ANT-X Simulation — MATLAB

LTV-MPC trajectory tracking for the ANT-X nano quadrotor (270 g) via differential flatness.
Euler 12-state and quaternion 13-state models · Circle, Lemniscate, Spiral trajectories · Full closed-loop results.

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🚀 Version 2 — Major Updates

V2 contains updated version of the original MATLAB simulation pipeline. Compared with v1, the main changes are:

• Trajectory recalibration — Updated trajectory parameters for more regular profiles and better compatibility with the laboratory flight volume.
• Improved PWC reference generation — Replaced pointwise input sampling with the exact integral average of the continuous input over each sampling interval, reducing bias in higher-order derivatives.
• 16-state nonlinear model — Introduced a new model including first-order motor dynamics and individual rotor thrust states.
• Motor allocation and mixer — Added explicit allocation and mixer matrices, including the conversion between virtual thrust/moments and individual rotor thrusts.
• Parameter identification framework — Added a preliminary parameter estimation procedure and a detailed experimental identification procedure intended for laboratory deployment.
• Updated MPC formulation — Replaced quadprog with OSQP, introduced direct physical constraints on individual rotor thrusts, systematically defined the weighting matrices, and adopted exact ZOH discretization.
• More realistic closed-loop simulation — Added actuation/computation latency and 3-second hover tails for Perturbation and Origin initial conditions, making the simulations more representative of the planned flight tests.
• Extended validation — Added quantitative tracking, computational-performance, and motor-allocation checks for the updated B01 formulation.

Introduction

This repository contains the first MATLAB simulation pipeline for the ANT-X UAV.

The pipeline covers the full stack from nonlinear modelling to closed-loop control:

1. Nonlinear rigid-body dynamic model (NED/FRD, Euler Z-Y-X or unit quaternion)
2. Differential flatness map — state and input reconstruction from flat outputs σ = [x, y, z, ψ]
3. Feasible trajectory generation (Circle, Lemniscate, Spiral) with offline PWC reference
4. Open-loop validation via ode45
5. LTV-MPC closed-loop control via sequential linearisation + quadprog
6. Per-run CSV export and quantitative tracking metrics

Detailed documentation is in documentation/.

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Prerequisites

Requirement	Notes
MATLAB R2024b or later	Earlier versions may work
Optimization Toolbox	Required for quadprog

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How to Use

1. Clone the repository

git clone https://github.com/tavaa/ANT_X_Simulation.git
cd ANT_X_Simulation

2. Open MATLAB and add the project to the path

addpath(genpath('.'))

3. Run open-loop tests (DoF validation)

% Individual degree-of-freedom tests
run('scripts/openloop/run_pitch_test.m')
run('scripts/openloop/run_roll_test.m')
run('scripts/openloop/run_thrust_test.m')
run('scripts/openloop/run_yaw_test.m')

4. Run open-loop trajectory tests

% Circle, Lemniscate, Spiral — continuous vs PWC comparison
run('scripts/openloop/run_circle_openloop.m')
run('scripts/openloop/run_lemniscate_openloop.m')
run('scripts/openloop/run_spiral_openloop.m')

5. Run closed-loop MPC simulations

% Full closed-loop: all trajectories × all initial conditions
run('scripts/MPCSimulationLoop.m')

Results are saved to results/<trajectory>/<initial_condition>/.

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Model

The quadrotor is modelled as a rigid body under Newton-Euler equations in NED/FRD frames.

12-state model (A00_SimplifiedModel.m) — Euler Z-Y-X attitude:

x = [xI  yI  zI  ẋB  ẏB  żB  φ  θ  ψ  p  q  r]ᵀ ∈ ℝ¹²
u = [T  L  M  N]ᵀ ∈ ℝ⁴

13-state model (A01_SimplifiedModel_quat.m) — unit quaternion attitude (singularity-free).

Parameters — m = 0.270 kg, b = 0.08 m, J = diag(0.00307, 0.00307, 0.00239) kg·m².
Inertia values from Cavagnini (2021); SysID aerodynamic parameters from El Omari (2023).

NED sign convention: z-down. Climb → negative zNED. Forward acceleration → θ < 0.

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Flatness Map

Flat outputs: σ = [xI, yI, zI, ψ]ᵀ

State and inputs are reconstructed algebraically from σ and its time derivatives up to snap (4th order), without integrating the rotational dynamics:

Quantity	Requires
Position, velocity	(position, velocities)
Attitude (R, Euler/quat)	(accellerations)
Body rates ωB	(jerk)
T, L, M, N	(snap)

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Trajectories

Trajectory	Shape	Parameters
Circle	Uniform circular, constant altitude	R = 1.0 m, ω = 1.0 rad/s, h = −1.0 m
Lemniscate (Gerono)	Figure-8 planar	R = 1.0 m, ω = 0.5 rad/s, h = −1.0 m
Spiral	Elliptic helix, descending	Rx = 2.0 m, Ry = 1.0 m, ω = 0.5 rad/s, vz = 0.2 m/s

Each trajectory provides position, velocity, acceleration, jerk, and snap in closed form. Offline PWC references are generated via GeneratePWC_reference.m using ode45.

Three initial conditions are tested per trajectory:

Condition	Description
OnReference	Drone initialised exactly at xref(0)
Perturbation	+0.20 m offset on x, −0.20 m on y; velocities/rates zeroed
Origin	Drone at ground (zI = −0.02 m), all states zero

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MPC

Method: LTV-MPC with sequential linearisation along the reference trajectory (Kunz, Huck & Summers, ECC 2013).

Parameter	Value
Sampling time Ts	0.05 s
Prediction horizon p	18 steps
State cost Q = Qf	diag(50,50,50, 8,8,12, 8,8,10, 1,1,1)
Input cost R	diag(2,2,2,2)
QP solver	quadprog (MATLAB Optimization Toolbox)

At each step: linearise → Euler-discretise → build sparse QP → solve → apply optimal u → warm-start next iteration.

State constraints (from El Omari Table 5.3):

-3 ≤ ẋB, ẏB ≤ 3 m/s      -3 ≤ żB ≤ 1 m/s
|φ|, |θ| ≤ 35°             |p|, |q|, |r| ≤ 600°/s
0 ≤ T ≤ 2mg ≈ 5.30 N      |L|, |M| ≤ 0.15 Nm      |N| ≤ 0.05 Nm

Steady-state tracking results (Ts = 0.05 s):

Scenario	Condition	RMSEpos SS [m]	tconv [s]	QP mean [ms]
Circle	Perturbation	0.0091	1.50	9.04
Lemniscate	Perturbation	0.0064	1.35	8.55
Spiral	Origin	0.0049	1.85	8.88

quadprog is not real-time embeddable.

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Repository Structure

antx-simulation/
├── models/               # non-linear models
├── flatness/             # flatness input/state reconstruction
├── trajectories/         # Shape classes (Circle, Lemniscate, Spiral, DoF tests)
├── control/
│   ├── MPC/              # QPBuilder, LTV-MPC controller
├── scripts/              # Runnable simulation scripts
├── results/              # Auto-generated output (CSV, plots, GIFs)
├── documentation/        # Full technical report (PDF)
└── README.md

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Known Limitations

• Rigid body only — no rotor drag, blade flapping, or motor/ESC dynamics
• Instantaneous actuation assumed
• Motor mixing matrix not implemented (CT, CQ unknown)
• Inertia values estimated from a similar platform; experimental re-identification needed
• No EKF — full noiseless state assumed
• Qf = Q is a pragmatic choice inherited from Kunz et al.; asymptotic stability not guaranteed

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References

1. ANT-X Development Team, ANT-X drone platform — software tools documentation, 2024.
2. G. Cavagnini, Identification and control of a nano quadrotor, MSc thesis, Politecnico di Milano, 2021.
3. L. Kunz, M. Huck, T. H. Summers, "Fast model predictive control of miniature helicopters," ECC 2013, pp. 1377–1382.
4. S. El Omari, Multirotor UAV control via dynamic inversion, MSc thesis, Politecnico di Milano, 2023.
5. D. Mellinger, V. Kumar, "Minimum snap trajectory generation and control for quadrotors," ICRA 2011, pp. 2520–2525.
6. M. Faessler, A. Franchi, D. Scaramuzza, "Differential flatness of quadrotor dynamics subject to rotor drag," RA-L, vol. 3, pp. 620–626, 2018.