MATLAB Answers

Help on trigonometric equation solution. Thanks a lot!

2 views (last 30 days)
Jack Dong
Jack Dong on 21 Jul 2021
Commented: Jack Dong on 22 Jul 2021
Hi Matlab Experts,
  • I built the "forward script" to calculate R1, R2 from initial inputs: X = 0.9425; Z = sin(X) + 2 = 2.809; beta = 0.531 and results for R1, R2 are showed as below.
ans =
7.7733501156970570278303966915701
ans =
11.261695645672471499665334704332
  • Now, I want to use calculated R1, R2 values to input to the "inverse script" for calculating backwards to initial values of X, Z, beta. Expectation is X = 0.9425; Z = sin(X)+2 = 2.809; beta = 0.531 as initial. But the returned beta value is 50.796485621 and even though this is true result in terms of trigonometry but it's not my wanted value (0.531). I think we should assign some conditions for beta, so that it can return the expected value 0.531 but I don't know how, plese help me. Thanks a lot!
ans =
0.94245504646393978078105437144795
ans =
2.8090036222785815735410394206806
ans =
50.796485621630240525665530786852
------------------------------------------------------------------------------------------
FORWARD SCRIPT TO CALCULATE R1, R2:
syms X Y Z alpha beta gama
deg2rad=pi/180;
theta_b=15*deg2rad;
theta_p=15*deg2rad;
R_b=1;
R_p=0.9;
Q0=1; %initial leg length
p=[X;Y;Z];
B1=[R_b*cos(15*deg2rad);R_b*sin(15*deg2rad);0];
B2=[R_b*cos(105*deg2rad);R_b*sin(105*deg2rad);0];
B3=[R_b*cos(135*deg2rad);R_b*sin(135*deg2rad);0];
B4=[R_b*cos(225*deg2rad);R_b*sin(225*deg2rad);0];
B5=[R_b*cos(255*deg2rad);R_b*sin(255*deg2rad);0];
B6=[R_b*cos(345*deg2rad);R_b*sin(345*deg2rad);0];
Bx1=B1(1,:);
Bx2=B2(1,:);
Bx3=B3(1,:);
Bx4=B4(1,:);
Bx5=B5(1,:);
Bx6=B6(1,:);
By1=B1(2,:);
By2=B2(2,:);
By3=B3(2,:);
By4=B4(2,:);
By5=B5(2,:);
By6=B6(2,:);
Bz1=B1(3,:);
Bz2=B2(3,:);
Bz3=B3(3,:);
Bz4=B4(3,:);
Bz5=B5(3,:);
Bz6=B6(3,:);
P1=[R_p*cos(75*deg2rad);R_b*sin(75*deg2rad);0];
P2=[R_p*cos(165*deg2rad);R_b*sin(165*deg2rad);0];
P3=[R_p*cos(195*deg2rad);R_b*sin(195*deg2rad);0];
P4=[R_p*cos(285*deg2rad);R_b*sin(285*deg2rad);0];
P5=[R_p*cos(315*deg2rad);R_b*sin(315*deg2rad);0];
P6=[R_p*cos(405*deg2rad);R_b*sin(405*deg2rad);0];
Px1=P1(1,:);
Px2=P2(1,:);
Px3=P3(1,:);
Px4=P4(1,:);
Px5=P5(1,:);
Px6=P6(1,:);
Py1=P1(2,:);
Py2=P2(2,:);
Py3=P3(2,:);
Py4=P4(2,:);
Py5=P5(2,:);
Py6=P6(2,:);
Pz1=P1(3,:);
Pz2=P2(3,:);
Pz3=P3(3,:);
Pz4=P4(3,:);
Pz5=P5(3,:);
Pz6=P6(3,:);
X=0.9425;
beta=0.531;
Y=0;
gama=0;
alpha=0;
Z=sin(X)+2;
R1=(X+((cos(gama)*cos(beta))*Px1)+(((cos(gama)*sin(beta)*sin(alpha))-(sin(gama)*cos(alpha)))*Py1)+(((cos(gama)*sin(beta)*cos(alpha))+(sin(gama)*sin(alpha)))*Pz1)-Bx1)^2+(Y+((sin(gama)*cos(beta))*Px1)+(((sin(gama)*sin(beta)*sin(alpha))+(cos(gama)*cos(alpha)))*Py1)+(((sin(gama)*sin(beta)*cos(alpha))-(cos(gama)*sin(alpha)))*Pz1)-By1)^2+(Z+((-sin(beta))*Px1)+((cos(beta)*sin(alpha))*Py1)+((cos(beta)*cos(alpha))*Pz1)-Bz1)^2;
L1=R1^0.5-Q0;
R2=(X+((cos(gama)*cos(beta))*Px2)+(((cos(gama)*sin(beta)*sin(alpha))-(sin(gama)*cos(alpha)))*Py2)+(((cos(gama)*sin(beta)*cos(alpha))+(sin(gama)*sin(alpha)))*Pz2)-Bx2)^2+(Y+((sin(gama)*cos(beta))*Px2)+(((sin(gama)*sin(beta)*sin(alpha))+(cos(gama)*cos(alpha)))*Py2)+(((sin(gama)*sin(beta)*cos(alpha))-(cos(gama)*sin(alpha)))*Pz2)-By2)^2+(Z+((-sin(beta))*Px2)+((cos(beta)*sin(alpha))*Py2)+((cos(beta)*cos(alpha))*Pz2)-Bz2)^2;
L2=R2^0.5-Q0;
vpa(R1.')
vpa(R2.')
INVERSE SCRIPT
syms X Y Z alpha beta gama
deg2rad=pi/180;
theta_b=15*deg2rad;
theta_p=15*deg2rad;
R_b=1;
R_p=0.9;
Q0=1; %initial leg length
p=[X;Y;Z];
B1=[R_b*cos(15*deg2rad);R_b*sin(15*deg2rad);0];
B2=[R_b*cos(105*deg2rad);R_b*sin(105*deg2rad);0];
B3=[R_b*cos(135*deg2rad);R_b*sin(135*deg2rad);0];
B4=[R_b*cos(225*deg2rad);R_b*sin(225*deg2rad);0];
B5=[R_b*cos(255*deg2rad);R_b*sin(255*deg2rad);0];
B6=[R_b*cos(345*deg2rad);R_b*sin(345*deg2rad);0];
Bx1=B1(1,:);
Bx2=B2(1,:);
Bx3=B3(1,:);
Bx4=B4(1,:);
Bx5=B5(1,:);
Bx6=B6(1,:);
By1=B1(2,:);
By2=B2(2,:);
By3=B3(2,:);
By4=B4(2,:);
By5=B5(2,:);
By6=B6(2,:);
Bz1=B1(3,:);
Bz2=B2(3,:);
Bz3=B3(3,:);
Bz4=B4(3,:);
Bz5=B5(3,:);
Bz6=B6(3,:);
P1=[R_p*cos(75*deg2rad);R_b*sin(75*deg2rad);0];
P2=[R_p*cos(165*deg2rad);R_b*sin(165*deg2rad);0];
P3=[R_p*cos(195*deg2rad);R_b*sin(195*deg2rad);0];
P4=[R_p*cos(285*deg2rad);R_b*sin(285*deg2rad);0];
P5=[R_p*cos(315*deg2rad);R_b*sin(315*deg2rad);0];
P6=[R_p*cos(405*deg2rad);R_b*sin(405*deg2rad);0];
Px1=P1(1,:);
Px2=P2(1,:);
Px3=P3(1,:);
Px4=P4(1,:);
Px5=P5(1,:);
Px6=P6(1,:);
Py1=P1(2,:);
Py2=P2(2,:);
Py3=P3(2,:);
Py4=P4(2,:);
Py5=P5(2,:);
Py6=P6(2,:);
Pz1=P1(3,:);
Pz2=P2(3,:);
Pz3=P3(3,:);
Pz4=P4(3,:);
Pz5=P5(3,:);
Pz6=P6(3,:);
Y=0;
gama=0;
alpha=0;
R1=(X+((cos(gama)*cos(beta))*Px1)+(((cos(gama)*sin(beta)*sin(alpha))-(sin(gama)*cos(alpha)))*Py1)+(((cos(gama)*sin(beta)*cos(alpha))+(sin(gama)*sin(alpha)))*Pz1)-Bx1)^2+(Y+((sin(gama)*cos(beta))*Px1)+(((sin(gama)*sin(beta)*sin(alpha))+(cos(gama)*cos(alpha)))*Py1)+(((sin(gama)*sin(beta)*cos(alpha))-(cos(gama)*sin(alpha)))*Pz1)-By1)^2+(Z+((-sin(beta))*Px1)+((cos(beta)*sin(alpha))*Py1)+((cos(beta)*cos(alpha))*Pz1)-Bz1)^2;
L1=R1^0.5-Q0;
R2=(X+((cos(gama)*cos(beta))*Px2)+(((cos(gama)*sin(beta)*sin(alpha))-(sin(gama)*cos(alpha)))*Py2)+(((cos(gama)*sin(beta)*cos(alpha))+(sin(gama)*sin(alpha)))*Pz2)-Bx2)^2+(Y+((sin(gama)*cos(beta))*Px2)+(((sin(gama)*sin(beta)*sin(alpha))+(cos(gama)*cos(alpha)))*Py2)+(((sin(gama)*sin(beta)*cos(alpha))-(cos(gama)*sin(alpha)))*Pz2)-By2)^2+(Z+((-sin(beta))*Px2)+((cos(beta)*sin(alpha))*Py2)+((cos(beta)*cos(alpha))*Pz2)-Bz2)^2;
L2=R2^0.5-Q0;
A=vpasolve(R1==7.7731884,R2==11.2615,Z==sin(X)+2);
vpa(A.X')
vpa(A.Z')
vpa(A.beta')

Accepted Answer

David Goodmanson
David Goodmanson on 21 Jul 2021
Edited: David Goodmanson on 21 Jul 2021
Hi Jack,
Fortunately the solver is coming up with the correct angle, except for the 2*pi ambiguity. So, after the solver does its job you can use
mod(50.796485621,2*pi)
ans = 0.5310
  1 Comment
Jack Dong
Jack Dong on 22 Jul 2021
Thanks David for your great answer but can we embed it into my script so that after running, I get the correct beta immediately. Thanks!
I tried this but failed!
vpa(A.X')
vpa(A.Z')
vpa(A.beta')
beta1=mod(A.beta,2*pi)
-----> Result is
ans =
0.94245504646393978078105437144795
ans =
2.8090036222785815735410394206806
ans =
50.796485621630240525665530786852
beta1 =
mod(50.796485621630240525665530786852, 2*pi)
------------------------------------------------------
Script
syms X Y Z alpha beta gama
deg2rad=pi/180;
theta_b=15*deg2rad;
theta_p=15*deg2rad;
R_b=1;
R_p=0.9;
Q0=1; %initial leg length
p=[X;Y;Z];
B1=[R_b*cos(15*deg2rad);R_b*sin(15*deg2rad);0];
B2=[R_b*cos(105*deg2rad);R_b*sin(105*deg2rad);0];
B3=[R_b*cos(135*deg2rad);R_b*sin(135*deg2rad);0];
B4=[R_b*cos(225*deg2rad);R_b*sin(225*deg2rad);0];
B5=[R_b*cos(255*deg2rad);R_b*sin(255*deg2rad);0];
B6=[R_b*cos(345*deg2rad);R_b*sin(345*deg2rad);0];
Bx1=B1(1,:);
Bx2=B2(1,:);
Bx3=B3(1,:);
Bx4=B4(1,:);
Bx5=B5(1,:);
Bx6=B6(1,:);
By1=B1(2,:);
By2=B2(2,:);
By3=B3(2,:);
By4=B4(2,:);
By5=B5(2,:);
By6=B6(2,:);
Bz1=B1(3,:);
Bz2=B2(3,:);
Bz3=B3(3,:);
Bz4=B4(3,:);
Bz5=B5(3,:);
Bz6=B6(3,:);
P1=[R_p*cos(75*deg2rad);R_b*sin(75*deg2rad);0];
P2=[R_p*cos(165*deg2rad);R_b*sin(165*deg2rad);0];
P3=[R_p*cos(195*deg2rad);R_b*sin(195*deg2rad);0];
P4=[R_p*cos(285*deg2rad);R_b*sin(285*deg2rad);0];
P5=[R_p*cos(315*deg2rad);R_b*sin(315*deg2rad);0];
P6=[R_p*cos(405*deg2rad);R_b*sin(405*deg2rad);0];
Px1=P1(1,:);
Px2=P2(1,:);
Px3=P3(1,:);
Px4=P4(1,:);
Px5=P5(1,:);
Px6=P6(1,:);
Py1=P1(2,:);
Py2=P2(2,:);
Py3=P3(2,:);
Py4=P4(2,:);
Py5=P5(2,:);
Py6=P6(2,:);
Pz1=P1(3,:);
Pz2=P2(3,:);
Pz3=P3(3,:);
Pz4=P4(3,:);
Pz5=P5(3,:);
Pz6=P6(3,:);
Y=0;
gama=0;
alpha=0;
R1=(X+((cos(gama)*cos(beta))*Px1)+(((cos(gama)*sin(beta)*sin(alpha))-(sin(gama)*cos(alpha)))*Py1)+(((cos(gama)*sin(beta)*cos(alpha))+(sin(gama)*sin(alpha)))*Pz1)-Bx1)^2+(Y+((sin(gama)*cos(beta))*Px1)+(((sin(gama)*sin(beta)*sin(alpha))+(cos(gama)*cos(alpha)))*Py1)+(((sin(gama)*sin(beta)*cos(alpha))-(cos(gama)*sin(alpha)))*Pz1)-By1)^2+(Z+((-sin(beta))*Px1)+((cos(beta)*sin(alpha))*Py1)+((cos(beta)*cos(alpha))*Pz1)-Bz1)^2;
L1=R1^0.5-Q0;
R2=(X+((cos(gama)*cos(beta))*Px2)+(((cos(gama)*sin(beta)*sin(alpha))-(sin(gama)*cos(alpha)))*Py2)+(((cos(gama)*sin(beta)*cos(alpha))+(sin(gama)*sin(alpha)))*Pz2)-Bx2)^2+(Y+((sin(gama)*cos(beta))*Px2)+(((sin(gama)*sin(beta)*sin(alpha))+(cos(gama)*cos(alpha)))*Py2)+(((sin(gama)*sin(beta)*cos(alpha))-(cos(gama)*sin(alpha)))*Pz2)-By2)^2+(Z+((-sin(beta))*Px2)+((cos(beta)*sin(alpha))*Py2)+((cos(beta)*cos(alpha))*Pz2)-Bz2)^2;
L2=R2^0.5-Q0;
A=vpasolve(R1==7.7731884,R2==11.2615,Z==sin(X)+2);
vpa(A.X')
vpa(A.Z')
vpa(A.beta')
beta1=mod(A.beta,2*pi)

Sign in to comment.

More Answers (0)

Products


Release

R2019b

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Translated by