IteraOptiRacing: Iterative Optimal Control

Updated 6 July 2026

IteraOptiRacing is a family of iterative optimal control methods that refine trajectories, speed profiles, and dynamics models to achieve minimum-lap-time solutions under practical constraints.
It employs diverse techniques including i2LQR, GP-augmented MPC, Bayesian optimization, and learning-informed initialization to enhance convergence and computational efficiency.
It integrates closed-loop feedback with adaptive safety constraints to balance real-time performance, model discrepancies, and computational tractability in autonomous racing.

In the literature collected under the name IteraOptiRacing, the shared concern is minimum-lap-time optimization through iterative refinement of trajectories, speed profiles, dynamics models, or terminal targets; in one explicit formulation, it is a unified planning-control strategy based on i2LQR for competing with other racing cars (Zeng et al., 13 Jul 2025). Across these formulations, the recurrent objective is the closed-lap minimum-time problem, typically written as $T=\int_{0}^{L}\frac{ds}{v(s)}$ , under track-boundary, actuator, and tire-force constraints, but the algorithmic realizations span free-trajectory optimal control, learning-informed initialization, GP-augmented MPC, spatial iterative learning, Bayesian optimization, and controller-guided adaptation (Garlick et al., 2021, Numerow et al., 2024, Nam et al., 28 Jan 2026).

1. Scope, lineage, and research setting

IteraOptiRacing emerged within a broader autonomous-racing literature in which repeated structure is exploited rather than ignored. One line of work formulates racing as a receding-horizon or full-lap optimal control problem, as in optimization-based autonomous racing of 1:43 scale RC cars, where a two-level planner–NMPC stack and model predictive contouring control were solved as convex QPs at 50 Hz on embedded platforms (Liniger et al., 2017). A second line replaces direct full-horizon search by iterative decomposition, as in the sequential two-step algorithm that alternates between a friction-limited speed profile and a convex curvature-minimization path update on the 4.5 km Thunderhill Raceway (Kapania et al., 2019). A third line exploits lap-to-lap repeatability through iterative learning control, using previous laps’ path-tracking errors to refine steering inputs for an Audi TTS at lateral accelerations of up to $8\,\mathrm{m/s^2}$ (Kapania et al., 2019).

Subsequent work extended the same iterative principle in several directions. Some methods learn a racing line or initialization prior from geometry, such as the feed-forward neural-network racing-line predictor trained on TUMFTM-generated optimal-control labels (Garlick et al., 2021) and the Formula 1 telemetry-based initializer that warms an IPOPT minimum-time solver (Shehadeh et al., 7 Mar 2026). Others learn model discrepancies rather than trajectories, as in double-iterative Gaussian Process residual compensation for both planner and controller (Su et al., 2023), active exploration for GP data acquisition in MPC (Benciolini et al., 2023), and semi-parametric neural residual dynamics updated online (Georgiev et al., 2020). More recent formulations move further toward unified loop structures: closed-loop spatial adaptation with NURBS, CMA-ES, and a Kalman-inspired constraint map (Wachter et al., 17 Feb 2026); full-lap trajectory search in a wavelet parameter space with BO and learned dynamics (Nam et al., 28 Jan 2026); and the explicitly titled IteraOptiRacing framework that merges obstacle avoidance and time optimization through i2LQR in multi-agent racing (Zeng et al., 13 Jul 2025).

This literature situates IteraOptiRacing at the intersection of optimal control, system identification, iterative learning, and real-time embedded computation. The unifying premise is not a single solver, but the use of repeated solves, repeated laps, or repeated historical data to move a racing policy closer to a minimum-time solution without paying the full cost of de novo global optimization at every step.

2. Common mathematical formulation

Most IteraOptiRacing formulations use curvilinear or Frenet coordinates anchored to a track centerline $c(s)$ , tangent $t(s)$ , normal $n(s)$ , and curvature $\kappa(s)$ . A racing line is represented by a lateral offset $y(s)$ or $d(s)$ , with global position given by $[X,Y](s,y)=c(s)+n(s)\,y$ in the reference-line formulations (Shehadeh et al., 7 Mar 2026). In the normal-line encoding used for learned raceline prediction, the path is represented as waypoint positions $w\in[0,1]$ along normals to the centerline, with $8\,\mathrm{m/s^2}$ 0 denoting the left boundary and $8\,\mathrm{m/s^2}$ 1 the right boundary (Garlick et al., 2021).

The shared cost is minimum lap time. Representative statements are

$8\,\mathrm{m/s^2}$ 2

for closed-path racing-line optimization (Garlick et al., 2021),

$8\,\mathrm{m/s^2}$ 3

for minimum-time optimal control along a reference line (Shehadeh et al., 7 Mar 2026), and

$8\,\mathrm{m/s^2}$ 4

for spatially discretized global trajectory optimization (Nam et al., 28 Jan 2026). Variants then add their own structure: MPCC formulations maximize progress while penalizing contouring and lag errors (Liniger et al., 2017, Gupta et al., 2022), i2LQR formulations choose terminal states from historical low-cost-to-go data (Zeng et al., 13 Jul 2025), and BO formulations treat lap time as a black-box objective over policy or trajectory parameters (Oliveira et al., 2018, Nam et al., 28 Jan 2026).

Vehicle models range from kinematic bicycle models to dynamic bicycle and double-track models. Representative continuous dynamics include

$8\,\mathrm{m/s^2}$ 5

$8\,\mathrm{m/s^2}$ 6

with linear-tire approximations $8\,\mathrm{m/s^2}$ 7 used for exposition and local refinements (Garlick et al., 2021). Other works use Pacejka or brush-tire models, explicit yaw dynamics, and rate-limited steering and torque states (Liniger et al., 2017, Numerow et al., 2024, Georgiev et al., 2020). For EV powertrains with discrete gears, the state may include battery energy, kinetic energy, and gear-ratio design variables, while the control includes motor and brake forces and a one-hot gear-selection vector (Cartignij et al., 2023).

Constraints are equally recurrent. Track feasibility appears as lateral-offset bounds $8\,\mathrm{m/s^2}$ 8 (Shehadeh et al., 7 Mar 2026) or $8\,\mathrm{m/s^2}$ 9 (Kapania et al., 2019). Tire-force feasibility is encoded through friction circles or ellipses such as

$c(s)$ 0

$c(s)$ 1

(Garlick et al., 2021, Shehadeh et al., 7 Mar 2026). Multi-agent formulations add obstacle-avoidance penalties or constraints; the i2LQR version uses

$c(s)$ 2

with an exponential barrier cost and a separate geometric collision check in curvilinear coordinates (Zeng et al., 13 Jul 2025).

3. Algorithmic families

A major branch uses learned priors to avoid solving the full minimum-time problem from scratch. The TUMFTM-based ANN predictor discretizes tracks at fixed 5 m spacing, encodes each normal by its length $c(s)$ 3, angular change $c(s)$ 4, and pseudo-normal deviation $c(s)$ 5, and maps a sliding window of $c(s)$ 6 features to local waypoint bundles through a four-layer feed-forward network with hidden sizes $c(s)$ 7, $c(s)$ 8, and $c(s)$ 9, Huber loss, and Nadam optimization (Garlick et al., 2021). The F1-initialization framework instead reconstructs expert lateral offsets from telemetry on 17 tracks, resamples at $t(s)$ 0 m, and predicts expert-like offsets with a dilated TCN, convolutional fusion, multi-head temporal attention, and a two-layer MLP, then converts the predicted offset into a solver warm start for IPOPT (Shehadeh et al., 7 Mar 2026).

A second branch retains classical optimization but redesigns the solver loop. The sequential two-step algorithm alternates between a forward–backward friction-limited speed-profile computation and a convex curvature-minimization path update until lap time ceases to improve (Kapania et al., 2019). The convex–PMP framework for multi-speed electric race cars splits a mixed-integer optimal control problem into a convex continuous optimization problem and a pointwise Hamiltonian gear-selection problem, with costate damping used to stabilize the alternation (Cartignij et al., 2023). Feasible SQP uses outer iterates that remain feasible for the original constraints at every interruption point, which makes suboptimal MPC “anytime feasible” rather than merely fast (Numerow et al., 2024). At the implementation level, multi-threaded MPCC reduces control-output latency by maintaining worker pools that solve optimization problems in parallel and discard stale solutions before publication (Gupta et al., 2022).

A third branch centers on learning model discrepancies and using the improved model inside planning or control. Double-GPR compensation augments both the planner’s nonlinear bicycle model and the controller’s linearized tracking model with GP means in the lateral and yaw channels (Su et al., 2023). Active exploration modifies the MPC reference to deliberately visit GP-uncertain regions early in training, then switches to pure racing once the dataset stabilizes (Benciolini et al., 2023). Semi-parametric residual learning uses a physics-based bicycle model plus a small neural network that predicts only $t(s)$ 1, $t(s)$ 2, and $t(s)$ 3, and updates the residual online through pseudo-rehearsal with a GMM and constrained gradients (Georgiev et al., 2020).

A fourth branch performs global search in lower-dimensional trajectory spaces. Track-centric iterative learning keeps only the coarsest db4 wavelet coefficients of $t(s)$ 4 and $t(s)$ 5, yielding $t(s)$ 6 decision variables for BO while the controller-in-the-loop simulation evaluates the actual lap time under learned residual dynamics (Nam et al., 28 Jan 2026). Coordinate-descent BO for acceleration policies represents $t(s)$ 7 in an RKHS with Matérn- $t(s)$ 8 basis functions and optimizes a GP surrogate of lap reward through randomized one-dimensional sweeps over policy parameters (Oliveira et al., 2018). Closed-loop spatial adaptation uses a cubic NURBS raceline, CMA-ES over control points, weights, and knots, and an adaptive map $t(s)$ 9 updated from controller tracking error over “blame regions” (Wachter et al., 17 Feb 2026).

The explicitly titled IteraOptiRacing framework places unified planning and control at the center. Its i2LQR formulation selects $n(s)$ 0 historical terminal targets near the current state, solves independent iLQR problems in parallel under affine time-varying bicycle dynamics, and reweights terminal, control, and barrier terms whenever a candidate trajectory violates collision constraints (Zeng et al., 13 Jul 2025).

4. Representations, priors, and closed-loop information

The distinctive feature of many IteraOptiRacing variants is not merely iteration, but what is being iterated. In learned raceline prediction, the crucial object is local track geometry. The normal-based representation used in the ANN predictor incorporates track width through $n(s)$ 1, curvature trend through $n(s)$ 2, and hairpin handling through the pseudo-normal deviation $n(s)$ 3; the chosen foresight $n(s)$ 4 corresponds to approximately 350 m look-ahead and look-behind, and the network outputs $n(s)$ 5 waypoints per window when $n(s)$ 6 (Garlick et al., 2021). This design embeds scale information directly in the input and explains why the same model can generalize to unseen tracks of different widths and lengths.

The Formula 1 initializer uses a different prior: expert driving behavior reconstructed from aligned GPS telemetry. Track geometries are taken from Assetto Corsa and the TUM Racetrack Database, telemetry is obtained via FastF1 at approximately 3 Hz, and each track uses 15 consecutive laps of a single driver after filtering for no rain, no safety car phases, and no nearby traffic (Shehadeh et al., 7 Mar 2026). The result is not a physics model but an expert-like geometric seed that places the solver closer to a favorable basin of attraction.

Other variants compress the trajectory itself. The wavelet formulation fixes all detail coefficients from a nominal trajectory and optimizes only the coarsest approximation coefficients for $n(s)$ 7 and $n(s)$ 8, which the reported implementation sets to five coefficients each using db4 at level $n(s)$ 9 with $\kappa(s)$ 0 samples (Nam et al., 28 Jan 2026). The NURBS formulation instead uses a clamped cubic curve with analytic $\kappa(s)$ 1 closure constraints that tie the last three control points to the first three, reducing six degrees of freedom in 2D while keeping analytic derivatives for curvature and time scaling (Wachter et al., 17 Feb 2026). The RKHS policy-search formulation compresses the control law rather than the path, representing acceleration as $\kappa(s)$ 2 over evenly spaced inducing points in normalized track progress (Oliveira et al., 2018).

Closed-loop feedback also becomes a structural prior. In the spatially adaptive NURBS method, tracking errors are not treated as nuisance terms but as measurements used to update a spatial acceleration-limit map through a scalar Kalman-inspired law (Wachter et al., 17 Feb 2026). In active-exploration GP MPC, posterior covariance determines where the controller should deliberately visit next (Benciolini et al., 2023). In i2LQR-based multi-agent racing, historical cost-to-go values serve as a terminal prior that biases local solves toward previously successful overtaking and time-optimal states (Zeng et al., 13 Jul 2025). This suggests that IteraOptiRacing is best understood as a family of methods that transforms repetition—of geometry, laps, targets, or solver structure—into usable prior information.

5. Reported performance

Empirical results are central to the literature because most formulations trade exact optimality for computational tractability, solver robustness, or online adaptability.

Paper	Reported performance	Context
(Garlick et al., 2021)	$\kappa(s)$ 3 m MAE overall, RMSE $\kappa(s)$ 4 m, apex error $\kappa(s)$ 5 m, $\kappa(s)$ 6 ms per full circuit, over $\kappa(s)$ 7 faster than traditional OCP	Feed-forward ANN raceline prediction
(Shehadeh et al., 7 Mar 2026)	F1-NN: $\kappa(s)$ 8 iterations, $\kappa(s)$ 9 s optimization runtime, $y(s)$ 0 s initialization, final lap time $y(s)$ 1 s	Formula 1 telemetry-informed initialization over 17 tracks
(Wachter et al., 17 Feb 2026)	$y(s)$ 2 lap-time reduction in simulation; $y(s)$ 3 improvement on hardware; convergence within approximately 8 laps in simulation and within $y(s)$ 4 laps on hardware	NURBS + CMA-ES + controller-guided adaptive constraint map
(Cartignij et al., 2023)	$y(s)$ 5 s average full-lap convergence for MGT; direct full-lap MISOCP with $y(s)$ 6 steps exceeds 10 hours	Convex–PMP co-optimization of gear design and control
(Numerow et al., 2024)	Mean FSQP computation time $y(s)$ 7 ms; success/convergence rates $y(s)$ 8– $y(s)$ 9; runtime speedup IPOPT/FSQP $d(s)$ 0– $d(s)$ 1	Anytime feasible SQP for suboptimal MPC
(Zeng et al., 13 Jul 2025)	Mean overtaking-phase solve time $d(s)$ 2 s versus $d(s)$ 3, $d(s)$ 4, and $d(s)$ 5 s for baselines	Unified i2LQR planning–control with 9 dynamic obstacles
(Nam et al., 28 Jan 2026)	Proposed method reaches $d(s)$ 6 s by iteration 10 and improves real lap time by up to $d(s)$ 7 over a nominal baseline	Wavelet BO with learned residual dynamics
(Lv et al., 2023)	AirSim shortest/average lap time $d(s)$ 8 s / $d(s)$ 9 s; real quadcopter from $[X,Y](s,y)=c(s)+n(s)\,y$ 0 s to $[X,Y](s,y)=c(s)+n(s)\,y$ 1 s by iteration 4	Spatial ILC within a virtual tube

Beyond these headline results, several comparisons clarify the trade space. The TUMFTM-trained ANN is reported to be comparable to human drivers’ line variability and autonomous racing map/planning accuracy, with a 95% confidence interval of lateral error of approximately $[X,Y](s,y)=c(s)+n(s)\,y$ 2 m (Garlick et al., 2021). The F1-based initializer preserves final optimized lap time while reducing iterations and total runtime relative to centerline and minimum-curvature seeds, and approaches the performance of true expert initialization at negligible inference cost (Shehadeh et al., 7 Mar 2026). The double-GPR planner–controller combination reaches an actual lap time of $[X,Y](s,y)=c(s)+n(s)\,y$ 3 s with a best planned–actual gap of $[X,Y](s,y)=c(s)+n(s)\,y$ 4 s, outperforming planner-only or controller-only GP compensation in the reported Gran Turismo Sport experiments (Su et al., 2023). The semi-parametric iterative learner reduces model MSE from approximately $[X,Y](s,y)=c(s)+n(s)\,y$ 5 to approximately $[X,Y](s,y)=c(s)+n(s)\,y$ 6 after online adaptation and produces lap times around $[X,Y](s,y)=c(s)+n(s)\,y$ 7 s in the modified-dynamics setting (Georgiev et al., 2020).

Earlier optimization-based baselines remain important reference points. On the 18.43 m RC-car track, MPCC achieves lap times around $[X,Y](s,y)=c(s)+n(s)\,y$ 8– $[X,Y](s,y)=c(s)+n(s)\,y$ 9 s versus $w\in[0,1]$ 0– $w\in[0,1]$ 1 s for the two-level planner–NMPC formulation, both at 50 Hz and speeds of more than $w\in[0,1]$ 2 m/s (Liniger et al., 2017). The Thunderhill two-step planner converges after four to five iterations, each taking about $w\in[0,1]$ 3– $w\in[0,1]$ 4 s over the full 4.5 km track, and yields a final experimental lap time of $w\in[0,1]$ 5 s versus $w\in[0,1]$ 6 s for a nonlinear gradient-descent solution and $w\in[0,1]$ 7 s for a professional driver (Kapania et al., 2019). These results explain why later IteraOptiRacing variants are typically evaluated not only on optimality but on initialization cost, solver sensitivity, and hardware viability.

6. Limitations, controversies, and future directions

The main unresolved issue is the balance between speed and guarantees. Learned priors can drastically accelerate optimization, but they do not eliminate sensitivity to domain shift. The ANN raceline predictor performs best on circuits whose features resemble the training distribution and incurs larger errors on rare geometries such as double-apex turns with varying radius and extremely long straights (Garlick et al., 2021). The F1-initialization method depends on noisy $w\in[0,1]$ 8 Hz telemetry, accurate registration to a standardized reference line, and vehicle-class similarity; it improves convergence without explicitly modeling dynamics or tire forces, but long straights followed by heavy-braking corners remain difficult to infer from geometry alone (Shehadeh et al., 7 Mar 2026).

A second issue is the distinction between soft and hard safety. Exponential barrier costs, probabilistic collision ellipsoids, or slackened track constraints are computationally attractive, but they do not provide the same guarantees as feasible-SQP or terminal-set constructions (Zeng et al., 13 Jul 2025, Numerow et al., 2024). This has produced two partially divergent design philosophies. One emphasizes unified, low-latency local optimization with soft penalties and adaptive reweighting; the other emphasizes recursive feasibility, time-varying terminal sets, and solver interruption safety. A plausible implication is that future IteraOptiRacing systems will combine both, using learned or historical priors to warm-start a solver whose intermediate iterates remain feasible.

A third limitation is model incompleteness. Multiple papers identify missing elevation, banking, friction variation, weather dependence, and spatially varying grip as persistent sources of suboptimality (Garlick et al., 2021, Wachter et al., 17 Feb 2026). Proposed remedies include banking and grade features, friction maps $w\in[0,1]$ 9, weather-conditioned inputs, richer tire models, physics-guided layers, differentiable vehicle models, GP-based local acceleration estimation, and multi-signal feedback that incorporates slip ratios, yaw-rate residuals, or actuator saturation (Garlick et al., 2021, Wachter et al., 17 Feb 2026, Benciolini et al., 2023). In the full-horizon BO framework, convergence toward the parameterized optimum depends on both residual-model convergence and BO suboptimality bounds, which means representational choices such as fixed wavelet detail coefficients remain a structural limitation rather than a mere implementation detail (Nam et al., 28 Jan 2026).

A final debate concerns what should be iterated: the controller, the model, the trajectory, or the task target. Controller-only learning reduces tracking error but may still follow a suboptimal reference; model-only learning improves simulation fidelity but may not exploit the new model globally; trajectory-only search may miss control limits or changing dynamics. The literature increasingly favors joint loops in which planning, tracking, and data collection are co-designed, whether through double GPR (Su et al., 2023), active-exploration MPC (Benciolini et al., 2023), track-centric BO (Nam et al., 28 Jan 2026), or unified i2LQR (Zeng et al., 13 Jul 2025). This suggests that IteraOptiRacing is evolving from a collection of acceleration tricks for optimal control into a more general iterative-performance architecture for racing systems that must learn, plan, and act under severe computational and dynamic uncertainty.