Advance Publication by J-STAGE
Journal of Biomechanical Science andEngineering

DOI:10.1299/jbse.20-00202

Received date : 17 April, 2020
Accepted date : 7 August, 2020
J-STAGE Advance Publication date : 20 August, 2020

© The Japan Society of Mechanical EngineersEffective position of the rotation axis of an ankle stretchingmachine and the effect of misalignment
Yuma SHIRAISHI*, Shogo OKAMOTO*, Naomi YAMADA*,**, Koki INOUE*, Yasuhiro AKIYAMA*
and Yoji YAMADA*∗ Graduate School of Engineering, Nagoya University

Nagoya, Japan
E-mail: shogo.okamoto@mae.nagoya-u.ac.jp∗∗ Aichi Medical College for Physical and Occupational Therapy

Kiyosu, Japan

Abstract
The mechanical rotation axes of joint exercisers are believed to operate better when they match the biomechanicalaxes of human joints. However, there are few studies regarding ankle stretching machines. Further, the maleﬀects
of rotation axis misalignments are not well known. Hence, we investigate the eﬀective positions of rotation axes forankle stretching machines and the eﬀects of misalignments using a pneumatic-driven stretching machine developedin our previous study (Shiraishi et al., 2020). Eight healthy young males (ages 23.3 ± 1.4 years) participated instretching exercises while the relative positions of the rotation axes between the machine and ankle were changed
via plates installed under the heel. The stretching machine dorsiﬂexed the feet of the participants, and the dorsiﬂexion angles and three-axial forces applied to the forefeet were recorded. The measured values at the maximum
dorsiﬂexion angle were evaluated by two-way analysis of variance and/or regression analysis. We determined thatthe rotation axis of the machine must be placed 7 mm above the lateral ankle because the normal force applied
to the forefoot and maximum dorsiﬂexion angle were large, whereas the friction force was moderate. Further,
the relationships among the dorsiﬂexion angle and contact forces were investigated via covariance selection. The
three-axial forces signiﬁcantly decreased as the axis of the machine was lowered below the ankle. Additionally,
the force normal to the sole had large positive eﬀects on the dorsiﬂexion angle and friction force of the sole, which
could damage the skin. The misalignment of the rotation axis increased the contact force at the sole when the axisof the machine was above the ankle or decreased the eﬃciency of force transmission from the stretching machine
to the user’s foot when the machine’s axis was below the ankle.
Keywords : Ankle contracture, Stretching, Ankle stretching machine, Rotation axis, At-home rehabilitation

1. Introduction
Ankle plantarﬂexion contracture is a common aftereﬀect of ankle fractures, stroke, and cerebral palsy, and it reduces
the elasticity of the soft tissues comprising the muscles and tendons around the ankle, thereby limiting the range of motion(ROM) of the ankle. Reports have estimated that 77% of ankle-fracture patients generally suﬀer from ankle contractures
immediately after cast removal (Pun et al., 1991), and approximately 84% of craniocerebral palsy patients (Yarkony and
Sahgal, 1987) and 50% of stroke patients (O’Dwyer et al., 1996) are also prone to these eﬀects. Stretching is the primary
physical therapy for treating ankle contractures and is eﬀective in improving the ROM of the ankle and reducing ankle
stiﬀness (Bressel and McNair, 2002; Medeiros and Martini, 2018; Harvey et al., 2017; Radford et al., 2006; Mizuno etal., 2013). Ankle contractures typically continue to progress after the patient is discharged from the hospital, so long-term
stretching is essential. However, medical insurance policies restrict the opportunities for rehabilitation after discharge,
and patients encounter diﬃculties in receiving long-term care from physical therapists in some countries. Therefore,
stretching machines developed for domestic usage are expected to address these problems.

© The Japan Society of Mechanical EngineersThus far, a few ankle stretching machines have been developed for treating or preventing ankle plantarﬂexion contractures. Machines for ankle dorsiﬂexion were developed using electromagnetic or linear motors (Waldman et al., 2013;
Ren et al., 2017; Toda et al., 2016, 2019; Zhou et al., 2016). Yamada et al. developed a machine for 3D foot motion for
dorsiﬂexion, abduction, and eversion using McKibben-type pneumatic actuators and wire-driven mechanisms (Yamada etal., 2017; Kimura et al., 2017). The authors of the present study have also developed an ankle stretching machine by modifying a commercial foot exerciser with a pneumatic actuator (Shiraishi et al., 2020) and demonstrated its eﬀectivenessin decreasing the passive dorsiﬂexion resistance in young healthy subjects. In the present study, we use this developed
machine. In addition, ankle rehabilitation machines for non-stretching purposes have been developed to prevent disuse
syndrome or thrombus. Homma et al. developed a machine for ankle plantar/dorsal ﬂexion using an electromagnetic
motor to ameliorate blood circulation (Homma et al., 2011). Other studies have developed ankle exercise machines using
pneumatic actuators (Sasanuma et al., 2018; Saga and Saito, 2008).
In this study, we focus on the rotation-axis position of an ankle stretching machine. It is believed that the rotation
axes of ankle rehabilitation machines should be aligned with the ankle joints of the users. However, accurate localizationof the biological rotation axis is visually impossible from external appearance, and the inclination may change during
ankle plantar/dorsiﬂexion (Hicks, 1953; Barnett and Napier, 1952). In other words, misalignments may necessarily occur
when using stretching machines. Studies on the eﬀects of misalignments of the rotation axes of the machine and biologicaljoints of the elbows, shoulders, and knees (Akiyama et al., 2012, 2015a, 2015b; Schiele and Van der Helm, 2006, Jarrasseand Morel, 2012; Zanotto et al., 2015) indicate that the contact forces increase in an undesirable manner. However, studies
using ankle stretching machines and those regarding the eﬀects of misalignments on stretching have not been conducted.
Although some believe that the eﬀects of misalignments are geometrically inferred because of the complex biomechanicalsystems, experimental measurements are necessary.
A study on the rotation axes of ankle stretching machines (Toda et al., 2019) investigated the position of the rotation
axis of the machine using a machine that involves substantially diﬀerent mechanisms from the one used in this study.
Toda et al. reported that the rotation axis (of the machine) positioned at the back of the ankle was the optimal position
considering that the stretching of the machine was similar to the techniques utilized by physical therapists. In this study,
however, we investigate the rotation axis of the ankle stretching machine from the perspective of the stretching eﬀect.
We previously investigated the eﬀects of the relative positions of the rotation axes between the ankle stretchingmachine and ankle in the sagittal plane using our pneumatic-driven ankle stretching machine. The results suggested thatthe force that did not contribute to ankle dorsiﬂexion diminished the stretching eﬀect (Shiraishi et al., 2019a). However,
the relationship between the dorsiﬂexion angle and force applied to the foot was not investigated. This study, therefore,
investigates the relationships between the three-axial forces applied on the foot and the dorsiﬂexion angle while changing
the relative positions of the axes. We then discuss the rotation axis positions of the ankle stretching machine for ensuringeﬀective stretching and lowering the risk of skin damage, as well as the eﬀects of misalignments of the rotation axes.
The experiments were conducted with the approval of the Institutional Review Board of the School of Engineering,
Nagoya University (#18-2).

2. Ankle stretching machine
The stretching machine (Fig. 1) was based on the chassis of a commercial foot exercise machine for ankle dorsi/
plantarﬂexion (Shiraishi et al., 2020). When the bellows (i.e., pneumatic actuator) were pressurized, it expanded in thelongitudinal direction, and the foot was dorsiﬂexed with rotation of the moving part of the machine (Fig. 2). The machine
repetitively moved the foot in the dorsi/plantarﬂexion direction within the ROM of the ankle. The air pressure was output
from an air compressor (CP-12Si, Minato Electrical Co., Ltd., Japan), and controlled using an electro-pneumatic regulator(ITV1030-312S, SMC Co., Ltd., Japan) and a microcomputer (mbed LPC1768, Arm Holdings plc, UK).
In the present study, the moving part of the machine lifted the forefoot using a foot contactor plate (7 × 2.5 cm)
attached to the eminence. This enabled measuring the contact force between the foot and machine, as will be describedlater in Section 3.3. Without this contactor, the force from the moving part would be applied to the entire forefoot. The
heel weakly contacted with the machine’s moving part. Hence, the force from the machine was primarily transmitted to
the forefoot and secondarily to the heel. The reaction force was largely concentrated on the posterior part of the calf.
The safety functions of the machine, although not the focus of this study, are brieﬂy mentioned here: To preventthe application of excessive forces on the foot, the output was restricted using a valve (JR-08, Astroproducts, Japan)
and an electro-pneumatic regulator. Additionally, the ROM of the ankle was restricted using mechanical limiters, thereby© The Japan Society of Mechanical EngineersFig. 1

Prototype of stretching machine for ankle dorsiﬂexion. The foot dorsiﬂexed slowly and maintained thedorsiﬂexed posture. Adapted from (Shiraishi et al., 2019b).

Fig. 2

Operational principles. Left: Initial stage. Right: Pressurization causes the bellows to expand
longitudinally, causing the moving part of the machine to rotate around the rotation center. Adaptedfrom (Shiraishi et al., 2019b).

preventing the ankle from moving out of the natural ROM by the excessive loads applied on the foot. The above-mentioned
upper limits of air pressure and the ROM of the ankle were determined based on the individual ankle stiﬀness and theROM. The machine was operated using an industrial three-position enabling switch (A4EG-BM2B041, OMRON Co.,
Ltd., Japan). While the user gripped the switch, the bellows continued to be pressurized and the foot was dorsiﬂexed.
When the user released or strongly gripped the switch, the air pressure in the bellows was released and the foot slowly
returned to the relaxed foot posture. Thus, stretching was continued only if the user intended so. The safety measures aredetailed in (Shiraishi et al., 2020).

3. Experiments
Stretching was performed using our stretching machine under diﬀerent conditions of relative rotation axis positionsof the machine and the ankle in the longitudinal axis of the lower leg. During stretching, the ankle dorsiﬂexion angle andthree-axial contact forces were measured, and the details are presented subsequently.
3.1. Participants
Eight healthy male participants in their twenties, with no injuries in their lower legs and feet, participated in theexperiments. The data of ﬁve participants among the eight were used in our previous study (Shiraishi et al., 2019a).
3.2. Experimental setup
The relative rotation axis positions between the stretching machine and the lateral ankle were changed by installing
plates of 7 mm thickness (Fig. 3). In Case 1 and 2, the rotation axis of the machine was 14 and 7 mm above the lateralankle, respectively; in Case 3, it was aligned with the lateral ankle; in Case 4–7, it was 7–28 mm below the lateral ankle. A
potential problem caused by this method is discussed in Section 4.4. The relative rotation axis position was determined at
5◦ of the ankle plantarﬂexion angle. To adjust to individual diﬀerences, the experimenter carefully selected the thicknessof the base plate beneath the sole; the 7 mm-thickness plates were placed upon this base plate. By changing the position of
the machine’s rotation axis relative to the human ankle, the direction of force received by the forefoot was also changed.
This change in contact force is considered to be attributable to the change in the maximum dorsiﬂexion angle as the© The Japan Society of Mechanical EngineersFig. 3

Rotation axis position of the machine relative to the ankle. Modiﬁcation from (Shiraishi et al., 2019b).

Fig. 4

Coordinate system of the contact force. x-axis is from medial to lateral; y-axis is from the heel to theforefoot; z-axis is from bottom to top.

Fig. 5Angle (ϕ) between the upper surface of the force sensor and the sole. Left: ϕ is 0◦ . Right: upper surfaceof the force sensor is inclined from the sole at ϕ deg. This is because the relative positions of the rotation
axes between the stretching machine and the ankle was changed by installing plates under the heel in ourstudy. Note that neither of these ﬁgures represent any case numbers.

direction of the applied force is a primary parameter of manual stretching (Yamada et al., 2014). Further, the dorsiﬂexion
angle and contact force may interact with each other.
3.3. Measurement of ankle dorsiﬂexion angle and forces
The ankle dorsiﬂexion angle was measured using two three-axial accelerometers (MMA7361, Freescale Semiconductor, USA). Each sensor was attached on the instep of the foot and the tibia, where the skin does not deform largely
during ankle movement. The variations in the inclination angles of the two sensors were measured, and a 0◦ dorsiﬂexion
angle was determined in the standing posture. The forces applied on the forefoot were measured using a three-axial force
sensor with a built-in ampliﬁer (USL08-H18-1KN-AP, TEC GIHAN Co., Ltd., Japan) (Fig. 3). The sensor was in contact
with the sole through an acrylic plate (70 × 25 × 3 mm). The positive directions of the forces were deﬁned as shown in
Fig. 4: x-axis is from the medial to the lateral side of the foot, y-axis is from the heel to the forefoot, and z-axis is fromthe sole to the instep of the foot. The relative position between the rotation axis of the machine and the lateral ankle waschanged by placing plates beneath the heel, thereby introducing diﬀerences in the inclination angles of the sole and uppersurface of the force sensor (acrylic plate) (Fig. 5). To calculate ϕ, a three-axial accelerometer was attached on the moving
part of the machine and the diﬀerences in angle between the moving part of the machine and the sole were calculated.
The contact force f ( = [ f x fy fz ]T ) was obtained by computing the rotation of the output of the force sensor f0 ( = [ f x0
fy0 fz0 ]T ):f = R x (ϕ) f0 ,(1)

where R x (ϕ) is a rotation coordinate transformation matrix around x-axis. The contact force at the dorsiﬂexion angle of5◦ was the base: f = [0 0 0]T . For example, the contact force components at the maximum dorsiﬂexion angle for Case 1© The Japan Society of Mechanical EngineersFig. 6

Results of typical stretching. After the start of stretching, dorsiﬂexion angle increased, and x-axial force
decreased and y-/z-axial forces increased simultaneously. The magnitude of all the forces attained the
maximum value at the maximum dorsiﬂexion angle.

were f x = −4.9 ± 0.5, fy = 16.0 ± 1.5, and fz = 68.2 ± 3.6 (N) (mean ± standard error).
According to Mao et al. (2017b), an inherently safe threshold of tangential traction for skin abrasion caused byrepeated rubbing motion is 40 kPa for the continuous stimulation of 300 s, which was the duration adopted in our previous
study (Shiraishi et al., 2020). Considering the surface area of the forefoot plate (1.75 × 10−3 m2 ), this threshold value
corresponds to fy = 70 (N), which is substantially greater than the value observed in the experiment. When using the
setup in the present study, none of the participants claimed any skin problems after the experiments.
3.4. Tasks
The participants sat on a chair in a relaxed state and wore the ankle stretching machine with their knees fully extended.
They wore a sock to prevent skin damage, and the foot was secured to the moving part of the machine with cloth bands tothe extent of non-excessive pressure applied on the foot. The calf was supported by the cushion attached to the machine,
as shown in Fig. 1.
The participants controlled the three-position enabling switch, and the foot was dorsiﬂexed. The initial foot posture
was slight plantar ﬂexion. The operational speed was less than 4◦ /s during ankle dorsiﬂexion. The participants released
the switch when they felt the calf muscle had stretched. One operation typically spent 30 s. Each participant conductedfour trials for each case. Considering the eﬀects of measurement order, two trials were ﬁrst conducted from Cases 1–7,
and two trials were then conducted from Cases 7–1.
3.5. Results
An example of the ankle dorsiﬂexion angle and three-axial forces applied on the foot is shown in Fig. 6. As the footwas dorsiﬂexed, the dorsiﬂexion angle increased slowly (Fig. 6 (a)) and the magnitude of each force increased (Fig. 6(b)–(d)). Similar trends were observed in most cases and participants. For the following analysis, the dorsiﬂexion angle
θ and the forces − f x , fy , and fz , which were all measured in the maximum dorsiﬂexed posture, were used; − f x was usedbecause f x was measured as negative. Fig. 7 shows θ, − f x , fy , and fz for each case after normalization with the mean value
among all trials for individuals, such that the individual mean was 1. Fig. 8 shows the mean contact forces for each case.
To investigate the variation of θ, − f x , fy , or fz between the cases, a two-way analysis of variance was performed© The Japan Society of Mechanical EngineersFig. 7

Means and standard errors of the dorsiﬂexion angles (θ) and forces applied to the foot (− f x , fy , fz ) in the
maximum dorsiﬂexion position. θ in Case 2 was signiﬁcantly larger than that in other cases and exhibitedinteraction with individuals. Three-axial forces signiﬁcantly decreased with the case number. ∗, ∗∗ and∗∗∗ represent p < 001, 0.05, and 0.001, respectively.

(a)

(c)

(b)

5
݂ (N)

െ݂ (N)

݂ (N)

50
10

௭

௬

௫

0

0
1

2

3

4

5

6

7

Above ankle ĸ Case ĺ Below ankle

Fig. 8

0
1
2
3
4
5
6
7
Above ankle ĸ Case ĺ Below ankle

1
2
3
4
5
6
7
Above ankle ĸ Case ĺ Below ankle

Means and standard errors of contact forces for each case.

using all cases and individuals as factors. In this analysis, the measured values before being normalized were used. The
F-values and p-values are listed in Table 1. The results of all quantities exhibited the eﬀects of cases, individuals, andinteraction.
Post-hoc tests were performed as follows:
As shown in Fig. 7 (a), the maximum dorsiﬂexion angle θ in Case 2 was markedly larger than that in other cases.
Therefore, Case 2 and other cases were compared via a two-way analysis of variance with no adjustment of p-values using
the cases (Case 2 and other cases) and individuals as the factors. The F-values and p-values are listed in Table 2. The
results exhibited the eﬀects of cases, individuals, and interaction for all the comparisons.
As shown in Fig. 7 (b)–(d), the forces − f x , fy , and fz decreased with the case number. The signiﬁcance of these trends
was investigated via a single regression analysis as a post-hoc test. In this analysis, normalized force values were used
to cancel the baseline diﬀerences between the participants. The results exhibited signiﬁcant trends of f x (t(223) = 6.4,
p = 1.1 × 10−9 ), fy (t(223) = 3.1, p = 2.5 × 10−3 ), and fz (t(223) = 4.9, p = 2.0 × 10−6 ).

4. Discussion
4.1. Causalities between dorsiﬂexion angle and forces
As noted in Section 3, it was found that the contact forces ( f x , fy , and fz ) and θ varied with the position of themachine’s rotation axis. A reasonable speculation for this is that the change in the rotation axis position varies the© The Japan Society of Mechanical Engineers

Table 1

Results of two-way analysis of variance: maximum dorsiﬂexion angle θ and applied forces f x , fy and fz
θ

FactorIndividuals
CasesInteraction
Table 2F(6, 144)
31.9
3.2
4.4

F(6, 144)
92.7
15.2
2.5

p
2.4×10−54
7.6×10−14
1.7×10−5fy
F(6, 144)
54.8
9.0
2.4

fz
p
3.8×10−40
1.6×10−8
3.0×10−5

F(6, 144)
147.0
11.2
3.5

p
3.3×10−68
1.7×10−10
5.6×10−9

Results of two-way analysis of variance: post-hoc test for maximum dorsiﬂexion angle θ compared with
Case 2
FactorIndividuals
CasesInteraction
FactorIndividuals
CasesInteraction

Fig. 9− fx
p
7.0×10−28
5.9×10−3
3.7×10−12

Case 2–1
F(1, 48)p
11.3
2.4 × 10−8
5.6
2.2 × 10−2
3.3
6.2 × 10−3
Case 2–5
F(1, 48)p
10.9
3.8 × 10−8
7.9
7.0 × 10−3
3.4
5.4 × 10−3

Case 2–3
F(1, 48)p
12.4
6.3 × 10−9
13.0
7.3 × 10−4
4.5
7.0 × 10−4
Case 2–6
F(1, 48)p
8.4
1.1 × 10−6
25.5
6.8 × 10−6
10.0
1.2 × 10−7

Case 2–4
F(1, 48)p
12.5
5.2 × 10−9
9.6
3.3 × 10−3
4.9
3.1 × 10−4
Case 2–7
F(1, 48)p
5.9
6.0 × 10−5
9.2
3.9 × 10−3
11.4
1.9 × 10−8

Causalities of dorsiﬂexion angle and three-axial forces applied on the foot at the maximum dorsiﬂexion
position. The values are partial correlation coeﬃcients. Red and blue arrows represent positive andnegative eﬀects, respectively. Black arrow represents partial correlation.

direction of the contact force at the forefoot and inﬂuences θ. Here, we investigate these relationships statistically.
Covariance selection was performed to clarify the relationship between θ, − f x , fy , and fz (Dempster, 1972). Startingwith the complete model that assumes meaningful connections between all variables, we removed the connections whose
partial correlations were small in the order and repeated this procedure unless the reduced model became diﬀerent from thecomplete model in the meaning of covariance matrix of four variables. The mean values for each case and participant were
calculated, and 56 samples (8 participants × 7 cases) were used. The reduced model obtained ﬁnally is shown in Fig. 9.
Whereas θ and f x exhibited a negative partial correlation, the other connections exhibited positive partial correlations.
The causalities of these variables were inferred as follows: fy and fz exhibited a positive partial correlation; fy and fz
exhibited a positive eﬀect on − f x ; − f x and fz exhibited negative and positive eﬀects on θ, respectively. These causalities
were interpreted as follows:
The measured forces could mainly be the reaction forces of ankle plantar ﬂexors and tendons. When the foot is dorsiﬂexed, the plantar ﬂexors generate a resistance in the plantarﬂexion direction. In other words, passive ankle dorsiﬂexion
applies a reaction force in the plantarﬂexion direction at the forefoot (red arrow in Fig. 10). At this time, the forces in
the positive direction of fy and fz are applied on the foot as the reaction force. Therefore, it is reasonable that fy and fz
exhibited a positive partial correlation.
During stretching for ankle dorsiﬂexion, the triceps surae mainly stretched, and so did the tibialis posterior. Tibialis
posterior is the main muscle of ankle inversion, as well as a collaborative muscle of plantarﬂexion, because it runs
diagonally around the ankle (Fig. 11). Therefore, when tibialis posterior stretches during ankle dorsiﬂexion, the tensionof the muscle works for both inversion and plantarﬂexion. In our experiments, the force in the inversion direction wasmeasured as − f x . Therefore, from an anatomical perspective, it is reasonable that fy and fz exhibited positive eﬀects on− fx .
It is clear that fz exhibits a positive eﬀect on θ because of the large contribution of fz to ankle dorsiﬂexion. In thestretching posture, fz mainly increases θ in the sagittal plane. However, fy could indirectly aﬀect θ through fz and could
still damage the skin.

© The Japan Society of Mechanical EngineersFig. 10Force generated by plantar ﬂexors in the sagittal plane. Red arrow represents the reactive force of theplantar ﬂexors, and black arrow represents the force applied on the foot in the positive direction of fy andfz .

Fig. 11

Tibialis posterior of right foot. Left: side view from medial side. Right: view from the heel. The picture
shows the part of the muscle whose originating location is actually at the crural interosseous membrane
and the adjacent face of the tibia and ﬁbula.

Fig. 12

Coupling motion of ankle. Abduction and eversion are coupled with dorsiﬂexion, thereby increasing the
maximum dorsiﬂexion angle.

The negative eﬀect of −fx on θ could be because of the disturbance of −fx for dorsiﬂexion. Ankle dorsiﬂexion occurs
coupled with eversion and abduction (Fig. 12), whereas plantarﬂexion occurs coupled with inversion and adduction. −fx
represents the force in the inversion direction, and disturbs eversion, which prevents ankle coupling motion and decreases
the maximum dorsiﬂexion angle. Therefore, it is understandable that − f x negatively aﬀects θ.
From the above analysis, fz largely determines θ. In this sense, the higher position of the machine’s rotation axis isfavored because fz increases with decrease in the case number. However, − f x and fy , which directly or indirectly disturb
the increase in θ, also increase with the case number. Hence, Case 2 where these force components are balanced wellmight have resulted in maximum θ.
4.2. Effective position of the rotation axis of the machineFirst, we focus on the stretching eﬀect. The causalities in Fig. 9 indicate that fz increases θ. Case 2, in which bothfz and θ are large, is determined the best for eﬀective stretching (Fig. 7 (a), (d)). Note that the eﬀective position has
individual diﬀerences because of the interactions between cases and individuals. Next, we focus on the shear force fy
during stretching as an index of risk of skin damage. fy exhibited a positive partial correlation with fz and signiﬁcantly
decreased with the case number (Fig. 7 (c)). Comparing the safe thresholds of tangential traction (Mao et al., 2017b), i.e.,
40 kPa, the mean fy values of all the cases were observed to be low. For example, the mean maximum fy was 16.0 N for© The Japan Society of Mechanical EngineersFig. 13

Position of the rotation center of the ankle. It can be located at the midpoint of the lateral and medial
ankle, or at a slightly lateral side of the midpoint.

Case 1 (Fig. 8), and this corresponded to a traction of 9.14 kPa by considering the area of the forefoot contactor used in
the study. The maximum force in one trial for Case 1 reached 44.0 N, leading to a traction of 28.6 kPa, which was still
below the safe threshold. Hence, we may determine the best case on the basis of θ, and Case 2 is suggested to be the mosteﬀective position.
The abovementioned discussion did not consider − f x . The causalities showed that − f x decreases θ. Following thistrend, θ should have increased with the case number because − f x signiﬁcantly decreased with the case number. However,the partial correlation coeﬃcient between − f x and θ was as small as −0.08 and − f x had only an insigniﬁcant eﬀect on θ.
Further, we consider the anatomical validity of Case 2—in which the rotation axis of the machine was positioned 7
mm above the lateral ankle—which was determined as the most eﬀective position. Studies on anatomical ankle rotation
axis have shown that its position changes with ankle motion, and it is inclined in the dorsiﬂexed posture to pass throughthe medial and lateral tip of the ankle (Hicks, 1953, Barnett and Napier, 1952). Additionally, it has been reported that therotation center of the ankle is at the midpoint of the medial and lateral tip of the ankle or slightly lateral side (Lundberg
et al., 1989) (Fig. 13). In other words, the rotation center of the ankle is positioned a little above the lateral ankle, and the3D ankle motion occurs around this rotation center. In our study, the heights of the medial and lateral tip of the ankle were
measured at the standing position, and their midpoint was deﬁned as the rotation center. Consequently, the rotation centeris estimated to be positioned 6.8 ± 2.2 mm (mean ± standard deviation) above the lateral ankle. This height approximately
matches the rotation axis position in Case 2. Therefore, in Case 2, the rotation axis of the machine can pass through thenear position of the rotation center of the ankle. This is consistent with the existing belief that the rotation axis of themachine should match that of the ankle.
4.3. Effect of misalignment of the rotation axes
As shown in Fig. 7, when the rotation axis of the machine was positioned above the lateral ankle, the shear force fy
was large and risk of skin damage was high. However, in the present study, we did not observe a tangential traction greaterthan the safe threshold estimated by (Mao et al., 2017b). When the rotation axis of the machine was positioned below thelateral ankle, the contact force applied on the foot was relatively small. Although the small shear force ( fy ) decreases therisk of skin damage, the small normal force fz decreases the stretching eﬀect.
Because of misalignment of the axes, the force from the machine might have not been eﬀectively transmitted toachieve foot dorsiﬂexion. For example, it was observed that the leg was lifted slightly during operation in cases withhigher numbers. The phenomenon of ﬂoating leg has been previously reported in a study on ankle stretching machines(Toda et al., 2019), where the authors showed that the relative positions of the rotation centers of the machine and ankle
result in ﬂoating of the Achilles tendon, i.e., leg. In our experimental setup, as shown in Fig. 8, the resultant forces at
the forefoot were smaller for the large case numbers than those for the small case numbers. This indicates that a part of
the generative force of the machine exerted to parts other than the forefoot, i.e., heel, when the case number was large.
The machine’s force was partly used to push the sole near the heel toward the frontal direction. Although we did notmeasure the force at the heel, this change could be subjectively felt. Such forces near the heel contributed only slightly
to the dorsiﬂexion of the foot because of the small moment arm. In other words, when the misalignment was signiﬁcant,the generative force was not eﬀectively utilized for foot dorsiﬂexion. Studies regarding wearable physical assistant robots
have similarly reported that misalignments increase the unnecessary contact forces (Akiyama et al., 2012, 2015a, 2015b;
Schiele and Van der Helm, 2006; Schiele, 2008).
© The Japan Society of Mechanical EngineersFig. 14

Geometrical relationship between the forces applied to the foot and the relative position of ankle androtation axis of the stretching machine. Rotation axis of the machine (a) matches the ankle, (b) is below
the ankle: fy is smaller and fz is larger, and (c) is above the ankle: fy is larger and fz is smaller. Note thatthe resultant force applied to the foot is assumed to be the same for all conditions.

Finally, we consider the eﬀect of misalignment from the perspective of geometrical relationship. As shown in Fig. 14(b), when the rotation axis of the machine is below the ankle, The direction of the force at the forefoot is slightly towards
the back compared with that for the matched case in (a). Contrarily, as shown in Fig. 14 (c), when the rotation axis of themachine is above the ankle, The direction of the force at the forefoot is slightly towards the front compared with that forthe matched case. This trend was mostly true for our experiments, as indicated by Fig. 8.
4.4. Limitations of the study
As described in Section 3.2, we inserted plates beneath the heel to change the relative positions between the ankle
and machine’s rotation centers. This manipulation also changed the relative positions between the machine and theparticipants’ lower legs that received the force reactions to stretch the ankles. We cannot decouple these two types ofchanges and specify the eﬀects of the latter changes on the contact forces at the forefoot and ankle dorsiﬂexion angles.
The human calf area is not cylindrical but complex, and the changes in the contact between the calf and machine might
have inﬂuenced the results of our experiment.
Further, our discussion and conclusions were based on the contact forces and foot angles at the maximum dorsiﬂexion
position, and we do not have the data during the motions. Such data may tell us how the foot is dorsiﬂexed and help us
specify eﬀective stretching measures with low power consumption.

5. Conclusions
We investigated the eﬀects of the rotation axis positions of a previously developed ankle stretching machine on
the dorsiﬂexion angles and contact forces at the forefoot. The eﬀective rotation axis of the machine is suggested to bepositioned 7 mm above the lateral ankle as in Case 2 for relatively small shear forces and large dorsiﬂexion angles. This
position can be close to the anatomical rotation center of the ankle, which corroborates the existing belief that the rotation
axis of the machine should be aligned with that of the ankle. Further, when the rotation axis of the machine was positioned
above the lateral ankle in Case 1, the contact force at the forefoot was large. This is a common eﬀect of misalignmentof the rotation axis, known from studies on wearable physical assistant robots. On the other hand, when the rotation axisof the machine was positioned below the ankle in Case 3–7, the forces applied on the forefoot were small. In this case,the misalignment decreases the eﬃcacy of both force transmission and stretching eﬀect. In a future study, this suggested
position of the machine’s rotation axis should be tested in clinical scenarios.

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