#### 3.1. Modeling of Four Fingers

Basically, the soft sensors were designed to measure linear displacement as the finger motions can be converted to linear displacement above the finger joints. Based on this, complex finger joints were modeled and the models were used to determine the sensor positions required to measure the complex finger motion, and extract joint angles.

The PIP and DIP joints have a single DOF, and therefore do not need complicated models. Also, the MCP joint was typically modeled as a universal joint with two DOFs, including the flexion/extension and abduction/adduction motions. The location of the soft sensor was determined by the rotation axis based on the models. Thus, the positioning of the soft sensors at the MCP, and PIP joints is obvious as shown in

Figure 6a.

For the flexion/extension of the MCP, PIP and DIP joints, cylindrical models, which mean the change in circular arc was converted into the angle, were applied as shown in

Figure 6a. The length change of the sensing unit at the MCP and PIP joints (Δ

L_{FE,MCP} and Δ

L_{FE,PIP}, respectively), were converted to the joint angle through a biomechanical model that was applied to calculate tendon excursions in the extensor muscles [

15]. The flexion/extension angles of the MCP and PIP joints were calculated assuming that each joint defines a circle with radii of

${r}_{MCP}$ and

${r}_{PIP}$, respectively. Please note that the DIP joint can be modeled similarly with the MCP and DIP joints as shown in

Figure 6a, but the DIP joint angle was not directly measured because the DIP joint angle can be estimated by the PIP joint angle due to the musculoskeletal dependency [

15]. As a result, the linear displacement on the skin at the MCP and PIP joints were converted with

${\theta}_{FE,MCP}$ and

${\theta}_{FE,PIP}$, and the DIP joint angle was calculated from the PIP joint angle, as follows:

As shown in

Figure 6b, the abduction/adduction motion was modeled as an arc trajectory between the fingers, which pivots on a central point between the capitate and trapezoid, shown in

Figure 2. Because the range of the angle was relatively small, the angle was measured through the measurement of the linear displacement using (5).

As a result, Type 1 sensors were attached on each phalanges except for the thumb, for the flexion/extension of the four fingers. Type 2 sensor was attached between the adjacent proximal phalanges to measure the abduction/adduction angles.

Please note that the exact

${r}_{MCP}$ and

${r}_{PIP}$ were not needed since the strain of the soft sensor could be directly mapped with the joint angles. Namely, the linear relationship between the joint angles and the linear displacement on the joint were obtained experimentally from the calibration process, which will be discussed in

Section 5, and the joint angles were calculated linearly with the strain of the soft sensor. Thus, the joint radii were only used to derive the relationship in (2)∼(5), but they were not actually masured and used in the system.

#### 3.2. Modeling of the Thumb

The MCP and IP joints of the thumb have similar models used for the four fingers shown in

Figure 6a. As the IP joint is formed on the proximal phalanx, its shape is similar with the PIP joint of the four fingers [

17]. The MCP joint of the thumb is known to have flexion/extension as well as abduction/adduction [

18]. As shown in

Figure 7a, while the thumb CMC joint was moved in a pure abduction/adduction from normal position, the abduction/adduction of the MCP joint is highly correlated with the abduction/adduction of the CMC joint because the two joints share a musculotendinous structure [

19]. To observe the relationship between the abduction/adduction of the CMC and MCP joint, comparison of two angles were conducted. Actual angles of the abduction/adduction at the CMC and MCP joint were calculated by MoCap data, while the thumb finger was induced to have a posture of clenching a fist. The abduction/adduction angle of MCP joint was assumed to be obtained linearly to the abduction/adduction angle of the CMC joint using least-squares, as follows:

where

${a}_{1}$ and

${a}_{2}$ were identified as 0.89 and 2.89 by the least square method. The RMS error was about 5.13 deg. The fitted and measured abduction/adduction angles of MCP joint are shown in

Figure 7b.

The complex musculoskeletal structure of the CMC joint requires modeling analysis to measure the 3-D finger motion. Cooney et al. introduced three axes of rotation that are orthogonal each other with regard to the trapezium and the first metacarpal bones as shown in

Figure 8a [

18]. In general, the thumb CMC joint is a saddle joint with two DOFs between the trapezoid and the first metacarpal bone shown in

Figure 8a. Although the saddle joint usually has two DOFs, it exhibits 3 different motion: the flexion/extension, abduction/adduction and axial rotation when ligaments of the bones are lax [

18].

Although it has three axes of rotation, the axial rotation angle shows dependency on flexion and abduction. Griffin et al. introduced a model that the CMC joint has roll motion instead of the flexion/extension, separating each joint as revolute joints with offsets as shown in

Figure 8b [

20]. However, as the rotation (flexion/extension) motion is fixed to the palm, it is not suitable to express changing flexion/extension axis. Many offsets of the CMC joint, which are distances between the adjacent joints, (e.g., between

${I}_{ABD}$ and

${T}_{TR}$,

${T}_{TR}$ and

${T}_{ABD}$,

${T}_{ABD}$ and

${T}_{MC-twist}$ in

Figure 8b), are included to the model, and the offsets are widely varied depending on the size of hand. Hollister et al. introduced a model which has non-intersecting and non-perpendicular axes of flexion/extension and abduction/adduction to express the dependent axial rotation [

21]. However, as the angle between the two axes and the offset are widely variable for individuals, this model is not suitable for the proposed soft sensor system. Therefore, a model that can be applied to the different hand and optimized with the soft sensor based system was needed.

The proposed model in this paper expresses the rotation axes as epidermal notation rather than the anatomic notation because the soft sensor was attached on the skin to measure the motion. The CMC joint was modeled to have two intersecting and perpendicular rotation axes with the flexion/extension and abduction/adduction, while keeping the orthogonality of the two axes [

18]. As the amount of the axial rotation angle is relatively small (17 deg for the normal movement) and the limited space for the phalanges on thumb finger, axial rotation was neglected and the number of soft sensor was reduced [

18,

22,

23]. The model for the CMC joint without the pronation rotation was suggested as shown in the

Figure 9.

For the verification of the proposed thumb model, the position from the CMC joint to the fingertip calculated by the proposed model was compared with the MoCap data. The reference fingertip position was obtained from the MoCap data of the CMC joint to the fingertip. The position based on the proposed model was calculated from the initial position with the defined rotation axes and joint angles. The rotation axes were defined from the reflective marker data of the MoCap as shown in

Figure 10a and they are

$\overrightarrow{AXI{S}_{AA}}$ and

$\overrightarrow{AXI{S}_{FE}}$, respectively, as follows:

The rotation axis for the abduction/adduction of the MCP joint was obtained parallel to the abduction/adduction of the CMC joint, and the axes for the flexion/extension of the MCP and IP joint were obtained parallel to the flexion/extension of the CMC joint. The joint angles were also calculated by the MoCap data. After calculating the vectors from the 3-D marker data, the angles between the vectors were calculated as shown in

Figure 10b. Root-mean-square (RMS) error of the

x,

y and

z positions was about 8.88 mm, 11.71 mm and 5.48 mm, respectively, which are in tolerable range for the finger motion measurement [

4,

5,

6,

7,

24].

As a result, two Type 2 sensors were attached between the trapezium and first metacarpal and between first and second metacarpal to measure the flexion/extension and the abduction/aduction joint angles, respectively, as shown in

Figure 3. As the MCP and IP joints of the thumb have a similar musculoskeletal structure with the PIP and MCP joint of the four fingers except the phalanges’ length, two Type 2 sensors were attached for the flexion/extension of the thumb MCP and IP joint.