# Industrial Robot DH Parameter Extraction and Transformation Matrices Let's solve the problem in structured steps: --- ## Step 1: Extract DH Parameters For each joint, fill the table with the Denavit-Hartenberg (DH) parameters, using the given lengths: - \( a_2 = 200\, \text{mm} \) - \( a_3 = 50\, \text{mm} \) - \( d_5 = 70\, \text{mm} \) - \( d_6 = 43\, \text{mm} \) | Link | θ | d | a | α | |------|------------|---------|---------|------------| | -1 | θ1 | | | 90° (π/2) | | 1-2 | θ2 | | \(a_2\) | ° | | 2-3 | θ3 | | \(a_3\) | 90° (π/2) | | 3-4 | θ4 | | | -90° (-π/2)| | 4-5 | θ5 | \(d_5\) | | 90° (π/2) | | 5-6 | θ6 | \(d_6\) | | ° | --- ## Step 2: Transformation Matrices The general DH transformation matrix is: \[ T_i^{i-1} = \begin{bmatrix} \cos\theta_i & -\sin\theta_i\cos\alpha_{i-1} & \sin\theta_i\sin\alpha_{i-1} & a_{i-1}\cos\theta_i \\ \sin\theta_i & \cos\theta_i\cos\alpha_{i-1} & -\cos\theta_i\sin\alpha_{i-1} & a_{i-1}\sin\theta_i \\ & \sin\alpha_{i-1} & \cos\alpha_{i-1} & d_i \\ & & & 1 \end{bmatrix} \] ### Example: First Two Transformation Matrices #### \( T_1^ \) (for joint 1): \[ T_1^ = \begin{bmatrix} \cos\theta_1 & & \sin\theta_1 & \\ \sin\theta_1 & & -\cos\theta_1 & \\ & 1 & & \\ & & & 1 \end{bmatrix} \] #### \( T_2^1 \) (for joint 2): \[ T_2^1 = \begin{bmatrix} \cos\theta_2 & -\sin\theta_2 & & a_2\cos\theta_2 \\ \sin\theta_2 & \cos\theta_2 & & a_2\sin\theta_2 \\ & & 1 & \\ & & & 1 \end{bmatrix} \] --- ## Final Answer: Complete Transformation Matrix The **overall transformation** from base to end-effector: \[ T_{06} = T_1^ \cdot T_2^1 \cdot T_3^2 \cdot T_4^3 \cdot T_5^4 \cdot T_6^5 \] Write each individual \( T \) as per the DH table above, using the provided \( a \) and \( d \) values. --- ### Example: Using Values Suppose \( \theta_1 = , \theta_2 = \): - \( T_1^ \) becomes: \[ \begin{bmatrix} 1 & & & \\ & & -1 & \\ & 1 & & \\ & & & 1 \end{bmatrix} \] - \( T_2^1 \) becomes: \[ \begin{bmatrix} 1 & & & 200 \\ & 1 & & \\ & & 1 & \\ & & & 1 \end{bmatrix} \] Multiply all 6 matrices for the final transformation. --- ## Graph (Robot Structure) Below is the reference image from your question (showing the robot structure with axes):  *Alt text: Industrial robot with axes and link lengths marked for DH parameter extraction.* --- ### **Summary Table (Final DH Table)** | Link | θ | d | a | α | |------|------|------|------|------| | -1 | θ1 | | | 90° | | 1-2 | θ2 | | 200 | ° | | 2-3 | θ3 | | 50 | 90° | | 3-4 | θ4 | | | -90° | | 4-5 | θ5 | 70 | | 90° | | 5-6 | θ6 | 43 | | ° | --- ### **Conclusion** 1. **Extracted DH parameters** in the table above. 2. **Transformation matrices** created for each axis. 3. **Overall transformation matrix** is the product of all six matrices. If you need the explicit multiplication for all six, let me know!

# Industrial Robot DH Parameter Extraction and Transformation Matrices Let's solve the problem in structured steps: ## Step 1: Extract DH Parameters For each joint, fill the table with the Denavit-Hartenberg (DH) parameters, using the given lengths: - \( a_2 = 200\, \text{mm} \) - \( a_3 = 50\, \text{mm} \) - \( d_5 = 70\, \text{mm} \) - \( d_6 = 43\, \text{mm} \) | Link | θ | d | a | α | |------|------------|---------|---------|------------| | -1 | θ1 | 0 | 0 | 90° (π/2) | | 1-2 | θ2 | 0 | \(a_2\) | 0° | | 2-3 | θ3 | 0 | \(a_3\) | 90° (π/2) | | 3-4 | θ4 | 0 | 0 | -90° (-π/2)| | 4-5 | θ5 | \(d_5\) | 0 | 90° (π/2) | | 5-6 | θ6 | \(d_6\) | 0 | 0° | ## Step 2: Transformation Matrices The general DH transformation matrix is given by: \[ T_i^{i-1} = \begin{bmatrix} \cos\theta_i & -\sin\theta_i\cos\alpha_{i-1} & \sin\theta_i\sin\alpha_{i-1} & a_{i-1}\cos\theta_i \\ \sin\theta_i & \cos\theta_i\cos\alpha_{i-1} & -\cos\theta_i\sin\alpha_{i-1} & a_{i-1}\sin\theta_i \\ 0 & \sin\alpha_{i-1} & \cos\alpha_{i-1} & d_i \\ 0 & 0 & 0 & 1 \end{bmatrix} \] ### Example: First Two Transformation Matrices #### Transformation Matrix for Joint 1 \( T_1^0 \): \[ T_1^0 = \begin{bmatrix} \cos\theta_1 & 0 & \sin\theta_1 & 0 \\ \sin\theta_1 & 0 & -\cos\theta_1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \] #### Transformation Matrix for Joint 2 \( T_2^1 \): \[ T_2^1 = \begin{bmatrix} \cos\theta_2 & -\sin\theta_2 & 0 & a_2\cos\theta_2 \\ \sin\theta_2 & \cos\theta_2 & 0 & a_2\sin\theta_2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \] ## Final Answer: Complete Transformation Matrix The **overall transformation** from base to end-effector: \[ T_{06} = T_1^0 \cdot T_2^1 \cdot T_3^2 \cdot T_4^3 \cdot T_5^4 \cdot T_6^5 \] ### Example: Using Values Assuming specific joint angles for simplicity: - \( \theta_1 = 0, \theta_2 = 0, \theta_3 = 90° \) This results in: - \( T_1^0 \): \[ T_1^0 = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \] - \( T_2^1 \): \[ T_2^1 = \begin{bmatrix} 1 & 0 & 0 & 200 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \] Multiply all 6 matrices for the final transformation. ## Graph (Robot Structure) Below is the reference image from your question, showing the robot structure with axes:  *Alt text: Industrial robot with axes and link lengths marked for DH parameter extraction.* ## Summary Table (Final DH Table) | Link | θ | d | a | α | |------|------|------|------|------| | -1 | θ1 | 0 | 0 | 90° | | 1-2 | θ2 | 0 | 200 | 0° | | 2-3 | θ3 | 0 | 50 | 90° | | 3-4 | θ4 | 0 | 0 | -90° | | 4-5 | θ5 | 70 | 0 | 90° | | 5-6 | θ6 | 43 | 0 | 0° | ## Conclusion 1. **Extracted DH parameters** are listed in the table above. 2. **Transformation matrices** for each joint have been created. 3. **Overall transformation matrix** is the product of all six matrices, which can be computed with specific joint angles for a final position in space. If you need further explicit multiplication for all six matrices, let me know!