ROBOOP::Quaternion Class Reference
#include <quaternion.h>
Detailed Description
Quaternion class definition.
Definition at line 82 of file quaternion.h.
Public Member Functions | |
| Quaternion () | |
| Constructor. | |
| Quaternion (const Quaternion &q) | |
| Constructor. | |
| Quaternion (const Real angle_in_rad, const ColumnVector &axis) | |
| Constructor. | |
| Quaternion (const Real s, const Real v1, const Real v2, const Real v3) | |
| Constructor. | |
| Quaternion (const Matrix &R) | |
| Constructor. | |
| Quaternion & | operator= (const Quaternion &q) |
| Overload = operator. | |
| Quaternion | operator+ (const Quaternion &q) const |
| Overload + operator. | |
| Quaternion | operator- (const Quaternion &q) const |
| Overload - operator. | |
| Quaternion | operator * (const Quaternion &q) const |
| Overload * operator. | |
| Quaternion | operator * (const Real c) const |
| Overload * operator, multiplication by a scalar. | |
| Quaternion | operator/ (const Quaternion &q) const |
| Overload / operator. | |
| Quaternion | operator/ (const Real c) const |
| Overload / operator, division by a scalar. | |
| Quaternion | conjugate () const |
| Conjugate. | |
| Quaternion | i () const |
| Quaternion inverse.
where | |
| Quaternion & | unit () |
| Normalize a quaternion. | |
| Quaternion | exp () const |
| Exponential of a quaternion. | |
| Quaternion | power (const Real t) const |
| Quaternion | Log () const |
| Logarithm of a unit quaternion. | |
| Quaternion | dot (const ColumnVector &w, const short sign) const |
| Quaternion time derivative. | |
| ReturnMatrix | E (const short sign) const |
| Matrix E. | |
| Real | norm () const |
| Return the quaternion norm. | |
| Real | dot_prod (const Quaternion &q) const |
| Quaternion dot product. | |
| Real | s () const |
| Return scalar part. | |
| void | set_s (const Real s) |
| Set scalar part. | |
| ReturnMatrix | v () const |
| Return vector part. | |
| void | set_v (const ColumnVector &v) |
| Set quaternion vector part. | |
| ReturnMatrix | R () const |
| Rotation matrix from a unit quaternion. | |
| ReturnMatrix | T () const |
| Transformation matrix from a quaternion. | |
Private Attributes | |
| Real | s_ |
| Quaternion scalar part. | |
| ColumnVector | v_ |
| Quaternion vector part. | |
Constructor & Destructor Documentation
| ROBOOP::Quaternion::Quaternion | ( | ) |
| ROBOOP::Quaternion::Quaternion | ( | const Quaternion & | q | ) |
| ROBOOP::Quaternion::Quaternion | ( | const Real | angle_in_rad, | |
| const ColumnVector & | axis | |||
| ) |
| ROBOOP::Quaternion::Quaternion | ( | const Matrix & | R | ) |
Constructor.
Cite_: Dam. The unit quaternion obtained from a matrix (see Quaternion::R())
![\[ R(s,v) = \left[ \begin{array}{ccc} s^2+v_1^2-v_2^2-v_3^2 & 2v_1v_2+2sv_3 & 2v_1v_3-2sv_2 \\ 2v_1v_2-2sv_3 & s^2-v_1^2+v_2^2-v_3^2 & 2v_2v_3+2sv_1 \\ 2v_1v_3+2sv_2 &2v_2v_3-2sv_1 & s^2-v_1^2-v_2^2+v_3^2 \end{array} \right] \]](form_125.png)
First we find 
:
![\[ R_{11} + R_{22} + R_{33} + R_{44} = 4s^2 \]](form_127.png)
Now the other values are:
![\[ s = \pm \frac{1}{2}\sqrt{R_{11} + R_{22} + R_{33} + R_{44}} \]](form_128.png)
![\[ v_1 = \frac{R_{32}-R_{23}}{4s} \]](form_129.png)
![\[ v_2 = \frac{R_{13}-R_{31}}{4s} \]](form_130.png)
![\[ v_3 = \frac{R_{21}-R_{12}}{4s} \]](form_131.png)
The sign of cannot be determined. Depending on the choice of the sign for s the sign of 
change as well. Thus the quaternions 
and 
represent the same rotation, but the interpolation curve changed with the choice of the sign. A positive sign has been chosen.
Definition at line 117 of file quaternion.cpp.
Member Function Documentation
| Quaternion ROBOOP::Quaternion::conjugate | ( | ) | const |
Conjugate.
The conjugate of a quaternion ![$q = [s, v]$](form_143.png)
is ![$q^{*} = [s, -v]$](form_146.png)
Definition at line 325 of file quaternion.cpp.
Referenced by i().
| Quaternion ROBOOP::Quaternion::dot | ( | const ColumnVector & | w, | |
| const short | sign | |||
| ) | const |
Quaternion time derivative.
The quaternion time derivative, quaternion propagation equation, is
![\[ \dot{s} = - \frac{1}{2}v^Tw_{_0} \]](form_155.png)
![\[ \dot{v} = \frac{1}{2}E(s,v)w_{_0} \]](form_156.png)
![\[ E = sI - S(v) \]](form_157.png)
where 
is the angular velocity vector expressed in the base frame. If the vector is expressed in the object frame, 
, the time derivative becomes
![\[ \dot{s} = - \frac{1}{2}v^Tw_{_b} \]](form_160.png)
![\[ \dot{v} = \frac{1}{2}E(s,v)w_{_b} \]](form_161.png)
![\[ E = sI + S(v) \]](form_162.png)
Definition at line 430 of file quaternion.cpp.
Referenced by ROBOOP::Impedance::control().
| Real ROBOOP::Quaternion::dot_prod | ( | const Quaternion & | q | ) | const |
Quaternion dot product.
The dot product of quaternion is defined by
![\[ q_1\cdot q_2 = s_1s_2 + v_1 \cdot v_2 \]](form_163.png)
Definition at line 487 of file quaternion.cpp.
Referenced by ROBOOP::Impedance::control(), ROBOOP::Slerp(), ROBOOP::Slerp_prime(), and ROBOOP::Resolved_acc::torque_cmd().
| ReturnMatrix ROBOOP::Quaternion::E | ( | const short | sign | ) | const |
Matrix E.
See Quaternion::dot for explanation.
Definition at line 468 of file quaternion.cpp.
Referenced by ROBOOP::Impedance::control(), and ROBOOP::Omega().
| Quaternion ROBOOP::Quaternion::exp | ( | ) | const |
Exponential of a quaternion.
Let a quaternion of the form ![$q = [0, \theta v]$](form_151.png)
, q is not necessarily a unit quaternion. Then the exponential function is defined by ![$q = [\cos(\theta),v \sin(\theta)]$](form_152.png)
.
Definition at line 378 of file quaternion.cpp.
Referenced by power().
| Quaternion ROBOOP::Quaternion::i | ( | ) | const |
Quaternion inverse.
![\[ q^{-1} = \frac{q^{*}}{N(q)} \]](form_148.png)
where 
and 
are the quaternion conjugate and the quaternion norm respectively.
Definition at line 365 of file quaternion.cpp.
Referenced by ROBOOP::Impedance::control(), operator/(), Quaternion(), ROBOOP::Slerp(), and ROBOOP::Squad_prime().
| Quaternion ROBOOP::Quaternion::Log | ( | ) | const |
Logarithm of a unit quaternion.
The logarithm function of a unit quaternion ![$q = [\cos(\theta), v \sin(\theta)]$](form_153.png)
is defined as ![$log(q) = [0, v\theta]$](form_154.png)
. The result is not necessary a unit quaternion.
Definition at line 407 of file quaternion.cpp.
Referenced by power().
| Real ROBOOP::Quaternion::norm | ( | ) | const |
Return the quaternion norm.
The norm of quaternion is defined by
![\[ N(q) = s^2 + v\cdot v \]](form_147.png)
Definition at line 340 of file quaternion.cpp.
| Quaternion ROBOOP::Quaternion::operator * | ( | const Real | c | ) | const |
Overload * operator, multiplication by a scalar.
and let 
. Then ![$rq = qr = [r, 0][s, v] = [rs, rv]$](form_145.png)
The result is not necessarily a unit quaternion even if is a unit quaternions.
Definition at line 272 of file quaternion.cpp.
| Quaternion ROBOOP::Quaternion::operator * | ( | const Quaternion & | rhs | ) | const |
Overload * operator.
The multiplication of two quaternions is
![\[ q = q_1q_2 = [s_1s_2 - v_1\cdot v_2, v_1 \times v_2 + s_1v_2 + s_2v_1] \]](form_139.png)
where 
and 
denote the scalar and vector product in 
respectively.
If 
and 
are unit quaternions, then q will also be a unit quaternion.
Definition at line 249 of file quaternion.cpp.
| Quaternion ROBOOP::Quaternion::operator+ | ( | const Quaternion & | rhs | ) | const |
Overload + operator.
The quaternion addition is
![\[ q_1 + q_2 = [s_1, v_1] + [s_2, v_2] = [s_1+s_2, v_1+v_2] \]](form_135.png)
The result is not necessarily a unit quaternion even if and are unit quaternions.
Definition at line 209 of file quaternion.cpp.
| Quaternion ROBOOP::Quaternion::operator- | ( | const Quaternion & | rhs | ) | const |
Overload - operator.
The quaternion soustraction is
![\[ q_1 - q_2 = [s_1, v_1] - [s_2, v_2] = [s_1-s_2, v_1-v_2] \]](form_138.png)
The result is not necessarily a unit quaternion even if and are unit quaternions.
Definition at line 229 of file quaternion.cpp.
| Quaternion ROBOOP::Quaternion::operator/ | ( | const Real | c | ) | const |
Overload / operator, division by a scalar.
Same explanation as multiplication by scaler.
Definition at line 302 of file quaternion.cpp.
| Quaternion ROBOOP::Quaternion::operator/ | ( | const Quaternion & | q | ) | const |
| Quaternion & ROBOOP::Quaternion::operator= | ( | const Quaternion & | q | ) |
| Quaternion ROBOOP::Quaternion::power | ( | const Real | t | ) | const |
| ReturnMatrix ROBOOP::Quaternion::R | ( | ) | const |
Rotation matrix from a unit quaternion.
where 
is a vector, 
a rotation matrix and q quaternion. The rotation matrix obtained from a quaternion is then
![\[ R(s,v) = (s^2 - v^Tv)I + 2vv^T - 2s S(v) \]](form_167.png)
where 
is the cross product matrix defined by
![\[ S(u) = \left[ \begin{array}{ccc} 0 & -u_3 & u_2 \\ u_3 &0 & -u_1 \\ -u_2 & u_1 & 0 \\ \end{array} \right] \]](form_169.png)
Definition at line 500 of file quaternion.cpp.
Referenced by Quaternion(), and ROBOOP::Dynamics::set_robot_on_first_point_of_splines().
| Real ROBOOP::Quaternion::s | ( | ) | const [inline] |
Return scalar part.
Definition at line 113 of file quaternion.h.
Referenced by ROBOOP::Clik::endeff_pos_ori_err(), ROBOOP::Integ_quat(), ROBOOP::Integ_Trap_quat_s(), and ROBOOP::Resolved_acc::torque_cmd().
| void ROBOOP::Quaternion::set_s | ( | const Real | s | ) | [inline] |
Set scalar part.
Definition at line 114 of file quaternion.h.
Referenced by ROBOOP::Integ_quat(), and ROBOOP::Integ_Trap_quat_s().
| void ROBOOP::Quaternion::set_v | ( | const ColumnVector & | v | ) |
Set quaternion vector part.
Definition at line 316 of file quaternion.cpp.
Referenced by ROBOOP::Integ_quat(), and ROBOOP::Integ_Trap_quat_v().
| ReturnMatrix ROBOOP::Quaternion::T | ( | ) | const |
Transformation matrix from a quaternion.
See Quaternion::R() for equations.
Definition at line 541 of file quaternion.cpp.
| Quaternion & ROBOOP::Quaternion::unit | ( | ) |
Normalize a quaternion.
Definition at line 353 of file quaternion.cpp.
Referenced by ROBOOP::Integ_quat().
| ReturnMatrix ROBOOP::Quaternion::v | ( | ) | const [inline] |
Return vector part.
Definition at line 115 of file quaternion.h.
Referenced by ROBOOP::Impedance::control(), ROBOOP::Clik::endeff_pos_ori_err(), ROBOOP::Integ_quat(), ROBOOP::Integ_Trap_quat_v(), ROBOOP::Omega(), set_v(), and ROBOOP::Resolved_acc::torque_cmd().
Member Data Documentation
Real ROBOOP::Quaternion::s_ [private] |
Quaternion scalar part.
Definition at line 121 of file quaternion.h.
Referenced by conjugate(), dot(), exp(), Log(), norm(), operator *(), operator+(), operator-(), operator/(), operator=(), Quaternion(), R(), s(), set_s(), T(), and unit().
ColumnVector ROBOOP::Quaternion::v_ [private] |
Quaternion vector part.
Definition at line 122 of file quaternion.h.
Referenced by conjugate(), dot(), exp(), Log(), norm(), operator *(), operator+(), operator-(), operator/(), operator=(), Quaternion(), R(), set_v(), T(), unit(), and v().
The documentation for this class was generated from the following files:
