## 1 Introduction  
Tracked vehicles have long been regarded as the optimal solution for traversing irregular and high-dynamic terrains. Their ability to maintain mobility and traction under extreme off-road conditions makes them indispensable in military, agricultural, and exploration applications [1]. Traditionally, torsion bar suspension systems have been widely adopted in such platforms due to their structural simplicity and durability. However, these systems present several limitations, including large installation volume, fixed stiffness, and poor adaptability to variable terrains. To address these shortcomings, hydropneumatic suspension units (HSUs) have been introduced as compact and modular alternatives. HSUs offer excellent nonlinear stiffness characteristics, high load-bearing capacity, and strong terrain adaptability, with the potential for rapid-response semi-active control [2, 3, 4]. Due to the superior performance of HSUs, such systems have received widespread attention in both research and industrial fields. For instance, [PERSON] and [PERSON] developed a dynamic model of a semi-active hydropneumatic spring-damper system for off-road vehicles and highlighted its effectiveness in improving vertical acceleration and ride comfort under rough terrain conditions [5]. [PERSON] and [PERSON] conducted nonlinear modeling of an in-arm hydropneumatic suspension (ISU) system for tracked vehicles, investigating the effects of orifice diameter and spring preload on vertical dynamic response. Their results demonstrated that structural tuning could effectively yield desirable support characteristics [6]. [PERSON] et al. designed and validated a multi-cylinder hydropneumatic suspension system for a novel road-rail vehicle. They proposed a nonlinear model using AMESim and evaluated various parameters through virtual tests, showing that a hydropneumatic suspension significantly improves adaptability to complex terrain and dynamic performance [7]. Additionally, [PERSON] et al. developed an integrated ISD (Inerter-Spring-Damper) hydropneumatic suspension system for heavy multi-axle vehicles, and experimental validation confirmed its advantages in vibration reduction and impact mitigation [8]. Depending on their structural integration, HSU can be categorized into an in-arm suspension unit (ISU) and on-arm suspension unit (OSU), with ISU offering enhanced compactness and integration capabilities. A key challenge in designing ISU systems is minimizing the side force exerted by the piston on the cylinder wall during its vertical motion. This side force can affect the durability, reliability, and efficiency of the suspension system. Recently, several cutting-edge works have extended the field of suspension system design. For instance, [PERSON] et al. [9] developed a robust static output feedback controller for MR-damper-based semi-active suspensions, achieving improved performance under uncertain conditions. In the context of tracked vehicles, nonlinear modeling of MR suspensions has also been explored to enhance dynamic response prediction accuracy [10]. Furthermore, fractional-order SH-GH control strategies have been introduced to ISD-type suspensions for broadband vibration optimization [11]. These approaches represent significant progress toward adaptable and intelligent suspension systems. However, most of these studies still rely on idealized working media or primarily focus on active control strategies. In industrial applications, the ideal gas (IG) equation or IG-based models are commonly adopted to describe the nonlinear behavior of gas within HSU structures due to their mathematical simplicity. Nevertheless, such simplified approaches fail to accurately capture the temperature-dependent behavior and hysteresis characteristics of real gases [12; 13; 14]. This study proposes a systematic model and design framework that integrates real gas thermodynamics and robust optimization strategies to overcome the above limitations.