Control & Guidance LabsInteractive course materialsRuns in your browser

English review collection. Expanded course notes and simulations are drafts for technical review. Activities are ungraded practice; model assumptions are stated in each guide.

Graduate robotics · nonlinear control · marine systems

Learn the theory.
See the dynamics.

A small collection of interactive control-engineering laboratories. Explore why equilibria are stable, then discover how multiple autonomous vehicles can follow different paths while coordinating their motion.

Explore simulators Lecture slides & notes ↓
03 learning modules05 stability examples2–5 coordinated USVs0 software installs
FROM SINGLE-SYSTEM STABILITY TO TEAM CONTROL● LIVE IDEAS
Stability analysisCooperative guidance
V̇(x) ≤ 0 · Stabilitysᵢ + dᵢ · Progress coordination
01 / Interactive tools

Choose a simulator

Each experiment runs independently in your browser. Change gains, initial conditions, vehicle settings and disturbances, then export data for further analysis.

Screenshot of the Lyapunov simulator showing a phase portrait, energy landscape and stability diagnosticsModule 01 · nonlinear control
FOUNDATIONS

Lyapunov stability
& control functions

Watch state trajectories move through a phase plane and across Lyapunov contours. Compare stability, convergence, LaSalle’s principle and a feedback law built from a control Lyapunov function.

Phase portraitsV and V̇ historiesCLF gain designCSV export
  • Stable center, asymptotic spiral, damped oscillator, unstable saddle
  • Adjust feedback gain and observe the CLF stability threshold
  • Modify initial state and introduce disturbances
Launch simulation ↗Student notes →
USVeᵧΔLOS pointModule 02 · single vehicle
MARINE PATH-FOLLOWING GUIDANCE

Line-of-sight
(LOS) guidance

Explore how a marine robot converges to a reference path using a geometric lookahead point. Watch steering, signed cross-track error and the Lyapunov candidate evolve in real time.

Lookahead ΔCross-track errorYaw responseOcean current
  • Visualize path tangent, LOS point, desired course and actual heading
  • Compare short versus long lookahead and tangent-only guidance
  • Observe uncompensated current drift and current compensation
Launch LOS simulator ↗Student notes →
target 1target 2target 3Module 03 · cooperative robotics
MULTI-ROBOT GUIDANCE

LOS & coordinated
virtual targets

Control a fleet of two to five USVs following straight, curved or sinusoidal routes. Compare independent guidance against coordinated virtual-target progression, path-station slots and leader–wingman behavior.

2–5 USVsWingman formationsVT speed & x-shiftsCommunication faults
  • Independent LOS and virtual-target modes, with or without coordination
  • Set path shapes, individual sine phase shifts and target speed
  • Evaluate synchronization versus actual vehicle separation
Launch simulation ↗Student notes →
02 / Suggested sequence

From theory to coordinated motion

A recommended progression for students who already know basic feedback control and linear systems.

STEP 01 · 15–20 MIN

Understand stability

Predict whether V decreases, stays constant or can grow. Compare phase portraits with the formal conditions for stability.

Read the concepts ↗
STEP 02 · 15–20 MIN

Design feedback

Choose a control gain for an unstable plant. Verify the sign of V̇ and find the threshold for convergence.

Open CLF experiment ↗
STEP 03 · 20–30 MIN

Follow a path, then coordinate a fleet

Use the dedicated LOS simulator to establish local path convergence. Then introduce a speed-limit fault. Compare following the path, following the virtual target and keeping the formation together.

Read the guidance notes ↗
Expanded English notes

Study the derivations

Student notes with assumptions, worked examples and limitations. English drafts for instructor review.

Single-vehicle teaching supplement

Virtual target & moving Frenet frame

Isolate target-rate and yaw-control terms, then resolve the displacement derivative into vehicle translation, target translation and frame rotation.

Open single-vehicle animation →
03 / Lecture material

Study resources

Original student handouts are available directly on this site; external slides and readings link to their authors’ original pages.

Marine path following & coordination

Guidance + slides
▤Quick guide: dedicated LOS guidanceGeometry, lookahead tuning, current and Lyapunov connection · on this site↗▤Quick guide: cooperative path followingLOS vs VT, wingman slots, speed consensus, exercises · on this site↗PDFGuidance Systems, Ch. 12 — Thor FossenOfficial handbook lecture-slides index; select Ch12.pdf↗LOSWorkshop: 2D/3D LOS guidanceFossen · follow the Workshop Lecture link↗PPTVirtual target & time coordination — I. KaminerUAV slides 13–17 and 35–39 · transferable concepts↗
04 / Practical activities

Try it, record it, explain it

Short activities for class or homework, with a prediction before each run and a quantitative result afterward.

20 MIN · STABILITY

Can a positive V guarantee convergence?

  1. Compare an undamped center and a stable spiral from the same initial state.
  2. Record the trajectories of V(t) and V̇(t).
  3. Find the range of feedback gains K for which the CLF example is asymptotically stable.
  4. Explain what changes at K = 1 and why a damped oscillator requires an invariance argument.
Deliverable: one plot and a short written justification.
25 MIN · MARINE ROBOTICS

Why doesn't path following imply coordination?

  1. Run three USVs with independent virtual targets and introduce a speed-limit fault.
  2. Repeat with coordinated virtual-target progression.
  3. Change VT speed and each sine path’s x-shift.
  4. Compare cross-track error, vehicle-to-target error and fleet progress error.
Deliverable: a comparison table and one key takeaway.
About the models and materials

The Lyapunov examples are idealized two-state dynamical systems. The marine simulator uses simplified kinematics and path-progress coordination; synchronization is not collision avoidance or a guarantee of a rigid Euclidean formation. The site itself uses static HTML/CSS and needs no runtime backend. External slides remain hosted by their original authors and require an internet connection.