Double Pendulum Simulation: SymPy

import numpy as np import matplotlib.pyplot as plt import sympy from matplotlib.animation import FFMpegWriter from matplotlib.patches import Circle

# Pendulum Parameters: m_1 = 1 # kg m_2 = 1 # kg l_1 = 1 # m l_2 = 1 # m g = 9.8 # m/s^2 # Create Symbols for Time: t = sympy.Symbol('t') # Creates symbolic variable t # Create Generalized Coordinates as a function of time: q = [theta_1, theta_2] th1 = sympy.Function('th1')(t) th2 = sympy.Function('th2')(t) # Position Equation: r = [x, y] r1 = np.array([l_1 * sympy.sin(th1), -l_1 * sympy.cos(th1)]) # Position of first pendulum r2 = np.array([l_2 * sympy.sin(th2) + r1[0], -l_2 * sympy.cos(th2) + r1[1]]) # Position of second pendulum # Velocity Equation: d/dt(r) = [dx/dt, dy/dt] v1 = np.array([r1[0].diff(t), r1[1].diff(t)]) # Velocity of first pendulum v2 = np.array([r2[0].diff(t), r2[1].diff(t)]) # Velocity of second pendulum # Energy Equations: T = 1/2 * m_1 * np.dot(v1, v1) + 1/2 * m_2 * np.dot(v2, v2) # Kinetic Energy V = m_1 * g * r1[1] + m_2 * g * r2[1] # Potential Energy L = T - V # Lagrangian # Lagrange Terms: dL_dth1 = L.diff(th1) dL_dth1_dt = L.diff(th1.diff(t)).diff(t) dL_dth2 = L.diff(th2) dL_dth2_dt = L.diff(th2.diff(t)).diff(t) # Euler-Lagrange Equations: dL/dq - d/dt(dL/ddq) = 0 th1_eqn = dL_dth1 - dL_dth1_dt th2_eqn = dL_dth2 - dL_dth2_dt # Replace Time Derivatives and Functions with Symbolic Variables: replacements = [(th1.diff(t).diff(t), sympy.Symbol('ddth1')), (th1.diff(t), sympy.Symbol('dth1')), (th1, sympy.Symbol('th1')), (th2.diff(t).diff(t), sympy.Symbol('ddth2')), (th2.diff(t), sympy.Symbol('dth2')), (th2, sympy.Symbol('th2'))] th1_eqn = th1_eqn.subs(replacements) th2_eqn = th2_eqn.subs(replacements) r1 = r1[0].subs(replacements), r1[1].subs(replacements) r2 = r2[0].subs(replacements), r2[1].subs(replacements) # Simplfiy then Force SymPy to Cancel factorization: [Sometimes needed to use .coeff()] th1_eqn = sympy.simplify(th1_eqn) th2_eqn = sympy.simplify(th2_eqn) th1_eqn = th1_eqn.cancel() th2_eqn = th2_eqn.cancel() # Solve for Coefficients for A * x = B where x = [ddth1 ddth2] A1 = th1_eqn.coeff(sympy.Symbol('ddth1')) A2 = th1_eqn.coeff(sympy.Symbol('ddth2')) A3 = th2_eqn.coeff(sympy.Symbol('ddth1')) A4 = th2_eqn.coeff(sympy.Symbol('ddth2')) # Multiply remaining terms by -1 to switch to other side of equation: A * x - B = 0 -> A * x = B remainder = [(sympy.Symbol('ddth1'), 0), (sympy.Symbol('ddth2'), 0)] B1 = -1 * th1_eqn.subs(remainder) B2 = -1 * th2_eqn.subs(remainder) # Generate Lambda Functions for A and B and Position Equations: replacements = (sympy.Symbol('th1'), sympy.Symbol('dth1'), sympy.Symbol('th2'), sympy.Symbol('dth2')) A1 = sympy.utilities.lambdify(replacements, A1, "numpy") A2 = sympy.utilities.lambdify(replacements, A2, "numpy") A3 = sympy.utilities.lambdify(replacements, A3, "numpy") A4 = sympy.utilities.lambdify(replacements, A4, "numpy") B1 = sympy.utilities.lambdify(replacements, B1, "numpy") B2 = sympy.utilities.lambdify(replacements, B2, "numpy") r1 = sympy.utilities.lambdify(replacements, r1, "numpy") r2 = sympy.utilities.lambdify(replacements, r2, "numpy")

# Simulate System: x0 = 1, 0, 0, 1 # th1, dth1, th2, dth2 dt = 0.001 sim_time = 10 time = np.arange(0, sim_time, dt) sim_length = len(time) # Initialize Arrays: th1_vec = np.zeros(sim_length) dth1_vec = np.zeros(sim_length) th2_vec = np.zeros(sim_length) dth2_vec = np.zeros(sim_length) x1_vec = np.zeros(sim_length) y1_vec = np.zeros(sim_length) x2_vec = np.zeros(sim_length) y2_vec = np.zeros(sim_length) # Evaluate Initial Conditions: th1_vec[0] = x0[0] dth1_vec[0] = x0[1] th2_vec[0] = x0[2] dth2_vec[0] = x0[3] x1_vec[0], y1_vec[0] = r1(x0[0], x0[1], x0[2], x0[3]) x2_vec[0], y2_vec[0] = r2(x0[0], x0[1], x0[2], x0[3]) # Initialize A and B: A = np.array([[0, 0], [0, 0]]) B = np.array([0, 0]) # Euler Integration: for i in range(1, sim_length): # Evaluate Dynamics: A[0, 0] = A1(th1_vec[i-1], dth1_vec[i-1], th2_vec[i-1], dth2_vec[i-1]) A[0, 1] = A2(th1_vec[i-1], dth1_vec[i-1], th2_vec[i-1], dth2_vec[i-1]) A[1, 0] = A3(th1_vec[i-1], dth1_vec[i-1], th2_vec[i-1], dth2_vec[i-1]) A[1, 1] = A4(th1_vec[i-1], dth1_vec[i-1], th2_vec[i-1], dth2_vec[i-1]) B[0] = B1(th1_vec[i-1], dth1_vec[i-1], th2_vec[i-1], dth2_vec[i-1]) B[1] = B2(th1_vec[i-1], dth1_vec[i-1], th2_vec[i-1], dth2_vec[i-1]) [ddth1, ddth2] = np.linalg.solve(A, B) # Euler Step Integration: th1_vec[i] = th1_vec[i-1] + dth1_vec[i-1] * dt dth1_vec[i] = dth1_vec[i-1] + ddth1 * dt th2_vec[i] = th2_vec[i-1] + dth2_vec[i-1] * dt dth2_vec[i] = dth2_vec[i-1] + ddth2 * dt # Animation States: x1_vec[i], y1_vec[i] = r1(th1_vec[i-1], dth1_vec[i-1], th2_vec[i-1], dth2_vec[i-1]) x2_vec[i], y2_vec[i] = r2(th1_vec[i-1], dth1_vec[i-1], th2_vec[i-1], dth2_vec[i-1])

# Create Animation: # Setup Figure: Initialize Figure / Axe Handles fig, ax = plt.subplots() p1, = ax.plot([], [], color='black', linewidth=2) p2, = ax.plot([], [], color='black', linewidth=2) lb, ub = -5, 5 ax.axis('equal') ax.set_xlim([lb, ub]) ax.set_xlabel('X') # X Label ax.set_ylabel('Y') # Y Label ax.set_title('Double Pendulum Simulation:') video_title = "simulation" # Setup Animation Writer: FPS = 20 sample_rate = int(1 / (dt * FPS)) dpi = 300 writerObj = FFMpegWriter(fps=FPS) # Initialize Patch: Pendulum 1 and 2 pendulum_1 = Circle((0, 0), radius=0.1, color='cornflowerblue', zorder=10) pendulum_2 = Circle((0, 0), radius=0.1, color='cornflowerblue', zorder=10) ax.add_patch(pendulum_1) ax.add_patch(pendulum_2) # Draw Static Objects: pin_joint = Circle((0, 0), radius=0.05, color='black', zorder=10) ax.add_patch(pin_joint) # Plot and Create Animation: with writerObj.saving(fig, video_title+".mp4", dpi): for i in range(0, sim_length, sample_rate): # Draw Pendulum Arm: x_pendulum_arm = [0, x1_vec[i], x2_vec[i]] y_pendulum_arm = [0, y1_vec[i], y2_vec[i]] p1.set_data(x_pendulum_arm, y_pendulum_arm) # Update Pendulum Patches: pendulum_1_center = x1_vec[i], y1_vec[i] pendulum_2_center = x2_vec[i], y2_vec[i] pendulum_1.center = pendulum_1_center pendulum_2.center = pendulum_2_center # Update Drawing: fig.canvas.draw() # Grab and Save Frame: writerObj.grab_frame()

from IPython.display import Video Video("/work/"+video_title+".mp4", embed=True, width=640, height=480)