C++ Vector3s::cross方法代码示例

void StaticPlaneSphereConstraint::computeContactBasis( const VectorXs& q, const VectorXs& v, MatrixXXsc& basis ) const
{
  const Vector3s n{ m_plane.n() };
  assert( fabs( n.norm() - 1.0 ) <= 1.0e-6 );

  // Compute the relative velocity to use as a direction for the tangent sample
  Vector3s s{ computeRelativeVelocity( q, v ) };
  // If the relative velocity is zero, any vector will do
  if( n.cross( s ).squaredNorm() < 1.0e-9 )
  {
    s = FrictionUtilities::orthogonalVector( n );
  }
  // Otherwise project out the component along the normal and normalize the relative velocity
  else
  {
    s = ( s - s.dot( n ) * n ).normalized();
  }
  // Invert the tangent vector in order to oppose
  s *= -1.0;

  // Create a second orthogonal sample in the tangent plane
  const Vector3s t{ n.cross( s ).normalized() }; // Don't need to normalize but it won't hurt

  assert( MathUtilities::isRightHandedOrthoNormal( n, s, t, 1.0e-6 ) );
  basis.resize( 3, 3 );
  basis.col( 0 ) = n;
  basis.col( 1 ) = s;
  basis.col( 2 ) = t;
}

void StaticPlaneSphereConstraint::evalH( const VectorXs& q, const MatrixXXsc& basis, MatrixXXsc& H0, MatrixXXsc& H1 ) const
{
  assert( H0.rows() == 3 );
  assert( H0.cols() == 6 );
  assert( H1.rows() == 3 );
  assert( H1.cols() == 6 );

  // Grab the contact normal
  const Vector3s n{ basis.col( 0 ) };
  // Grab the tangent basis
  const Vector3s s{ basis.col( 1 ) };
  const Vector3s t{ basis.col( 2 ) };
  assert( MathUtilities::isRightHandedOrthoNormal( n, s, t, 1.0e-6 ) );

  // Compute the displacement from the center of mass to the point of contact
  assert( m_r >= 0.0 );
  const Vector3s r_world{ - m_r * n };

  H0.block<1,3>(0,0) = n;
  H0.block<1,3>(0,3).setZero();

  H0.block<1,3>(1,0) = s;
  H0.block<1,3>(1,3) = r_world.cross( s );

  H0.block<1,3>(2,0) = t;
  H0.block<1,3>(2,3) = r_world.cross( t );
}

// This method and the smooth version share the second half of code. Abstract that out.
void StaticPlaneSphereConstraint::computeGeneralizedFrictionDisk( const VectorXs& q, const VectorXs& v, const int start_column, const int num_samples, SparseMatrixsc& D, VectorXs& drel ) const
{
  assert( start_column >= 0 );
  assert( start_column < D.cols() );
  assert( num_samples > 0 );
  assert( start_column + num_samples - 1 < D.cols() );
  assert( q.size() % 12 == 0 );
  assert( q.size() == 2 * v.size() );

  const Vector3s n{ m_plane.n() };
  assert( fabs( n.norm() - 1.0 ) <= 1.0e-6 );

  std::vector<Vector3s> friction_disk( static_cast<std::vector<Vector3s>::size_type>( num_samples ) );
  assert( friction_disk.size() == std::vector<Vector3s>::size_type( num_samples ) );
  {
    // Compute the relative velocity
    Vector3s tangent_suggestion{ computeRelativeVelocity( q, v ) };
    if( tangent_suggestion.cross( n ).squaredNorm() < 1.0e-9 )
    {
      tangent_suggestion = FrictionUtilities::orthogonalVector( n );
    }
    tangent_suggestion *= -1.0;

    // Sample the friction disk
    FrictionUtilities::generateOrthogonalVectors( n, friction_disk, tangent_suggestion );
  }
  assert( unsigned( num_samples ) == friction_disk.size() );

  // Compute the displacement from the center of mass to the point of contact
  assert( fabs( n.norm() - 1.0 ) <= 1.0e-10 ); assert( m_r >= 0.0 );
  const Vector3s r_world{ - m_r * n };

  // Cache the velocity of the collision point on the plane
  const Vector3s plane_collision_point_vel{ computePlaneCollisionPointVelocity( q ) };

  // For each sample of the friction disk
  const unsigned nbodies{ static_cast<unsigned>( q.size() / 12 ) };
  for( unsigned friction_sample = 0; friction_sample < unsigned( num_samples ); ++friction_sample )
  {
    const unsigned cur_col{ start_column + friction_sample };
    assert( cur_col < unsigned( D.cols() ) );

    // Effect on center of mass
    D.insert( 3 * m_sphere_idx + 0, cur_col ) = friction_disk[friction_sample].x();
    D.insert( 3 * m_sphere_idx + 1, cur_col ) = friction_disk[friction_sample].y();
    D.insert( 3 * m_sphere_idx + 2, cur_col ) = friction_disk[friction_sample].z();

    // Effect on orientation
    {
      const Vector3s ntilde{ r_world.cross( friction_disk[friction_sample] ) };
      D.insert( 3 * ( nbodies + m_sphere_idx ) + 0, cur_col ) = ntilde.x();
      D.insert( 3 * ( nbodies + m_sphere_idx ) + 1, cur_col ) = ntilde.y();
      D.insert( 3 * ( nbodies + m_sphere_idx ) + 2, cur_col ) = ntilde.z();
    }

    // Relative velocity contribution from kinematic scripting
    assert( cur_col < drel.size() );
    drel( cur_col ) = - friction_disk[friction_sample].dot( plane_collision_point_vel );
  }
}

void StaticPlaneSphereConstraint::computeGeneralizedFrictionGivenTangentSample( const VectorXs& q, const VectorXs& t, const unsigned column, SparseMatrixsc& D ) const
{
  assert( t.size() == 3 );
  assert( column < unsigned( D.cols() ) );
  assert( q.size() % 12 == 0 );
  assert( fabs( t.norm() - 1.0 ) <= 1.0e-6 );
  assert( fabs( m_plane.n().dot( t ) ) <= 1.0e-6 );

  // Effect on center of mass
  D.insert( 3 * m_sphere_idx + 0, column ) = t.x();
  D.insert( 3 * m_sphere_idx + 1, column ) = t.y();
  D.insert( 3 * m_sphere_idx + 2, column ) = t.z();

  // Effect on orientation
  {
    const unsigned nbodies{ static_cast<unsigned>( q.size() / 12 ) };
    // Compute the displacement from the center of mass to the point of contact
    assert( fabs( m_plane.n().norm() - 1.0 ) <= 1.0e-10 );
    assert( m_r >= 0.0 );
    const Vector3s r_world{ - m_r * m_plane.n() };
    const Vector3s ntilde{ r_world.cross( Eigen::Map<const Vector3s>( t.data() ) ) };
    D.insert( 3 * ( nbodies + m_sphere_idx ) + 0, column ) = ntilde.x();
    D.insert( 3 * ( nbodies + m_sphere_idx ) + 1, column ) = ntilde.y();
    D.insert( 3 * ( nbodies + m_sphere_idx ) + 2, column ) = ntilde.z();
  }
}

void FrictionUtilities::generateOrthogonalVectors( const Vector3s& n, std::vector<Vector3s>& vectors, const Vector3s& suggested_tangent )
{
  if( vectors.empty() )
  {
    return;
  }
  assert( ( suggested_tangent.cross( n ) ).squaredNorm() != 0.0 );

  // Make sure the first vector is orthogonal to n and unit length
  vectors[0] = ( suggested_tangent - n.dot( suggested_tangent ) * n ).normalized();
  assert( fabs( vectors[0].norm() - 1.0 ) <= 1.0e-10 );
  assert( fabs( n.dot( vectors[0] ) ) <= 1.0e-10 );

  // Generate the remaining vectors by rotating the first vector about n
  const scalar dtheta{ 2.0 * MathDefines::PI<scalar>() / scalar( vectors.size() ) };
  for( std::vector<Vector3s>::size_type i = 1; i < vectors.size(); ++i )
  {
    vectors[i] = Eigen::AngleAxis<scalar>{ i * dtheta, n } * vectors[0];
    assert( fabs( vectors[i].norm() - 1.0 ) <= 1.0e-10 );
    assert( fabs( n.dot( vectors[i] ) ) <= 1.0e-10 );
  }

  #ifndef NDEBUG
  if( vectors.size() == 1 )
  {
    return;
  }
  // Check that the angle between each vector is the one we expect
  for( std::vector<Vector3s>::size_type i = 0; i < vectors.size(); ++i )
  {
    assert( fabs( angleBetweenVectors( vectors[i], vectors[( i + 1 ) % vectors.size()] ) - dtheta ) < 1.0e-6 );
  }
  #endif
}

bool MathUtilities::isRightHandedOrthoNormal( const Vector3s& a, const Vector3s& b, const Vector3s& c, const scalar& tol )
{
  // All basis vectors should be unit
  if( fabs( a.norm() - 1.0 ) > tol ) { return false; }
  if( fabs( b.norm() - 1.0 ) > tol ) { return false; }
  if( fabs( c.norm() - 1.0 ) > tol ) { return false; }
  // All basis vectors should be mutually orthogonal
  if( fabs( a.dot( b ) ) > tol ) { return false; }
  if( fabs( a.dot( c ) ) > tol ) { return false; }
  if( fabs( b.dot( c ) ) > tol ) { return false; }
  // Coordinate system should be right handed
  if( ( a.cross( b ) - c ).lpNorm<Eigen::Infinity>() > tol ) { return false; }
  return true;
}

// Unsigned angle
scalar angleBetweenVectors( const Vector3s& v0, const Vector3s& v1 )
{
  const scalar s = v0.cross( v1 ).norm();
  const scalar c = v0.dot( v1 );
  return atan2( s, c );
}