SpatialAlgebra.h

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#ifndef SimTK_SIMMATRIX_SPATIAL_ALGEBRA_H_
#define SimTK_SIMMATRIX_SPATIAL_ALGEBRA_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: 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 <iostream>

namespace SimTK {


typedef Vec<2,   Vec3>  SpatialVec;
typedef Row<2,   Row3>  SpatialRow;
typedef Mat<2,2, Mat33> SpatialMat;


// Pre-declare methods here so that we can list them in whatever order we'd
// like them to appear in Doxygen.
inline SpatialVec findRelativeVelocity( const Transform&  X_FA,
                                        const SpatialVec& V_FA,
                                        const Transform&  X_FB,
                                        const SpatialVec& V_FB);
inline SpatialVec findRelativeVelocityInF(  const Vec3&       p_AB_F,
                                            const SpatialVec& V_FA,
                                            const SpatialVec& V_FB);

inline SpatialVec findRelativeAcceleration( const Transform&  X_FA,
                                            const SpatialVec& V_FA,
                                            const SpatialVec& A_FA,
                                            const Transform&  X_FB,
                                            const SpatialVec& V_FB,
                                            const SpatialVec& A_FB);
inline SpatialVec findRelativeAccelerationInF(  const Vec3&       p_AB_F,
                                                const SpatialVec& V_FA,
                                                const SpatialVec& A_FA,
                                                const SpatialVec& V_FB,
                                                const SpatialVec& A_FB);

inline SpatialVec reverseRelativeVelocity(const Transform&  X_AB,
                                          const SpatialVec& V_AB);
inline SpatialVec reverseRelativeVelocityInA(const Transform&  X_AB,
                                             const SpatialVec& V_AB);

inline SpatialVec shiftVelocityBy(const SpatialVec& V_AB, const Vec3& r_A);
inline SpatialVec shiftVelocityFromTo(const SpatialVec& V_A_BP, 
                                      const Vec3&       fromP_A,
                                      const Vec3&       toQ_A);

inline SpatialVec shiftForceBy(const SpatialVec& F_AP, const Vec3& r_A);
inline SpatialVec shiftForceFromTo(const SpatialVec& F_AP, 
                                   const Vec3&       fromP_A,
                                   const Vec3&       toQ_A);



//==============================================================================
//                           FIND RELATIVE VELOCITY
//==============================================================================
inline SpatialVec findRelativeVelocity(const Transform&  X_FA,
                                       const SpatialVec& V_FA,
                                       const Transform&  X_FB,
                                       const SpatialVec& V_FB)
{
    const Vec3 p_AB_F = X_FB.p() - X_FA.p();                    //  3 flops
    return ~X_FA.R()*findRelativeVelocityInF(p_AB_F,V_FA,V_FB); // 48 flops
}



//==============================================================================
//                         FIND RELATIVE VELOCITY IN F
//==============================================================================
inline SpatialVec findRelativeVelocityInF(const Vec3&       p_AB_F,
                                          const SpatialVec& V_FA,
                                          const SpatialVec& V_FB)
{
    // Relative angular velocity of B in A, expressed in F.
    const Vec3 w_AB_F     = V_FB[0] - V_FA[0];              // 3 flops
    // Relative linear velocity of B in A, taken and expressed in F.
    const Vec3 p_AB_F_dot = V_FB[1] - V_FA[1];              // 3 flops
    // Get linear velocity taken in A by removing the component due
    // to A's rotation in F (still expressed in F).
    const Vec3 v_AB_F = p_AB_F_dot - V_FA[0] % p_AB_F;      // 12 flops

    return SpatialVec(w_AB_F, v_AB_F);
}



//==============================================================================
//                        FIND RELATIVE ACCELERATION
//==============================================================================
inline SpatialVec findRelativeAcceleration( const Transform&  X_FA,
                                            const SpatialVec& V_FA,
                                            const SpatialVec& A_FA,
                                            const Transform&  X_FB,
                                            const SpatialVec& V_FB,
                                            const SpatialVec& A_FB)
{
    const Vec3 p_AB_F = X_FB.p() - X_FA.p();                        //  3 flops
    return ~X_FA.R() *                                              // 30 flops
           findRelativeAccelerationInF(p_AB_F,V_FA,A_FA,V_FB,A_FB); // 72 flops
}



//==============================================================================
//                       FIND RELATIVE ACCELERATION IN F
//==============================================================================
inline SpatialVec findRelativeAccelerationInF(  const Vec3&       p_AB_F,
                                                const SpatialVec& V_FA,
                                                const SpatialVec& A_FA,
                                                const SpatialVec& V_FB,
                                                const SpatialVec& A_FB)
{
    const Vec3& w_FA = V_FA[0];     // aliases for convenience
    const Vec3& w_FB = V_FB[0];
    const Vec3& b_FA = A_FA[0];
    const Vec3& b_FB = A_FB[0];

    const Vec3 p_AB_F_dot    = V_FB[1] - V_FA[1]; // d/dt p taken in F   (3 flops)
    const Vec3 p_AB_F_dotdot = A_FB[1] - A_FA[1]; // d^2/dt^2 taken in F (3 flops)

    const Vec3 w_AB_F =     // relative angvel of B in A, exp. in F
        w_FB - w_FA;        // (3 flops)
    const Vec3 v_AB_F =              // d/dt p taken in A, exp in F
        p_AB_F_dot - w_FA % p_AB_F;  // (12 flops)

    const Vec3 w_AB_F_dot = b_FB - b_FA; // d/dt of w_AB_F taken in F (3 flops)
    const Vec3 v_AB_F_dot =              // d/dt v_AB_F taken in F
        p_AB_F_dotdot - (b_FA % p_AB_F + w_FA % p_AB_F_dot); // (24 flops)
    
    // We have the derivative in F; change it to derivative in A by adding in 
    // contribution caused by motion of F in A, that is w_AF X w_AB_F. (Note 
    // that w_AF=-w_FA.)
    const Vec3 b_AB_F =             // ang. accel. of B in A, exp. in F
        w_AB_F_dot - w_FA % w_AB_F; // (12 flops)
    const Vec3 a_AB_F =             // taken in A, exp. in F
        v_AB_F_dot - w_FA % v_AB_F; // (12 flops)

    return SpatialVec(b_AB_F, a_AB_F); // taken in A, expressed in F
}



//==============================================================================
//                         REVERSE RELATIVE VELOCITY
//==============================================================================
inline SpatialVec reverseRelativeVelocity(const Transform&  X_AB,
                                          const SpatialVec& V_AB)
{
    // Reverse the velocity but with the result still expressed in A.
    const SpatialVec V_BA_A = reverseRelativeVelocityInA(X_AB,V_AB); 
                                                                // 21 flops
    // Then reexpress in B.
    return ~X_AB.R()*V_BA_A;                                    // 30 flops
}



//==============================================================================
//                       REVERSE RELATIVE VELOCITY IN A
//==============================================================================
inline SpatialVec reverseRelativeVelocityInA(const Transform&  X_AB,
                                             const SpatialVec& V_AB)
{
    // Change the measurement point from a point coincident with OB
    // to a point coincident with OA, and negate since we want A's velocity
    // in B rather than the other way around.
    const SpatialVec V_BA_A = -shiftVelocityBy(V_AB, -X_AB.p()); // 21 flops
    return V_BA_A;
}



//==============================================================================
//                           SHIFT VELOCITY BY
//==============================================================================
inline SpatialVec shiftVelocityBy(const SpatialVec& V_AB, const Vec3& r_A)
{   return SpatialVec( V_AB[0], V_AB[1] + V_AB[0] % r_A ); } // vp=v + wXr


//==============================================================================
//                          SHIFT VELOCITY FROM TO
//==============================================================================
inline SpatialVec shiftVelocityFromTo(const SpatialVec& V_A_BP, 
                                      const Vec3&       fromP_A,
                                      const Vec3&       toQ_A)
{   return shiftVelocityBy(V_A_BP, toQ_A - fromP_A); }



//==============================================================================
//                         SHIFT ACCELERATION BY
//==============================================================================
inline SpatialVec shiftAccelerationBy(const SpatialVec& A_AB, 
                                      const Vec3&       w_AB, 
                                      const Vec3&       r_A)
{   return SpatialVec( A_AB[0],   
                       A_AB[1] + A_AB[0] % r_A  + w_AB % (w_AB % r_A) ); } 



//==============================================================================
//                        SHIFT ACCELERATION FROM TO
//==============================================================================
inline SpatialVec shiftAccelerationFromTo(const SpatialVec& A_A_BP, 
                                          const Vec3&       w_AB,
                                          const Vec3&       fromP_A,
                                          const Vec3&       toQ_A)
{   return shiftAccelerationBy(A_A_BP, w_AB, toQ_A - fromP_A); }



//==============================================================================
//                              SHIFT FORCE BY
//==============================================================================
inline SpatialVec shiftForceBy(const SpatialVec& F_AP, const Vec3& r_A)
{   return SpatialVec(F_AP[0] -  r_A % F_AP[1], F_AP[1]); } // mq = mp - r X f



//==============================================================================
//                           SHIFT FORCE FROM TO
//==============================================================================
inline SpatialVec shiftForceFromTo(const SpatialVec& F_AP, 
                                   const Vec3&       fromP_A,
                                   const Vec3&       toQ_A)
{   return shiftForceBy(F_AP, toQ_A - fromP_A); }

//==============================================================================
//                                  PHI MATRIX
//==============================================================================
// support for efficient matrix multiplication involving the special phi
// matrix

class PhiMatrixTranspose;

class PhiMatrix {
public:
    typedef PhiMatrixTranspose TransposeType;

    PhiMatrix() { setToNaN(); }
    explicit PhiMatrix(const Vec3& l) : l_(l) {}

    void setToZero() { l_ = 0; }
    void setToNaN()  { l_.setToNaN(); }

    SpatialMat toSpatialMat() const {
        return SpatialMat(Mat33(1), crossMat(l_),
                          Mat33(0),   Mat33(1));
    }

    const Vec3& l() const { return l_; }
private:
    Vec3 l_;
};

class PhiMatrixTranspose {
public:
    explicit PhiMatrixTranspose(const PhiMatrix& phi) : phi(phi) {}

    SpatialMat toSpatialMat() const {
        return SpatialMat(   Mat33(1)    , Mat33(0),
                          crossMat(-l()) , Mat33(1));
    }

    const Vec3& l() const {return phi.l();}
private:
  const PhiMatrix& phi;
};

inline PhiMatrixTranspose
transpose(const PhiMatrix& phi)
{
    PhiMatrixTranspose ret(phi);
    return ret;
}

inline PhiMatrixTranspose
operator~(const PhiMatrix& phi) {return transpose(phi);}

inline SpatialVec
operator*(const PhiMatrix&  phi,
          const SpatialVec& v)
{
    return SpatialVec(v[0] + phi.l() % v[1], // 12 flops
                      v[1]);
}

inline SpatialMat
operator*(const PhiMatrix&  phi,
          const SpatialMat& m)
{
    const Mat33 x = crossMat(phi.l());  // 3 flops
    return SpatialMat( m(0,0) + x*m(1,0), m(0,1) + x*m(1,1), // 108 flops
                           m(1,0)       ,     m(1,1));
}

inline SpatialMat
operator*(const SpatialMat& m,
          const PhiMatrix&  phi)
{
    const Mat33 x = crossMat(phi.l());  // 3 flops
    return SpatialMat( m(0,0),    m(0,0)*x + m(0,1),   // 54 flops
                       m(1,0),    m(1,0)*x + m(1,1) ); // 54 flops
}

inline SpatialVec
operator*(const PhiMatrixTranspose& phiT,
          const SpatialVec&         v)
{
    return SpatialVec(v[0],
                      v[1] + v[0] % phiT.l());  // 12 flops
}

inline SpatialMat
operator*(const PhiMatrixTranspose& phiT,
          const SpatialMat&         m)
{
    const Mat33 x = crossMat(phiT.l());  // 3 flops
    return SpatialMat(    m(0,0)     ,     m(0,1),      
                      m(1,0)-x*m(0,0), m(1,1)-x*m(0,1)); // 108 flops
}

inline SpatialMat
operator*(const SpatialMat::THerm&  m,
          const PhiMatrixTranspose& phiT)
{
    const Mat33 x = crossMat(phiT.l()); // 3 flops
    return SpatialMat( m(0,0) - m(0,1) * x, m(0,1),     // 54 flops
                       m(1,0) - m(1,1) * x, m(1,1) );   // 54 flops
}

inline SpatialMat
operator*(const SpatialMat&         m,
          const PhiMatrixTranspose& phiT)
{
    const Mat33 x = crossMat(phiT.l()); // 3 flops
    return SpatialMat( m(0,0) - m(0,1) * x, m(0,1),     // 54 flops
                       m(1,0) - m(1,1) * x, m(1,1) );   // 54 flops
}

inline bool
operator==(const PhiMatrix& p1, const PhiMatrix& p2)
{
    return p1.l() == p2.l();
}

inline bool
operator==(const PhiMatrixTranspose& p1, const PhiMatrixTranspose& p2)
{
    return p1.l() == p2.l();
}
} // namespace SimTK

#endif // SimTK_SIMMATRIX_SPATIAL_ALGEBRA_H_