Assessment Package: Kinematics Fundamentals

Overview

Component	Weight	Format
Theory Quiz	15%	Multiple choice + calculations
Lab Assessment	35%	Jupyter notebooks with auto-grading
Simulation Project	35%	Code submission with test harness
Ethics Component	15%	Written reflection

Theory Quiz (15%)

Time Limit: 30 minutes Passing Score: 70% Attempts Allowed: 2

Questions

Q1 (Multiple Choice, 2 points)

Question: In the Denavit-Hartenberg (DH) convention, the parameter 'a' represents:

Options: a) Rotation about the z-axis b) Translation along the z-axis c) Translation along the x-axis (link length) d) Rotation about the x-axis

Correct Answer: c Explanation: In DH convention, 'a' is the link length—the distance along the x-axis from z_{i-1} to z_i. Learning Objective: Apply DH parameters to derive transformation matrices

Q2 (Calculation, 4 points)

Question: A 2-DOF planar arm has link lengths L1 = 0.5m and L2 = 0.3m. If θ1 = 60° and θ2 = 45°, calculate the (x, y) position of the end-effector.

Sample Answer:

Elbow: x1 = 0.5 × cos(60°) = 0.25m, y1 = 0.5 × sin(60°) = 0.433m
End-effector: x = 0.25 + 0.3 × cos(105°) = 0.25 + (-0.078) = 0.172m
y = 0.433 + 0.3 × sin(105°) = 0.433 + 0.290 = 0.723m
Position: (0.172m, 0.723m)

Rubric:

2 points: Correct elbow position
2 points: Correct end-effector position Learning Objective: Implement forward kinematics equations for serial manipulators

Q3 (Multiple Choice, 2 points)

Question: For a 2-DOF planar arm with L1 = L2 = 0.5m, what is the shape of the reachable workspace?

Options: a) A circle with radius 1.0m b) An annular region (donut) with outer radius 1.0m and inner radius 0m c) A square with side length 1.0m d) An annular region with outer radius 1.0m and inner radius 0.5m

Correct Answer: b Explanation: When L1 = L2, the inner radius is |L1 - L2| = 0, so the workspace is a full disk (degenerate annulus) with radius L1 + L2 = 1.0m. Learning Objective: Analyze workspace and reachability for manipulators

Q4 (Short Answer, 4 points)

Question: Explain why inverse kinematics can have multiple solutions and how a robot might choose between them.

Sample Answer: Multiple solutions exist because different joint configurations can place the end-effector at the same location (e.g., elbow-up vs. elbow-down). Robots choose between solutions based on criteria such as:

Minimizing joint motion from current configuration
Avoiding obstacles or singularities
Staying within joint limits
Selecting the configuration closest to a "home" pose
Minimizing energy consumption

Rubric:

2 points: Correctly explains why multiple solutions exist
2 points: Provides reasonable selection criteria Learning Objective: Solve inverse kinematics with multiple solution handling

Q5 (Multiple Choice, 2 points)

Question: What happens at a kinematic singularity?

Options: a) The robot gains additional degrees of freedom b) The Jacobian matrix becomes singular (loses rank) c) The robot cannot reach any targets d) All inverse kinematics solutions become identical

Correct Answer: b Explanation: At a singularity, the Jacobian loses rank, meaning the robot loses a degree of freedom in Cartesian space. Small Cartesian motions may require infinite joint velocities. Learning Objective: Identify and handle kinematic singularities

Q6 (Calculation, 4 points)

Question: For the 2-DOF arm in Q2, compute the Jacobian matrix elements J11 and J21 (first column) when θ1 = 90° and θ2 = 0°.

The Jacobian for a 2-DOF arm is:

J = [−L1·sin(θ1) − L2·sin(θ1+θ2), −L2·sin(θ1+θ2)]
    [ L1·cos(θ1) + L2·cos(θ1+θ2),  L2·cos(θ1+θ2)]

Sample Answer:

θ1 = 90°, θ2 = 0° → θ1 + θ2 = 90°
J11 = −0.5·sin(90°) − 0.3·sin(90°) = −0.5 − 0.3 = −0.8
J21 = 0.5·cos(90°) + 0.3·cos(90°) = 0 + 0 = 0

Rubric:

2 points: Correct J11 calculation
2 points: Correct J21 calculation Learning Objective: Compute the Jacobian matrix for velocity kinematics

Q7 (Multiple Choice, 2 points)

Question: A humanoid robot arm has 7 DOF but only needs 6 DOF to specify arbitrary end-effector position and orientation. This extra DOF is called:

Options: a) A redundant degree of freedom b) A singular degree of freedom c) A constrained degree of freedom d) A virtual degree of freedom

Correct Answer: a Explanation: When a manipulator has more DOF than needed for the task, the extra DOF are called redundant. Redundancy allows the robot to optimize secondary objectives (obstacle avoidance, joint limits) while achieving the primary task. Learning Objective: Understand kinematic redundancy

Total Quiz Points: 20

Lab Assessment (35%)

Grading Rubric for Labs 03-01, 03-02, 03-03

Criterion	Weight	Excellent (90-100%)	Good (70-89%)	Satisfactory (50-69%)	Needs Work (<50%)
FK Implementation	30%	Accurate, well-documented	Minor errors	Basic functionality	Major errors
IK Implementation	30%	Multiple solutions handled	Single solution works	Partial IK	Doesn't work
Workspace Analysis	20%	Complete with visualization	Good analysis	Basic analysis	Incomplete
Code Quality	20%	Clean, documented, tested	Good code	Functional	Messy or buggy

Auto-Grading Tests

# tests/lab_03_tests.py
import numpy as np

def test_fk_2dof():
    """Test forward kinematics accuracy."""
    # θ1=0, θ2=0 should give (L1+L2, 0)
    x, y = forward_kinematics_2dof(0, 0, 0.5, 0.3)
    assert abs(x - 0.8) < 1e-6
    assert abs(y) < 1e-6

def test_ik_reaches_target():
    """Test that IK solution reaches target."""
    target = (0.5, 0.3)
    solution = inverse_kinematics_2dof(*target)
    if solution:
        achieved = forward_kinematics_2dof(*solution)
        error = np.sqrt(
            (achieved[0] - target[0])**2 +
            (achieved[1] - target[1])**2
        )
        assert error < 1e-4

def test_ik_unreachable():
    """Test that unreachable points return None."""
    solution = inverse_kinematics_2dof(2.0, 0)
    assert solution is None

def test_jacobian_dimensions():
    """Test Jacobian matrix has correct shape."""
    J = jacobian_2dof(0, 0)
    assert J.shape == (2, 2)

Simulation Project (35%)

Project: Humanoid Reaching Task

Description: Implement a complete reaching task for a humanoid robot arm, including workspace analysis, IK solving, trajectory planning, and collision checking.

Deliverables

Deliverable	Format	Points
Workspace visualization	PNG/PDF	15
IK solver module	Python	30
Trajectory planner	Python	25
Demo notebook	Jupyter	20
Documentation	Markdown	10

Requirements

Workspace Analysis (15 points)
- Compute reachable workspace for the right arm
- Visualize in 3D with color-coded reachability
- Identify singularity regions
IK Solver (30 points)
- Implement numerical IK for 7-DOF arm
- Handle joint limits
- Return multiple solutions when applicable
- Include convergence diagnostics
Trajectory Planning (25 points)
- Generate smooth trajectories between poses
- Implement both joint-space and Cartesian-space interpolation
- Verify collision-free paths
Demonstration (20 points)
- Interactive notebook showing complete pipeline
- Multiple reaching scenarios
- Performance metrics (solve time, accuracy)
Documentation (10 points)
- Clear API documentation
- Usage examples
- Limitations and known issues

Rubric

Criterion	Excellent	Good	Satisfactory	Needs Work
Correctness	All features work	Minor bugs	Partial functionality	Major issues
Completeness	All requirements met	Most met	Some missing	Many missing
Code Quality	Publication-ready	Good style	Functional	Poor quality
Analysis	Deep insights	Good observations	Basic analysis	Superficial

Ethics Component (15%)

Format: Written Reflection

Length: 400-600 words

Prompt

Consider a humanoid robot designed to work in a hospital environment, reaching for objects on shelves, opening doors, and assisting patients.

Address the following:

How would you define the robot's workspace boundaries to ensure patient safety?
When multiple IK solutions exist for reaching a target, what criteria should guide the selection? Consider:
- A solution that keeps the elbow away from bedridden patients
- A solution that minimizes motion through high-traffic areas
- A solution that is most energy-efficient
How should the robot behave when approaching the edge of its workspace (near singularities)?
What kinematic constraints would you impose specifically for a healthcare environment?

Rubric

Criterion	Points	Description
Safety Analysis	30	Identifies key workspace safety concerns
Solution Selection	25	Thoughtful criteria for IK selection
Singularity Handling	20	Practical approach to edge cases
Healthcare Context	15	Appropriate for healthcare environment
Writing Quality	10	Clear, organized presentation

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Alignment with Learning Objectives

Learning Objective	Quiz	Lab	Project	Ethics
DH parameters	Q1	✓	✓
Forward kinematics	Q2	✓	✓
Workspace analysis	Q3	✓	✓	✓
Inverse kinematics	Q4	✓	✓	✓
Singularities	Q5	✓	✓	✓
Jacobian	Q6	✓	✓
Redundancy	Q7		✓

Overview​

Theory Quiz (15%)​

Questions​

Q1 (Multiple Choice, 2 points)​

Q2 (Calculation, 4 points)​

Q3 (Multiple Choice, 2 points)​

Q4 (Short Answer, 4 points)​

Q5 (Multiple Choice, 2 points)​

Q6 (Calculation, 4 points)​

Q7 (Multiple Choice, 2 points)​

Lab Assessment (35%)​

Grading Rubric for Labs 03-01, 03-02, 03-03​

Auto-Grading Tests​

Simulation Project (35%)​

Project: Humanoid Reaching Task​

Deliverables​

Requirements​

Rubric​

Ethics Component (15%)​

Format: Written Reflection​

Prompt​

Rubric​

Export Formats​

Alignment with Learning Objectives​

Overview

Theory Quiz (15%)

Questions

Q1 (Multiple Choice, 2 points)

Q2 (Calculation, 4 points)

Q3 (Multiple Choice, 2 points)

Q4 (Short Answer, 4 points)

Q5 (Multiple Choice, 2 points)

Q6 (Calculation, 4 points)

Q7 (Multiple Choice, 2 points)

Lab Assessment (35%)

Grading Rubric for Labs 03-01, 03-02, 03-03

Auto-Grading Tests

Simulation Project (35%)

Project: Humanoid Reaching Task

Deliverables

Requirements

Rubric

Ethics Component (15%)

Format: Written Reflection

Prompt

Rubric

Export Formats

Alignment with Learning Objectives