Kane’s Method & Lagrange’s Method (Docstrings)

class sympy.physics.mechanics.kane.KanesMethod(frame, q_ind, u_ind, kd_eqs=None, q_dependent=None, configuration_constraints=None, u_dependent=None, velocity_constraints=None, acceleration_constraints=None, u_auxiliary=None, bodies=None, forcelist=None, explicit_kinematics=True, kd_eqs_solver='LU', constraint_solver='LU')

Kane’s method object.

Parameters:

frame : ReferenceFrame

The inertial reference frame for the system.

q_ind : iterable of dynamicsymbols

Independent generalized coordinates.

u_ind : iterable of dynamicsymbols

Independent generalized speeds.

kd_eqs : iterable of Expr, optional

Kinematic differential equations, which linearly relate the generalized speeds to the time-derivatives of the generalized coordinates.

q_dependent : iterable of dynamicsymbols, optional

Dependent generalized coordinates.

configuration_constraints : iterable of Expr, optional

Constraints on the system’s configuration, i.e. holonomic constraints.

u_dependent : iterable of dynamicsymbols, optional

Dependent generalized speeds.

velocity_constraints : iterable of Expr, optional

Constraints on the system’s velocity, i.e. the combination of the nonholonomic constraints and the time-derivative of the holonomic constraints.

acceleration_constraints : iterable of Expr, optional

Constraints on the system’s acceleration, by default these are the time-derivative of the velocity constraints.

u_auxiliary : iterable of dynamicsymbols, optional

Auxiliary generalized speeds.

bodies : iterable of Particle and/or RigidBody, optional

The particles and rigid bodies in the system.

forcelist : iterable of tuple[Point | ReferenceFrame, Vector], optional

Forces and torques applied on the system.

explicit_kinematics : bool

Boolean whether the mass matrices and forcing vectors should use the explicit form (default) or implicit form for kinematics. See the notes for more details.

kd_eqs_solver : str, callable

Method used to solve the kinematic differential equations. If a string is supplied, it should be a valid method that can be used with the sympy.matrices.matrixbase.MatrixBase.solve(). If a callable is supplied, it should have the format f(A, rhs), where it solves the equations and returns the solution. The default utilizes LU solve. See the notes for more information.

constraint_solver : str, callable

Method used to solve the velocity constraints. If a string is supplied, it should be a valid method that can be used with the sympy.matrices.matrixbase.MatrixBase.solve(). If a callable is supplied, it should have the format f(A, rhs), where it solves the equations and returns the solution. The default utilizes LU solve. See the notes for more information.

Explanation

This object is used to do the “book-keeping” as you go through and form equations of motion in the way Kane presents in: Kane, T., Levinson, D. Dynamics Theory and Applications. 1985 McGraw-Hill

The attributes are for equations in the form [M] udot = forcing.

Notes

The mass matrices and forcing vectors related to kinematic equations are given in the explicit form by default. In other words, the kinematic mass matrix is \(\mathbf{k_{k\dot{q}}} = \mathbf{I}\). In order to get the implicit form of those matrices/vectors, you can set the explicit_kinematics attribute to False. So \(\mathbf{k_{k\dot{q}}}\) is not necessarily an identity matrix. This can provide more compact equations for non-simple kinematics.

Two linear solvers can be supplied to KanesMethod: one for solving the kinematic differential equations and one to solve the velocity constraints. Both of these sets of equations can be expressed as a linear system Ax = rhs, which have to be solved in order to obtain the equations of motion.

The default solver 'LU', which stands for LU solve, results relatively low number of operations. The weakness of this method is that it can result in zero division errors.

If zero divisions are encountered, a possible solver which may solve the problem is "CRAMER". This method uses Cramer’s rule to solve the system. This method is slower and results in more operations than the default solver. However it only uses a single division by default per entry of the solution.

While a valid list of solvers can be found at sympy.matrices.matrixbase.MatrixBase.solve(), it is also possible to supply a \(callable\). This way it is possible to use a different solver routine. If the kinematic differential equations are not too complex it can be worth it to simplify the solution by using lambda A, b: simplify(Matrix.LUsolve(A, b)). Another option solver one may use is sympy.solvers.solveset.linsolve(). This can be done using \(lambda A, b: tuple(linsolve((A, b)))[0]\), where we select the first solution as our system should have only one unique solution.

Examples

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

In this example, we first need to do the kinematics. This involves creating generalized speeds and coordinates and their derivatives. Then we create a point and set its velocity in a frame.

>>> from sympy import symbols
>>> from sympy.physics.mechanics import dynamicsymbols, ReferenceFrame
>>> from sympy.physics.mechanics import Point, Particle, KanesMethod
>>> q, u = dynamicsymbols('q u')
>>> qd, ud = dynamicsymbols('q u', 1)
>>> m, c, k = symbols('m c k')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, u * N.x)

Next we need to arrange/store information in the way that KanesMethod requires. The kinematic differential equations should be an iterable of expressions. A list of forces/torques must be constructed, where each entry in the list is a (Point, Vector) or (ReferenceFrame, Vector) tuple, where the Vectors represent the Force or Torque. Next a particle needs to be created, and it needs to have a point and mass assigned to it. Finally, a list of all bodies and particles needs to be created.

>>> kd = [qd - u]
>>> FL = [(P, (-k * q - c * u) * N.x)]
>>> pa = Particle('pa', P, m)
>>> BL = [pa]

Finally we can generate the equations of motion. First we create the KanesMethod object and supply an inertial frame, coordinates, generalized speeds, and the kinematic differential equations. Additional quantities such as configuration and motion constraints, dependent coordinates and speeds, and auxiliary speeds are also supplied here (see the online documentation). Next we form FR* and FR to complete: Fr + Fr* = 0. We have the equations of motion at this point. It makes sense to rearrange them though, so we calculate the mass matrix and the forcing terms, for E.o.M. in the form: [MM] udot = forcing, where MM is the mass matrix, udot is a vector of the time derivatives of the generalized speeds, and forcing is a vector representing “forcing” terms.

>>> KM = KanesMethod(N, q_ind=[q], u_ind=[u], kd_eqs=kd)
>>> (fr, frstar) = KM.kanes_equations(BL, FL)
>>> MM = KM.mass_matrix
>>> forcing = KM.forcing
>>> rhs = MM.inv() * forcing
>>> rhs
Matrix([[(-c*u(t) - k*q(t))/m]])
>>> KM.linearize(A_and_B=True)[0]
Matrix([
[   0,    1],
[-k/m, -c/m]])

Please look at the documentation pages for more information on how to perform linearization and how to deal with dependent coordinates & speeds, and how do deal with bringing non-contributing forces into evidence.

Attributes

q, u	(Matrix) Matrices of the generalized coordinates and speeds
bodies	(iterable) Iterable of Particle and RigidBody objects in the system.
loads	(iterable) Iterable of (Point, vector) or (ReferenceFrame, vector) tuples describing the forces on the system.
auxiliary_eqs	(Matrix) If applicable, the set of auxiliary Kane’s equations used to solve for non-contributing forces.
mass_matrix	(Matrix) The system’s dynamics mass matrix: [k_d; k_dnh]
forcing	(Matrix) The system’s dynamics forcing vector: -[f_d; f_dnh]
mass_matrix_kin	(Matrix) The “mass matrix” for kinematic differential equations: k_kqdot
forcing_kin	(Matrix) The forcing vector for kinematic differential equations: -(k_ku*u + f_k)
mass_matrix_full	(Matrix) The “mass matrix” for the u’s and q’s with dynamics and kinematics
forcing_full	(Matrix) The “forcing vector” for the u’s and q’s with dynamics and kinematics

property auxiliary_eqs

A matrix containing the auxiliary equations.

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.

property forcing_kin

The kinematic “forcing vector” of the system.

kanes_equations(bodies=None, loads=None)

Method to form Kane’s equations, Fr + Fr* = 0.

Parameters:

bodies : iterable

An iterable of all RigidBody’s and Particle’s in the system. A system must have at least one body.

loads : iterable

Takes in an iterable of (Particle, Vector) or (ReferenceFrame, Vector) tuples which represent the force at a point or torque on a frame. Must be either a non-empty iterable of tuples or None which corresponds to a system with no constraints.

Explanation

Returns (Fr, Fr*). In the case where auxiliary generalized speeds are present (say, s auxiliary speeds, o generalized speeds, and m motion constraints) the length of the returned vectors will be o - m + s in length. The first o - m equations will be the constrained Kane’s equations, then the s auxiliary Kane’s equations. These auxiliary equations can be accessed with the auxiliary_eqs property.

kindiffdict()

Returns a dictionary mapping q’ to u.

linearize(*, new_method=None, linear_solver='LU', **kwargs)

Linearize the equations of motion about a symbolic operating point.

Parameters:

new_method

Deprecated, does nothing and will be removed.

linear_solver : str, callable

Method used to solve the several symbolic linear systems of the form A*x=b in the linearization process. If a string is supplied, it should be a valid method that can be used with the sympy.matrices.matrixbase.MatrixBase.solve(). If a callable is supplied, it should have the format x = f(A, b), where it solves the equations and returns the solution. The default is 'LU' which corresponds to SymPy’s A.LUsolve(b). LUsolve() is fast to compute but will often result in divide-by-zero and thus nan results.

**kwargs

Extra keyword arguments are passed to sympy.physics.mechanics.linearize.Linearizer.linearize().

Explanation

If kwarg A_and_B is False (default), returns M, A, B, r for the linearized form, M*[q’, u’]^T = A*[q_ind, u_ind]^T + B*r.

If kwarg A_and_B is True, returns A, B, r for the linearized form dx = A*x + B*r, where x = [q_ind, u_ind]^T. Note that this is computationally intensive if there are many symbolic parameters. For this reason, it may be more desirable to use the default A_and_B=False, returning M, A, and B. Values may then be substituted in to these matrices, and the state space form found as A = P.T*M.inv()*A, B = P.T*M.inv()*B, where P = Linearizer.perm_mat.

In both cases, r is found as all dynamicsymbols in the equations of motion that are not part of q, u, q’, or u’. They are sorted in canonical form.

The operating points may be also entered using the op_point kwarg. This takes a dictionary of {symbol: value}, or a an iterable of such dictionaries. The values may be numeric or symbolic. The more values you can specify beforehand, the faster this computation will run.

For more documentation, please see the Linearizer class.

property mass_matrix

The mass matrix of the system.

property mass_matrix_full

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

property mass_matrix_kin

The kinematic “mass matrix” \(\mathbf{k_{k\dot{q}}}\) of the system.

rhs(inv_method=None)

Returns the system’s equations of motion in first order form. The output is the right hand side of:

x' = |q'| =: f(q, u, r, p, t)
     |u'|

The right hand side is what is needed by most numerical ODE integrators.

Parameters:

inv_method : str

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

to_linearizer(linear_solver='LU')

Returns an instance of the Linearizer class, initiated from the data in the KanesMethod class. This may be more desirable than using the linearize class method, as the Linearizer object will allow more efficient recalculation (i.e. about varying operating points).

Parameters:

linear_solver : str, callable

Method used to solve the several symbolic linear systems of the form A*x=b in the linearization process. If a string is supplied, it should be a valid method that can be used with the sympy.matrices.matrixbase.MatrixBase.solve(). If a callable is supplied, it should have the format x = f(A, b), where it solves the equations and returns the solution. The default is 'LU' which corresponds to SymPy’s A.LUsolve(b). LUsolve() is fast to compute but will often result in divide-by-zero and thus nan results.

Returns:

Linearizer

An instantiated sympy.physics.mechanics.linearize.Linearizer.

class sympy.physics.mechanics.lagrange.LagrangesMethod(Lagrangian, qs, forcelist=None, bodies=None, frame=None, hol_coneqs=None, nonhol_coneqs=None)

Lagrange’s method object.

Explanation

This object generates the equations of motion in a two step procedure. The first step involves the initialization of LagrangesMethod by supplying the Lagrangian and the generalized coordinates, at the bare minimum. If there are any constraint equations, they can be supplied as keyword arguments. The Lagrange multipliers are automatically generated and are equal in number to the constraint equations. Similarly any non-conservative forces can be supplied in an iterable (as described below and also shown in the example) along with a ReferenceFrame. This is also discussed further in the __init__ method.

Examples

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

In this example, we first need to do the kinematics. This involves creating generalized coordinates and their derivatives. Then we create a point and set its velocity in a frame.

>>> from sympy.physics.mechanics import LagrangesMethod, Lagrangian
>>> from sympy.physics.mechanics import ReferenceFrame, Particle, Point
>>> from sympy.physics.mechanics import dynamicsymbols
>>> from sympy import symbols
>>> q = dynamicsymbols('q')
>>> qd = dynamicsymbols('q', 1)
>>> m, k, b = symbols('m k b')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, qd * N.x)

We need to then prepare the information as required by LagrangesMethod to generate equations of motion. First we create the Particle, which has a point attached to it. Following this the lagrangian is created from the kinetic and potential energies. Then, an iterable of nonconservative forces/torques must be constructed, where each item is a (Point, Vector) or (ReferenceFrame, Vector) tuple, with the Vectors representing the nonconservative forces or torques.

>>> Pa = Particle('Pa', P, m)
>>> Pa.potential_energy = k * q**2 / 2.0
>>> L = Lagrangian(N, Pa)
>>> fl = [(P, -b * qd * N.x)]

Finally we can generate the equations of motion. First we create the LagrangesMethod object. To do this one must supply the Lagrangian, and the generalized coordinates. The constraint equations, the forcelist, and the inertial frame may also be provided, if relevant. Next we generate Lagrange’s equations of motion, such that: Lagrange’s equations of motion = 0. We have the equations of motion at this point.

>>> l = LagrangesMethod(L, [q], forcelist = fl, frame = N)
>>> print(l.form_lagranges_equations())
Matrix([[b*Derivative(q(t), t) + 1.0*k*q(t) + m*Derivative(q(t), (t, 2))]])

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

>>> print(l.rhs())
Matrix([[Derivative(q(t), t)], [(-b*Derivative(q(t), t) - 1.0*k*q(t))/m]])

Please refer to the docstrings on each method for more details.

Attributes

q, u	(Matrix) Matrices of the generalized coordinates and speeds
loads	(iterable) Iterable of (Point, vector) or (ReferenceFrame, vector) tuples describing the forces on the system.
bodies	(iterable) Iterable containing the rigid bodies and particles of the system.
mass_matrix	(Matrix) The system’s mass matrix
forcing	(Matrix) The system’s forcing vector
mass_matrix_full	(Matrix) The “mass matrix” for the qdot’s, qdoubledot’s, and the lagrange multipliers (lam)
forcing_full	(Matrix) The forcing vector for the qdot’s, qdoubledot’s and lagrange multipliers (lam)

property forcing

Returns the forcing vector from ‘lagranges_equations’ method.

property forcing_full

Augments qdots to the forcing vector above.

form_lagranges_equations()

Method to form Lagrange’s equations of motion.

Returns a vector of equations of motion using Lagrange’s equations of the second kind.

linearize(q_ind=None, qd_ind=None, q_dep=None, qd_dep=None, linear_solver='LU', **kwargs)

Linearize the equations of motion about a symbolic operating point.

Parameters:

linear_solver : str, callable

Method used to solve the several symbolic linear systems of the form A*x=b in the linearization process. If a string is supplied, it should be a valid method that can be used with the sympy.matrices.matrixbase.MatrixBase.solve(). If a callable is supplied, it should have the format x = f(A, b), where it solves the equations and returns the solution. The default is 'LU' which corresponds to SymPy’s A.LUsolve(b). LUsolve() is fast to compute but will often result in divide-by-zero and thus nan results.

**kwargs

Extra keyword arguments are passed to sympy.physics.mechanics.linearize.Linearizer.linearize().

Explanation

If kwarg A_and_B is False (default), returns M, A, B, r for the linearized form, M*[q’, u’]^T = A*[q_ind, u_ind]^T + B*r.

If kwarg A_and_B is True, returns A, B, r for the linearized form dx = A*x + B*r, where x = [q_ind, u_ind]^T. Note that this is computationally intensive if there are many symbolic parameters. For this reason, it may be more desirable to use the default A_and_B=False, returning M, A, and B. Values may then be substituted in to these matrices, and the state space form found as A = P.T*M.inv()*A, B = P.T*M.inv()*B, where P = Linearizer.perm_mat.

In both cases, r is found as all dynamicsymbols in the equations of motion that are not part of q, u, q’, or u’. They are sorted in canonical form.

The operating points may be also entered using the op_point kwarg. This takes a dictionary of {symbol: value}, or a an iterable of such dictionaries. The values may be numeric or symbolic. The more values you can specify beforehand, the faster this computation will run.

For more documentation, please see the Linearizer class.

property mass_matrix

Returns the mass matrix, which is augmented by the Lagrange multipliers, if necessary.

Explanation

If the system is described by ‘n’ generalized coordinates and there are no constraint equations then an n X n matrix is returned.

If there are ‘n’ generalized coordinates and ‘m’ constraint equations have been supplied during initialization then an n X (n+m) matrix is returned. The (n + m - 1)th and (n + m)th columns contain the coefficients of the Lagrange multipliers.

property mass_matrix_full

Augments the coefficients of qdots to the mass_matrix.

rhs(inv_method=None, **kwargs)

Returns equations that can be solved numerically.

Parameters:

inv_method : str

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

solve_multipliers(op_point=None, sol_type='dict')

Solves for the values of the lagrange multipliers symbolically at the specified operating point.

Parameters:

op_point : dict or iterable of dicts, optional

Point at which to solve at. The operating point is specified as a dictionary or iterable of dictionaries of {symbol: value}. The value may be numeric or symbolic itself.

sol_type : str, optional

Solution return type. Valid options are: - ‘dict’: A dict of {symbol : value} (default) - ‘Matrix’: An ordered column matrix of the solution

to_linearizer(q_ind=None, qd_ind=None, q_dep=None, qd_dep=None, linear_solver='LU')

Returns an instance of the Linearizer class, initiated from the data in the LagrangesMethod class. This may be more desirable than using the linearize class method, as the Linearizer object will allow more efficient recalculation (i.e. about varying operating points).

Parameters:

q_ind, qd_ind : array_like, optional

The independent generalized coordinates and speeds.

q_dep, qd_dep : array_like, optional

The dependent generalized coordinates and speeds.

linear_solver : str, callable

Method used to solve the several symbolic linear systems of the form A*x=b in the linearization process. If a string is supplied, it should be a valid method that can be used with the sympy.matrices.matrixbase.MatrixBase.solve(). If a callable is supplied, it should have the format x = f(A, b), where it solves the equations and returns the solution. The default is 'LU' which corresponds to SymPy’s A.LUsolve(b). LUsolve() is fast to compute but will often result in divide-by-zero and thus nan results.

Returns:

Linearizer

An instantiated sympy.physics.mechanics.linearize.Linearizer.