## 1 Introduction  
The weight of an object is defined as the product of its mass and gravitational acceleration. However, humans do not always perceive weight according to its physical definition. Instead, weight is often estimated based on generated motor commands and sensations during motions [1; 2; 3; 4] and various types of physical quantities, such as density and inertia [5; 6; 7]. By controlling specific factors that affect weight perception, an object can feel heavier or lighter than its actual weight. These phenomena are referred to as weight illusions. Investigating weight illusions leads to an understanding of weight perception mechanisms. The size-weight illusion is a representative example of weight illusions, in which an object with a smaller volume is perceived as heavier than an object with a larger volume, despite weighing the same [8]. The underlying principle behind this illusion however is still under debate. Nevertheless, a gap between the perceived weight of the lifted object and its expected weight before lifting is considered to be part of the reason for the illusion [1; 9]. Furthermore, the material, color, and brightness also cause weight illusions, in which an object that is expected to be heavy from its appearance is perceived as being light [10; 11; 12]. As evident from the above examples, the appearance of an object is one of the factors underlying the weight illusion. The presentation of tactile stimuli to the hand also leads to the weight illusion. Deep sensations primarily influence weight perception, but cutaneous sensations are also influential, especially for light objects [13; 14; 15; 16]. When lifting an object, the skin deforms in the direction of the gravity or force acting on the held object. Therefore, an additional skin deformation in the direction of gravity makes objects feel heavier [15; 17; 18]. Some researchers have devised methods to increase the perceived weight using vibratory stimuli. When asymmetric vibration is presented to the hands, an additional force is felt in the direction of greater acceleration [19; 20; 21; 22]. Therefore, asymmetric vibration in the direction of gravity makes objects feel heavier [20]. Furthermore, vibratory stimuli coinciding with the acceleration of moving hands can increase the perceived weight of an object [23; 24]. Our research team previously identified a weight illusion caused by a continuous sinusoidal vibration to the fingertips [25]. When a vibrating object is lifted, it feels heavier than a still object with the same mass. In our previous study, 13 out of 15 participants reported this illusory weight perception [25]. We call this phenomenon the \"vibration-weight illusion\". Continuous vibrations over a wide range of frequencies from 30 Hz to 300 Hz generated the vibration-weight illusion [25]. As mentioned above, earlier studies used asymmetric vibration [19; 20] or vibration coinciding with moving hands [23; 24] to present the weight. In contrast, the vibration-weight illusion is caused by symmetric and continuous vibrations. The characteristics of the stimuli in the vibration-weight illusion differ from those of other weight presentation methods reported by earlier researchers. Therefore, investigating the mechanisms underlying this illusion may lead to new insights into the perceptual effects of vibratory stimuli and the mechanisms of weight perception. We hypothesized that an increase in the grip force will be the main factor underlying the vibration-weight illusion. Humans feel their own motions and forces intensely when the related motor commands are large [26; 27; 28; 29]. In the case of the vibration-weight illusion, an additional grip force may be used to stabilize the vibrating object, and the additional force may make the object feel heavier than its actual weight. Therefore, this study investigated the relationship between the grip force adjustment and vibration-weight illusion.