FreeBody Class Reference

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.

• [Kane, 1983] "Spacecraft Dynamics," McGraw-Hill Book Co., New York, 1983. (with P. W. Likins and D. A. Levinson). Available for free .pdf download: https://ecommons.cornell.edu/handle/1813/637

#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 body B's moment of inertia about any axis that passes through Bcm (B's center of mass) and is perpendicular to B's inertia symmetry axis. More...
double	get_J () const
	Returns body's moment of inertia about the axis that passes through Bcm (B's center of mass) and is parallel to B's inertia symmetry axis. 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...
std::pair< double, double >	CalcAngularRates_s_p () const
	Returns angular rates associated with spin s and precession p from the analytical solution [Kane, 1983] for rotational motion (angular velocity and quaternion) for torque-free motion of an axis-symmetric rigid body B in a Newtonian frame (World). More...
Implements CopyConstructible, CopyAssignable, MoveConstructible, MoveAssignable
	FreeBody (const FreeBody &)=default
FreeBody &	operator= (const FreeBody &)=default
	FreeBody (FreeBody &&)=default
FreeBody &	operator= (FreeBody &&)=default

Constructor & Destructor Documentation

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
	)

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

Member Function Documentation

CalcAngularRates_s_p()

std::pair<double, double> CalcAngularRates_s_p

(

)

const

Returns angular rates associated with spin s and precession p from the analytical solution [Kane, 1983] for rotational motion (angular velocity and quaternion) for torque-free motion of an axis-symmetric rigid body B in a Newtonian frame (World).

Kane's solution for B's angular velocity wx*Bx + wy*By + wz*Bz is in terms of initial values wx0, wy0, wz0 as wx = wx0 * cos(s * t) + wy0 * sin(s * t) wy = -wx0 * sin(s * t) + wy0 * cos(s * t) wz = wz0 For more information, see [Kane, 1983] Pages 60-62 and 159-169.

Note

the return value of s may be negative, zero, or positive, whereas the return value of p is nonnegative. The values of s and p are returned in units of radian/second.

• [Kane, 1983] "Spacecraft Dynamics," McGraw-Hill Book Co., New York,
  1. (with P. W. Likins and D. A. Levinson). Available for free .pdf download: https://ecommons.cornell.edu/handle/1813/637

CalcInitial_v_NBcm_N()

Eigen::Vector3d CalcInitial_v_NBcm_N

(

)

const

CalcInitial_w_NB_N()

Eigen::Vector3d CalcInitial_w_NB_N

(

)

const

CalculateExactRotationalSolutionNB()

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.

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.

• [Kane, 1983] "Spacecraft Dynamics," McGraw-Hill Book Co., New York,
  1. (with P. W. Likins and D. A. Levinson). Available for free .pdf download: https://ecommons.cornell.edu/handle/1813/637

CalculateExactTranslationalSolution()

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.

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

Returns body B's moment of inertia about any axis that passes through Bcm (B's center of mass) and is perpendicular to B's inertia symmetry axis.

For example, for a cylinder of radius r, length h and uniformly distributed mass m with its cylindrical axis aligned 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

get_initial_quat_NB()

const Eigen::Quaterniond& get_initial_quat_NB

(

)

const

get_initial_w_NB_B()

const Eigen::Vector3d& get_initial_w_NB_B

(

)

const

get_J()

double get_J

(

)

const

Returns body's moment of inertia about the axis that passes through Bcm (B's center of mass) and is parallel to B's inertia symmetry axis.

For example, for a cylinder of radius r, length h and uniformly distributed mass m with its cylindrical axis aligned 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

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

)

set_initial_quat_NB()

void set_initial_quat_NB

(

const Eigen::Quaterniond &

quat_NB

)

set_initial_v_NBcm_B()

void set_initial_v_NBcm_B

(

const Eigen::Vector3d &

v_NBcm_B

)

set_initial_w_NB_B()

void set_initial_w_NB_B

(

const Eigen::Vector3d &

w_NB_B

)

SetUniformGravityExpressedInWorld()

void SetUniformGravityExpressedInWorld

(

const Eigen::Vector3d &

gravity

)

---

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

• drake/multibody/benchmarks/free_body/free_body.h