Online Constraint Optimization-Based Indirect-Adaptive Robust Control for Autonomous Vehicles with Unknown Friction

I. INTRODUCTION

• Offline tire models can lose accuracy when operating conditions differ from identification data, while prior online methods do not establish explicit transient and steady-state tracking-error bounds.Repeated friction measurements are costly, and training data covering every road and tire condition are impractical.Source: §I · p. 1
• The upper OCP plans constrained, time-optimal motion, while lower-layer IARC adapts tire-model parameters online and uses robust feedback to bound tracking error without requiring parameter convergence.The lower layer uses measurements gathered during driving without prior friction measurements or offline training data.Source: §I · pp. 1–2
• An estimation monitor suspends nominal learning at model mismatch, and a smooth reset lowers estimated tire capacity before updated estimates revise planner constraints.Monitoring is needed because saturated data can invalidate the estimator’s regression model and compromise planning feasibility.Source: §I · pp. 1–2
• Across µ ∈ [0.2, 1.05], simulations report speed errors within Le = 0.05 m/s and 24.4% less time than the conservative baseline.The proposed method provides online friction estimation and adaptation under varying uncertain friction conditions.Source: §I · p. 2

A. Vehicle Dynamics Model

• Vehicle Dynamics Model: planning uses a rear-drive single-track model in Frenet coordinates, while IARC simplifies speed control to longitudinal SISO dynamics and groups unmodeled effects into acceleration uncertainty.The model lumps left and right tracks into one centered track; the planning layer retains coupled longitudinal, lateral, and yaw dynamics.Source: A. Vehicle Dynamics Model · p. 2

B. Tire Model

• Normal-driving tire constraints require unsaturated operation and bound each axle’s longitudinal force by |Fx,i| ≤ µFz,i.The passage defines µ as tire-road friction and Fz,i as normal tire load.Source: B. Tire Model · p. 2
• The lateral tire-force model uses cornering stiffness and lateral slip angle, with a stated applicability condition involving the sliding slip angle.The passage gives a force-dependent expression for the sliding-slip angle and states the model’s condition at zero lateral-force capacity.Source: B. Tire Model · p. 2
• For IARC, rear longitudinal force uses the unsaturated branch of an isotropic brush model, with its domain defined by slip, friction, normal load, and cornering stiffness.Rear-wheel longitudinal slip is defined using measured wheel speed and vehicle speed.Source: B. Tire Model · pp. 2–3

C. Problem Formulation

• The unknown parameter vector depends on rear normal load and tire-road friction; its exact values are unknown, but its variations are assumed limited.Source: C. Problem Formulation · p. 3
• The formulation assumes the unknown parameter vector and uncertain nonlinearity lie within a known bounded convex set.The bounds are represented using known minimum and maximum parameter vectors and a known uncertainty bound.Source: C. Problem Formulation · p. 3
• The objective is time-optimal motion along the desired trajectory with accurate tracking under bounded parametric uncertainty and nonlinearities, subject to the stated constraints.Source: C. Problem Formulation · p. 3

A. Control Framework

• At 50 Hz, the upper-layer OCP uses updated parameter estimates to plan feasible, time-optimal reference motion for tracking the desired trajectory.The plan includes longitudinal motion and steering.Source: A. Control Framework · p. 3
• At 200 Hz, the lower-layer IARC tracks the planned longitudinal speed using model compensation and robust feedback against tracking errors and uncertainty.Its tracking-error bound and feedback-force reserve set the planner’s constraint margins.Source: A. Control Framework · p. 3
• Vehicle-state feedback supports lower-layer estimation, and updated parameter estimates return to the upper layer to revise its model and constraints for replanning.Source: A. Control Framework · p. 3

B. Lower Layer Indirect-Adaptive Robust Control

• The estimator uses filtered acceleration regression and normalized least-squares adaptation, with projection at parameter-set boundaries.Filtering measured signals avoids differentiating measured velocity; the projection uses the set boundary’s outward normal.Source: 1) Adaptation Law · pp. 3–4
• The monitor rejects learning data when the estimated local force–slip slope falls below a threshold, because saturated-tire responses mismatch the unsaturated model.The threshold specifies the minimum slope accepted for learning.Source: 2) Estimator Monitoring Design · p. 4
• When learning is suspended, a smooth parameter reset moves estimates toward θmax to reduce the risk that abrupt constraint tightening makes the OCP infeasible.The reset preserves parameter bounds; smaller γr gives faster correction.Source: 2) Estimator Monitoring Design · p. 4
• Model compensation handles reference motion and known dynamics, while bounded robust feedback corrects speed-tracking errors under parameter uncertainty and unmodeled effects.The robust feedback is selected to satisfy the stated ARC performance conditions.Source: 3) Control Law Design · p. 4
• The speed-tracking error is guaranteed to remain within the prescribed bound when the complete control command is realizable.The guaranteed ultimate error radius is stated as √(ε1/k1); the prescribed bound conditions are given for Le.Source: 3) Control Law Design · p. 5
• Parameter convergence is conditional: under Lemma 2’s stated assumptions, the parameter-estimation error tends to zero.Source: 3) Control Law Design · p. 5
• The feedback force needed to maintain the tracking bound is reserved from attainable longitudinal force limits, which must leave a nonempty tightened interval.The reserve covers both acceleration and braking and is independent of the current parameter estimate for fixed design constants.Source: 3) Control Law Design · p. 5

C. Upper Layer Optimization

• At each planning update, the planner uses current vehicle state and estimated tire capacities to replan feasible motion.The finite-horizon OCP is solved using acados.Source: C. Upper Layer Optimization · p. 5
• The prediction horizon is represented by states and inputs at nodes t0 through tN across N prescribed intervals.Source: C. Upper Layer Optimization · p. 5
• Speed limits are tightened by the IARC error allowance, while rear-force limits reserve feedback authority and steering, force-rate, and corridor constraints bound planned motion.Slip limits use estimated tire capacity; nonnegative slack can relax them for additional model mismatch or tight feasibility.Source: C. Upper Layer Optimization · p. 5
• The OCP rewards forward progress while penalizing reference-path deviation, rapid control changes, and slip- and friction-limit violations.Positive weights are used for friction slacks.Source: C. Upper Layer Optimization · pp. 5–6
• The planner supplies speed, nominal rear-force, and steering references, which the faster IARC reconstructs and uses to compute compensation and feedback force commands.Source: C. Upper Layer Optimization · p. 6

A. Experiment Setup

• The 2411 m Monteblanco track simulation includes acceleration, braking, and cornering across friction patches from µ = 0.3 to 0.8 after a dry launch section.The vehicle starts at 12.44 m/s, and the drivable lane is 4.5 m wide.Source: A. Experiment Setup · p. 6
• The proposed OCP–IARC method includes online parameter estimation, estimator monitoring, and feedback of estimates to the planner, initialized with µ̂(0) = 1.0.Source: A. Experiment Setup · p. 6
• Three nominal NMPC baselines hold friction fixed at µ = 1.0, µ = 0.6, or µ = 0.3 throughout the run.These represent high-, moderate-, and low-friction assumptions, respectively.Source: A. Experiment Setup · p. 6

• The friction coefficient is bounded by µ ∈[0.2, 1.05], with initial estimate ˆµ(0) = 1.0.Source: B. Parameter Selection · p. 6
• Accounting for load transfer, rear tire force capacity ranges from 1592 to 12367 N, with corresponding bounds for θ1 and θ2.θ1 ∈[2.69 × 10−5, 2.79 × 10−4] and θ2 ∈[2.42 × 10−10, 2.60 × 10−8].Source: B. Parameter Selection · p. 6
• The planner runs at 50 Hz over a 5.04-second horizon divided into 40 intervals, while IARC runs at 200 Hz.Source: B. Parameter Selection · p. 6
• For tracking allowance Le = 0.05, the simulation sets h1 = 0.74, k1 = 7.4, ε1 ≈ 0.02, and feedback force reserve FsM ≈ 1385.5 N.Source: B. Parameter Selection · p. 6

• Nominal NMPC with priors µ = 1.0 and 0.6 crashes in the initial µ = 0.3 patch at 200 m and 262 m, respectively, with maximum lateral errors of at least 6 m.Peak yaw errors are 22.747° and 21.021°.Source: C. Results · p. 6
• 202.7 s; ICE with prior µ = 0.3 takes 32% longer than the proposed method, completing only 1697 m (70.4% of the lap) by 153.2 s.Source: C. Results · p. 6
• 153.2 s; the proposed method completes the full lap with lateral RMSE 0.208 m, maximum lateral error 0.772 m, and maximum speed error 0.688 mm/s.Figure 2 shows close tracking of replanned speed during acceleration and braking, with no lane-bound violations.Source: C. Results · p. 6, Figure 2 · p. 7
• At transitions from µ = 1.0 to 0.3 and 0.8 to 0.4, estimated rear tire capacity drops sharply; it then increases progressively in higher-grip sections during acceleration and braking.Figure 3 identifies the estimated and true capacities, static-load capacity reference, driving/braking periods, and friction transitions.Source: Figure 3 · p. 7, C. Results · p. 7
• The estimation monitor suspends nominal learning on force–slip mismatch, and smooth parameter reset reduces tire capacity used by the planner to support feasible replanning under changing friction.The responses are reported alongside accurate speed tracking and lane-bound satisfaction, with the prescribed tracking allowance and feedback force reserve.Source: C. Results · p. 7

• The two-layer framework uses an upper OCP for time-optimal motion and lower IARC for explicit transient and steady-state longitudinal speed-tracking bounds without parameter convergence.Tracking-error and feedback-force margins enter the OCP constraints to account for control effort during adaptation.Source: §V · p. 7
• An estimation monitor suspends nominal learning when force–slip data indicate mismatch, and a smooth reset reduces estimated tire capacity while limiting abrupt constraint tightening.Monitored estimates update the planner’s tire model and friction constraints for online replanning without prior friction measurements or offline training data.Source: §V · p. 7
• 153.2 s; high-fidelity simulations complete the full lap with lateral tracking RMSE 0.208 m, while longitudinal speed-tracking errors remain below Le = 0.05 m/s.Optimistic fixed-friction NMPC crashes in the low-friction section; the conservative baseline takes 202.7 s, 32% longer.Source: §V · p. 7