How to find the transformation matrix for a plat knowing the old and new coordinates of 3 points on it ?

Hi All

How do I define the transformation matrix of a plate that moves in space , knowing the old and new coordinates of 3 points on the plate ? (assume a circular plate and reference coordinate system in the center of the plate )

[farzad]

Any ideas ?

Answer 1

[Matt J]

You could use this FEX file

https://www.mathworks.com/matlabcentral/fileexchange/26186-absolute-orientation-horn-s-method

[farzad]

Thank you very much

just a question about calling the function, should I nest this in my code, or I can call the function when it is in a separate file ?

and in the second case , how do I do that ?

[Matt J]

You can call absor like any Matlab command. You can also obtain usage instructions using help or doc like in any Matlab command

>> doc absor

[farzad]

Thank you , that was written also in the code.

and one thing : I needed to have the angles with the axes of reference system.

for 2D problems it gives

regParams.theta

is there something equivalent for 3D I could get from this code ?

[Matt J]

Attached are some functions for decomposing a 3x3 rotation matrix into angles in various ways.

[farzad]

Thank you !

is the following correct ?

rotm= regParams.R;
            
eulZYX = rotm2eul(rotm);

[Matt J]

It is correct if that is the decomposition that you want.

[farzad]

the three angles with principle axes ?

[Matt J]

The three angles of what with the principal axes?

[farzad]

I assumed that if I have a plate with 3 known points on it, and I have the initial and final position of them in a movement of the plate , I could know the angles of the normal of the plate with principal axes

[Matt J]

That would be,

angles=acosd(regParams.R*InitialNormal)

where InitialNormal is the initial normal to the plate as a 3x1 vector:

P=[point1(:),point2(:),point3(:)].';
[~,~,V]=svd(P-mean(P));

InitialNormal=V(:,end);

[farzad]

Thank you so much! What is the ~ sign?

[Matt J]

You are welcome, but please accept-click the answer if it was what you were looking for.

The ~ tells Matlab that an ouput argument will not be used and that it need not assign it to any output variable.

[farzad]

what is the shortcut key for that sign and what is it called ?

[farzad]

and another doubt :

if the initial vector is like [0,0,1] , then is the regParams.R gives the angles in the same order ?

XYZ ?

[farzad]

also one thing , how do i force the angle results be smaller than 90° starting from 0 ° ?

[Matt J]

By positioning the plate appropriately. Make sure that the normal to the plate makes the angles with the principal axes that you want.

[farzad]

just one doubt :

should the point1 be a one row or one column matrix ?

P=[point1(:),point2(:),point3(:)].';

[Matt J]

It won't matter because point1(:) is always a column vector.

[farzad]

Thank you , still sorry for the question : is the result of eulZYX in Radians or degrees ?

[Matt J]

What is eulZYX?

>> which eulZYX
'eulZYX' not found.

[Jan]

@farzad: Do not assume, that the users of the forum have installed the same 3rd party tools or Matlab toolboxes as you have. Check the location of the functions by which . If it is not included in the standard toolboxes, post a link to the documentation or source of the function.

By the way: A simple test reveals, what the output is.

[farzad]

ok thank you

[farzad]

Ok . I tried for this triangle, and it does not work !!

A1= [10 -70 0; 60 30 0; -40 50 0]

A2= [10 -61.812 -22.498; 60 26.491 9.642; -40 44.151 16.07]

these are the coordinates of a triangle that has rotated around the X axis for 20° , with no rotation around the other axes. the triangle is in XY plane by the way.

so when I use absor, I get :

>> Alfa= eulZYX(1)

Alfa =

-0.0371

>> Beta= eulZYX(2)

Beta =

-0.0198

>> gamma=eulZYX(3)

gamma =

-0.0345

[Matt J]

Here's what I get:

>> reg=absor(A1.',A2.','doTrans',0);

>> rot2taitbryan(reg.R,'xyz')

ans =

   20.0003    0.0001    0.2375
   
   
>> acosd(reg.R(:,3))

ans =

   89.9999
  110.0003
   20.0003

[farzad]

Thank you so much!!! but I had no idea if I had to use the functions this way, were these in absor documents ?

but it does not work for me !! I don't have rot2taitbryan function, what is it ?

[farzad]

And why should the method change ? was the previous one not valid ?

[farzad]

apparently the second method is not right, why is it different ?

[farzad]

@Mat J

shall you please explain me kindly , why did you use this new way ? and why the previous suggestion was not working ?

[Matt J]

I have not proposed anything different than before. I gave you the function rot2taitbryan back on Jun 17,

https://www.mathworks.com/matlabcentral/answers/467055-how-to-find-the-transformation-matrix-for-a-plat-knowing-the-old-and-new-coordinates-of-3-points-on#comment_715585

[farzad]

ok, then what is the difference of this decomposition with the above ?

rotm= regParams.R;
            
eulZYX = rotm2eul(rotm);

[Matt J]

rot2eul returns results in radians and rot2taitbryan returns results in degrees. Also, rot2eul is only available to people who have the Robotics Toolbox (which I do not).

[farzad]

@Matt J Hi Mat. reading Wikipedia, I still have doubt about the roll angle.

cause what I need as Roll , is the rotation around the original X axis, Yaw , the rotation around the original Z axis, and pitch: the rotaion around the original Y axis

all of the Euler methods give the rotation around a new axis at some point

[Matt J]

cause what I need as Roll , is the rotation around the original X axis, Yaw , the rotation around the original Z axis, and pitch: the rotaion around the original Y axis

That is what rot2taitbryan gives you.

[farzad]

Thank you, exactly the script you wrote in the comments above? Intrinsic and extrinsic confuse me

There are six possibilities of choosing the rotation axes for Tait–Bryan angles. The six possible sequences are:

x-y′-z″ (intrinsic rotations) or z-y-x (extrinsic rotations) y-z′-x″ (intrinsic rotations) or x-z-y (extrinsic rotations) z-x'-y″ (intrinsic rotations) or y-x-z (extrinsic rotations) x-z′-y″ (intrinsic rotations) or y-z-x (extrinsic rotations) z-y′-x″ (intrinsic rotations) or x-y-z (extrinsic rotations): the intrinsic rotations are known as: yaw, pitch and roll y-x′-z″ (intrinsic rotations) or z-x-y (extrinsic rotations)

[farzad]

@Mat J how does this code recognize what was the order of the rotations applied?

Cause the stste of a plane is different if you apply Rx first then Ry, then Rz With the case that you first apply Rz, then Rx and then Ry

[Matt J]

As mentioned in the help doc, the code only does extrinsic rotations and the order of the rotations is specified by the second input argument:

>> help rot2taitbryan

  Calculates extrinsic Tait-Bryan angles from rotation matrix
 
   ang = rot2taitbryan(R,order)
 
  * `R`: Rotation matrix (3x3)
  * `order`: Order of angles to return; any of:
               {'XYZ','XZY','YXZ','YZX','ZYX','ZXY'} %case insensitive
            
  * `ANG`: Angles (degrees) in specified order
 
 NOTE: order of application of these rotations is right-to-left, both as
 indicated by `order` and `ANG'. So, as an example,
 
         >> R=Rz(30)*Ry(20)*Rx(60)
  
          R =
  
              0.8138    0.0065    0.5811
              0.4698    0.5811   -0.6645
             -0.3420    0.8138    0.4698
  
          >> rot2taitbryan(R,'zyx')
  
          ans =
  
             30.0000   20.0000   60.0000

[farzad]

Dear Matt : thank you for the above answer : but what if I don't know the order of rotation ?? this command asks for the order actually

second question : Does Absor.m give me the rotation matrix ? how can I get it ?

[Matt J]

but what if I don't know the order of rotation

The order is something you choose not something you determine. All 6 decompositions are equivalent - it just depends on your preference.

Does Absor.m give me the rotation matrix

Yes, it is in the regParams.R output.

Answer 2

[Jan]

You can define the motion by a translation of the center of the 3 points and a rotation of the local coordinate system.

PointsA = [x1, y1, z1; ...
           x2, y2, z2; ...
           x3, y3, z3];
PointsB = ...

Translation = mean(PointsB, 1) - mean(PointsA, 1);

% For the local coordinate systems find an orthonormal tripod:
v1 = PointsA(1, :) - PointsA(2, :);
n1 = v1 ./ norm(v1);
v2 = PointsA(2, :) - PointsA(3, :);
n2 = v2 ./ norm(v2);
c2 = cross(n1, n2);
n12 = c2 ./ norm(c2);
CoorA = [n1, n12, cross(n1, n12)];

% The same for B...
Rotation = CoorA * CoorB'

Anotehr approach would be the "Helical Axis": Any motion can be defined by an axis and some rotation around it and translation along it. See http://www.kwon3d.com/theory/jkinem/helical.html

[farzad]

thank you

my main question is the rotations. I was using something similar to your suggestion, but it was giving not satisfying results.

so you are saying that using absor and rcParams.R is not a good idea ?

[Matt J]

so you are saying that using absor and rcParams.R is not a good idea ?

Jan's approach will be sub-optimal if you have non-neglible measurement errors in A and B. Also, if you decide to use more than 3 points (for data redundancy), absor will better accommodate that case.

[Jan]

@farzad: Of course I did not say that "using adsor is not a good idea". Especially if you have more than 3 points on a rigid body, an optimization appraoch is the correct approach.

[farzad]

Thank you very much Jan