Kane’s Method in Physics/Mechanics

mechanics provides functionality for deriving equations of motion using Kane’s method [Kane1985]. This document will describe Kane’s method as used in this module, but not how the equations are actually derived.

Structure of Equations

In mechanics we are assuming there are 5 basic sets of equations needed to describe a system. They are: holonomic constraints, non-holonomic constraints, kinematic differential equations, dynamic equations, and differentiated non-holonomic equations.

\[\begin{split}\mathbf{f_h}(q, t) &= 0\\ \mathbf{k_{nh}}(q, t) u + \mathbf{f_{nh}}(q, t) &= 0\\ \mathbf{k_{k\dot{q}}}(q, t) \dot{q} + \mathbf{k_{ku}}(q, t) u + \mathbf{f_k}(q, t) &= 0\\ \mathbf{k_d}(q, t) \dot{u} + \mathbf{f_d}(q, \dot{q}, u, t) &= 0\\ \mathbf{k_{dnh}}(q, t) \dot{u} + \mathbf{f_{dnh}}(q, \dot{q}, u, t) &= 0\\\end{split}\]

In mechanics holonomic constraints are only used for the linearization process; it is assumed that they will be too complicated to solve for the dependent coordinate(s). If you are able to easily solve a holonomic constraint, you should consider redefining your problem in terms of a smaller set of coordinates. Alternatively, the time-differentiated holonomic constraints can be supplied.

Kane’s method forms two expressions, \(F_r\) and \(F_r^*\), whose sum is zero. In this module, these expressions are rearranged into the following form:

\(\mathbf{M}(q, t) \dot{u} = \mathbf{f}(q, \dot{q}, u, t)\)

For a non-holonomic system with \(o\) total speeds and \(m\) motion constraints, we will get o - m equations. The mass-matrix/forcing equations are then augmented in the following fashion:

\[\begin{split}\mathbf{M}(q, t) &= \begin{bmatrix} \mathbf{k_d}(q, t) \\ \mathbf{k_{dnh}}(q, t) \end{bmatrix}\\ \mathbf{_{(forcing)}}(q, \dot{q}, u, t) &= \begin{bmatrix} - \mathbf{f_d}(q, \dot{q}, u, t) \\ - \mathbf{f_{dnh}}(q, \dot{q}, u, t) \end{bmatrix}\\\end{split}\]

Kane’s Method in Physics/Mechanics

The formulation of the equations of motion in mechanics starts with creation of a KanesMethod object. Upon initialization of the KanesMethod object, an inertial reference frame needs to be supplied. along with some basic system information, such as coordinates and speeds

>>> from sympy.physics.mechanics import *
>>> N = ReferenceFrame('N')
>>> q1, q2, u1, u2 = dynamicsymbols('q1 q2 u1 u2')
>>> q1d, q2d, u1d, u2d = dynamicsymbols('q1 q2 u1 u2', 1)
>>> KM = KanesMethod(N, [q1, q2], [u1, u2])

It is also important to supply the order of coordinates and speeds properly if there are dependent coordinates and speeds. They must be supplied after independent coordinates and speeds or as a keyword argument; this is shown later.

>>> q1, q2, q3, q4 = dynamicsymbols('q1 q2 q3 q4')
>>> u1, u2, u3, u4 = dynamicsymbols('u1 u2 u3 u4')
>>> # Here we will assume q2 is dependent, and u2 and u3 are dependent
>>> # We need the constraint equations to enter them though
>>> KM = KanesMethod(N, [q1, q3, q4], [u1, u4])

Additionally, if there are auxiliary speeds, they need to be identified here. See the examples for more information on this. In this example u4 is the auxiliary speed.

>>> KM = KanesMethod(N, [q1, q3, q4], [u1, u2, u3], u_auxiliary=[u4])

Kinematic differential equations must also be supplied; there are to be provided as a list of expressions which are each equal to zero. A trivial example follows:

>>> kd = [q1d - u1, q2d - u2]

Turning on mechanics_printing() makes the expressions significantly shorter and is recommended. Alternatively, the mprint and mpprint commands can be used.

If there are non-holonomic constraints, dependent speeds need to be specified (and so do dependent coordinates, but they only come into play when linearizing the system). The constraints need to be supplied in a list of expressions which are equal to zero, trivial motion and configuration constraints are shown below:

>>> N = ReferenceFrame('N')
>>> q1, q2, q3, q4 = dynamicsymbols('q1 q2 q3 q4')
>>> q1d, q2d, q3d, q4d = dynamicsymbols('q1 q2 q3 q4', 1)
>>> u1, u2, u3, u4 = dynamicsymbols('u1 u2 u3 u4')
>>> #Here we will assume q2 is dependent, and u2 and u3 are dependent
>>> speed_cons = [u2 - u1, u3 - u1 - u4]
>>> coord_cons = [q2 - q1]
>>> q_ind = [q1, q3, q4]
>>> q_dep = [q2]
>>> u_ind = [u1, u4]
>>> u_dep = [u2, u3]
>>> kd = [q1d - u1, q2d - u2, q3d - u3, q4d - u4]
>>> KM = KanesMethod(N, q_ind, u_ind, kd,
...           q_dependent=q_dep,
...           configuration_constraints=coord_cons,
...           u_dependent=u_dep,
...           velocity_constraints=speed_cons)

A dictionary returning the solved \(\dot{q}\)‘s can also be solved for:

>>> mechanics_printing(pretty_print=False)
>>> KM.kindiffdict()
{q1': u1, q2': u2, q3': u3, q4': u4}

The final step in forming the equations of motion is supplying a list of bodies and particles, and a list of 2-tuples of the form (Point, Vector) or (ReferenceFrame, Vector) to represent applied forces and torques.

>>> N = ReferenceFrame('N')
>>> q, u = dynamicsymbols('q u')
>>> qd, ud = dynamicsymbols('q u', 1)
>>> P = Point('P')
>>> P.set_vel(N, u * N.x)
>>> Pa = Particle('Pa', P, 5)
>>> BL = [Pa]
>>> FL = [(P, 7 * N.x)]
>>> KM = KanesMethod(N, [q], [u], [qd - u])
>>> (fr, frstar) = KM.kanes_equations(BL, FL)
>>> KM.mass_matrix
Matrix([[5]])
>>> KM.forcing
Matrix([[7]])

When there are motion constraints, the mass matrix is augmented by the \(k_{dnh}(q, t)\) matrix, and the forcing vector by the \(f_{dnh}(q, \dot{q}, u, t)\) vector.

There are also the “full” mass matrix and “full” forcing vector terms, these include the kinematic differential equations; the mass matrix is of size (n + o) x (n + o), or square and the size of all coordinates and speeds.

>>> KM.mass_matrix_full
Matrix([
[1, 0],
[0, 5]])
>>> KM.forcing_full
Matrix([
[u],
[7]])

Exploration of the provided examples is encouraged in order to gain more understanding of the KanesMethod object.