Assessment Package: Module 05 - Dynamics and Control

Assessment Overview

Component	Weight	Format	Duration
Theory Quiz	15%	Multiple choice + derivations	45 minutes
Lab Exercises	35%	Python implementations	3 labs
Simulation Project	35%	Controller design + analysis	1 week
Ethics Discussion	15%	Written reflection	500 words
Total	100%

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Theory Quiz

Time Limit: 45 minutes Passing Score: 70% Attempts: 2

Section A: Multiple Choice (40 points)

Q1. In a PID controller, the integral term primarily serves to:

• a) Increase response speed
• b) Reduce overshoot
• c) Eliminate steady-state error
• d) Dampen oscillations

Q2. For a second-order system with natural frequency ωn, critical damping occurs when the damping ratio ζ equals:

• a) 0
• b) 0.5
• c) 1.0
• d) 2.0

Q3. The mass matrix M(q) in the robot dynamics equation M(q)q̈ + C(q,q̇)q̇ + g(q) = τ is:

• a) Always diagonal
• b) Always symmetric positive definite
• c) Constant for all configurations
• d) Independent of joint positions

Q4. Computed torque control achieves linearized dynamics by:

• a) Ignoring nonlinear terms
• b) Using high gains to dominate nonlinearities
• c) Canceling nonlinear dynamics with model-based feedforward
• d) Approximating the system as linear around an operating point

Q5. Anti-windup in PID control prevents:

• a) Integral term from growing unboundedly during saturation
• b) Derivative term from amplifying noise
• c) Proportional term from causing overshoot
• d) The controller from responding to step inputs

Q6. In the Ziegler-Nichols tuning method, Ku represents:

• a) The ultimate (critical) gain causing sustained oscillation
• b) The gain at which the system becomes unstable
• c) The optimal proportional gain
• d) The gain margin of the system

Q7. The Coriolis matrix C(q, q̇) in robot dynamics represents forces due to:

• a) Gravity alone
• b) Friction in joints
• c) Motion-dependent coupling between joints
• d) External disturbances

Q8. An overdamped system response:

• a) Oscillates before settling
• b) Returns to equilibrium without oscillation
• c) Takes infinite time to settle
• d) Becomes unstable

Q9. Model-based control requires:

• a) Perfect knowledge of system dynamics
• b) An approximate model of system dynamics
• c) Only input-output data
• d) Real-time system identification

Q10. The primary advantage of computed torque over PID for multi-joint robots is:

• a) Simpler implementation
• b) No need for sensor feedback
• c) Decoupled joint dynamics
• d) Lower computational cost

Section B: Short Answer and Derivations (60 points)

Q11. (15 points) Given a 1-DOF system with transfer function G(s) = K/(s² + 2ζωₙs + ωₙ²), derive the relationship between settling time (2% criterion) and the damping ratio ζ and natural frequency ωₙ for an underdamped system.

Q12. (15 points) For a 2-DOF robot arm, the mass matrix is:

M(q) = [a + b*cos(q₂)    c + d*cos(q₂)]
       [c + d*cos(q₂)    e            ]

Show that M(q) is symmetric. Under what conditions is M(q) positive definite?

Q13. (15 points) A joint controlled by PID with Kp=100, Ki=10, Kd=20 has a steady-state position of 0.98 rad when the setpoint is 1.0 rad. A constant disturbance torque is causing the error. Calculate the magnitude of the disturbance torque and explain why the integral term hasn't eliminated the error.

Q14. (15 points) Explain why derivative gain Kd on the error signal can cause "derivative kick" when the setpoint changes suddenly. Describe an alternative implementation that avoids this problem.

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Lab Exercises

Lab 05-01: PID Control (30% of lab grade)

Grading Rubric:

Criterion	Excellent (90-100%)	Proficient (70-89%)	Developing (50-69%)	Beginning ( <50%)
PID Implementation	Complete with anti-windup and proper discretization	Working without anti-windup	Partial implementation	Non-functional
Step Response Analysis	Accurate metrics with proper definitions	Minor calculation errors	Incomplete metrics	Missing analysis
Gain Tuning	Systematic exploration with clear conclusions	Working comparison	Limited exploration	No tuning study
Visualization	Publication-quality with annotations	Functional plots	Basic plots	No visualization
Code Quality	Well-documented, modular, tested	Working code	Partially working	Non-functional

Lab 05-02: Computed Torque Control (35% of lab grade)

Grading Rubric:

Criterion	Excellent (90-100%)	Proficient (70-89%)	Developing (50-69%)	Beginning ( <50%)
Dynamics Extraction	Correctly extracts M and bias from MuJoCo	Minor issues	Partial extraction	Cannot extract
CTC Implementation	Proper inverse dynamics with feedforward	Working without feedforward	Partial implementation	Non-functional
Comparison Study	Fair comparison with PID, clear conclusions	Basic comparison	Limited comparison	No comparison
Trajectory Tracking	Excellent tracking with feedforward	Tracking without feedforward	Poor tracking	Non-functional
Analysis	Insightful interpretation of results	Basic analysis	Incomplete	Missing

Lab 05-03: Adaptive Control (35% of lab grade)

Grading Rubric:

Criterion	Excellent (90-100%)	Proficient (70-89%)	Developing (50-69%)	Beginning (Below 50%)
Model Perturbation	Correctly creates significant mismatch	Minor perturbation	Too small effect	Cannot perturb
Adaptation Law	Proper gradient update with bounds	Working update	Partial implementation	Non-functional
Integration	Smooth integration of adaptation with CTC	Working integration	Partial integration	Not integrated
Convergence	Parameter converges to correct value	Converges approximately	Slow/partial convergence	Does not converge
Analysis	Comprehensive stability and convergence analysis	Basic analysis	Incomplete	Missing

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Simulation Project

Project: Multi-Joint Robot Arm Controller Design

Objective: Design, implement, and compare control strategies for a 3-DOF robot arm performing a pick-and-place task.

Duration: 1 week Deliverables: Code repository + 4-page technical report

Requirements

1. System Modeling (20%)

   • Extract and analyze dynamics matrices for the 3-DOF arm
   • Characterize the system (natural frequencies, coupling)
   • Document model assumptions and limitations

2. PID Controller (20%)

   • Implement independent joint PID controllers
   • Tune using systematic method (Ziegler-Nichols or similar)
   • Achieve setpoint regulation within 5% error in 2 seconds

3. Computed Torque Controller (30%)

   • Implement CTC for the 3-DOF system
   • Demonstrate improved coupling rejection compared to PID
   • Achieve trajectory tracking with RMS error < 2 degrees

4. Comparative Analysis (30%)

   • Test both controllers on identical trajectories
   • Quantify: settling time, overshoot, tracking error, energy consumption
   • Analyze robustness to payload changes (±20% mass)
   • Provide recommendations for deployment

Grading Rubric

Criterion	Points	Description
System Analysis	20	Correct dynamics extraction and characterization
PID Implementation	20	Working controller meeting specifications
CTC Implementation	30	Correct implementation with trajectory tracking
Comparison Quality	20	Fair, thorough comparison with clear metrics
Report Quality	10	Clear writing, proper figures, professional presentation
Total	100

Test Trajectories

1. Step Response: Simultaneous step in all joints
2. Sinusoidal Tracking: Each joint follows sin wave at different frequency
3. Pick-and-Place: Move from home → pick position → place position → home
4. Robustness Test: Repeat with 20% increased payload mass

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Ethics Discussion

Prompt

In a 500-word reflection, address the following scenario:

You are the control systems engineer for a collaborative robot deployed in a small manufacturing shop. The robot performs assembly tasks alongside human workers.

After six months of safe operation, a new production manager requests that you increase the robot's speed by 40% to meet a large order deadline. Your analysis shows this would:

• Increase peak velocities from 0.5 m/s to 0.7 m/s
• Reduce stopping distance from 15 cm to 25 cm
• Increase collision force from 80N to 140N (still below the 150N safety threshold)

The workers have not been consulted about this change.

Address the following in your reflection:

• What are your professional obligations as the control engineer in this situation?
• How should the trade-off between productivity and safety margin be decided, and by whom?
• What would you do if management insisted on the change over your objections?
• How might you approach the workers affected by this decision?

Rubric

Criterion	Excellent (90-100%)	Proficient (70-89%)	Developing (50-69%)	Beginning ( <50%)
Professional Ethics	Clear application of engineering ethics codes	Sound ethical reasoning	Basic ethical awareness	Ignores ethical dimension
Stakeholder Analysis	Considers all affected parties	Most stakeholders	Limited consideration	Single perspective
Technical Analysis	Correctly interprets safety implications	Basic technical understanding	Some technical errors	Misunderstands situation
Actionable Recommendations	Specific, feasible proposals	General recommendations	Vague suggestions	No recommendations
Writing Quality	Clear, well-organized, persuasive	Clear with minor issues	Some clarity problems	Unclear

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Answer Key (Instructor Access Only)

Quiz Answers

Section A:

1. c) Eliminate steady-state error
2. c) 1.0
3. b) Always symmetric positive definite
4. c) Canceling nonlinear dynamics with model-based feedforward
5. a) Integral term from growing unboundedly during saturation
6. a) The ultimate (critical) gain causing sustained oscillation
7. c) Motion-dependent coupling between joints
8. b) Returns to equilibrium without oscillation
9. b) An approximate model of system dynamics
10. c) Decoupled joint dynamics

Section B:

Q11: For underdamped second-order system (0 < ζ < 1):

• Envelope decays as exp(-ζωₙt)
• 2% settling when exp(-ζωₙtₛ) = 0.02
• -ζωₙtₛ = ln(0.02) ≈ -3.9
• tₛ ≈ 4/(ζωₙ) for 2% criterion

Q12: Symmetry: M₁₂ = c + d*cos(q₂) = M₂₁ ✓ Positive definite conditions:

• M₁₁ > 0: a + b*cos(q₂) > 0 → a > |b|
• det(M) > 0: (a + b*cos(q₂))e - (c + dcos(q₂))² > 0
• Typically satisfied by physical mass/inertia constraints

Q13: At steady state, Kpe + Ki∫e dt = τ_disturbance With e = 0.02 rad steady-state, integral keeps growing (not saturating), so:

• If Ki*∫e dt has reached a limit due to anti-windup, τ_disturbance ≈ 0.02*100 = 2 Nm
• Integral hasn't eliminated error because it's hitting anti-windup limits OR hasn't had enough time to accumulate

Q14: Derivative kick: When setpoint changes from r₁ to r₂ instantaneously:

• Error jumps from (r₁-y) to (r₂-y)
• de/dt → ∞ (impulse)
• Kd*de/dt produces large torque spike

Solution: Derivative on measurement (not error):

• Instead of Kdd(r-y)/dt, use -Kddy/dt
• Setpoint changes don't affect derivative term
• Only actual system velocity is differentiated

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