Solver Performance & Exact Ising Formulation
Reducing 2D multilateration latency from 111.4 ms to 0.81 ms on an embedded companion computer, and machine-checking the binary branch selection as an exact Ising Hamiltonian.
The Real-Time Budget on an Airframe
In high-speed FPV swarm operations, algorithmic latency directly translates into physical collision hazard. An interceptor cruising at 15 m/s travels 1.5 cm every single millisecond.
Early prototype implementations that utilized naive coordinate grid searches or non-convergent Nelder-Mead simplex routines suffered from execution times averaging 111.4 ms per frame. In a dynamic flight scenario, a 111 ms lag creates 1.67 m of unmodeled spatial displacement, causing the follower's proportional-derivative flight controller to induce severe phase lag and pilot-induced oscillation (PIO).
Solver Latency: Benchmark on ARM Cortex-A76 (European compute unit)
Naive Grid Search
111.4 ms
Unsafe for flight (> 50 ms)
Luftschar Solver
0.81 ms
137x speedup · N = 1,000
Control Loop Margin
> 96%
Fits inside 200 ms UWB frame
How We Brought Runtime to 0.81 ms
In the formation solver implementation, the localization problem is formulated as an analytical non-linear least squares optimization using a specialized damped Levenberg-Marquardt solver with warm-start covariance seeding:
- Analytic Jacobian Computation: Rather than approximating gradients via finite differencing (which scales quadratically with unit count), the partial derivatives of range equations with respect to follower coordinates
(x, y)are evaluated in closed form. - Velocity Vector Seeding: The initial estimate for frame
kis seeded directly from the Kalman filter's forward-propagated constant-velocity state, reducing the required iteration count to fewer than 4 convergence steps under nominal flight.
The Exact Ising Spin Mapping
In academic literature, quantum or combinatorial optimization is frequently misapplied as an approximate relaxation (QUBO) to problems that classical algorithms handle trivially.
The dual-tag mirror ambiguity is fundamentally different: each follower aircraft has exactly two candidate branches—its true position (+1) or its reflection about the anchor axis (-1). This is an exact physical Ising spin:
In the mirror symmetry solver, we proved this mapping is mathematically exact, not an approximation. Because an edge's cost depends strictly on the spins of its two endpoints, it has four degrees of freedom. The basis {1, s_i, s_j, s_i · s_j}spans this space completely.
Our automated self-test evaluates all 2^N configurations twice—once through the Ising Hamiltonian coefficients, and once by physically reflecting coordinates in 3D space and summing residual errors:
- Max Relative Disagreement: < 7E-15 (machine-precision agreement).
- Anchor Edges Coupling:
max |J| = 0.000E+000(asserted exact zero). - Peer Edge Coupling:
max |J| = 2685.7(three orders of magnitude above noise).
This ensures that evaluating branch combinations is both instantaneous (< 50 microseconds for swarms up to 12 airframes) and mathematically rigorous.