0.0212,-1.5646,0.0114,-0.0070,0.0000,0.0001,0.0000,0.0000,0.0000,-0.0001,0.2250,-0.6876,-0.2825,-0.1342,-0.1899,-0.1162,-0.0000,-0.2137,-0.1500,-0.0115,0.0034,0.0000,-0.0000,-2.5597,-1.4762,-0.1235,0.0364,0.0000
+781.8310,0.0063,-1.5576,0.0212,-1.5646,0.0114,-0.0070,0.0000,0.0001,0.0002,0.0000,0.0000,-0.0001,0.2173,-0.7979,-0.2646,-0.1487,-0.1665,-0.1405,-0.0000,-0.2140,-0.1502,-0.0115,0.0034,0.0000,-0.0000,-2.5631,-1.4776,-0.1236,0.0365,0.0000
diff --git a/regressorWithMotorDynamics.m b/regressorWithMotorDynamics.m
index b71d524..1aa3b4b 100644
--- a/regressorWithMotorDynamics.m
+++ b/regressorWithMotorDynamics.m
@@ -6,9 +6,13 @@ function Y = regressorWithMotorDynamics(q,qd,q2d)
 % parameter is added to existing vector of each link [pi_i I_rflctd_i]
 % so that each link has 11 parameters
 % ----------------------------------------------------------------------
+if size(q,1)==6 && size(q,2)==1 && size(qd,1)==6 && size(qd,2)==1 ...
+        && size(q2d,1)==6 && size(q2d,2)==1
     Y_rgd_bdy = full_regressor_UR10E(q,qd,q2d);
     Y_mtrs = diag(q2d);
     Y = [Y_rgd_bdy(:,1:10), Y_mtrs(:,1), Y_rgd_bdy(:,11:20), Y_mtrs(:,2),...
          Y_rgd_bdy(:,21:30), Y_mtrs(:,3), Y_rgd_bdy(:,31:40), Y_mtrs(:,4),...
          Y_rgd_bdy(:,41:50), Y_mtrs(:,5), Y_rgd_bdy(:,51:60), Y_mtrs(:,6)];
-end
\ No newline at end of file
+else
+    error('Input dimension error!')
+end
diff --git a/ur10_indntfcn_real.m b/ur10_indntfcn_real.m
index a48d4b0..409b98f 100644
--- a/ur10_indntfcn_real.m
+++ b/ur10_indntfcn_real.m
@@ -8,51 +8,123 @@ run('data_prcsng.m')
 % Generate Regressors based on data
 % ------------------------------------------------------------------------
 % Load matrices that map standard set of paratmers to base parameters
-load('full2base_mapping.mat');
+% load('full2base_mapping.mat');
+load('baseQR.mat'); % load mapping from full parameters to base parameters
+E1 = baseQR.permutationMatrix(:,1:baseQR.numberOfBaseParameters);
 m_load = 1.069;
 
-%Constracting regressor matrix
+%Constracting regressor matrix for unloaded case
 Wb_uldd = []; I_uldd = []; 
-for i = 1:2:length(t_msrd)
-    Yb_ulddi = base_regressor_UR10E(q_msrd(i,:)',...
-                qd_fltrd(i,:)',q2d_est(i,:)');
-    Yfrctni = ur10_frctn_rgsr(qd_fltrd(i,:)');
-    Ydrvi = ur10_drv_rgsr(q2d_est(i,:)');
-    Wb_uldd = vertcat(Wb_uldd,[Yb_ulddi, Ydrvi, Yfrctni]);
-    I_uldd = vertcat(I_uldd, diag(i_fltrd(i,:)));
+for i = 1:1:length(unloadedTrajectory.t)
+    Y_ulddi = regressorWithMotorDynamics(unloadedTrajectory.q(i,:)',...
+                                         unloadedTrajectory.qd_fltrd(i,:)',...
+                                         unloadedTrajectory.q2d_est(i,:)');
+                                     
+    Yfrctni = frictionRegressor(unloadedTrajectory.qd_fltrd(i,:)');
+    Ybi_uldd = [Y_ulddi*E1, Yfrctni];
+    
+    Wb_uldd = vertcat(Wb_uldd, Ybi_uldd);
+    I_uldd = vertcat(I_uldd, diag(unloadedTrajectory.i_fltrd(i,:)));
 end
 
+% Constracting regressor matrix for loaded case
 Wb_ldd = []; Wl = []; I_ldd = [];
-for i = 1:2:length(t_msrd_ldd)
-    Yb_lddi = base_regressor_UR10E(q_msrd_ldd(i,:)',...
-                qd_fltrd_ldd(i,:)',q2d_est_ldd(i,:)');
-    Yfrctni = ur10_frctn_rgsr(qd_fltrd_ldd(i,:)');
-    Ydrvi = ur10_drv_rgsr(q2d_est_ldd(i,:)');
+for i = 1:1:length(loadedTrajectory.t)
+    Y_lddi = regressorWithMotorDynamics(loadedTrajectory.q(i,:)',...
+                                        loadedTrajectory.qd_fltrd(i,:)',...
+                                        loadedTrajectory.q2d_est(i,:)');
+                                    
+    Yfrctni = frictionRegressor(loadedTrajectory.qd_fltrd(i,:)');
     
-    Yli = load_regressor_UR10E(q_msrd_ldd(i,:)',...
-                qd_fltrd_ldd(i,:)',q2d_est_ldd(i,:)');
+    Yli = load_regressor_UR10E(loadedTrajectory.q(i,:)',...
+                               loadedTrajectory.qd_fltrd(i,:)',...
+                               loadedTrajectory.q2d_est(i,:)');
+                           
+    Ybi_ldd = [Y_lddi*E1, Yfrctni];
     
-    Wb_ldd = vertcat(Wb_ldd,[Yb_lddi, Ydrvi, Yfrctni]);
+    Wb_ldd = vertcat(Wb_ldd, Ybi_ldd);
     Wl = vertcat(Wl,Yli); 
-    I_ldd = vertcat(I_ldd, diag(i_fltrd_ldd(i,:)));
+    I_ldd = vertcat(I_ldd, diag(loadedTrajectory.i_fltrd(i,:)));
+end
+Wl_uknown = Wl(:,1:9);
+Wl_known = Wl(:,10); % mass of the load is known 
+
+
+%% Using total least squares
+Wb_tls = [I_uldd   -Wb_uldd   zeros(size(I_uldd,1), size(Wl,2));
+          I_ldd    -Wb_ldd    -Wl_uknown    -Wl_known*m_load];
+
+% SVD decompostion of Wb_tls to solve total least squares
+[~,~,V] = svd(Wb_tls,'econ');
+% Scaling of the solution
+lmda = 1/V(end,end);
+pi_tls = lmda*V(:,end);
+% drive gains
+drvGains = pi_tls(1:6)
+
+% Finding weighting matrix, joint by joint
+G = zeros(6);
+for i = 1:6
+    Wib_tls = Wb_tls(i:6:end,:);
+    [~,Si,Vi] = svd(Wib_tls,'econ');
+    sgmai = Si(end,end)/sqrt((size(Wib_tls,1)-size(Wib_tls,2)));
+    G(i,i) = 1/sgmai^2;
 end
 
-% ----------------------------------------------------------------------
-% Set-up SDP optimization procedure
-% -----------------------------------------------------------------------
+for i = 1:6:size(Wb_tls,1)
+    Wb_tls(i:i+5,:) = G*Wb_tls(i:i+5,:);
+end
+[~,~,V] = svd(Wb_tls,'econ');
+lmda = 1/V(end,end);
+pi_tls = lmda*V(:,end);
+drvGains = pi_tls(1:6)
+
+
+
+%% Identification of parameters including drive gains
+Wb_ls = [I_uldd     -Wb_uldd    zeros(size(I_uldd,1), size(Wl_uknown,2));
+         I_ldd      -Wb_ldd     -Wl_uknown];
+     
+Yb_ts = [zeros(size(I_uldd,1),1); Wl_known*m_load];
+
+% Compute least squares solution
+pi_ls = ((Wb_ls'*Wb_ls)\Wb_ls')*Yb_ts;
+drvGains = pi_ls(1:6)
+
+
+G = zeros(6);
+for i = 1:6
+    Wib_ls = Wb_ls(i:6:end,:);
+    Yib_ls = Yb_ts(i:6:end);
+    sgmai_sqrd = norm(Yib_ls - Wib_ls*pi_ls,2)^2/(size(Wib_ls,1)-size(Wib_ls,2));
+    G(i,i) = 1/sgmai_sqrd;
+end
+
+for i = 1:6:size(Wb_ls,1)
+    Wb_ls(i:i+5,:) = G*Wb_ls(i:i+5,:);
+    Yb_ts(i:i+5) = G*Yb_ts(i:i+5);
+end
+pi_tot = ((Wb_ls'*Wb_ls)\Wb_ls')*Yb_ts;
+drvGains = pi_tot(1:6)
+
+
+
+%% Set-up SDP optimization procedure
 drv_gns = sdpvar(6,1); % variables for base paramters
 pi_load_unknw = sdpvar(9,1); % varaibles for unknown load paramters
 pi_frctn = sdpvar(18,1);
-pi_rtr = sdpvar(4,1);
-pi_b = sdpvar(36,1); % variables for base paramters
-pi_d = sdpvar(24,1); % variables for dependent paramters
+pi_b = sdpvar(baseQR.numberOfBaseParameters,1); % variables for base paramters
+pi_d = sdpvar(26,1); % variables for dependent paramters
 
 % Bijective mapping from [pi_b; pi_d] to standard parameters pi
-pii = [Pb' Pd']*[ eye(36) -Kd; zeros(24,36) eye(24) ]*[pi_b; pi_d];
+pii = baseQR.permutationMatrix*[ eye(baseQR.numberOfBaseParameters), ...
+                                -baseQR.beta; ...
+                                zeros(26,baseQR.numberOfBaseParameters), ... 
+                                eye(26) ]*[pi_b; pi_d];
 
-% Feasibility contrraints of the link paramteres
+% Feasibility contrraints of the link paramteres and rotor inertia
 cnstr = diag(drv_gns)>0;
-for i = 1:10:60
+for i = 1:11:66
     link_inertia_i = [pii(i), pii(i+1), pii(i+2); ...
                       pii(i+1), pii(i+3), pii(i+4); ...
                       pii(i+2), pii(i+4), pii(i+5)];
@@ -60,7 +132,7 @@ for i = 1:10:60
     frst_mmnt_i = vec2skewSymMat(pii(i+6:i+8));
     
     Di = [link_inertia_i, frst_mmnt_i'; frst_mmnt_i, pii(i+9)*eye(3)];
-    cnstr = [cnstr, Di>0];
+    cnstr = [cnstr, Di>0, pii(i+10)>0];
 end
 
 % Feasibility constraints on the load paramters
@@ -77,23 +149,22 @@ for i = 1:6
    cnstr = [cnstr, pi_frctn(3*i-2)>0, pi_frctn(3*i-1)>0];  
 end
 
-% Feasibiliy of the rotor inertia
-cnstr = [cnstr, diag(pi_rtr)>0];
-
 % Defining pbjective function
 t1 = [zeros(size(I_uldd,1),1); -Wl(:,end)*m_load];
 
 t2 = [-I_uldd, Wb_uldd, zeros(size(Wb_uldd,1), size(Wl,2)-1); ...
       -I_ldd, Wb_ldd, Wl(:,1:9) ];
   
-obj = norm(t1 - t2*[drv_gns; pi_b; pi_rtr; pi_frctn; pi_load_unknw]);
+obj = norm(t1 - t2*[drv_gns; pi_b; pi_frctn; pi_load_unknw]);
 
 % Solving sdp problem
 sol = optimize(cnstr,obj,sdpsettings('solver','sdpt3'));
 
 % Getting values of the estimated patamters
-drv_gns = value(drv_gns);
+drv_gns = value(drv_gns)
 
+
+return
 % -----------------------------------------------------------------------
 % When drive gains are known we optimize for paramters
 % -----------------------------------------------------------------------
diff --git a/ur10_indntfcn_smltn.m b/ur10_indntfcn_smltn.m
index ab3d18c..bb5eafb 100644
--- a/ur10_indntfcn_smltn.m
+++ b/ur10_indntfcn_smltn.m
@@ -1,38 +1,224 @@
-% clc; clear all; close all;
-% run('main_ur10.m');
-run('ur10_base_params_sym.m');
+% ---------------------------------------------------------------------
+% In this script identification of dynamic parameters as well as
+% drive gains are performed on ideal trajectory found from
+% optimization procedure.
+% ---------------------------------------------------------------------
+clc; clear all; close all;
 
-load('opt_sltn_N5T20.mat'); % optimization trajectory paramters
-load('pi_base.mat'); % base parameters given by manufacturer URDF
+% get robot parameters from urdf
+run('main_ur.m');
+
+% add reflected rotor inertia to parameters
+reflectedRotorInertia = rand([1,6]);
+ur10.pi = [ur10.pi; reflectedRotorInertia];
+ur10.pi = reshape(ur10.pi,[size(ur10.pi,1)*size(ur10.pi,2), 1]);
+
+% get base parameters
+load('baseQR.mat'); 
+E1 = baseQR.permutationMatrix(:,1:baseQR.numberOfBaseParameters);
+ur10.pi_base = [eye(baseQR.numberOfBaseParameters) baseQR.beta]*...
+                    baseQR.permutationMatrix'*ur10.pi;
+                
+% add friction parmeters to the vector of base parameters
+frictionParameters = rand([18,1]);
+ur10.pi_base = [ur10.pi_base; frictionParameters];
+
+% optimized trajectory
+load('ptrnSrch_N5T20QR.mat'); 
 traj_par.t_smp = 1e-1; % sampling time
 traj_par.t = 0:traj_par.t_smp:traj_par.T; % time
 
 % generilized corrdinates, velocities and acelerations of the trajectory
 [q,qd,q2d] = mixed_traj(traj_par.t,c_pol,a,b,traj_par.wf,traj_par.N);
 
-% load paramters
+% add noise to data
+noiseVariance = 0.01;
+q = q + sqrt(noiseVariance)*rand(size(q));
+qd = qd + sqrt(noiseVariance)*rand(size(qd));
+q2d = q2d + sqrt(2*noiseVariance)*rand(size(q2d));
+
+% load and drive paramters
 pi_load =  [0.2,0.05,0.07,0.15,0.01,0.4, 0.05, 0, 0, 1.069]';
 K_drv = [11 13 15 10 12 14]';
 m_load = 1.069;
 
-% building regressor based on the trajectory
-W = []; Wl = []; I_uldd = []; I_ldd = [];  
-for i = 1:length(traj_par.t)
-    Yi = base_regressor_UR10E(q(:,i),qd(:,i),q2d(:,i));
-    Yli = load_regressor_UR10E(q(:,i),qd(:,i),q2d(:,i));
-    
-    tau_ulddi = Yi*pir_base_vctr;
-    tau_lddi = [Yi Yli]*[pir_base_vctr; pi_load];
-    
-    I_uldd = vertcat(I_uldd, diag(diag(K_drv)\tau_ulddi));
-    I_ldd = vertcat(I_ldd, diag(diag(K_drv)\tau_lddi));
-    
-    W = vertcat(W,Yi);
-    Wl = vertcat(Wl,Yli);
+
+% Constracting regressor matrix for unloaded case
+Wb_uldd = []; I_uldd = []; 
+for i = 1:1:length(traj_par.t)
+    Y_ulddi = regressorWithMotorDynamics(q(:,i),qd(:,i),q2d(:,i));                              
+    Yfrctni = frictionRegressor(qd(:,i));
+    Ybi_uldd = [Y_ulddi*E1, Yfrctni];
+    taui_uldd = Ybi_uldd*ur10.pi_base;
+        
+    Wb_uldd = vertcat(Wb_uldd, Ybi_uldd);
+    I_uldd = vertcat(I_uldd, diag(diag(K_drv)\taui_uldd + ...
+                                sqrt(noiseVariance)*rand(6,1)));
 end
 
-% getting torques based on the parameters given in URDF
-Tau = W*pir_base_vctr;
+% Constracting regressor matrix for loaded case
+Wb_ldd = []; Wl = []; I_ldd = [];
+for i = 1:1:length(traj_par.t)
+    Y_lddi = regressorWithMotorDynamics(q(:,i),qd(:,i),q2d(:,i));                                
+    Yfrctni = frictionRegressor(qd(:,i));
+    Yli = load_regressor_UR10E(q(:,i),qd(:,i),q2d(:,i));
+    Ybi_ldd = [Y_lddi*E1, Yfrctni];
+    taui_ldd = [Ybi_ldd Yli]*[ur10.pi_base; pi_load];
+    
+    Wb_ldd = vertcat(Wb_ldd, Ybi_ldd);
+    Wl = vertcat(Wl, Yli); 
+    I_ldd = vertcat(I_ldd, diag(diag(K_drv)\taui_ldd + ... 
+                                sqrt(0)*rand(6,1)));
+end
+Wl_uknown = Wl(:,1:9);
+Wl_known = Wl(:,10); % mass of the load is known 
+
+
+%% Identification of parameters including drive gains
+Wb_ls = [I_uldd     -Wb_uldd    zeros(size(I_uldd,1), size(Wl_uknown,2));
+         I_ldd      -Wb_ldd     -Wl_uknown];
+     
+Yb_ts = [zeros(size(I_uldd,1),1); Wl_known*m_load];
+
+% Compute least squares solution
+pi_ls = ((Wb_ls'*Wb_ls)\Wb_ls')*Yb_ts;
+drvGains = [pi_ls(1:6) K_drv]
+
+G = zeros(6);
+for i = 1:6
+    Wib_ls = Wb_ls(i:6:end,:);
+    Yib_ls = Yb_ts(i:6:end);
+    sgmai_sqrd = norm(Yib_ls - Wib_ls*pi_ls,2)^2/(size(Wib_ls,1)-size(Wib_ls,2));
+    G(i,i) = 1/sqrt(sgmai_sqrd);
+end
+
+for i = 1:6:size(Wb_ls,1)
+    Wb_ls(i:i+5,:) = G*Wb_ls(i:i+5,:);
+    Yb_ts(i:i+5) = G*Yb_ts(i:i+5);
+end
+pi_tot = ((Wb_ls'*Wb_ls)\Wb_ls')*Yb_ts;
+drvGains = pi_tot(1:6)
+
+
+nmbrRwsUnldd = size(Wb_uldd,1);
+G_uldd = zeros(6);
+G_ldd = zeros(6);
+for i = 1:6
+    Wib_uldd = Wb_ls(i:6:nmbrRwsUnldd,:);
+    Yib_uldd = Yb_ts(i:6:nmbrRwsUnldd);
+    sgmai_sqrd = norm(Yib_uldd - Wib_uldd*pi_ls,2)^2/(size(Wib_uldd,1)-rank(Wib_uldd));
+    G_uldd(i,i) = 1/sqrt(sgmai_sqrd);
+    
+    Wib_ldd = Wb_ls(nmbrRwsUnldd+i:6:end,:);
+    Yib_ldd = Yb_ts(nmbrRwsUnldd+i:6:end);
+    sgmai_sqrd = norm(Yib_ldd - Wib_ldd*pi_ls,2)^2/(size(Wib_ldd,1)-rank(Wib_ldd));
+    G_ldd(i,i) = 1/sqrt(sgmai_sqrd);
+end
+
+
+for i = 1:6:size(Wb_uldd,1)
+    Wb_ls(i:i+5,:) = G_uldd*Wb_ls(i:i+5,:);
+    Yb_ts(i:i+5) = G_uldd*Yb_ts(i:i+5);
+end
+
+for i = nmbrRwsUnldd+1:6:nmbrRwsUnldd+size(Wb_ldd,1)   
+    Wb_ls(i:i+5,:) = G_ldd*Wb_ls(i:i+5,:);
+    Yb_ts(i:i+5) = G_ldd*Yb_ts(i:i+5);
+end
+
+pi_tot = ((Wb_ls'*Wb_ls)\Wb_ls')*Yb_ts;
+drvGains = pi_tot(1:6)
+
+
+
+
+return
+%% Using total least squares
+Wb_tls = [I_uldd   -Wb_uldd   zeros(size(I_uldd,1), size(Wl,2));
+          I_ldd    -Wb_ldd    -Wl_uknown    -Wl_known*m_load];
+
+% SVD decompostion of Wb_tls to solve total least squares
+[~,~,V] = svd(Wb_tls,'econ');
+% Scaling of the solution
+lmda = 1/V(end,end);
+pi_tls = lmda*V(:,end);
+% drive gains
+drvGains = [pi_tls(1:6) K_drv]
+
+% Finding weighting matrix, joint by joint
+G = zeros(6);
+for i = 1:6
+    Wib_tls = Wb_tls(i:6:end,:);
+    [~,Si,Vi] = svd(Wib_tls,'econ');
+    sgmai = Si(end,end)/sqrt((size(Wib_tls,1)-size(Wib_tls,2)));
+    G(i,i) = 1/sgmai^2;
+end
+
+for i = 1:6:size(Wb_tls,1)
+    Wb_tls(i:i+5,:) = G*Wb_tls(i:i+5,:);
+end
+[~,~,V] = svd(Wb_tls,'econ');
+lmda = 1/V(end,end);
+pi_tls = lmda*V(:,end);
+drvGains = pi_tls(1:6)
+
+
+%% Set-up SDP optimization procedure
+drv_gns = sdpvar(6,1); % variables for base paramters
+pi_load_unknw = sdpvar(9,1); % varaibles for unknown load paramters
+pi_frctn = sdpvar(18,1);
+pi_b = sdpvar(baseQR.numberOfBaseParameters,1); % variables for base paramters
+pi_d = sdpvar(26,1); % variables for dependent paramters
+
+% Bijective mapping from [pi_b; pi_d] to standard parameters pi
+pii = baseQR.permutationMatrix*[ eye(baseQR.numberOfBaseParameters), ...
+                                -baseQR.beta; ...
+                                zeros(26,baseQR.numberOfBaseParameters), ... 
+                                eye(26) ]*[pi_b; pi_d];
+
+% Feasibility contrraints of the link paramteres and rotor inertia
+cnstr = diag(drv_gns)>0;
+for i = 1:11:66
+    link_inertia_i = [pii(i), pii(i+1), pii(i+2); ...
+                      pii(i+1), pii(i+3), pii(i+4); ...
+                      pii(i+2), pii(i+4), pii(i+5)];
+                  
+    frst_mmnt_i = vec2skewSymMat(pii(i+6:i+8));
+    
+    Di = [link_inertia_i, frst_mmnt_i'; frst_mmnt_i, pii(i+9)*eye(3)];
+    cnstr = [cnstr, Di>0, pii(i+10)>0];
+end
+
+% Feasibility constraints on the load paramters
+load_inertia = [pi_load_unknw(1), pi_load_unknw(2), pi_load_unknw(3); ...
+                pi_load_unknw(2), pi_load_unknw(4), pi_load_unknw(5); ...
+                pi_load_unknw(3), pi_load_unknw(5), pi_load_unknw(6)];                  
+load_frst_mmnt = vec2skewSymMat(pi_load_unknw(7:9));    
+Dl = [load_inertia, load_frst_mmnt'; load_frst_mmnt, m_load*eye(3)];
+
+cnstr = [cnstr, Dl>0];
+
+% Feasibility constraints on the friction prameters 
+for i = 1:6
+   cnstr = [cnstr, pi_frctn(3*i-2)>0, pi_frctn(3*i-1)>0];  
+end
+
+% Defining pbjective function
+t1 = [zeros(size(I_uldd,1),1); -Wl(:,end)*m_load];
+
+t2 = [-I_uldd, Wb_uldd, zeros(size(Wb_uldd,1), size(Wl,2)-1); ...
+      -I_ldd, Wb_ldd, Wl(:,1:9) ];
+  
+obj = norm(t1 - t2*[drv_gns; pi_b; pi_frctn; pi_load_unknw]);
+
+% Solving sdp problem
+sol = optimize(cnstr,obj,sdpsettings('solver','sdpt3'));
+
+% Getting values of the estimated patamters
+drv_gns = [value(drv_gns) K_drv]
+
+
+return
 
 % ------------------------------------------------------------------------
 % Using SDP to find load parmeters along with drive gains
diff --git a/ur10_indntfcn_smltn.m~ b/ur10_indntfcn_smltn.m~
new file mode 100644
index 0000000..b14017a
--- /dev/null
+++ b/ur10_indntfcn_smltn.m~
@@ -0,0 +1,464 @@
+% ---------------------------------------------------------------------
+% In this script identification of dynamic parameters as well as
+% drive gains are performed on ideal trajectory found from
+% optimization procedure.
+% ---------------------------------------------------------------------
+clc; clear all; close all;
+
+% get robot parameters from urdf
+run('main_ur.m');
+
+% add reflected rotor inertia to parameters
+reflectedRotorInertia = rand([1,6]);
+ur10.pi = [ur10.pi; reflectedRotorInertia];
+ur10.pi = reshape(ur10.pi,[size(ur10.pi,1)*size(ur10.pi,2), 1]);
+
+% get base parameters
+load('baseQR.mat'); 
+E1 = baseQR.permutationMatrix(:,1:baseQR.numberOfBaseParameters);
+ur10.pi_base = [eye(baseQR.numberOfBaseParameters) baseQR.beta]*...
+                    baseQR.permutationMatrix'*ur10.pi;
+                
+% add friction parmeters to the vector of base parameters
+frictionParameters = rand([18,1]);
+ur10.pi_base = [ur10.pi_base; frictionParameters];
+
+% optimized trajectory
+load('ptrnSrch_N5T20QR.mat'); 
+traj_par.t_smp = 1e-1; % sampling time
+traj_par.t = 0:traj_par.t_smp:traj_par.T; % time
+
+% generilized corrdinates, velocities and acelerations of the trajectory
+[q,qd,q2d] = mixed_traj(traj_par.t,c_pol,a,b,traj_par.wf,traj_par.N);
+
+% add noise to data
+noiseVariance = 0.01;
+q = q + sqrt(noiseVariance)*rand(size(q));
+qd = qd + sqrt(noiseVariance)*rand(size(qd));
+q2d = q2d + sqrt(2*noiseVariance)*rand(size(q2d));
+
+% load and drive paramters
+pi_load =  [0.2,0.05,0.07,0.15,0.01,0.4, 0.05, 0, 0, 1.069]';
+K_drv = [11 13 15 10 12 14]';
+m_load = 1.069;
+
+
+% Constracting regressor matrix for unloaded case
+Wb_uldd = []; I_uldd = []; 
+for i = 1:1:length(traj_par.t)
+    Y_ulddi = regressorWithMotorDynamics(q(:,i),qd(:,i),q2d(:,i));                              
+    Yfrctni = frictionRegressor(qd(:,i));
+    Ybi_uldd = [Y_ulddi*E1, Yfrctni];
+    taui_uldd = Ybi_uldd*ur10.pi_base;
+        
+    Wb_uldd = vertcat(Wb_uldd, Ybi_uldd);
+    I_uldd = vertcat(I_uldd, diag(diag(K_drv)\taui_uldd + ...
+                                sqrt(noiseVariance)*rand(6,1)));
+end
+
+% Constracting regressor matrix for loaded case
+Wb_ldd = []; Wl = []; I_ldd = [];
+for i = 1:1:length(traj_par.t)
+    Y_lddi = regressorWithMotorDynamics(q(:,i),qd(:,i),q2d(:,i));                                
+    Yfrctni = frictionRegressor(qd(:,i));
+    Yli = load_regressor_UR10E(q(:,i),qd(:,i),q2d(:,i));
+    Ybi_ldd = [Y_lddi*E1, Yfrctni];
+    taui_ldd = [Ybi_ldd Yli]*[ur10.pi_base; pi_load];
+    
+    Wb_ldd = vertcat(Wb_ldd, Ybi_ldd);
+    Wl = vertcat(Wl, Yli); 
+    I_ldd = vertcat(I_ldd, diag(diag(K_drv)\taui_ldd + ... 
+                                sqrt(0)*rand(6,1)));
+end
+Wl_uknown = Wl(:,1:9);
+Wl_known = Wl(:,10); % mass of the load is known 
+
+
+%% Identification of parameters including drive gains
+Wb_ls = [I_uldd     -Wb_uldd    zeros(size(I_uldd,1), size(Wl_uknown,2));
+         I_ldd      -Wb_ldd     -Wl_uknown];
+     
+Yb_ts = [zeros(size(I_uldd,1),1); Wl_known*m_load];
+
+% Compute least squares solution
+pi_ls = ((Wb_ls'*Wb_ls)\Wb_ls')*Yb_ts;
+drvGains = [pi_ls(1:6) K_drv]
+
+G = zeros(6);
+for i = 1:6
+    Wib_ls = Wb_ls(i:6:end,:);
+    Yib_ls = Yb_ts(i:6:end);
+    sgmai_sqrd = norm(Yib_ls - Wib_ls*pi_ls,2)^2/(size(Wib_ls,1)-size(Wib_ls,2));
+    G(i,i) = 1/sgmai_sqrd;
+end
+
+for i = 1:6:size(Wb_ls,1)
+    Wb_ls(i:i+5,:) = G*Wb_ls(i:i+5,:);
+    Yb_ts(i:i+5) = G*Yb_ts(i:i+5);
+end
+pi_tot = ((Wb_ls'*Wb_ls)\Wb_ls')*Yb_ts;
+drvGains = pi_tot(1:6)
+
+
+nmbrRwsUnldd = size(Wb_uldd,1);
+G_uldd = zeros(6);
+G_ldd = zeros(6);
+for i = 1:6
+    Wib_uldd = Wb_ls(i:6:nmbrRwsUnldd,:);
+    Yib_uldd = Yb_ts(i:6:nmbrRwsUnldd);
+    sgmai_sqrd = norm(Yib_uldd - Wib_uldd*pi_ls,2)^2/(size(Wib_uldd,1)-rank(Wib_uldd));
+    G_uldd(i,i) = 1/sgmai_sqrd;
+    
+    Wib_ldd = Wb_ls(nmbrRwsUnldd+i:6:end,:);
+    Yib_ldd = Yb_ts(nmbrRwsUnldd+i:6:end);
+    sgmai_sqrd = norm(Yib_ldd - Wib_ldd*pi_ls,2)^2/(size(Wib_ldd,1)-rank(Wib_ldd));
+    G_ldd(i,i) = 1/sgmai_sqrd;
+end
+
+
+for i = 1:6:size(Wb_uldd,1)
+    Wb_ls(i:i+5,:) = G_uldd*Wb_ls(i:i+5,:);
+    Yb_ts(i:i+5) = G_uldd*Yb_ts(i:i+5);
+    
+    Wb_ls(size(Wb_uldd,1)+i:i+5,:) = G_ldd*Wb_ls(size(Wb_uldd,1)+i:i+5,:);
+    Yb_ts(size(Wb_uldd,1)+i:i+5) = G_ldd*Yb_ts(size(Wb_uldd,1)+i:i+5);
+end
+pi_tot = ((Wb_ls'*Wb_ls)\Wb_ls')*Yb_ts;
+drvGains = pi_tot(1:6)
+
+
+
+
+return
+%% Using total least squares
+Wb_tls = [I_uldd   -Wb_uldd   zeros(size(I_uldd,1), size(Wl,2));
+          I_ldd    -Wb_ldd    -Wl_uknown    -Wl_known*m_load];
+
+% SVD decompostion of Wb_tls to solve total least squares
+[~,~,V] = svd(Wb_tls,'econ');
+% Scaling of the solution
+lmda = 1/V(end,end);
+pi_tls = lmda*V(:,end);
+% drive gains
+drvGains = [pi_tls(1:6) K_drv]
+
+% Finding weighting matrix, joint by joint
+G = zeros(6);
+for i = 1:6
+    Wib_tls = Wb_tls(i:6:end,:);
+    [~,Si,Vi] = svd(Wib_tls,'econ');
+    sgmai = Si(end,end)/sqrt((size(Wib_tls,1)-size(Wib_tls,2)));
+    G(i,i) = 1/sgmai^2;
+end
+
+for i = 1:6:size(Wb_tls,1)
+    Wb_tls(i:i+5,:) = G*Wb_tls(i:i+5,:);
+end
+[~,~,V] = svd(Wb_tls,'econ');
+lmda = 1/V(end,end);
+pi_tls = lmda*V(:,end);
+drvGains = pi_tls(1:6)
+
+
+%% Set-up SDP optimization procedure
+drv_gns = sdpvar(6,1); % variables for base paramters
+pi_load_unknw = sdpvar(9,1); % varaibles for unknown load paramters
+pi_frctn = sdpvar(18,1);
+pi_b = sdpvar(baseQR.numberOfBaseParameters,1); % variables for base paramters
+pi_d = sdpvar(26,1); % variables for dependent paramters
+
+% Bijective mapping from [pi_b; pi_d] to standard parameters pi
+pii = baseQR.permutationMatrix*[ eye(baseQR.numberOfBaseParameters), ...
+                                -baseQR.beta; ...
+                                zeros(26,baseQR.numberOfBaseParameters), ... 
+                                eye(26) ]*[pi_b; pi_d];
+
+% Feasibility contrraints of the link paramteres and rotor inertia
+cnstr = diag(drv_gns)>0;
+for i = 1:11:66
+    link_inertia_i = [pii(i), pii(i+1), pii(i+2); ...
+                      pii(i+1), pii(i+3), pii(i+4); ...
+                      pii(i+2), pii(i+4), pii(i+5)];
+                  
+    frst_mmnt_i = vec2skewSymMat(pii(i+6:i+8));
+    
+    Di = [link_inertia_i, frst_mmnt_i'; frst_mmnt_i, pii(i+9)*eye(3)];
+    cnstr = [cnstr, Di>0, pii(i+10)>0];
+end
+
+% Feasibility constraints on the load paramters
+load_inertia = [pi_load_unknw(1), pi_load_unknw(2), pi_load_unknw(3); ...
+                pi_load_unknw(2), pi_load_unknw(4), pi_load_unknw(5); ...
+                pi_load_unknw(3), pi_load_unknw(5), pi_load_unknw(6)];                  
+load_frst_mmnt = vec2skewSymMat(pi_load_unknw(7:9));    
+Dl = [load_inertia, load_frst_mmnt'; load_frst_mmnt, m_load*eye(3)];
+
+cnstr = [cnstr, Dl>0];
+
+% Feasibility constraints on the friction prameters 
+for i = 1:6
+   cnstr = [cnstr, pi_frctn(3*i-2)>0, pi_frctn(3*i-1)>0];  
+end
+
+% Defining pbjective function
+t1 = [zeros(size(I_uldd,1),1); -Wl(:,end)*m_load];
+
+t2 = [-I_uldd, Wb_uldd, zeros(size(Wb_uldd,1), size(Wl,2)-1); ...
+      -I_ldd, Wb_ldd, Wl(:,1:9) ];
+  
+obj = norm(t1 - t2*[drv_gns; pi_b; pi_frctn; pi_load_unknw]);
+
+% Solving sdp problem
+sol = optimize(cnstr,obj,sdpsettings('solver','sdpt3'));
+
+% Getting values of the estimated patamters
+drv_gns = [value(drv_gns) K_drv]
+
+
+return
+
+% ------------------------------------------------------------------------
+% Using SDP to find load parmeters along with drive gains
+% ------------------------------------------------------------------------
+%{
+drv_gns = sdpvar(6,1); % variables for base paramters
+pi_load_unknw = sdpvar(9,1); % varaibles for unknown load paramters
+% u = sdpvar(1,1); % variable for cost
+% 
+% % Writing |Tau - W*pi_b| using Schur compliment
+% U1 = [u, (-Wl(:,end)*m_load - [-(I_ldd-I_uldd) Wl(:,1:9)]*[drv_gns;pi_load_unknw])'; ...
+%       -Wl(:,end)*m_load - [-(I_ldd-I_uldd) Wl(:,1:9)]*[drv_gns;pi_load_unknw], sparse(eye(size(Wl,1)))];
+
+% Feasibility contrraints
+% U2 = diag(drv_gns);
+load_inertia = [pi_load_unknw(1), pi_load_unknw(2), pi_load_unknw(3); ...
+                pi_load_unknw(2), pi_load_unknw(4), pi_load_unknw(5); ...
+                pi_load_unknw(3), pi_load_unknw(5), pi_load_unknw(6)];
+                  
+load_frst_mmnt = vec2skewSymMat(pi_load_unknw(7:9));
+    
+D = [load_inertia, load_frst_mmnt'; load_frst_mmnt, m_load*eye(3)];
+% U2 = blkdiag(U2,D);
+% Overall constraints
+% cnstr = blkdiag(U1,U2)>0;
+cnstr = [drv_gns>0, D>0];
+
+% Objective function
+obj = norm(-Wl(:,end)*m_load - [-(I_ldd-I_uldd) Wl(:,1:9)]*[drv_gns;pi_load_unknw]);
+
+% Solving sdp problem
+sol = optimize(cnstr,obj,sdpsettings('solver','sdpt3'));
+
+%}
+
+% ------------------------------------------------------------------------
+% Computing drive gains using second method
+% ------------------------------------------------------------------------
+drv_gns = sdpvar(6,1); % variables for base paramters
+pi_load_unknw = sdpvar(9,1); % varaibles for unknown load paramters
+pi_b = sdpvar(36,1); % variables for base paramters
+pi_d = sdpvar(24,1); % variables for dependent paramters
+
+% Bijective mapping from [pi_b; pi_d] to standard parameters pi
+pii = [Pb' Pd']*[ eye(36) -double(Kd); zeros(24,36) eye(24)]*[pi_b; pi_d];
+
+% Feasibility contrraints
+cnstr = drv_gns>0;
+for i = 1:10:60
+    link_inertia_i = [pii(i), pii(i+1), pii(i+2); ...
+                      pii(i+1), pii(i+3), pii(i+4); ...
+                      pii(i+2), pii(i+4), pii(i+5)];
+                  
+    frst_mmnt_i = vec2skewSymMat(pii(i+6:i+8));
+    
+    Di = [link_inertia_i, frst_mmnt_i'; frst_mmnt_i, pii(i+9)*eye(3)];
+    cnstr = [cnstr, Di>0];
+end
+
+load_inertia = [pi_load_unknw(1), pi_load_unknw(2), pi_load_unknw(3); ...
+                pi_load_unknw(2), pi_load_unknw(4), pi_load_unknw(5); ...
+                pi_load_unknw(3), pi_load_unknw(5), pi_load_unknw(6)];
+                  
+load_frst_mmnt = vec2skewSymMat(pi_load_unknw(7:9));
+    
+Dl = [load_inertia, load_frst_mmnt'; load_frst_mmnt, m_load*eye(3)];
+cnstr = [cnstr, Dl>0];
+
+t1 = [zeros(size(I_uldd,1),1); -Wl(:,end)*m_load];
+t2 = [-I_uldd, W, zeros(size(W,1), size(Wl,2)-1); ...
+      -I_ldd, W, Wl(:,1:9) ];
+obj = norm(t1 - t2*[drv_gns;pi_b;pi_load_unknw]);
+
+% Solving sdp problem
+sol = optimize(cnstr,obj,sdpsettings('solver','sdpt3'));
+return
+% ------------------------------------------------------------------------
+% Least square
+% ------------------------------------------------------------------------
+pir_hat = (W'*W)\(W'*Tau);
+
+% ------------------------------------------------------------------------
+% Ridge regression
+% ------------------------------------------------------------------------
+% pi_hat = mldivide(W'*W + 0.001*eye(size(W'*W)),W'*tau);
+
+% ------------------------------------------------------------------------
+% Using SDP to find inertial parameters
+% ------------------------------------------------------------------------
+pi_b = sdpvar(36,1); % variables for base paramters
+pi_d = sdpvar(24,1); % variables for dependent paramters
+u = sdpvar(1,1); % variable for cost
+
+% Writing |Tau - W*pi_b| using Schur compliment
+U1 = [u, (Tau - W*pi_b)'; Tau - W*pi_b, eye(size(W,1))];
+
+% Bijective mapping from [pi_b; pi_d] to standard parameters pi
+pii = [Pb' Pd']*[ eye(36) -double(Kd); zeros(24,36) eye(24)]*[pi_b; pi_d];
+
+% Feasibility contrraints
+U2 = [];
+for i = 1:10:60
+    link_inertia_i = [pii(i), pii(i+1), pii(i+2); ...
+                      pii(i+1), pii(i+3), pii(i+4); ...
+                      pii(i+2), pii(i+4), pii(i+5)];
+                  
+    frst_mmnt_i = vec2skewSymMat(pii(i+6:i+8));
+    
+    Di = [link_inertia_i, frst_mmnt_i'; frst_mmnt_i, pii(i+9)*eye(3)];
+    U2 = blkdiag(U2,Di);
+end
+% Overall constraints
+cnstr = blkdiag(U1,U2)>0;
+
+% Objective function
+obj = u;
+
+% Solving sdp problem
+sol = optimize(cnstr,obj,sdpsettings('solver','sdpt3'));
+
+% Getting numerical values of optimization paramters
+base_prms = value(pi_b);
+dpndn_prms = value(pi_d);
+
+
+
+
+
+return
+% ------------------------------------------------------------------------
+% DREM
+% ------------------------------------------------------------------------
+alpha = [1, 1.3, 1.6, 1.3, 1.15, 1.5];
+beta = [1, 1.25, 1.5, 1.75, 2, 2.25];
+dly = [3, 6, 9, 12, 15];
+
+% tau(:,1) = base_regressor(q(:,1),qd(:,1),q2d(:,1))*pir_base_vctr;
+% tau_f1(:,1) = tau(:,1); tau_f2(:,1) = tau(:,1); tau_f3(:,1) = tau(:,1);
+% tau_f4(:,1) = tau(:,1); tau_f5(:,1) = tau(:,1); tau_f6(:,1) = tau(:,1);
+% 
+% for i = 2:length(traj_par.t)
+%     tau(:,i) = base_regressor(q(:,i),qd(:,i),q2d(:,i))*pir_base_vctr;
+%     for j = 1:6
+%        tau_f1(j,i) = low_pass_filter(tau(j,i),tau_f1(j,i-1),alpha(1),beta(1),dt); 
+%        tau_f2(j,i) = low_pass_filter(tau(j,i),tau_f2(j,i-1),alpha(2),beta(2),dt); 
+%        tau_f3(j,i) = low_pass_filter(tau(j,i),tau_f3(j,i-1),alpha(3),beta(3),dt); 
+%        tau_f4(j,i) = low_pass_filter(tau(j,i),tau_f4(j,i-1),alpha(4),beta(4),dt); 
+%        tau_f5(j,i) = low_pass_filter(tau(j,i),tau_f5(j,i-1),alpha(5),beta(5),dt); 
+%     end
+% end
+% tau_e1 = vertcat(tau,tau_f1,tau_f2,tau_f3,tau_f4,tau_f5);
+
+for i = 1:length(traj_par.t)
+    tau(:,i) = base_regressor(q(:,i),qd(:,i),q2d(:,i))*pir_base_vctr;
+    if i>dly(1); tau_f1(:,i) = tau(:,i-dly(1)); else; tau_f1(:,i) = zeros(6,1); end
+    if i>dly(2); tau_f2(:,i) = tau(:,i-dly(2)); else; tau_f2(:,i) = zeros(6,1); end
+    if i>dly(3); tau_f3(:,i) = tau(:,i-dly(3)); else; tau_f3(:,i) = zeros(6,1); end
+    if i>dly(4); tau_f4(:,i) = tau(:,i-dly(4)); else; tau_f4(:,i) = zeros(6,1); end
+    if i>dly(5); tau_f5(:,i) = tau(:,i-dly(5)); else; tau_f5(:,i) = zeros(6,1); end
+end
+tau_e2 = vertcat(tau,tau_f1,tau_f2,tau_f3,tau_f4,tau_f5);
+
+%{
+jnt = 4;
+figure
+plot(traj_par.t,tau(jnt,:))
+hold on
+plot(traj_par.t,tau_f1(jnt,:))
+plot(traj_par.t,tau_f2(jnt,:))
+plot(traj_par.t,tau_f3(jnt,:))
+plot(traj_par.t,tau_f4(jnt,:))
+plot(traj_par.t,tau_f5(jnt,:))
+%}
+
+% Y(:,:,1) = base_regressor(q(:,1),qd(:,1),q2d(:,1));
+% Y_f1(:,:,1) = Y(:,:,1); Y_f2(:,:,1) = Y(:,:,1); Y_f3(:,:,1) = Y(:,:,1);
+% Y_f4(:,:,1) = Y(:,:,1); Y_f5(:,:,1) = Y(:,:,1); Y_f6(:,:,1) = Y(:,:,1);
+% for i = 2:length(traj_par.t)
+%    Y(:,:,i) = base_regressor(q(:,i),qd(:,i),q2d(:,i));
+%    for j = 1:6
+%        for k = 1:36
+%            Y_f1(j,k,i) = low_pass_filter(Y(j,k,i),Y_f1(j,k,i-1),alpha(1),beta(1),dt);
+%            Y_f2(j,k,i) = low_pass_filter(Y(j,k,i),Y_f2(j,k,i-1),alpha(2),beta(2),dt);
+%            Y_f3(j,k,i) = low_pass_filter(Y(j,k,i),Y_f3(j,k,i-1),alpha(3),beta(3),dt);
+%            Y_f4(j,k,i) = low_pass_filter(Y(j,k,i),Y_f4(j,k,i-1),alpha(4),beta(4),dt);
+%            Y_f5(j,k,i) = low_pass_filter(Y(j,k,i),Y_f5(j,k,i-1),alpha(5),beta(5),dt);
+%        end
+%    end
+%     
+% end
+% Y_e1 = vertcat(Y, Y_f1, Y_f2, Y_f3, Y_f4, Y_f5);
+
+for i = 1:length(traj_par.t)
+   Y(:,:,i) = base_regressor(q(:,i),qd(:,i),q2d(:,i));
+   if i>dly(1); Y_f1(:,:,i) = Y(:,:,i-dly(1)); else; Y_f1(:,:,i) = zeros(6,36); end
+   if i>dly(2); Y_f2(:,:,i) = Y(:,:,i-dly(2)); else; Y_f2(:,:,i) = zeros(6,36); end
+   if i>dly(3); Y_f3(:,:,i) = Y(:,:,i-dly(3)); else; Y_f3(:,:,i) = zeros(6,36); end
+   if i>dly(4); Y_f4(:,:,i) = Y(:,:,i-dly(4)); else; Y_f4(:,:,i) = zeros(6,36); end
+   if i>dly(5); Y_f5(:,:,i) = Y(:,:,i-dly(5)); else; Y_f5(:,:,i) = zeros(6,36); end
+end
+Y_e2 = vertcat(Y, Y_f1, Y_f2, Y_f3, Y_f4, Y_f5);
+
+%{
+row_Y = 1;
+column_Y = 8;
+figure
+plot(traj_par.t, reshape(Y(row_Y,column_Y,:),[size(Y,3),1]))
+hold on
+plot(traj_par.t, reshape(Y_f1(row_Y,column_Y,:),[size(Y_f1,3),1]))
+plot(traj_par.t, reshape(Y_f2(row_Y,column_Y,:),[size(Y_f2,3),1]))
+plot(traj_par.t, reshape(Y_f3(row_Y,column_Y,:),[size(Y_f3,3),1]))
+plot(traj_par.t, reshape(Y_f4(row_Y,column_Y,:),[size(Y_f4,3),1]))
+plot(traj_par.t, reshape(Y_f5(row_Y,column_Y,:),[size(Y_f5,3),1]))
+%}
+
+gma1 = 1e-2;
+pi_drem(:,1) = zeros(36,1);
+for i = 2:length(traj_par.t)
+   Yei = Y_e2(:,:,i);
+   tauei = tau_e2(:,i);
+   
+   adjYei =  adjoint(Yei);
+   detYei = det(Yei);
+   YYi = adjYei*tauei;
+   erri = YYi - detYei*pi_drem(:,i-1);
+   gradi = detYei/(gma1 + detYei^2)*erri;
+   pi_drem(:,i) = pi_drem(:,i-1) + gma1*gradi;
+end
+
+figure
+hold on
+for i = 1:5
+    plot(traj_par.t,pi_drem(i,1:end))
+end
+
+return
+
+
+% ------------------------------------------------------------------------
+% Low pass filter for DREM
+% ------------------------------------------------------------------------
+function y_cur = low_pass_filter(u_cur,y_prev,alpha,beta,T)
+    t1 = 1+beta*T;
+    t2 = alpha*T;
+    y_cur = 1/t1*y_prev + t2/t1*u_cur;
+end
\ No newline at end of file
diff --git a/ur_exprmt_dsgn.m b/ur_exprmt_dsgn.m
index 0aab9a5..b22f544 100644
--- a/ur_exprmt_dsgn.m
+++ b/ur_exprmt_dsgn.m
@@ -37,8 +37,8 @@ traj_par.t = 0:traj_par.t_smp:traj_par.T;  % time
 traj_par.N = 5;          % number of harmonics
 traj_par.q0 = deg2rad([0 -90 0 -90 0 0 ]');
 % Use different limit for positions for safety
-traj_par.q_min = deg2rad([-180 -180 -90 -180 -90 -90]');
-traj_par.q_max = deg2rad([180 0 90 0 90 90]');
+traj_par.q_min = -deg2rad([180  180  160   180  90   90]');
+traj_par.q_max =  deg2rad([180  0    160   0    90   90]');
 traj_par.qd_max = qd_max;
 traj_par.q2d_max = 2*ones(6,1);
 
@@ -124,7 +124,7 @@ subplot(3,1,3)
 pathToFolder = 'trajectory_optmzn/';
 t1 = strcat('N',num2str(traj_par.N),'T',num2str(traj_par.T));
 if strcmp(optmznAlgorithm, 'patternsearch')
-    filename = strcat(pathToFolder,'ptrnSrch_',t1,'QR.mat');
+    filename = strcat(pathToFolder,'ptrnSrch_',t1,'QR2.mat');
 elseif strcmp(optmznAlgorithm, 'ga')
     filename = strcat(pathToFolder,'ga_',t1,'.mat');
 elseif strcmp(optmznAlgorithm, 'fmincon')