# Joint Angle Estimation of a Tendon-Driven Soft Wearable Robot through a Tension and Stroke Measurement

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. System Design

#### 2.1.1. Glove Design

#### 2.1.2. Actuation and Control System Design

#### 2.2. Kinematic and Stiffness Parameter Estimation

#### 2.2.1. Kinematic System Identification: The Relationship between the Joint Angle and the Fingertip Position

**{F}**should be transformed with respect to a moving frame

**{M}**, which moves along with the hand, since it is the relative position of the finger with respect to the hand that is meaningful. We defined this reference moving frame

**{M}**to be located at the back of the hand (base frame in Figure 3), since this part does not move relative to other parts in a hand while an index finger is in motion. The coordinate systems of the fixed frame

**{F}**and reference frame

**{M}**are shown in Figure 7a. Using this concept, transformation of a marker position ${\mathit{X}}_{\mathit{M}}\in {\mathbb{R}}^{3\times 1}$ from frame

**{F}**into frame

**{M}**can be written as Equation (4), using a transformation matrix ${\mathit{T}}_{\mathit{MF}}$$\in {\mathbb{R}}^{4\times 4}$. ${\mathit{X}}_{\mathit{M}}$, ${\mathit{X}}_{\mathit{F}}$, and ${\mathit{P}}_{\mathit{M}}$ are illustrated in Figure 7a, and ${\mathit{R}}_{\mathit{MF}}$ is a rotation matrix from frame

**{M}**to

**{F}**.

**{F}**, as shown in Equation (6), while ${\widehat{\mathit{x}}}_{\mathit{F}}$, ${\widehat{\mathit{y}}}_{\mathit{F}}$, ${\widehat{\mathit{z}}}_{\mathit{F}}$ can be written as Equation (5). From the relationship between the two coordinates shown in Equation (5), the position of the markers in frame

**{M}**can be expressed in a matrix form, as shown in Equation (6). Other details about coordinate transformation can be found in previous works about robotics [30]. One thing different from traditional robotics is that usually a transformation matrix shows different forms because the matrices are defined to transform a vector from a moving frame to a fixed frame.

**T**using Vicon data and matrix

**T**using the POE method in Equation (7), we can finally estimate the center of rotation in a finger joint as a function of the rotation angle. The angle of rotation $\theta $ and rotational axis $\widehat{\mathit{w}}$ can be obtained from Equation (10) and Equation (11). Here, ${r}_{ii}$ in Equation (10) is an ith component in the main diagonal of a rotation matrix and ${w}_{x},{w}_{y},{w}_{z}$ in Equation (11) are x, y, and z components of $\widehat{\mathit{w}}$. From Equation (8), the linear velocity is equal to ${\mathit{G}}^{-\mathbf{1}}\left(\mathit{\theta}\right)\mathit{p}$, while ${\mathit{G}}^{-\mathbf{1}}\left(\mathit{\theta}\right)$ can be written as Equation (9). Knowing that the linear velocity is equal to $-\mathit{w}\times \mathit{Q}$, we can determine the direction and magnitude of the $\mathit{Q}$, which points at the joint center, as in Equation (12) and Equation (13). Equation (12) and Equation (13) are based on an assumption that we are looking for the vector $\mathit{Q}$ that is perpendicular to the rotational axis $\widehat{\mathit{w}}$. This method can be applied to finding the center of the finger joints.

#### 2.2.2. Stiffness Parameter Estimation: Relationship between Tension and Joint Angle

#### 2.3. Experimental Methodology

## 3. Results

#### 3.1. Kinematic System Identification: Estimation of the Relationship between Joint Angle and Fingertip Posture

#### 3.2. Stiffness Parameter Estimation—Estimation of the Relationship between Tension and Joint Angle

#### 3.3. Grasp Posture and Range of Motion

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

MCP joint | Metacarpophalangeal joint |

PIP joint | Proximal interphalangeal joint |

DIP joint | Distal interphalangeal joint |

PoE | Product of exponential |

A.F. | All Flexor |

A.E. | All Extensor |

M.F. | MCP Flexor |

RMSE | Root mean square error |

DOF | Degree of freedom |

GPR | Gaussian Process Regression |

## Appendix A. Anatomic Expressions of the Human Hand

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**Figure 1.**Schematic view of the Exo-Index system; (

**a**) shows an external device for synchronizing load cell data measured in the MCU and posture data measured by the Vicon; (

**b**) shows overall control and actuation system for the robot; (

**c**) shows how the tension sensor is designed using loadcell; (

**d**) provides a brief look at how the three tendons of the actuator are connected to the glove.

**Figure 2.**Schematic view of the tendon routing method in the Exo-Index; (

**a**) and (

**b**) show a schematic of the flexor router, while (

**c**) and (

**d**) show a schematic of the extensor router; (

**e**) shows aspects of how the moment arm changes with respect to variation of the joint angle. Here, length of the flexor router and extensor (S) router (${a}_{i}$, ${b}_{i-1}$, ${m}_{i-1}$, and ${n}_{i}$) each are 5, 5, 3, and 3 mm, while ${m}_{i-1}$, and ${n}_{i}$ of the extensor (L) router is increased to 5 mm.

**Figure 3.**Overall view of Exo-Index: (

**a**) shows two flexors (yellow line means All Flexor (A.F.) while the white line shows MCP Flexor (M.F.)) in a hand-open position, (

**b**) shows an extensor named as All Extensor (blue line) in a hand-held position. In addition, the method of attaching vicon markers is shown in (

**c**). Three markers attached on the back side of the hand are considered as the base frame.

**Figure 6.**Schematic diagram to show purpose of kinematic identification and stiffness parameters estimation. Stiffness parameter estimation elucidates the relationship between the joint angle and the actuation data. The main purpose of stiffness parameter estimation is to obtain the relationship between the joint and the end-effector; this is highly related to kinematic analysis.

**Figure 8.**Schematic view of the experimental protocol: (

**a**) shows a systemic view of vicon and the control system. By transmitting a sync signal to the ADC converter of the Vicon system, the loadcell data and encoder data were synchronized with the Vicon data. (

**b**) shows the location where vicon markers are attached. A Total 14 markers are used in this experimental setup; 12 markers are used for the index finger and two markers are attached to the thumb to measure the thumb position.

**Figure 9.**Experimental results about the position of the finger joints found using a vicon motion capture system. (

**a**) The marker position used to measure the joint position. With this marker setup, the position of the MCP, PIP, and DIP joint are measured, as shown in (

**b**–

**d**). Since the joints move as the joint angle changes, the positions of the joints are expressed in terms of the angle of the joints.

**Figure 10.**Experimental results of the stiffness parameter estimation that shows a relationship between the joint angle and the wire tension. (

**a**,

**c**,

**e**) are plots of the ground truth angles measured by Vicon cameras and estimated angles using the wire tension and wire stroke. (

**b**,

**d**,

**f**) are comparisons between the estimation and the ground truth. The RMS error indicates the disparity from the y = x relationship.

**Figure 11.**Results of estimation that shows relationship between joint angle and wire tension. (

**a**) shows comparison of the ground truth angle with the estimated angle from data-driven method and the estimated angle from model-driven method; (

**b**) and (

**c**) each show response plot of the data-driven method and model-driven method respectively.

**Figure 12.**Various postures with the assistance of Exo-Index. With assistance of the proposed robot, three types of grasp postures are established depending on the object size and shape. (

**a**) and (

**d**) show the posture using the Exo-Index. For more detail, (

**b**) shows how card is grasped using narrow pinch while (

**e**) shows how glue is grasped with caging. In addition, (

**c**) and (

**f**) show wide pinch and the grasped objects are paper cup and bottle.

**Figure 13.**Workspace of the distal phalange under three different motions. Black workspace indicates a kinematically available workspace which is calculated by all possible values in the ROM of the MCP, PIP, and DIP joints. Blue workspace indicates a workspace which is measured by a spontaneous motion of a non-disabled person. The workspace reached by actuating Exo-Index is represented in red. The X and Y axes are coordinates located in the MCP joint and are co-planar with the finger, where X axis is the direction of the proximal phalange when MCP is at zero position and Y axis points at the palmar direction.

**Table 1.**Index finger joint position (mm) and finger phalange length (mm). G in the table indicates a gradient of the regression result and the Y means Y-intercept of the regression.

Joint | X | Y | Z | Length | ||||||
---|---|---|---|---|---|---|---|---|---|---|

G | Y | RMSE | G | Y | RMSE | G | Y | RMSE | (mm) | |

MCP | −6.05 | 48.78 | 9.19 | −16.04 | 23.73 | 5.37 | −0.27 | −18.13 | 5.97 | 42.64 |

PIP | −2.72 | 33.58 | 0.92 | 1.91 | −1.94 | 1.40 | −0.13 | −20.64 | 0.80 | 20.14 |

DIP | −0.71 | 20.41 | 0.97 | 0.26 | 2.05 | 1.71 | 0.13 | −19.19 | 0.83 | 18.72 |

**Table 2.**Range of motion (ROM) and workspace of each motion. In the table, K.M. means kinematically possible motion, S.M. means spontaneous motion, and A.M. means assisted motion. Since concept of K.M. is only used in workspace analysis, the ROM of K.M. is blank.

MCP (rad) | PIP (rad) | DIP (rad) | Workspace (mm^{2}) | ||||
---|---|---|---|---|---|---|---|

Max | Min | Max | Min | Max | Min | ||

K.M. | - | - | - | - | - | - | 8712.30 |

S.M. | 1.34 | −0.35 | 1.56 | 0.08 | 0.92 | 0.07 | 5884.10 |

A.M. | 32.90 | 21.27 | 0.88 | 3.11 | 4.55 | 1.48 | 3770.80 |

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## Share and Cite

**MDPI and ACS Style**

Kim, B.; Ryu, J.; Cho, K.-J.
Joint Angle Estimation of a Tendon-Driven Soft Wearable Robot through a Tension and Stroke Measurement. *Sensors* **2020**, *20*, 2852.
https://doi.org/10.3390/s20102852

**AMA Style**

Kim B, Ryu J, Cho K-J.
Joint Angle Estimation of a Tendon-Driven Soft Wearable Robot through a Tension and Stroke Measurement. *Sensors*. 2020; 20(10):2852.
https://doi.org/10.3390/s20102852

**Chicago/Turabian Style**

Kim, Byungchul, Jiwon Ryu, and Kyu-Jin Cho.
2020. "Joint Angle Estimation of a Tendon-Driven Soft Wearable Robot through a Tension and Stroke Measurement" *Sensors* 20, no. 10: 2852.
https://doi.org/10.3390/s20102852