Tube-Based Robust Control Strategy for Vision-Guided Autonomous Vehicles

Der-Hau Lee

eess.SYMar 24, 2025

Tube-Based Robust Control Strategy for Vision-Guided Autonomous Vehicles

Authors: Der-Hau Lee

Organizations: Department of Electrophysics, National Yang Ming Chiao Tung University, Hsinchu 300, Taiwan

Abstract

A robust control strategy for autonomous vehicles can improve system stability, enhance riding comfort, and prevent driving accidents. This paper presents a novel interpolation-tubebased constrained iterative linear quadratic regulator (itubeCILQR) algorithm for autonomous computer-vision-based vehicle lane-keeping. The goal of the algorithm is to enhance robustness during high-speed cornering on tight turns. Compared with standard tube-based approaches, the proposed itube-CILQR algorithm reduces system conservatism and exhibits higher computational speed. Numerical simulations and vision-based experiments were conducted to examine the feasibility of using the proposed algorithm for controlling autonomous vehicles. The results indicated that the proposed algorithm achieved superior vehicle lane-keeping performance to variational CILQRbased methods and model predictive control (MPC) approaches involving the use of a classical interior-point optimizer. Specifically, itube-CILQR required an average runtime of 3.45 ms to generate a control signal for guiding a self-driving vehicle. By comparison, itube-MPC typically required a 4.32 times longer computation time to complete the same task. Moreover, the influence of conservatism on system behavior was investigated by exploring the variations in the interpolation variables derived using the proposed itube-CILQR algorithm during lane-keeping maneuvers.

Figures & tables

$dt$	0.01 s	${C_{\alpha f}}$	80000 N/rad
$v_{x}$	{20.0, 22.2} m/s	${C_{\alpha r}}$	80000 N/rad
$I_{z}$	2000 kgm 2	$l_{f}$	1.27 m
$m$	1150 kg	$l_{r}$	1.37 m

TABLE I: Model and vehicle parameters used in this study

$N$	30	${\bf Q}$	diag[20, 1, 20, 1]
$\Delta$	[-2.0, 2.0] m	${\bf R}$	60 $\bf I$
$\dot{\Delta}$	[-9.0, 9.0] m/s	${\bf S}$	50 $\bf I$
$\theta$	[- $\pi/2$ , $\pi/2$ ] rad	$q_{s}$	100
$\dot{\theta}$	[-4.0, 4.0] rad/s	$q_{u}$	10
$\delta$	[- $\pi/6$ , $\pi/6$ ] rad	$q_{l1}$	80
$\kappa$	[-0.1, 0.1] 1/m	$q_{l2}$	20

TABLE II: Control parameter values and constraints

Fig. 1: Visualization of sets

\mathbb{X}^{\prime}

\mathbb{\bar{X}}^{\prime}

, and

\mathbb{S}_{RPI}

when

v_{x}

= 20.0 or 22.2 m/s and

\kappa

\pm

0.05 and

\pm

0.1 1/m. Here,

x_{1}\equiv\dot{\Delta}

and

x_{3}\equiv\dot{\theta}

Fig. 2: Visualization of sets

\mathbb{\bar{T}}^{\prime}

and

\mathbb{\bar{T}}^{\prime}_{lir}

when

v_{x}

= 20.0 or 22.2 m/s and

\kappa

\pm

0.1 1/m. In (a),

\mathbb{\bar{T}}^{\prime}

is defined by 14 inequalities, and the corresponding vertex coordinates in the first and third quadrants are identical to those of

\mathbb{\bar{X}}^{\prime}

in Fig. 1(a).

Fig. 3: Profile of an example exponential barrier function

B\left(x\right)=\exp\left[{q\left({x-1}\right)}\right]+\exp\left[{q\left({-1-x}\right)}\right]

, where

x

can be a state or control variable. As the penalty parameter

q

increases,

B(x)

increases rapidly when

x>1

and

x<-1

but approaches zero when

x=0

Fig. 4: Numerical simulation results obtained with different CILQR algorithms when

v_{x}

= 20.0 m/s,

D

= 0.22,

\left|{x_{1}}\right|\leq

9.0 m/s, and

\left|{x_{3}}\right|\leq

4.0 rad/s. (a) Trajectory of the actual system state

x_{0}

. (b) Trajectory of the actual system control input. (c) Trajectories of the first components of the optimal interpolation variable sequences for

x_{0}\equiv\Delta

and

u\equiv\delta

. In (a), the

x_{0}

values obtained with the tube-CILQR-up and itube-CILQR algorithms at

t

= 700 are -0.2221 and -0.2201 m, respectively. In (c),

\Delta\lambda

= 0.0331, 0.1283, and 0.0834 at

t

= 0, 600, and 1100, respectively. The experimental results obtained at different

D

values are presented in Table III.

Fig. 5: Numerical simulation results obtained with different CILQR algorithms when

v_{x}

= 20.0 m/s and

D

= 0.22. Parts (a) and (b) were obtained when

\left|{x_{1}}\right|\leq

10.0 m/s and

\left|{x_{3}}\right|\leq

4.5 rad/s. Parts (c) and (d) were acquired when

\left|{x_{1}}\right|\leq

8.0 m/s and

\left|{x_{3}}\right|\leq

3.5 rad/s. In (a), the

x_{0}

values obtained with the tube-CILQR-up and itube-CILQR algorithms at

t

= 700 are -0.2078 and -0.2069 m, respectively. In (b),

\Delta\lambda

= 0.0136, 0.0821, and 0.0493 at

t

= 0, 600, and 1100, respectively. In (c), the

x_{0}

values obtained with the tube-CILQR-up and itube-CILQR algorithms at

t

= 700 are -0.2411 and -0.2376 m, respectively. In (d),

\Delta\lambda

= 0.0402, 0.1691, and 0.1103 at

t

= 0, 600, and 1100, respectively.

Fig. 6: Numerical simulation results obtained with different MPC algorithms when

v_{x}

= 20.0 m/s,

D

= 0.22,

\left|{x_{1}}\right|\leq

9.0 m/s, and

\left|{x_{3}}\right|\leq

4.0 rad/s. (a) Trajectory of the actual system state

x_{0}

. (b) Trajectory of the actual system control input. (c) Trajectories of the first components of the optimal interpolation variable sequences. In (a), the

x_{0}

value obtained at

t

= 700 with both the tube-MPC-up and for itube-MPC algorithms is -0.2162 m.

Fig. 7: Computation times of the (a) tube-CILQR-up, (b) itube-CILQR, (c) tube-MPC-up, and (d) itube-MPC algorithms at each time step.

a Parameters $v_{x}$ = 20.0 m/s, $\left\|{x_{1}}\right\|\leq$ 9.0 m/s, and $\left\|{x_{3}}\right\|\leq$ 4.0 rad/s were used.
$D$	0.160	0.190	0.220	0.230	0.240	0.250	0.255	0.258	0.260	0.270
$\lambda_{d}^{*}$	0.680	0.620	0.560	0.540	0.520	0.500	0.490	0.484	0.480	0.460
$x_{0}$ $(t=700)$	-0.2209	-0.2206	-0.2201	-0.2200	-0.2198	-0.2197	-0.2200	-0.2207	-0.2217	-0.2220
$\Delta\lambda$ $(t=600)$	0.0926	0.1105	0.1283	0.1341	0.1399	0.1453	0.1480	0.1496	0.1507	0.1561

TABLE III: Experimental results obtained with the itube-CILQR algorithm a under different

D

values

Method	tube-CILQR-up	itube-CILQR	tube-MPC-up	itube-MPC
Latency (ms)	0.76	3.45	5.89	14.92
a All algorithms were implemented in C++.

TABLE IV: Average computation times for the tube-CILQR-up, itube-CILQR, tube-MPC-up, and itube-MPC algorithms a

Panel of Figure 12 — Fig. 8: Adopted vision perception system (left) [ 25 ] , and an example traffic situation and maps of track-A and track-B in TORCS for autonomous driving (right). The input of the adopted deep neural network model was an RGB image with a resolution of 228 $\times$ 228 pixels. The ego vehicle was designed to begin maneuvering at the locations marked by red symbols on the track maps, and the driving direction was counterclockwise. Both tracks contained several turns [Fig. 10(a) and Fig. 11(a)] and had different road friction values, which increased the difficulty of high-speed cornering.

Fig. 9: Trajectories obtained for

\Delta

by using the (a) itube-CILQR, (b) tube-MPC-up, and (c) itube-MPC algorithms when

fr

= 1.1 or 1.0 for an ego vehicle traveling on track-A at 72 or 80 km/h.

Fig. 10: Lane curvature results (

\kappa

) and computed gaps (

\Delta\lambda

) for track-A. (a) Curvature profile of track-A. Trajectories obtained for

\Delta\lambda

by using the itube-CILQR algorithm when

fr

was 0.8 and the ego vehicle was traveling at (b) 72 and (c) 80 km/h. The results of the associated quantitative analyses are presented in Table V.

Fig. 11: Lane curvature results (

\kappa

) and computed gaps (

\Delta\lambda

) for track-B. (a) Curvature profile of track-B. Trajectories obtained for

\Delta\lambda

by using the itube-CILQR algorithm when

fr

was 0.8 and the ego vehicle was traveling at (b) 72 and (c) 80 km/h. The results of the associated quantitative analyses are presented in Table V.

$fr$	track-A				track-B
	80 km/h		72 km/h		80 km/h		72 km/h
	Avg	SD	Avg	SD	Avg	SD	Avg	SD
0.80	0.040587	0.013882	0.040110	0.009718	0.037401	0.009966	0.036320	0.006903
0.85	0.040086	0.009908	0.039750	0.009818	0.037047	0.008608	0.036401	0.007932
0.90	0.039868	0.009830	0.039938	0.009995	0.036809	0.008079	0.036516	0.008138
1.00	0.040076	0.010097	0.039445	0.008625	0.036859	0.009265	0.036303	0.007021

TABLE V: Average and standard deviation values of

\Delta\lambda

during lane-keeping maneuvers performed using the itube-CILQR algorithm

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Abstract

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