3-DOF Robot (DH Representation) Assignment: Frames, DH Table, and 𝑈 𝑇 𝐻 U T H ​

Problem 1: 3-DOF Robot — DH Representation & Forward Kinematics (Assignment Prompt)

You are given a 3-DOF robot (as shown in the figure provided in your coursework). Your task is to use the Denavit–Hartenberg (DH) representation to systematically define the robot’s coordinate frames and derive its forward kinematics.

This problem is designed to test your understanding of coordinate frame assignment, DH parameterization, and homogeneous transformations.

✅ What You Must Deliver

(a) Assign DH Coordinate Reference Frames

Using the DH convention:

1. Assign a base frame UUU.
2. Assign a frame to each joint (Frame 1, Frame 2, Frame 3).
3. Assign the end-effector frame HHH.
4. Clearly indicate:
5. The direction of each ziz_izi​ axis (must align with joint axis iii)
6. The direction of each xix_ixi​ axis (along the common normal between ziz_izi​ and zi+1z_{i+1}zi+1​)
7. The origin placement for each frame

✅ Tip: Your frame assignment must be consistent with the right-hand rule and the standard DH rules.

(b) Write the DH Parameter Table

Create a DH table with the four standard DH parameters for each joint/link:

1. aia_iai​ (link length)
2. αi\alpha_iαi​ (link twist)
3. did_idi​ (link offset)
4. θi\theta_iθi​ (joint angle)

Your table should have one row per joint, typically in the format:

Be sure to use the correct values based on the geometry shown in the figure and specify which terms are constants and which are variables (e.g., θi=qi\theta_i = q_iθi​=qi​ for revolute joints, di=qid_i = q_idi​=qi​ for prismatic joints).

(c) Find the Transformation Matrix UTH^{U}T_{H}UTH​

Using your DH parameters, compute the homogeneous transformation from the base frame to the end-effector frame:

UTH= UT1 1T2 2T3 3TH^{U}T_{H} =\ ^{U}T_{1}\ ^{1}T_{2}\ ^{2}T_{3}\ ^{3}T_{H}UTH​= UT1​ 1T2​ 2T3​ 3TH​For each link transform, use the standard DH transformation:

i−1Ti=[cos⁡θi−sin⁡θicos⁡αisin⁡θisin⁡αiaicos⁡θisin⁡θicos⁡θicos⁡αi−cos⁡θisin⁡αiaisin⁡θi0sin⁡αicos⁡αidi0001]^{i-1}T_i = \begin{bmatrix} \cos\theta_i & -\sin\theta_i\cos\alpha_i & \sin\theta_i\sin\alpha_i & a_i\cos\theta_i \\ \sin\theta_i & \cos\theta_i\cos\alpha_i & -\cos\theta_i\sin\alpha_i & a_i\sin\theta_i \\ 0 & \sin\alpha_i & \cos\alpha_i & d_i \\ 0 & 0 & 0 & 1 \end{bmatrix}i−1Ti​=​cosθi​sinθi​00​−sinθi​cosαi​cosθi​cosαi​sinαi​0​sinθi​sinαi​−cosθi​sinαi​cosαi​0​ai​cosθi​ai​sinθi​di​1​​Then multiply the matrices in order to obtain the final expression for UTH^{U}T_{H}UTH​. Your final matrix should be a 4×4 homogeneous transformation matrix that includes:

1. Rotation (top-left 3×3)
2. Position vector (top-right 3×1)
3. The bottom row [0 0 0 1][0\ 0\ 0\ 1][0 0 0 1]

✅ Submission Expectations (What Your Instructor Will Look For)

To earn full marks, your submission should include:

1. A clear and correct DH frame diagram (labels + axes directions)
2. A complete DH table with correct parameter values
3. Correct step-by-step DH transformation setup
4. A final, simplified expression for UTH^{U}T_{H}UTH​

🚀 Need the Full Worked Solution (Frames + DH Table + Matrix Multiplication)?

If you only have the question but need the complete solution (including the correctly assigned frames for your exact figure, the DH parameter table, and the fully multiplied UTH^{U}T_{H}UTH​ matrix), you can order a full solution from Onpoint Essays.

✅ Accurate DH frame assignment for your diagram

✅ Correct DH table (standard or modified DH as required)

✅ Full forward-kinematics derivation

✅ Clean presentation suitable for submission