Inverted Pendulum on a Cart: Modelling, PID Control and Simscape Simulation

A cart–pendulum system modelled from first principles, stabilised with a PID controller in MATLAB/Simulink, and validated against a CAD-based Simscape Multibody model built from a SolidWorks assembly. Course project for Dynamics.

SolidWorks assembly of the cart and pendulum

Overview

Phase	What was done	Tools
1. Analytical model	Equations of motion with the Lagrangian method, linearisation about upright, state-space model and transfer function	MATLAB, Symbolic Math Toolbox
2. Control design	PID controller with filtered derivative, tuned for a 60° phase margin, simulated on the linear and nonlinear models	MATLAB, Simulink, Control System Toolbox
3. CAD validation	SolidWorks assembly imported into Simscape Multibody, same PID gains, results compared with the analytical model	SolidWorks, Simscape Multibody Link

Key results

The upright pendulum is open-loop unstable (pole at +4.72 rad/s). The PID controller (Kp = −4192.9, Ki = −8365.6, Kd = −645.6, Tf = 0.01 s) balances it from the 1.37° tilt of the CAD pose:

Metric	Simulink (analytical)	Simscape (CAD)	Difference
Settling time (2 %)	1.219 s	1.224 s	+0.4 %
Undershoot past upright	27.43 %	27.05 %	−1.4 %
Peak control force	100.12 N	99.97 N	−0.15 %
Cart peak / final position	31.05 / 29.26 mm	30.98 / 29.41 mm	−0.2 / +0.5 %

The two models agree to within 0.0067° in pendulum angle (RMS 0.0014°). Phase margin 60°, peak sensitivity 1.16, and the loop stays stable for ±20 % changes in pendulum and cart mass.

Simscape vs Simulink comparison

More figures

**Simulink model and closed-loop response (Phase 2)**

**Simscape Multibody model and scope output (Phase 3)** Simscape model

Repository structure

matlab/   MATLAB scripts, Simulink and Simscape models, data files, STEP geometry
cad/      SolidWorks assembly and parts
report/   Project report (PDF) and its LaTeX source
docs/     Figures used in this README

Keep every file in matlab/ in the same folder: the Simscape model loads the *_sldprt.STEP geometry files by name.

How to run

Requires MATLAB R2025b with Simulink, Control System Toolbox, Symbolic Math Toolbox and Simscape Multibody.

Open MATLAB and set matlab/ as the current folder.
derive_symbolic – derives the equations of motion and the A, B matrices.
main_pid_inverted_pendulum – designs the PID controller, saves pid_gains.mat, plots the response.
build_simulink_pid_model – builds and runs pendulum_pid.slx (scopes + results figure).
PendulumAssy_DataFile, then open PendulumAssy.slx and press Run – CAD-based simulation with the 3-D animation.
compare_pid_vs_assy – runs both models and produces the comparison figure and table.
Optional: optimize_pid re-derives the gains; PendulumAssyDisturb.slx adds a torque disturbance.

Assumptions and limitations

Rigid bodies, planar motion, no friction or actuator limits, ideal sensors. The PID acts on the pendulum angle only, so the cart position is not regulated: after a disturbance the cart drifts. A cascade loop on cart position, or LQR, would fix this.

Author

Mary Sam: Dynamics course project, 2026.

Mary Kuru Sam