V_ROTEU2QR converts a sequence of Euler angles to a rotation matrix
Inputs:
M a string of n characters from the set determining the order of rotation axes
as listed below:
'x','y','z' rotate around the given axis by the corresponding angle
given in e()
'1','2','3' 90 degree rotation around x,y or z axis; doesn't use a value from e()
'4','5','6' 180 degree rotation around x,y or z axis; doesn't use a value from e()
'7','8','9' 270 degree rotation around x,y or z axis; doesn't use a value from e()
'r','d' all angles are given in radians or degrees [default='r']
'R','D' all angles are given in radians or degrees and are negated
'o','O','a','A' selects whether to rotate the object or the coordinate axes and
whether the rotation axes remain fixed in space for consecutive
rotations (extrinsic) or else move with each rotation (intrinsic).
'o' = object-extrinsic [default]
'O' = object-intrinsic
'a' = axes-extrinsic
'A' = axes-intrinsic
E(K,...) K Euler angles in radians (or degrees if 'd' or 'D' specified) per quaternion where K is the number of 'xyz' characters in M.
A positive rotation is clockwise if looking along the +ve axis away from the origin or anti-clockwise if 'R' or 'D' is given.
The x, y, z axes form a right-handed triple.
Outputs:
R(3,3,...) Output rotation matrix
Plots a diagram if no output specified
The string M specifies the seqeunce of axes about which the rotations are performed. There are 12
possible 3-character sequences that avoid consecutive repetitions. These are 'Euler angles' if
there is a repeated axis or 'Tait-Bryan angles' if not. Common choices are:
(1) 'zxz' the most common Euler angle set
(2) 'xyz' corresponds to 'roll, pitch, yaw' for an aeroplane heading in the x direction with y to
the right and z down. The intrinsic equivalent is 'Ozyx' corresponding to 'yaw, pitch, roll'.
(3) 'z1z1z' involves 5 rotations, in which all the non-fixed rotations are around the z axis.
The Euler angles are not, in general, unique. In particular the following equivalences exist:
(1) v_roteu2ro('zxz',[a b c]) = v_roteu2ro('zxz',[a+pi -b c+pi])
(2) v_roteu2ro('xyz',[a b c]) = v_roteu2ro('zxz',[a+pi pi-b c+pi])
(3) v_roteu2ro('456',[]) = eye(3) % also true for any ordering of '456'
(4) v_roteu2ro('x',a) = v_roteu2ro('5x5',-a) = v_roteu2ro('5x6',pi-a) % also true if 5,6 are interchanged0001 function r=v_roteu2ro(varargin) 0002 %V_ROTEU2QR converts a sequence of Euler angles to a rotation matrix 0003 % Inputs: 0004 % 0005 % M a string of n characters from the set determining the order of rotation axes 0006 % as listed below: 0007 % 'x','y','z' rotate around the given axis by the corresponding angle 0008 % given in e() 0009 % '1','2','3' 90 degree rotation around x,y or z axis; doesn't use a value from e() 0010 % '4','5','6' 180 degree rotation around x,y or z axis; doesn't use a value from e() 0011 % '7','8','9' 270 degree rotation around x,y or z axis; doesn't use a value from e() 0012 % 'r','d' all angles are given in radians or degrees [default='r'] 0013 % 'R','D' all angles are given in radians or degrees and are negated 0014 % 'o','O','a','A' selects whether to rotate the object or the coordinate axes and 0015 % whether the rotation axes remain fixed in space for consecutive 0016 % rotations (extrinsic) or else move with each rotation (intrinsic). 0017 % 'o' = object-extrinsic [default] 0018 % 'O' = object-intrinsic 0019 % 'a' = axes-extrinsic 0020 % 'A' = axes-intrinsic 0021 % 0022 % E(K,...) K Euler angles in radians (or degrees if 'd' or 'D' specified) per quaternion where K is the number of 'xyz' characters in M. 0023 % A positive rotation is clockwise if looking along the +ve axis away from the origin or anti-clockwise if 'R' or 'D' is given. 0024 % The x, y, z axes form a right-handed triple. 0025 % 0026 % Outputs: 0027 % 0028 % R(3,3,...) Output rotation matrix 0029 % Plots a diagram if no output specified 0030 % 0031 % The string M specifies the seqeunce of axes about which the rotations are performed. There are 12 0032 % possible 3-character sequences that avoid consecutive repetitions. These are 'Euler angles' if 0033 % there is a repeated axis or 'Tait-Bryan angles' if not. Common choices are: 0034 % (1) 'zxz' the most common Euler angle set 0035 % (2) 'xyz' corresponds to 'roll, pitch, yaw' for an aeroplane heading in the x direction with y to 0036 % the right and z down. The intrinsic equivalent is 'Ozyx' corresponding to 'yaw, pitch, roll'. 0037 % (3) 'z1z1z' involves 5 rotations, in which all the non-fixed rotations are around the z axis. 0038 % 0039 % The Euler angles are not, in general, unique. In particular the following equivalences exist: 0040 % (1) v_roteu2ro('zxz',[a b c]) = v_roteu2ro('zxz',[a+pi -b c+pi]) 0041 % (2) v_roteu2ro('xyz',[a b c]) = v_roteu2ro('zxz',[a+pi pi-b c+pi]) 0042 % (3) v_roteu2ro('456',[]) = eye(3) % also true for any ordering of '456' 0043 % (4) v_roteu2ro('x',a) = v_roteu2ro('5x5',-a) = v_roteu2ro('5x6',pi-a) % also true if 5,6 are interchanged 0044 0045 % Copyright (C) Mike Brookes 2007-2020 0046 % Version: $Id: v_roteu2ro.m 11260 2020-07-18 20:07:58Z dmb $ 0047 % 0048 % VOICEBOX is a MATLAB toolbox for speech processing. 0049 % Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html 0050 % 0051 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0052 % This program is free software; you can redistribute it and/or modify 0053 % it under the terms of the GNU General Public License as published by 0054 % the Free Software Foundation; either version 2 of the License, or 0055 % (at your option) any later version. 0056 % 0057 % This program is distributed in the hope that it will be useful, 0058 % but WITHOUT ANY WARRANTY; without even the implied warranty of 0059 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 0060 % GNU General Public License for more details. 0061 % 0062 % You can obtain a copy of the GNU General Public License from 0063 % http://www.gnu.org/copyleft/gpl.html or by writing to 0064 % Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA. 0065 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0066 if nargout 0067 r=v_rotqr2ro(v_roteu2qr(varargin{:})); 0068 else 0069 v_rotqr2ro(v_roteu2qr(varargin{:})); % draw a cube 0070 end