○Draft — not verified

Lab 2: Velocity Kinematics — The 2-DOF Jacobian

Singularities and Manipulability for a 2-Link Planar Arm

Component: Case Study / Labs 2 of 3 (10 % of 30 %) CLO: CLO3 (P5) Released: Week 6 — 18 May 2026 Due: Week 8 — 5 June 2026, 23:59 MYTType: Individual lab report

Objective

This lab converts the theory in Module 3.1 (Jacobian for 2-DOF systems) into running MATLAB/Python code you can interrogate. You will derive the Jacobian of a 2-link planar arm by hand, compute it numerically, verify them against each other, draw the manipulability ellipsoid at three poses, and identify singular configurations. You will then cross-check your derivation against the live 3-DOF arm shipped with ArbiterROS (/robot-arm, implemented in frontend/lib/Robot/arm3dof.ts) so the 2-DOF theory is anchored to a running simulation you can drive in the browser.

The 2-DOF scope is deliberate: it is the maximum manipulator complexity retained in the revised MEC781 syllabus. The 3-DOF platform extension in Task 6 is observation-only — it demonstrates that the same Jacobian machinery scales to the anthropomorphic configuration without re-deriving it. Full n-DOF treatment is available in the Supplementary/ folder (S7–S9).

Learning Objectives

Derive the analytic Jacobian of a 2-link planar arm from its forward kinematics
Compute a numerical Jacobian via finite differences and verify agreement
Visualise the manipulability ellipsoid at a chosen pose
Classify configurations as regular, singular, or near-singular using $det (J)$ and $κ (J)$
Explain the physical meaning of singularity in terms of velocity transmission

Lab Setup

Use the following 2-link planar arm:

Parameter	Value
Link 1 length $ℓ_{1}$	$1.0$ m
Link 2 length $ℓ_{2}$	$0.8$ m
Joint 1 angle $q_{1}$	variable
Joint 2 angle $q_{2}$	variable
Base frame	origin at the shoulder

Forward kinematics of the end-effector:

x y = ℓ_{1} cos q_{1} + ℓ_{2} cos (q_{1} + q_{2}) = ℓ_{1} sin q_{1} + ℓ_{2} sin (q_{1} + q_{2})

Task 1 — Analytical Jacobian (20 %)

Derive $J (q) = \partial p / \partial q$ where $p = [x, y]^{⊤}$ and $q = [q_{1}, q_{2}]^{⊤}$ . Show the derivation step-by-step.
Write $J$ symbolically (in terms of $s_{1} = sin q_{1}$ , $c_{1} = cos q_{1}$ , etc.).
Compute $det J$ and simplify. State the singular condition in terms of $q_{2}$ .

Task 2 — Numerical Jacobian (20 %)

Implement a central-difference numerical Jacobian with step size $h = 1 0^{- 6}$ :

J_{ij} (q) \approx \frac{p _{i} ( q + h e _{j} ) - p _{i} ( q - h e _{j} )}{2 h}

Evaluate both the analytic and numerical Jacobians at:

- $q^A = (30^\circ,\,45^\circ)$

- $q^B = (0^\circ,\,0^\circ)$ (singular)

- $q^C = (60^\circ,\,90^\circ)$

Report the Frobenius-norm difference $∥ J_{analytic} - J_{numerical} ∥_{F}$ for each pose. Should be $\sim 1 0^{- 9}$ for regular poses.

Task 3 — Manipulability (20 %)

Compute the manipulability measure $w = det (J J^{⊤})$ at poses $q^{A}$ , $q^{B}$ , $q^{C}$ .
Plot the manipulability ellipsoid at pose $q^{A}$ . Use the eigen-decomposition of $J J^{⊤}$ and scale the axes by $λ_{i}$ .
Plot the ellipsoid at $q^{B}$ . What shape do you get? Interpret.
Plot the ellipsoid at $q^{C}$ . Comment on the ratio of major to minor axes.

Task 4 — Singularity Sweep (20 %)

Fix $q_{1} = 3 0^{\circ}$ . Sweep $q_{2}$ from $- 18 0^{\circ}$ to $+ 18 0^{\circ}$ in $1^{\circ}$ increments.
Plot $∣ det J (q) ∣$ versus $q_{2}$ . Mark the singular angles.
Plot the condition number $κ (J) = σ_{m a x} / σ_{m i n}$ versus $q_{2}$ on a log scale.
Discuss: workspace boundary vs. interior singularities. Which singular angle(s) correspond to which?

Task 5 — Physical Interpretation (10 %)

Answer briefly:

At $q_{2} = 0^{\circ}$ , in which direction cannot the end-effector move instantaneously? Justify using the null space of $J^{⊤}$ .
Why does the condition number blow up near but not exactly at the singularity?
How would you use the manipulability measure $w$ as part of a Jacobian-based controller’s cost function? (Hint: see Module-3/3.2 § 3.2.4 for the task-space velocity-resolved motion control formulation.)

Task 6 — Platform Cross-Check on the 3-DOF Arm (10 %)

Open /robot-arm on https://arbiter.txio.live. The live arm is a 3R anthropomorphic configuration implemented in frontend/lib/Robot/arm3dof.ts with reactive state in frontend/composables/Robot/useArmKinematics.ts. The analytical $3 \times 3$ position Jacobian is exported as jacobianAnalytical(q, p) and the manipulability / condition diagnostics as analyseJacobian(q, p).

Set $q_{1} = 0$ (base). Fixing base rotation reduces the visible motion to a 2-DOF shoulder–elbow plane. Confirm that the shoulder–elbow sub-block of the platform’s Jacobian matches your Task 1 derivation up to the $L_{0}, L_{1}, L_{2}$ parameter substitution. Report the maximum element-wise difference.
Drive the arm to a fully extended pose ( $q_{3} \to 0$ ). Record the manipulability $w$ reported by the UI and compare qualitatively to your Task 3 plot for $q^{B}$ .
Drive the arm slowly through the manipulability-threshold boundary ( $w < 1 0^{- 2}$ , default w0 in DEFAULT_DLS). Observe how the damped-least-squares solver activates in the UI. Explain in one sentence what the damping does to the Jacobian inverse and why this prevents the control input from blowing up.

Deliverable: three UI screenshots (one per sub-task) plus the numerical comparison table from sub-task 1.

Deliverables

#	Item	Format
1	Lab report	PDF, max 5 pages
2	Source code	`.m` or `.py`, single reproducible entry point
3	Figures from Tasks 3–4	Embedded in report

File name format: MEC781-Lab2-<StudentID>-<Surname>.{pdf,zip}

Starter code: Assessments/20261/starter-kits/Lab2/Lab2-starter.py

Evaluation Criteria

Criterion	Weight	Description
Analytic Jacobian derivation	20 %	Correctness, clarity, singular condition identified
Numerical verification	20 %	Correct implementation, matches analytic at regular poses
Manipulability plots	20 %	Correct ellipsoid shapes, correct interpretation
Singularity sweep	20 %	Plots correct, singular angles identified
Physical interpretation	10 %	Depth of reasoning on null space and controller use
Platform cross-check	10 %	2-DOF $\leftrightarrow$ 3-DOF match, DLS observation documented

Resources

Module-3/3.1-Jacobian for 2-DOF Systems.pdf — primary reference
Module-3/3.2-Jacobian of Mobile Robots and Velocity Control.pdf — mobile-robot analogue
Supplementary/S7-Jacobian Overview (Full n-DOF).pdf — optional extension
Spong, Hutchinson, Vidyasagar — Robot Modeling and Control, Ch. 4 (in References/)
ArbiterROS source (Task 6): frontend/lib/Robot/arm3dof.ts (pure-math, unit-testable), frontend/composables/Robot/useArmKinematics.ts (reactive state), frontend/components/Simulations/Arm3DOF.vue (Three.js renderer)
Live platform: https://arbiter.txio.live/robot-arm and the /robot-arm-guide explainer

Practical Guidance

Use symbolic math (MATLAB syms or Python sympy) for Task 1 if you want — cleaner derivation.
For Task 3, the ellipsoid axes are the eigenvectors of $J J^{⊤}$ , not $J$ .
If your ellipsoid at $q^{B}$ looks like a line segment, you did it right — that is exactly what a singular Jacobian looks like geometrically.