skydy.rigidbody package

Submodules

skydy.rigidbody.Body module

class skydy.rigidbody.Body.Body(name=None)

Bases: skydy.configuration.Configuration.Configuration

__init__(name=None)

A Body is a collection of particles. It has a mass, and some dimensions (length, width and height), and by default has six degrees of freedom (DOFs). As such, it extends the Configuration class, with the addition of the inertial properties (InertiaMatrix and MassMatrix) and some dimensions (DimensionSymbols).

We also instantite two empty lists for the forces and torques that might be applied to the body.

Parameters

name (int, float or string) – the name of the body.

Returns

None

Example

>>> from skydy.rigidbody import Body
>>> # Empty initializer
>>> b_1 = Body()
>>> # Define name by a number
>>> b_2 = Body(2)
>>> # Empty initializer
>>> b_3 = Body("car")

add_force(force_vector, force_location)

Add, or apply a force to the body. For a force to do anything, the force must have a direction AND a location.

Parameters

• force_vector (BodyForce) – the force direction, as defined in the body’s coordinate frame.

• force_location (BodyCoordinate) – the location of the force, as define in the body’s coordinate frame.

Returns

None

Examples

>>> from skydy.Body import Body, BodyForce, BodyCoordinate
>>> # Define the body
>>> b = Body()
>>> # Apply a force in the x-direction
>>> f_1 = BodyForce("1", x_dir=True)
>>> # Apply the force at the origin of the Body
>>> f_loc = BodyCoordinate("PF1")
>>> # We can now apply the force
>>> b.apply_force(f_1, f_loc)

add_torque(torque_vector, torque_location)

Add, or apply a torque to the body. For a torque to f do anything, the torque must have a direction AND a location.

Parameters

• torque_vector (BodyTorque) – the torque direction, as defined in the body’s coordinate frame.

• torque_location (BodyCoordinate) – the location of the torque, as define in the body’s coordinate frame.

Returns

None

Examples

>>> from skydy.Body import Body, BodyTorque, BodyCoordinate
>>> # Define the body
>>> b = Body()
>>> # Apply a torque in the x-direction
>>> t_1 = BodyTorque("1", x_dir=True)
>>> # Apply the torque at the origin of the Body
>>> t_loc = BodyCoordinate("PT1")
>>> # We can now apply the torque
>>> b.apply_torque(t_1, t_loc)

body_twists()

Calculate the body twists to ultimately determine the translational and rotational velocities.

omega is the body rotational velocity. v_body is the body translational velocity.

Mathematical expressions are:

Let r in R^3 be the position of the COM, and R in SO(3) the body orientation.

Then,

omega_hat = R^(-1) @ (d/dt) R, omega = unhat_map(omega_hat), v_body = (d/dt) r.

Refer to Bullo and Lewis for more information.

Parameters

None –

Returns

the body translational velocities omega_body (sympy.matrix): the body rotational velocities

Return type

v_body (sympy.matrix)

draw(ax=None, ref_body=None, ref_joint=None, sub_vals=None)

Draw the body.

Parameters

• ax (matplotlib.axes._subplots.AxesSubplot) – the axis to plot the connection on.

• ref_body (None or Body) – the reference Body to propagate dimensions, coordinates etc.

• ref_joint (None or numpy.ndarray) – the location of the joint the body is connected to.

• sub_vals (dict) – symbol-value pairs required to go from symbolic to numeric expression. It is important to note, that all symbols each body is dependent on, for example, for upstream bodies and joints, are included.

Returns

updated axes, with plots.

Return type

ax (matplotlib.axes._subplots.AxesSubplot)

Example

>>> import matplotlib.pyplot as plt
>>> from skydy.rigidbody import Body
>>> # Define the body
>>> b = Body()
>>> # Define the axes
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111, projection='3d')
>>> ax = b.draw(ax)
>>> plt.show()

id_counter = 1

kinetic_energy()

Determine the kinetic energy of the body.

Let v_body in R^3, w_body in R^3 be the translational and rotational velocities, M in R^(3x3), I in R^(3x3) be the mass and inertia matrices, then,

KE_trans = (1/2) * v_vody^(T) @ M @ v_body KE_rot = (1/2) * w_vody^(T) @ I @ w_body

KE_tot = KE_trans + KE_rot.

Parameters

None –

Returns

the symbol expression of the kinetic energy.

Return type

KE_tot (sympy.symbol)

potential_energy(gravity)

Determine the potential energy of the body.

The PE is the component of the position of the body in the z-coordinate (height), times gravity (g), times mass, i.e.,

PE = m * dot(g, r)

Parameters

gravity (sympy.matrix) – the gravity vector in the global coordinte frame, typically [0, 0, g]

Returns

the symbol expression of the potential energy

Return type

potential_energy (sympy.symbols)

Examples

>>> from skydy.rigidbody import Body
>>> b = Body()
>>> # Define the gravity vector
>>> g = sym.Matrix([0, 0, sym.Symbol('g')])
>>> # Calculate the potential energy
>>> b.potential_energy(g)

class skydy.rigidbody.Body.Ground

Bases: skydy.rigidbody.Body.Body

__init__()

The base Body for every system.

The Ground defines the global coordinate frame, that cannot move or rotate. All coordinates are constrained, with zero constant value.

Parameters

None –

Returns

None

Example

>>> from skydy.rigidbody import Ground
>>> b_gnd = Ground()

skydy.rigidbody.Body.symbols_to_latex(symbols, prefix=None)

Convert symbols to latex.

Parameters

• symbols (sympy.matrix or sympy.Symbol) – a list of symbols or symbol.

• prefix (str) – an “equals” prefix.

Returns

sympy object turned into a latex-able string.

Return type

latex_str (str)

skydy.rigidbody.BodyCoordinate module

class skydy.rigidbody.BodyCoordinate.BodyCoordinate(name, x=0, y=0, z=0)

Bases: skydy.configuration.BaseSymbols.DimensionSymbols

__init__(name, x=0, y=0, z=0)

A coordinate of a point on a body. Inherits from DimensionSymbols.

Parameters

• name (int, float or string) – the name of the body or object we want to

• for. (generate dimensions) –

• x (int or float) – the x-coordinate to a point from the body’s COM

• y (int or float) – the y-coordinate to a point from the body’s COM

• z (int or float) – the z-coordinate to a point from the body’s COM

Returns

None

Example

>>> from skydy.configuration import BodyCoordinate
>>> # Create a set of symbols
>>> body_name = 1
>>> # Default constructor
>>> body_coord = BodyCoordinate(body_name)
>>> # Assign some distances
>>> body_coord = BodyCoordinate(body_name, 1, 2, 3)
>>> body_coord = BodyCoordinate(body_name, 3, -4, 4)

class skydy.rigidbody.BodyCoordinate.GroundCoordinate

Bases: skydy.rigidbody.BodyCoordinate.BodyCoordinate

__init__()

A BodyCoordinate with a default name of O, for origin, and (x, y, z) = (0, 0, 0)

Example

>>> from skydy.rigidbody import GroundCoordinate
>>> gnd_coord = GroundCoordinate()

skydy.rigidbody.BodyForce module

class skydy.rigidbody.BodyForce.BodyForce(name, x_dir=False, y_dir=False, z_dir=False)

Bases: skydy.configuration.BaseSymbols.ForceSymbols

__init__(name, x_dir=False, y_dir=False, z_dir=False)

Define a force that acts on a Body. The direction of the force is defined the the relevant body’s coordinate frame.

By default, a force is assgined a default value of 1, if the force is defined in that direction, or 0 otherwise.

Parameters

• name (int or str) – the name for the force.

• x_dir (bool) – if the defined force acts in the body x-direction

• y_dir (bool) – if the defined force acts in the body y-direction

• z_dir (bool) – if the defined force acts in the body z-direction

Returns

None

Example

>>> from skydy.rigidbody import BodyForce
>>> # Instantiate a force in the body x-direction
>>> f_1 = BodyForce("1", x_dir=True)
>>> # Instantiate another force in the body y-direction
>>> f_2 = BodyForce("2", y_dir=True)
>>> # Instantiate another force in all directions
>>> f_3 = BodyForce("3", True, True, True)

class skydy.rigidbody.BodyForce.BodyTorque(name, x_dir=False, y_dir=False, z_dir=False)

Bases: skydy.configuration.BaseSymbols.TorqueSymbols

__init__(name, x_dir=False, y_dir=False, z_dir=False)

Define a torque that acts on a Body. The direction of the torque is defined the the relevant body’s coordinate frame, and a rotation ABOUT that coordinate direction (as per the right hand rule).

By default, a torque is assgined a default value of 1, if the torque is defined in that direction, or 0 otherwise.

Parameters

• name (int or str) – the name for the torque.

• x_dir (bool) – if the defined torque acts around the body x-direction

• y_dir (bool) – if the defined torque acts around the body y-direction

• z_dir (bool) – if the defined torque acts around the body z-direction

Returns

None

Example

>>> from skydy.rigidbody import BodyTorque
>>> # Instantiate a torque about the body x-direction
>>> t_1 = BodyTorque("1", x_dir=True)
>>> # Instantiate another torque about the body y-direction
>>> t_2 = BodyTorque("2", y_dir=True)
>>> # Instantiate another torque about all directions
>>> t_3 = BodyTorque("3", True, True, True)

Module contents