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DynamicCal.m
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DynamicCal.m
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%Dynamics for crustcrawler
%Gruppe 364 3. semester 2020
%% Setting up symbolic values (for symbolic calculation)
syms g I_xx_1 I_xy_1 I_xz_1 I_xy_1 I_yy_1 I_yz_1 I_xz_1 I_yz_1 I_zz_1
syms I_xx_2 I_xy_2 I_xz_2 I_xy_2 I_yy_2 I_yz_2 I_xz_2 I_yz_2 I_zz_2
syms I_xx_3 I_xy_3 I_xz_3 I_xy_3 I_yy_3 I_yz_3 I_xz_3 I_yz_3 I_zz_3
syms l1 lc1 l2 lc2 l3 lc3
syms m1 m2 m3
syms th1 th2 th3
syms thd1 thd2 thd3
syms thdd1 thdd2 thdd3
syms thd12 thd13 thd23
syms thd1sq thd2sq thd3sq
syms tau1 tau2 tau3
%% Gravity vector
g_vec = [
0
0
-g];
%% Inertia tensors
IC_1 = [
I_xx_1 I_xy_1 I_xz_1
I_xy_1 I_yy_1 I_yz_1
I_xz_1 I_yz_1 I_zz_1];
IC_2 = [
I_xx_2 I_xy_2 I_xz_2
I_xy_2 I_yy_2 I_yz_2
I_xz_2 I_yz_2 I_zz_2];
IC_3 = [
I_xx_3 I_xy_3 I_xz_3
I_xy_3 I_yy_3 I_yz_3
I_xz_3 I_yz_3 I_zz_3];
%% Ratation matrices;
R_01 = [
cos(th1) -sin(th1) 0
sin(th1) cos(th1) 0
0 0 1];
R_12 = [
-sin(th2) -cos(th2) 0
0 0 -1
cos(th2) -sin(th2) 0];
R_23 = [
cos(th3) -sin(th3) 0
sin(th3) cos(th3) 0
0 0 1];
R_02 = R_01 * R_12;
R_03 = R_01 * R_12 * R_23;
%% Unit vectors
e1 = [
1
0
0];
e3 = [
0
0
1];
%% Position vectors
Pv1 = R_01 * (l1 * e3);
Pv2 = R_02 * (l2 * e1);
PvC1 = R_01 * lc1 * e3;
PvC2 = R_02 * lc2 * e1;
PvC3 = R_03 * lc3 * e1;
PC1 = PvC1;
PC2 = PvC2 + Pv1;
PC3 = PvC3 + Pv1 + Pv2;
%% Angular velocities
Om11 = thd1 * e3;
Om22 = inv(R_12) * Om11 + thd2 * e3;
Om33 = inv(R_23) * Om22 + thd3 * e3;
Om01 = R_01 * thd1 * e3;
Om02 = R_01 * Om11 + R_02 * thd2 * e3;
Om03 = R_02 * Om22 + R_03 * thd3 * e3;
%% Linear velocities
Vc1 = cross(Om01,PC1);
Vc2 = cross(Om02,PC2);
Vc3 = cross(Om03,PC3);
%% Kinetic energies
T1 = 1/2 * m1 * transpose(Vc1) * Vc1 + 1/2 * transpose(Om11) * IC_1 ...
* Om11;
T2 = 1/2 * m2 * transpose(Vc2) * Vc2 + 1/2 * transpose(Om22) * IC_2 ...
* Om22;
T3 = 1/2 * m3 * transpose(Vc3) * Vc3 + 1/2 * transpose(Om33) * IC_3 ...
* Om33;
%% potential energies
V1 = -m1 * transpose(g_vec) * PC1;
V2 = -m2 * transpose(g_vec) * PC2;
V3 = -m3 * transpose(g_vec) * PC3;
%% The lagrangian
T_total = simplify(T1 + T2 + T3);
V_total = simplify(V1 + V2 + V3);
Langragian_system = T_total-V_total;
%% Torques
%tau 1
diff_L_th1 = diff(Langragian_system,th1);
diff_L_thd1 = diff(Langragian_system,thd1);
diff_time_diff_L_th1_d...
= diff(diff_L_thd1,th1) * thd1 ...
+ diff(diff_L_thd1,th2) * thd2 ...
+ diff(diff_L_thd1,th3) * thd3 ...
+ diff(diff_L_thd1,thd1) * thdd1 ...
+ diff(diff_L_thd1,thd2) * thdd2 ...
+ diff(diff_L_thd1,thd3) * thdd3;
tau_1 = simplify(diff_time_diff_L_th1_d - diff_L_th1)
%tau 2
diff_L_th2 = diff(Langragian_system,th2);
diff_L_thd2 = diff(Langragian_system,thd2);
diff_time_diff_L_th2_d...
= diff(diff_L_thd2,th1) * thd1 ...
+ diff(diff_L_thd2,th2) * thd2 ...
+ diff(diff_L_thd2,th3) * thd3 ...
+ diff(diff_L_thd2,thd1) * thdd1 ...
+ diff(diff_L_thd2,thd2) * thdd2 ...
+ diff(diff_L_thd2,thd3) * thdd3;
tau_2 = simplify(diff_time_diff_L_th2_d - diff_L_th2)
%tau 3
diff_L_th3 = diff(Langragian_system,th3);
diff_L_thd3 = diff(Langragian_system,thd3);
diff_time_diff_L_th3_d...
= diff(diff_L_thd3,th1) * thd1 ...
+ diff(diff_L_thd3,th2) * thd2 ...
+ diff(diff_L_thd3,th3) * thd3 ...
+ diff(diff_L_thd3,thd1) * thdd1 ...
+ diff(diff_L_thd3,thd2) * thdd2 ...
+ diff(diff_L_thd3,thd3) * thdd3;
tau_3 = simplify(diff_time_diff_L_th3_d - diff_L_th3)
%% Statespace equation terms
% Mass Matrix
MassMatrix = [
simplify(diff(tau_1,thdd1)),...
simplify(diff(tau_1,thdd2)),...
simplify(diff(tau_1,thdd3));...
...
simplify(diff(tau_2,thdd1)),...
simplify(diff(tau_2,thdd2)),...
simplify(diff(tau_2,thdd3));...
...
simplify(diff(tau_3,thdd1)),...
simplify(diff(tau_3,thdd2)),...
simplify(diff(tau_3,thdd3))
]
%Print Mass Matrix to .txt file
fid = fopen('MassMatrix.txt', 'wt');
fprintf(fid, '%s\n', char(MassMatrix));
fclose(fid);
% "subs" function is used to substitute ex thd1*thd2 to thd12, for partial
% differentiation in MATLAB
t1_thdthd_sub=...
subs(tau_1,[thd1*thd2,thd1*thd2,thd2*thd3],[thd12,thd13,thd23]);
t2_thdthd_sub=...
subs(tau_2,[thd1*thd2,thd1*thd2,thd2*thd3],[thd12,thd13,thd23]);
t3_thdthd_sub=...
subs(tau_3,[thd1*thd2,thd1*thd2,thd2*thd3],[thd12,thd13,thd23]);
% Coriolis coeficients
CoriolisMatrix = [
simplify(diff(t1_thdthd_sub,thd12)),...
simplify(diff(t1_thdthd_sub,thd13)),...
simplify(diff(t1_thdthd_sub,thd23));...
...
simplify(diff(t2_thdthd_sub,thd12)),...
simplify(diff(t2_thdthd_sub,thd13)),...
simplify(diff(t2_thdthd_sub,thd23));...
...
simplify(diff(t3_thdthd_sub,thd12)),...
simplify(diff(t3_thdthd_sub,thd13)),...
simplify(diff(t3_thdthd_sub,thd23))
]
% "subs" function is used to substitute ex thd1^2 to thd1sq, for partial
% differentiation in MATLAB
t1_thdsq_sub=subs(tau_1,[thd1^2,thd2^2,thd3^2],[thd1sq,thd2sq,thd3sq]);
t2_thdsq_sub=subs(tau_2,[thd1^2,thd2^2,thd3^2],[thd1sq,thd2sq,thd3sq]);
t3_thdsq_sub=subs(tau_3,[thd1^2,thd2^2,thd3^2],[thd1sq,thd2sq,thd3sq]);
% centrifugal coeficients
CentrifugalMatrix = [
simplify(diff(t1_thdsq_sub,thd1sq)),...
simplify(diff(t1_thdsq_sub,thd2sq)),...
simplify(diff(t1_thdsq_sub,thd3sq));...
...
simplify(diff(t2_thdsq_sub,thd1sq)),...
simplify(diff(t2_thdsq_sub,thd2sq)),...
simplify(diff(t2_thdsq_sub,thd3sq));...
...
simplify(diff(t3_thdsq_sub,thd1sq))...
simplify(diff(t3_thdsq_sub,thd2sq))...
simplify(diff(t3_thdsq_sub,thd3sq))
]
thdthdVector = [thd1*thd2;thd1*thd3;thd2*thd3];
thdsqVector = [thd1^2;thd2^2;thd3^2];
% VelocityVector
VelocityVector = CoriolisMatrix*thdthdVector+CentrifugalMatrix*thdsqVector
%Print VelocityVector to .txt file
fid = fopen('VelocityVector.txt', 'wt');
fprintf(fid, '%s\n', char(VelocityVector));
fclose(fid);
% GravityVector
GravityVector_no_g = [
simplify(diff(tau_1,g));
simplify(diff(tau_2,g));
simplify(diff(tau_3,g))
];
GravityVector = g*GravityVector_no_g
%Print GravityVector_no_g to .txt file
fid = fopen('GravityVector_no_g.txt', 'wt');
fprintf(fid, '%s\n', char(GravityVector_no_g));
fclose(fid);
%% Statespace equation
jointAngleAccVector = [
thdd1;
thdd2;
thdd3
];
%Inverse Dynamics Equation
tauVector = simplify(MassMatrix*jointAngleAccVector+VelocityVector...
+GravityVector)
%Forward Dynamics Equation
tauVector = [
tau1;
tau2;
tau3
];
jointAngleAccVector = inv(MassMatrix)*(tauVector-...
(VelocityVector)+GravityVector)