# Constraint Study for a Hand Exoskeleton: Human Hand Kinematics and Dynamics

Authors: Fai Chen Chen; Silvia Appendino; Alessandro Battezzato; Alain Favetto; Mehdi Mousavi; Francesco Pescarmona
Journal: Journal of Robotics
 (2013)

---

## Abstract

In the last few years, the number of projects studying the human hand from the robotic point of view has increased rapidly, due to the growing interest in academic and industrial applications. Nevertheless, the complexity of the human hand given its large number of degrees of freedom (DoF) within a significantly reduced space requires an exhaustive analysis, before proposing any applications. The aim of this paper is to provide a complete summary of the kinematic and dynamic characteristics of the human hand as a preliminary step towards the development of hand devices such as prosthetic/robotic hands and exoskeletons imitating the human hand shape and functionality. A collection of data and constraints relevant to hand movements is presented, and the direct and inverse kinematics are solved for all the fingers as well as the dynamics; anthropometric data and dynamics equations allow performing simulations to understand the behavior of the finger.


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## Body

## 1. Introduction

The human hand is a complex mechanism; it has a wide range of DoFs, allowing a great variety of movements. In recent years, as robotics has advanced, significant efforts have been devoted to the development of hand devices. The two main related application fields are prosthetic/robotic hands and exoskeletons. On one side, robotic hands are developed with the characteristics complying to those of the human hand, taking advantage of its variety of movements, thereby avoiding the use of a large number of end effectors when performing tasks with different objects (e.g., Eurobot [1], Robonaut [2]). On the other side, exoskeletons are designed to fit onto the human hand, aiming at enhancing performance in the carrying out of daily activities (e.g., improving astronauts’ hand performance during extravehicular activity [3]) or supporting the rehabilitation stage of hand injury recovery.There are currently many different projects underway. Schabowsky et al. [4] introduced a newly developed Hand Exoskeleton Rehabilitation Robot (HEXORR), which was designed to provide a full range of motion for all fingers. NASA and General Motors presented a prototype of the Human Grasp Assist device [5] (K-Glove). Worsnopp et al. [6] introduced a finger exoskeleton for hand rehabilitation following strokes, to facilitate movement, especially pinch. Another project is being developed by Ho et al. [7]: their exoskeleton hand is EMG driven, again for rehabilitation, but working on all the fingers. All of these projects present a different number of DoFs and different structures, but in general they are developed with the objective of mimicking the main characteristics of the human hand. This implies a complete understanding of these characteristics, involving the anthropometric dimensions of the human hand, its kinematics, and its dynamics.This paper aims at analyzing all the aforementioned aspects. It is organized as follows: Section2 presents anthropometric data about the hand, such as dimensions of the hand and phalanges; Section 3 contains the main constraints of finger movements, explaining each type of constraint in the natural movement of the human hand; Section 4 describes the kinematical model of the hand; direct and inverse kinematics are developed step by step. Section 5 presents the dynamics of a single finger; Section 6 shows the implementation of the dynamic equations on a practical example. Finally, conclusions are presented in Section 7.


## 2. Human Hand Data

The human hand is composed of 5 digits: 4 fingers (index, ring, middle, and little fingers) and the thumb. The thumb is characterized by three articulations and three phalanges. The fingers also comprise three different articulations and four phalanges. Figure1 shows names and acronyms of each articulation and phalanx. The wrist has two functional DoFs. The TMC joint of the thumb holds two DoFs (flexion/extension and adduction/abduction). A single DoF (flexion/extension) characterizes the MCP and IP joints of the thumb as well as the PIP and DIP joints of the fingers. Whereas the eight bones of the carpus articulate finely with each other producing small deformations, their representation in a single rigid segment is a consistent approximation [8].Figure 1
Anatomical details of the hand skeleton.The analysis of kinematics and dynamics requires knowledge regarding the dimensions of the fingers and of the palm and their respective range of motion (RoM); those data are reported below. Tables1 and 2 show the results of Garrett’s studies [9, 10] for finger lengths and palm dimensions, measurements that were taken from the right hands of 148 men and 211 women. In Table 1, crotch to tip is the distance along the axis of the digit from the midpoint of its tip to the level of the corresponding webbed crotch between two digit; wrist crease to tip is the distance along the axis of the digit from the midpoint of its tip to the wrist crease baseline.Table 1
Mean finger lengths and palm dimensions of USAF male (M)/female (F) flying personnel [9, 10] (cm).
Finger length (crotch to tip)
Finger length (wrist crease to tip)
Mean
s.d.
<5%
<95%
Mean
s.d.
<5%
<95%
M
Thumb
5.87
0.45
5.07
6.57
12.70
1.13
11.05
14.68
Index
7.53
0.46
6.83
8.19
18.52
0.88
17.33
20.06
Middle
8.57
0.51
7.82
9.74
19.52
0.92
18.10
21.04
Ring
8.0
0.47
7.44
8.93
18.72
0.91
17.52
20.28
Little
6.14
0.47
5.44
6.99
16.61
0.91
15.11
18.10
F
Thumb
5.37
0.44
4.68
6.12
11.05
1.00
9.51
12.83
Index
6.90
0.52
6.10
7.80
16.67
0.89
15.21
18.14
Middle
7.79
0.51
7.01
8.68
17.65
0.87
16.22
19.05
Ring
7.31
0.52
6.52
8.22
16.76
8.94
15.28
18.20
Little
5.46
0.44
4.80
6.24
14.64
0.92
13.11
16.12Table 2
Average hand dimensions of USAF male (M)/female (F) flying personnel [9, 10] (cm).
Joint
Mean length
Hand length
M
19.72
F
17.93
Hand breadth
M
8.96
F
7.71
Hand circumference
M
21.59
F
18.71
Hand thickness
M
3.29
F
2.76
Hand depth
M
6.19
F
5.17A few researchers have measured the length of each phalanx separately. A study with a variety of candidates is the one performed by Habib and Kamal [11]. The results of their study for each phalanx of index, middle, ring, and little fingers are in Table 3 (I1 means distal phalanx, I2 middle phalanx, and I3 proximal phalanx of the index finger. The same notation is used for other digits). A similar survey is also present in Jasuja and Singh’s study [12]. As shown in Table 3, the average dimensions of the hand are quite similar to the ones presented in Table 2, with a maximum difference of 2.18%. The data presented here is the sample distribution over geographic regions and Air Force Commands, which may be quite representative of possible EVA glove users.Table 3
Mean length of the hand and of the phalanges of index, middle, ring, and little fingers (cm).
Hand
I1
I2
I3
M1
M2
M3
R1
R2
R3
L1
L2
L3
Male right hand
19.29
2.32
2.37
2.65
2.60
2.78
2.80
2.29
2.56
2.76
1.96
1.92
2.51
Male left hand
19.36
2.32
2.39
2.61
2.60
2.82
2.75
2.30
2.59
2.78
1.95
1.98
2.49
Female right hand
17.60
2.23
2.24
2.45
2.44
2.55
2.56
2.12
2.34
2.52
1.79
1.74
2.26
Female left hand
17.62
2.20
2.24
2.35
2.24
2.43
2.53
2.13
2.36
2.49
1.77
1.77
2.26As mentioned above, those data can give us an idea of the mean values of the length of each element that composes the human hand. With proper modifications, this data can also be used to compose a model to simulate its movements: in particular, it must be taken into account that the distance between a joint and a digit webbing is not representative of the equivalent link length. A simulation was performed, and the results are presented in Section6.


## 3. Constraints Overview

Hand and digit motions are subject to several constraints that limit the range of the natural movements of human fingers. Constraints can be roughly divided into three types: static, intrafinger, and interfinger constraints. Intra- and inter-finger, constraints are often called dynamic constraints that in the fingers of the hand are essentially constraints between joint motions. However, this range of movement is somewhat ambiguous because the range depends on various factors involving human hand biomechanics; therefore they are difficult to be expressed in closed forms (i.e., equations) and how to model such constraints still needs further investigation. The significance of taking finger constraints into consideration is that this causes the DoFs of the human hand to decrease. It is important to underline the fact that different individuals can show significant variations regarding the following constraints.


### 3.1. Intrafinger Constraints

Intrafinger constraints are constraints between different joints in the same finger. Cobos et al. [14] presented several constraints for fingers and thumb. Equation (1) presents the relation between the joints of a finger, as first proposed by Rijpkema and Girard [15]:
(1)θDIP≈23θPIPθPIP≈34θMCPf/e.In (2), the constraint relations between the joints when flexing or extending the thumb are
(2)θIP≈12θMCPf/e,θMCPf/e≈54θTMCf/e.These constraints are not strict. In fact, there are individuals who are more able than others to control their DIP joint, and anybody can force a behavior slightly out of these equations for any joint. However, in normal conditions those constraints are respected quite faithfully.It is important to underline that the constraints mentioned above are due to the physiologic nature of the human hand. Additionally, other constraints can be considered, due to ergonomics. In this case, the relations of the constraints are related to a specific task, such as grasping a circular or a prismatic object.Table4 shows the typical ergonomic constraints relative to a circular grasping.Table 4
Intrafinger constraints for a circular grasp.
Thumb
Index
Middle
Ring
Little
No constraint
θ
CMC
f
/
e
Index
=
θ
CMC
f
/
e
Middle
θ
CMC
f
/
e
Middle
=
(
1
/
2
)
θ
CMC
f
/
e
Ring
θ
CMC
f
/
e
Ring
=
(
2
/
3
)
θ
CMC
f
/
e
Little
No constraint
θ
TMC
f
/
e
=
(
11
/
10
)
θ
MCP
f
/
e
No constraint
θ
MCP
a
b
/
a
d
Middle
=
(
1
/
5
)
θ
MCP
a
b
/
a
d
Index
θ
MCP
a
b
/
a
d
Ring
=
(
1
/
2
)
θ
MCP
a
b
/
a
d
Little
No constraint
θ
MCP
f
/
e
=
(
4
/
5
)
θ
IP
f
/
e
θ
MCP
f
/
e
Index
=
(
4
/
3
)
θ
PIP
f
/
e
Index
θ
MCP
f
/
e
Middle
=
(
4
/
3
)
θ
PIP
f
/
e
Middle
θ
MCP
f
/
e
Ring
=
(
4
/
3
)
θ
PIP
f
/
e
Ring
θ
MCP
f
/
e
Little
=
(
4
/
3
)
θ
PIP
f
/
e
Little
No constraint
θ
PIP
f
/
e
Index
=
(
3
/
2
)
θ
DIP
f
/
e
Index
θ
PIP
f
/
e
Middle
=
(
3
/
2
)
θ
DIP
f
/
e
Middle
θ
PIP
f
/
e
Ring
=
(
3
/
2
)
θ
DIP
f
/
e
Ring
θ
PIP
f
/
e
Little
=
(
3
/
2
)
θ
DIP
f
/
e
LittleTherefore the thumb is defined by 2 DoFs, index finger by 2 DoFs, middle finger by 1 DoF, ring finger by 1 DoF, and little finger by 3 DoFs. The ring finger is calculated on the basis of the little and middle fingers joints. Therefore, the thumb, the index finger, and the little finger are the most important fingers when defining circular grasps [13].Table5 shows the ergonomic intrafinger constraints for a prismatic grasp. As a result, the thumb is defined by 2 DoFs, index finger by 2 DoFs, middle finger by 1 DoF, ring finger by 1 DoF, and little finger by 3 DoFs. Comparing the circular grasping and the prismatic one, the latter presents fewer constraints among fingers.Table 5
Intrafinger constraints for a prismatic grasp.
Thumb
Index
Middle
Ring
Little
No constraint
θ
CMC
f
/
e
Index
=
θ
CMC
f
/
e
Middle
θ
CMC
f
/
e
Middle
=
(
1
/
2
)
θ
CMC
f
/
e
Ring
θ
CMC
f
/
e
Ring
=
(
2
/
3
)
θ
CMC
f
/
e
Little
No constraint
θ
TMC
f
/
e
=
(
11
/
10
)
θ
MCP
f
/
e
No constraint
θ
MCP
a
b
/
a
d
Middle
=
(
1
/
5
)
θ
MCP
a
b
/
a
d
Index
θ
MCP
a
b
/
a
d
Ring
=
(
1
/
2
)
θ
MCP
a
b
/
a
d
Little
No constraint
θ
MCP
f
/
e
=
(
6
/
5
)
θ
IP
f
/
e
θ
MCP
f
/
e
Index
=
(
3
/
2
)
θ
PIP
f
/
e
Index
θ
MCP
f
/
e
Middle
=
(
3
/
2
)
θ
PIP
f
/
e
Middle
θ
MCP
f
/
e
Ring
=
(
3
/
2
)
θ
PIP
f
/
e
Ring
θ
MCP
f
/
e
Little
=
(
3
/
2
)
θ
PIP
f
/
e
Little
No constraint
θ
PIP
f
/
e
Index
=
2
θ
DIP
f
/
e
Index
θ
PIP
f
/
e
Middle
=
2
θ
DIP
f
/
e
Middle
θ
PIP
f
/
e
Ring
=
2
θ
DIP
f
/
e
Ring
θ
PIP
f
/
e
Little
=
2
θ
DIP
f
/
e
Little


### 3.2. Interfinger Constraints

Interfinger constraints correlate two joints belonging to different fingers. For instance, with the hand in the open position (at rest), when one bends the index finger at the MCP joint, the MCP joint of the middle finger bends automatically as well, with respective certain proportionality.As for intra-finger constraints, different individuals can show important differences. Moreover, also in this case one can forcefully overcome some of these constraints.On the other hand, there are constraints that cannot be explicitly represented in equations. A few cases are explained below.Some joints belonging to different fingers seem to be “naturally” interconnected. By that, we mean that the respective angles vary more or less proportionally, that is, unless a voluntary counter force is applied, imposing a different type of motion. However, trying to move a finger in such an unnatural way often results in oddly fatiguing efforts.For instance, there is a coupled movement when the index finger and little finger are at rest. The flexion of the middle finger is equal to the flexion of the ring finger as described in(3)θMCPf/eMiddle≈θMCPf/eRing.Another coupled movement is produced when flexing the ring finger, thereby causing a slight flexion on the middle finger and on the little finger, that happens to be the same angle rotation as described in(4)θMCPf/eMiddle≈θMCPf/eLittle.There is also a coupled movement in abduction/adduction, which is generated among ring and little fingers. In most cases, the movement is similar to(5)θMCPab/adRing≈θMCPab/adLittle.Additionally, some joints can be tied by more intricate relations, divided into two types. The first type occurs when there is a unique motion, like the flexion in the MCP joint of the little finger. Equation (6) represents this type of relation between the fingers
(6)θMCPf/eRing≈712θMCPf/eLittle,θMCPf/eMiddle≈23θMCPf/eRing,θMCPf/eRing-θMCPf/eMiddle<60°,θMCPf/eLittle-θMCPf/eRing<50°.These equations indicate that when there is flexion, the ring and middle fingers will also flex in a certain range with respect to the little finger. For instance, when the little finger is flexed, the middle finger will flex 2/3 times the ring finger.The second relationship is when there is a simple MCP flexion as in the case of the index finger, thereby causing some additional natural movement. Equation (7) represents this type of relation between the index and the middle fingers
(7)θMCPf/eMiddle≈15θMCPf/eIndex.This relation is only present when there is a single flexion on the index finger MCP joint, so when it occurs, the middle finger MCP joint will flex naturally and passively with this proportionality.These constraints are important because they express the natural relation between single joints. Thus devices designed to mimic the human hand must comply as precisely as possible with these constraints.


### 3.3. Static Constraints

The normal range of motion (ROM) of human hand joints corresponds to static constraints on joint angles in the model. These constraints are limits on the values that theθ parameters can assume. Main static constraints (Table 6) were collected by Cobos et al. [13]. By applying these constraints to the inverse kinematics presented later in Section 4.2, some DoFs can be neglected, and therefore the complexity of the system can be reduced.Table 6
Statics constraints [13].
Finger
Flexion
Extension
Abd./add.
Thumb
TMC
50°–90°
15°
45°–60°
MCP
75°–80°
0°
5°
IP
75°–80°
5°–10°
5°
Index
CMC
5°
0°
0°
MCP
90°
30°–40°
60°
PIP
110°
0°
0°
DIP
80°–90°
5°
0°
Middle
CMC
5°
0°
0°
MCP
90°
30°–40°
45°
PIP
110°
0°
0°
DIP
80°–90°
5°
0°
Ring
CMC
10°
0°
0°
MCP
90°
30°–40°
45°
PIP
120°
0°
0°
DIP
80°–90°
5°
0°
Little
CMC
15°
0°
0°
MCP
90°
30°–40°
50°
PIP
135°
0°
0°
DIP
90°
5°
0°


## 3.1. Intrafinger Constraints

Intrafinger constraints are constraints between different joints in the same finger. Cobos et al. [14] presented several constraints for fingers and thumb. Equation (1) presents the relation between the joints of a finger, as first proposed by Rijpkema and Girard [15]:
(1)θDIP≈23θPIPθPIP≈34θMCPf/e.In (2), the constraint relations between the joints when flexing or extending the thumb are
(2)θIP≈12θMCPf/e,θMCPf/e≈54θTMCf/e.These constraints are not strict. In fact, there are individuals who are more able than others to control their DIP joint, and anybody can force a behavior slightly out of these equations for any joint. However, in normal conditions those constraints are respected quite faithfully.It is important to underline that the constraints mentioned above are due to the physiologic nature of the human hand. Additionally, other constraints can be considered, due to ergonomics. In this case, the relations of the constraints are related to a specific task, such as grasping a circular or a prismatic object.Table4 shows the typical ergonomic constraints relative to a circular grasping.Table 4
Intrafinger constraints for a circular grasp.
Thumb
Index
Middle
Ring
Little
No constraint
θ
CMC
f
/
e
Index
=
θ
CMC
f
/
e
Middle
θ
CMC
f
/
e
Middle
=
(
1
/
2
)
θ
CMC
f
/
e
Ring
θ
CMC
f
/
e
Ring
=
(
2
/
3
)
θ
CMC
f
/
e
Little
No constraint
θ
TMC
f
/
e
=
(
11
/
10
)
θ
MCP
f
/
e
No constraint
θ
MCP
a
b
/
a
d
Middle
=
(
1
/
5
)
θ
MCP
a
b
/
a
d
Index
θ
MCP
a
b
/
a
d
Ring
=
(
1
/
2
)
θ
MCP
a
b
/
a
d
Little
No constraint
θ
MCP
f
/
e
=
(
4
/
5
)
θ
IP
f
/
e
θ
MCP
f
/
e
Index
=
(
4
/
3
)
θ
PIP
f
/
e
Index
θ
MCP
f
/
e
Middle
=
(
4
/
3
)
θ
PIP
f
/
e
Middle
θ
MCP
f
/
e
Ring
=
(
4
/
3
)
θ
PIP
f
/
e
Ring
θ
MCP
f
/
e
Little
=
(
4
/
3
)
θ
PIP
f
/
e
Little
No constraint
θ
PIP
f
/
e
Index
=
(
3
/
2
)
θ
DIP
f
/
e
Index
θ
PIP
f
/
e
Middle
=
(
3
/
2
)
θ
DIP
f
/
e
Middle
θ
PIP
f
/
e
Ring
=
(
3
/
2
)
θ
DIP
f
/
e
Ring
θ
PIP
f
/
e
Little
=
(
3
/
2
)
θ
DIP
f
/
e
LittleTherefore the thumb is defined by 2 DoFs, index finger by 2 DoFs, middle finger by 1 DoF, ring finger by 1 DoF, and little finger by 3 DoFs. The ring finger is calculated on the basis of the little and middle fingers joints. Therefore, the thumb, the index finger, and the little finger are the most important fingers when defining circular grasps [13].Table5 shows the ergonomic intrafinger constraints for a prismatic grasp. As a result, the thumb is defined by 2 DoFs, index finger by 2 DoFs, middle finger by 1 DoF, ring finger by 1 DoF, and little finger by 3 DoFs. Comparing the circular grasping and the prismatic one, the latter presents fewer constraints among fingers.Table 5
Intrafinger constraints for a prismatic grasp.
Thumb
Index
Middle
Ring
Little
No constraint
θ
CMC
f
/
e
Index
=
θ
CMC
f
/
e
Middle
θ
CMC
f
/
e
Middle
=
(
1
/
2
)
θ
CMC
f
/
e
Ring
θ
CMC
f
/
e
Ring
=
(
2
/
3
)
θ
CMC
f
/
e
Little
No constraint
θ
TMC
f
/
e
=
(
11
/
10
)
θ
MCP
f
/
e
No constraint
θ
MCP
a
b
/
a
d
Middle
=
(
1
/
5
)
θ
MCP
a
b
/
a
d
Index
θ
MCP
a
b
/
a
d
Ring
=
(
1
/
2
)
θ
MCP
a
b
/
a
d
Little
No constraint
θ
MCP
f
/
e
=
(
6
/
5
)
θ
IP
f
/
e
θ
MCP
f
/
e
Index
=
(
3
/
2
)
θ
PIP
f
/
e
Index
θ
MCP
f
/
e
Middle
=
(
3
/
2
)
θ
PIP
f
/
e
Middle
θ
MCP
f
/
e
Ring
=
(
3
/
2
)
θ
PIP
f
/
e
Ring
θ
MCP
f
/
e
Little
=
(
3
/
2
)
θ
PIP
f
/
e
Little
No constraint
θ
PIP
f
/
e
Index
=
2
θ
DIP
f
/
e
Index
θ
PIP
f
/
e
Middle
=
2
θ
DIP
f
/
e
Middle
θ
PIP
f
/
e
Ring
=
2
θ
DIP
f
/
e
Ring
θ
PIP
f
/
e
Little
=
2
θ
DIP
f
/
e
Little


## 3.2. Interfinger Constraints

Interfinger constraints correlate two joints belonging to different fingers. For instance, with the hand in the open position (at rest), when one bends the index finger at the MCP joint, the MCP joint of the middle finger bends automatically as well, with respective certain proportionality.As for intra-finger constraints, different individuals can show important differences. Moreover, also in this case one can forcefully overcome some of these constraints.On the other hand, there are constraints that cannot be explicitly represented in equations. A few cases are explained below.Some joints belonging to different fingers seem to be “naturally” interconnected. By that, we mean that the respective angles vary more or less proportionally, that is, unless a voluntary counter force is applied, imposing a different type of motion. However, trying to move a finger in such an unnatural way often results in oddly fatiguing efforts.For instance, there is a coupled movement when the index finger and little finger are at rest. The flexion of the middle finger is equal to the flexion of the ring finger as described in(3)θMCPf/eMiddle≈θMCPf/eRing.Another coupled movement is produced when flexing the ring finger, thereby causing a slight flexion on the middle finger and on the little finger, that happens to be the same angle rotation as described in(4)θMCPf/eMiddle≈θMCPf/eLittle.There is also a coupled movement in abduction/adduction, which is generated among ring and little fingers. In most cases, the movement is similar to(5)θMCPab/adRing≈θMCPab/adLittle.Additionally, some joints can be tied by more intricate relations, divided into two types. The first type occurs when there is a unique motion, like the flexion in the MCP joint of the little finger. Equation (6) represents this type of relation between the fingers
(6)θMCPf/eRing≈712θMCPf/eLittle,θMCPf/eMiddle≈23θMCPf/eRing,θMCPf/eRing-θMCPf/eMiddle<60°,θMCPf/eLittle-θMCPf/eRing<50°.These equations indicate that when there is flexion, the ring and middle fingers will also flex in a certain range with respect to the little finger. For instance, when the little finger is flexed, the middle finger will flex 2/3 times the ring finger.The second relationship is when there is a simple MCP flexion as in the case of the index finger, thereby causing some additional natural movement. Equation (7) represents this type of relation between the index and the middle fingers
(7)θMCPf/eMiddle≈15θMCPf/eIndex.This relation is only present when there is a single flexion on the index finger MCP joint, so when it occurs, the middle finger MCP joint will flex naturally and passively with this proportionality.These constraints are important because they express the natural relation between single joints. Thus devices designed to mimic the human hand must comply as precisely as possible with these constraints.


## 3.3. Static Constraints

The normal range of motion (ROM) of human hand joints corresponds to static constraints on joint angles in the model. These constraints are limits on the values that theθ parameters can assume. Main static constraints (Table 6) were collected by Cobos et al. [13]. By applying these constraints to the inverse kinematics presented later in Section 4.2, some DoFs can be neglected, and therefore the complexity of the system can be reduced.Table 6
Statics constraints [13].
Finger
Flexion
Extension
Abd./add.
Thumb
TMC
50°–90°
15°
45°–60°
MCP
75°–80°
0°
5°
IP
75°–80°
5°–10°
5°
Index
CMC
5°
0°
0°
MCP
90°
30°–40°
60°
PIP
110°
0°
0°
DIP
80°–90°
5°
0°
Middle
CMC
5°
0°
0°
MCP
90°
30°–40°
45°
PIP
110°
0°
0°
DIP
80°–90°
5°
0°
Ring
CMC
10°
0°
0°
MCP
90°
30°–40°
45°
PIP
120°
0°
0°
DIP
80°–90°
5°
0°
Little
CMC
15°
0°
0°
MCP
90°
30°–40°
50°
PIP
135°
0°
0°
DIP
90°
5°
0°


## 4. Kinematic Model

The kinematic model proposed here is composed of 19 links corresponding to the human bones and 24 DoFs modeled by rotational joints. Two different kinematic configurations are considered for the fingers, one for the thumb modeled as 3 links and 4 joints and another for the other fingers (index, middle, ring, and little fingers). Each of them is modeled using 4 links and 5 joints, as in Figure2. Note that the CMC joint represents the deformation of the palm, for instance, when the hand is grasping a ball, while the MCP abduction/adduction joint is defined before MCP flexion/extension.Figure 2
Kinematic configuration of the human hand. The thumb is defined by 3 links and 4 degrees of freedom whereas index, middle, ring, and little fingers are defined by 4 links and 5 DoFs.


### 4.1. Direct Kinematics

Direct kinematic equations are used to obtain the fingertip position and orientation according to the joint angles. The model equations are calculated by means of modified Denavit-Hartenberg (MDH) parameters, introduced by Craig [16]. The difference from the Denavit-Hartenberg convention is the fact that, on MDH, the z-axis of the reference frame {j}, called zj, is coincident with the axis of joint j. In comparison to MD the z-axis of the frame {j} is coincident with the axis of joint j+1. The advantage of using MDH is that there is no need for further transformation of the references in order to work with the dynamics of the rigid bodies.


#### 4.1.1. Direct Kinematics of the Index, Middle, Ring, and Little Fingers

Each of the fingers contains four bones: metacarpal, proximal, middle, and distal (Figure1). These bones correspond approximately to the links of the serial kinematic chain (Figure 2). Each articulation presented above for these four fingers corresponds to the joints: CMC, MCP, PIP, and DIP. The MCP joint can be split into 2 DoFs, which carry out the adduction/abduction and flexion/extension movements, respectively. All the other joints only allow flexion/extension movements. Having defined a numbering of the fingers, from 0 to 4 where finger 0 is the thumb and finger 4 is the little finger, Table 7 shows the MDH parameters for fingers 1 to 4.Table 7
Modified D–H parameters for fingers 1 to 4.
Fingeri(i=1,…,4)
Joint
α
j
-
1
a
j
-
1
d
j
θ
j
j
=
1
π
/
2
0
0
θ
CMC
f
/
e
j
=
2
-
π
/
2
L
0
i
0
θ
MCP
a
b
/
a
d
j
=
3
π
/
2
0
0
θ
MCP
f
/
e
j
=
4
0
L
1
i
0
θ
PIP
f
/
e
j
=
5
0
L
2
i
0
θ
DIP
f
/
eEquation (8) shows the direct kinematics from index (i=1) to the little finger (i=4)(8)Qi=Ti00·T60i(θj)=Ti00·∏j=15Tij(j-1)(θj)·Tti65,Qi=Ti00·T10i(θCMCf/e)·T21i(θMCPab/ad)·T32i(θMCPf/e)·T43i(θPIPf/e)·T54i(θDIPf/e)·Tti65,
where(i)
Qi represents a matrix containing the position and orientation of the fingertip of each finger;(ii)
Ti00 represents a rototranslation matrix taking into account the fact that the fingers are slightly fanned out and making it possible to pass from the initial base reference frame (R0) to the alignment of the ith finger first reference frame (R0i);(iii)
Ti(θj)60 is a matrix containing the geometrical transformation between the ith finger first reference frame and the ith fingertip (fti). The matrix is composed of the concatenation of the transformation matrices of each finger link;(iv)
Tij(j-1)(θj) is a matrix containing the geometrical transformation between the (j-1)th reference frame and the jth reference frame of the ith finger;(v)
Tti65 represents the position pti=[ltixltiyltiz]T of the fingertip with respect to the distal (5th) reference frame.i corresponds to index(1), middle(2), ring(3), and little(4) fingers. j corresponds to each finger’s joint CMCf/e, MCPab/ad, MCPf/e, PIPf/e, and DIPf/e. ti stands for the tip of the ith finger.The coefficientsQist, which are the elements of the sth row and tth column of the matrix expressed in (8), relative to the ith finger, are given in Appendix A.


#### 4.1.2. Direct Kinematics of the Thumb

The thumb presents three bones (Figure1): metacarpal, proximal, and distal. These bones correspond approximately to the length of each link. The respective joints are TMC, MCP, and IP. The TMC joint presents 2 DoFs, allowing adduction/abduction and flexion/extension. Table 8 shows the MDH parameters. Equation (9), with the same notation scheme as (8), shows the direct kinematics for the thumb
(9)Q0=T000·T0(θk)50,Q0=T000·T100(θTMCab/ad)·T210(θTMCf/e)·T320(θMCPf/e)·T430(θIPf/e)·T54t0,
where k corresponds to the thumb joint TMCab/ad, TMCf/e, MCPf/e, and IPf/e. The position of the fingertip is pt0=[lt0xlt0ylt0z]T. The coefficients Q0st of the matrix expressed in (9) are given in Appendix A.Table 8
Modified D–H parameters of the thumb.
Joint
α
j
-
1
a
j
-
1
d
j
θ
j
j
=
1
0
0
0
θ
TMC
a
b
/
a
d
j
=
2
π
/
2
0
0
θ
TMC
f
/
e
j
=
3
0
L
00
0
θ
MCP
f
/
e
j
=
4
0
L
10
0
θ
IP
f
/
e


### 4.2. Inverse Kinematics

Inverse kinematics is used to obtain the joint angle values according to the fingertip position and orientation. The inverse kinematics will be solved separately for the thumb and for the other fingers. The model of the human hand is a redundant case; therefore several solutions to the inverse kinematic problem exist. In order to find a unique solution, it is necessary to take into account the constraints presented in Section3.


#### 4.2.1. Inverse Kinematics of a Finger (Index(i=1) to Little Finger (i=4))

The anglesθi_CMCf/e, θi_MCPab/ad, θi_MCPf/e, and θi_PIPf/e, θi_DIPf/e are obtained from (8), where the matrix Qi, that is, the left member of the equation, is known. Hence, algebraically we solve the inverse kinematics for the CMCf/e, MCPab/ad, and DIPf/e joints. The procedure here shown solves the kinematics of the index finger (i=1); however, it is valid for every other finger except the thumb (i=2,3,4). For sake of brevity in the following, the i index is omitted.The duplication of solutions, when solving an arctangent, for example, is eliminated thanks to the constraints of Section3.3, because the ranges of all physiological angles are smaller than 180° and most of them are smaller than 90°. This fact is implicitly taken into account in the following. 										(10)θCMCf/e=atan(Q33Q13),θMCPab/ad=atan[Q13-Q23cos(θCMCf/e)],θDIPf/e=atan(ϵ1ϵ2),
where ϵ1 and ϵ2 are expressed as
(11)ϵ1=(-1)·(mQ22+nQ21)(m2-n2),ϵ2=Q21m+nm·(-1)·(mQ22+nQ21)(m2-n2),m=sin(θMCPab/ad)cos(θMCPf/e)cos(θPIPf/e)-sin(θMCPab/ad)sin(θMCPf/e)sin(θPIPf/e),n=sin(θMCPab/ad)cos(θMCPf/e)sin(θPIPf/e)-sin(θMCPab/ad)sin(θMCPf/e)cos(θPIPf/e).TheMCPf/e, PIPf/e joints are solved through a geometric method; see Figure 3.Figure 3
Inverse kinematics for the index finger and thumb.Starting from the vectorF1→ which contains the position of the fingertip, it is possible to obtain H1→ with the following expression:
(12)H1→=F1→-[L31*i^].
Then, the vector P1→ is expressed as
(13)P1→=[P1xP1yP1z]T,
where
(14)P1x=L01cos(θCMC),P1y=L01sin(θCMC),P1z=0.
Knowing H1→ and P1→ from (12) and (13), the vector u1→ is expressed as
(15)u1→=H1→-P1→.
Angles φ1 and φ2 are expressed as
(16)φ1=acos(r22+L012-r122r2l01),φ2=acos(r22+L112-L2122r2L11)
with variables r1 and r2 equal to
(17)r1=∥H1→∥,r2=∥u1→∥.
So θMCPf/e is obtained as
(18)θMCPf/e=π-φ1-φ2.
And θPIPf/e is obtained as
(19)θPIPf/e=π-φ3,
where
(20)φ3=acos(L212+L112-r222L21L11).


#### 4.2.2. Inverse Kinematics of the Thumb

A similar procedure can be applied to the thumb, in whichQ0, that is, the left side of (9), is known; then the joint angle θTMCab/ad is obtained algebraically as follows:
(21)θTMCab/ad=atan(Q13-Q23),
where the Qst term in (21) is the element of the sth row and tth column of the Q0 matrix. Then, using the geometrical method (Figure 3), the joint angles θMCPf/e and θIPf/e can be obtained:
(22)θMCPf/e=π-γ1,θIPf/e=π-γ2-γ3,
where
(23)γ1=acos(L102+L002-r322L10L00),γ2=acos(L202+s32-r422L20r3),γ3=acos(r32+L102-L0022rsL10),r3=∥H0→∥,r4=∥F0→∥,H0→=F0→-[L20*i^].
The joint θTMCf/e is obtained algebraically as follows:
(24)θTMCf/e=atan(ϵ4ϵ3),
where
(25)ϵ3=(Q32+Q31)cosμ,ϵ4=A31-(Q31+Q32)cosμ·sinμcosμ,μ=θMCP+θIP.


## 4.1. Direct Kinematics

Direct kinematic equations are used to obtain the fingertip position and orientation according to the joint angles. The model equations are calculated by means of modified Denavit-Hartenberg (MDH) parameters, introduced by Craig [16]. The difference from the Denavit-Hartenberg convention is the fact that, on MDH, the z-axis of the reference frame {j}, called zj, is coincident with the axis of joint j. In comparison to MD the z-axis of the frame {j} is coincident with the axis of joint j+1. The advantage of using MDH is that there is no need for further transformation of the references in order to work with the dynamics of the rigid bodies.


### 4.1.1. Direct Kinematics of the Index, Middle, Ring, and Little Fingers

Each of the fingers contains four bones: metacarpal, proximal, middle, and distal (Figure1). These bones correspond approximately to the links of the serial kinematic chain (Figure 2). Each articulation presented above for these four fingers corresponds to the joints: CMC, MCP, PIP, and DIP. The MCP joint can be split into 2 DoFs, which carry out the adduction/abduction and flexion/extension movements, respectively. All the other joints only allow flexion/extension movements. Having defined a numbering of the fingers, from 0 to 4 where finger 0 is the thumb and finger 4 is the little finger, Table 7 shows the MDH parameters for fingers 1 to 4.Table 7
Modified D–H parameters for fingers 1 to 4.
Fingeri(i=1,…,4)
Joint
α
j
-
1
a
j
-
1
d
j
θ
j
j
=
1
π
/
2
0
0
θ
CMC
f
/
e
j
=
2
-
π
/
2
L
0
i
0
θ
MCP
a
b
/
a
d
j
=
3
π
/
2
0
0
θ
MCP
f
/
e
j
=
4
0
L
1
i
0
θ
PIP
f
/
e
j
=
5
0
L
2
i
0
θ
DIP
f
/
eEquation (8) shows the direct kinematics from index (i=1) to the little finger (i=4)(8)Qi=Ti00·T60i(θj)=Ti00·∏j=15Tij(j-1)(θj)·Tti65,Qi=Ti00·T10i(θCMCf/e)·T21i(θMCPab/ad)·T32i(θMCPf/e)·T43i(θPIPf/e)·T54i(θDIPf/e)·Tti65,
where(i)
Qi represents a matrix containing the position and orientation of the fingertip of each finger;(ii)
Ti00 represents a rototranslation matrix taking into account the fact that the fingers are slightly fanned out and making it possible to pass from the initial base reference frame (R0) to the alignment of the ith finger first reference frame (R0i);(iii)
Ti(θj)60 is a matrix containing the geometrical transformation between the ith finger first reference frame and the ith fingertip (fti). The matrix is composed of the concatenation of the transformation matrices of each finger link;(iv)
Tij(j-1)(θj) is a matrix containing the geometrical transformation between the (j-1)th reference frame and the jth reference frame of the ith finger;(v)
Tti65 represents the position pti=[ltixltiyltiz]T of the fingertip with respect to the distal (5th) reference frame.i corresponds to index(1), middle(2), ring(3), and little(4) fingers. j corresponds to each finger’s joint CMCf/e, MCPab/ad, MCPf/e, PIPf/e, and DIPf/e. ti stands for the tip of the ith finger.The coefficientsQist, which are the elements of the sth row and tth column of the matrix expressed in (8), relative to the ith finger, are given in Appendix A.


### 4.1.2. Direct Kinematics of the Thumb

The thumb presents three bones (Figure1): metacarpal, proximal, and distal. These bones correspond approximately to the length of each link. The respective joints are TMC, MCP, and IP. The TMC joint presents 2 DoFs, allowing adduction/abduction and flexion/extension. Table 8 shows the MDH parameters. Equation (9), with the same notation scheme as (8), shows the direct kinematics for the thumb
(9)Q0=T000·T0(θk)50,Q0=T000·T100(θTMCab/ad)·T210(θTMCf/e)·T320(θMCPf/e)·T430(θIPf/e)·T54t0,
where k corresponds to the thumb joint TMCab/ad, TMCf/e, MCPf/e, and IPf/e. The position of the fingertip is pt0=[lt0xlt0ylt0z]T. The coefficients Q0st of the matrix expressed in (9) are given in Appendix A.Table 8
Modified D–H parameters of the thumb.
Joint
α
j
-
1
a
j
-
1
d
j
θ
j
j
=
1
0
0
0
θ
TMC
a
b
/
a
d
j
=
2
π
/
2
0
0
θ
TMC
f
/
e
j
=
3
0
L
00
0
θ
MCP
f
/
e
j
=
4
0
L
10
0
θ
IP
f
/
e


## 4.1.1. Direct Kinematics of the Index, Middle, Ring, and Little Fingers

Each of the fingers contains four bones: metacarpal, proximal, middle, and distal (Figure1). These bones correspond approximately to the links of the serial kinematic chain (Figure 2). Each articulation presented above for these four fingers corresponds to the joints: CMC, MCP, PIP, and DIP. The MCP joint can be split into 2 DoFs, which carry out the adduction/abduction and flexion/extension movements, respectively. All the other joints only allow flexion/extension movements. Having defined a numbering of the fingers, from 0 to 4 where finger 0 is the thumb and finger 4 is the little finger, Table 7 shows the MDH parameters for fingers 1 to 4.Table 7
Modified D–H parameters for fingers 1 to 4.
Fingeri(i=1,…,4)
Joint
α
j
-
1
a
j
-
1
d
j
θ
j
j
=
1
π
/
2
0
0
θ
CMC
f
/
e
j
=
2
-
π
/
2
L
0
i
0
θ
MCP
a
b
/
a
d
j
=
3
π
/
2
0
0
θ
MCP
f
/
e
j
=
4
0
L
1
i
0
θ
PIP
f
/
e
j
=
5
0
L
2
i
0
θ
DIP
f
/
eEquation (8) shows the direct kinematics from index (i=1) to the little finger (i=4)(8)Qi=Ti00·T60i(θj)=Ti00·∏j=15Tij(j-1)(θj)·Tti65,Qi=Ti00·T10i(θCMCf/e)·T21i(θMCPab/ad)·T32i(θMCPf/e)·T43i(θPIPf/e)·T54i(θDIPf/e)·Tti65,
where(i)
Qi represents a matrix containing the position and orientation of the fingertip of each finger;(ii)
Ti00 represents a rototranslation matrix taking into account the fact that the fingers are slightly fanned out and making it possible to pass from the initial base reference frame (R0) to the alignment of the ith finger first reference frame (R0i);(iii)
Ti(θj)60 is a matrix containing the geometrical transformation between the ith finger first reference frame and the ith fingertip (fti). The matrix is composed of the concatenation of the transformation matrices of each finger link;(iv)
Tij(j-1)(θj) is a matrix containing the geometrical transformation between the (j-1)th reference frame and the jth reference frame of the ith finger;(v)
Tti65 represents the position pti=[ltixltiyltiz]T of the fingertip with respect to the distal (5th) reference frame.i corresponds to index(1), middle(2), ring(3), and little(4) fingers. j corresponds to each finger’s joint CMCf/e, MCPab/ad, MCPf/e, PIPf/e, and DIPf/e. ti stands for the tip of the ith finger.The coefficientsQist, which are the elements of the sth row and tth column of the matrix expressed in (8), relative to the ith finger, are given in Appendix A.


## 4.1.2. Direct Kinematics of the Thumb

The thumb presents three bones (Figure1): metacarpal, proximal, and distal. These bones correspond approximately to the length of each link. The respective joints are TMC, MCP, and IP. The TMC joint presents 2 DoFs, allowing adduction/abduction and flexion/extension. Table 8 shows the MDH parameters. Equation (9), with the same notation scheme as (8), shows the direct kinematics for the thumb
(9)Q0=T000·T0(θk)50,Q0=T000·T100(θTMCab/ad)·T210(θTMCf/e)·T320(θMCPf/e)·T430(θIPf/e)·T54t0,
where k corresponds to the thumb joint TMCab/ad, TMCf/e, MCPf/e, and IPf/e. The position of the fingertip is pt0=[lt0xlt0ylt0z]T. The coefficients Q0st of the matrix expressed in (9) are given in Appendix A.Table 8
Modified D–H parameters of the thumb.
Joint
α
j
-
1
a
j
-
1
d
j
θ
j
j
=
1
0
0
0
θ
TMC
a
b
/
a
d
j
=
2
π
/
2
0
0
θ
TMC
f
/
e
j
=
3
0
L
00
0
θ
MCP
f
/
e
j
=
4
0
L
10
0
θ
IP
f
/
e


## 4.2. Inverse Kinematics

Inverse kinematics is used to obtain the joint angle values according to the fingertip position and orientation. The inverse kinematics will be solved separately for the thumb and for the other fingers. The model of the human hand is a redundant case; therefore several solutions to the inverse kinematic problem exist. In order to find a unique solution, it is necessary to take into account the constraints presented in Section3.


### 4.2.1. Inverse Kinematics of a Finger (Index(i=1) to Little Finger (i=4))

The anglesθi_CMCf/e, θi_MCPab/ad, θi_MCPf/e, and θi_PIPf/e, θi_DIPf/e are obtained from (8), where the matrix Qi, that is, the left member of the equation, is known. Hence, algebraically we solve the inverse kinematics for the CMCf/e, MCPab/ad, and DIPf/e joints. The procedure here shown solves the kinematics of the index finger (i=1); however, it is valid for every other finger except the thumb (i=2,3,4). For sake of brevity in the following, the i index is omitted.The duplication of solutions, when solving an arctangent, for example, is eliminated thanks to the constraints of Section3.3, because the ranges of all physiological angles are smaller than 180° and most of them are smaller than 90°. This fact is implicitly taken into account in the following. 										(10)θCMCf/e=atan(Q33Q13),θMCPab/ad=atan[Q13-Q23cos(θCMCf/e)],θDIPf/e=atan(ϵ1ϵ2),
where ϵ1 and ϵ2 are expressed as
(11)ϵ1=(-1)·(mQ22+nQ21)(m2-n2),ϵ2=Q21m+nm·(-1)·(mQ22+nQ21)(m2-n2),m=sin(θMCPab/ad)cos(θMCPf/e)cos(θPIPf/e)-sin(θMCPab/ad)sin(θMCPf/e)sin(θPIPf/e),n=sin(θMCPab/ad)cos(θMCPf/e)sin(θPIPf/e)-sin(θMCPab/ad)sin(θMCPf/e)cos(θPIPf/e).TheMCPf/e, PIPf/e joints are solved through a geometric method; see Figure 3.Figure 3
Inverse kinematics for the index finger and thumb.Starting from the vectorF1→ which contains the position of the fingertip, it is possible to obtain H1→ with the following expression:
(12)H1→=F1→-[L31*i^].
Then, the vector P1→ is expressed as
(13)P1→=[P1xP1yP1z]T,
where
(14)P1x=L01cos(θCMC),P1y=L01sin(θCMC),P1z=0.
Knowing H1→ and P1→ from (12) and (13), the vector u1→ is expressed as
(15)u1→=H1→-P1→.
Angles φ1 and φ2 are expressed as
(16)φ1=acos(r22+L012-r122r2l01),φ2=acos(r22+L112-L2122r2L11)
with variables r1 and r2 equal to
(17)r1=∥H1→∥,r2=∥u1→∥.
So θMCPf/e is obtained as
(18)θMCPf/e=π-φ1-φ2.
And θPIPf/e is obtained as
(19)θPIPf/e=π-φ3,
where
(20)φ3=acos(L212+L112-r222L21L11).


### 4.2.2. Inverse Kinematics of the Thumb

A similar procedure can be applied to the thumb, in whichQ0, that is, the left side of (9), is known; then the joint angle θTMCab/ad is obtained algebraically as follows:
(21)θTMCab/ad=atan(Q13-Q23),
where the Qst term in (21) is the element of the sth row and tth column of the Q0 matrix. Then, using the geometrical method (Figure 3), the joint angles θMCPf/e and θIPf/e can be obtained:
(22)θMCPf/e=π-γ1,θIPf/e=π-γ2-γ3,
where
(23)γ1=acos(L102+L002-r322L10L00),γ2=acos(L202+s32-r422L20r3),γ3=acos(r32+L102-L0022rsL10),r3=∥H0→∥,r4=∥F0→∥,H0→=F0→-[L20*i^].
The joint θTMCf/e is obtained algebraically as follows:
(24)θTMCf/e=atan(ϵ4ϵ3),
where
(25)ϵ3=(Q32+Q31)cosμ,ϵ4=A31-(Q31+Q32)cosμ·sinμcosμ,μ=θMCP+θIP.


## 4.2.1. Inverse Kinematics of a Finger (Index(i=1) to Little Finger (i=4))

The anglesθi_CMCf/e, θi_MCPab/ad, θi_MCPf/e, and θi_PIPf/e, θi_DIPf/e are obtained from (8), where the matrix Qi, that is, the left member of the equation, is known. Hence, algebraically we solve the inverse kinematics for the CMCf/e, MCPab/ad, and DIPf/e joints. The procedure here shown solves the kinematics of the index finger (i=1); however, it is valid for every other finger except the thumb (i=2,3,4). For sake of brevity in the following, the i index is omitted.The duplication of solutions, when solving an arctangent, for example, is eliminated thanks to the constraints of Section3.3, because the ranges of all physiological angles are smaller than 180° and most of them are smaller than 90°. This fact is implicitly taken into account in the following. 										(10)θCMCf/e=atan(Q33Q13),θMCPab/ad=atan[Q13-Q23cos(θCMCf/e)],θDIPf/e=atan(ϵ1ϵ2),
where ϵ1 and ϵ2 are expressed as
(11)ϵ1=(-1)·(mQ22+nQ21)(m2-n2),ϵ2=Q21m+nm·(-1)·(mQ22+nQ21)(m2-n2),m=sin(θMCPab/ad)cos(θMCPf/e)cos(θPIPf/e)-sin(θMCPab/ad)sin(θMCPf/e)sin(θPIPf/e),n=sin(θMCPab/ad)cos(θMCPf/e)sin(θPIPf/e)-sin(θMCPab/ad)sin(θMCPf/e)cos(θPIPf/e).TheMCPf/e, PIPf/e joints are solved through a geometric method; see Figure 3.Figure 3
Inverse kinematics for the index finger and thumb.Starting from the vectorF1→ which contains the position of the fingertip, it is possible to obtain H1→ with the following expression:
(12)H1→=F1→-[L31*i^].
Then, the vector P1→ is expressed as
(13)P1→=[P1xP1yP1z]T,
where
(14)P1x=L01cos(θCMC),P1y=L01sin(θCMC),P1z=0.
Knowing H1→ and P1→ from (12) and (13), the vector u1→ is expressed as
(15)u1→=H1→-P1→.
Angles φ1 and φ2 are expressed as
(16)φ1=acos(r22+L012-r122r2l01),φ2=acos(r22+L112-L2122r2L11)
with variables r1 and r2 equal to
(17)r1=∥H1→∥,r2=∥u1→∥.
So θMCPf/e is obtained as
(18)θMCPf/e=π-φ1-φ2.
And θPIPf/e is obtained as
(19)θPIPf/e=π-φ3,
where
(20)φ3=acos(L212+L112-r222L21L11).


## 4.2.2. Inverse Kinematics of the Thumb

A similar procedure can be applied to the thumb, in whichQ0, that is, the left side of (9), is known; then the joint angle θTMCab/ad is obtained algebraically as follows:
(21)θTMCab/ad=atan(Q13-Q23),
where the Qst term in (21) is the element of the sth row and tth column of the Q0 matrix. Then, using the geometrical method (Figure 3), the joint angles θMCPf/e and θIPf/e can be obtained:
(22)θMCPf/e=π-γ1,θIPf/e=π-γ2-γ3,
where
(23)γ1=acos(L102+L002-r322L10L00),γ2=acos(L202+s32-r422L20r3),γ3=acos(r32+L102-L0022rsL10),r3=∥H0→∥,r4=∥F0→∥,H0→=F0→-[L20*i^].
The joint θTMCf/e is obtained algebraically as follows:
(24)θTMCf/e=atan(ϵ4ϵ3),
where
(25)ϵ3=(Q32+Q31)cosμ,ϵ4=A31-(Q31+Q32)cosμ·sinμcosμ,μ=θMCP+θIP.


## 5. Dynamics of a Single Finger

This section provides the dynamics equation system of a generic single finger, neglecting the MCP abduction/adduction motion. Theith index is again omitted. Based on the convention of Figure 4, the dynamic model is determined using Euler-Lagrange equations. It is important to highlight the fact that on the model the metacarpus is fixed, while only the finger phalanges are moving parts. Thus, R2 is the base reference system, and all the equations are written with respect to R2.Figure 4
Dynamic model of index(1).The following equations are applicable to any 3R planar robot, after making the necessary adaptations to the variables.The equations require the estimation of the kinetic energy and potential energy, as shown in the following.In order to simplify the expressed vectors and further developments, the following notation is used:(26)θ3=θMCPf/e,θ4=θPIPf/e,θ5=θDIPf/e.The kinetic energy is calculated starting from the position vectors of the center of mass of each phalanx with respect to the base reference frameR2; in general, the position of the center of mass of the jth phalanx with respect to the j reference frame is
(27)G→g=[bgxbgy0];g=1,2,3.The mass of thejth phalanx is equal to mg, and the respective moment of inertia with respect to the axis z is equal to Ig.Thus, each position vector with respect to the base reference frameR2 is presented below, where cθj and sθj stand for cosine and sine of θj, respectively. Consider
(28)G→1=[b1xcθ3-b1ysθ3b1xsθ3+b1ycθ30],G→2=[L1cθ3L1sθ30]+[b2xc(θ3+θ4)-b2ys(θ3+θ4)b2xs(θ3+θ4)+b2yc(θ3+θ4)0],G→3=[L1cθ3L1sθ30]+[L2c(θ3+θ4)L2s(θ3+θ4)0]+[b3xc(θ3+θ4+θ5)-b3ys(θ3+θ4+θ5)b3xs(θ3+θ4+θ5)+b3yc(θ3+θ4+θ5)0].The velocities can be obtained by differentiating the position vectors with respect to time, taking into account the following notation of (29) and representing time derivatives through dot notation:
(29)θ3+θ4=θ34,θ˙3+θ˙4=θ˙34,θ3+θ4+θ5=θ345,θ˙3+θ˙4+θ˙5=θ˙345.The velocities of each center of mass are obtained:(30)vG1=[-b1xθ˙3sθ3-b1yθ˙3cθ3b1xθ˙3cθ3-b1yθ˙3sθ30],vG2=[-L1θ˙3sθ3-b2xθ˙34sθ34-b2yθ˙34cθ34L1θ˙3cθ3+b2xθ˙34cθ34-b2yθ˙34sθ340],vG3=[-L1θ˙3sθ3-L2θ˙34sθ34-b3xθ˙345sθ345-b3yθ˙345cθ345L1θ˙3cθ3+L2θ˙34cθ34+b3xθ˙345cθ345-b3yθ˙345sθ3450].The kinetic energy is expressed as(31)T=12m1(vG1)2+12m2(vG2)2+12m3(vG3)2+12I1(θ˙3)2+12I2(θ˙34)2+12I3(θ˙345)2.There are two forms of potential energy considered in this study: the gravitational potential energy and the elastic potential energy. Regarding the elastic potential energy, the value of the stiffness is considered as an average value (constant valuekg) in this study, as a close approximation of the nonlinear and anisotropic finger stiffness (i.e., it varies with the direction). k1, k2, and k3 are the stiffness values for the MCP, DIP, and PIP joints, respectively [17].The potential energy for the finger is expressed as(32)U=m1g(b1xsθ3+b1ycθ3)+m2g(L1sθ3+b2xsθ34+b2ycθ34)+m3g(L1sθ3+L2sθ34+b3xsθ345+b3ycθ345)+12k1(θ3)2+12k2(θ34-θ3)2+12k3(θ345-θ34)2.Moreover, introducing a function of the generalized velocities, usually referred to as the Rayleigh dissipation functionF, for the damping forces, this function is expressed as
(33)F=12c1(θ˙3)2+12c2(θ˙34-θ˙3)2+12c3(θ˙345-θ˙34)2,
where the damping constant (cg) stands for the nonconservative contribution caused by the muscles, actuating the finger. Nonconservative forces contributed less than 15% to the total force response to static displacement. Muscle viscosity is dissipative and, hence, non-conservative, resulting in a force field with nonzero curl [17]. To be more precise, values c1, c2, and c3 are the damping values for the MCP, DIP, and PIP joints, respectively.The Euler-Lagrange equations thus become, considering the three generalized coordinatesθ3, θ34, and θ345,
(34)ddt(∂(T-U)∂θ˙3)-∂(T-U)∂θ3+∂F∂θ˙3=τ3,ddt(∂(T-U)∂θ˙34)-∂(T-U)∂θ34+∂F∂θ˙34=τ4,ddt(∂(T-U)∂θ˙345)-∂(T-U)∂θ345+∂F∂θ˙345=τ5
in which the τj(j=3,4,5) terms contain the forces applied through the muscles in order to actuate the phalanges and the contact forces, shown in Figure 5.Figure 5
Contact forces.According to the virtual work principle, the equation to calculate the generalized force can be expressed as(35)τj=(∑j=35δWj)δθj;j=3,4,5,
where δWj is the virtual work done by the force applied to the system. In the current case, it is
(36)τ3=(F3ye3x-F3xe3y)+[-L1sθ3L1cθ30]·[F4xcθ34-F4ysθ34F4xsθ34+F4ycθ340]+[-L1sθ3L1cθ30]·[F5xcθ345-F5ysθ345F5xsθ345+F5ycθ3450]+Cm1-Cm2,τ4=(F4ye4x-F4xe4y)+[-L2sθ34L2cθ340]·[F5xcθ345-F5ysθ345F5xsθ345+F5ycθ3450]+Cm2-Cm3,τ5=(F5ye5x-F5xe5y)+Cm3,
where the term Cmg refers to the torque produced by the muscles on the jth joint, and Fj=[FjxFjy0]T is the contact force applied to the jth phalanx at the point defined by the position vector ej=[ejxeiy0]T.Calculating each element of the Euler-Lagrange equations, the dynamical system (33) becomes
(37)A1θ¨3+A2θ¨34+A3θ¨345=A4,B1θ¨3+B2θ¨34+B3θ¨345=B4,C1θ¨3+C2θ¨34+C3θ¨345=C4.The termsA1, A2, and A3 contain the coefficients of the accelerations and A4 the remaining terms; the same goes for the other Euler-Lagrange equations.Equation (37) allows the direct dynamics of the finger to be solved, where, given the torques exerted by the muscles on each phalange, the movement of the finger can be calculated. If, on the other side, an inverse dynamics problem is set, it is simple to rearrange the equation system (33) to obtain the trend of the unknown muscle torques from the phalanges motion laws.An expansion of the coefficients of equation (37) can be found in Appendix B, equation (B.1).


## 6. No-Load Joint Velocities

A test campaign was carried out, in order to evaluate the maximum angular velocities of the finger joints. Ten male and ten female subjects were asked to perform the following operation as fast as possible during 10 seconds (i.e., before any fatigue phenomenon arises to deteriorate the subject performance): starting from the completely extended hand, they have to flex the fingers to a fist and then to extend them again. The maximum flexion angles for the index finger, corresponding to the fist configuration on each cycle, have been statically measured on the subjects as 85 deg for the MCP joint, 105 deg for the PIP joint, and 70 deg for the DIP one. In the following, those angles will be considered negative in accordance with the notations of Section4.1. Ten male and ten female subjects were involved in the investigation: the statistical output of the test is reported in Table 9.Table 9
Joint velocity tests.
Males
FP
AB
AF
MP
AL
NC
MS
AL
DM
RC
Mean
st.dev.
Average freq. (Hz)
Omega max. MCP (rad/s)
Omega max. PIP (rad/s)
Omega max. DIP (rad/s)
Cycles in 10 s
27
27
31
25
20
29
23
27
21
31
26.1
3.843
2.61
12.164
15.026
10.018
Females
SA
TT
NG
FC
EA
GM
VC
SP
MQ
DH
Mean
st.dev.
Average freq. (Hz)
Omega max. MCP (rad/s)
Omega max. PIP (rad/s)
Omega max. DIP (rad/s)
Cycles in 10 s
27
24
17
34
29
27
30
24
21
27
26
4.784
2.6
12.118
14.969
9.979There are no significant differences between the male and the female subjects. The results obtained by our test campaign show velocities in accordance with previous tests, for instance, in [18].The trend of the phalange angles was supposed to follow a sinusoidal law, in the formθj=(Dj/2)(cos(πt/t1)-1), where j=3,4,5. The term Dj is the range of the movement on the jth joint, while t1=10s/numcycles is the period of each cycle. The time differentiation of this law permits obtaining the trend of the velocities, as well as the maximum angular velocity of each joint, equal to θ˙j_max=|-πDj/2t1|.Here, such experimental motion law has been implemented in the dynamic modeling of the finger. The simulation deals with an inverse dynamics case study: given the motion law of the system (i.e., the kinematic angles of the phalanges and their time derivatives), the torques exerted by the muscles on each phalanx are calculated. In order to evaluate correctly such torques in absence of load, it is necessary to insert also the viscoelastic term that can be preponderant with respect to the dynamical term [19]. The values of the stiffness and the damping coefficients for the MCP joint are obtained through an extrapolation of the values of MCP stiffness and damping, reported in a series of tests in [20], for the no-load case, where the applied external force is null. Then, given the fact that the tests in [20] compute translational stiffness and damping coefficients, they are transformed into rotational ones, to be coherent with the dynamic modeling. The resulting coefficients are shown in Table 11, reported in the appendix, together with the anthropometric data and numerical constants imposed in the model to simulate the behavior of a human index finger.The trends of the muscle torquesCm1, Cm2, and Cm3 are reported in Figure 6. First of all, these values can be compared with the maximum torque capability of each phalange: Hasser [18] reports these maximum values as Cmmax=[46302280775]T. The current test, without external load, correctly produces a set of torques that are quite lower than these limits. Moreover, the results shown in [19] state that the average total torque for a complete flexion-extension movement of the MCP joint is 100.8 Nmm for the female subjects and 194.9 Nmm for the male ones: these values are quite near to the ones produced here, where the joint velocity is higher. The same paper states that the contribute of the viscoelastic terms is preponderant with respect to the dynamical one: the same is true for the current analysis, where the coefficients fkj=|Ckj|/|Cmj|, fcj=|Ccj|/|Cmj| and fdj=|Cdj|/|Cmj|, that are, respectively, the torque portion due to the stiffness term, the damping term, and the dynamic term for the jth phalange, are reported in Table 10. However, if [19] shows that the stiffness term is about the double of the damping one, in the current analysis the situation is reversed: it is presumably due to the higher velocity investigated here.Table 10
Percentages of stiffness, damping, and inertial torques with respect to the total torque.
MCP
PIP
DIP
f
k
j
38.05%
38.48%
38.58%
f
c
j
57.58%
60.59%
61%
f
d
j
4.37%
0.93%
0.42%Table 11
Anthropometric data and numerical constants used for simulation.
m
1
=
7.05
g
m
2
=
3.97
g
m
3
=
2.70
g
I
1
=
1000
g
mm
2
I
2
=
240
g
mm
2
I
3
=
120
g
mm
2
k
1
=
0
.
0902
Nm
k
2
=
0.0902
Nm
k
3
=
0.0902
Nm
c
1
=
0
.
0225
Nms
c
2
=
0.0225
Nms
c
3
=
0.0225
Nms
L
1
=
[
50
0
0
]
T
mm
L
2
=
[
20
0
0
]
T
mm
L
3
=
[
25
0
0
]
T
mm
b
1
=
[
25
0
0
]
T
mm
b
2
=
[
10
0
0
]
T
mm
b
3
=
[
12.5
0
0
]
T
mm
e
1
=
[
25
-
8
0
]
T
mm
e
2
=
[
10
-
7
0
]
T
mm
e
3
=
[
12
-
6
0
]
T
mmFigure 6
Trend of the muscle torques.In conclusion, the results of the current simulation are in good accordance with the literature data. More precise results could be obtained as a consequence of a more accurate estimation of the dynamical parameters inserted into the model, in particular the stiffness and the damping coefficients. For example, due to lack of data, in the current simulation these parameters have been set as equal for the MCP, PIP, and DIP joints: in reality, the PIP and DIP joints have different characteristics, which should not be neglected for a deeper investigation.Finally, given the previously described test and the related muscle torque, shown in Figure6, then the mean power associated with the opening and closing of the index finger at maximum velocity in the absence of load can be calculated: it is equal to 1.3 W. The maximum power for the MCP joint in the absence of load is 1.24 W, the maximum power for the PIP joint is 1.94 W, and for the DIP joint it is 0.86 W. This value constitutes a correction of the maximum power estimation for the finger joints, reported by Hasser [18], which states that the absence of external load corresponds to zero muscle torque.


## 7. Conclusions

An exhaustive study of the human hand was performed, dealing with the kinematics of the human hand and showing the matrices with the MDH parameters and detailed equations for both the direct and inverse kinematics of the hand. In addition, the dynamics of a single finger was analyzed and finally written using the same reference systems as per the kinematics. A case study was proposed, and a simulation was completed using the provided anthropometric data, in order to investigate the capabilities of the proposed analytical system.The results of the current study can be exploited to conceive future hand devices. On one side, direct and inverse kinematics constitute preliminary stages for the development of any structure similar to the human hand, for instance, in robotic or rehabilitation hands projects. On the other side, dynamics equations allow the behavior of fingers to be simulated. Hence, the value of these equations is twofold: to model the finger itself, for the use in a control scheme model of a human-machine interface, such as a hand exoskeleton, or to model a finger-like architecture, such as a robotic hand.



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*Source: 910961-2013-09-09.xml*