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SINDy + MPC for Multirotor Collision Avoidance : When Data-Driven Identification Meets Predictive Control

A closer look at sparse dynamics identification and model predictive control for multirotor collision avoidance.

SINDy + MPC for multirotor collision avoidance: overview

Paper: Lee et al., Sparse Identification of Nonlinear Dynamics‐Based Model Predictive Control for Multirotor Collision Avoidance, IET Control Theory & Applications, 2025. DOI: 10.1049/cth2.70049 · arXiv: 2412.06388

All figures in this post are reproduced directly from the paper above (with the author’s involvement).

Why this paper?

Strap a payload to a multirotor and the vehicle is no longer the one you modeled. Mass and inertia change, aerodynamic disturbances enter, and the controller starts to wobble. Now demand that the same vehicle track a reference trajectory while avoiding obstacles : and the problem becomes genuinely interesting.

This paper attacks the problem with a data-driven identification + model predictive control (MPC) combo. The identifier of choice is SINDy (Sparse Identification of Nonlinear Dynamics).

System model: a multirotor with payload

The 6-DOF dynamics considered in the paper are

\[\dot{\eta} = \nu, \qquad \dot{\nu} = \tfrac{1}{m_t}(R_{BI} f_t + f_a) - g\] \[\dot{\Omega} = R_\omega \omega, \qquad \dot{\omega} = I_t^{-1}\left(\tau_t + \tau_a - \omega \times I_t \omega \right)\]

Multirotor with payload Fig. 1. Frame definitions for the payload-carrying multirotor: body frame ${B}$, inertial frame ${I}$, and rotor thrusts $T_1$-$T_4$. The payload changes both the mass and the inertia of the vehicle. (Source: Lee et al., 2025, Fig. 1)

The catch is that the payload changes mass and inertia:

  • $m_t = m_m + m_p$
  • $I_t = I_m + I_p$

Aerodynamic effects are modeled as linear damping:

  • $f_a = [-K_F \dot{x},\ -K_F \dot{y},\ -K_F \dot{z}]^\top$
  • $\tau_a = [-K_{M,p}\, p,\ -K_{M,q}\, q,\ -K_{M,r}\, r]^\top$

The practical issue is that $m_p$, $I_p$, $K_F$, and $K_M$ are all unknown. They must be estimated from data.

Why SINDy?

There is no shortage of ways to learn dynamics from data : neural networks, Gaussian processes, Koopman embeddings. They share one drawback: the resulting model is a black box. You cannot easily say which term in the dynamics caused a given output, nor can you give the model a physical meaning.

SINDy is different.

  • It assumes that the governing equations of the system are sparse : only a small number of terms matter.
  • A candidate function library $\Psi$ is regressed against state derivatives, and L1 regularization keeps only the meaningful coefficients.
  • The result is a human-readable ODE that requires modest amounts of data.

For safety-critical aerospace applications, the interpretability of SINDy is a real asset.

SINDy concept Fig. 2. The SINDy regression in pictures. From measured time signals $\dot x(t), \dot y(t), \dot z(t)$ we form $\dot X = \Psi(X,U)\Sigma$, where the library $\Psi$ holds candidate monomials and cross-products (e.g.\ $1, xu, y, x^2u, \ldots, z^5$) and the coefficient matrix $\Sigma$ is driven sparse by L1 regularization. (Source: Lee et al., 2025, Fig. 2)

SINDy in one equation

Stack the state snapshots into $X$ and their numerical derivatives into $\dot X$. Then regress

\[\dot X = \Psi(X,U)\, \Sigma\]

with an L1 penalty on the coefficient matrix:

\[\Sigma^\star = \arg\min_\Sigma \, \| \dot X - \Psi(X,U)\Sigma \|_2^2 + \lambda \|\Sigma\|_1\]

Larger $\lambda$ produces a sparser $\Sigma$, leaving only the dominant dynamics terms.

Embedding physical priors in the library

A clever element of the paper is that physics-informed terms are baked into the candidate library:

\[\Psi_{\text{tr}} = \big[\,\Psi_{\text{prior}}(\text{thrust, gravity}) \;\big|\; \Psi_{\text{poly}}(\text{velocity polynomials})\,\big]\] \[\Psi_{\text{ro}} = \big[\,\Psi_{\text{prior}}(\text{control, rotational coupling}) \;\big|\; \Psi_{\text{poly}}(\text{angular-rate polynomials})\,\big]\]

This guides the regression toward physically plausible solutions even with limited data.

Where does the data come from?

Hovering yields almost no information. The authors instead fly a PID baseline controller along a rectangular reference trajectory for 100 s, logging states and inputs at $\Delta t = 0.002\,\text{s}$. Translation and rotation models are then identified separately.

SINDy-MPC: closing the loop with the identified model

The identified $\dot x = \hat f(x,u)$ is plugged directly into an MPC formulation:

\[\min_{u_1,\dots,u_N}\; \sum_{i=1}^{N} \big[\, (r_i - y_i)^\top Q (r_i - y_i) + u_i^\top R\, u_i \,\big]\]

subject to

  • $\dot x = \hat f(x,u)$ : SINDy-identified dynamics,
  • $u_{\min} \le u \le u_{\max}$ : input bounds,
  • $x_1 = x_{\text{init}}$ : initial state,
  • $\sqrt{(x_{\text{ob}}-x)^2 + (y_{\text{ob}}-y)^2 + (z_{\text{ob}}-z)^2} \ge D_{\min}$ : collision avoidance.

The last constraint is what makes this work: a simple distance inequality forces the optimizer to bend the trajectory around obstacles.

SINDy-MPC architecture Fig. 3. The full SINDy-MPC pipeline. (Left) Offline stage : baseline-flight data $\dot X, X, U$ feeds a sparse regression that yields the library $\Psi(X,U)$ and a sparse $\Sigma$. (Right) Online stage : the identified model is plugged into an MPC that tracks $r(t)$ while closing the loop with the plant $\dot x = f(x,u)$. (Source: Lee et al., 2025, Fig. 3)

Identification accuracy (Tables II-III)

The paper reports true vs identified coefficients for the dominant terms of both translational and rotational dynamics. The key entries:

Term True $\Sigma$ Identified $\hat\Sigma$ Error
$R_{(1,3)} F_B \to \dot{x}$ 0.7692 0.7707 0.20%
$R_{(2,3)} F_B \to \dot{y}$ 0.7692 0.7704 0.16%
$L \to \dot{p}$ 32.258 32.248 0.03%
$M \to \dot{q}$ 26.316 26.316 0.00%
$N \to \dot{r}$ 15.873 15.879 0.04%

Every dominant coefficient is recovered to within 1%. The exception is yaw aerodynamic damping ($-K_{M,r} r$), where the rectangular reference simply does not excite yaw enough to identify the term reliably (>30% error). Excitation is part of the design problem.

Closed-loop performance (Table IV)

A direct comparison between the SINDy-MPC and the nominal-model MPC under the same payload + aerodynamic uncertainty:

RMSE [m] $x$ $y$ $z$
SINDy-MPC 0.813 0.381 0.399
Nominal-model MPC 0.909 0.452 0.628

The largest improvement is on the z axis (36% reduction, $0.628 \to 0.399$), which is exactly where unmodeled mass and aerodynamic drag hurt the most.

Position tracking Fig. 4. 3D trajectory tracking. With a rectangular reference and two obstacles, SINDy-MPC follows the reference more closely while still keeping the safety distance from each obstacle. (Source: Lee et al., 2025, Fig. 8)

Position error Fig. 5. Position-tracking error time histories. SINDy-MPC (blue) shows consistently smaller error than nominal-model MPC (red) on every axis, with the gap most pronounced on the z axis. (Source: Lee et al., 2025, Fig. 9)

Contributions in one paragraph

  1. A meaningful multirotor model that explicitly accounts for payload and aerodynamic uncertainty.
  2. The first SINDy + MPC framework specifically targeted at multirotor collision avoidance.
  3. Embedding physical priors in the candidate library to combine data efficiency with model interpretability.

My take

In a world increasingly dominated by neural identifiers, this paper is a reminder that sparsity and interpretability still matter : especially in aerospace, where being able to explain why a model produces a given output is half the value.

The honest limitations:

  • The result depends heavily on the chosen library : domain knowledge is unavoidable.
  • Data distribution matters; the yaw case shows what happens when it does not.
  • Online re-identification and adaptive SINDy are natural follow-up directions.

A real-flight implementation, plus an adaptive SINDy variant for changing payloads, would be the obvious next step.

References

Figure provenance

All figures in this post are reproduced directly from Lee et al. (2025) : the original paper of the post’s author. The original figure number from the paper is given in each caption.

Corrections welcome. As always, the paper itself is the best source.