Decoupled Modeling and Cascaded Control of a Two-Degrees of Freedom Pendulum-Driven Spherical Robot

Simulink model accompanying the paper

Decoupled Modeling and Cascaded Control of a Two-Degrees of Freedom Pendulum-Driven Spherical Robot (authors, venue, year — to be added)

Heading control on the real robot

spherical_robot_demo_plots.mp4

Recorded run replayed at 2× speed: position from the overhead camera (Luxonis OAK 4 D Pro), measured heading from the IMU (red arrow) and heading reference (yellow arrow), together with a separate run filmed with a phone.

Experimental setup

 

The robot (top: in the arena, bottom: with the upper half of the shell removed) is driven by a brushless motor on the shaft and tilted laterally by a servo; two IMUs on the pendulum measure the heading and the tilt. The position is measured by an overhead camera (Luxonis OAK 4 D Pro), and the controllers run on a host computer (ROS 2, 50 Hz) that communicates with the robot over ESP-NOW. The robot was designed and built by Hanjo Claassen at Delft University of Technology.

Contents

File	Description
simulink_model/spherical_robot_2DOF.slx	Simulink model with the decoupled model and the multibody (Simscape) model of the robot, controlled by the same cascaded controller
docs/	Figures used in this README

The model contains two plants that are simulated in parallel with the same references and controllers:

• System model: the decoupled model of the paper, i.e. two planar Lagrangian models (longitudinal and lateral axis) closed by the heading kinematics, $\dot\psi = (v/r)\tan\varphi_y$, $\dot X = v\cos\psi$, $\dot Y = v\sin\psi$.
• Simscape model: a three-dimensional multibody model of the shell and the two-axis pendulum with compliant ground contact and friction, in which turning emerges from the dynamics.

Control structure

• Longitudinal axis: speed loop $C_v$ → shaft-rate loop $C_\omega$ → motor torque $u_x$
• Lateral axis: heading loop $C_\psi$ → relative pendulum roll loop $C_\gamma$ → servo torque $u_y$

Requirements

• MATLAB / Simulink R2026b (the model can be saved for older releases with Save As)
• Simscape
• Simscape Multibody

Running the model

1. Open simulink_model/spherical_robot_2DOF.slx.
2. Select the reference with the manual switches at the top level of the model (speed step, tilt step, heading step, heading ramp, circle, figure-eight).
3. Run the simulation. The scopes and XY graphs show the responses and paths of both models.

Citation

If you use this model, please cite the paper:

@article{TODO,
  title   = {Decoupled Modeling and Cascaded Heading Control of a
             Pendulum-Driven Spherical Robot},
  author  = {TODO},
  journal = {TODO},
  year    = {TODO}
}

License

MIT, see LICENSE.