Let's break this down **step-by-step**, explaining the process to implement the simulation for the two-link manipulator in Python (you can easily adapt to Matlab if needed). **Final Python code is provided at the end.** --- ## **Step 1: Equations of Motion** For a two-link manipulator, the equations of motion (from the Euler-Lagrange method) are typically: \[ M(q)\ddot{q} + C(q, \dot{q})\dot{q} + G(q) = \tau \] Where: - \( q = [q_1, q_2]^T \): joint angles - \( \dot{q} = [\dot{q}_1, \dot{q}_2]^T \): joint velocities - \( \ddot{q} = [\ddot{q}_1, \ddot{q}_2]^T \): joint accelerations - \( M(q) \): inertia matrix - \( C(q, \dot{q}) \): Coriolis/centrifugal matrix - \( G(q) \): gravity vector - \( \tau = [\tau_1, \tau_2]^T \): joint torques --- ## **Step 2: Plug in Parameters** Given values: - \( m_1 = m_2 = 0.5 \) - \( l_1 = l_2 = 0.2 \) - \( l_{c1} = l_{c2} = 0.1 \) - \( I_1 = I_2 = m \cdot l^2 / 12 \) - \( g = 9.81 \) - \( dt = 0.01 \) - Initial: \( q = [-\pi/2, 0], \dot{q} = [0, 0] \) - Torques: \(\tau_1 = 0.2, \tau_2 = 0.2\) --- ## **Step 3: Write Out Matrices** **Inertia Matrix \( M(q) \):** \[ M_{11} = I_1 + I_2 + m_2 l_1^2 + 2 m_2 l_1 l_{c2} \cos(q_2) + m_1 l_{c1}^2 + m_2 l_{c2}^2 \] \[ M_{12} = I_2 + m_2 l_1 l_{c2} \cos(q_2) + m_2 l_{c2}^2 \] \[ M_{21} = M_{12} \] \[ M_{22} = I_2 + m_2 l_{c2}^2 \] **Coriolis and Centrifugal Matrix \( C(q, \dot{q}) \):** \[ C_{11} = -m_2 l_1 l_{c2} \sin(q_2) \dot{q}_2 \] \[ C_{12} = -m_2 l_1 l_{c2} \sin(q_2) (\dot{q}_1 + \dot{q}_2) \] \[ C_{21} = m_2 l_1 l_{c2} \sin(q_2) \dot{q}_1 \] \[ C_{22} = 0 \] **Gravity Vector \( G(q) \):** \[ G_1 = (m_1 l_{c1} + m_2 l_1) g \cos(q_1) + m_2 l_{c2} g \cos(q_1 + q_2) \] \[ G_2 = m_2 l_{c2} g \cos(q_1 + q_2) \] --- ## **Step 4: Euler Integration** Given \(\ddot{q}\), you update as: \[ \dot{q}_{n+1} = \dot{q}_n + \ddot{q}_n dt \\ q_{n+1} = q_n + \dot{q}_{n+1} dt \] --- ## **Step 5: Implementation in Python** Here is a simple Python implementation: ```python import numpy as np import matplotlib.pyplot as plt # Parameters m1 = 0.5 m2 = 0.5 l1 = 0.2 l2 = 0.2 lc1 = 0.1 lc2 = 0.1 I1 = m1 * l1**2 / 12 I2 = m2 * l2**2 / 12 g = 9.81 dt = 0.01 steps = 1000 # Initial conditions q = np.array([[-np.pi/2], [0.0]]) # q1, q2 qd = np.array([[0.0], [0.0]]) # q1_dot, q2_dot # Constant torques tau = np.array([[0.2], [0.2]]) # Storage Q = [q.flatten()] QD = [qd.flatten()] for i in range(steps): q1, q2 = q.flatten() q1d, q2d = qd.flatten() # Inertia Matrix M(q) m11 = I1 + I2 + m2 * l1**2 + 2 * m2 * l1 * lc2 * np.cos(q2) + m1 * lc1**2 + m2 * lc2**2 m12 = I2 + m2 * l1 * lc2 * np.cos(q2) + m2 * lc2**2 m22 = I2 + m2 * lc2**2 M = np.array([[m11, m12], [m12, m22]]) # Coriolis/Centrifugal h = -m2 * l1 * lc2 * np.sin(q2) c1 = h * q2d c2 = h * (q1d + q2d) C = np.array([[c1, c2], [-c1, 0]]) # Gravity g1 = (m1 * lc1 + m2 * l1) * g * np.cos(q1) + m2 * lc2 * g * np.cos(q1 + q2) g2 = m2 * lc2 * g * np.cos(q1 + q2) G = np.array([[g1], [g2]]) # Equation of motion: M*ddq + C*qd + G = tau qdd = np.linalg.inv(M) @ (tau - (C @ qd) - G) # Euler integration qd = qd + qdd * dt q = q + qd * dt Q.append(q.flatten()) QD.append(qd.flatten()) Q = np.array(Q) QD = np.array(QD) # Plot plt.figure(figsize=(10,5)) plt.plot(Q[:, 0], label='q1 (rad)') plt.plot(Q[:, 1], label='q2 (rad)') plt.xlabel('Time step') plt.ylabel('Joint Angles (rad)') plt.legend() plt.title('Two-Link Manipulator Joint Angles') plt.show() ``` --- ## **Summary** - **Equations of motion** are set up using the given parameters. - **Euler integration** is used to update positions and velocities step-by-step. - **Python code** provided can be run as is. - **Final Answer:** The code simulates the two-link manipulator's motion under constant torque, starting from the specified initial conditions. --- **Let me know if you want Matlab code or further explanation on any step!**