Measurement of the Transfer Function of Kinesthetic Illusion Induced by Antagonistic Tendon Vibration

The kinesthetic illusion induced by tendon vibration has the potential to be applied to virtual reality because it can provide the sensation of motion without the need to actually move the body. The psychophysical and neurophysiological properties of this phenomenon have been studied for a long time, and more recently, research has been conducted to realize the presentation of complex motion. However, there is still a lack of knowledge that quantitatively relates time-varying vibratory stimuli to the resulting kinesthesia. In response to this situation, we experimentally quantify the relationship between the time-varying frequency of the vibratory stimuli and the perceived joint angle, as a transfer function which is directly applicable to the presentation of kinesthetic illusions. To minimize temporal error, we presented vibration stimuli to one arm and a physical motion to the other arm, and asked participants to adjust the amplitude and phase of the physical motion so that the sensations of the two stimuli matched. The transfer function obtained from the experiment was nearly constant between the presented reciprocating frequencies of 0.05 Hz and 0.30 Hz, with a flat amplitude response and a phase advance of approximately 180 degrees. This transfer function represents motion in the opposite direction to that expected from existing knowledge. The most plausible explanation for this is the generation of an illusory force sensation by activation of the Golgi tendon organ. However, there is a small possibility that this was due to tonic vibration reflexes or a misunderstanding of the experimental task.


I. INTRODUCTION
Even with their eyes closed, people can tell what position their arms and legs are in and how they are moving.This is called proprioception or kinesthesia (collectively referred to as proprioception in this paper).
Proprioception is an essential sense for humans to live, but in virtual reality (VR) it can be an obstacle.That is, in order to experience virtual body movements, it is necessary The associate editor coordinating the review of this manuscript and approving it for publication was Xiaojie Su .
to actually move the body, and if the avatar moves but the body does not actually move, a sense of discomfort arises.This creates problems such as the need for a large space and large equipment (such as an omnidirectional treadmill), and the risk of injury.
Fortunately, however, there is a ''back door'' to proprioception, i.e., a way to generate it without physical movement.A typical example of such a ''back door'' is the kinesthetic illusion induced by tendon vibration stimulation discovered by Goodwin et al. [1].In this phenomenon, when vibrations are applied from above the skin to tendons, which are tissues that connect muscles and bones, the sensation that the body is moving is produced even when the body is not actually moving.This phenomenon is thought to be due to the activation of the muscle spindle, which is one of the organs responsible for proprioception and detects the length of muscles, by the vibration.
Attempts have been made to exploit this phenomenon to solve problems in VR.For example, Hagimori et al. [2] proposed a method to create the perception of greater motion in VR than in reality by combining tendon vibration with visual stimulation using a head-mounted display (HMD).Leonardis et al. [3] proposed a chair-like device that combines tendon vibration stimuli and a motion platform to create the sensation of walking in a virtual space.
However, most of the existing applications are limited to presenting qualitative kinesthesia or sensory presentation combined with vision, etc., which can be easily presented quantitatively, and it is still difficult to present quantitative kinesthesia using tendon vibration alone.Some studies, e.g., by Albert et al. [4] or Thyrion and Roll [5], have succeeded in presenting a complex kinesthetic illusion by modulating the vibration frequency, but the resulting sensations are not stable.
There are several reasons for this, but one of the most important is that the knowledge of the relationship between the parameters of the vibratory stimulus and the generated kinesthetic illusion is insufficient for VR application.In particular, there is a significant lack of knowledge regarding the characteristics of responses to dynamically changing vibration stimuli.Although there are studies on the temporal resolution of the kinesthetic illusion [6] and on the relationship between muscle length and neural activity (e.g., [7]), there is still no knowledge that allows us to determine the vibration parameters to produce the desired kinesthetic illusion.
The contribution of this paper is to quantitatively measure the temporal characteristics of kinesthetic illusion in response to dynamically varying tendon vibration stimuli in a form that can be easily applied to the presentation of kinesthetic illusion.The temporal characteristics are represented as a transfer function under the assumption that the relationship between vibration frequency and perceived motion is a linear time-invariant system and are quantified by a psychophysical experiment.In the experiment, a frequency-modulated vibration presents the kinesthetic illusion of reciprocating motion at speeds ranging from 0.05 Hz to 0.3 Hz, and participants adjust the speed and amplitude of the simultaneously presented physical motion to match the sensations.This experimental protocol does not require participants to reproduce the kinesthetic illusion with the opposite arm, thus minimizing the time lag between recognition and reproduction of the illusion and allowing for more accurate phase measurements.Quantitative presentation of kinesthetic illusion can be achieved by passing the trajectory of the movement to be presented through the inverse of this transfer function.

II. RELATED WORK A. PROPRIOCEPTION AND KINESTHETIC ILLUSION
Proprioception is the sense of body posture and movement that is essential for a person to control their own body [8] and is also related to the sense of body ownership [9].This sense is a complex system formed by information from various receptors.Two of the most important types of receptors are known to be the muscle spindle, which detects the state of muscle extension, and the Golgi tendon organ, which detects the force applied to a muscle [10].
As with other senses, proprioception is known to be subject to illusions.A typical example is the kinesthetic illusion induced by tendon vibration [1].In this phenomenon, when a vibration of about 100 Hz is applied to a tendon of a muscle, the muscle spindle inside the muscle is activated and movement is perceived as if the muscle were being stretched.It has been reported that the firing frequency of the nerves (muscle spindle primary endings) caused by the vibration matches the frequency of the vibration at least up to 80 Hz [11].
The sensation produced by tendon vibration depends on a variety of conditions, including vibration parameters and posture.For example, as the amplitude of vibration increases, the perceived illusory motion becomes stronger and faster [12].Regarding frequency, perceived motion increases with increasing frequency up to about 70 Hz [13], but not at higher frequencies [12].Gilhodes et al. [14] also found that the direction and magnitude of the kinesthetic illusion changes with the difference in vibration frequency given to the two muscles when vibration is presented to an antagonistic muscle pair (biceps and triceps brachii).
Kinesthetic illusion is known to be influenced by sensory information other than that from muscle spindles.Presenting visual information that coincides with the kinesthetic illusion at the same time as the tendon vibration enhances the kinesthetic illusion [15], [16].On the other hand, some visual stimuli may inhibit kinesthetic illusion [17].

B. APPLICATION OF KINESTHETIC ILLUSION IN ENGINEERING
Due to the nature of the kinesthetic illusion, which can only present the sensation of motion without any physical movement, it has been applied to VR as a method to solve problems caused by physical movement.The most typical of these is the work of Hagimori et al. [2].They applied tendon vibrations to the forearm while experiencing a VR environment through an HMD to induce a perception of motion that is approximately 20 degrees greater than the actual elbow angle.In addition, we [18] proposed a VR system that provides visual stimuli and tendon vibration in response to the force exerted by the user, giving the user the sensation of freely moving the arm even when the body is completely fixed.Leonardis et al. [3] proposed a method to present the sensation of walking in a seated position by combining tendon vibration in the legs with visual and vestibular stimuli.Ushiyama et al. [19] proposed a system that presents kinesthetic illusions of the arms while grasping a fixed bar to create the sensation of relative body tilt in combination with visual presentation using an HMD.
In addition to VR, there have been attempts to apply kinesthetic illusion to various engineering fields.For example, Marasco et al. [20] proposed a method to provide feedback of the opening and closing of a robotic prosthetic hand using kinesthetic illusion.Barsotti et al. [21] proposed a method to support the brain-computer interface based on motor imagery by using feedback through tendon vibration.
However, most of these applications use methods based on simple assumptions to control the kinesthetic illusion, such as simple vibration on/off or, even in advanced cases, vibration at a frequency proportional to the target quantity.As a result, it is either impossible to present a quantitative kinesthetic illusion such as ''the elbow bends 10 degrees'', or the quantitative potion of the sensation is compensated by other senses such as vision.

C. QUANTITATIVE PRESENTATION OF KINESTHETIC ILLUSION
Using other senses, such as vision, in conjunction with tendon vibration stimulation is not always possible in VR.For example, situations in which a moving part such as a leg or arm is out of sight are common in activities such as walking.
In addition, research on the virtual hand illusion, in which an avatar's hand is perceived as one's own, has shown that the time difference between vision and proprioception interferes with the sense of body ownership [22].Therefore, it is of great importance to quantitatively present dynamic kinesthetic illusions in VR applications.
Currently, the most promising method for quantitative control of the kinesthetic illusion is to control the sensation produced by modulating the vibration frequency, based on the fact that the speed and magnitude of the kinesthetic illusion varies with the vibration frequency [12], [13], [14].This is also based on the aforementioned physiological knowledge of the correspondence between vibration frequency and neural firing frequency [11].In fact, Albert et al. [4] have shown that it is possible to reproduce recorded motion as a kinesthetic illusion by modulating the vibration frequency based on the recorded neural firing patterns.Thyrion and Roll [5] have also achieved the presentation of three-dimensional motion trajectories by decomposing the velocity vector into the directions of motion to which the muscle spindles of each muscle are most sensitive and converting them into vibration frequency.However, although these studies were able to reproduce the qualitative characteristics of the target motion trajectory, the actual perceived motion varied greatly from subject to subject and from trial to trial.Therefore, the quantitative presentation of kinesthetic illusion remains difficult.

D. TEMPORAL CHARACTERISTICS OF KINESTHETIC ILLUSION
The temporal characteristics of the kinesthetic illusion, i.e., the characteristics of the sensation produced by time-varying vibratory stimuli, have also been studied, although to a lesser extent.Fuentes et al. [6] have shown that the temporal resolution of the kinesthetic illusion is about 0.3 seconds, based on experiments to determine the switching period during which the two directions of the kinesthetic illusion become indistinguishable when the extensor and flexor muscles are alternately stimulated for different periods.
Although not a property of the kinesthetic illusion itself, the properties of neural firing in response to actual movement and changes in muscle length have been studied extensively.For example, Matthews and Stein [23] measured the response of cat muscle spindles to sinusoidal changes in muscle length.In a study to construct a model to predict neural firing, Hasan [7] modeled the frequency of neural firing as a function of the time-varying length of muscle fibers as a nonlinear differential equation and found agreement with measured values at constant velocity or sinusoidal muscle stretch.In addition, Mileusnic et al. [24] built a physical model of muscle spindles based on anatomical and physiological findings and fitted parameters based on nerve firing data in cats.

III. THEORY A. DEFINITION OF THE TRANSFER FUNCTION
The transfer function H (s) of a linear time-invariant system is the function such that Y (s) = H (s)X (s), where X (s) and Y (s) are the Laplace transforms of the input signal X (t) and the output signal Y (t).This represents the temporal characteristics of the system.
In this study, the input signal x(t) is twice the difference in stimulation frequency in the triceps brachii (TB) and biceps brachii (BB) muscles involved in the elbow joint movement.That is, the TB and BB tendons are stimulated at frequencies respectively, where f c is the center frequency.This is based on the aforementioned experiment by Gilhodes et al. [14] and has the advantage of not using low frequencies, which are difficult to output due to the characteristics of the transducer and cause little illusion.The output signal y(t) represents the perceived elbow joint angle, with flexion being positive.When using this for presentation of kinesthetic illusion, we can obtain x(t) by applying the motion trajectory y(t) to the inverse filter of H (s), and drive the vibrators with the frequencies f TB (t) and f BB (t) determined by x(t).

B. MEASUREMENT OF THE TRANSFER FUNCTION
This study uses an experimental paradigm in which a kinesthetic illusion of reciprocating motion is presented to the participant's left arm with vibratory stimuli, while at the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
same time an actual reciprocating motion (physical stimuli) is presented to the right arm, and the participant is asked to adjust the parameters of the physical stimuli until the two sensations are identical.This differs from the experimental paradigm commonly used in kinesthetic illusion studies (e.g., [1], [14]), in which participants are asked to reproduce the perceived motion with the opposite arm.This is to avoid the delay between the participant's perception of the motion and its reproduction, and to make the measurement of the phase difference between input and output more accurate.
The vibratory stimuli presented to the left arm correspond to the input x(t) of the transfer function, and frequency modulation of the vibratory stimuli with x(t) produces a reciprocating kinesthetic illusion.The difference in vibration frequency between TB and BB, which is the input signal, is where A vib is the amplitude of the vibration frequency change and f is the frequency of the frequency change.
The physical stimulus of the right arm corresponds to the output y(t) of the transfer function.This is a sinusoidal reciprocating motion, where amplitude A phy and phase φ are adjusted by the participant.The transfer function H (s) is obtained by recording the parameters A phy and φ adjusted by the participant at different input frequencies f .

A. EXPERIMENTAL SETUP
The system configuration is shown in Fig. 1.A photograph of the experiment is shown in Fig. 2. To minimize the time delay between the vibration and the physical stimuli, all control of the vibration frequencies and the motor was performed in a microcontroller (ESP32, Espressif Systems, China).The vibration stimuli were presented by two self-developed wearable vibration modules.Each module was based on a microcontroller development kit (ATOM Matrix, M5Stack, China).The microcontroller (ESP32) in the development kit generated sinusoidal vibrations that drove a voice-coil transducer (Vp408, Acouve Laboratory, Japan) via a motor driver.Each vibration module was controlled via serial communication from the main microcontroller, with a maximum theoretical time delay of 3 ms.This is well below the reported temporal resolution of 0.3 s for the kinesthetic illusion [6].An accelerometer (MPU-6886, InvenSense, USA) built into the development kit was used to measure the vibration amplitude.In addition, a simple force sensor called a force-sensing resistor (FSR X 402, Interlink Electronics, USA) was included to measure the force applied to the participant's body.The physical stimuli were presented by driving the participant's forearm with a geared motor (4697, Pololu, USA).The motor was controlled by a proportional-integral-derivative (PID) controller based on feedback from the encoder, and the error between the  target angle and the actual angle was kept within 1 • .The mechanical backlash of this mechanism was 2.5 • .Note that a gamepad wirelessly connected to the PC was used by the participant to operate the experimental system.

B. EXPERIMENTAL PROCEDURE
Ten participants were initially recruited for the experiment, but one of them could not be confirmed to have a kinesthetic illusion during the pre-measurement check (see below), so the experiment was stopped and measurements were taken on the remaining nine participants.The experiment was approved by the ethics committee of our institution, and informed consent was obtained from all ten participants.The nine participants ranged in age from 21 to 28 years, eight were male and one female, six were right-handed and three were left-handed.However, regardless of the participants' dominant hand, the vibratory stimuli were presented to the left arm and the physical stimuli were presented to the right arm.The pressing force of the vibration modules was adjusted between 2 N and 5 N using force-sensing resistors.
After attaching the vibration module to the participant and securing the right arm to the device, appropriate vibrations were presented to each muscle and the participant was verbally checked for the production of a kinesthetic illusion in both extension and flexion directions.If no kinesthetic illusion was produced in even one direction, the vibration module was readjusted and the check was repeated.After passing the check, the vibration modules were calibrated to present vibrations of constant amplitude independent of frequency using the built-in accelerometers.
The transfer function was measured by presenting reciprocating motions of different frequencies f using kinesthetic illusions and physical stimuli, and asking participants to find the parameters A phy and φ of the physical stimuli such that the motions felt by both arms were identical.The frequency f of the motion was set to 11 steps from 0.05 Hz to 0.3 Hz in 0.025 Hz increments, and the trials were performed in a random order.In each condition, stimulation and parameter adjustment were repeated until participants judged that the sensations of the two stimuli were the same.The amplitude of the physical stimuli, A phy , was adjustable from 5 • to 30 • in 5 • increments, and the phase, φ, was adjustable from 0 • to 360 • in 10 • increments.Once the participant judged the two sensations to be identical, the similarity of the sensations in the final parameters was rated on a 7-point Likert scale (1: completely different, 7: completely the same) before moving on to the next condition.The amplitude of the vibration frequency change (A vib ) was set to 30 Hz, the center frequency of the vibration (f c ) to 80 Hz, and the root-mean-square acceleration amplitude of the vibration to 3 G (29.4 m/s 2 ).The stimulus duration was equal to two reciprocating movements (2/f [s]).To make the angle of the physical stimulus 0 • at the beginning and end of the stimulation, the phase of the vibratory stimuli (input x(t)) was delayed by φ instead of being advanced by φ to create an equivalent phase difference.During the stimulation, white noise was played through headphones and participants were instructed to keep their eyes closed during the stimulation.After the stimulation stopped, the white noise stopped, and the participants opened their eyes and followed the on-screen instructions as they adjusted the parameters.
It should be noted that the clock and notification messages on the PC screen were not hidden during the experiment.Thus, it is possible that participants, especially those who had plans after the experiment, were aware of the time spent on the experiment and were biased to finish it faster.

C. DATA ANALYSIS
The transfer function must be in a form that can be easily handled by computer programs in order to be used for quantitative presentation of the kinesthetic illusion.Therefore, in this study, a transfer function expressed as a mathematical expression in the Laplace domain was fitted to the data obtained as a table of amplitude and phase response with respect to input frequency.The fitted transfer function is This equation was obtained based on the finding that the output of the muscle spindle encodes the length of the muscle and its temporal variation [8], and further by neglecting nonlinear factors, including the relationship between muscle length and joint angle, and assuming the relationship x(t) = aẏ(t) + by(t).
Fitting was performed by converting the amplitude and phase characteristics of each participant for each condition into a phasor representation as H (s) = A phy e jφ (6) and minimizing the sum of squared errors with Ĥ (s) for all data.The function scipy.optimize.least_squaresfrom the scientific computing library SciPy [25] was used to compute this nonlinear least squares problem, and the initial values of the solution were a = 0 and b = 1.

V. RESULTS
Fig. 3 shows the amplitude response of the transfer function, i.e., the magnitude of the perceived reciprocating motion.The mean value (thick dashed line) shows a low-pass filter-like tendency (the higher the reciprocation frequency, the smaller the perceived motion).However, the fitted transfer function (thick solid line) is almost flat.Fig. 4 shows the phase response, i.e. the phase difference of the perceived reciprocating motion y(t) with respect to the input signal x(t) (vibration frequency variation).The thick dashed line represents the circular mean (calculated with SciPy's scipy.stats.circmeanfunction), i.e. the mean taking into account the periodicity of the phase (angle).Here, both the circular mean and the fitted transfer function have a phase difference of about 180 • at all frequencies.Fig. 5 shows the final similarity ratings for each condition.It can be seen that for many frequencies, the median is at an intermediate level among the seven levels.In fact, a Wilcoxon signed-rank test shows that the median of the similarity ratings is greater than 4 (p < .001,using  SciPy's scipy.stats.wilcoxonfunction, as a onesample, one-tailed test, testing that the answer minus 4 is greater than 0).
The parameters obtained by fitting Equation ( 5) were a = −0.0215and b = −0.1315.In other words, the transfer function obtained in the experiment is approximated as Figures 6 and 7 show the transfer functions obtained by performing the same fitting for each participant.The values of parameters a and b are given in Table 1.

A. ON THE MEASURED TRANSFER FUNCTION
Given the relatively good results of the similarity assessment, including one of the statistical test, the experimental tasks of matching physical motion with kinesthetic illusion were successfully performed.However, the transfer function obtained showed little frequency dependence, with most frequencies producing a phase advance of about 180 • .The   fitting results for each participant, with the exception of a few outliers, show a transfer function similar to the whole, with a constant amplitude and a phase difference of about 180 • , regardless of frequency.However, the magnitude of the amplitude response varies from participant to participant.
This result is unexpected given the neurophysiological finding that muscle spindle output encodes muscle length and its rate of change [8], and the previous study [5] that used vibration frequencies corresponding to the speed of the presented movement.Furthermore, the experiment by Gilhodes et al. [14], in which vibration stimuli were applied with a constant frequency difference that did not change over time, produced a kinesthetic illusion in the direction of muscle elongation on the side with the higher frequency.With this knowledge, the transfer function should have the characteristics of a low-pass filter and the phase difference should be about 0 • at about 0 Hz.However, the results of the present experiment indicate that a kinesthetic illusion was produced in the direction of muscle contraction on the side with the higher frequency.
A possible reason for this result is that not only the kinesthetic illusion but also the force sensation was produced, and this may have contributed significantly to the transfer function.It is known that when a muscle is contracted, not only muscle spindles but also Golgi tendon organs, which are force receptors, respond to vibration stimuli [11], [26].Since Golgi tendon organs are normally activated when the corresponding muscle is under load, the resulting illusory force is thought to be in the opposite direction of the kinesthetic illusion.In the previous study [14], participants actively moved the non-vibrated arm to reproduce the kinesthetic illusion, so the muscles of the movable arm contracted isotonically, and there was little information from the Golgi tendon organ.In contrast, in the present study, the non-vibrated arm was moved by an external force, resulting in more Golgi tendon organ activity, and this activity was compared with the Golgi tendon organ activity induced by the vibration stimulus, which is thought to have increased the contribution of force perception.
Another explanation could be the influence of the tonic vibration reflex (TVR).TVR is a phenomenon in which muscle spindles activated by vibratory stimulation cause the muscle on the stimulated side to contract [27], producing a physical (i.e., not illusory) movement in the opposite direction of the kinesthetic illusion.However, in the present study, the arm to be vibrated was placed on a table (see Fig. 2), and when the tendon of insertion is stimulated as in the present study, a reflex in the same direction as the kinesthetic illusion (antagonist vibratory response) is more likely to occur than TVR [28].Therefore, the possibility that TVR is the cause of the 180 • phase difference is small.
It is also possible that the participant simply misunderstood the instructions for the experiment and thought that the angles were reversed when comparing the sensations in the left and right arms.However, this possibility is also small because the participants were adequately warned at the beginning of the experiment.
As for the fact that the transfer functions have similar values at most frequencies, it may be that the range of reciprocating frequencies used in this experiment was too small for the frequency dependence to be apparent.

B. LIMITATIONS
It should be noted, however, that the data obtained in this experiment were obtained under limited conditions.First, this experiment measured the transfer function when a reciprocating motion illusion was presented.As mentioned above, there is a possibility that the perceptual properties of reciprocating and unidirectional kinesthetic illusions differ.In this study, we followed Gilhodes et al. [14] and applied vibration stimuli of different frequencies to two muscles in an antagonistic relationship, but a quantitative presentation method for kinesthetic illusions could also include, for example, changing the intensity of the kinesthetic illusion by changing the amplitude of the vibration.In such a case, the transfer function is likely to be different.
Furthermore, due to technical limitations in this study, the force with which the transducer is pressed against the skin was set in a relatively wide range, from 2 N to 5 N.According to Ferrari et al. [29], the pressing force affects the perceptual threshold (the smallest vibration amplitude at which the kinesthetic illusion occurs).It is not clear whether the pressing force also affected the magnitude of the illusion when the amplitude exceeded the threshold, but if so, this may have contributed to the individual differences.
It should also be noted that the number and population of participants was limited.To the best of our knowledge, there is no consensus on the number and distribution of participants in this type of experiment, but the distribution of 21 to 28 years old, 8 males and 1 female, is not sufficiently representative of the general population.Therefore, the results of this study should be considered as data for this limited population.

VII. CONCLUSION
In this paper, we experimentally measured the relationship between the frequencies of temporally varying tendon vibration stimuli and the kinesthetic illusion perceived by participants as a transfer function, in order to provide useful data for realizing a quantitative presentation of the kinesthetic illusion with tendon vibration stimuli.Vibration stimuli producing a reciprocating motion illusion at different frequencies were presented to one arm of the participant while a physical reciprocating motion was presented to the other arm.The result was a transfer function with a nearly constant amplitude response and a phase advance of approximately 180 • at reciprocating frequencies of 0.05 Hz to 0.30 Hz.This is inconsistent with previous findings and suggests the possibility of activation of the Golgi tendon organ.However, this study only measured the transfer function in a limited situation where the reciprocating motion illusion of the forearm was presented by modulating the vibration frequencies.In the future, it will be necessary to investigate more advanced models that account for nonlinearity and to evaluate kinesthetic illusion rendering using the data obtained, as well as to measure the transfer function of other body regions and other stimulation methods.

FIGURE
FIGUREConfiguration of the experimental system.

FIGURE 2 .
FIGURE 2. Photograph of the experiment.

FIGURE 3 .
FIGURE 3. Amplitude response.The thin dashed line is the data for each participant, the thick dashed line is the mean of all participants, and the solid line is the fitted transfer function.

FIGURE 4 .
FIGURE 4. Phase response.The thin dashed line is each participant's data, the thick dashed line is the circular mean, and the solid line is the fitted transfer function.

FIGURE 5 .
FIGURE 5. Similarity ratings of final sensations.The thin dashed line is each participant's rating, and the solid line is the median.

FIGURE
FIGURETransfer functions obtained by per-participant fitting (phase).

TABLE 1 .
Values of the parameters obtained by per-participant fitting.