Source-linked AI summary
IMU-Centric Moving Horizon Estimation for Lateral Dynamics Estimation Across Vehicles and Grip Conditions
Seuffo Akouan ha Ngoune, Alessandro Toschi, Paolo Burgio, Marko Bertogna
TL;DR
The paper addresses the limited availability of direct lateral-velocity measurements and the sensitivity of model-based estimation to tire and road variability. It develops an IMU-centric constrained MHE with a compact SST tire surrogate and evaluates it across sports cars and an autonomous race car, reporting accurate and robust lateral-state reconstruction.
Problem
Lateral velocity is rarely measured directly because dedicated sensors are costly, while model-based estimators remain sensitive to tire and road variability.
Method
The framework uses constrained IMU-centric MHE with a planar single-track model, compact SST surrogate, and bounded online force-capacity coefficients.
Results
Experimental evaluations report accurate lateral-velocity and lateral-acceleration estimates across human-driven sports cars and an autonomous open-wheel race car.
Takeaways & Limitations
The approach provides a physics-based lateral-dynamics estimator with limited vehicle-specific tuning and no external odometry updates.
Abstract
from arXiv · showhide
Accurate estimation of lateral vehicle dynamics near the adhesion limit is important for stability control and high-performance driving, but lateral velocity is rarely measured directly because sensors such as optical sensors are costly. This paper presents an inertial measurement unit (IMU)-centric Moving Horizon Estimation framework that reconstructs lateral velocity using standard onboard signals, without relying on exteroceptive odometry or detailed tire-parameter tuning. Experimental validation on human-driven sports cars and an autonomous open-wheel race car across tracks, maneuvers, and conditions demonstrates accurate and robust lateral velocity and lateral acceleration estimates. The proposed framework is available at https://github.com/Aseuffo/IMU-Centric-MHE
I. INTRODUCTION
The paper addresses the difficulty of estimating lateral velocity without costly dedicated sensors or external odometry. It proposes an IMU-centric constrained MHE framework designed to accommodate tire and road variability with limited calibration.
- Lateral velocity is rarely measured directly in production vehicles because dedicated sensors are costly and impractical.
- Model-based estimators remain sensitive to tire and road variability and to assumptions embedded in the tire model.
- The proposed framework reconstructs lateral velocity and lateral acceleration from inertial measurements and available onboard signals using IMU-centric MHE.
- The estimator combines a planar single-track model with a compact SST surrogate and bounded, regularized force-capacity variables.
- The framework is evaluated on two human-driven sports cars and an autonomous open-wheel race car across multiple validation settings.
- Unlike external-odometry or high-dimensional tire-model pipelines, the approach uses low-dimensional bounded coefficients and does not require prior lateral-velocity measurements for full tire-parameter fitting.
III. SINE SATURATION TIRE MODEL
The vehicle model uses planar single-track dynamics with lateral acceleration included as an estimator state. Its force-based dynamics are made differentiable so acceleration can be constrained and regularized within the MHE.
- A. Planar single-track dynamics: The formulation uses a differentiable planar single-track model because MHE repeatedly solves a nonlinear program.
- A. Planar single-track dynamics: The vehicle is modeled as a rigid body in the horizontal plane with lumped front and rear axles on the centerline.
- A. Planar single-track dynamics: The baseline formulation neglects pitch, roll, vertical dynamics, combined slip, camber, and road banking unless additional information is available.
- A. Planar single-track dynamics: The state and input variables include longitudinal and lateral center-of-gravity velocities, yaw rate, lateral acceleration, steering angle, and longitudinal acceleration.
- A. Planar single-track dynamics: The lateral tire forces enter the equations of motion through the front and rear force model, with mass, yaw inertia, and center-of-gravity axle distances defining the vehicle dynamics.
- A. Planar single-track dynamics: Lateral acceleration is included as a state and defined through the lateral force balance so the IMU acceleration measurement can enter the MHE cost directly.
- A. Planar single-track dynamics: Under piecewise-constant inputs, the lateral-acceleration dynamics are approximated by the chain rule to remain consistent with the force-based model.
B. Simplified lateral tire-force model
The SST tire model is a compact differentiable surrogate that represents lateral-force capacity with fixed curvature and online-estimated bounded amplitude coefficients. This improves conditioning while retaining a smooth bounded force envelope.
- B. Simplified lateral tire-force model: High-dimensional Magic-Formula estimation can become strongly coupled with slip-angle estimation and ill-conditioned when lateral excitation is weak.
- B. Simplified lateral tire-force model: The SST model uses a differentiable sine-based force mapping with slip angles computed from the standard single-track model.
- B. Simplified lateral tire-force model: Each axle uses a fixed curvature factor B_i and an online-estimated positive coefficient η_y,i that scales effective lateral force capacity.
- B. Simplified lateral tire-force model: The coefficient η_y,i absorbs residual modeling errors and unrepresented variability, but is not interpreted as a Magic-Formula parameter.
- B. Simplified lateral tire-force model: The model provides smooth bounded force behavior and a global cap through the imposed upper bound on η_y,i.
- B. Simplified lateral tire-force model: The sine mapping is selected to remain monotone over the slip-angle envelope observed in the experimental datasets.
- B. Simplified lateral tire-force model: Bounding and regularizing η_y,i prevents drift when slip excitation is weak.
IV. OBSERVABILITY CONSIDERATIONS
Observability determines whether the internal state can be uniquely inferred from measured outputs over a finite interval under known inputs. In this framework, MHE performance therefore depends on the information carried by measured trajectories over its estimation window.
- IV. OBSERVABILITY CONSIDERATIONS: In nonlinear systems, local observability means neighboring initial states produce distinguishable output trajectories for some admissible input.
- IV. OBSERVABILITY CONSIDERATIONS: Because MHE reconstructs unmeasured components by fitting a dynamic model to measured trajectories over a window, practical estimation depends on local observability.
A. Nonlinear local observability and OI rank conditions
The paper assesses local observability and parameter identifiability using Lie-derivative rank conditions applied to an augmented vehicle model. The analysis treats lateral velocity and tire force-capacity coefficients as unknown quantities alongside measured outputs and inputs.
- Lie derivatives of the output map are stacked into a nonlinear observability matrix, whose full column rank establishes local observability.
- Unknown constant parameters are added as augmented states with zero dynamics, producing an observability–identifiability matrix.
- The measured outputs are longitudinal velocity, yaw rate, and lateral acceleration, while lateral velocity and front/rear force-capacity coefficients are treated as unknowns.
- With n+q = 5 unknowns and m = 3 outputs, the analysis determines the minimum derivative order at which the OI matrix may become full rank.
- Derivative blocks are added until full rank is reached or the rank stops increasing, since higher orders cannot increase rank beyond n + q − 1.
C. Local observability of lateral velocity
Lateral velocity is locally observable when measured lateral acceleration carries information about it through slip-angle-dependent tire forces. Tire-parameter identifiability additionally requires sufficiently distinct front- and rear-axle excitation, while weak excitation creates degenerate regimes.
- Because longitudinal velocity and yaw rate are measured directly, local observability of lateral velocity depends on lateral acceleration sensitivity through the slip angles.
- For vx > 0 and a locally monotone lateral-force map, both axle slip angles depend on vy, making the chain-rule sensitivity generically nonzero.
- Front and rear force capacities are locally identifiable when both axles have non-negligible slip excitation and cos δ ≠ 0.
- Lateral acceleration supplies a force-sum constraint, while yaw dynamics supply a force-moment constraint that separates front and rear contributions under sufficient excitation.
- Very low speed or near-straight driving collapses the parameter sensitivities, so bounded and regularized coefficients prevent drift when measurements are temporarily uninformative.
A. MHE versus Kalman Filtering
MHE and Kalman filtering are equivalent in the linear–Gaussian unconstrained case when the arrival cost is chosen appropriately. The paper motivates MHE for this application because it directly handles physical constraints, nonlinear saturation, and uncertainty in measured inputs.
- In the linear–Gaussian unconstrained case, MHE can reproduce the Kalman-filter estimate through an appropriate arrival cost.
- MHE directly enforces physically meaningful bounds on states and SST parameters, whereas standard EKF and UKF handle them heuristically.
- Near-limit driving and bounded tire-force behavior weaken local-linearization assumptions underlying EKF methods.
- Repeated constrained nonlinear optimization allows MHE to accommodate nonlinear and saturated operating regimes.
- The estimation model uses a discrete-time transition map with measured inputs, states, outputs, parameters, and process and measurement disturbances.
- A penalized input correction lets the estimator compensate for delay, calibration error, or measurement noise in steering and longitudinal-acceleration signals.
- Fourth-order Runge–Kutta discretization and multiple shooting improve numerical robustness and warm-starting in receding-horizon operation.
C. MAP Viewpoint and MHE Cost
The MHE is formulated as a maximum a posteriori estimate over a sliding window, jointly optimizing the state trajectory, input corrections, and constant parameters. Weighted residuals and physical constraints encode model consistency, measurement fit, parameter regularization, and input fidelity.
- MHE estimates the state trajectory, input-correction sequence, and constant parameter vector jointly over a sliding window.
- The MAP formulation yields a weighted least-squares problem subject to dynamic and physical constraints.
- The state transition constraint propagates each state using the discretized model and a model-consistency residual.
- Input and state bounds constrain corrected inputs and estimated states throughout the horizon.
- The cost weights arrival information, measurement residuals, model-consistency residuals, parameter deviation, and input corrections with positive-definite matrices.
- Larger input-correction weights keep corrected inputs close to measurements, while smaller weights permit greater compensation for input uncertainty.
D. Implementation, Solver, and Warm Start
The implementation validates the MHE across three vehicles and racing scenarios, using warm starts to reduce iterations and support real-time feasibility. Offline experiments show accurate estimation during aggressive cornering and low-excitation conditions.
- Solver implementation: The nonlinear program is implemented in CasADi and solved with IPOPT using warm-start settings.The implementation uses automatic differentiation and an interior-point optimizer.
- Warm start: Warm starting shifts the previous state and input trajectories and reuses Lagrange multipliers, reducing iterations during receding-horizon operation.Consecutive MHE problems differ by a one-step window shift.
- Experimental validation: 3 vehicle platforms and 2 racing tracks are used to validate stability across modeling assumptions and operating conditions.The results include the Ferrari 250 LM, Corvette Grand Sport, and autonomous Super Formula EAV-25.
- Runtime: 3.9 ms average solve time at N = 10 and 6.3 ms at N = 30 leave margin within the 0.01 s sampling period.These runtimes were measured offline on an Intel Core i9-14900HX; the authors describe real-time feasibility as suggested by the measurements.
- Case 1 results: The MHE matches post-processed lateral-velocity ground truth during aggressive cornering and remains stable when |ay| < 1 m/s2.Estimated lateral acceleration and yaw rate follow measured signals while filtering IMU noise.
B. Case 2: Corvette Tire Analysis
The Corvette analysis tests whether MHE-derived states preserve nonlinear tire behavior, while the Super Formula case examines adaptation under incorrect tire parameters and reduced grip. Results show accurate force reconstruction, rapid convergence, and sensitivity to horizon and curvature tuning.
- Corvette tire analysis: MHE-based tire-force reconstruction captures both linear and saturation regimes despite the simplified vehicle model.Independent Pacejka fits to ground-truth-based and MHE-based scatters show strong overlap.
- Parameter adaptation: The effective force-capacity coefficient ηy,i adapts online from an incorrect initialization to match the measured vehicle response.The coefficient represents available lateral force capacity under current operating conditions and is bounded and regularized.
- Super Formula convergence: Within a fraction of a second, MHE converges to the correct lateral velocity after ηy,i is initialized substantially below its effective level.In the low-grip qualifying session, convergence is achieved by the apex of Turn 1 despite initial overestimation at corner entry.
- Benchmark comparison: MHE reaches comparable accuracy to a UKF with correctly identified tire parameters after one corner, without prior knowledge of the exact friction level.A UKF initialized with an underestimated Pacejka peak factor D retains a persistent bias that cannot be compensated online.
- Sensitivity analysis: Both excessively long and overly short horizons increase lateral-velocity RMSE, while curvature-factor deviations also increase RMSE but preserve qualitative trends.Long horizons accumulate simplified-model mismatch, whereas short horizons are more sensitive to measurement noise and less constrained by dynamics; changing B by ±20% is evaluated.
VII. CONCLUSION AND FUTURE WORK
The IMU-centric MHE reconstructs lateral velocity and acceleration without external odometry or high-dimensional tire-parameter identification, while accommodating changing grip. Results across sports cars and an autonomous race car support limited vehicle-specific tuning, and measured runtimes suggest real-time feasibility; future work targets broader adaptation and validation.
- Conclusion: The framework reconstructs lateral velocity and lateral acceleration without external odometry or high-dimensional tire-parameter identification.It uses bounded and regularized variables to accommodate effective force-capacity variations instead of assuming a fixed tire model.
- Conclusion: Reduced sensitivity to tire-parameter uncertainty and changing grip was observed against the evaluated filtering baselines.The reported explanation is adaptation of effective force-capacity variations through bounded and regularized variables.
- Practical advantages: The reduced formulation mainly depends on geometric parameters and mass, limiting vehicle-specific tuning requirements.These quantities are typically available or can be measured with limited effort.
- Validation: Validation covered two human-driven sports cars in the REVS database and autonomous Super Formula runs with the EAV-25 platform.These results support the approach across different vehicle platforms and driving settings.
- Implementation: Measured solver runtimes on the tested high-end CPU platform suggest that real-time implementation is feasible.
- Future work: Future work will adapt the SST curvature parameter online, pursue embedded implementation, and broaden validation across vehicles and environments.Proposed extensions include hierarchical estimation and testing in urban and off-road scenarios.