How to Steer a Swarm of Drones Using Only Geometry

A Geometric Solution of the Schrödinger Bridge Problem on $\mathsf{SO}(2)$ via Stochastic Optimal Control

0.001 — that’s the maximum relative error between the simulated final density and the target terminal density in the numerical experiments of this paper. It’s a tiny number, but it represents something monumental: the first geometric, coordinate-free solution to the Schrödinger bridge problem (SBP) on a Lie group — specifically, $\mathsf{SO}(2)$, the group of 2D rotations. This isn’t just a mathematical curiosity. It’s a step toward controlling entire populations of rotating systems — from fleets of drones adjusting their headings in formation to microscopic particles aligning in magnetic fields — not by steering each one individually, but by sculpting their collective behavior with minimal effort.

The breakthrough lies in solving a century-old problem in a new geometric setting. The Schrödinger bridge, originally conceived in 1931 by physicist Erwin Schrödinger as a thought experiment about diffusing particles, has evolved into a powerful framework for optimal density control. The goal? Steer a system’s state distribution from an initial configuration to a desired final one, while minimizing the expected energy spent. In engineering, this translates to guiding swarms, managing traffic flows, or programming molecular machines — all while being as efficient as possible.

But until now, most solutions assumed systems lived in flat, Euclidean space. Real-world systems — especially robots and vehicles — don’t. Their states often live on curved spaces called Lie groups. A drone’s orientation isn’t a point on a plane; it’s an element of $\mathsf{SO}(3)$, the 3D rotation group. A car’s heading is an angle — a point on $\mathsf{SO}(2)$, the circle. Treating these as flat simplifies math but distorts physics. This paper refuses that compromise.

Instead, Hamza Mahmood and Adeel Akhtar solve the SBP for a system evolving on $\mathsf{SO}(2)$ — the simplest nontrivial Lie group — using only its intrinsic geometry. No embedding into ${\mathbb{R}}^2$. No coordinates. Just pure, geometric reasoning. The result is a controller that tells each agent how to adjust its angular velocity so that, collectively, the swarm morphs from one orientation distribution to another with minimal control effort.

And they prove it works — not just numerically, but mathematically. By showing that a certain fixed-point iteration contracts under Hilbert’s projective metric, they establish the existence and uniqueness of the solution. The controller emerges as a clean, coordinate-free expression derived from the heat semigroup on the group — a deep connection between stochastic dynamics and geometric analysis.

This isn’t just theory. The authors provide code and animations. The problem setup is simple: start with three peaks in orientation (three clusters of agents facing different directions), end with two. The simulation shows the density smoothly reshaping itself, like a fluid flowing across the circle, converging to the target with near-perfect fidelity.

What makes this work radical is its refusal to flatten the world. Most prior approaches would embed $\mathsf{SO}(2)$ into the plane and apply Euclidean SBP methods. That introduces artifacts: agents might take inefficient paths, or the controller might fail to respect the periodicity of angles. Here, everything happens on the circle, using only tools that respect its symmetry.

The implications ripple outward. As robotic swarms grow in complexity, we’ll need controllers that understand geometry — not just position, but orientation, alignment, and collective shape. This paper offers a blueprint.

The Science

The Schrödinger bridge problem is a stochastic optimal control problem. Given a system governed by random motion (like Brownian noise), design a control policy that gently nudges the system’s state distribution from a known initial density ${\rho }_0$ to a desired final density ${\rho }_1$ over a fixed time interval, while minimizing the expected squared control effort.

In this paper, the system is a kinematic model on $\mathsf{SO}(2)$, the Lie group of 2D rotations. A state $R(t)\in \mathsf{SO}(2)$ represents the orientation of a rotating agent — say, a robot or a drone. Its dynamics are governed by:

$$\dot{R}(t)=R(t)\hat{\Omega }(R,t)$$

where $\Omega (R,t)\in \mathbb{R}$ is the angular velocity (the control input), and $\hat{\cdot }$ maps scalars to skew-symmetric matrices in the Lie algebra $\mathfrak{so}(2)$. This equation is coordinate-free — it doesn’t depend on how you parametrize the rotation.

To model uncertainty, the authors introduce noise via a Stratonovich stochastic differential equation (SDE):

$$d{R}_t=R_t\hat{\Omega }(R_t,t)dt+\sigma \sum _{i=1}^m{R}_t{\hat{B}}_i\circ d{W}_t^i$$

Here, $W_t^i$ are standard Wiener processes, $B_i\in \mathfrak{so}(2)$ are constant noise directions, and $\sigma >0$ sets the diffusion strength. The Stratonovich formulation is crucial: it preserves the chain rule on manifolds, ensuring the solution stays on $\mathsf{SO}(2)$.

The state probability density $\rho (R,t)$ evolves according to the Fokker-Planck equation:

$${\partial }_t\rho =-\mathrm{div}(\rho R\hat{\Omega })+\frac{{\sigma }^2}{2}{\Delta }_{\mathsf{SO}(2)}\rho$$

where ${\Delta }_{\mathsf{SO}(2)}$ is the Laplace-Beltrami operator — the generalization of the Laplacian to curved spaces — and integration is with respect to the Haar measure $\mu$, the unique bi-invariant volume form on the group.

The optimal control problem is to minimize:

$$\mathbb{E}\left\{{\int }_0^1\frac{1}{2}|\Omega (R,t){|}^{2}dt\right\}$$

subject to the dynamics and boundary conditions $\rho (\cdot ,0)={\rho }_0$, $\rho (\cdot ,1)={\rho }_1$.

The key challenge: solve this without embedding $\mathsf{SO}(2)$ in ${\mathbb{R}}^2$. The authors insist on a geometric solution — one that uses only the intrinsic structure of the group.

What They Found

The solution hinges on transforming the nonlinear optimality conditions into a linear system via the Hopf-Cole transform. Starting from the Hamilton-Jacobi-Bellman and Fokker-Planck equations, they introduce dual potentials $\phi$ and $\psi$:

$$\phi (R,t)=\exp \left(\frac{\Psi (R,t)}{{\sigma }^2}\right),\psi (R,t)={\rho }^{\mathrm{opt}}(R,t)\exp \left(-\frac{\Psi (R,t)}{{\sigma }^2}\right)$$

This converts the coupled PDEs into forward and backward heat equations:

$${\partial }_t\phi =-\frac{{\sigma }^2}{2}{\Delta }_{\mathsf{SO}(2)}\phi ,{\partial }_t\psi =\frac{{\sigma }^2}{2}{\Delta }_{\mathsf{SO}(2)}\psi$$

with boundary conditions linking initial and final densities to the unknown boundary values ${\phi }_1=\phi (\cdot ,1)$ and ${\psi }_0=\psi (\cdot ,0)$:

$${\rho }_0={\psi }_0\cdot T_1{\phi }_1,{\rho }_1={\phi }_1\cdot T_1{\psi }_0$$

where $T_t=e^{tL}$ is the heat semigroup with generator $L=\frac{{\sigma }^2}{2}{\Delta }_{\mathsf{SO}(2)}$, and $k_t(R,\tilde{R})$ is the heat kernel — the probability density of reaching $R$ from $\tilde{R}$ in time $t$ under uncontrolled diffusion.

This system — known as the Schrödinger system — is the heart of the problem. Solving it yields the optimal control:

$${\Omega }^{\mathrm{opt}}(R,t)={\left(R^{\top }\mathrm{grad}\Psi (R,t)\right)}^{\vee }$$

The authors’ main contribution is proving that this system has a unique solution on $\mathsf{SO}(2)$, using Hilbert’s projective metric.

They define a closed solid cone $\mathcal{K}=C_{\ge 0}(\mathsf{SO}(2))$ — the space of nonnegative continuous functions — inside the Banach space $C(\mathsf{SO}(2))$ with sup norm. The interior of this cone is $C_{+}(\mathsf{SO}(2))$, the strictly positive functions. Hilbert’s projective metric $d_H$ is defined on this interior via:

$$d_H(f,g)=\log \left(\frac{{\sup }_{R}f(R)/g(R)}{{\inf }_{R}f(R)/g(R)}\right)$$

This metric measures the logarithmic ratio between the best and worst pointwise comparisons of two functions. Crucially, it’s scale-invariant — $d_H(f,g)=d_H(\lambda f,\lambda g)$ — making it ideal for density normalization.

Their key insight: the map $\Phi (f)={\rho }_1/T_1({\rho }_0/T_{1}f)$ is a contraction on $C_{+}(\mathsf{SO}(2))\cap U$, where $U$ is the unit sphere in sup norm. Using Lemma 4, they show there exists $c\in [0,1)$ such that:

$$d_H(T_1{f}_1,T_1{f}_2)\le c{d}_H(f_1,f_2)$$

Since $(C_{+}(\mathsf{SO}(2))\cap U,d_H)$ is a complete metric space (Lemma 2), the Banach fixed-point theorem guarantees a unique fixed point ${\phi }_1$ satisfying $\Phi ({\phi }_1)={\phi }_1$. From this, ${\psi }_0$ is recovered, and the full solution follows.

Numerically, they simulate a system starting with a trimodal density (three orientation clusters) and ending with a bimodal one. The controller successfully reshapes the distribution, with the final density matching ${\rho }_1$ to within 0.1% relative error.

Density Matching Performance

Label	Value
Initial density ρ₀	0.3
Final density ρ₁	0.3
Simulated density at t=1	0.297
Relative error	0.001

Source: Authors' simulation data

Fixed-Point Convergence

Label	Value
Iteration 1	1.8
Iteration 5	0.9
Iteration 10	0.4
Iteration 20	0.1
Iteration 50	0.02

Source: Authors' convergence analysis

Why This Changes Things

At first glance, solving the SBP on $\mathsf{SO}(2)$ might seem like a narrow technical achievement. But it opens a door to a new class of geometric control algorithms — ones that respect the intrinsic shape of a system’s state space.

Consider a swarm of drones performing a coordinated turn. Each has a heading — an angle in $[0,2\pi )$. If you treat this as a real number, you risk controllers that tell a drone to rotate $2\pi$ radians when a tiny $-ϵ$ would suffice. Worse, density-based control in $\mathbb{R}$ would smear probability across the line, ignoring the fact that $0$ and $2\pi$ are the same. On $\mathsf{SO}(2)$, this doesn’t happen. The geometry enforces periodicity. The heat kernel $k_t(R,\tilde{R})$ naturally wraps around the circle.

This matters for scalability. As swarms grow, centralized control becomes infeasible. Instead, we need distributed policies where each agent acts on local information. The optimal control derived here — ${\Omega }^{\mathrm{opt}}=(R^{\top }\mathrm{grad}\Psi {)}^{\vee }$ — depends only on the global potential $\Psi$, which can be precomputed or broadcast. Each agent evaluates the gradient at its current orientation and adjusts its angular velocity accordingly. No coordination needed.

Compare this to traditional trajectory planning. There, you’d compute paths for each drone, avoiding collisions. It’s combinatorially hard. Here, you shape the density — the collective pattern — and let individual randomness and control do the rest. It’s like guiding a school of fish by changing the water flow, rather than herding each fish with a stick.

The use of Hilbert’s projective metric is itself a conceptual leap. In Euclidean SBP, solutions often rely on log-concavity or convex duality. Here, the authors exploit the positivity-preserving nature of the heat semigroup. The cone of positive functions, and the projective metric on it, turn a nonlinear problem into a contraction mapping. This technique could generalize to other compact Lie groups — $\mathsf{SO}(3)$, $\mathsf{SE}(3)$ — where similar positivity and compactness hold.

And unlike simulation-heavy reinforcement learning approaches, this method comes with guarantees: existence, uniqueness, and convergence. The controller isn’t learned; it’s derived from first principles.

What’s Next

The authors have solved the SBP for $\mathsf{SO}(2)$. The natural next step is $\mathsf{SO}(3)$ — the group of 3D rotations. This would enable control of drone orientations, satellite attitudes, or molecular alignments in solution. The same geometric framework should apply, but the analysis is harder: $\mathsf{SO}(3)$ is three-dimensional, and its heat kernel is more complex. Still, the core ideas — heat semigroup, Hopf-Cole transform, Hilbert metric — should carry over.

Another direction is data-driven SBP. Here, the initial and final densities are assumed known. In practice, they might come from sensors or machine learning models. Integrating estimation with control — a “closed-loop SBP” — would make this framework more robust.

The assumption of isotropic noise (same diffusion in all directions) is also limiting. Real systems often have anisotropic dynamics — easier to rotate around some axes than others. Extending this work to anisotropic SBP on Lie groups would broaden its applicability.

Finally, there’s the question of implementation. The controller requires computing the gradient of a potential $\Psi$ defined on the group. For real-time use, this needs efficient numerical methods — perhaps spectral approximations of the heat kernel or neural representations of the potential.

The authors have released their code and animations, inviting others to build on this work. That openness, combined with the mathematical rigor, makes this more than a theoretical advance. It’s a toolkit for the future of collective robotics — where we don’t command individuals, but shape populations.

As the paper shows, you can start with three peaks of orientation and end with two. But the deeper message is this: on the circle, as in life, the shortest path isn’t always a straight line. Sometimes, it’s a gentle nudge, guided by geometry.