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main_tf_11.m
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main_tf_11.m
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clear all
close all
clc
%%
syms Mh Jh % Mass and Moment Inertia of Trunk
syms Ms_int Js_int % Mass and Moment of inertia of torque sensor (Internal Ring)
syms Ms_ext Js_ext % Mass and Moment of inertia of torque sensor (External Ring)
syms M_act J_act % Mass and Moment Inertia of Actuator
syms g % Gravity acceleration
syms rh % Raidal Displacement of Central mass of trunk with respect rotational axis
syms Kh dh dhs % Trunk Stiffness , Trunk Damping , Trk_trq Damping
syms Ks ds % Stiffness of torque sensor , Damping of torque sensor
syms Kg dg dgs % Gearbox Stiffness , Gearbox Damping , Gea_Trq Damping
syms teta_h tetad_h tetadd_h % Trunk Position , Trunk Velocity , Trunk Acceleration with respect to vertical axis
syms teta_ext tetad_ext tetadd_ext % torque sensor Position, Velocity , Acceleration (external ring)
syms teta_int tetad_int tetadd_int % torque sensor Position, Velocity , Acceleration (internal ring)
syms teta_act tetad_act tetadd_act % BDC motor shaft Position , BDC motor shaft Velocity , BDC motor shaft Acceleration, data from encoder
syms tau_h tau_m % Torque by Trunk , Torque by BDC motor
syms rg % Gearbox ratio
syms bm % Friction between motor and gearbox
syms Tm % Motor Torque
%%
syms TETA_H TETAD_H TETADD_H
syms TETA_EXT TETAD_EXT TETADD_EXT
syms TETA_INT TETAD_INT TETADD_INT
syms TETA_ACT TETAD_ACT TETADD_ACT
%% Equations of Motion
% eqns = [ Tm - Bf*tetadm - Jtot*TETADDM - (Kg/rg)*( (tetam/rg) - tetas ) - (Dg/rg)*( (tetadm/rg) - tetads ) == 0 ,
% Kg*( (tetam/rg) - tetas ) + Dg*( (tetadm/rg) - tetads ) - Js*TETADDS - Ks*(tetas - tetal) - Ds*(tetads - tetadl) == 0 ,
% Ks*( tetas - tetal ) + Ds*( tetads - tetadl ) - Jl*TETADDL - Kl*(tetal - tetah) - Dl*(tetadl - tetadh) == 0 ];
eqns = [ Tm - bm*tetad_act - J_act*TETADD_ACT - (Kg/rg)*( (teta_act/rg) - teta_int ) - (dgs/rg)*( (tetad_act/rg) - tetad_int ) - (dg/rg)*(tetad_act/rg) == 0 ,
Kg*( (teta_act/rg) - teta_int ) + dgs*( (tetad_act/rg) - tetad_int ) - Js_int*TETADD_INT - Ks*(teta_int - teta_ext) - ds*(tetad_int - tetad_ext) == 0 ,
Ks*(teta_int - teta_ext) + ds*(tetad_int - tetad_ext) - Js_ext*TETADD_EXT - Kh*(teta_ext - teta_h) - dhs*(tetad_ext - tetad_h) == 0 ,
Kh*(teta_ext - teta_h) + dhs*(tetad_ext - tetad_h) - TETADD_H*Jh - tetad_h*dh == 0 ];
vars = [ TETADD_ACT TETADD_INT TETADD_EXT TETADD_H ];
%%
eqns_mat = [ eqns ,
TETAD_ACT == tetad_act ,
TETAD_INT == tetad_int ,
TETAD_EXT == tetad_ext ,
TETAD_H == tetad_h ];
vars_mat = [ TETAD_ACT TETADD_ACT TETAD_INT TETADD_INT TETAD_EXT TETADD_EXT TETAD_H TETADD_H];
[TETAD_ACT TETADD_ACT TETAD_INT TETADD_INT TETAD_EXT TETADD_EXT TETAD_H TETADD_H] = solve(eqns_mat, vars_mat)
%%
%STATE SPACE MODEL
%state = [ motor position , motor velocity , Internal torque sensor position , Internal torque sensor velocity , External torque sensor position , External torque sensor velocity , Trunk position , Trunk Velocity ]
%inputs = [ motor torque ]
eqns_mat_form = [ TETAD_ACT TETADD_ACT TETAD_INT TETADD_INT TETAD_EXT TETADD_EXT TETAD_H ];
vars_mat_form = [ teta_act tetad_act teta_int tetad_int teta_ext tetad_ext teta_h ];
ns = length(vars_mat_form); % Number of States
[A,nB] = equationsToMatrix( eqns_mat_form , vars_mat_form );
A_eq = A;
Bt = -nB;
[B_eq,~] = equationsToMatrix( Bt , [ Tm , tetad_h ] );
C_eq = [ 0 , 0 , Ks , ds , -Ks , -ds, 0 ] ;
D_eq = [ 0 , 0 ];
%% Transfer Function
syms s
% H(s) = Y(s) / U(s) = C*(sI-A)^(-1)*B + D
disp(' ')
disp('The symbolic transfer function is:')
TF = simplify( C_eq*[ (s*eye(ns,ns) - A_eq )^(-1) ]*B_eq + D_eq , 'Steps',100 );
pretty(TF)
disp(' ')
%% System Parametes from Data sheet
J_act = [0.000306 + 0.28200e-04]; % Inertia of motor rotor + harmonic drive , kilogram metre squared [kg. m2]
rg = 160; % Gear Ratio of harmonic drive
Js_int = 1.1e-4 ; % Inertia of Internal Torque Sensor Ring + Metal Coupling
Js_ext = 9.58e-4 ; % Inertia of External Torque Sensor Ring
Kg = 2.7e4; % Harmonic Drive Stiffness [Nm/rad]
ds = 0; % Torque Sensor Damping Ratio [Nm.Sec/rad]
Ks = 8.1853e4*1.4; % Torque Sensor Stiffness [Nm/rad]
dg = 0.65; % Harmonic Drive Damping Ratio [Nm.Sec/rad]
dgs = 6; % Damping Ratio between torque sensor and harmonic drive [Nm.Sec/rad]
bm = 4.7698e-04; % Friction between motor and harmonic drive [Nm.Sec/rad]
dhs = 0; % Damping between trunk and torque sensor [Nm.Sec/rad]
Kh = 1125; % Trunk Stiffness [Nm/rad]
W = 75; % Human Weight
Jh = 5.02; % Inertia of Trunk [kg. m2]
dh = 0.75; % Damping of Trunk [Nm.Sec/rad]
Vel_mot_nom = 263; % Nominal motor speed [rad/sec]
Trq_mot_nom = .56; % Nominal motor torque [Nm]
Ts = -1; % Sampling Time
%%
A_eq
A_lin = subs(A_eq);
B_eq
B_lin = subs(B_eq);
C_eq
C_lin = subs(C_eq);
D_lin = zeros(1,2)
A = double(A_lin)
A(7,7) = - 1e6;
B = double(B_lin)
C = [ 0 , 1 , 0 , 0 , 0 , 0 , 0;
0 , 0 , Ks , ds , -Ks , -ds, 0 ]
Ct = [ 0 , 0 , 0 , dhs , Kh , -dhs , -Kh ]
C = double( subs( C) )
Ct = double( subs( Ct) )
D = zeros(2,2)
Ai = [ A , zeros(7,1);
-Ct , zeros(1,1)]
Bi = [ B(:,1) ; zeros(1,1) ]
sys_sys = ss(A,B,C,D) % Steady Space for the Model with 7 states
%% Observability
Ab = A;
Ab(7,7) = 1e-12;
Cb = [ C ; Ct];
Ob = [ Cb ; Cb*Ab ; Cb*(Ab^2) ; Cb*(Ab^3) ; Cb*(Ab^4) ; Cb*(Ab^5) ; Cb*(Ab^6) ]
tol = 1e-6;
rank_Ob = rank(Ob,tol)
%% Kalman Gain
v_vel = .1e-6;
v_trq = 35e-3;
w_trq = .1e-3;
w_vel = .15e-3;
Qk = diag([ 10e-14 , 1.225e-3]);
Rk = diag([ 10e-8 , 2.25e-8]);
Tsk = .001;
%% Controllability
Co = ctrb(Ai,Bi);
unco = length(A) - rank(Co,tol)
Ai = [ Ab , zeros(7,1);
-Ct , zeros(1,1)]
%%
Qk = 1;
Rk = 1;
%%
Q = eye(8,8);
Q(1,1) = 1e1;
Q(2,2) = 1e1;
Q(3,3) = 1e1;
Q(4,4) = 1e1;
Q(5,5) = 1e1;
Q(6,6) = 1e1;
Q(7,7) = 1e1;
Q(8,8) = 1e2;
R = .6;
N = zeros(8,1);
% PSM = [Q N;N' R];
% eig(PSM);
% eig(A);
[K,S,P] = lqr(Ai,Bi,Q,R)
% chol(PSM);
%%