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openchain.py
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openchain.py
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import numpy as np
import sympy as smp
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
from typing import TypeAlias
from tqdm import tqdm
from sympy import lambdify
from scipy import integrate
from visualization import AnimatableKSegmentData, Animator, AnimatableLinePlot
# simulation constants
class Params:
def __init__(self, n: int, # number of links
ls: list[float], # length of each link
ms: list[float], # mass of each link
inertias: list[float], # inertia of each link
rcom: list[float], # ratio of center of mass. com is p[i] + (p[i+1] - p[i]) * rcom[i]
g: float # gravity
):
self.n = n
self.ls = ls
self.ms = ms
self.inertias = inertias
self.rcom = rcom
self.g = g
Vector: TypeAlias = np.array
class Dynamics:
def __init__(self, params: Params):
self.params = params
def forward_dynamics(self, q: Vector, dq: Vector, tau: Vector):
""" calculates ddq/dt2 """
raise NotImplementedError
def forward_dynamics(self, q: Vector, dq: Vector, ddq: Vector):
""" calculates tau to create ddq """
raise NotImplementedError
class LagrangianDynamics(Dynamics):
def __init__(self, params: Params):
super().__init__(params)
self.M, self.h = self.formulate_dynamics()
# M(q) ddq + h(q, dq) = tau
def formulate_dynamics(self):
n = self.params.n
q = np.array(smp.MatrixSymbol('q', n, 1)).reshape((n,))
dq = np.array(smp.MatrixSymbol('dq', n, 1)).reshape((n,))
rcom = self.params.rcom
ls = self.params.ls
ms = self.params.ms
g = self.params.g
inertias = self.params.inertias
p_joint = np.zeros(2)
K = smp.numer(0)
U = smp.numer(0)
for i in tqdm(range(n), 'construct kinematics'):
qsum = q[0:i + 1].sum()
vec = np.block([smp.cos(qsum) * ls[i],
smp.sin(qsum) * ls[i]]
)
p_com = p_joint + rcom[i] * vec
v_com_0 = sum(smp.diff(p_com[0], q[j]) * dq[j] for j in range(i + 1)) # jacobian
v_com_1 = sum(smp.diff(p_com[1], q[j]) * dq[j] for j in range(i + 1)) # jacobian
v_com = np.array([v_com_0, v_com_1])
w_com = dq[0:i + 1].sum()
U += ms[i] * g * p_com[1]
K += 0.5 * ((v_com @ v_com) * ms[i] + w_com * w_com * inertias[i])
p_joint = p_joint + vec
K = K.simplify()
M = [
[K.diff(dq[i]).diff(dq[j]).simplify()
for j in range(n)]
for i in tqdm(range(n), 'calculate mass matrix')] # M = ddK / dq^2
C = [
sum(
(0.5 * (M[i][k].diff(q[j]) + M[i][j].diff(q[k]) - M[j][k].diff(q[i])) * dq[j] * dq[k]).simplify()
for j in range(n) for k in range(n)
)
for i in tqdm(range(n), 'calculate coriolis matrix'
'ix')] # C = dqT Gamma dq
G = [
smp.diff(U, q[i])
for i in tqdm(range(n), 'calculating g(theta)')
]
M_lambified = [
[lambdify([q], M[i][j], 'numpy')
for j in range(n)]
for i in range(n)
]
M_func = lambda q: np.array([
[M_lambified[i][j](q)
for j in range(n)]
for i in range(n)
])
C_lambified = [lambdify([q, dq], C[i], 'numpy')
for i in range(n)]
G_lambified = [lambdify([q], G[i], 'numpy')
for i in range(n)]
h_func = lambda q, dq: np.array([
G_lambified[i](q) + C_lambified[i](q, dq)
for i in range(n)
])
return M_func, h_func
def forward_dynamics(self, q: Vector, dq: Vector, tau: Vector):
""" calculates ddq/dt2 """
# M(q) ddq + h(q, dq) = tau
return np.linalg.solve(
self.M(q),
tau - self.h(q, dq)
)
def mass_matrix(self, q):
return self.M(q)
class Kinematics:
def __init__(self, params: Params):
self.params = params
def get_positions(self, q): # join state
n = self.params.n
ls = self.params.ls
p_joint = np.zeros(2)
ps = [p_joint]
for i in range(n):
qsum = q[0:i + 1].sum()
vec = np.block([np.cos(qsum) * ls[i],
np.sin(qsum) * ls[i]])
p_joint = p_joint + vec
ps.append(p_joint)
return ps
def get_com_positions(self, q):
n = self.params.n
ls = self.params.ls
rs = self.params.rcom
p_joint = np.zeros(2)
p_coms = []
for i in range(n):
qsum = q[0:i + 1].sum()
vec = np.block([np.cos(qsum) * ls[i],
np.sin(qsum) * ls[i]])
p_com = p_joint + vec * rs[i]
p_joint = p_joint + vec
p_coms.append(p_com)
return p_coms
def simulate_no_torque(params: Params, dt: float, duration: float, q0: Vector, dq0: Vector):
cur_t = 0
point_history = []
mech_e_hist = []
potential_e_hist = []
kinetic_e_hist = []
ts = []
kinematics = Kinematics(params)
dynamics = LagrangianDynamics(params)
q = q0
dq = dq0
for _frame in tqdm(range(round(duration/dt)), "simulating open chain"):
points = kinematics.get_positions(q)
point_history.append(points)
ts.append(cur_t)
potential = sum(
m * pos[1] * params.g
for m, pos in zip(params.ms, kinematics.get_com_positions(q))
)
kinetic = 0.5 * (dq.T @ dynamics.mass_matrix(q) @ dq)
potential_e_hist.append(potential)
kinetic_e_hist.append(kinetic)
mech_e_hist.append(potential + kinetic)
zero_tau = np.zeros_like(q)
# X = q, dq
x_shape = np.array([q, dq]).shape # we have to flatten it since ode only works with flat vectors
def dstate_dt(t, X):
q, dq = list(X.reshape(x_shape))
dX = np.array([dq, dynamics.forward_dynamics(q, dq, zero_tau)])
return dX.reshape((-1,))
cur_t += dt
x0 = np.array([q, dq]).reshape((-1,))
x_new = integrate.odeint(dstate_dt, x0, [0, dt], tfirst=True)[1]
q, dq = list(x_new.reshape(x_shape))
return point_history, {
"potential_e_hist": potential_e_hist,
"mech_e_hist": mech_e_hist,
"kinetic_e_hist": kinetic_e_hist,
"ts": ts}
def plot_animation(points_history, logs, total_time, speed):
# plot
fig = plt.figure()
fig.set_figheight(8)
fig.set_figwidth(8)
spec = plt.GridSpec(ncols=1, nrows=4,
width_ratios=[1], wspace=0.5,
hspace=0.5, height_ratios=[4, 1, 1, 1])
main_ax = fig.add_subplot(spec[0])
u_ax = fig.add_subplot(spec[1])
k_ax = fig.add_subplot(spec[2])
e_ax = fig.add_subplot(spec[3])
segments = AnimatableKSegmentData(main_ax, points_data=points_history, name="open chain", fix_scale=True)
potential_e_plot = AnimatableLinePlot(u_ax, x_data=logs["ts"], y_data=logs["potential_e_hist"], name="potential energy", fix_scale=False)
kinetic_e_plot = AnimatableLinePlot(k_ax, x_data=logs["ts"], y_data=logs["kinetic_e_hist"], name="kinetic energy", fix_scale=False)
mech_e_plot = AnimatableLinePlot(e_ax, x_data=logs["ts"], y_data=logs["mech_e_hist"], name="mechanical energy", fix_scale=False)
animator = Animator(fig=fig, interval=30, total_time=total_time,
animatable_datas=[segments, mech_e_plot, potential_e_plot, kinetic_e_plot],
speed=speed)
anim = FuncAnimation(animator.fig,
animator.animate,
frames=animator.num_frames,
interval=animator.interval)
plt.show()
if __name__ == "__main__":
params = Params(
n=3,
ls=[1, 1, 1],
ms=[1, 1, 1],
inertias=[0.1, 0.1, 0.1],
rcom=[0.3, 0.3, 0.3],
g=9.8)
q0 = np.array([0, np.pi/2, -np.pi/2])
dq0 = np.array([0, 0, 0])
total_time = 10
points_history, logs = simulate_no_torque(params=params, dt=0.001, duration=total_time, q0=q0, dq0=dq0)
points_history = np.array(points_history)
plot_animation(points_history, logs, total_time=total_time, speed=1)