/** * $Id: MT_ExpMap.h,v 1.4 2003/05/01 19:52:20 sirdude Exp $ * ***** BEGIN GPL/BL DUAL LICENSE BLOCK ***** * * This program is free software; you can redistribute it and/or * modify it under the terms of the GNU General Public License * as published by the Free Software Foundation; either version 2 * of the License, or (at your option) any later version. The Blender * Foundation also sells licenses for use in proprietary software under * the Blender License. See http://www.blender.org/BL/ for information * about this. * * This program is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software Foundation, * Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. * * The Original Code is Copyright (C) 2001-2002 by NaN Holding BV. * All rights reserved. * * The Original Code is: all of this file. * * Contributor(s): none yet. * * ***** END GPL/BL DUAL LICENSE BLOCK ***** */ /** * $Id: MT_ExpMap.h,v 1.4 2003/05/01 19:52:20 sirdude Exp $ * Copyright (C) 2001 NaN Technologies B.V. * * @author Laurence */ #ifndef MT_ExpMap_H #define MT_ExpMap_H #include <MT_assert.h> #include "MT_Vector3.h" #include "MT_Quaternion.h" #include "MT_Matrix4x4.h" const MT_Scalar MT_EXPMAP_MINANGLE (1e-7); /** * MT_ExpMap an exponential map parameterization of rotations * in 3D. This implementation is derived from the paper * "F. Sebastian Grassia. Practical parameterization of * rotations using the exponential map. Journal of Graphics Tools, * 3(3):29-48, 1998" Please go to http://www.acm.org/jgt/papers/Grassia98/ * for a thorough description of the theory and sample code used * to derive this class. * * Basic overview of why this class is used. * In an IK system we need to paramterize the joint angles in some * way. Typically 2 parameterizations are used. * - Euler Angles * These suffer from singularities in the parameterization known * as gimbal lock. They also do not interpolate well. For every * set of euler angles there is exactly 1 corresponding 3d rotation. * - Quaternions. * Great for interpolating. Only unit quaternions are valid rotations * means that in a differential ik solver we often stray outside of * this manifold into invalid rotations. Means we have to do a lot * of nasty normalizations all the time. Does not suffer from * gimbal lock problems. More expensive to compute partial derivatives * as there are 4 of them. * * So exponential map is similar to a quaternion axis/angle * representation but we store the angle as the length of the * axis. So require only 3 parameters. Means that all exponential * maps are valid rotations. Suffers from gimbal lock. But it's * possible to detect when gimbal lock is near and reparameterize * away from it. Also nice for interpolating. * Exponential maps are share some of the useful properties of * euler and quaternion parameterizations. And are very useful * for differential IK solvers. */ 00086 class MT_ExpMap { public: /** * Default constructor * @warning there is no initialization in the * default constructor */ 00095 MT_ExpMap() {} MT_ExpMap(const MT_Vector3& v) : m_v(v) {} MT_ExpMap(const float v[3]) : m_v(v) {} MT_ExpMap(const double v[3]) : m_v(v) {} MT_ExpMap(MT_Scalar x, MT_Scalar y, MT_Scalar z) : m_v(x, y, z) {} /** * Construct an exponential map from a quaternion */ 00108 MT_ExpMap( const MT_Quaternion &q ) { setRotation(q); }; /** * Accessors * Decided not to inherit from MT_Vector3 but rather * this class contains an MT_Vector3. This is because * it is very dangerous to use MT_Vector3 functions * on this class and some of them have no direct meaning. */ MT_Vector3 & 00123 vector( ) { return m_v; }; const MT_Vector3 & vector( ) const { return m_v; }; /** * Set the exponential map from a quaternion */ void setRotation( const MT_Quaternion &q ); /** * Convert from an exponential map to a quaternion * representation */ MT_Quaternion getRotation( ) const; /** * Convert the exponential map to a 3x3 matrix */ MT_Matrix3x3 getMatrix( ) const; /** * Force a reparameterization check of the exponential * map. * @param theta returns the new axis-angle. * @return true iff a reParameterization took place. * Use this function whenever you adjust the vector * representing the exponential map. */ bool reParameterize( MT_Scalar &theta ); /** * Compute the partial derivatives of the exponential * map (dR/de - where R is a 4x4 matrix formed * from the map) and return them as a 4x4 matrix */ MT_Matrix4x4 partialDerivatives( const int i ) const ; private : MT_Vector3 m_v; // private methods // Compute partial derivatives dR (3x3 rotation matrix) / dVi (EM vector) // given the partial derivative dQ (Quaternion) / dVi (ith element of EM vector) void compute_dRdVi( const MT_Quaternion &q, const MT_Quaternion &dQdV, MT_Matrix4x4 & dRdVi ) const; // compute partial derivatives dQ/dVi void compute_dQdVi( int i, MT_Quaternion & dQdX ) const ; }; #endif

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