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* Copyright (C) 2015 Open Source Robotics Foundation
*
* 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.
*
*/
#ifndef IGNITION_MATH_MASSMATRIX3_HH_
#define IGNITION_MATH_MASSMATRIX3_HH_
#include <algorithm>
#include <string>
#include <vector>
#include <ignition/math/config.hh>
#include "ignition/math/Helpers.hh"
#include "ignition/math/Quaternion.hh"
#include "ignition/math/Vector2.hh"
#include "ignition/math/Vector3.hh"
#include "ignition/math/Matrix3.hh"
namespace ignition
{
namespace math
{
inline namespace IGNITION_MATH_VERSION_NAMESPACE
{
/// \class MassMatrix3 MassMatrix3.hh ignition/math/MassMatrix3.hh
/// \brief A class for inertial information about a rigid body
/// consisting of the scalar mass and a 3x3 symmetric moment
/// of inertia matrix stored as two Vector3's.
template<typename T>
class MassMatrix3
{
/// \brief Default Constructor
public: MassMatrix3() : mass(0)
{}
/// \brief Constructor.
/// \param[in] _mass Mass value in kg if using metric.
/// \param[in] _ixxyyzz Diagonal moments of inertia.
/// \param[in] _ixyxzyz Off-diagonal moments of inertia
public: MassMatrix3(const T &_mass,
const Vector3<T> &_ixxyyzz,
const Vector3<T> &_ixyxzyz)
: mass(_mass), Ixxyyzz(_ixxyyzz), Ixyxzyz(_ixyxzyz)
{}
/// \brief Copy constructor.
/// \param[in] _massMatrix MassMatrix3 element to copy
public: MassMatrix3(const MassMatrix3<T> &_m)
: mass(_m.Mass()), Ixxyyzz(_m.DiagonalMoments()),
Ixyxzyz(_m.OffDiagonalMoments())
{}
/// \brief Destructor.
public: virtual ~MassMatrix3() {}
/// \brief Set the mass.
/// \param[in] _m New mass value.
/// \return True if the MassMatrix3 is valid.
public: bool Mass(const T &_m)
{
this->mass = _m;
return this->IsValid();
}
/// \brief Get the mass
/// \return The mass value
public: T Mass() const
{
return this->mass;
}
/// \brief Set the moment of inertia matrix.
/// \param[in] _ixx X second moment of inertia (MOI) about x axis.
/// \param[in] _iyy Y second moment of inertia about y axis.
/// \param[in] _izz Z second moment of inertia about z axis.
/// \param[in] _ixy XY inertia.
/// \param[in] _ixz XZ inertia.
/// \param[in] _iyz YZ inertia.
/// \return True if the MassMatrix3 is valid.
public: bool InertiaMatrix(const T &_ixx, const T &_iyy, const T &_izz,
const T &_ixy, const T &_ixz, const T &_iyz)
{
this->Ixxyyzz.Set(_ixx, _iyy, _izz);
this->Ixyxzyz.Set(_ixy, _ixz, _iyz);
return this->IsValid();
}
/// \brief Get the diagonal moments of inertia (Ixx, Iyy, Izz).
/// \return The diagonal moments.
public: Vector3<T> DiagonalMoments() const
{
return this->Ixxyyzz;
}
/// \brief Get the off-diagonal moments of inertia (Ixy, Ixz, Iyz).
/// \return The off-diagonal moments of inertia.
public: Vector3<T> OffDiagonalMoments() const
{
return this->Ixyxzyz;
}
/// \brief Set the diagonal moments of inertia (Ixx, Iyy, Izz).
/// \param[in] _ixxyyzz diagonal moments of inertia
/// \return True if the MassMatrix3 is valid.
public: bool DiagonalMoments(const Vector3<T> &_ixxyyzz)
{
this->Ixxyyzz = _ixxyyzz;
return this->IsValid();
}
/// \brief Set the off-diagonal moments of inertia (Ixy, Ixz, Iyz).
/// \param[in] _ixyxzyz off-diagonal moments of inertia
/// \return True if the MassMatrix3 is valid.
public: bool OffDiagonalMoments(const Vector3<T> &_ixyxzyz)
{
this->Ixyxzyz = _ixyxzyz;
return this->IsValid();
}
/// \brief Get IXX
/// \return IXX value
public: T IXX() const
{
return this->Ixxyyzz[0];
}
/// \brief Get IYY
/// \return IYY value
public: T IYY() const
{
return this->Ixxyyzz[1];
}
/// \brief Get IZZ
/// \return IZZ value
public: T IZZ() const
{
return this->Ixxyyzz[2];
}
/// \brief Get IXY
/// \return IXY value
public: T IXY() const
{
return this->Ixyxzyz[0];
}
/// \brief Get IXZ
/// \return IXZ value
public: T IXZ() const
{
return this->Ixyxzyz[1];
}
/// \brief Get IYZ
/// \return IYZ value
public: T IYZ() const
{
return this->Ixyxzyz[2];
}
/// \brief Set IXX
/// \param[in] _v IXX value
/// \return True if the MassMatrix3 is valid.
public: bool IXX(const T &_v)
{
this->Ixxyyzz.X(_v);
return this->IsValid();
}
/// \brief Set IYY
/// \param[in] _v IYY value
/// \return True if the MassMatrix3 is valid.
public: bool IYY(const T &_v)
{
this->Ixxyyzz.Y(_v);
return this->IsValid();
}
/// \brief Set IZZ
/// \param[in] _v IZZ value
/// \return True if the MassMatrix3 is valid.
public: bool IZZ(const T &_v)
{
this->Ixxyyzz.Z(_v);
return this->IsValid();
}
/// \brief Set IXY
/// \param[in] _v IXY value
/// \return True if the MassMatrix3 is valid.
public: bool IXY(const T &_v)
{
this->Ixyxzyz.X(_v);
return this->IsValid();
}
/// \brief Set IXZ
/// \param[in] _v IXZ value
/// \return True if the MassMatrix3 is valid.
public: bool IXZ(const T &_v)
{
this->Ixyxzyz.Y(_v);
return this->IsValid();
}
/// \brief Set IYZ
/// \param[in] _v IYZ value
/// \return True if the MassMatrix3 is valid.
public: bool IYZ(const T &_v)
{
this->Ixyxzyz.Z(_v);
return this->IsValid();
}
/// \brief returns Moments of Inertia as a Matrix3
/// \return Moments of Inertia as a Matrix3
public: Matrix3<T> MOI() const
{
return Matrix3<T>(
this->Ixxyyzz[0], this->Ixyxzyz[0], this->Ixyxzyz[1],
this->Ixyxzyz[0], this->Ixxyyzz[1], this->Ixyxzyz[2],
this->Ixyxzyz[1], this->Ixyxzyz[2], this->Ixxyyzz[2]);
}
/// \brief Sets Moments of Inertia (MOI) from a Matrix3.
/// Symmetric component of input matrix is used by averaging
/// off-axis terms.
/// \param[in] Moments of Inertia as a Matrix3
/// \return True if the MassMatrix3 is valid.
public: bool MOI(const Matrix3<T> &_moi)
{
this->Ixxyyzz.Set(_moi(0, 0), _moi(1, 1), _moi(2, 2));
this->Ixyxzyz.Set(
0.5*(_moi(0, 1) + _moi(1, 0)),
0.5*(_moi(0, 2) + _moi(2, 0)),
0.5*(_moi(1, 2) + _moi(2, 1)));
return this->IsValid();
}
/// \brief Equal operator.
/// \param[in] _massMatrix MassMatrix3 to copy.
/// \return Reference to this object.
public: MassMatrix3 &operator=(const MassMatrix3<T> &_massMatrix)
{
this->mass = _massMatrix.Mass();
this->Ixxyyzz = _massMatrix.DiagonalMoments();
this->Ixyxzyz = _massMatrix.OffDiagonalMoments();
return *this;
}
/// \brief Equality comparison operator.
/// \param[in] _m MassMatrix3 to copy.
/// \return true if each component is equal within a default tolerance,
/// false otherwise
public: bool operator==(const MassMatrix3<T> &_m) const
{
return equal<T>(this->mass, _m.Mass()) &&
(this->Ixxyyzz == _m.DiagonalMoments()) &&
(this->Ixyxzyz == _m.OffDiagonalMoments());
}
/// \brief Inequality test operator
/// \param[in] _m MassMatrix3<T> to test
/// \return True if not equal (using the default tolerance of 1e-6)
public: bool operator!=(const MassMatrix3<T> &_m) const
{
return !(*this == _m);
}
/// \brief Verify that inertia values are positive definite
/// \return True if mass is positive and moment of inertia matrix
/// is positive definite.
public: bool IsPositive() const
{
// Check if mass and determinants of all upper left submatrices
// of moment of inertia matrix are positive
return (this->mass > 0) &&
(this->IXX() > 0) &&
(this->IXX()*this->IYY() - std::pow(this->IXY(), 2) > 0) &&
(this->MOI().Determinant() > 0);
}
/// \brief Verify that inertia values are positive definite
/// and satisfy the triangle inequality.
/// \return True if IsPositive and moment of inertia satisfies
/// the triangle inequality.
public: bool IsValid() const
{
return this->IsPositive() && ValidMoments(this->PrincipalMoments());
}
/// \brief Verify that principal moments are positive
/// and satisfy the triangle inequality.
/// \param[in] _moments Principal moments of inertia.
/// \return True if moments of inertia are positive
/// and satisfy the triangle inequality.
public: static bool ValidMoments(const Vector3<T> &_moments)
{
return _moments[0] > 0 &&
_moments[1] > 0 &&
_moments[2] > 0 &&
_moments[0] + _moments[1] > _moments[2] &&
_moments[1] + _moments[2] > _moments[0] &&
_moments[2] + _moments[0] > _moments[1];
}
/// \brief Compute principal moments of inertia,
/// which are the eigenvalues of the moment of inertia matrix.
/// \param[in] _tol Relative tolerance given by absolute value
/// of _tol.
/// Negative values of _tol are interpreted as a flag that
/// causes principal moments to always be sorted from smallest
/// to largest.
/// \return Principal moments of inertia.
/// If the matrix is already diagonal and _tol is positive,
/// they are returned in the existing order.
/// Otherwise, the moments are sorted from smallest to largest.
public: Vector3<T> PrincipalMoments(const T _tol = 1e-6) const
{
// Compute tolerance relative to maximum value of inertia diagonal
T tol = _tol * this->Ixxyyzz.Max();
if (this->Ixyxzyz.Equal(Vector3<T>::Zero, tol))
{
// Matrix is already diagonalized, return diagonal moments
return this->Ixxyyzz;
}
// Algorithm based on http://arxiv.org/abs/1306.6291v4
// A Method for Fast Diagonalization of a 2x2 or 3x3 Real Symmetric
// Matrix, by Maarten Kronenburg
Vector3<T> Id(this->Ixxyyzz);
Vector3<T> Ip(this->Ixyxzyz);
// b = Ixx + Iyy + Izz
T b = Id.Sum();
// c = Ixx*Iyy - Ixy^2 + Ixx*Izz - Ixz^2 + Iyy*Izz - Iyz^2
T c = Id[0]*Id[1] - std::pow(Ip[0], 2)
+ Id[0]*Id[2] - std::pow(Ip[1], 2)
+ Id[1]*Id[2] - std::pow(Ip[2], 2);
// d = Ixx*Iyz^2 + Iyy*Ixz^2 + Izz*Ixy^2 - Ixx*Iyy*Izz - 2*Ixy*Ixz*Iyz
T d = Id[0]*std::pow(Ip[2], 2)
+ Id[1]*std::pow(Ip[1], 2)
+ Id[2]*std::pow(Ip[0], 2)
- Id[0]*Id[1]*Id[2]
- 2*Ip[0]*Ip[1]*Ip[2];
// p = b^2 - 3c
T p = std::pow(b, 2) - 3*c;
// At this point, it is important to check that p is not close
// to zero, since its inverse is used to compute delta.
// In equation 4.7, p is expressed as a sum of squares
// that is only zero if the matrix is diagonal
// with identical principal moments.
// This check has no test coverage, since this function returns
// immediately if a diagonal matrix is detected.
if (p < std::pow(tol, 2))
return b / 3.0 * Vector3<T>::One;
// q = 2b^3 - 9bc - 27d
T q = 2*std::pow(b, 3) - 9*b*c - 27*d;
// delta = acos(q / (2 * p^(1.5)))
// additionally clamp the argument to [-1,1]
T delta = acos(clamp<T>(0.5 * q / std::pow(p, 1.5), -1, 1));
// sort the moments from smallest to largest
T moment0 = (b + 2*sqrt(p) * cos(delta / 3.0)) / 3.0;
T moment1 = (b + 2*sqrt(p) * cos((delta + 2*IGN_PI)/3.0)) / 3.0;
T moment2 = (b + 2*sqrt(p) * cos((delta - 2*IGN_PI)/3.0)) / 3.0;
sort3(moment0, moment1, moment2);
return Vector3<T>(moment0, moment1, moment2);
}
/// \brief Compute rotational offset of principal axes.
/// \param[in] _tol Relative tolerance given by absolute value
/// of _tol.
/// Negative values of _tol are interpreted as a flag that
/// causes principal moments to always be sorted from smallest
/// to largest.
/// \return Quaternion representing rotational offset of principal axes.
/// With a rotation matrix constructed from this quaternion R(q)
/// and a diagonal matrix L with principal moments on the diagonal,
/// the original moment of inertia matrix MOI can be reconstructed
/// with MOI = R(q).Transpose() * L * R(q)
public: Quaternion<T> PrincipalAxesOffset(const T _tol = 1e-6) const
{
// Compute tolerance relative to maximum value of inertia diagonal
T tol = _tol * this->Ixxyyzz.Max();
Vector3<T> moments = this->PrincipalMoments(tol);
if (moments.Equal(this->Ixxyyzz, tol) ||
(math::equal<T>(moments[0], moments[1], std::abs(tol)) &&
math::equal<T>(moments[0], moments[2], std::abs(tol))))
{
// matrix is already aligned with principal axes
// or all three moments are approximately equal
// return identity rotation
return Quaternion<T>::Identity;
}
// Algorithm based on http://arxiv.org/abs/1306.6291v4
// A Method for Fast Diagonalization of a 2x2 or 3x3 Real Symmetric
// Matrix, by Maarten Kronenburg
// A real, symmetric matrix can be diagonalized by an orthogonal matrix
// (due to the finite-dimensional spectral theorem
// https://en.wikipedia.org/wiki/Spectral_theorem
// #Hermitian_maps_and_Hermitian_matrices ),
// and another name for orthogonal matrix is rotation matrix.
// Section 5 of the paper shows how to compute Euler angles
// phi1, phi2, and phi3 that map to a rotation matrix.
// In some cases, there are multiple possible values for a given angle,
// such as phi1, that are denoted as phi11, phi12, phi11a, phi12a, etc.
// Similar variable names are used to the paper so that the paper
// can be used as an additional reference.
// f1, f2 defined in equations 5.5, 5.6
Vector2<T> f1(this->Ixyxzyz[0], -this->Ixyxzyz[1]);
Vector2<T> f2(this->Ixxyyzz[1] - this->Ixxyyzz[2],
-2*this->Ixyxzyz[2]);
// Check if two moments are equal, since different equations are used
// The moments vector is already sorted, so just check adjacent values.
Vector2<T> momentsDiff(moments[0] - moments[1],
moments[1] - moments[2]);
// index of unequal moment
int unequalMoment = -1;
if (equal<T>(momentsDiff[0], 0, std::abs(tol)))
unequalMoment = 2;
else if (equal<T>(momentsDiff[1], 0, std::abs(tol)))
unequalMoment = 0;
if (unequalMoment >= 0)
{
// moments[1] is the repeated value
// it is not equal to moments[unequalMoment]
// momentsDiff3 = lambda - lambda3
T momentsDiff3 = moments[1] - moments[unequalMoment];
// eq 5.21:
// s = cos(phi2)^2 = (A_11 - lambda3) / (lambda - lambda3)
// s >= 0 since A_11 is in range [lambda, lambda3]
T s = (this->Ixxyyzz[0] - moments[unequalMoment]) / momentsDiff3;
// set phi3 to zero for repeated moments (eq 5.23)
T phi3 = 0;
// phi2 = +- acos(sqrt(s))
// start with just the positive value
// also clamp the acos argument to prevent NaN's
T phi2 = acos(clamp<T>(ClampedSqrt(s), -1, 1));
// The paper defines variables phi11 and phi12
// which are candidate values of angle phi1.
// phi12 is straightforward to compute as a function of f2 and g2.
// eq 5.25:
Vector2<T> g2(momentsDiff3 * s, 0);
// combining eq 5.12 and 5.14, and subtracting psi2
// instead of multiplying by its rotation matrix:
math::Angle phi12(0.5*(Angle2(g2, tol) - Angle2(f2, tol)));
phi12.Normalize();
// The paragraph prior to equation 5.16 describes how to choose
// the candidate value of phi1 based on the length
// of the f1 and f2 vectors.
// * When |f1| != 0 and |f2| != 0, then one should choose the
// value of phi2 so that phi11 = phi12
// * When |f1| == 0 and f2 != 0, then phi1 = phi12
// phi11 can be ignored, and either sign of phi2 can be used
// * The case of |f2| == 0 can be ignored at this point in the code
// since having a repeated moment when |f2| == 0 implies that
// the matrix is diagonal. But this function returns a unit
// quaternion for diagonal matrices, so we can assume |f2| != 0
// See MassMatrix3.ipynb for a more complete discussion.
//
// Since |f2| != 0, we only need to consider |f1|
// * |f1| == 0: phi1 = phi12
// * |f1| != 0: choose phi2 so that phi11 == phi12
// In either case, phi1 = phi12,
// and the sign of phi2 must be chosen to make phi11 == phi12
T phi1 = phi12.Radian();
bool f1small = f1.SquaredLength() < std::pow(tol, 2);
if (!f1small)
{
// a: phi2 > 0
// eq. 5.24
Vector2<T> g1a(0, 0.5*momentsDiff3 * sin(2*phi2));
// combining eq 5.11 and 5.13, and subtracting psi1
// instead of multiplying by its rotation matrix:
math::Angle phi11a(Angle2(g1a, tol) - Angle2(f1, tol));
phi11a.Normalize();
// b: phi2 < 0
// eq. 5.24
Vector2<T> g1b(0, 0.5*momentsDiff3 * sin(-2*phi2));
// combining eq 5.11 and 5.13, and subtracting psi1
// instead of multiplying by its rotation matrix:
math::Angle phi11b(Angle2(g1b, tol) - Angle2(f1, tol));
phi11b.Normalize();
// choose sign of phi2
// based on whether phi11a or phi11b is closer to phi12
// use sin and cos to account for angle wrapping
T erra = std::pow(sin(phi1) - sin(phi11a.Radian()), 2)
+ std::pow(cos(phi1) - cos(phi11a.Radian()), 2);
T errb = std::pow(sin(phi1) - sin(phi11b.Radian()), 2)
+ std::pow(cos(phi1) - cos(phi11b.Radian()), 2);
if (errb < erra)
{
phi2 *= -1;
}
}
// I determined these arguments using trial and error
Quaternion<T> result = Quaternion<T>(-phi1, -phi2, -phi3).Inverse();
// Previous equations assume repeated moments are at the beginning
// of the moments vector (moments[0] == moments[1]).
// We have the vectors sorted by size, so it's possible that the
// repeated moments are at the end (moments[1] == moments[2]).
// In this case (unequalMoment == 0), we apply an extra
// rotation that exchanges moment[0] and moment[2]
// Rotation matrix = [ 0 0 1]
// [ 0 1 0]
// [-1 0 0]
// That is equivalent to a 90 degree pitch
if (unequalMoment == 0)
result *= Quaternion<T>(0, IGN_PI_2, 0);
return result;
}
// No repeated principal moments
// eq 5.1:
T v = (std::pow(this->Ixyxzyz[0], 2) + std::pow(this->Ixyxzyz[1], 2)
+(this->Ixxyyzz[0] - moments[2])
*(this->Ixxyyzz[0] + moments[2] - moments[0] - moments[1]))
/ ((moments[1] - moments[2]) * (moments[2] - moments[0]));
// value of w depends on v
T w;
if (v < std::abs(tol))
{
// first sentence after eq 5.4:
// "In the case that v = 0, then w = 1."
w = 1;
}
else
{
// eq 5.2:
w = (this->Ixxyyzz[0] - moments[2] + (moments[2] - moments[1])*v)
/ ((moments[0] - moments[1]) * v);
}
// initialize values of angle phi1, phi2, phi3
T phi1 = 0;
// eq 5.3: start with positive value
T phi2 = acos(clamp<T>(ClampedSqrt(v), -1, 1));
// eq 5.4: start with positive value
T phi3 = acos(clamp<T>(ClampedSqrt(w), -1, 1));
// compute g1, g2 for phi2,phi3 >= 0
// equations 5.7, 5.8
Vector2<T> g1(
0.5* (moments[0]-moments[1])*ClampedSqrt(v)*sin(2*phi3),
0.5*((moments[0]-moments[1])*w + moments[1]-moments[2])*sin(2*phi2));
Vector2<T> g2(
(moments[0]-moments[1])*(1 + (v-2)*w) + (moments[1]-moments[2])*v,
(moments[0]-moments[1])*sin(phi2)*sin(2*phi3));
// The paragraph prior to equation 5.16 describes how to choose
// the candidate value of phi1 based on the length
// of the f1 and f2 vectors.
// * The case of |f1| == |f2| == 0 implies a repeated moment,
// which should not be possible at this point in the code
// * When |f1| != 0 and |f2| != 0, then one should choose the
// value of phi2 so that phi11 = phi12
// * When |f1| == 0 and f2 != 0, then phi1 = phi12
// phi11 can be ignored, and either sign of phi2, phi3 can be used
// * When |f2| == 0 and f1 != 0, then phi1 = phi11
// phi12 can be ignored, and either sign of phi2, phi3 can be used
bool f1small = f1.SquaredLength() < std::pow(tol, 2);
bool f2small = f2.SquaredLength() < std::pow(tol, 2);
if (f1small && f2small)
{
// this should never happen
// f1small && f2small implies a repeated moment
// return invalid quaternion
/// \todo Use a mock class to test this line
return Quaternion<T>::Zero;
}
else if (f1small)
{
// use phi12 (equations 5.12, 5.14)
math::Angle phi12(0.5*(Angle2(g2, tol) - Angle2(f2, tol)));
phi12.Normalize();
phi1 = phi12.Radian();
}
else if (f2small)
{
// use phi11 (equations 5.11, 5.13)
math::Angle phi11(Angle2(g1, tol) - Angle2(f1, tol));
phi11.Normalize();
phi1 = phi11.Radian();
}
else
{
// check for when phi11 == phi12
// eqs 5.11, 5.13:
math::Angle phi11(Angle2(g1, tol) - Angle2(f1, tol));
phi11.Normalize();
// eqs 5.12, 5.14:
math::Angle phi12(0.5*(Angle2(g2, tol) - Angle2(f2, tol)));
phi12.Normalize();
T err = std::pow(sin(phi11.Radian()) - sin(phi12.Radian()), 2)
+ std::pow(cos(phi11.Radian()) - cos(phi12.Radian()), 2);
phi1 = phi11.Radian();
math::Vector2<T> signsPhi23(1, 1);
// case a: phi2 <= 0
{
Vector2<T> g1a = Vector2<T>(1, -1) * g1;
Vector2<T> g2a = Vector2<T>(1, -1) * g2;
math::Angle phi11a(Angle2(g1a, tol) - Angle2(f1, tol));
math::Angle phi12a(0.5*(Angle2(g2a, tol) - Angle2(f2, tol)));
phi11a.Normalize();
phi12a.Normalize();
T erra = std::pow(sin(phi11a.Radian()) - sin(phi12a.Radian()), 2)
+ std::pow(cos(phi11a.Radian()) - cos(phi12a.Radian()), 2);
if (erra < err)
{
err = erra;
phi1 = phi11a.Radian();
signsPhi23.Set(-1, 1);
}
}
// case b: phi3 <= 0
{
Vector2<T> g1b = Vector2<T>(-1, 1) * g1;
Vector2<T> g2b = Vector2<T>(1, -1) * g2;
math::Angle phi11b(Angle2(g1b, tol) - Angle2(f1, tol));
math::Angle phi12b(0.5*(Angle2(g2b, tol) - Angle2(f2, tol)));
phi11b.Normalize();
phi12b.Normalize();
T errb = std::pow(sin(phi11b.Radian()) - sin(phi12b.Radian()), 2)
+ std::pow(cos(phi11b.Radian()) - cos(phi12b.Radian()), 2);
if (errb < err)
{
err = errb;
phi1 = phi11b.Radian();
signsPhi23.Set(1, -1);
}
}
// case c: phi2,phi3 <= 0
{
Vector2<T> g1c = Vector2<T>(-1, -1) * g1;
Vector2<T> g2c = g2;
math::Angle phi11c(Angle2(g1c, tol) - Angle2(f1, tol));
math::Angle phi12c(0.5*(Angle2(g2c, tol) - Angle2(f2, tol)));
phi11c.Normalize();
phi12c.Normalize();
T errc = std::pow(sin(phi11c.Radian()) - sin(phi12c.Radian()), 2)
+ std::pow(cos(phi11c.Radian()) - cos(phi12c.Radian()), 2);
if (errc < err)
{
err = errc;
phi1 = phi11c.Radian();
signsPhi23.Set(-1, -1);
}
}
// apply sign changes
phi2 *= signsPhi23[0];
phi3 *= signsPhi23[1];
}
// I determined these arguments using trial and error
return Quaternion<T>(-phi1, -phi2, -phi3).Inverse();
}
/// \brief Get dimensions and rotation offset of uniform box
/// with equivalent mass and moment of inertia.
/// To compute this, the Matrix3 is diagonalized.
/// The eigenvalues on the diagonal and the rotation offset
/// of the principal axes are returned.
/// \param[in] _size Dimensions of box aligned with principal axes.
/// \param[in] _rot Rotational offset of principal axes.
/// \param[in] _tol Relative tolerance.
/// \return True if box properties were computed successfully.
public: bool EquivalentBox(Vector3<T> &_size,
Quaternion<T> &_rot,
const T _tol = 1e-6) const
{
if (!this->IsPositive())
{
// inertia is not positive, cannot compute equivalent box
return false;
}
Vector3<T> moments = this->PrincipalMoments(_tol);
if (!ValidMoments(moments))
{
// principal moments don't satisfy the triangle identity
return false;
}
// The reason for checking that the principal moments satisfy
// the triangle inequality
// I1 + I2 - I3 >= 0
// is to ensure that the arguments to sqrt in these equations
// are positive and the box size is real.
_size.X(sqrt(6*(moments.Y() + moments.Z() - moments.X()) / this->mass));
_size.Y(sqrt(6*(moments.Z() + moments.X() - moments.Y()) / this->mass));
_size.Z(sqrt(6*(moments.X() + moments.Y() - moments.Z()) / this->mass));
_rot = this->PrincipalAxesOffset(_tol);
if (_rot == Quaternion<T>::Zero)
{
// _rot is an invalid quaternion
/// \todo Use a mock class to test this line
return false;
}
return true;
}
/// \brief Set inertial properties based on mass and equivalent box.
/// \param[in] _mass Mass to set.
/// \param[in] _size Size of equivalent box.
/// \param[in] _rot Rotational offset of equivalent box.
/// \return True if inertial properties were set successfully.
public: bool SetFromBox(const T _mass,
const Vector3<T> &_size,
const Quaternion<T> &_rot = Quaternion<T>::Identity)
{
// Check that _mass and _size are strictly positive
// and that quatenion is valid
if (_mass <= 0 || _size.Min() <= 0 || _rot == Quaternion<T>::Zero)
{
return false;
}
this->Mass(_mass);
return this->SetFromBox(_size, _rot);
}
/// \brief Set inertial properties based on equivalent box
/// using the current mass value.
/// \param[in] _size Size of equivalent box.
/// \param[in] _rot Rotational offset of equivalent box.
/// \return True if inertial properties were set successfully.
public: bool SetFromBox(const Vector3<T> &_size,
const Quaternion<T> &_rot = Quaternion<T>::Identity)
{
// Check that _mass and _size are strictly positive
// and that quatenion is valid
if (this->Mass() <= 0 || _size.Min() <= 0 ||
_rot == Quaternion<T>::Zero)
{
return false;
}
// Diagonal matrix L with principal moments
Matrix3<T> L;
T x2 = std::pow(_size.X(), 2);
T y2 = std::pow(_size.Y(), 2);
T z2 = std::pow(_size.Z(), 2);
L(0, 0) = this->mass / 12.0 * (y2 + z2);
L(1, 1) = this->mass / 12.0 * (z2 + x2);
L(2, 2) = this->mass / 12.0 * (x2 + y2);
Matrix3<T> R(_rot);
return this->MOI(R * L * R.Transposed());
}
/// \brief Set inertial properties based on mass and equivalent cylinder
/// aligned with Z axis.
/// \param[in] _mass Mass to set.
/// \param[in] _length Length of cylinder along Z axis.
/// \param[in] _radius Radius of cylinder.
/// \param[in] _rot Rotational offset of equivalent cylinder.
/// \return True if inertial properties were set successfully.
public: bool SetFromCylinderZ(const T _mass,
const T _length,
const T _radius,
const Quaternion<T> &_rot = Quaternion<T>::Identity)
{
// Check that _mass, _radius and _length are strictly positive
// and that quatenion is valid
if (_mass <= 0 || _length <= 0 || _radius <= 0 ||
_rot == Quaternion<T>::Zero)
{
return false;
}
this->Mass(_mass);
return this->SetFromCylinderZ(_length, _radius, _rot);
}
/// \brief Set inertial properties based on equivalent cylinder
/// aligned with Z axis using the current mass value.
/// \param[in] _length Length of cylinder along Z axis.
/// \param[in] _radius Radius of cylinder.
/// \param[in] _rot Rotational offset of equivalent cylinder.
/// \return True if inertial properties were set successfully.
public: bool SetFromCylinderZ(const T _length,
const T _radius,
const Quaternion<T> &_rot)
{
// Check that _mass and _size are strictly positive
// and that quatenion is valid
if (this->Mass() <= 0 || _length <= 0 || _radius <= 0 ||
_rot == Quaternion<T>::Zero)
{
return false;
}
// Diagonal matrix L with principal moments
T radius2 = std::pow(_radius, 2);
Matrix3<T> L;
L(0, 0) = this->mass / 12.0 * (3*radius2 + std::pow(_length, 2));
L(1, 1) = L(0, 0);
L(2, 2) = this->mass / 2.0 * radius2;
Matrix3<T> R(_rot);
return this->MOI(R * L * R.Transposed());
}
/// \brief Set inertial properties based on mass and equivalent sphere.
/// \param[in] _mass Mass to set.
/// \param[in] _radius Radius of equivalent, uniform sphere.
/// \return True if inertial properties were set successfully.
public: bool SetFromSphere(const T _mass, const T _radius)
{
// Check that _mass and _radius are strictly positive
if (_mass <= 0 || _radius <= 0)
{
return false;
}
this->Mass(_mass);
return this->SetFromSphere(_radius);
}
/// \brief Set inertial properties based on equivalent sphere
/// using the current mass value.
/// \param[in] _radius Radius of equivalent, uniform sphere.
/// \return True if inertial properties were set successfully.
public: bool SetFromSphere(const T _radius)
{
// Check that _mass and _radius are strictly positive
if (this->Mass() <= 0 || _radius <= 0)
{
return false;
}
// Diagonal matrix L with principal moments
T radius2 = std::pow(_radius, 2);
Matrix3<T> L;
L(0, 0) = 0.4 * this->mass * radius2;
L(1, 1) = 0.4 * this->mass * radius2;
L(2, 2) = 0.4 * this->mass * radius2;
return this->MOI(L);
}
/// \brief Square root of positive numbers, otherwise zero.
/// \param[in] _x Number to be square rooted.
/// \return sqrt(_x) if _x > 0, otherwise 0
private: static inline T ClampedSqrt(const T &_x)
{
if (_x <= 0)
return 0;
return sqrt(_x);
}
/// \brief Angle formed by direction of a Vector2.
/// \param[in] _v Vector whose direction is to be computed.
/// \param[in] _eps Minimum length of vector required for computing angle.
/// \return Angle formed between vector and X axis,
/// or zero if vector has length less than 1e-6.
private: static T Angle2(const Vector2<T> &_v, const T _eps = 1e-6)
{
if (_v.SquaredLength() < std::pow(_eps, 2))
return 0;
return atan2(_v[1], _v[0]);
}
/// \brief Mass of the object. Default is 0.0.
private: T mass;
/// \brief Principal moments of inertia. Default is (0.0 0.0 0.0)
/// These Moments of Inertia are specified in the local frame.
/// Where Ixxyyzz.x is Ixx, Ixxyyzz.y is Iyy and Ixxyyzz.z is Izz.
private: Vector3<T> Ixxyyzz;
/// \brief Product moments of inertia. Default is (0.0 0.0 0.0)
/// These MOI off-diagonals are specified in the local frame.
/// Where Ixyxzyz.x is Ixy, Ixyxzyz.y is Ixz and Ixyxzyz.z is Iyz.
private: Vector3<T> Ixyxzyz;
};
typedef MassMatrix3<double> MassMatrix3d;
typedef MassMatrix3<float> MassMatrix3f;
}
}
}
#endif
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