inverse geometric model arm robot : system equations non linear

Mar 2016
3
0
france
Hi,

i'm studying a robot arm

i have done davenit and direct geometric model and i obtain this equations system



eq1=335*cos(t2)* sin(t3) -77*sin(t2)-260*sin(t2)*sin(t4)+260*cos(t2)*cos(t3)*cos(t4)+85=x;
eq2=335*cos(t3)-260* sin(t3)*cos(t4)=y;
eq3=0-335*sin(t2)* sin(t3) -77*cos(t2) -260*cos(t2)*sin(t4)-260*sin(t2)*cos(t3)*cos(t4) =z;
which is equal to

(1)S+(2)*C2 =>eq1=260*sin(2*t2)*sin(t4)== 8-x*sin(t2)-z*cos(t2);
eq2=335*cos(t3)-260*sin(t3)*cos(t4)-y==0;
eq3=0-335*sin(t2)* sin(t3) -77*cos(t2) -260*cos(t2)*sin(t4)-260*sin(t2)*cos(t3)*cos(t4) =z;

avec T2 [ -PI/4. PI/2] T3[ -PI/4 PI/4] T4 [0 PI/2 ]



Now i want to solve it and express t2 = f(x,y,z) t3=g(x,y,z) t4 = h(x,y,z)

it try many ways (paul method susbstitution cos=1-u²/1+u²) but i don't manage to find the results

If someone can help me even with a computing solution from mapple and so on

thankscinematique.png
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
Hey dim22.

It's probably going to be useful to use a numerical solving scheme by doing what Newtons method does to find zero's of a function except you do it for multiple variables.

After you code that up then focus on whether you can get a symbolic answer because you may not.

Take a look at generalizations of iterative root finding algorithms to multiple dimensions based on matrix techniques.
 
Mar 2016
3
0
france
Thanks for your help,
I have ever used this method with the inversion of jacobian : Basicly
i calculate J the jacobian and i obtain delta x = J delta t
After i calculate the monrose pseudo invers and i obtain delta t = J* deltax
But this relation is a kinematic relation not a geometric one
My problem is to fine an anlytic solution which looks like t= f(x,y,z)
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
If you need an analytical solution you should try and represent the system of equations in matrix form (including tensor form) and post the matrices.

Also - if you are using a specific co-ordinate system transformation (which it looks like you are but I'm not completely sure) then say which one (i.e. spherical, cylindrical, etc).

Basically this will be a basis conversion with differential geometry.

Understand that you should organize your work so that others who want to help you have an easier time of doing so.
 
Mar 2016
3
0
france
oK my problem consists in determining functions which link end effector positions (x,y,z) to joint angles ones (t2,t3,t4)
To do that i have folowed this method :
1) i have modelised the arm with davenit parameters and find the joint matix (attachment : matproblem1&2)
2) i have written the position equation (attachment : matproblem3)
3) i have obtained this equations sytem
eq1=335*cos(t2)* sin(t3) -77*sin(t2)-260*sin(t2)*sin(t4)+260*cos(t2)*cos(t3)*cos(t4)+85 =x;
eq2=335*cos(t3)-260* sin(t3)*cos(t4)=y;
eq3=0-335*sin(t2)* sin(t3) -77*cos(t2) -260*cos(t2)*sin(t4)-260*sin(t2)*cos(t3)*cos(t4) =z;
which is equal to

(1)S+(2)*C2 =>eq1=260*sin(2*t2)*sin(t4)== 8-x*sin(t2)-z*cos(t2);
eq2=335*cos(t3)-260*sin(t3)*cos(t4)-y==0;
eq3=0-335*sin(t2)* sin(t3) -77*cos(t2) -260*cos(t2)*sin(t4)-260*sin(t2)*cos(t3)*cos(t4) =z;

avec T2 [ -PI/4. PI/2] T3[ -PI/4 PI/4] T4 [0 PI/2 ]



Now i want to solve it and express t2 = f(x,y,z) t3=g(x,y,z) t4 = h(x,y,z)
 

Attachments

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
Can you factor out the non rotational forms in each matrix?

This would mean that you have a matrix as a function of each angle.

Once you have this you have a way of representing a matrix (based on an angle) relative to the other ones and doing some matrix algebra will allow you to get that matrix on one side of the equation.

Look at elementary row operation matrices to remove variables in a matrix and balance these operations on both sides.