## 1 Introduction  
A tracked vehicle uses a track system composed of track plates and grousers as its mobility system. While the track system has lower drive autonomy than the wheel system, it provides a large contact area and lowers ground contact pressure, enabling better traction [1]. The track system is generally used in heavy vehicles (e.g., excavator, crawler tractor, and armed vehicle). Unlike road vehicles, which utilize the entire engine torque through sufficient ground strength as the driving force, the tractive performance of tracked vehicles, which drive on unpaved off-roads/terrains (hereinafter \"off-road tracked vehicles\"), is hindered by the shearing action and ground sinkage that occur on soil-track interfaces [2]. As the ground sinkage causes motion resistance that directly interferes with driving the off-road tracked vehicle, it is important to assess and limit ground sinkage [3]. Ground sinkage is categorized into static sinkage, which is induced by the weight of a vehicle, and slip sinkage, which occurs when the track system imposes a shearing load on the ground. Since [PERSON] [4] pioneered the soil-track interaction theory, research for predicting ground sinkage has mainly focused on static sinkage. [PERSON] [4] and [PERSON] et al. [5] demonstrated that it is possible to calculate static sinkage through an elastic settlement assessment method, assuming the track system as the shallow foundation that supports the off-road vehicle's weight. In addition,various researchers [2; 3; 6; 7] have proposed empirical equations for ground contact pressure and static sinkage based on the results of bevameter pressure-sinkage experiments performed on various types of ground. [PERSON] and [PERSON] [8] and [PERSON] et al. [9] used these equations to calculate the static sinkage of off-road tracked vehicles under various ground and vehicle conditions, which were applied to assess motion resistance. References to slip sinkage are first found in [PERSON]'s terramechanic book. [PERSON] [4] noted that the trim effect (or slip sinkage effect), first considered a result of soil pumping under the track system, was estimated at \(5^{\circ}\). Similarly, [PERSON] and [PERSON] [10] stated that the sinkage is linearly proportional with the horizontal distance between an arbitrarily chosen point on the track and the leading point of the interface, and adopted the equation proposed by [PERSON] [11] to calculate the maximum sinkage (i.e., sinkage at the end point of the interface). These studies [4; 10] imply the presence of slip sinkage, but the effect of slip ratio (or slip displacement) is not taken into account in the calculation of the sinkage. Several researchers [12; 13; 14; 15] developed slip sinkage models that include the effect of the slip ratio. [PERSON] [14] proposed a slip sinkage formula as a function of slip ratio and grouser height. [PERSON] et al. [15] developed an analytical model that took into account soil deformation propagation depth, particles trajectories, and volume changes under the track system in the shearing process. [PERSON] [13] extended the work of [PERSON] et al. [12], and expressed the slip sinkage using slip ratio and static sinkage based on the law of conservation energy. However, these slip sinkage models [12; 13; 14; 15] do not account for changes in slip sinkage along the track system; the slip sinkage increases proportionally with the horizontal distance from the leading point of the soil-track interface [10]. To overcome these limitations, this study conducted model track experiments using different surcharge stresses (7.0 kPa, 10.6 kPa, and 14.0 kPa) in silty sand comprising three different relative densities (40%, 60%, and 80%). This allowed a comprehensive assessment of the effect of various ground and load conditions on slip sinkage. From the experiment results, we then proposed an equation for calculating the slip sinkage of off-road tracked vehicles as a function of static sinkage, slip ratio, and the horizontal distance from the leading point of the soil-track interface.