System (Docstrings)¶

class sympy.physics.mechanics.system.SymbolicSystem( coord_states, right_hand_side, speeds=None, mass_matrix=None, coordinate_derivatives=None, alg_con=None, output_eqns={}, coord_idxs=None, speed_idxs=None, bodies=None, loads=None, )[source]¶

SymbolicSystem is a class that contains all the information about a system in a symbolic format such as the equations of motions and the bodies and loads in the system.

There are three ways that the equations of motion can be described for Symbolic System:

[1] Explicit form where the kinematics and dynamics are combined
x’ = F_1(x, t, r, p)

[2] Implicit form where the kinematics and dynamics are combined
M_2(x, p) x’ = F_2(x, t, r, p)

[3] Implicit form where the kinematics and dynamics are separate
M_3(q, p) u’ = F_3(q, u, t, r, p) q’ = G(q, u, t, r, p)

where

x : states, e.g. [q, u] t : time r : specified (exogenous) inputs p : constants q : generalized coordinates u : generalized speeds F_1 : right hand side of the combined equations in explicit form F_2 : right hand side of the combined equations in implicit form F_3 : right hand side of the dynamical equations in implicit form M_2 : mass matrix of the combined equations in implicit form M_3 : mass matrix of the dynamical equations in implicit form G : right hand side of the kinematical differential equations

Parameters:

coord_states : ordered iterable of functions of time

This input will either be a collection of the coordinates or states of the system depending on whether or not the speeds are also given. If speeds are specified this input will be assumed to be the coordinates otherwise this input will be assumed to be the states.

right_hand_sideMatrix
This variable is the right hand side of the equations of motion in any of the forms. The specific form will be assumed depending on whether a mass matrix or coordinate derivatives are given.

speedsordered iterable of functions of time, optional
This is a collection of the generalized speeds of the system. If given it will be assumed that the first argument (coord_states) will represent the generalized coordinates of the system.

mass_matrixMatrix, optional
The matrix of the implicit forms of the equations of motion (forms [2] and [3]). The distinction between the forms is determined by whether or not the coordinate derivatives are passed in. If they are given form [3] will be assumed otherwise form [2] is assumed.

coordinate_derivativesMatrix, optional
The right hand side of the kinematical equations in explicit form. If given it will be assumed that the equations of motion are being entered in form [3].

alg_conIterable, optional
The indexes of the rows in the equations of motion that contain algebraic constraints instead of differential equations. If the equations are input in form [3], it will be assumed the indexes are referencing the mass_matrix/right_hand_side combination and not the coordinate_derivatives.

output_eqnsDictionary, optional
Any output equations that are desired to be tracked are stored in a dictionary where the key corresponds to the name given for the specific equation and the value is the equation itself in symbolic form

coord_idxsIterable, optional
If coord_states corresponds to the states rather than the coordinates this variable will tell SymbolicSystem which indexes of the states correspond to generalized coordinates.

speed_idxsIterable, optional
If coord_states corresponds to the states rather than the coordinates this variable will tell SymbolicSystem which indexes of the states correspond to generalized speeds.

bodiesiterable of Body/Rigidbody objects, optional
Iterable containing the bodies of the system

loadsiterable of load instances (described below), optional
Iterable containing the loads of the system where forces are given by (point of application, force vector) and torques are given by (reference frame acting upon, torque vector). Ex [(point, force), (ref_frame, torque)]

Example

As a simple example, the dynamics of a simple pendulum will be input into a SymbolicSystem object manually. First some imports will be needed and then symbols will be set up for the length of the pendulum (l), mass at the end of the pendulum (m), and a constant for gravity (g).

>>> from sympy import Matrix, sin, symbols
>>> from sympy.physics.mechanics import dynamicsymbols, SymbolicSystem
>>> l, m, g = symbols('l m g')

The system will be defined by an angle of theta from the vertical and a generalized speed of omega will be used where omega = theta_dot.

>>> theta, omega = dynamicsymbols('theta omega')

Now the equations of motion are ready to be formed and passed to the SymbolicSystem object.

>>> kin_explicit_rhs = Matrix([omega])
>>> dyn_implicit_mat = Matrix([l**2 * m])
>>> dyn_implicit_rhs = Matrix([-g * l * m * sin(theta)])
>>> symsystem = SymbolicSystem([theta], dyn_implicit_rhs, [omega],
...                            dyn_implicit_mat)

Notes

m : number of generalized speeds n : number of generalized coordinates o : number of states

Attributes

coordinates	(Matrix, shape(n, 1)) This is a matrix containing the generalized coordinates of the system
speeds	(Matrix, shape(m, 1)) This is a matrix containing the generalized speeds of the system
states	(Matrix, shape(o, 1)) This is a matrix containing the state variables of the system
alg_con	(List) This list contains the indices of the algebraic constraints in the combined equations of motion. The presence of these constraints requires that a DAE solver be used instead of an ODE solver. If the system is given in form [3] the alg_con variable will be adjusted such that it is a representation of the combined kinematics and dynamics thus make sure it always matches the mass matrix entered.
dyn_implicit_mat	(Matrix, shape(m, m)) This is the M matrix in form [3] of the equations of motion (the mass matrix or generalized inertia matrix of the dynamical equations of motion in implicit form).
dyn_implicit_rhs	(Matrix, shape(m, 1)) This is the F vector in form [3] of the equations of motion (the right hand side of the dynamical equations of motion in implicit form).
comb_implicit_mat	(Matrix, shape(o, o)) This is the M matrix in form [2] of the equations of motion. This matrix contains a block diagonal structure where the top left block (the first rows) represent the matrix in the implicit form of the kinematical equations and the bottom right block (the last rows) represent the matrix in the implicit form of the dynamical equations.
comb_implicit_rhs	(Matrix, shape(o, 1)) This is the F vector in form [2] of the equations of motion. The top part of the vector represents the right hand side of the implicit form of the kinemaical equations and the bottom of the vector represents the right hand side of the implicit form of the dynamical equations of motion.
comb_explicit_rhs	(Matrix, shape(o, 1)) This vector represents the right hand side of the combined equations of motion in explicit form (form [1] from above).
kin_explicit_rhs	(Matrix, shape(m, 1)) This is the right hand side of the explicit form of the kinematical equations of motion as can be seen in form [3] (the G matrix).
output_eqns	(Dictionary) If output equations were given they are stored in a dictionary where the key corresponds to the name given for the specific equation and the value is the equation itself in symbolic form
bodies	(Tuple) If the bodies in the system were given they are stored in a tuple for future access
loads	(Tuple) If the loads in the system were given they are stored in a tuple for future access. This includes forces and torques where forces are given by (point of application, force vector) and torques are given by (reference frame acted upon, torque vector).

property alg_con¶: Returns a list with the indices of the rows containing algebraic constraints in the combined form of the equations of motion

property bodies¶: Returns the bodies in the system

property comb_explicit_rhs¶: Returns the right hand side of the equations of motion in explicit form, x’ = F, where the kinematical equations are included

property comb_implicit_mat¶: Returns the matrix, M, corresponding to the equations of motion in implicit form (form [2]), M x’ = F, where the kinematical equations are included

property comb_implicit_rhs¶: Returns the column matrix, F, corresponding to the equations of motion in implicit form (form [2]), M x’ = F, where the kinematical equations are included

compute_explicit_form()[source]¶: If the explicit right hand side of the combined equations of motion is to provided upon initialization, this method will calculate it. This calculation can potentially take awhile to compute.

constant_symbols()[source]¶: Returns a column matrix containing all of the symbols in the system that do not depend on time

property coordinates¶: Returns the column matrix of the generalized coordinates

property dyn_implicit_mat¶: Returns the matrix, M, corresponding to the dynamic equations in implicit form, M x’ = F, where the kinematical equations are not included

property dyn_implicit_rhs¶: Returns the column matrix, F, corresponding to the dynamic equations in implicit form, M x’ = F, where the kinematical equations are not included

dynamic_symbols()[source]¶: Returns a column matrix containing all of the symbols in the system that depend on time

property kin_explicit_rhs¶: Returns the right hand side of the kinematical equations in explicit form, q’ = G

property loads¶: Returns the loads in the system

property speeds¶: Returns the column matrix of generalized speeds

property states¶: Returns the column matrix of the state variables

class sympy.physics.mechanics.system.System(frame=None, fixed_point=None)[source]¶

Class to define a multibody system and form its equations of motion.

Explanation

A System instance stores the different objects associated with a model, including bodies, joints, constraints, and other relevant information. With all the relationships between components defined, the System can be used to form the equations of motion using a backend, such as KanesMethod. The System has been designed to be compatible with third-party libraries for greater flexibility and integration with other tools.

Examples

In the example below a cart with a pendulum is created. The cart moves along the x axis of the rail and the pendulum rotates about the z axis. The length of the pendulum is l with the pendulum represented as a particle. To move the cart a time dependent force F is applied to the cart.

We first need to import some functions and create some of our variables.

>>> from sympy import symbols, simplify
>>> from sympy.physics.mechanics import (
...     mechanics_printing, dynamicsymbols, RigidBody, Particle,
...     ReferenceFrame, PrismaticJoint, PinJoint, System)
>>> mechanics_printing(pretty_print=False)
>>> g, l = symbols('g l')
>>> F = dynamicsymbols('F')

The next step is to create bodies. It is also useful to create a frame for locating the particle with respect to the pin joint later on, as a particle does not have a body-fixed frame.

>>> rail = RigidBody('rail')
>>> cart = RigidBody('cart')
>>> bob = Particle('bob')
>>> bob_frame = ReferenceFrame('bob_frame')

Initialize the system, with the rail as the Newtonian reference. The body is also automatically added to the system.

>>> system = System.from_newtonian(rail)
>>> print(system.bodies[0])
rail

Create the joints, while immediately also adding them to the system.

>>> system.add_joints(
...     PrismaticJoint('slider', rail, cart, joint_axis=rail.x),
...     PinJoint('pin', cart, bob, joint_axis=cart.z,
...              child_interframe=bob_frame,
...              child_point=l * bob_frame.y)
... )
>>> system.joints
(PrismaticJoint: slider  parent: rail  child: cart,
PinJoint: pin  parent: cart  child: bob)

While adding the joints, the associated generalized coordinates, generalized speeds, kinematic differential equations and bodies are also added to the system.

>>> system.q
Matrix([
[q_slider],
[   q_pin]])
>>> system.u
Matrix([
[u_slider],
[   u_pin]])
>>> system.kdes
Matrix([
[u_slider - q_slider'],
[      u_pin - q_pin']])
>>> [body.name for body in system.bodies]
['rail', 'cart', 'bob']

With the kinematics established, we can now apply gravity and the cart force F.

>>> system.apply_uniform_gravity(-g * system.y)
>>> system.add_loads((cart.masscenter, F * rail.x))
>>> system.loads
((rail_masscenter, - g*rail_mass*rail_frame.y),
 (cart_masscenter, - cart_mass*g*rail_frame.y),
 (bob_masscenter, - bob_mass*g*rail_frame.y),
 (cart_masscenter, F*rail_frame.x))

With the entire system defined, we can now form the equations of motion. Before forming the equations of motion, one can also run some checks that will try to identify some common errors.

>>> system.validate_system()
>>> system.form_eoms()
Matrix([
[bob_mass*l*u_pin**2*sin(q_pin) - bob_mass*l*cos(q_pin)*u_pin'
 - (bob_mass + cart_mass)*u_slider' + F],
[                   -bob_mass*g*l*sin(q_pin) - bob_mass*l**2*u_pin'
 - bob_mass*l*cos(q_pin)*u_slider']])
>>> simplify(system.mass_matrix)
Matrix([
[ bob_mass + cart_mass, bob_mass*l*cos(q_pin)],
[bob_mass*l*cos(q_pin),         bob_mass*l**2]])
>>> system.forcing
Matrix([
[bob_mass*l*u_pin**2*sin(q_pin) + F],
[          -bob_mass*g*l*sin(q_pin)]])

The complexity of the above example can be increased if we add a constraint to prevent the particle from moving in the horizontal (x) direction. This can be done by adding a holonomic constraint. After which we should also redefine what our (in)dependent generalized coordinates and speeds are.

>>> system.add_holonomic_constraints(
...     bob.masscenter.pos_from(rail.masscenter).dot(system.x)
... )
>>> system.q_ind = system.get_joint('pin').coordinates
>>> system.q_dep = system.get_joint('slider').coordinates
>>> system.u_ind = system.get_joint('pin').speeds
>>> system.u_dep = system.get_joint('slider').speeds

With the updated system the equations of motion can be formed again.

>>> system.validate_system()
>>> system.form_eoms()
Matrix([[-bob_mass*g*l*sin(q_pin)
         - bob_mass*l**2*u_pin'
         - bob_mass*l*cos(q_pin)*u_slider'
         - l*(bob_mass*l*u_pin**2*sin(q_pin)
         - bob_mass*l*cos(q_pin)*u_pin'
         - (bob_mass + cart_mass)*u_slider')*cos(q_pin)
         - l*F*cos(q_pin)]])
>>> simplify(system.mass_matrix)
Matrix([
[bob_mass*l**2*sin(q_pin)**2, -cart_mass*l*cos(q_pin)],
[               l*cos(q_pin),                       1]])
>>> simplify(system.forcing)
Matrix([
[-l*(bob_mass*g*sin(q_pin) + bob_mass*l*u_pin**2*sin(2*q_pin)/2
 + F*cos(q_pin))],
[
l*u_pin**2*sin(q_pin)]])

Attributes

frame	(ReferenceFrame) Inertial reference frame of the system.
fixed_point	(Point) A fixed point in the inertial reference frame.
x	(Vector) Unit vector fixed in the inertial reference frame.
y	(Vector) Unit vector fixed in the inertial reference frame.
z	(Vector) Unit vector fixed in the inertial reference frame.
q	(ImmutableMatrix) Matrix of all the generalized coordinates, i.e. the independent generalized coordinates stacked upon the dependent.
u	(ImmutableMatrix) Matrix of all the generalized speeds, i.e. the independent generealized speeds stacked upon the dependent.
q_ind	(ImmutableMatrix) Matrix of the independent generalized coordinates.
q_dep	(ImmutableMatrix) Matrix of the dependent generalized coordinates.
u_ind	(ImmutableMatrix) Matrix of the independent generalized speeds.
u_dep	(ImmutableMatrix) Matrix of the dependent generalized speeds.
u_aux	(ImmutableMatrix) Matrix of auxiliary generalized speeds.
kdes	(ImmutableMatrix) Matrix of the kinematical differential equations as expressions equated to the zero matrix.
bodies	(tuple of BodyBase subclasses) Tuple of all bodies that make up the system.
joints	(tuple of Joint) Tuple of all joints that connect bodies in the system.
loads	(tuple of LoadBase subclasses) Tuple of all loads that have been applied to the system.
actuators	(tuple of ActuatorBase subclasses) Tuple of all actuators present in the system.
holonomic_constraints	(ImmutableMatrix) Matrix with the holonomic constraints as expressions equated to the zero matrix.
nonholonomic_constraints	(ImmutableMatrix) Matrix with the nonholonomic constraints as expressions equated to the zero matrix.
velocity_constraints	(ImmutableMatrix) Matrix with the velocity constraints as expressions equated to the zero matrix. These are by default derived as the time derivatives of the holonomic constraints extended with the nonholonomic constraints.
eom_method	(subclass of KanesMethod or LagrangesMethod) Backend for forming the equations of motion.

property actuators¶: Tuple of actuators present in the system.

add_actuators(*actuators)[source]¶

Add actuator(s) to the system.

Parameters:: *actuators : subclass of ActuatorBase

One or more actuators.

add_auxiliary_speeds(*speeds)[source]¶

Add auxiliary speed(s) to the system.

Parameters:: *speeds : dynamicsymbols

One or more auxiliary speeds to be added to the system.

add_bodies(*bodies)[source]¶

Add body(ies) to the system.

Parameters:: bodies : Particle or RigidBody

One or more bodies.

add_coordinates( *coordinates, independent=True, )[source]¶

Add generalized coordinate(s) to the system.

Parameters:

*coordinates : dynamicsymbols

One or more generalized coordinates to be added to the system.

independent : bool or list of bool, optional

Boolean whether a coordinate is dependent or independent. The default is True, so the coordinates are added as independent by default.

add_holonomic_constraints(*constraints)[source]¶

Add holonomic constraint(s) to the system.

Parameters:: *constraints : Expr

One or more holonomic constraints, which are expressions that should be zero.

add_joints(*joints)[source]¶

Add joint(s) to the system.

Parameters:: *joints : subclass of Joint

One or more joints.

Explanation

This methods adds one or more joints to the system including its associated objects, i.e. generalized coordinates, generalized speeds, kinematic differential equations and the bodies.

Notes

For the generalized coordinates, generalized speeds and bodies it is checked whether they are already known by the system instance. If they are, then they are not added. The kinematic differential equations are however always added to the system, so you should not also manually add those on beforehand.

add_kdes(*kdes)[source]¶

Add kinematic differential equation(s) to the system.

Parameters:: *kdes : Expr

One or more kinematic differential equations.

add_loads(*loads)[source]¶

Add load(s) to the system.

Parameters:: *loads : Force or Torque

One or more loads.

add_nonholonomic_constraints(*constraints)[source]¶

Add nonholonomic constraint(s) to the system.

Parameters:: *constraints : Expr

One or more nonholonomic constraints, which are expressions that should be zero.

add_speeds(*speeds, independent=True)[source]¶

Add generalized speed(s) to the system.

Parameters:

*speeds : dynamicsymbols

One or more generalized speeds to be added to the system.

independent : bool or list of bool, optional

Boolean whether a speed is dependent or independent. The default is True, so the speeds are added as independent by default.

apply_uniform_gravity(acceleration)[source]¶

Apply uniform gravity to all bodies in the system by adding loads.

Parameters:: acceleration : Vector

The acceleration due to gravity.

property bodies¶: Tuple of all bodies that have been added to the system.

property eom_method¶: Backend for forming the equations of motion.

property fixed_point¶: Fixed point in the inertial reference frame.

property forcing¶: The forcing vector of the system.

property forcing_full¶: The forcing vector of the system, augmented by the kinematic differential equations in explicit or implicit form.

form_eoms(

eom_method=<class 'sympy.physics.mechanics.kane.KanesMethod'>,

**kwargs,

)[source]¶

Form the equations of motion of the system.

Parameters:: eom_method : subclass of KanesMethod or LagrangesMethod

Backend class to be used for forming the equations of motion. The default is KanesMethod.
Returns:: ImmutableMatrix

Vector of equations of motions.

Examples

This is a simple example for a one degree of freedom translational spring-mass-damper.

>>> from sympy import S, symbols
>>> from sympy.physics.mechanics import (
...     LagrangesMethod, dynamicsymbols, PrismaticJoint, Particle,
...     RigidBody, System)
>>> q = dynamicsymbols('q')
>>> qd = dynamicsymbols('q', 1)
>>> m, k, b = symbols('m k b')
>>> wall = RigidBody('W')
>>> system = System.from_newtonian(wall)
>>> bob = Particle('P', mass=m)
>>> bob.potential_energy = S.Half * k * q**2
>>> system.add_joints(PrismaticJoint('J', wall, bob, q, qd))
>>> system.add_loads((bob.masscenter, b * qd * system.x))
>>> system.form_eoms(LagrangesMethod)
Matrix([[-b*Derivative(q(t), t) + k*q(t) + m*Derivative(q(t), (t, 2))]])

We can also solve for the states using the ‘rhs’ method.

>>> system.rhs()
Matrix([
[               Derivative(q(t), t)],
[(b*Derivative(q(t), t) - k*q(t))/m]])

property frame¶: Inertial reference frame of the system.

classmethod from_newtonian(newtonian)[source]¶: Constructs the system with respect to a Newtonian body.

get_body(name)[source]¶

Retrieve a body from the system by name.

Parameters:: name : str

The name of the body to retrieve.
Returns:: RigidBody or Particle

The body with the given name, or None if no such body exists.

get_joint(name)[source]¶

Retrieve a joint from the system by name.

Parameters:: name : str

The name of the joint to retrieve.
Returns:: subclass of Joint

The joint with the given name, or None if no such joint exists.

property holonomic_constraints¶: Matrix with the holonomic constraints as expressions equated to the zero matrix.

property joints¶: Tuple of all joints that have been added to the system.

property kdes¶: Kinematical differential equations as expressions equated to the zero matrix. These equations describe the coupling between the generalized coordinates and the generalized speeds.

property loads¶: Tuple of loads that have been applied on the system.

property mass_matrix¶

The mass matrix of the system.

Explanation

The mass matrix \(M_d\) and the forcing vector \(f_d\) of a system describe the system’s dynamics according to the following equations:

\[M_d \dot{u} = f_d\]

where \(\dot{u}\) is the time derivative of the generalized speeds.

property mass_matrix_full¶

The mass matrix of the system, augmented by the kinematic differential equations in explicit or implicit form.

Explanation

The full mass matrix \(M_m\) and the full forcing vector \(f_m\) of a system describe the dynamics and kinematics according to the following equation:

\[M_m \dot{x} = f_m\]

where \(x\) is the state vector stacking \(q\) and \(u\).

property nonholonomic_constraints¶: Matrix with the nonholonomic constraints as expressions equated to the zero matrix.

property q¶: Matrix of all the generalized coordinates with the independent stacked upon the dependent.

property q_dep¶: Matrix of the dependent generalized coordinates.

property q_ind¶: Matrix of the independent generalized coordinates.

rhs(inv_method=None)[source]¶

Compute the equations of motion in the explicit form.

Parameters:: inv_method : str

The specific sympy inverse matrix calculation method to use. For a list of valid methods, see inv()
Returns:: ImmutableMatrix

Equations of motion in the explicit form.