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Furuta Pendulum — Hybrid Control System

MCT411: Hybrid Control | Fall 2025 | Team 14
Ain Shams University — Faculty of Engineering, Mechatronics Department

📖 Overview

This repository contains the MATLAB/Simulink implementation of a hybrid control system for the Furuta (Rotary Inverted) Pendulum — a nonlinear, underactuated mechatronic benchmark system. The control architecture combines two distinct strategies:

  1. Energy-Based Swing-Up Controller — pumps energy into the pendulum to drive it from the natural downward-stable position to the upright equilibrium.
  2. Linear Quadratic Regulator (LQR) — takes over once the pendulum is near upright, maintaining stability against small disturbances.

The full system is validated through both MATLAB/Simulink simulation and Hardware-in-the-Loop (HIL) testing on a physical prototype.

🧑‍💻 Team

Name ID Contribution
Abdelghaffar Essam Abdelghaffar 2100428 Mechanical Design, LQR Implementation
Ingy Ahmed Sherif 2100648 LQR Implementation, Simulation Setup
Kareem Saleh Hassan 2100909 Mechanical Design, Swing-Up Implementation
Mariam Mohamed El Sebaey 2100260 Swing-Up Implementation, Simulation Setup

⚙️ System Description

The Furuta Pendulum consists of two arms whose rotary axes are perpendicular to each other:

  • Arm 1 (Table/Rotor): Horizontal arm driven by a DC motor. Angle: θ₁
  • Arm 2 (Pendulum): Vertical free-swinging arm attached to the end of Arm 1. Angle: θ₂

The system is underactuated — only one torque input (the motor) controls two degrees of freedom.

Hardware Components

Component Specification
Microcontroller ESP32
Motor Driver H-Bridge (bidirectional)
Motor Encoder 600 PPR (horizontal arm θ₁)
Pendulum Encoder High-resolution (pendulum angle θ₂)
Power Supply 12 V DC
Host Interface USB/Serial → MATLAB/Simulink HIL

Physical Parameters

Parameter Symbol Value
Pendulum mass mᵣ 0.018 kg
Pendulum COM to rotation axis lᵣ 0.0638 m
Pendulum moment of inertia Jᵣ 9.83 × 10⁻⁵ kg·m²
Table rotation axis to pendulum pivot lₜ 0.07 m
Rotor moment of inertia J_motor 0.0024 kg·m²
Motor resistance Rₐ 5.8466 Ω
Motor constants kₜ = k_b 0.048 V/rpm
Armature inductance Lₐ 9.15 × 10⁻⁴ H

Friction coefficients (motor, pendulum, table) were identified via parameter estimation rather than theoretical derivation.

🎮 Control Architecture

State Vector

x = [θ₁  θ̇₁  θ₂  θ̇₂]ᵀ

1. Swing-Up Controller (Energy-Based)

The controller injects energy by oscillating the horizontal arm, guided by the difference between the current pendulum energy and the target energy at the upright equilibrium:

E  = ½ Jᵣ θ̇₁² + mᵣ g lᵣ (1 + cos θ₁)
Eᵣ = 2 mᵣ g lᵣ

u  = sat( k_swing · (Eᵣ − E) · θ̇₁ · cos θ₁ )

2. Balancing Controller (LQR)

The nonlinear Simscape Multibody model is linearized at the upright equilibrium using Simulink Model Linearizer, yielding the state-space matrices used for LQR design:

ẋ = Ax + Bu

LQR Cost Function:

J = ∫₀^∞ (xᵀQx + uᵀRu) dt

Tuned Weights:

Q = diag([1, 0.1, 100, 10])
R = 100

Resulting Gain:

K = [−0.0100  −0.0857  1.5880  0.1478]
u(t) = −K x(t)

3. Supervisory Logic (Stateflow)

A Stateflow chart manages transitions between control modes:

Swing_Up ──(|θ₂| ≤ 15°, |θ̇₂| ≤ 7)──► LQR
   ▲                                      │
   │                                      │ (|θ̇₂| ≥ 25 or timeout)
   │                                      ▼
   └────────(safe)────────────────── Recovery
State Condition
Swing_Up Default; energy injection active
LQR |θ₂| ≤ 15° and |θ̇₂| ≤ 7 rad/s
Recovery |θ₂| ≥ 110° or speed exceeded; motor halted

📁 Repository Structure

furuta-pendulum/
├── Simulink/
│   ├── SwingUp_Model.slx          # Energy-based swing-up subsystem
│   ├── LQR_Model.slx              # LQR balancing subsystem
│   ├── FullSystem_Model.slx       # Complete hybrid model with Stateflow
│   └── Pendulum_SimMechanics.slx  # Simscape Multibody physical model
├── MATLAB/
│   ├── parameters.m               # System physical parameters
│   ├── lqr_design.m               # LQR gain computation (K, Q, R)
│   └── linearize_model.m          # Linearization script
├── ESP32/
│   └── firmware/                  # Microcontroller code for HIL
├── Results/
│   ├── simulation_results.png
│   └── real_life_results.png
├── Report/
│   └── T14_Hybrid_Project_Report.pdf
└── README.md

🚀 Getting Started

Requirements

  • MATLAB R2022b or later
  • Simulink
  • Simscape & Simscape Multibody toolboxes
  • Control System Toolbox (for lqr())
  • Stateflow

Running the Simulation

  1. Clone the repository:

    git clone https://github.com/<your-org>/furuta-pendulum.git
cd furuta-pendulum

  2. Open MATLAB and run the parameter initialization script:

    run('MATLAB/parameters.m')

  3. Open and run the full Simulink model:

    open('Simulink/FullSystem_Model.slx')

  4. Press Run in Simulink. Observe the four scopes: PendAngle, PendVel, MotorAngle, MotorVel.

HIL Deployment (ESP32)

  1. Flash the ESP32 firmware from ESP32/firmware/.
  2. Connect via USB to the host PC.
  3. In Simulink, configure the serial port in the ESP32 communication block.
  4. Run the model in External Mode for real-time HIL control.

📊 Results Summary

Metric Simulation Real-Life
Swing-up success ✅ ✅
LQR stabilization ✅ ✅
Settling time ~7 s ~12 s
Post-stabilization oscillations Low Slightly higher

The discrepancies between simulation and real-life results are attributed to unmodeled joint backlash, high-frequency friction, and minor actuation delays.

📐 Controller Requirements

Requirement Specification
Swing-Up Move pendulum to upright
Stabilization Maintain upright via LQR
Sampling rate ≥ 0.1 kHz
Torque limits Motor saturation constraints
Robustness Against small disturbances

📚 References

  • B. Yao, "Modeling of Rotary Inverted Pendulum," 2022.
  • K. J. Åström & K. Furuta, "Swinging up a pendulum by energy control," Automatica, 2000.

📄 License

This project is submitted as academic coursework for MCT411 at Ain Shams University. All rights reserved by the authors.