## 1 Introduction  
Urban population growth and car ownership have put enormous pressure on public transport [1]. Although it can be alleviated by subways, trans, and other ways, there are shortcomings such as high construction costs, long construction periods, and high maintenance costs [2]. To overcome the above problems, a kind of public transportation system called the virtual track train (VTT) has been developed rapidly in recent years. The VTT is a multi-articulated public transportation system using a rail transit management mode. The so-called \"virtual track\" differs from traditional rail systems' track. It does not have a strong constraint between the track and the vehicle but is composed of ground markers or magnetic spikes and is a digital track. The VTT adopts rubber tire running gears and uses environmental perception technology to recognize the virtual track and environment. Then, path-tracking control technology is used to control the wheel steering angles of each axle and the driving torque of the driving wheels, allowing the vehicle to autonomously track the target path and speed with high accuracy. Compared with traditional rail transit systems, the construction costs, construction cycle, and maintenance costs of the VTT can be significantly reduced [2, 3]. Compared with bus rapid transit (BRT), the VTT has some advantages. The longer body can carry more passengers, and its capacity is equivalent to trans. The all-axle steering and distributed drive technology make it more flexible and have a smaller sweep width, which can pass through curves with smaller radii. The VTT has received widespread attention recently, and multiple operational lines have been built. At present, there are mainly three types of VTTs in commercial operation: Super autonomous Rail rapid Transit (SRT) released by CRRC Zhuzhou Locomotive Co., Ltd. in Zhuzhou of China, Automatic Rail rapid Transit (ART) released by the CRRC Zhuzhou Institute in Zhuzhou of China, and Digital Rail guided Train (DRT) released by CRRC Nanjing Puzhen Co., Ltd in Nanjing of China. The SRT adopts a four-module marshalling. The first and fourth modules each have two steerable rubber wheel running gears and the middle module has one steerable rubber wheel running gear. The first axle of the first module and the second axle of the fourth car (the first and sixth axles of the entire vehicle) are equipped with drive motors, and adjacent modules are connected through hinge plates. The ART adopts a three-module marshalling. Each module is equipped with two independent steering rubber wheel running gears. The DRT train also has a three-module marshalling, but adjacent modules are connected through articulated double-axle rubber wheel bogies. In addition, there are also some transportation systems with operational characteristics and control methods similar to the VTT, for example, the AutoTram Extra Grand series developed by the Fraunhofer Institute in Germany [4], the Civis optical guided bus developed by Siemens and Iveco, the Phileas guided bus system developed by the VDL Group in the Netherlands [5], and the TEOR system operated commercially in Rouen, the capital of the Normandy, in France [6]. Among them, the SRT has a higher carrying capacity and curve passing flexibility due to its larger number of marshalling, making it more advantageous than other transportation systems. The path-tracking control technology is the key to the safe and stable operation of the VTT along the target path with minimal error. Refs. [7; 8; 9; 10; 11] introduce some control strategies for single articulated vehicles with tractor-trailer combinations to ensure mobility at low speed and medium- and high-speed stability. These methods use the articulated point of the tractor as the following point of the trailer for tracking control based on the vehicle kinematics model.