UR5 Robot Manipulator – Modeling & Simulation 🦾

A simulation model of a UR5 robotic manipulator with PD + Gravity compensation control.

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📖 Overview

This project models and simulates the UR5 robotic arm manipulator using a modular approach based on:

• Link and joint submodels
• Harmonic drive gear dynamics
• PD + Gravity compensation control

The model has been developed using Bond Graph and custom submodels.

The goal is to validate the dynamic model and control strategy through simulation of a pick-and-place trajectory in joint space.

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🛠 Features

• 6-DOF manipulator model (UR5)
• Denavit–Hartenberg (DH) parameterization for frame transformations
• Link dynamics implemented via Euler equations
• Revolute joint constraints for 1-DOF motion
• Harmonic drive submodel with stiffness, damping, and gear reduction ratio
• PD + Gravity compensation controller in the joint space
• Trajectory planning with cycloidal interpolation

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📐 Modeling Details

Denavit–Hartenberg Convention

Frame transformations are implemented using Modulated Transformers, ensuring correct reference frames during simulation.

Link Submodel

Each link is treated as a rigid body with two-port connections, exchanging forces and torques. Dynamics are computed using Euler’s equations:

$$ \dot{p} = F - \omega \times p, \quad \dot{h} = \tau - \omega \times h $$

Joint Submodel

Each revolute joint constrains motion to a single axis and computes joint angle θ as the integral of angular velocity.

Harmonic Drive

Implements:

• Gear reduction ratio (e.g., HFUS-20: -1/100)
• Nonlinear stiffness with three constant regions $K_1, K_2, K_3$
• Viscous damping and static friction

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🎛 Controller

The control law is:

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

where:

• $K_p, K_d$ are PD gains
• $G(q)$ is gravity torque compensation
• $e$ is the position error

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📊 Simulation Results

• Trajectory: Three-point pick-and-place task

• Trajectory profile: Cycloidal in joint space

• Results:

  • Joint positions follow reference closely
  • Errors are small
  • Gravity compensation effective

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✅ Conclusions

• The model accurately reproduces UR5 dynamics
• The PD + gravity compensation controller yields expected performance
• The framework is suitable for further control design and testing
• Future improvements: experimental validation with a real UR5 robot

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📚 References

• A. Macchelli – Modeling and Simulation of Mechatronic Systems M, DEI – University of Bologna
• C. Melchiorri – Industrial Robotics, DEI – University of Bologna

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👥 Contributors

This project was developed as part of the Automation Engineering course at the University of Bologna (AY 2024/2025).

Name	Role
Battistini Enrico	Modeling & Simulation
Samorì Filippo	Controller Design & PD Tuning
Subini Jacopo	Harmonic Drive Modeling & Report Editing

💡 Contributions were equally important in modeling, simulation, and documentation.

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📜 License

MIT License – See LICENSE for details.