Certainly! Here's a step-by-step explanation for deriving the forward kinematic equations of the three-link planar manipulator using the DH (Denavit-Hartenberg) convention: ### Step 1: Assign coordinate frames - Assign coordinate frames to each joint, starting from the base. - Frame is at the base, Frame 1 at joint 1, Frame 2 at joint 2, and Frame 3 at joint 3 (end-effector). ### Step 2: Identify joint variables - For a planar manipulator, assume all joints are revolute. - Let \( \theta_1, \theta_2, \theta_3 \) be the joint angles. ### Step 3: Determine DH parameters For each link \( i \): | Link \( i \) | \( a_i \) | \( \alpha_i \) | \( d_i \) | \( \theta_i \) | |--------------|-----------|----------------|-----------|----------------| | 1 | \( a_1 \) | | | \( \theta_1 \) | | 2 | \( a_2 \) | | | \( \theta_2 \) | | 3 | \( a_3 \) | | | \( \theta_3 \) | - \( a_i \): link length - \( \alpha_i \): link twist ( for planar) - \( d_i \): link offset ( for revolute joints in a planar setup) ### Step 4: Write the transformation matrices The transformation matrix from frame \( i-1 \) to \( i \): \[ T_i^{i-1} = \begin{bmatrix} \cos \theta_i & -\sin \theta_i & & a_i \cos \theta_i \\ \sin \theta_i & \cos \theta_i & & a_i \sin \theta_i \\ & & 1 & \\ & & & 1 \end{bmatrix} \] ### Step 5: Compute the overall transformation matrix Multiply the individual transformation matrices: \[ T_^3 = T_1^ \times T_2^1 \times T_3^2 \] which results in the position of the end-effector: \[ \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \text{position part of } T_^3 \] ### Step 6: Derive the forward kinematic equations Express the position of the end-effector in terms of joint angles and link lengths: \[ x = a_1 \cos \theta_1 + a_2 \cos (\theta_1 + \theta_2) + a_3 \cos (\theta_1 + \theta_2 + \theta_3) \] \[ y = a_1 \sin \theta_1 + a_2 \sin (\theta_1 + \theta_2) + a_3 \sin (\theta_1 + \theta_2 + \theta_3) \] ### Final answer: The forward kinematic equations for the position of the end-eff are: \[ x = a_1 \cos \theta_1 + a_2 \cos (\theta_1 + \theta_2) + a_3 \cos (\theta_1 + \theta_2 + \theta_3) \] \[ y = a_1 \sin \theta_1 + a_2 \sin (\theta_1 + \theta_2) + a_3 \sin (\theta_1 + \theta_2 + \theta_3) \] This describes the position of the end-effector in the plane based on joint angles and link lengths.