// Approach
The 12 simulations are structured across three levels of increasing complexity. Each level builds on the previous one by introducing a new variable — control strategy, state observer, or tyre model — while keeping all other parameters identical, so that performance differences can be directly attributed to the change under test.
// 01 — Level 1
Command Strategies
4 Tests · QuartAdams Trajectory · Linear tyre · Rear axle ON · No Observer · Progress Reference : Test 1.1
PD Commande
Test 1.1
Lateral Errors
Angle braquage (rad)
RMSE Lat
0.4058 m
Max Errors
0.5 m
Stability
Stable
Progress
−
Purely reactive by design: the controller corrects only after an error has built up, with no knowledge of upcoming curvature.
This inherent delay explains the large lateral deviation observed in corners.
Exact Linearization via Backstepping
Test 1.2
Lateral Errors
RMSE Lat
0.3726 m
Max Errors
0.9645 m
Stability
Stable
Progress
+8%
The Backstepping law performs exact feedback linearization of the kinematic model, analytically cancelling its nonlinearities via a change of coordinates.
Unlike the PD, the controller is aware of the current curvature and applies the appropriate steering angle without waiting for an error to develop.
This reduces the steady-state lateral error in the curve significantly. However, the maximum error is slightly larger than the PD baseline — the transient at curve entry is not yet anticipated, only reacted to.
Backstepping and FeedForward
Test 1.3
Lateral Errors
RMSE Lat
0.8605 m
Max Errors
3.160 m
Stability
Stable
Progress
+112%
The Feedforward extension relies on a prediction loop that explicitly uses β_F and β_R to simulate the vehicle's future trajectory.
Without an observer, both are assumed zero — the controller predicts a drift-free vehicle and applies premature, incorrect steering corrections.
This is not a design flaw but a dependency: the Feedforward requires accurate slip angle estimates to function.
That missing piece is introduced in Test 1.4.
Generalized Predicted Control
Test 1.4
Lateral Errors
RMSE Lat
0.0977 m
Max Errors
0.50 m
Stability
Stable
Progress
+76%
The GPC adds an explicit model of the steering actuator's first-order dynamics on top of the Backstepping with Feedforward.
Instead of issuing a single instantaneous command, it computes an optimal sequence over a prediction horizon that drives the mechanical steering angle toward its desired future value, compensating for the actuator's inherent lag.
Curve entries and exits become noticeably crisper, and the residual phase error left by the Feedforward alone disappears. This is the most accurate controller of the four.
// 02 — Level 2
Observer
5 Tests · QuartAdams Trajectory · GPC · Progress Reference : Test 1.4
Rear Axle Off
Test 2.1
Lateral Errors
RMSE Lat
0.1174 m
Max Errors
0.5 m
Stability
Stable
Progress
−20%
With the rear axle disabled, the front axle takes full responsibility for both lateral and heading correction.
The coupling between the two error states is no longer damped, and the controller must work harder to maintain the trajectory.
However, at 2 m/s on a smooth circular path, the effect remains moderate. The QuartAdams trajectory simply does not generate enough transient stress to fully expose the limitation.
The real cost of disabling the rear axle will become apparent in Level 3, where abrupt heading changes push the front-axle-only configuration to its limits.
Pacejka
Test 2.2
Lateral Errors
RMSE Lat
0.2975 m
Max Errors
0.5103 m
Stability
Stable
Progress
-204%
With the Pacejka tire model and no observer, the vehicle remains stable but develops a persistent oscillation around the reference trajectory — visible both on the path plot and the error signal.
The controller assumes zero slip angles, while the nonlinear tire forces generate real, non-negligible β_F and β_R.
This unmodeled drift introduces a continuous mismatch between the assumed and actual vehicle dynamics, which the GPC cannot correct since it cannot see it.
The trajectory is followed in a broad sense, but the RMSE degrades noticeably compared to the linear tire baseline. This is precisely the gap that the observer is designed to close.
Kinematic Inversion
Test 2.3
Lateral Errors
Steering Angle (rad)
RMSE Lat
0.1106 m
Max Errors
0.50 m
Stability
Stable
Progress
-13%
Adding the kinematic inversion observer immediately recovers most of the degradation introduced by the Pacejka model — the RMSE drops significantly compared to Test 2.2, confirming that the slip angle estimates are meaningful and correctly fed into the controller.
However, the steering angle plot reveals a critical issue: the front axle command oscillates violently at high frequency, saturating repeatedly between its mechanical limits.
In a real vehicle, this chattering would cause rapid wear of the hydraulic actuator and steering linkage, and would likely trigger a safety cutoff.
The kinematic inversion observer is effective in terms of tracking accuracy, but its estimates are noisy enough to destabilize the actuator.
Luenberger
Test 2.4
Lateral Errors
Steering Angle (rad)
RMSE Lat
0.1422 m
Max Errors
0.50 m
Stability
Stable
Progress
-45%
The Luenberger observer trades instantaneous accuracy for smoothness.
Unlike the kinematic inversion, it progressively corrects its slip angle estimates through a parallel simulation loop, filtering out the high-frequency noise that caused the actuator chattering in Test 2.3.
The steering command is now clean and mechanically acceptable. However, this filtering introduces a small convergence delay — the estimated β lags slightly behind the real slip angle in the curve, leaving a modest residual error.
The RMSE is slightly higher than Test 2.3, but the actuator behavior is dramatically more realistic. This is the expected precision-vs-smoothness tradeoff inherent to observer design.
Lyapunov-based Adaptive Observer
Test 2.5
Lateral Errors
Steering Angle (rad)
RMSE Lat
0.1022 m
Max Errors
0.50 m
Stability
Stable
Progress
-4%
The Lyapunov-based adaptive observer delivers the best results of the three estimation strategies.
Unlike the Luenberger, its adaptation law is derived directly from a Lyapunov stability proof — the slip angle estimates are continuously adjusted to guarantee error convergence, not just filtered.
The steering command retains the smoothness of Test 2.4 while recovering the precision lost by the filtering delay.
With only a 4% degradation relative to the linear tire baseline, the observer almost entirely compensates the nonlinear Pacejka effects.
This is the configuration that justifies the full observer architecture — accurate, smooth, and theoretically grounded.
// 03 — Level 3
Sigmoid
3 Tests · ZAdams Trajectory · Progress Reference : Test 3.1
GPC & Lyapunov & Pacejka & Rear ON
Test 3.1
Lateral Errors
Steering Angle (rad)
RMSE Lat
0.0893 m
Max Errors
0.50 m
Stability
Stable
Progress
-
The sigmoid is a standard benchmark in mobile robotics, chosen precisely because its continuously varying curvature stress-tests the anticipation capabilities of any path-tracking controller.
The GPC begins steering before the error develops, while the Lyapunov observer feeds accurate slip angle estimates into the prediction loop throughout the maneuver.
After the transition, the error oscillates briefly then converges rapidly on the final straight.
The lower RMSE compared to QuartAdams is expected — a single transient followed by straight lines is inherently less demanding than a sustained circular regime.
GPC & Lyapunov & Pacejka & Rear OFF
Test 3.2
Lateral Errors
Steering Angle (rad)
RMSE Lat
6.3539 m
Max Errors
18.4307 m
Stability
NOT Stable
Progress
-
Disabling the rear axle on the ZAdams trajectory produces a catastrophic failure.
Without the rear axle to decouple lateral and heading errors, the sigmoid transition triggers an unrecoverable oscillation — the front axle saturates repeatedly at its mechanical limits, the heading error compounds at each correction attempt, and the vehicle never re-establishes contact with the reference path.
The system is marked unstable, with a peak error exceeding 18 meters.
This result contrasts sharply with Test 2.1, where the same configuration on QuartAdams produced only a modest 20% degradation. The sigmoid exposes what the smooth circular regime could not: abrupt heading changes are precisely the scenario where rear axle damping is not optional — it is the difference between recovery and divergence.
PD & Lyapunov & Pacejka & Rear ON
Test 3.3
Lateral Errors
Steering Angle (rad)
RMSE Lat
0.3057 m
Max Errors
0.7020 m
Stability
Stable
Progress
-242%
Despite appearing visually acceptable, the numbers tell a different story.
The PD controller follows the general shape of the sigmoid but reacts entirely after the fact — the error peaks at 0.70 m at the curve apex, where the full GPC configuration stays below 0.50 m throughout.
The steering angle confirms the reactive nature of the controller: a single large correction is applied once the error has already developed, with no anticipation of the heading change.
The RMSE is three times higher than Test 3.1. At low speed on a forgiving trajectory the PD is adequate — but adequate is not precise, and this gap would widen significantly at higher speeds or on more demanding paths.
// Conclusion
Takeaway
Every component in this architecture earns its place — but only under the right conditions. The PD is adequate when the trajectory is forgiving and the speed is low. The Backstepping removes the steady-state error the PD cannot. The Feedforward anticipates what the Backstepping only reacts to — but demands accurate slip angle estimates to do so. The GPC closes the loop on the actuator's own lag. The observer makes the nonlinear tire model transparent to the controller. The rear axle decouples what the front axle alone cannot separate.
None of these additions is redundant. Each one addresses a specific failure mode that the previous layer exposes. The gap between the simplest and the most complete configuration is invisible on easy paths — and decisive exactly when it matters most.