Non-Linear Control of a 2-DOF Robotic Arm

Complete design, simulation, and hardware implementation of a non-linear controlled two-degree-of-freedom (2-DOF) planar robotic arm. The project covers mathematical modeling using the Euler–Lagrange formulation, non-linear stability analysis via Lyapunov methods, feedback linearization control synthesis, MATLAB/Simulink simulation, and real-time embedded implementation on an Arduino microcontroller.

Paper — Non-Linear Control of a 2-DOF Robotic Arm: Complete Design, Simulation, and Hardware Implementation (PDF)

Fully assembled 2-DOF robotic arm prototype with 3D-printed PLA+ links, 25GA370 DC gear motors, L298N driver, and Arduino Mega 2560.

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Table of Contents

• Overview
• Key Results
• Hardware
• Simulation Results
• Hardware Design
• Repository Structure
• MATLAB Workflow
• Arduino Workflow
• Authors
• References
• License

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Overview

The system controls a planar 2-DOF arm with joint variables $q = [q_1, q_2]^T$ governed by the Euler–Lagrange equation:

$$M(q)\ddot{q} + C(q, \dot{q})\dot{q} + G(q) + D\dot{q} = \tau$$

A feedback linearization controller cancels all non-linear terms, producing a linearized closed-loop system. The control law computes:

$$\tau_{\text{des}} = M(q)\ddot{q}_r + C(q,\dot{q})\dot{q} + G(q) + (D + \beta)\dot{q}$$

with reference acceleration $\ddot{q}_r = \ddot{q}_d + K_p e + K_d \dot{e}$, where $e = q_d - q_m$.

Lyapunov stability is guaranteed via $V = \tfrac{1}{2}e^T K_p e + \tfrac{1}{2}\dot{e}^T\dot{e}$, yielding $\dot{V} = -\dot{e}^T K_d \dot{e} \leq 0$.

Key Results

Metric	Value
Steady-state position error	< 0.01 rad
Circular trajectory phase error	< 0.02 rad
Amplitude error	< 1 %
Control loop rate	100 Hz (10 ms)
Controllability rank	4 (full) across entire workspace

Hardware

Component	Specification
Motors	25GA370 DC gear motor, 12 V, 50:1 gear ratio, 0.44 N·m stall torque
Encoders	Integrated quadrature, 550 counts/rev
Controller	Arduino Mega 2560
Driver	L298N dual H-bridge (2.5–46 V, 2 A/channel)
Links	3D-printed PLA+, SolidWorks-designed

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Simulation Results

Position Regulation

Two gain configurations were compared for step response ($q_d = [0.5, 0.5]$ rad):

Characteristic	Low Gains ($K_p = 5$, $K_d = 1$)	High Gains ($K_p = 10$, $K_d = 3$)
Overshoot	~0.05 rad	~0.01 rad
Settling time	~3 s	~1.5 s
Control effort	Low	High
Noise sensitivity	Low	High

Circular Trajectory Tracking

End-effector commanded to trace a circle of radius $R = 0.3$ m at $\omega = 0.5$ rad/s. Measured joint angles track desired references with phase error < 0.02 rad.

Circular trajectory tracking: measured joint angles (solid) closely follow desired references (dashed) with minimal phase lag.

Hybrid Mode (Position → Trajectory)

The full sequence (OFF → POSITION → CIRCLE) demonstrates seamless mode transitions managed by a Stateflow state machine:

Stateflow state machine governing mode transitions: OFF → POSITION → CIRCLE.

Closed-loop equilibrium response: joint angles converge to zero (top), velocities decay (middle), and control voltages settle (bottom).

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Hardware Design

SolidWorks CAD assembly showing both links, motor housings, and mounting points.

 

Left: 25GA370 DC gear motor with integrated quadrature encoder. Right: 6-pin encoder interface (Motor±, Encoder±, channels A/B).

Embedded Control Loop (Arduino)

1. Read encoder counts → compute joint angles: q = 2π · count / 550
2. Estimate velocities via filtered finite differences:
   q̇_filt[k] = 0.7·q̇_filt[k-1] + 0.3·Δq/Ts
3. Evaluate state machine (OFF / POS / CIRCLE), generate references
4. Execute feedback linearization → compute τ_des → convert to PWM
5. Apply saturation (±12 V) → output to L298N

Repository Structure

.
|-- matlab/
|   |-- Advanced_trial.slx
|   |-- init_params.m
|   |-- check_controllability.m
|   `-- check_nonlinear_controllability_lie.m
|-- arduino/
|   |-- end_effector_circle/
|   |   `-- end_effector_circle.ino
|   |-- joint_space/
|   |   `-- joint_space.ino
|   |-- nonlinear_trajectory/
|   |   `-- nonlinear_trajectory.ino
|   `-- nonlinear_trajectory_alt/
|       `-- nonlinear_trajectory_alt.ino
`-- docs/
    |-- Nonlinear_Control_2DOF_Arm.pdf
    `-- figures/                        # Extracted from the report

MATLAB Workflow

1. Open MATLAB in the matlab/ directory.
2. Run init_params.m to initialize arm and motor parameters.
3. Execute:
   • check_controllability.m
   • check_nonlinear_controllability_lie.m
4. Open Advanced_trial.slx for simulation and controller testing.

Arduino Workflow

1. Open one sketch folder under arduino/ in Arduino IDE.
2. Select the board and COM port matching your hardware.
3. Upload one of the control sketches:
   • end_effector_circle.ino
   • joint_space.ino
   • nonlinear_trajectory.ino
   • nonlinear_trajectory_alt.ino

nonlinear_trajectory_alt is preserved as an alternate trajectory/controller variant.

Notes

• This repository intentionally keeps multiple controller implementations to compare behavior across trajectory spaces and tuning choices.
• Update pin definitions and hardware constants in each sketch before deployment if your wiring differs.

Authors

Name	Affiliation
Ahmed Mostafa	Mechatronics Engineering, GUC
Andrew Abdelmalak	Mechatronics Engineering, GUC
Hazim Alwakad	Mechatronics Engineering, GUC
Mazen Amr	Mechatronics Engineering, GUC
Samir Yacoub	Mechatronics Engineering, GUC
Youssef Youssry	Mechatronics Engineering, GUC

Acknowledgments

The authors thank Prof. Ayman A. El-Badawy and Eng. Karim Abdelsalam for their guidance and instruction throughout this project.

Report

The full project report is available in docs/Nonlinear_Control_2DOF_Arm.pdf.

References

1. J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice-Hall, 1991.
2. M. W. Spong, S. Hutchinson, and M. Vidyasagar, Robot Modeling and Control, 2nd ed., Wiley, 2020.
3. B. Siciliano et al., Robotics: Modelling, Planning and Control, Springer, 2009.
4. A. A. El-Badawy, "Advanced Mechatronics Tutorial Notes," GUC, 2025.
5. Ampere Electronics, "25GA370 DC gear motor with encoder specifications," 2023.
6. V. H. Benitez et al., "Design of an affordable IoT open-source robot arm for online teaching," HardwareX, 8, 2020.

License

Released under the MIT License. See LICENSE.