## 1 Introduction  
Recently, there has been an increased focus on seabed resources [1], which has spurred the development of related mining technologies, such as the piping lifting system (e.g., [2; 3]). In seabed mining, vehicles like backhoes, bulldozers, and excavators are essential for collecting and transporting resources, significantly impacting the efficiency of the process. Typically, these mining vehicles are heavy and equipped with a track system that offers better flotation and traction than wheels on the seabed, which typically consists of soft clay. Consequently, tracked vehicles are predominantly used in deep-sea mining as part of the piping lifting system, and their performance is extensively evaluated (e.g., [4; 5; 6; 7]). When a tracked vehicle moves, its weight and engine torque are transmitted to the ground via the track system as vertical and horizontal forces. These forces induce shear deformation in the ground, generating a shear force known as soil thrust [8]. The vehicle advances as a reaction force overcomes this shear resistance; thus, soil thrust serves as a key performance indicator for the tracked vehicle, and its accurate estimation is important in seabed mining. [PERSON] [9] initially introduced a soil thrust solution by assuming a rectangular block failure in the ground caused by the track system. Based on this, various researchers have conducted model experiments to demonstrate that block failure typically occurs in sandy soils and have developed an improved method for evaluating soil thrust (e.g., [10; 11; 12; 13]). Additionally, further studies have both experimentally and numerically analyzed soil thrust considering the properties of the soil and the shape of the track system on sandy ground (e.g., [14; 15]). For a track system on clay ground, numerous experimental and numerical studies have focused on evaluating soil thrust based on the conditions of the tracked vehicle and the ground [16; 17; 18; 19; 20; 21]. [PERSON] et al. [22] experimentally demonstrated that clay under a track system can experience a wedge failure mode as well as the block failure mode. [PERSON] et al. [23] conducted a numerical analysis considering variables such as the strength parameters of clay and specifications of tracked vehicles, confirming the occurrence of the wedge failure mode. These findings suggest that under a track system, the failure mode in clay varies with the specifications of the track vehicle and the undrained shear strength of clay; thus, the estimation of the reasonable failure mode is one of the key factors in the evaluation of soil thrust of a track system over clay ground. [PERSON] and [PERSON] [24] proposed upper bound solutions for soil thrust of a track system over clay ground, focusing on different failure modes (e.g., block, triangular wedge, and trapezoidal wedge) under the plane strain condition; subsequently, [PERSON] and [PERSON] [25] expanded on this research by developing a three-dimensional theoretical solution for soil thrust of an individual track system over clay ground; however, these studies only focused on tracked vehicles operating on flat terrain. To address this limitation, [PERSON] and [PERSON] [26] introduced a theoretical solution for the soil thrust of a track system operating over a clay slope; however, this study only considered the soil thrust in the two-dimensional plane strain condition. This study introduces a three-dimensional upper bound solution for soil thrust on a clay slope. The present study considered block, triangular wedge, and trapezoidal wedge failure modes, developing theoretical solutions for each. The seabed is typically composed of clayey deposits with undrained shear strengths ranging from 1.9 to 40 kPa [27]; thus, this study assumed the full saturation of clay. As the upper vehicle typically moves faster than the rate at which pore water drains in the clay, this study assumes undrained conditions within clay ground.