Rotation.h

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//-----------------------------------------------------------------------------
// File:     Rotation.h
// Class:    Rotation and InverseRotation 
// Parent:   Mat33
// Purpose:  3x3 rotation class relating two right-handed orthogonal bases
//-----------------------------------------------------------------------------
#ifndef SIMTK_ROTATION_H 
#define SIMTK_ROTATION_H 

/* -------------------------------------------------------------------------- *
 *                      SimTK Core: SimTK Simmatrix(tm)                       *
 * -------------------------------------------------------------------------- *
 * This is part of the SimTK Core biosimulation toolkit originating from      *
 * Simbios, the NIH National Center for Physics-Based Simulation of           *
 * Biological Structures at Stanford, funded under the NIH Roadmap for        *
 * Medical Research, grant U54 GM072970. See https://simtk.org.               *
 *                                                                            *
 * Portions copyright (c) 2005-7 Stanford University and the Authors.         *
 * Authors: Paul Mitiguy and Michael Sherman                                  *
 * Contributors:                                                              *
 *                                                                            *
 * Permission is hereby granted, free of charge, to any person obtaining a    *
 * copy of this software and associated documentation files (the "Software"), *
 * to deal in the Software without restriction, including without limitation  *
 * the rights to use, copy, modify, merge, publish, distribute, sublicense,   *
 * and/or sell copies of the Software, and to permit persons to whom the      *
 * Software is furnished to do so, subject to the following conditions:       *
 *                                                                            *
 * The above copyright notice and this permission notice shall be included in *
 * all copies or substantial portions of the Software.                        *
 *                                                                            *
 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR *
 * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,   *
 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL    *
 * THE AUTHORS, CONTRIBUTORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM,    *
 * DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR      *
 * OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE  *
 * USE OR OTHER DEALINGS IN THE SOFTWARE.                                     *
 * -------------------------------------------------------------------------- */

//-----------------------------------------------------------------------------

#include "SimTKcommon/SmallMatrix.h"
#include "SimTKcommon/internal/UnitVec.h"
#include "SimTKcommon/internal/Quaternion.h"
#include "SimTKcommon/internal/CoordinateAxis.h"

//-----------------------------------------------------------------------------
#include <iosfwd>  // Forward declaration of iostream
//-----------------------------------------------------------------------------

//-----------------------------------------------------------------------------
namespace SimTK {


enum BodyOrSpaceType { BodyRotationSequence=0, SpaceRotationSequence=1 };

//-----------------------------------------------------------------------------
// Forward declarations
class InverseRotation;


//-----------------------------------------------------------------------------
//------------------------------------------------------------------------------
class Rotation : public Mat33 {
public:
    // Default constructor and constructor-like methods
    Rotation() : Mat33(1) {}    
    Rotation&  setRotationToIdentityMatrix()  { Mat33::operator=(1);  return *this; }
    Rotation&  setRotationToNaN()             { Mat33::setToNaN();    return *this; } 

    // Default copy constructor and assignment operator
    Rotation( const Rotation& R ) : Mat33(R)  {}
    Rotation&  operator=( const Rotation& R )  { Mat33::operator=( R.asMat33() );  return *this; }


    Rotation( Real angle, const CoordinateAxis& axis )             { setRotationFromAngleAboutAxis( angle, axis ); }
    Rotation( Real angle, const CoordinateAxis::XCoordinateAxis )  { setRotationFromAngleAboutX( std::cos(angle), std::sin(angle) ); }
    Rotation( Real angle, const CoordinateAxis::YCoordinateAxis )  { setRotationFromAngleAboutY( std::cos(angle), std::sin(angle) ); }
    Rotation( Real angle, const CoordinateAxis::ZCoordinateAxis )  { setRotationFromAngleAboutZ( std::cos(angle), std::sin(angle) ); }


    Rotation&  setRotationFromAngleAboutAxis( Real angle, const CoordinateAxis& axis )  { return axis.isXAxis() ? setRotationFromAngleAboutX(angle) : (axis.isYAxis() ? setRotationFromAngleAboutY(angle) : setRotationFromAngleAboutZ(angle) ); }
    Rotation&  setRotationFromAngleAboutX( Real angle )  { return setRotationFromAngleAboutX( std::cos(angle), std::sin(angle) ); }
    Rotation&  setRotationFromAngleAboutY( Real angle )  { return setRotationFromAngleAboutY( std::cos(angle), std::sin(angle) ); }
    Rotation&  setRotationFromAngleAboutZ( Real angle )  { return setRotationFromAngleAboutZ( std::cos(angle), std::sin(angle) ); }
    Rotation&  setRotationFromAngleAboutX( Real cosAngle, Real sinAngle )  { Mat33& R = *this;  R[0][0] = 1;   R[0][1] = R[0][2] = R[1][0] = R[2][0] = 0;   R[1][1] = R[2][2] = cosAngle;  R[1][2] = -(R[2][1] = sinAngle);  return *this; }
    Rotation&  setRotationFromAngleAboutY( Real cosAngle, Real sinAngle )  { Mat33& R = *this;  R[1][1] = 1;   R[0][1] = R[1][0] = R[1][2] = R[2][1] = 0;   R[0][0] = R[2][2] = cosAngle;  R[2][0] = -(R[0][2] = sinAngle);  return *this; }
    Rotation&  setRotationFromAngleAboutZ( Real cosAngle, Real sinAngle )  { Mat33& R = *this;  R[2][2] = 1;   R[0][2] = R[1][2] = R[2][0] = R[2][1] = 0;   R[0][0] = R[1][1] = cosAngle;  R[0][1] = -(R[1][0] = sinAngle);  return *this; }


    Rotation( Real angle, const UnitVec3& unitVector ) { setRotationFromAngleAboutUnitVector(angle,unitVector); }
    Rotation( Real angle, const Vec3& nonUnitVector )  { setRotationFromAngleAboutNonUnitVector(angle,nonUnitVector); }


    Rotation&  setRotationFromAngleAboutNonUnitVector( Real angle, const Vec3& nonUnitVector )  { return setRotationFromAngleAboutUnitVector( angle, UnitVec3(nonUnitVector) ); }
    SimTK_SimTKCOMMON_EXPORT Rotation&  setRotationFromAngleAboutUnitVector( Real angle, const UnitVec3& unitVector );

    Rotation( BodyOrSpaceType bodyOrSpace, Real angle1, const CoordinateAxis& axis1, Real angle2, const CoordinateAxis& axis2 )                                            { setRotationFromTwoAnglesTwoAxes(    bodyOrSpace,angle1,axis1,angle2,axis2); }
    Rotation( BodyOrSpaceType bodyOrSpace, Real angle1, const CoordinateAxis& axis1, Real angle2, const CoordinateAxis& axis2, Real angle3, const CoordinateAxis& axis3 )  { setRotationFromThreeAnglesThreeAxes(bodyOrSpace,angle1,axis1,angle2,axis2,angle3,axis3); }
    SimTK_SimTKCOMMON_EXPORT Rotation&  setRotationFromTwoAnglesTwoAxes(     BodyOrSpaceType bodyOrSpace, Real angle1, const CoordinateAxis& axis1, Real angle2, const CoordinateAxis& axis2 ); 
    SimTK_SimTKCOMMON_EXPORT Rotation&  setRotationFromThreeAnglesThreeAxes( BodyOrSpaceType bodyOrSpace, Real angle1, const CoordinateAxis& axis1, Real angle2, const CoordinateAxis& axis2, Real angle3, const CoordinateAxis& axis3 );

    void setRotationToBodyFixedXY( const Vec2& v)   { setRotationFromTwoAnglesTwoAxes(     BodyRotationSequence, v[0], XAxis, v[1], YAxis ); }
    void setRotationToBodyFixedXYZ( const Vec3& v)  { setRotationFromThreeAnglesThreeAxes( BodyRotationSequence, v[0], XAxis, v[1], YAxis, v[2], ZAxis ); }

    explicit Rotation( const Quaternion& q )  { setRotationFromQuaternion(q); }
    SimTK_SimTKCOMMON_EXPORT Rotation&  setRotationFromQuaternion( const Quaternion& q );

    Rotation( const Mat33& m, bool ) : Mat33(m) {}

    explicit Rotation( const Mat33& m )  { setRotationFromApproximateMat33(m); }
    SimTK_SimTKCOMMON_EXPORT Rotation&  setRotationFromApproximateMat33( const Mat33& m );


    Rotation( const UnitVec3& uvec, const CoordinateAxis axis )  { setRotationFromOneAxis(uvec,axis); }
    SimTK_SimTKCOMMON_EXPORT Rotation&  setRotationFromOneAxis( const UnitVec3& uvec, const CoordinateAxis axis );


    Rotation( const UnitVec3& uveci, const CoordinateAxis& axisi, const Vec3& vecjApprox, const CoordinateAxis& axisjApprox )  { setRotationFromTwoAxes(uveci,axisi,vecjApprox,axisjApprox); }
    SimTK_SimTKCOMMON_EXPORT Rotation&  setRotationFromTwoAxes( const UnitVec3& uveci, const CoordinateAxis& axisi, const Vec3& vecjApprox, const CoordinateAxis& axisjApprox );

    // Converts rotation matrix to one or two or three orientation angles.
    // Note:  The result is most meaningful if the Rotation matrix is one that can be produced by such a sequence.
    // Use1:  someRotation.convertOneAxisRotationToOneAngle( XAxis );
    // Use2:  someRotation.convertTwoAxesRotationToTwoAngles(     SpaceRotationSequence, YAxis, ZAxis );
    // Use3:  someRotation.convertThreeAxesRotationToThreeAngles( SpaceRotationSequence, ZAxis, YAxis, XAxis );
    // Use4:  someRotation.convertRotationToAngleAxis();   Return: [angleInRadians, unitVectorX, unitVectorY, unitVectorZ].

    SimTK_SimTKCOMMON_EXPORT Real  convertOneAxisRotationToOneAngle( const CoordinateAxis& axis1 ) const;
    SimTK_SimTKCOMMON_EXPORT Vec2  convertTwoAxesRotationToTwoAngles(     BodyOrSpaceType bodyOrSpace, const CoordinateAxis& axis1, const CoordinateAxis& axis2 ) const;
    SimTK_SimTKCOMMON_EXPORT Vec3  convertThreeAxesRotationToThreeAngles( BodyOrSpaceType bodyOrSpace, const CoordinateAxis& axis1, const CoordinateAxis& axis2, const CoordinateAxis& axis3 ) const;
    SimTK_SimTKCOMMON_EXPORT Quaternion  convertRotationToQuaternion() const;
    Vec4  convertRotationToAngleAxis() const  { return convertRotationToQuaternion().convertQuaternionToAngleAxis(); }

    Vec2  convertRotationToBodyFixedXY() const   { return convertTwoAxesRotationToTwoAngles( BodyRotationSequence, XAxis, YAxis ); }
    Vec3  convertRotationToBodyFixedXYZ() const  { return convertThreeAxesRotationToThreeAngles( BodyRotationSequence, XAxis, YAxis, ZAxis ); }


    SimTK_SimTKCOMMON_EXPORT bool  isSameRotationToWithinAngle( const Rotation& R, Real okPointingAngleErrorRads ) const;
    bool  isSameRotationToWithinAngleOfMachinePrecision( const Rotation& R) const      { return isSameRotationToWithinAngle( R, SignificantReal ); }
    Real  getMaxAbsDifferenceInRotationElements( const Rotation& R ) const             { const Mat33& A = asMat33();  const Mat33& B = R.asMat33();  Real maxDiff = 0.0;  for( int i=0;  i<=2; i++ ) for( int j=0; j<=2; j++ ) { Real absDiff = std::fabs(A[i][j] - B[i][j]);  if( absDiff > maxDiff ) maxDiff = absDiff; }  return maxDiff; } 
    bool  areAllRotationElementsSameToEpsilon( const Rotation& R, Real epsilon ) const { return getMaxAbsDifferenceInRotationElements(R) <= epsilon ; }
    bool  areAllRotationElementsSameToMachinePrecision( const Rotation& R ) const      { return areAllRotationElementsSameToEpsilon( R, SignificantReal ); } 

    inline Rotation( const InverseRotation& );
    inline Rotation& operator=( const InverseRotation& );

    const InverseRotation&  invert() const  { return *reinterpret_cast<const InverseRotation*>(this); }
    InverseRotation&        updInvert()     { return *reinterpret_cast<InverseRotation*>(this); }

    // Transpose, and transpose operators (override Mat33 versions of transpose).
    const InverseRotation&  transpose() const  { return invert(); }
    const InverseRotation&  operator~() const  { return invert(); }
    InverseRotation&        updTranspose()     { return updInvert(); }
    InverseRotation&        operator~()        { return updInvert(); }

    // Multiply-equals and divide-equals operators 
    inline Rotation&  operator*=( const Rotation& R );
    inline Rotation&  operator/=( const Rotation& R );
    inline Rotation&  operator*=( const InverseRotation& );
    inline Rotation&  operator/=( const InverseRotation& );


    const Mat33&  asMat33() const  { return *static_cast<const Mat33*>(this); }
    Mat33         toMat33() const  { return asMat33(); }

    // Note: This does not have unit stride.
    typedef  UnitVec<Mat33::RowSpacing>  ColType;
    typedef  UnitRow<Mat33::ColSpacing>  RowType;
    const RowType&  row( int i ) const         { return reinterpret_cast<const RowType&>(asMat33()[i]); }
    const ColType&  col( int j ) const         { return reinterpret_cast<const ColType&>(asMat33()(j)); }
    const ColType&  x() const                  { return col(0); }
    const ColType&  y() const                  { return col(1); }
    const ColType&  z() const                  { return col(2); }
    const RowType&  operator[]( int i ) const  { return row(i); }
    const ColType&  operator()( int j ) const  { return col(j); }


    Rotation&  setRotationFromMat33TrustMe( const Mat33& m )  { Mat33& R = *this;  R[0][0]=m[0][0];  R[0][1]=m[0][1];  R[0][2]=m[0][2];  R[1][0]=m[1][0];  R[1][1]=m[1][1];  R[1][2]=m[1][2];  R[2][0]=m[2][0];  R[2][1]=m[2][1];  R[2][2]=m[2][2];  return *this; }   
    Rotation&  setRotationColFromUnitVecTrustMe( int coli, const UnitVec3& uveci )  { Mat33& R = *this;   R[0][coli]=uveci[0];  R[1][coli]=uveci[1];  R[2][coli]=uveci[2];  return *this; }   
    Rotation&  setRotationFromUnitVecsTrustMe( const UnitVec3& colA, const UnitVec3& colB, const UnitVec3& colC )  { setRotationColFromUnitVecTrustMe(0,colA);  setRotationColFromUnitVecTrustMe(1,colB);  return setRotationColFromUnitVecTrustMe(2,colC); }   

//----------------------------------------------------------------------------------------------------
// The following code is obsolete - it is here temporarily for backward compatibility (Mitiguy 9/5/2007)
//----------------------------------------------------------------------------------------------------
private:
    // These static methods are like constructors with friendlier names.
    static Rotation zero() { return Rotation(); }
    static Rotation NaN()  { Rotation r;  r.setRotationToNaN();  return r; }

    Rotation&  setToZero()            { return setRotationToIdentityMatrix(); }
    Rotation&  setToIdentityMatrix()  { return setRotationToIdentityMatrix(); }
    Rotation&  setToNaN()             { return setRotationToNaN(); }
    static Rotation  trustMe( const Mat33& m )  { return Rotation(m,true); }

    // One-angle rotations.
    static Rotation aboutX( const Real& angleInRad ) { return Rotation( angleInRad, XAxis ); }
    static Rotation aboutY( const Real& angleInRad ) { return Rotation( angleInRad, YAxis ); }
    static Rotation aboutZ( const Real& angleInRad ) { return Rotation( angleInRad, ZAxis ); }
    static Rotation aboutAxis( const Real& angleInRad, const UnitVec3& axis ) { return Rotation(angleInRad,axis); }
    static Rotation aboutAxis( const Real& angleInRad, const Vec3& axis )     { return Rotation(angleInRad,axis); }
    void  setToRotationAboutZ( const Real& q ) { setRotationFromAngleAboutZ( q ); }

    // Two-angle space-fixed rotations.
    static Rotation aboutXThenOldY(const Real& xInRad, const Real& yInRad) { return Rotation( SpaceRotationSequence, xInRad, XAxis, yInRad, YAxis ); }
    static Rotation aboutYThenOldX(const Real& yInRad, const Real& xInRad) { return Rotation( SpaceRotationSequence, yInRad, YAxis, xInRad, XAxis ); }
    static Rotation aboutXThenOldZ(const Real& xInRad, const Real& zInRad) { return Rotation( SpaceRotationSequence, xInRad, XAxis, zInRad, ZAxis ); }
    static Rotation aboutZThenOldX(const Real& zInRad, const Real& xInRad) { return Rotation( SpaceRotationSequence, zInRad, ZAxis, xInRad, XAxis ); }
    static Rotation aboutYThenOldZ(const Real& yInRad, const Real& zInRad) { return Rotation( SpaceRotationSequence, yInRad, YAxis, zInRad, ZAxis ); }
    static Rotation aboutZThenOldY(const Real& zInRad, const Real& yInRad) { return Rotation( SpaceRotationSequence, zInRad, ZAxis, yInRad, YAxis ); }

    // Two-angle body fixed rotations (reversed space-fixed ones).
    static Rotation aboutXThenNewY(const Real& xInRad, const Real& yInRad) { return Rotation( BodyRotationSequence, xInRad, XAxis, yInRad, YAxis ); }
    static Rotation aboutYThenNewX(const Real& yInRad, const Real& xInRad) { return aboutXThenOldY(xInRad, yInRad); }
    static Rotation aboutXThenNewZ(const Real& xInRad, const Real& zInRad) { return aboutZThenOldX(zInRad, xInRad); }
    static Rotation aboutZThenNewX(const Real& zInRad, const Real& xInRad) { return aboutXThenOldZ(xInRad, zInRad); }
    static Rotation aboutYThenNewZ(const Real& yInRad, const Real& zInRad) { return aboutZThenOldY(zInRad, yInRad); }
    static Rotation aboutZThenNewY(const Real& zInRad, const Real& yInRad) { return aboutYThenOldZ(yInRad, zInRad); }

    explicit Rotation( const UnitVec3& uvecZ )  { setRotationFromOneAxis(uvecZ,ZAxis); }

    //  We will take x seriously after normalizing, but use the y only to create z = normalize(x X y), 
    //  then y = z X x. Bad things happen if x and y are aligned but we may not catch it.
    Rotation( const Vec3& x, const Vec3& yish )  { setRotationFromTwoAxes( UnitVec3(x), XAxis, yish, YAxis ); }

    void setToQuaternion( const Quaternion& q )  { setRotationFromQuaternion(q); }

    //  Similarly for BodyFixed123.
    void setToBodyFixed321( const Vec3& v)  { setRotationFromThreeAnglesThreeAxes( BodyRotationSequence, v[0], ZAxis, v[1], YAxis, v[2], XAxis ); }
    void setToBodyFixed123( const Vec3& v)  { setRotationToBodyFixedXYZ(v); }

    Vec4  convertToAngleAxis() const  { return convertRotationToAngleAxis(); }

    Quaternion  convertToQuaternion() const  { return convertRotationToQuaternion(); }

    void setToSpaceFixed12( const Vec2& q ) { setRotationFromTwoAnglesTwoAxes( SpaceRotationSequence, q[0], XAxis, q[1], YAxis ); }

    Vec3  convertToBodyFixed123() const  { return convertRotationToBodyFixedXYZ(); }
    Vec2  convertToBodyFixed12() const   { return convertRotationToBodyFixedXY(); }
    Vec2  convertToSpaceFixed12() const  { return convertTwoAxesRotationToTwoAngles( SpaceRotationSequence, XAxis, YAxis ); }

//--------------------------------- PAUL CONTINUE FROM HERE ----------------------------------
public:
//--------------------------------------------------------------------------------------------
    static Vec3 convertAngVelToBodyFixed321Dot(const Vec3& q, const Vec3& w_PB_B) {
        const Real s1 = std::sin(q[1]), c1 = std::cos(q[1]);
        const Real s2 = std::sin(q[2]), c2 = std::cos(q[2]);
        const Real ooc1 = Real(1)/c1;
        const Real s2oc1 = s2*ooc1, c2oc1 = c2*ooc1;

        const Mat33 E( 0,    s2oc1  ,  c2oc1  ,
                       0,      c2   ,   -s2   ,
                       1,  s1*s2oc1 , s1*c2oc1 );
        return E * w_PB_B;
    }

    static Vec3 convertBodyFixed321DotToAngVel(const Vec3& q, const Vec3& qd) {
        const Real s1 = std::sin(q[1]), c1 = std::cos(q[1]);
        const Real s2 = std::sin(q[2]), c2 = std::cos(q[2]);
        const Real ooc1  = 1/c1;
        const Real s2oc1 = s2*ooc1, c2oc1 = c2*ooc1;

        const Mat33 Einv(  -s1  ,  0  ,  1 ,
                          c1*s2 ,  c2 ,  0 ,
                          c1*c2 , -s2 ,  0 );
        return Einv*qd;
    }

    // TODO: sherm: is this right? Warning: everything is measured in the
    // *PARENT* frame, but has to be expressed in the *BODY* frame.
    static Vec3 convertAngVelDotToBodyFixed321DotDot
        (const Vec3& q, const Vec3& w_PB_B, const Vec3& wdot)
    {
        const Real s1 = std::sin(q[1]), c1 = std::cos(q[1]);
        const Real s2 = std::sin(q[2]), c2 = std::cos(q[2]);
        const Real ooc1  = 1/c1;
        const Real s2oc1 = s2*ooc1, c2oc1 = c2*ooc1;

        const Mat33 E( 0 ,   s2oc1  ,  c2oc1  ,
                       0 ,     c2   ,   -s2   ,
                       1 , s1*s2oc1 , s1*c2oc1 );
        const Vec3 qdot = E * w_PB_B;

        const Real t =  qdot[1]*qdot[2]*s1*ooc1;
        const Real a =  t*c2oc1; // d/dt s2oc1
        const Real b = -t*s2oc1; // d/dt c2oc1

        const Mat33 Edot( 0 ,       a           ,         b         ,
                          0 ,   -qdot[2]*s2     ,    -qdot[2]*c2    ,
                          0 , s1*a + qdot[1]*s2 , s1*b + qdot[1]*c2 );

        return E*wdot + Edot*w_PB_B;
    }
    
    static Mat33 calcQBlockForBodyXYZInBodyFrame(const Vec3& q) {
        const Real s1 = std::sin(q[1]), c1 = std::cos(q[1]);
        const Real s2 = std::sin(q[2]), c2 = std::cos(q[2]);
        const Real ooc1  = 1/c1;
        const Real s2oc1 = s2*ooc1, c2oc1 = c2*ooc1;

        return Mat33(    c2oc1  , -s2oc1  , 0,
                           s2   ,    c2   , 0,
                      -s1*c2oc1 , s1*s2oc1, 1 );
    }

    static Mat33 calcQInvBlockForBodyXYZInBodyFrame(const Vec3& q) {
        const Real s1 = std::sin(q[1]), c1 = std::cos(q[1]);
        const Real s2 = std::sin(q[2]), c2 = std::cos(q[2]);

        return Mat33( c1*c2 ,  s2 , 0 ,
                     -c1*s2 ,  c2 , 0 ,
                       s1   ,  0  , 1 );
    }

    static Vec3 convertAngVelToBodyFixed123Dot(const Vec3& q, const Vec3& w_PB_B) {
        return calcQBlockForBodyXYZInBodyFrame(q)*w_PB_B;
    }

    static Vec3 convertBodyFixed123DotToAngVel(const Vec3& q, const Vec3& qd) {
        return calcQInvBlockForBodyXYZInBodyFrame(q)*qd;
    }


    // TODO: sherm: is this right? Warning: everything is measured in the
    // *PARENT* frame, but has to be expressed in the *BODY* frame.
    static Vec3 convertAngVelDotToBodyFixed123DotDot
        (const Vec3& q, const Vec3& w_PB_B, const Vec3& wdot)
    {
        const Real s1 = std::sin(q[1]), c1 = std::cos(q[1]);
        const Real s2 = std::sin(q[2]), c2 = std::cos(q[2]);
        const Real ooc1  = 1/c1;
        const Real s2oc1 = s2*ooc1, c2oc1 = c2*ooc1;

        const Mat33 Q(    c2oc1  , -s2oc1  , 0,
                            s2   ,    c2   , 0,
                       -s1*c2oc1 , s1*s2oc1, 1 );
        const Vec3 qdot = Q * w_PB_B;

        const Real t =  qdot[1]*qdot[2]*s1*ooc1;
        const Real a =  t*c2oc1; // d/dt s2oc1
        const Real b = -t*s2oc1; // d/dt c2oc1

        const Mat33 QDot(       b           ,        -a         , 0,
                             qdot[2]*c2     ,    -qdot[2]*s2    , 0,
                         -s1*b - qdot[1]*c2 , s1*a + qdot[1]*s2 , 0 );

        return Q*wdot + QDot*w_PB_B;
    }

    static Mat43 calcUnnormalizedQBlockForQuaternion(const Vec4& q) {
        const Vec4 e = q/2;
        return Mat43(-e[1],-e[2],-e[3],
                      e[0], e[3],-e[2],
                     -e[3], e[0], e[1],
                      e[2],-e[1], e[0]);
    }

    static Mat34 calcUnnormalizedQInvBlockForQuaternion(const Vec4& q) {
        const Vec4 e = 2*q;
        return Mat34(-e[1], e[0],-e[3], e[2],
                     -e[2], e[3], e[0],-e[1],
                     -e[3],-e[2], e[1], e[0]);
    }


    static Vec4 convertAngVelToQuaternionDot(const Vec4& q, const Vec3& w_PB_P) {
        return calcUnnormalizedQBlockForQuaternion(q)*w_PB_P;
    }

    static Vec3 convertQuaternionDotToAngVel(const Vec4& q, const Vec4& qd) {
        return calcUnnormalizedQInvBlockForQuaternion(q)*qd;
    }

    static Vec4 convertAngVelDotToQuaternionDotDot
        (const Vec4& q, const Vec3& w_PB_P, const Vec3& wdot)
    {
        const Mat43 Q = calcUnnormalizedQBlockForQuaternion(q);
        const Vec4  edot = (Q*w_PB_P)/2; // i.e., edot=qdot/2
        const Mat43 QDot(-edot[1],-edot[2],-edot[3],
                          edot[0], edot[3],-edot[2],
                         -edot[3], edot[0], edot[1],
                          edot[2],-edot[1], edot[0]);

        return  Q*wdot + QDot*w_PB_P;
    }


private:
    // This is only for the most trustworthy of callers, that is, methods of the Rotation class. 
    // There are a lot of ways for this NOT to be a legitimate rotation matrix -- be careful!!
    // Note that these are supplied in rows.
    Rotation( const Real& xx, const Real& xy, const Real& xz,
              const Real& yx, const Real& yy, const Real& yz,
              const Real& zx, const Real& zy, const Real& zz )
            : Mat33( xx,xy,xz, yx,yy,yz, zx,zy,zz ) {}

    // These next methods are highly-efficient power-user methods.  Read the documentation to understand them.
    SimTK_SimTKCOMMON_EXPORT Rotation&  setTwoAngleTwoAxesBodyFixedForwardCyclicalRotation(     Real cosAngle1, Real sinAngle1, const CoordinateAxis& axis1, Real cosAngle2, Real sinAngle2, const CoordinateAxis& axis2 );
    SimTK_SimTKCOMMON_EXPORT Rotation&  setThreeAngleTwoAxesBodyFixedForwardCyclicalRotation(   Real cosAngle1, Real sinAngle1, const CoordinateAxis& axis1, Real cosAngle2, Real sinAngle2, const CoordinateAxis& axis2, Real cosAngle3, Real sinAngle3 );
    SimTK_SimTKCOMMON_EXPORT Rotation&  setThreeAngleThreeAxesBodyFixedForwardCyclicalRotation( Real cosAngle1, Real sinAngle1, const CoordinateAxis& axis1, Real cosAngle2, Real sinAngle2, const CoordinateAxis& axis2, Real cosAngle3, Real sinAngle3, const CoordinateAxis& axis3 );

    // These next methods highly-efficient power-user methods convert Rotation matrices to orientation angles are highly-efficient power-user methods.  Read the documentation to understand them.
    SimTK_SimTKCOMMON_EXPORT Vec2  convertTwoAxesBodyFixedRotationToTwoAngles(     const CoordinateAxis& axis1, const CoordinateAxis& axis2 ) const;
    SimTK_SimTKCOMMON_EXPORT Vec3  convertTwoAxesBodyFixedRotationToThreeAngles(   const CoordinateAxis& axis1, const CoordinateAxis& axis2 ) const;
    SimTK_SimTKCOMMON_EXPORT Vec3  convertThreeAxesBodyFixedRotationToThreeAngles( const CoordinateAxis& axis1, const CoordinateAxis& axis2, const CoordinateAxis& axis3 ) const;

};


class InverseRotation : public Mat33::TransposeType {
public:
    // Convenient shortcut name
    typedef  Mat33::TransposeType  BaseMat;

    // Should not usually construct one of these as they should only occur as expression intermediates.
    // But if you must ...
    InverseRotation() : BaseMat(1) {}

    // Default constructor and copy constructor
    InverseRotation( const InverseRotation& R ) : BaseMat(R) {}
    InverseRotation&  operator=( const InverseRotation& R )  { BaseMat::operator=( R.asMat33() );  return *this; }

    // Convert from InverseRotation to Rotation (no cost)
    const Rotation&  invert() const { return *reinterpret_cast<const Rotation*>(this); }
    Rotation&  updInvert()          { return *reinterpret_cast<Rotation*>(this); }

    // Transpose, and transpose operators (override BaseMat versions of transpose).
    const Rotation&  transpose() const  { return invert(); }
    const Rotation&  operator~() const  { return invert(); }
    Rotation&        updTranspose()     { return updInvert(); }
    Rotation&        operator~()        { return updInvert(); }

    // Note that this does not have unit stride.
    typedef  UnitVec<BaseMat::RowSpacing>  ColType;
    typedef  UnitRow<BaseMat::ColSpacing>  RowType;
    const RowType&  row( int i ) const         { return reinterpret_cast<const RowType&>(asMat33()[i]); }
    const ColType&  col( int j ) const         { return reinterpret_cast<const ColType&>(asMat33()(j)); }
    const ColType&  x() const                  { return col(0); }
    const ColType&  y() const                  { return col(1); }
    const ColType&  z() const                  { return col(2); }
    const RowType&  operator[]( int i ) const  { return row(i); }
    const ColType&  operator()( int j ) const  { return col(j); }

    const BaseMat&  asMat33() const  { return *static_cast<const BaseMat*>(this); }
    BaseMat         toMat33() const  { return asMat33(); }

};


SimTK_SimTKCOMMON_EXPORT std::ostream&  operator<<( std::ostream& o, const Rotation& m );

template <int S> inline UnitVec<1>  operator*( const Rotation& R,        const UnitVec<S>& v )       { return UnitVec<1>(R.asMat33()* v.asVec3(),  true); }
template <int S> inline UnitRow<1>  operator*( const UnitRow<S>& r,      const Rotation& R   )       { return UnitRow<1>(r.asRow3() * R.asMat33(), true); }
template <int S> inline UnitVec<1>  operator*( const InverseRotation& R, const UnitVec<S>& v )       { return UnitVec<1>(R.asMat33()* v.asVec3(),  true); }
template <int S> inline UnitRow<1>  operator*( const UnitRow<S>& r,      const InverseRotation& R )  { return UnitRow<1>(r.asRow3() * R.asMat33(), true); }

inline Rotation::Rotation( const InverseRotation& R) : Mat33( R.asMat33() )  {}

inline Rotation&  Rotation::operator=(  const InverseRotation& R )  { static_cast<Mat33&>(*this)  = R.asMat33();    return *this; }
inline Rotation&  Rotation::operator*=( const Rotation& R )         { static_cast<Mat33&>(*this) *= R.asMat33();    return *this; }
inline Rotation&  Rotation::operator/=( const Rotation& R )         { static_cast<Mat33&>(*this) *= (~R).asMat33(); return *this; }
inline Rotation&  Rotation::operator*=( const InverseRotation& R )  { static_cast<Mat33&>(*this) *= R.asMat33();    return *this; }
inline Rotation&  Rotation::operator/=( const InverseRotation& R )  { static_cast<Mat33&>(*this) *= (~R).asMat33(); return *this; }

inline Rotation  operator*( const Rotation&        R1, const Rotation&        R2 )  { return Rotation(R1) *= R2; }
inline Rotation  operator*( const Rotation&        R1, const InverseRotation& R2 )  { return Rotation(R1) *= R2; }
inline Rotation  operator*( const InverseRotation& R1, const Rotation&        R2 )  { return Rotation(R1) *= R2; }
inline Rotation  operator*( const InverseRotation& R1, const InverseRotation& R2 )  { return Rotation(R1) *= R2; }

inline Rotation operator/( const Rotation&        R1, const Rotation&        R2 )  {return Rotation(R1) /= R2;}
inline Rotation operator/( const Rotation&        R1, const InverseRotation& R2 )  {return Rotation(R1) /= R2;}
inline Rotation operator/( const InverseRotation& R1, const Rotation&        R2 )  {return Rotation(R1) /= R2;}
inline Rotation operator/( const InverseRotation& R1, const InverseRotation& R2 )  {return Rotation(R1) /= R2;}


//------------------------------------------------------------------------------
}  // End of namespace SimTK

//--------------------------------------------------------------------------
#endif // SIMTK_ROTATION_H_
//--------------------------------------------------------------------------