rt_tr2eul - homogeneous transform to Euler angles[view code]

Calling Sequence

[a b c] = rt_tr2eul(T)

Parameters

• T : 4-by-4 matrix. The homogeneous transform.
• a : scalar. The phi angle in radians, i.e. a rotation about the Z-axis of current reference frame.
• b : scalar. The theta angle in radians, i.e. a rotation about the Y-axis of current reference frame.
• c : scalar. The psi angle in radians, i.e. a rotation about the Z-axis of current reference frame.

Description

Return a vector of Euler angles, in radians, corresponding to the rotational part of the homogeneous transform T.

Examples

   // The following example shows how rt_tr2eul() can be used to get the
   // orientation of a coordinate frame, described by a given homogeneous
   // transform, expressed in terms of Euler angles.

   // generate a homogeneous transform
   e = [%pi/6, %pi/4, -2/5*%pi],    // Euler angles set
   T = rt_eul2tr(e);

   // corresponding Euler angles set
   e_comp = rt_tr2eul(T),

   // Remember that the solution to the problem of computing a set of
   // Euler angles corresponding to the rotational part of a homogeneous
   // transform is not univoque.
   // rt_tr2eul() computes the solution reducing the range to whom theta
   // belongs to (0, %pi) radians.
   // Therefore, if you generate a coordinate frame from a set of Euler
   // angles with theta belonging to (-%pi, 0) and then you try to get
   // its orientation by using rt_tr2eul() function, be aware that you
   // won't obtain the starting set of angles.

   e1 = [0.6, -1.2, -1.75],         // Euler angles set
   T = rt_eul2tr(e1);
   e2 = rt_tr2eul(T),               // e2 is different from e1, but they
                                    // are equivalent in their effects

   // Lastly, note that rt_tr2eul() provides a single solution when the
   // computed set of Euler angles has theta = 0 or theta = %pi. But you
   // should remember that in these cases infinitely many solutions are
   // corresponding to the rotational part of the homogeneous transform.

   e1 = [0.6, 0, -1.75];            // Euler angles set with theta = 0
   T = rt_eul2tr(e1),
   // all solutions for which phi + psi = 0.6 + (-1.75) are equivalent in
   // their effects
   e1_a = rt_tr2eul(T);             Ta = rt_eul2tr(e1_a), 
   e1_b = [0.25, 0, -1.15-0.25];    Tb = rt_eul2tr(e1_b),
   e1_c = [2.6, 0, -1.15-2.6];      Tc = rt_eul2tr(e1_c),
   e1_d = [-1.35, 0, -1.15+1.35];   Td = rt_eul2tr(e1_d),
   e1_e = [-0.9, 0, -1.15+0.9];     Te = rt_eul2tr(e1_e),

   e2 = [-2.2, %pi, 0.95];          // Euler angles set with theta = %pi
   T = rt_eul2tr(e2),
   // all solutions for which psi - phi = 0.95 - (-2.2) are equivalent in
   // their effects
   e2_a = rt_tr2eul(T);             Ta = rt_eul2tr(e2_a), 
   e2_b = [0, %pi, 3.15];           Tb = rt_eul2tr(e2_b),
   e2_c = [-2.1, %pi, 3.15-2.1];    Tc = rt_eul2tr(e2_c),
   e2_d = [1.25, %pi, 3.15+1.25];   Td = rt_eul2tr(e2_d),
   e2_e = [0.5, %pi, 3.15+0.5];     Te = rt_eul2tr(e2_e),

  

See Also

rt_eul2tr,  rt_tr2rpy,  

Authors

original Matlab version by

Peter I. Corke CSIRO Manufacturing Science and Technology

Scilab implementation by

Matteo Morelli Interdepartmental Research Center "E. Piaggio", University of Pisa

Bibliography

Corke, P.I. "A Robotics Toolbox for MATLAB", IEEE Robotics and Automation Magazine, Volume 3(1), March 1996, pp. 24-32

R. P. Paul, Robot Manipulators: Mathematics, Programming and Control. Cambridge, Massachusetts: MIT Press, 1981.

L. Sciavicco, B. Siciliano, Modelling and Control of Robot Manipulators. 2nd Edition, Springer-Verlag Advanced Textbooks in Control and Signal Processing Series, London, UK, 2000.