## 2 Related Studies and Hypothesis of the Vibration-Weight Illusion  
An asymmetric vibration is an example of force presentation by vibratory stimuli [19; 20]. It comprises a faster motion of mass in one direction and a slower motion in the other, and the illusory force felt in the former direction. When an object held in the hand vibrates asymmetrically, it generates higher forces in the direction of a greater vibratory amplitude than in the opposite direction [21]. Moreover, the skin deforms more in that direction [22]. The nonlinearity between the physical stimuli, that is, the force and skin deformation, and their subjective magnitudes is a potential cause of the illusory force perception. An asymmetrically vibrating object feels heavier when the vibration is in the direction of gravity [20]. However, the vibration-weight illusion discussed in the present study uses a sinusoidal vibration, which is symmetric. Moreover, the vibration-weight illusion occurs even when the vibration is presented in a direction perpendicular to gravity.  
[PERSON] et al. and [PERSON] et al. proposed a method to increase the perceived inertia or mass by controlling the presentation timing of vibratory stimuli [23; 24]. The inertial force generated by swaying an object causes finger skin deformations that activate mechanoereptive units. They intended to additionally stimulate the mechanoereceptive units of the fingertips by vibration and cause an illusory sense of a magnified skin deformation. They thus made a swayed object feel heavier by presenting vibrations [23; 24]. However, the vibration-weight illusion discussed in the present study is generated by a continuous vibration while the object is statically grasped. It thus differs from their findings.  
The tonic vibration reflex is a muscle contraction evoked by the vibration applied to the muscle below the skin that stimulates muscle spindles, leading to a muscle contraction reflex [30; 31]. The stimulation of agonist muscles makes an object feel lighter, and conversely, the stimulation of antagonist muscles makes the object feel heavier [26]. There are no reports of a tonic vibration at the fingers because fingers lack muscles; however, in the case of the forearm, a vibration with an amplitude of 0.2-0.3 mm at a frequency of 100-200 Hz was found to effectively generate a tonic vibration reflex [32]. If the tonic vibration reflex causes the vibration-weight illusion, the vibration delivered to the fingertips should be transferred to the antagonist muscles of the wrist or forearm. However, vibrations with amplitudes of less than 0.1 mm to the fingertips generated the vibration-weight illusion [25]. This stimulus is small when compared to that of a previous study [32]. Therefore, we speculated that the tonic vibration reflex does not cause the vibration-weight illusion.  
We hypothesized that the vibration applied to an object will cause the grip force to increase as a result of the automatic grip force adjustment for stable grasping. Motions that generate more motor commands are perceived as higher intensity [26; 27; 28; 29]. When the motor commands related to motions are larger, the object's weight is perceived as heavier [1; 2; 3; 4]. Therefore, even if the weight of the object does not change, the increase in the grip force may make the grasped object feel heavier.