Assignment 1: Mathematical Modeling of Differential-Drive Mobile Robots

Lagrangian Dynamics and Velocity Kinematics

Component: Assignment 1 of 3 (7 % of 20 % Assignment component) CLO: CLO2 (C5) Released: Week 3 — 27 April 2026 Due: Week 5 — 19 May 2026, 23:59 MYTType: Individual literature review + derivation

Background

Every mobile-robot controller you build in Modules 4–6 of this course stands on a mathematical model of the robot. If the model is wrong, the PID gains don’t transfer, the SLAM odometry drifts, and the figure-eight trajectory diverges. This assignment asks you to pin down that model rigorously for a differential- drive robot — the platform used by ArbiterROS and by the Pi 4B hardware you will touch in Module 6. You will read the provided papers on mobile-robot kinematics and Lagrangian dynamics, derive the equations of motion for a two-wheel differential-drive robot, and explain how the resulting model connects to the velocity-kinematics (Jacobian) treatment in Module 3.

Learning Objectives

• Extract industrial engineering principles (actuator economics, kinematic abstraction, TCO, production base) from a current industry briefing and tie each to a specific term in the robot EOM
• Distinguish kinematic from dynamic models and identify when each is appropriate
• Derive the Lagrangian equations of motion for a two-wheel differential-drive robot starting from kinetic and potential energy
• Identify the state vector, generalised coordinates, non-holonomic constraints, and control inputs
• Map the kinematic model of a differential-drive robot to its Jacobian and interpret singularities in the velocity map
• Communicate the modelling workflow as a postgraduate-level technical review (IEEE style)

Part 0 — Principles of Industrialised Robotics (15 %)

Required reading: Module-1/Industrializing the Humanoid.pdf (or .pptx). This briefing deconstructs why Boston Dynamics’ Atlas became an industrial product in 2026 after a 30-year research run, and why the same technology failed at DARPA, Google, and SoftBank.

Extract four engineering principles from the reading and map each one to a specific symbol or term in the EOM you will derive in Part B. The point of this section is not business strategy — it is to ground the dynamics you are about to derive in the commercial constraints that make mobile/humanoid robots viable at scale. Expected length: 600–800 words.

Suggested mapping (you are free to propose your own, but each principle must land on a concrete symbol in $M(q)\ddot{q}+C(q,\dot{q})\dot{q}=B(q)\tau$ or in $\dot{q}=S(q)\nu$):

1. Actuator-cost bottleneck (60 % of material cost). How does this constrain the achievable $\tau$ in $B(q)\tau$, and why does the ${\dot{q}}_{\text{max}}$ limit matter more than peak torque in practice?
2. Kinematic abstraction (/cmd_vel as the universal interface). Why does the reduction $\dot{q}=S(q)\nu$ with $\nu =[v,\omega {]}^{\top }$ enable the Orbit-style “instant replication” and fleet scaling the reading highlights?
3. TCO economics (10-year lifespan, <\90$/day).Whichtermsin$M(q)$and$C(q,\dot q)$ are actually sized by worst-case load cases, and how does payload variation (your Bonus Task) feed back into this?
4. Production base over code (“the greatest software could not solve a manufacturing problem”). What does it mean for the model parameters $m,r,b,I$ to be identified once on a Hyundai Mobis-style mass-produced actuator set and then reused across a fleet?

Cite the PPTX and at least one additional industry source (a whitepaper, a company technical post, or a peer-reviewed paper on actuator design at scale).

Part A — Literature Review (25 %)

Write a 1200–1500 word technical review that:

1. Explains the systematic process of building a mathematical model for a mobile robot (state vector $\to$ coordinate frames $\to$ constraints $\to$ EOM).
2. Contrasts kinematic and dynamic modelling approaches. Be concrete: give examples of control problems where one is sufficient and the other is required.
3. Identifies the key variables, parameters, and constraints for a differential-drive platform (wheel radius $r$, wheelbase $2b$, mass $m$, moment of inertia $I$, non-holonomic constraint, wheel torques).
4. Cites at least three peer-reviewed references in IEEE format. At least one must be a primary Lagrangian treatment (Spong / Khatib / Siciliano-level textbook or equivalent paper).
5. Cross-references at least one equation from Module-2/2.1-Lagrange Equations.pdf and at least one equation from Module-2/2.2-Dynamics of Mobile Robotic Systems.pdf.

Part B — Lagrangian Derivation (40 %)

Derive the equations of motion for a planar two-wheel differential-drive robot from first principles using the Lagrangian method:

1. Define generalised coordinates $q=[x,y,\theta {]}^{\top }$ and the pseudo-velocities $v$ (linear) and $\omega$ (angular).
2. Write the kinetic energy $T(q,\dot{q})$ including translational and rotational contributions.
3. Apply the non-holonomic rolling constraint

and show how you handle it (Lagrange multiplier *or* reduced form — state your choice and justify).

4. Derive the Euler–Lagrange equations and express the final EOM in the form

5. Give numerical values for $M$, $C$, $B$ using the ArbiterROS default platform parameters as shipped in frontend/lib/Robot/MobileRobotPhysics.js: $m=2.0\mathrm{kg}$, $r=0.033\mathrm{m}$ (wheel radius), $b=0.1\mathrm{m}$ (half-wheelbase), $I=0.01\mathrm{kg}{\mathrm{m}}^2$. Verify dimensions and state which constructor option you would pass to override each parameter.

Part C — Kinematic-to-Jacobian Appendix (20 %)

Write a short appendix (one to two pages) that:

1. States the forward velocity kinematics $\dot{q}=S(q)\nu$ of the differential-drive robot, with $\nu =[v,\omega {]}^{\top }$.
2. Computes the analytical Jacobian mapping $\nu \to \dot{q}$.
3. Identifies under which configuration(s) the Jacobian becomes rank-deficient and explains what this means physically.
4. Cross-references Module-3/3.2-Jacobian of Mobile Robots and Velocity Control.pdf.

Bonus Task (+2 marks, optional) — Payload Variation on ArbiterROS

Students who want an additional challenge may run the following experiment directly on the ArbiterROS web platform. It is optional and capped at +2 bonus marks.

1. Open /figure-eight-nav on https://arbiter.txio.live. This page drives the MobileRobotPhysics dynamic model through the FigureEightTrajectory reference path using useFigureEightNavigation.ts.
2. From the admin panel (/admin, password arbiter2024) or by editing the physics constructor options locally, raise the robot mass from $m=2.0$ kg to $m=4.0$ kg at the $1/4$ point of the figure-eight and restore it at the $3/4$ point (simulate a pickup/drop event). Apply a matching change to $I$ consistent with your Part B derivation.
3. Record useAnalyticsEngine CSV traces for three configurations: (a) kinematic-only go-to-goal controller, (b) Pure Pursuit with fixed gains, (c) Pure Pursuit with gains re-tuned for the heavier payload.
4. Compare cross-track-error RMS and peak across the three configurations.
5. Submit a two-page appendix plus the raw CSV files.

Connect the result back to your Part B EOM: which term in $M(q)\ddot{q}+C(q,\dot{q})\dot{q}=B\tau$ is actually affected when the payload changes, and why does the kinematic-only controller remain insensitive to it?

Deliverables

File name format: MEC781-A1-<StudentID>-<Surname>.pdf
Submission channel: UFUTURE / UiTM LMS
Citation style: IEEE
Plagiarism: standard UiTM postgraduate rules; Turnitin $<20\%$

Evaluation Criteria

Master’s-level difficulty split: C3–C4 = 2 marks (29 %), C5–C6 = 5 marks (71 %) (meets the $\ge 60\%$ C5–C6 requirement).

Resources

• Module-1_Introduction-and-Basics/Industrializing the Humanoid.pdf (required, Part 0)
• Module-1_Introduction-and-Basics/1.3-Introduction to Mobile Robotic Systems.pdf
• Module-2_Robot-Dynamics/2.1-Lagrange Equations.pdf
• Module-2_Robot-Dynamics/2.2-Dynamics of Mobile Robotic Systems.pdf
• Module-3_Jacobian-Matrix/3.2-Jacobian of Mobile Robots and Velocity Control.pdf
• References/ folder (Spong, Siciliano, Khatib chapters)
• Literature/ folder (assigned papers)
• ArbiterROS source: frontend/lib/Robot/MobileRobotPhysics.js, frontend/lib/Robot/FigureEightTrajectory.js, frontend/composables/Robot/useFigureEightNavigation.ts (bonus task)
• Live platform: https://arbiter.txio.live/figure-eight-nav

Questions

Office hours Monday 10:00–12:00 and Wednesday 14:00–16:00, or email haniframli@uitm.edu.my. Please do not wait until the final weekend.