Difference between example Acrobot plant A matrix and standard form - drake

Question

In section 3.4.1 of the Underactuated Robotics notes (https://underactuated.mit.edu/acrobot.html#section4), the manipulator equations are linearized around a fixed point and the matrix A_lin is derived.
While verifying the linearization of my own attempt at making an acrobot, I used the python notebook provided in Example 3.5 (LQR for the Acrobot and Cart-pole) to obtain the A matrix of the linearized Acrobot (Plant from the Examples module). I did this by simply adding 'print(linearized_acrobot.A())' on line 21 of the LQR for Acrobot block. Interestingly, I noticed that the bottom right 2x2 block is nonzero, which is different from the form derived in the notes. What is the reason behind the difference? For convenience I'll leave the code below:
import matplotlib.pyplot as plt
import mpld3
import numpy as np
from IPython.display import HTML, display
from pydrake.all import (AddMultibodyPlantSceneGraph, ControllabilityMatrix,
DiagramBuilder, Linearize, LinearQuadraticRegulator,
MeshcatVisualizerCpp, Parser, Saturation, SceneGraph,
Simulator, StartMeshcat, WrapToSystem)
from pydrake.examples.acrobot import (AcrobotGeometry, AcrobotInput,
AcrobotPlant, AcrobotState)
from pydrake.solvers.mathematicalprogram import MathematicalProgram, Solve
from underactuated import FindResource, running_as_notebook
from underactuated.meshcat_cpp_utils import MeshcatSliders
from underactuated.quadrotor2d import Quadrotor2D, Quadrotor2DVisualizer
if running_as_notebook:
mpld3.enable_notebook()
def UprightState():
state = AcrobotState()
state.set_theta1(np.pi)
state.set_theta2(0.)
state.set_theta1dot(0.)
state.set_theta2dot(0.)
return state
def acrobot_controllability():
acrobot = AcrobotPlant()
context = acrobot.CreateDefaultContext()
input = AcrobotInput()
input.set_tau(0.)
acrobot.get_input_port(0).FixValue(context, input)
context.get_mutable_continuous_state_vector()\
.SetFromVector(UprightState().CopyToVector())
linearized_acrobot = Linearize(acrobot, context)
print(linearized_acrobot.A())
print(
f"The singular values of the controllability matrix are: {np.linalg.svd(ControllabilityMatrix(linearized_acrobot), compute_uv=False)}"
)
acrobot_controllability()

Answer 1

Great question. The AcrobotPlant in Drake has default parameters which include some joint friction, which leads to non-zero elements in the bottom corner. If you amend your code with
acrobot = AcrobotPlant()
context = acrobot.CreateDefaultContext()
params = acrobot.get_mutable_parameters(context)
print(params)
params.set_b1(0)
params.set_b2(0)
then the bottom-right 2x2 elements of the linearized A are zero as expected.