# Design and Stability Analysis of Six-Degree-of-Freedom Hydro-Pneumatic Spring Wheel-Leg
## 1 Introduction  
The suspension is a key component of a vehicle. It not only connects the chassis and wheels but also transmits forces and torques between them. Road excitations are transmitted to the suspension through the wheels. Then, the suspension absorbs shocks and filters vibrations, ensuring vehicle ride comfort and reducing damage to vehicle components due to excessive vibrations [1]. A hydro-pneumatic suspension is a special suspension that combines elastic elements and dampers into a single unit. It uses inert gas inside an accumulator as the elastic medium and hydraulic oil as the transmission medium for damping [2; 3]. This system has excellent nonlinear damping characteristics and strong environmental adaptability. Thus, it is widely used in heavy-duty vehicles operating under complex conditions [4; 5; 6; 7]. In-wheel motor technology integrates the motor into the wheel hub of the vehicle. It replaces the traditional engine-transmission-drive shaft configuration and achieves a distributed electric drive [8]. This not only simplifiesthe vehicle structure but also provides greater design flexibility for the body structure and functional layout. Although the in-wheel motor drive mode has many advantages in the structure and design, it also causes changes in the distribution of vehicle mass, especially a significant increase in unsprung mass. It also caused significant changes in the vertical dynamic characteristics of the vehicle [9]. Moreover, the introduction of in-wheel motors alters the chassis structure and lateral dimensions of the vehicle, significantly changing its lateral and longitudinal dynamics. These changes impose new design requirements on the chassis suspension systems of in-wheel motor vehicles [10; 11; 12]. Many researchers have explored this topic. For instance, the author of [13] established an objective function for evaluating vehicle ride comfort by analyzing the configuration of electric wheel suspension. By optimizing the suspension stiffness and damping coefficients between the in-wheel motor and the vehicle body, and between the motor and the wheels, to improve the smoothness of the vehicle. The authors of [14] established an 11-degree-of-freedom dynamic model to study the dynamic impact of electromagnetic vibrations of a reducer-free in-wheel motor on the vertical and longitudinal dynamics of electric vehicles. The results indicate that electromagnetic vibrations negatively affected the vehicle's vertical and longitudinal dynamics under all conditions. The authors of [15] proposed a controllable suspension to address the increased unsprung mass of in-wheel motor vehicles. The optimal design control strategy was adopted, and a sliding mode controller was designed to effectively suppress vehicle vibration. The authors of [16] developed an imbalance radial force equation for in-wheel motors and an electromechanical coupling model for electromagnetic active suspension. By optimizing the suspension control parameters, the negative effects of vehicle vibration can be effectively suppressed. The authors of [17] designed an active suspension dynamic vibration absorber for wheel-side drive systems. It adopts a finite frequency H\({}_{\infty}\) state feedback control strategy, which achieves good ride comfort and reduces the dynamic loads on the in-wheel motor bearings. Considering the impact of motor electromagnetic excitation on vehicle dynamics, the authors of [18] established a dynamic model of the in-wheel motor suspension system. Obtained the optimal parameters of the active suspension system by using a particle swarm optimization algorithm. To address the issue of reduced ride comfort in in-wheel motor-driven vehicles due to the increased unsprung mass, the authors of [19] designed a suspension configuration that adds a damper to the steering knuckles of the vehicle. By establishing a mathematical model of the system and optimizing the parameters of the vibration absorber, the problem of vertical negative effects in automobiles has been significantly improved.