Drake
FreeBody Class Reference

The purpose of the FreeBody class is to provide the data (initial values and gravity) and methods for calculating the exact analytical solution for the translational and rotational motion of a torque-free rigid body B with axially symmetric inertia, in a Newtonian frame (World) N. More...

#include <drake/multibody/benchmarks/free_body/free_body.h>

## Public Member Functions

FreeBody (const Eigen::Quaterniond &initial_quat_NB, const Eigen::Vector3d &initial_w_NB_B, const Eigen::Vector3d &initial_p_NoBcm_N, const Eigen::Vector3d &initial_v_NBcm_B, const Eigen::Vector3d &gravity_N)
Constructs a class that can be queried for exact values of orientation, position, and motion of a torque-free rigid body at time t. More...

~FreeBody ()=default

double get_I () const
Returns the body's moment of inertia about an axis perpendicular to its axis of rotation and passing through its center of mass. More...

double get_J () const
Returns body's moment of inertia about its axis of rotation. More...

const Eigen::Quaterniond & get_initial_quat_NB () const

const Eigen::Vector3d & get_initial_w_NB_B () const

Eigen::Vector3d CalcInitial_w_NB_N () const

const Eigen::Vector3d & get_initial_p_NoBcm_N () const

const Eigen::Vector3d & get_uniform_gravity_expressed_in_world () const

Eigen::Vector3d CalcInitial_v_NBcm_N () const

void set_initial_quat_NB (const Eigen::Quaterniond &quat_NB)

void set_initial_w_NB_B (const Eigen::Vector3d &w_NB_B)

void set_initial_p_NoBcm_N (const Eigen::Vector3d &p_NoBcm_N)

void set_initial_v_NBcm_B (const Eigen::Vector3d &v_NBcm_B)

void SetUniformGravityExpressedInWorld (const Eigen::Vector3d &gravity)

std::tuple< Eigen::Quaterniond, Eigen::Vector4d, Eigen::Vector3d, Eigen::Vector3d > CalculateExactRotationalSolutionNB (const double t) const
Calculates exact solutions for quaternion and angular velocity expressed in body-frame, and their time derivatives for torque-free rotational motion of axis-symmetric rigid body B in Newtonian frame (World) N, where torque-free means the moment of forces about B's mass center is zero. More...

std::tuple< Eigen::Vector3d, Eigen::Vector3d, Eigen::Vector3d > CalculateExactTranslationalSolution (const double t) const
Calculates exact solutions for translational motion of an arbitrary rigid body B in a Newtonian frame (world) N. More...

Implements CopyConstructible, CopyAssignable, MoveConstructible, MoveAssignable
FreeBody (const FreeBody &)=default

FreeBodyoperator= (const FreeBody &)=default

FreeBody (FreeBody &&)=default

FreeBodyoperator= (FreeBody &&)=default

## Detailed Description

The purpose of the FreeBody class is to provide the data (initial values and gravity) and methods for calculating the exact analytical solution for the translational and rotational motion of a torque-free rigid body B with axially symmetric inertia, in a Newtonian frame (World) N.

Examples of bodies with axially symmetric inertia include cylinders, rods or bars with a circular or square cross section and spinning tops. Since the only external forces on B are uniform gravitational forces, there exists an exact closed-form analytical solution for B's motion. The closed- form rotational solution is available since B is "torque-free", i.e., the moment of all forces about B's mass center is zero. This class calculates the body B's quaternion, angular velocity and angular acceleration expressed in B (body-frame) as well as the position, velocity, acceleration of Bcm (B's center of mass) in N (World). Algorithm from [Kane, 1983] Sections 1.13 and 3.1, Pages 60-62 and 159-169.

## ◆ FreeBody() [1/3]

 FreeBody ( const FreeBody & )
default

## ◆ FreeBody() [2/3]

 FreeBody ( FreeBody && )
default

## ◆ FreeBody() [3/3]

 FreeBody ( const Eigen::Quaterniond & initial_quat_NB, const Eigen::Vector3d & initial_w_NB_B, const Eigen::Vector3d & initial_p_NoBcm_N, const Eigen::Vector3d & initial_v_NBcm_B, const Eigen::Vector3d & gravity_N )
inline

Constructs a class that can be queried for exact values of orientation, position, and motion of a torque-free rigid body at time t.

Parameters
 [in] initial_quat_NB Value at time t = 0 of the quaternion relating right-handed orthonormal vectors Nx, Ny, Nz fixed in N (world) to right-handed orthonormal unit vectors Bx, By, Bz fixed in B (body). Note: The unit vector Bz is parallel to body B's symmetry axis. Note: The quaternion should already be normalized before it is passed. [in] initial_W_NB_B Value at time t = 0 of the angular velocity in N of body B, expressed in N. [in] initial_p_NoBcm_N Value at time t = 0 of the position vector from No (origin of world N) to Bcm (B's center of mass), expressed in N. [in] initial_v_NBcm_N Value at time t = 0 of the velocity in N of Bcm (B's center of mass), expressed in N. [in] gravity_N Local gravitational acceleration, expressed in N.

## ◆ ~FreeBody()

 ~FreeBody ( )
default

## ◆ CalcInitial_v_NBcm_N()

 Eigen::Vector3d CalcInitial_v_NBcm_N ( ) const
inline

## ◆ CalcInitial_w_NB_N()

 Eigen::Vector3d CalcInitial_w_NB_N ( ) const
inline

## ◆ CalculateExactRotationalSolutionNB()

 std::tuple< Quaterniond, Vector4d, Vector3d, Vector3d > CalculateExactRotationalSolutionNB ( const double t ) const

Calculates exact solutions for quaternion and angular velocity expressed in body-frame, and their time derivatives for torque-free rotational motion of axis-symmetric rigid body B in Newtonian frame (World) N, where torque-free means the moment of forces about B's mass center is zero.

The quaternion characterizes the orientation between right-handed orthogonal unit vectors Nx, Ny, Nz fixed in N and right-handed orthogonal unit vectors Bx, By, Bz fixed in B, where Bz is parallel to B's symmetry axis.

Note
CalculateExactRotationalSolutionABInitiallyAligned() implements the algorithm from [Kane, 1983] Sections 1.13 and 3.1, Pages 60-62 and 159-169.
Parameters
 t Current value of time.
Returns
Machine-precision values at time t are returned as defined below.
Note
This function allows for initial misalignment of Nx, Ny, Nz and Bx, By, Bz.
std::tuple Description
quat_NB Quaternion relating frame N to frame B: [e0, e1, e2, e3]

| Note: quat_NB is analogous to the rotation matrix R_NB. quatDt | Time-derivative of `quat_NB', i.e., [ė0, ė1, ė2, ė3]. w_NB_B | B's angular velocity in N, expressed in B. alpha_NB_B | B's angular acceleration in N, expressed in B.

## ◆ CalculateExactTranslationalSolution()

 std::tuple< Vector3d, Vector3d, Vector3d > CalculateExactTranslationalSolution ( const double t ) const

Calculates exact solutions for translational motion of an arbitrary rigid body B in a Newtonian frame (world) N.

Algorithm from high-school physics.

Parameters
 t Current value of time.
Returns
Machine-precision values at time t are returned as defined below.
std::tuple Description
xyz Vector3d [x, y, z], Bcm's position from No, expressed in N.
xyzDt Vector3d [ẋ, ẏ, ż] Bcm's velocity in N, expressed in N.
xyzDDt Vector3d [ẍ ÿ z̈], Bcm's acceleration in N, expressed in N.

## ◆ get_I()

 double get_I ( ) const
inline

Returns the body's moment of inertia about an axis perpendicular to its axis of rotation and passing through its center of mass.

For example, for a cylinder of radius r, length h and uniformly distributed mass m with its rotational axis aligined along its body frame z-axis this would be: I = Ixx = Iyy = m / 12 (3 r² + h²)

## ◆ get_initial_p_NoBcm_N()

 const Eigen::Vector3d& get_initial_p_NoBcm_N ( ) const
inline

## ◆ get_initial_quat_NB()

 const Eigen::Quaterniond& get_initial_quat_NB ( ) const
inline

## ◆ get_initial_w_NB_B()

 const Eigen::Vector3d& get_initial_w_NB_B ( ) const
inline

## ◆ get_J()

 double get_J ( ) const
inline

Returns body's moment of inertia about its axis of rotation.

For example, for a cylinder of radius r, length h and uniformly distributed mass m with its rotational axis aligined along its body frame z-axis this would be: J = Izz = m r² / 2

## ◆ get_uniform_gravity_expressed_in_world()

 const Eigen::Vector3d& get_uniform_gravity_expressed_in_world ( ) const
inline

## ◆ operator=() [1/2]

 FreeBody& operator= ( FreeBody && )
default

## ◆ operator=() [2/2]

 FreeBody& operator= ( const FreeBody & )
default

## ◆ set_initial_p_NoBcm_N()

 void set_initial_p_NoBcm_N ( const Eigen::Vector3d & p_NoBcm_N )
inline

## ◆ set_initial_quat_NB()

 void set_initial_quat_NB ( const Eigen::Quaterniond & quat_NB )
inline

## ◆ set_initial_v_NBcm_B()

 void set_initial_v_NBcm_B ( const Eigen::Vector3d & v_NBcm_B )
inline

## ◆ set_initial_w_NB_B()

 void set_initial_w_NB_B ( const Eigen::Vector3d & w_NB_B )
inline

## ◆ SetUniformGravityExpressedInWorld()

 void SetUniformGravityExpressedInWorld ( const Eigen::Vector3d & gravity )
inline

The documentation for this class was generated from the following files: