Rotation.h

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#ifndef SimTK_SimTKCOMMON_ROTATION_H_ 
#define SimTK_SimTKCOMMON_ROTATION_H_ 

/* -------------------------------------------------------------------------- *
 *                       Simbody(tm): SimTKcommon                             *
 * -------------------------------------------------------------------------- *
 * This is part of the SimTK 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/home/simbody.  *
 *                                                                            *
 * Portions copyright (c) 2005-12 Stanford University and the Authors.        *
 * Authors: Paul Mitiguy, Michael Sherman                                     *
 * Contributors:                                                              *
 *                                                                            *
 * Licensed under the Apache License, Version 2.0 (the "License"); you may    *
 * not use this file except in compliance with the License. You may obtain a  *
 * copy of the License at http://www.apache.org/licenses/LICENSE-2.0.         *
 *                                                                            *
 * Unless required by applicable law or agreed to in writing, software        *
 * distributed under the License is distributed on an "AS IS" BASIS,          *
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.   *
 * See the License for the specific language governing permissions and        *
 * limitations under the License.                                             *
 * -------------------------------------------------------------------------- */

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

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

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

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


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

//------------------------------------------------------------------------------
// Forward declarations
template <class P> class Rotation_;
template <class P> class InverseRotation_;

typedef Rotation_<Real>             Rotation;
typedef Rotation_<float>           fRotation;
typedef Rotation_<double>          dRotation;

typedef InverseRotation_<Real>      InverseRotation;
typedef InverseRotation_<float>    fInverseRotation;
typedef InverseRotation_<double>   dInverseRotation;

//------------------------------------------------------------------------------
//------------------------------------------------------------------------------
template <class P> // templatized by precision
class Rotation_ : public Mat<3,3,P> {
    typedef P               RealP;
    typedef Mat<2,2,P>      Mat22P;
    typedef Mat<3,2,P>      Mat32P;
    typedef Mat<3,3,P>      Mat33P;
    typedef Mat<4,3,P>      Mat43P;
    typedef Mat<3,4,P>      Mat34P;
    typedef Vec<2,P>        Vec2P;
    typedef Vec<3,P>        Vec3P;
    typedef Vec<4,P>        Vec4P;
    typedef UnitVec<P,1>    UnitVec3P; // stride is 1 here, length is always 3
    typedef SymMat<3,P>     SymMat33P;
    typedef Quaternion_<P>  QuaternionP;
public:
    // Default constructor and constructor-like methods
    Rotation_() : Mat33P(1) {}    
    Rotation_&  setRotationToIdentityMatrix()  { Mat33P::operator=(RealP(1));  return *this; }
    Rotation_&  setRotationToNaN()             { Mat33P::setToNaN();    return *this; } 

    // Default copy constructor and assignment operator
    Rotation_( const Rotation_& R ) : Mat33P(R)  {}
    Rotation_&  operator=( const Rotation_& R )  { Mat33P::operator=( R.asMat33() );  return *this; }


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

    Rotation_&  setRotationFromAngleAboutAxis( RealP angle, const CoordinateAxis& axis )  { return axis.isXAxis() ? setRotationFromAngleAboutX(angle) : (axis.isYAxis() ? setRotationFromAngleAboutY(angle) : setRotationFromAngleAboutZ(angle) ); }
    Rotation_&  setRotationFromAngleAboutX( RealP angle )  { return setRotationFromAngleAboutX( std::cos(angle), std::sin(angle) ); }
    Rotation_&  setRotationFromAngleAboutY( RealP angle )  { return setRotationFromAngleAboutY( std::cos(angle), std::sin(angle) ); }
    Rotation_&  setRotationFromAngleAboutZ( RealP angle )  { return setRotationFromAngleAboutZ( std::cos(angle), std::sin(angle) ); }
    Rotation_&  setRotationFromAngleAboutX( RealP cosAngle, RealP sinAngle )  { Mat33P& 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( RealP cosAngle, RealP sinAngle )  { Mat33P& 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( RealP cosAngle, RealP sinAngle )  { Mat33P& 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_( RealP angle, const UnitVec3P& unitVector ) { setRotationFromAngleAboutUnitVector(angle,unitVector); }
    Rotation_( RealP angle, const Vec3P& nonUnitVector )  { setRotationFromAngleAboutNonUnitVector(angle,nonUnitVector); }

    Rotation_&  setRotationFromAngleAboutNonUnitVector( RealP angle, const Vec3P& nonUnitVector )  { return setRotationFromAngleAboutUnitVector( angle, UnitVec3P(nonUnitVector) ); }
    SimTK_SimTKCOMMON_EXPORT Rotation_&  setRotationFromAngleAboutUnitVector( RealP angle, const UnitVec3P& unitVector );

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

    void setRotationToBodyFixedXY( const Vec2P& v)   { setRotationFromTwoAnglesTwoAxes(     BodyRotationSequence, v[0], XAxis, v[1], YAxis ); }
    void setRotationToBodyFixedXYZ( const Vec3P& v)  { setRotationFromThreeAnglesThreeAxes( BodyRotationSequence, v[0], XAxis, v[1], YAxis, v[2], ZAxis ); }

    explicit Rotation_( const QuaternionP& q )  { setRotationFromQuaternion(q); }
    SimTK_SimTKCOMMON_EXPORT Rotation_&  setRotationFromQuaternion( const QuaternionP& q );

    Rotation_( const Mat33P& m, bool ) : Mat33P(m) {}

    explicit Rotation_( const Mat33P& m )  { setRotationFromApproximateMat33(m); }
    SimTK_SimTKCOMMON_EXPORT Rotation_&  setRotationFromApproximateMat33( const Mat33P& m );


    Rotation_( const UnitVec3P& uvec, const CoordinateAxis axis )  { setRotationFromOneAxis(uvec,axis); }
    SimTK_SimTKCOMMON_EXPORT Rotation_&  setRotationFromOneAxis( const UnitVec3P& uvec, const CoordinateAxis axis );


    Rotation_( const UnitVec3P& uveci, const CoordinateAxis& axisi, const Vec3P& vecjApprox, const CoordinateAxis& axisjApprox )  { setRotationFromTwoAxes(uveci,axisi,vecjApprox,axisjApprox); }
    SimTK_SimTKCOMMON_EXPORT Rotation_&  setRotationFromTwoAxes( const UnitVec3P& uveci, const CoordinateAxis& axisi, const Vec3P& 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 RealP  convertOneAxisRotationToOneAngle( const CoordinateAxis& axis1 ) const;
    SimTK_SimTKCOMMON_EXPORT Vec2P  convertTwoAxesRotationToTwoAngles(     BodyOrSpaceType bodyOrSpace, const CoordinateAxis& axis1, const CoordinateAxis& axis2 ) const;
    SimTK_SimTKCOMMON_EXPORT Vec3P  convertThreeAxesRotationToThreeAngles( BodyOrSpaceType bodyOrSpace, const CoordinateAxis& axis1, const CoordinateAxis& axis2, const CoordinateAxis& axis3 ) const;
    SimTK_SimTKCOMMON_EXPORT QuaternionP convertRotationToQuaternion() const;
    Vec4P  convertRotationToAngleAxis() const  { return convertRotationToQuaternion().convertQuaternionToAngleAxis(); }

    Vec2P  convertRotationToBodyFixedXY() const   { return convertTwoAxesRotationToTwoAngles( BodyRotationSequence, XAxis, YAxis ); }
    Vec3P  convertRotationToBodyFixedXYZ() const  { return convertThreeAxesRotationToThreeAngles( BodyRotationSequence, XAxis, YAxis, ZAxis ); }

    SimTK_SimTKCOMMON_EXPORT SymMat33P reexpressSymMat33(const SymMat33P& S_BB) const;


    SimTK_SimTKCOMMON_EXPORT bool  isSameRotationToWithinAngle( const Rotation_& R, RealP okPointingAngleErrorRads ) const;
    bool isSameRotationToWithinAngleOfMachinePrecision( const Rotation_& R) const       
    {   return isSameRotationToWithinAngle( R, NTraits<P>::getSignificant() ); }
    RealP  getMaxAbsDifferenceInRotationElements( const Rotation_& R ) const {            
        const Mat33P& A = asMat33();  const Mat33P& B = R.asMat33();  RealP maxDiff = 0;  
        for( int i=0;  i<=2; i++ ) for( int j=0; j<=2; j++ ) 
        {   RealP absDiff = std::fabs(A[i][j] - B[i][j]);  
            if( absDiff > maxDiff ) maxDiff = absDiff; }  
        return maxDiff; 
    } 

    bool  areAllRotationElementsSameToEpsilon( const Rotation_& R, RealP epsilon ) const 
    {   return getMaxAbsDifferenceInRotationElements(R) <= epsilon ; }
    bool  areAllRotationElementsSameToMachinePrecision( const Rotation_& R ) const       
    {   return areAllRotationElementsSameToEpsilon( R, NTraits<P>::getSignificant() ); } 

    inline Rotation_( const InverseRotation_<P>& );
    inline Rotation_& operator=( const InverseRotation_<P>& );

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


    const InverseRotation_<P>&  transpose() const  { return invert(); }
    const InverseRotation_<P>&  operator~() const  { return invert(); }
    InverseRotation_<P>&        updTranspose()     { return updInvert(); }
    InverseRotation_<P>&        operator~()        { return updInvert(); }


    inline Rotation_&  operator*=( const Rotation_<P>& R );
    inline Rotation_&  operator/=( const Rotation_<P>& R );
    inline Rotation_&  operator*=( const InverseRotation_<P>& );
    inline Rotation_&  operator/=( const InverseRotation_<P>& );


    const Mat33P&  asMat33() const  { return *static_cast<const Mat33P*>(this); }
    Mat33P         toMat33() const  { return asMat33(); }

    typedef  UnitVec<P,Mat33P::RowSpacing>  ColType;
    typedef  UnitRow<P,Mat33P::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 Mat33P& m )  
    {   Mat33P& R = *this; R=m;  return *this; }   
    Rotation_&  setRotationColFromUnitVecTrustMe( int colj, const UnitVec3P& uvecj )  
    {   Mat33P& R = *this; R(colj)=uvecj.asVec3(); return *this; }   
    Rotation_&  setRotationFromUnitVecsTrustMe( const UnitVec3P& colA, const UnitVec3P& colB, const UnitVec3P& colC )  
    {   Mat33P& R = *this; R(0)=colA.asVec3(); R(1)=colB.asVec3(); R(2)=colC.asVec3(); return *this; }  

//--------------------------- PAUL CONTINUE FROM HERE --------------------------
public:
//------------------------------------------------------------------------------

    // These are ad hoc routines that don't match the nice API Paul Mitiguy
    // implemented above.


    void setRotationToBodyFixedXYZ(const Vec3P& c, const Vec3P& s) {
        Mat33P& R = *this;
        const RealP s0s1=s[0]*s[1], s2c0=s[2]*c[0], c0c2=c[0]*c[2], nc1= -c[1];

        R = Mat33P(     c[1]*c[2]      ,         s[2]*nc1       ,    s[1]  ,
                    s2c0 + s0s1*c[2]   ,     c0c2 - s0s1*s[2]   , s[0]*nc1 ,
                 s[0]*s[2] - s[1]*c0c2 ,  s[0]*c[2] + s[1]*s2c0 , c[0]*c[1] );
    }


    static Vec3P convertAngVelToBodyFixed321Dot(const Vec3P& q, const Vec3P& w_PB_B) {
        const RealP s1 = std::sin(q[1]), c1 = std::cos(q[1]);
        const RealP s2 = std::sin(q[2]), c2 = std::cos(q[2]);
        const RealP ooc1 = RealP(1)/c1;
        const RealP s2oc1 = s2*ooc1, c2oc1 = c2*ooc1;

        const Mat33P E( 0,    s2oc1  ,  c2oc1  ,
                        0,      c2   ,   -s2   ,
                        1,  s1*s2oc1 , s1*c2oc1 );
        return E * w_PB_B;
    }

    static Vec3P convertBodyFixed321DotToAngVel(const Vec3P& q, const Vec3P& qd) {
        const RealP s1 = std::sin(q[1]), c1 = std::cos(q[1]);
        const RealP s2 = std::sin(q[2]), c2 = std::cos(q[2]);

        const Mat33P 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 angular velocities and accelerations are
    // expressed in the *BODY* frame.
    // TODO: this is not an efficient way to do this computation.
    static Vec3P convertAngVelDotToBodyFixed321DotDot
        (const Vec3P& q, const Vec3P& w_PB_B, const Vec3P& wdot_PB_B)
    {
        const RealP s1 = std::sin(q[1]), c1 = std::cos(q[1]);
        const RealP s2 = std::sin(q[2]), c2 = std::cos(q[2]);
        const RealP ooc1  = 1/c1;
        const RealP s2oc1 = s2*ooc1, c2oc1 = c2*ooc1, s1oc1 = s1*ooc1;

        const Mat33P E( 0 ,   s2oc1  ,  c2oc1  ,
                       0 ,     c2   ,   -s2   ,
                       1 , s1*s2oc1 , s1*c2oc1 );
        const Vec3P qdot = E * w_PB_B;

        const RealP t = qdot[1]*s1oc1;
        const RealP a = t*s2oc1 + qdot[2]*c2oc1; // d/dt s2oc1
        const RealP b = t*c2oc1 - qdot[2]*s2oc1; // d/dt c2oc1

        const Mat33P 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_PB_B + Edot*w_PB_B;
    }
    
    static Mat33P calcNForBodyXYZInBodyFrame(const Vec3P& q) {
        // Note: q[0] is not referenced so we won't waste time calculating
        // its cosine and sine here.
        return calcNForBodyXYZInBodyFrame
           (Vec3P(0, std::cos(q[1]), std::cos(q[2])),
            Vec3P(0, std::sin(q[1]), std::sin(q[2])));
    }

    static Mat33P calcNForBodyXYZInBodyFrame(const Vec3P& cq, const Vec3P& sq) {
        const RealP s1 = sq[1], c1 = cq[1];
        const RealP s2 = sq[2], c2 = cq[2];
        const RealP ooc1  = 1/c1;
        const RealP s2oc1 = s2*ooc1, c2oc1 = c2*ooc1;

        return Mat33P(    c2oc1  , -s2oc1  , 0,
                            s2   ,    c2   , 0,
                       -s1*c2oc1 , s1*s2oc1, 1 );
    }

    static Mat33P calcNForBodyXYZInParentFrame(const Vec3P& q) {
        // Note: q[2] is not referenced so we won't waste time calculating
        // its cosine and sine here.
        return calcNForBodyXYZInParentFrame
           (Vec3P(std::cos(q[0]), std::cos(q[1]), 0),
            Vec3P(std::sin(q[0]), std::sin(q[1]), 0));
    }

    static Mat33P calcNForBodyXYZInParentFrame(const Vec3P& cq, const Vec3P& sq) {
        const RealP s0 = sq[0], c0 = cq[0];
        const RealP s1 = sq[1], c1 = cq[1];
        const RealP ooc1  = 1/c1;
        const RealP s0oc1 = s0*ooc1, c0oc1 = c0*ooc1;

        return Mat33P( 1 , s1*s0oc1 , -s1*c0oc1,
                       0 ,    c0    ,    s0,
                       0 ,  -s0oc1  ,  c0oc1 );
    }

    static Vec3P multiplyByBodyXYZ_N_P(const Vec2P& cosxy,
                                       const Vec2P& sinxy,
                                       RealP        oocosy,
                                       const Vec3P& w_PB)
    {
        const RealP s0 = sinxy[0], c0 = cosxy[0];
        const RealP s1 = sinxy[1];
        const RealP w0 = w_PB[0], w1 = w_PB[1], w2 = w_PB[2];

        const RealP t = (s0*w1-c0*w2)*oocosy;
        return Vec3P( w0 + t*s1, c0*w1 + s0*w2, -t ); // qdot
    }

    static Vec3P multiplyByBodyXYZ_NT_P(const Vec2P& cosxy,
                                        const Vec2P& sinxy,
                                        RealP        oocosy,
                                        const Vec3P& q)
    {
        const RealP s0 = sinxy[0], c0 = cosxy[0];
        const RealP s1 = sinxy[1];
        const RealP q0 = q[0], q1 = q[1], q2 = q[2];

        const RealP t = (q0*s1-q2) * oocosy;
        return Vec3P( q0, c0*q1 + t*s0, s0*q1 - t*c0 ); // v_P
    }

    static Vec3P convertAngVelInParentToBodyXYZDot
       (const Vec2P& cosxy,  
        const Vec2P& sinxy,  
        RealP        oocosy, 
        const Vec3P& w_PB)   
    {
        return multiplyByBodyXYZ_N_P(cosxy,sinxy,oocosy,w_PB);
    }

    static Vec3P convertAngAccInParentToBodyXYZDotDot
       (const Vec2P& cosxy,  
        const Vec2P& sinxy,  
        RealP        oocosy, 
        const Vec3P& qdot,   
        const Vec3P& b_PB)   
    {
        const RealP s1 = sinxy[1], c1 = cosxy[1];
        const RealP q0 = qdot[0], q1 = qdot[1], q2 = qdot[2];

        // 10 flops
        const Vec3P Nb = multiplyByBodyXYZ_N_P(cosxy,sinxy,oocosy,b_PB);

        const RealP q1oc1 = q1*oocosy;
        const Vec3P NDotw((q0*s1-q2)*q1oc1,     //   NDot_P*w_PB
                           q0*q2*c1,            // = NDot_P*(NInv_P*qdot)
                          (q2*s1-q0)*q1oc1 );   // (9 flops)

        return Nb + NDotw; // 3 flops
    }


    static Vec3P multiplyByBodyXYZ_NInv_P(const Vec2P& cosxy,
                                          const Vec2P& sinxy,
                                          const Vec3P& qdot)
    {
        const RealP s0 = sinxy[0], c0 = cosxy[0];
        const RealP s1 = sinxy[1], c1 = cosxy[1];
        const RealP q0 = qdot[0], q1 = qdot[1], q2 = qdot[2];
        const RealP c1q2 = c1*q2;

        return Vec3P( q0 + s1*q2,           // w_PB
                      c0*q1 - s0*c1q2, 
                      s0*q1 + c0*c1q2 );
    }

    static Vec3P multiplyByBodyXYZ_NInvT_P(const Vec2P& cosxy,
                                           const Vec2P& sinxy,
                                           const Vec3P& v_P)
    {
        const RealP s0 = sinxy[0], c0 = cosxy[0];
        const RealP s1 = sinxy[1], c1 = cosxy[1];
        const RealP w0 = v_P[0], w1 = v_P[1], w2 = v_P[2];

        return Vec3P( w0,                           // qdot-like
                      c0*w1 + s0*w2,
                      s1*w0 - s0*c1*w1 + c0*c1*w2);
    }

    static Mat33P calcNDotForBodyXYZInBodyFrame
       (const Vec3P& q, const Vec3P& qdot) {
        // Note: q[0] is not referenced so we won't waste time calculating
        // its cosine and sine here.
        return calcNDotForBodyXYZInBodyFrame
           (Vec3P(0, std::cos(q[1]), std::cos(q[2])),
            Vec3P(0, std::sin(q[1]), std::sin(q[2])),
            qdot);
    }

    static Mat33P calcNDotForBodyXYZInBodyFrame
       (const Vec3P& cq, const Vec3P& sq, const Vec3P& qdot) 
    {
        const RealP s1 = sq[1], c1 = cq[1];
        const RealP s2 = sq[2], c2 = cq[2];
        const RealP ooc1  = 1/c1;
        const RealP s2oc1 = s2*ooc1, c2oc1 = c2*ooc1;

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

        return Mat33P(       b             ,        -a         , 0,
                          qdot[2]*c2       ,    -qdot[2]*s2    , 0,
                      -(s1*b + qdot[1]*c2) , s1*a + qdot[1]*s2 , 0 );
    }

    static Mat33P calcNDotForBodyXYZInParentFrame
       (const Vec3P& q, const Vec3P& qdot) {
        // Note: q[2] is not referenced so we won't waste time calculating
        // its cosine and sine here.
        const RealP cy = std::cos(q[1]); // cos(y)
        return calcNDotForBodyXYZInParentFrame
           (Vec2P(std::cos(q[0]), cy), 
            Vec2P(std::sin(q[0]), std::sin(q[1])),
            1/cy, qdot);
    }

    static Mat33P calcNDotForBodyXYZInParentFrame
       (const Vec2P& cq, const Vec2P& sq, RealP ooc1, const Vec3P& qdot) {
        const RealP s0 = sq[0], c0 = cq[0];
        const RealP s1 = sq[1], c1 = cq[1];
        const RealP s0oc1 = s0*ooc1, c0oc1 = c0*ooc1;

        const RealP t = qdot[1]*s1*ooc1;
        const RealP a = t*s0oc1 + qdot[0]*c0oc1; // d/dt s0oc1
        const RealP b = t*c0oc1 - qdot[0]*s0oc1; // d/dt c0oc1

        return Mat33P( 0,  s1*a + qdot[1]*s0, -(s1*b + qdot[1]*c0), 
                       0,    -qdot[0]*s0    ,     qdot[0]*c0      ,
                       0,        -a         ,         b            );
    }

    static Mat33P calcNInvForBodyXYZInBodyFrame(const Vec3P& q) {
        // Note: q[0] is not referenced so we won't waste time calculating
        // its cosine and sine here.
        return calcNInvForBodyXYZInBodyFrame
           (Vec3P(0, std::cos(q[1]), std::cos(q[2])),
            Vec3P(0, std::sin(q[1]), std::sin(q[2])));
    }

    static Mat33P calcNInvForBodyXYZInBodyFrame
       (const Vec3P& cq, const Vec3P& sq) {
        const RealP s1 = sq[1], c1 = cq[1];
        const RealP s2 = sq[2], c2 = cq[2];

        return Mat33P( c1*c2 ,  s2 , 0 ,
                      -c1*s2 ,  c2 , 0 ,
                        s1   ,  0  , 1 );
    }

    static Mat33P calcNInvForBodyXYZInParentFrame(const Vec3P& q) {
        // Note: q[0] is not referenced so we won't waste time calculating
        // its cosine and sine here.
        return calcNInvForBodyXYZInParentFrame
           (Vec3P(std::cos(q[0]), std::cos(q[1]), 0),
            Vec3P(std::sin(q[0]), std::sin(q[1]), 0));
    }

    static Mat33P calcNInvForBodyXYZInParentFrame
       (const Vec3P& cq, const Vec3P& sq) {
        const RealP s0 = sq[0], c0 = cq[0];
        const RealP s1 = sq[1], c1 = cq[1];

        return Mat33P( 1 ,  0  ,   s1   ,
                       0 ,  c0 , -s0*c1 ,
                       0 ,  s0 ,  c0*c1 );
    }

    static Vec3P convertAngVelInBodyFrameToBodyXYZDot
       (const Vec3P& q, const Vec3P& w_PB_B) {  
        return convertAngVelInBodyFrameToBodyXYZDot
           (Vec3P(0, std::cos(q[1]), std::cos(q[2])),
            Vec3P(0, std::sin(q[1]), std::sin(q[2])),
            w_PB_B); 
    }

    //TODO: reimplement
    static Vec3P convertAngVelInBodyFrameToBodyXYZDot
       (const Vec3P& cq, const Vec3P& sq, const Vec3P& w_PB_B) 
    {   return calcNForBodyXYZInBodyFrame(cq,sq)*w_PB_B; }

    static Vec3P convertBodyXYZDotToAngVelInBodyFrame
       (const Vec3P& q, const Vec3P& qdot) {   
           return convertBodyXYZDotToAngVelInBodyFrame
                       (Vec3P(0, std::cos(q[1]), std::cos(q[2])),
                        Vec3P(0, std::sin(q[1]), std::sin(q[2])),
                        qdot); 
    }

    // TODO: reimplement
    static Vec3P convertBodyXYZDotToAngVelInBodyFrame
       (const Vec3P& cq, const Vec3P& sq, const Vec3P& qdot) 
    {   return calcNInvForBodyXYZInBodyFrame(cq,sq)*qdot; }

    static Vec3P convertAngVelDotInBodyFrameToBodyXYZDotDot
        (const Vec3P& q, const Vec3P& w_PB_B, const Vec3P& wdot_PB_B)
    {
        // Note: q[0] is not referenced so we won't waste time calculating
        // its cosine and sine here.
        return convertAngVelDotInBodyFrameToBodyXYZDotDot
                   (Vec3P(0, std::cos(q[1]), std::cos(q[2])),
                    Vec3P(0, std::sin(q[1]), std::sin(q[2])),
                    w_PB_B, wdot_PB_B);
    }

    // TODO: reimplement
    static Vec3P convertAngVelDotInBodyFrameToBodyXYZDotDot
       (const Vec3P& cq, const Vec3P& sq, const Vec3P& w_PB_B, const Vec3P& wdot_PB_B)
    {
        const Mat33P N      = calcNForBodyXYZInBodyFrame(cq,sq);
        const Vec3P  qdot   = N * w_PB_B; // 15 flops
        const Mat33P NDot   = calcNDotForBodyXYZInBodyFrame(cq,sq,qdot);

        return N*wdot_PB_B + NDot*w_PB_B; // 33 flops
    }

    static Mat43P calcUnnormalizedNForQuaternion(const Vec4P& q) {
        const Vec4P e = q/2;
        const RealP ne1 = -e[1], ne2 = -e[2], ne3 = -e[3];
        return Mat43P( ne1,  ne2,  ne3,
                       e[0], e[3], ne2,
                       ne3,  e[0], e[1],
                       e[2], ne1,  e[0]);
    }

    static Mat43P calcUnnormalizedNDotForQuaternion(const Vec4P& qdot) {
        const Vec4P ed = qdot/2;
        const RealP ned1 = -ed[1], ned2 = -ed[2], ned3 = -ed[3];
        return Mat43P( ned1,  ned2,  ned3,
                       ed[0], ed[3], ned2,
                       ned3,  ed[0], ed[1],
                       ed[2], ned1,  ed[0]);
    }

    static Mat34P calcUnnormalizedNInvForQuaternion(const Vec4P& q) {
        const Vec4P e = 2*q;
        const RealP ne1 = -e[1], ne2 = -e[2], ne3 = -e[3];
        return Mat34P(ne1, e[0], ne3,  e[2],
                      ne2, e[3], e[0], ne1,
                      ne3, ne2,  e[1], e[0]);
    }


    static Vec4P convertAngVelToQuaternionDot(const Vec4P& q, const Vec3P& w_PB_P) {
        return calcUnnormalizedNForQuaternion(q)*w_PB_P;
    }

    static Vec3P convertQuaternionDotToAngVel(const Vec4P& q, const Vec4P& qdot) {
        return calcUnnormalizedNInvForQuaternion(q)*qdot;
    }

    // TODO: reimplement
    static Vec4P OLDconvertAngVelDotToQuaternionDotDot
       (const Vec4P& q, const Vec3P& w_PB_P, const Vec3P& wdot_PB_P)
    {
        const Mat43P N    = calcUnnormalizedNForQuaternion(q);
        const Vec4P      qdot = N*w_PB_P;   // 20 flops
        const Mat43P NDot = calcUnnormalizedNDotForQuaternion(qdot);

        return  N*wdot_PB_P + NDot*w_PB_P;  // 44 flops
    }

    static Vec4P convertAngVelDotToQuaternionDotDot
       (const Vec4P& q, const Vec3P& w_PB, const Vec3P& b_PB)
    {
        const Mat43P N     = calcUnnormalizedNForQuaternion(q); //  7 flops
        const Vec4P  Nb    = N*b_PB;                            // 20 flops
        const Vec4P  NDotw = RealP(-.25)*w_PB.normSqr()*q;      // 10 flops
        return Nb + NDotw;                                      //  4 flops
    }


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 RealP& xx, const RealP& xy, const RealP& xz,
               const RealP& yx, const RealP& yy, const RealP& yz,
               const RealP& zx, const RealP& zy, const RealP& zz )
    :   Mat33P( xx,xy,xz, yx,yy,yz, zx,zy,zz ) {}

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

    // These next methods are highly-efficient power-user methods to convert 
    // Rotation matrices to orientation angles.  Read the code to understand them.
    SimTK_SimTKCOMMON_EXPORT Vec2P  convertTwoAxesBodyFixedRotationToTwoAngles(     const CoordinateAxis& axis1, const CoordinateAxis& axis2 ) const;
    SimTK_SimTKCOMMON_EXPORT Vec3P  convertTwoAxesBodyFixedRotationToThreeAngles(   const CoordinateAxis& axis1, const CoordinateAxis& axis2 ) const;
    SimTK_SimTKCOMMON_EXPORT Vec3P  convertThreeAxesBodyFixedRotationToThreeAngles( const CoordinateAxis& axis1, const CoordinateAxis& axis2, const CoordinateAxis& axis3 ) const;

//------------------------------------------------------------------------------
// These are obsolete names from a previous release, listed here so that 
// users will get a decipherable compilation error. (sherm 091101)
//------------------------------------------------------------------------------
private:
    // REPLACED BY: calcNForBodyXYZInBodyFrame()
    static Mat33P calcQBlockForBodyXYZInBodyFrame(const Vec3P& a)
    {   return calcNForBodyXYZInBodyFrame(a); }
    // REPLACED BY: calcNInvForBodyXYZInBodyFrame()
    static Mat33P calcQInvBlockForBodyXYZInBodyFrame(const Vec3P& a)
    {   return calcNInvForBodyXYZInBodyFrame(a); }
    // REPLACED BY: calcUnnormalizedNForQuaternion()
    static Mat<4,3,P> calcUnnormalizedQBlockForQuaternion(const Vec4P& q)
    {   return calcUnnormalizedNForQuaternion(q); }
    // REPLACED BY: calcUnnormalizedNInvForQuaternion()
    static Mat<3,4,P> calcUnnormalizedQInvBlockForQuaternion(const Vec4P& q)
    {   return calcUnnormalizedNInvForQuaternion(q); }
    // REPLACED BY: convertAngVelInBodyFrameToBodyXYZDot
    static Vec3P convertAngVelToBodyFixed123Dot(const Vec3P& q, const Vec3P& w_PB_B) 
    {   return convertAngVelInBodyFrameToBodyXYZDot(q,w_PB_B); }
    // REPLACED BY: convertBodyXYZDotToAngVelInBodyFrame
    static Vec3P convertBodyFixed123DotToAngVel(const Vec3P& q, const Vec3P& qdot) 
    {   return convertBodyXYZDotToAngVelInBodyFrame(q,qdot); }
    // REPLACED BY: convertAngVelDotInBodyFrameToBodyXYZDotDot
    static Vec3P convertAngVelDotToBodyFixed123DotDot
        (const Vec3P& q, const Vec3P& w_PB_B, const Vec3P& wdot_PB_B)
    {   return convertAngVelDotInBodyFrameToBodyXYZDotDot(q,w_PB_B,wdot_PB_B); }

//------------------------------------------------------------------------------
// 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 Mat33P& m )  { return Rotation_(m,true); }

    // One-angle rotations.
    static Rotation_ aboutX( const RealP& angleInRad ) { return Rotation_( angleInRad, XAxis ); }
    static Rotation_ aboutY( const RealP& angleInRad ) { return Rotation_( angleInRad, YAxis ); }
    static Rotation_ aboutZ( const RealP& angleInRad ) { return Rotation_( angleInRad, ZAxis ); }
    static Rotation_ aboutAxis( const RealP& angleInRad, const UnitVec3P& axis ) { return Rotation_(angleInRad,axis); }
    static Rotation_ aboutAxis( const RealP& angleInRad, const Vec3P& axis )     { return Rotation_(angleInRad,axis); }
    void  setToRotationAboutZ( const RealP& q ) { setRotationFromAngleAboutZ( q ); }

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

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

    explicit Rotation_( const UnitVec3P& 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 Vec3P& x, const Vec3P& yish )  { setRotationFromTwoAxes( UnitVec3P(x), XAxis, yish, YAxis ); }

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

    //  Similarly for BodyFixed123.
    void setToBodyFixed321( const Vec3P& v)  { setRotationFromThreeAnglesThreeAxes( BodyRotationSequence, v[0], ZAxis, v[1], YAxis, v[2], XAxis ); }
    void setToBodyFixed123( const Vec3P& v)  { setRotationToBodyFixedXYZ(v); }

    Vec4P convertToAngleAxis() const  { return convertRotationToAngleAxis(); }

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

    void setToSpaceFixed12( const Vec2P& q ) { setRotationFromTwoAnglesTwoAxes( SpaceRotationSequence, q[0], XAxis, q[1], YAxis ); }

    Vec3P  convertToBodyFixed123() const  { return convertRotationToBodyFixedXYZ(); }
    Vec2P  convertToBodyFixed12() const   { return convertRotationToBodyFixedXY(); }
    Vec2P  convertToSpaceFixed12() const  { return convertTwoAxesRotationToTwoAngles( SpaceRotationSequence, XAxis, YAxis ); }
};


template <class P>
class InverseRotation_ : public Mat<3,3,P>::TransposeType {
    typedef P               RealP;
    typedef Rotation_<P>    RotationP;
    typedef Mat<3,3,P>      Mat33P; // not the base type!
    typedef SymMat<3,P>     SymMat33P;
    typedef Mat<2,2,P>      Mat22P;
    typedef Mat<3,2,P>      Mat32P;
    typedef Vec<2,P>        Vec2P;
    typedef Vec<3,P>        Vec3P;
    typedef Vec<4,P>        Vec4P;
    typedef Quaternion_<P>  QuaternionP;
public:
    typedef typename Mat<3,3,P>::TransposeType  BaseMat;


    typedef  UnitVec<P,BaseMat::RowSpacing>  ColType;
    typedef  UnitRow<P,BaseMat::ColSpacing>  RowType;

    InverseRotation_() : BaseMat(1) {}

    InverseRotation_( const InverseRotation_& R ) : BaseMat(R) {}
    InverseRotation_&  operator=( const InverseRotation_& R )  
    {   BaseMat::operator=(R.asMat33());  return *this; }

    SimTK_SimTKCOMMON_EXPORT SymMat33P reexpressSymMat33(const SymMat33P& S_BB) const;


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


    const RotationP&  transpose() const  { return invert(); }
    const RotationP&  operator~() const  { return invert(); }
    RotationP&        updTranspose()     { return updInvert(); }
    RotationP&        operator~()        { return updInvert(); }


    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(); }
};

template <class P> SimTK_SimTKCOMMON_EXPORT std::ostream& 
operator<<(std::ostream&, const Rotation_<P>&);
template <class P> SimTK_SimTKCOMMON_EXPORT std::ostream& 
operator<<(std::ostream&, const InverseRotation_<P>&);


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

// Couldn't implement these Rotation_ methods until InverseRotation_ was defined.
template <class P> inline
Rotation_<P>::Rotation_(const InverseRotation_<P>& R) : Mat<3,3,P>( R.asMat33() ) {}
template <class P> inline Rotation_<P>&  
Rotation_<P>::operator=(const InverseRotation_<P>& R)  {static_cast<Mat<3,3,P>&>(*this)  = R.asMat33();    return *this;}
template <class P> inline Rotation_<P>&  
Rotation_<P>::operator*=(const Rotation_<P>& R)        {static_cast<Mat<3,3,P>&>(*this) *= R.asMat33();    return *this;}
template <class P> inline Rotation_<P>&  
Rotation_<P>::operator/=(const Rotation_<P>& R)        {static_cast<Mat<3,3,P>&>(*this) *= (~R).asMat33(); return *this;}
template <class P> inline Rotation_<P>&  
Rotation_<P>::operator*=(const InverseRotation_<P>& R) {static_cast<Mat<3,3,P>&>(*this) *= R.asMat33();    return *this;}
template <class P> inline Rotation_<P>&  
Rotation_<P>::operator/=(const InverseRotation_<P>& R) {static_cast<Mat<3,3,P>&>(*this) *= (~R).asMat33(); return *this;}


template <class P> inline Rotation_<P>
operator*(const Rotation_<P>&        R1, const Rotation_<P>&        R2)  {return Rotation_<P>(R1) *= R2;}
template <class P> inline Rotation_<P>
operator*(const Rotation_<P>&        R1, const InverseRotation_<P>& R2)  {return Rotation_<P>(R1) *= R2;}
template <class P> inline Rotation_<P>
operator*(const InverseRotation_<P>& R1, const Rotation_<P>&        R2)  {return Rotation_<P>(R1) *= R2;}
template <class P> inline Rotation_<P>
operator*(const InverseRotation_<P>& R1, const InverseRotation_<P>& R2)  {return Rotation_<P>(R1) *= R2;}


template <class P> inline Rotation_<P>
operator/( const Rotation_<P>&        R1, const Rotation_<P>&        R2 )  {return Rotation_<P>(R1) /= R2;}
template <class P> inline Rotation_<P>
operator/( const Rotation_<P>&        R1, const InverseRotation&     R2 )  {return Rotation_<P>(R1) /= R2;}
template <class P> inline Rotation_<P>
operator/( const InverseRotation_<P>& R1, const Rotation_<P>&        R2 )  {return Rotation_<P>(R1) /= R2;}
template <class P> inline Rotation_<P>
operator/( const InverseRotation_<P>& R1, const InverseRotation_<P>& R2 )  {return Rotation_<P>(R1) /= R2;}


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

//--------------------------------------------------------------------------
#endif // SimTK_SimTKCOMMON_ROTATION_H_
//--------------------------------------------------------------------------