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Nonminimal Coordinates Pendulum

In this example we demonstrate the use of the functionality provided in mechanics for deriving the equations of motion (EOM) for a pendulum with a nonminimal set of coordinates. As the pendulum is a one degree of freedom system, it can be described using one coordinate and one speed (the pendulum angle, and the angular velocity respectively). Choosing instead to describe the system using the \(x\) and \(y\) coordinates of the mass results in a need for constraints. The system is shown below:

The system will be modeled using both Kane’s and Lagrange’s methods, and the resulting EOM linearized. While this is a simple problem, it should illustrate the use of the linearization methods in the presence of constraints.

Kane’s Method

First we need to create the dynamicsymbols needed to describe the system as shown in the above diagram. In this case, the generalized coordinates \(q_1\) and \(q_2\) represent the mass \(x\) and \(y\) coordinates in the inertial \(N\) frame. Likewise, the generalized speeds \(u_1\) and \(u_2\) represent the velocities in these directions. We also create some symbols to represent the length and mass of the pendulum, as well as gravity and time.

>>> from sympy.physics.mechanics import *
>>> from sympy import symbols, atan, Matrix, solve
>>> # Create generalized coordinates and speeds for this non-minimal realization
>>> # q1, q2 = N.x and N.y coordinates of pendulum
>>> # u1, u2 = N.x and N.y velocities of pendulum
>>> q1, q2 = dynamicsymbols('q1:3')
>>> q1d, q2d = dynamicsymbols('q1:3', level=1)
>>> u1, u2 = dynamicsymbols('u1:3')
>>> u1d, u2d = dynamicsymbols('u1:3', level=1)
>>> L, m, g, t = symbols('L, m, g, t')

Next, we create a world coordinate frame \(N\), and its origin point \(N^*\). The velocity of the origin is set to 0. A second coordinate frame \(A\) is oriented such that its x-axis is along the pendulum (as shown in the diagram above).

>>> # Compose world frame
>>> N = ReferenceFrame('N')
>>> pN = Point('N*')
>>> pN.set_vel(N, 0)

>>> # A.x is along the pendulum
>>> theta1 = atan(q2/q1)
>>> A = N.orientnew('A', 'axis', [theta1, N.z])

Locating the pendulum mass is then as easy as specifying its location with in terms of its x and y coordinates in the world frame. A Particle object is then created to represent the mass at this location.

>>> # Locate the pendulum mass
>>> P = pN.locatenew('P1', q1*N.x + q2*N.y)
>>> pP = Particle('pP', P, m)

The kinematic differential equations (KDEs) relate the derivatives of the generalized coordinates to the generalized speeds. In this case the speeds are the derivatives, so these are simple. A dictionary is also created to map \(\dot{q}\) to \(u\):

>>> # Calculate the kinematic differential equations
>>> kde = Matrix([q1d - u1,
...               q2d - u2])
>>> dq_dict = solve(kde, [q1d, q2d])

The velocity of the mass is then the time derivative of the position from the origin \(N^*\):

>>> # Set velocity of point P
>>> P.set_vel(N, P.pos_from(pN).dt(N).subs(dq_dict))

As this system has more coordinates than degrees of freedom, constraints are needed. The configuration constraints relate the coordinates to each other. In this case the constraint is that the distance from the origin to the mass is always the length \(L\) (the pendulum doesn’t get longer). Likewise, the velocity constraint is that the mass velocity in the A.x direction is always 0 (no radial velocity).

>>> f_c = Matrix([P.pos_from(pN).magnitude() - L])
>>> f_v = Matrix([P.vel(N).express(A).dot(A.x)])
>>> f_v.simplify()

The force on the system is just gravity, at point P.

>>> # Input the force resultant at P
>>> R = m*g*N.x

With the problem setup, the equations of motion can be generated using the KanesMethod class. As there are constraints, dependent and independent coordinates need to be provided to the class. In this case we’ll use \(q_2\) and \(u_2\) as the independent coordinates and speeds:

>>> # Derive the equations of motion using the KanesMethod class.
>>> KM = KanesMethod(N, q_ind=[q2], u_ind=[u2], q_dependent=[q1],
...                  u_dependent=[u1], configuration_constraints=f_c,
...                  velocity_constraints=f_v, kd_eqs=kde)
>>> (fr, frstar) = KM.kanes_equations([(P, R)], [pP])

For linearization, operating points can be specified on the call, or be substituted in afterwards. In this case we’ll provide them in the call, supplied in a list. The A_and_B=True kwarg indicates to solve invert the \(M\) matrix and solve for just the explicit linearized \(A\) and \(B\) matrices. The simplify=True kwarg indicates to simplify inside the linearize call, and return the presimplified matrices. The cost of doing this is small for simple systems, but for larger systems this can be a costly operation, and should be avoided.

>>> # Set the operating point to be straight down, and non-moving
>>> q_op = {q1: L, q2: 0}
>>> u_op = {u1: 0, u2: 0}
>>> ud_op = {u1d: 0, u2d: 0}
>>> # Perform the linearization
>>> A, B, inp_vec = KM.linearize(op_point=[q_op, u_op, ud_op], A_and_B=True,
...                              new_method=True, simplify=True)
>>> A
Matrix([
[   0, 1],
[-g/L, 0]])
>>> B
Matrix(0, 0, [])

The resulting \(A\) matrix has dimensions 2 x 2, while the number of total states is len(q) + len(u) = 2 + 2 = 4. This is because for constrained systems the resulting A_and_B form has a partitioned state vector only containing the independent coordinates and speeds. Written out mathematically, the system linearized about this point would be written as:

\[\begin{split}\begin{bmatrix} \dot{q_2} \\ \dot{u_2} \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ \frac{-g}{L} & 0 \end{bmatrix} \begin{bmatrix} q_2 \\ u_2 \end{bmatrix}\end{split}\]

Lagrange’s Method

The derivation using Lagrange’s method is very similar to the approach using Kane’s method described above. As before, we first create the dynamicsymbols needed to describe the system. In this case, the generalized coordinates \(q_1\) and \(q_2\) represent the mass \(x\) and \(y\) coordinates in the inertial \(N\) frame. This results in the time derivatives \(\dot{q_1}\) and \(\dot{q_2}\) representing the velocities in these directions. We also create some symbols to represent the length and mass of the pendulum, as well as gravity and time.

>>> from sympy.physics.mechanics import *
>>> from sympy import symbols, atan, Matrix
>>> q1, q2 = dynamicsymbols('q1:3')
>>> q1d, q2d = dynamicsymbols('q1:3', level=1)
>>> L, m, g, t = symbols('L, m, g, t')

Next, we create a world coordinate frame \(N\), and its origin point \(N^*\). The velocity of the origin is set to 0. A second coordinate frame \(A\) is oriented such that its x-axis is along the pendulum (as shown in the diagram above).

>>> # Compose World Frame
>>> N = ReferenceFrame('N')
>>> pN = Point('N*')
>>> pN.set_vel(N, 0)
>>> # A.x is along the pendulum
>>> theta1 = atan(q2/q1)
>>> A = N.orientnew('A', 'axis', [theta1, N.z])

Locating the pendulum mass is then as easy as specifying its location with in terms of its x and y coordinates in the world frame. A Particle object is then created to represent the mass at this location.

>>> # Create point P, the pendulum mass
>>> P = pN.locatenew('P1', q1*N.x + q2*N.y)
>>> P.set_vel(N, P.pos_from(pN).dt(N))
>>> pP = Particle('pP', P, m)

As this system has more coordinates than degrees of freedom, constraints are needed. In this case only a single holonomic constraints is needed: the distance from the origin to the mass is always the length \(L\) (the pendulum doesn’t get longer).

>>> # Holonomic Constraint Equations
>>> f_c = Matrix([q1**2 + q2**2 - L**2])

The force on the system is just gravity, at point P.

>>> # Input the force resultant at P
>>> R = m*g*N.x

With the problem setup, the Lagrangian can be calculated, and the equations of motion formed. Note that the call to LagrangesMethod includes the Lagrangian, the generalized coordinates, the constraints (specified by hol_coneqs or nonhol_coneqs), the list of (body, force) pairs, and the inertial frame. In contrast to the KanesMethod initializer, independent and dependent coordinates are not partitioned inside the LagrangesMethod object. Such a partition is supplied later.

>>> # Calculate the lagrangian, and form the equations of motion
>>> Lag = Lagrangian(N, pP)
>>> LM = LagrangesMethod(Lag, [q1, q2], hol_coneqs=f_c, forcelist=[(P, R)], frame=N)
>>> lag_eqs = LM.form_lagranges_equations()

Next, we compose the operating point dictionary, set in the hanging at rest position:

>>> # Compose operating point
>>> op_point = {q1: L, q2: 0, q1d: 0, q2d: 0, q1d.diff(t): 0, q2d.diff(t): 0}

As there are constraints in the formulation, there will be corresponding Lagrange Multipliers. These may appear inside the linearized form as well, and thus should also be included inside the operating point dictionary. Fortunately, the LagrangesMethod class provides an easy way of solving for the multipliers at a given operating point using the solve_multipliers method.

>>> # Solve for multiplier operating point
>>> lam_op = LM.solve_multipliers(op_point=op_point)

With this solution, linearization can be completed. Note that in contrast to the KanesMethod approach, the LagrangesMethod.linearize method also requires the partitioning of the generalized coordinates and their time derivatives into independent and dependent vectors. This is the same as what was passed into the KanesMethod constructor above:

>>> op_point.update(lam_op)
>>> # Perform the Linearization
>>> A, B, inp_vec = LM.linearize([q2], [q2d], [q1], [q1d],
...                             op_point=op_point, A_and_B=True)
>>> A
Matrix([
[     0, 1],
[-g/L, 0]])
>>> B
Matrix(0, 0, [])

The resulting \(A\) matrix has dimensions 2 x 2, while the number of total states is 2*len(q) = 4. This is because for constrained systems the resulting A_and_B form has a partitioned state vector only containing the independent coordinates and their derivatives. Written out mathematically, the system linearized about this point would be written as:

\[\begin{split}\begin{bmatrix} \dot{q_2} \\ \ddot{q_2} \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ \frac{-g}{L} & 0 \end{bmatrix} \begin{bmatrix} q_2 \\ \dot{q_2} \end{bmatrix}\end{split}\]

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