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Abstract

Since human hands have complex anatomical structure and are hard to be duplicated, lots of researches have been done to simplify it by gesture analysis, functional workspace evaluation and many other approaches. In this article, a novel light-duty four-finger hand driven by tendons is presented. The design is based on principal components analysis theory in order to reduce the minor degrees of freedom and lower the mechanism complexity. Meanwhile, optimization of the structure parameters is carried out with respect to the analyses of grasp ability and the single finger’s dexterity. A few simple and explicit indices are defined to evaluate the hand properties. In addition, this article also discusses the tendon routing method to realize the accuracy control of the tendon-driven hand’s joints. The approach presented in this article can be easily implemented on other mechanical hand platforms and assist the grasp planning in further works.

Keywords

Dexterous hand grasp evaluation optimization design

Introduction

The human hand is often regarded as the most dexterous end effector in existence because it could afford great grasp power with very light structure and realize a wide range of manipulations at the same time. Many researchers have investigated anatomical structures of human hands, aiming to copy the abilities of human hands. These studies suggested several general disciplines and methods to build models according to the skeleton structure.^1
–3 Furthermore, a well-known anatomically correct test bed (ACT) hand was created fully following the structure of human hands.^4
–6 Based on this hand, several human hand’s force properties were discussed.⁷ However, the complex structure of human hand is still very difficult to reproduce, and the numerous degrees of freedom (DoFs) will increase the complexity and instability of the control system. Therefore, lots of researches have tried to simplify the whole hand system by focusing on the grasp ability and reducing the dexterous manipulation ability.^8
–10 Besides, soft joints are also used to increase the grasp universality and stability. This kind of mechanical hands have been widely used in the prosthetic area nowadays.^11
–13 On the other hand, in order to retain limited dexterity on the foundation of reliable grasp, manipulating modes and motion patterns were abstracted by studying the daily grasp posture and the anatomical structure of human hands.^14,15 Two classic sorting methods, principal components analysis (PCA) and Cutkosky taxonomy, were proposed,^16,17 which are currently widely used in posture planning, mechanical structure designing and other fields of robot research. The PCA taxonomy proposed a new coordinate system to describe human hand’s daily postures and the description is a dynamic procedure, which indicates the continuous relationship between different joints, instead of dividing posture into several static states. Depending on this theory, researchers presented several kinds of dexterous hands^18
–20 which could realize lots of anthropopathic postures with few drive motors. Except these mechanical hands with less active driven DoFs, the most advanced dexterous hands with mass of drivers, such as the Awiwi Hand,²¹ which covers most DoFs of human hands, are also designed based on the in-depth study of human hand’s movement patterns.

However, these hands are basically designed based on qualitative analysis and fewer researchers have discussed about the optimization of hand’s mechanism. In this research, a light-duty four-finger tendon-driven hand that is able to realize stable power grasp is designed and introduced based on optimization analysis. Quantitative analyses based on dexterity and posture theories are discussed to obtain optimal geometry parameters of the hand. The evaluation of the grasp ability is also investigated to improve the whole composition. At last, the structure of the whole hand system is presented in detail.

Optimization of hand’s geometry parameters

Deciding the general structure configuration according to setting missions is a prerequisite of creating a mechanical hand. For the sake of discussion, the joints and the phalanges of hand are defined as shown in Figure 1. In this article, realizing stable grasp is the primary goal, as well as light scale, high controllability and certain dexterity of the whole system. Nowadays, hands with two or three fingers are considered as the most preferable and stable configuration for grasping manipulations. But this kind of hands loses lots of dexterities and generally needs to be mounted on reconfigurable bases^22,23 to widen operation abilities. Meanwhile they are more similar to claw-type end effectors than human hands. Referring to anthropopathic hands, those with high dexterity almost fully imitate the structure of human hand. And in order to improve the grasp force, their joints are generally driven by multi-tendons. Thus, driven system enlarges the number of motors and leads to the instability, complexity and integration difficulty of the whole system as the first three hands listed in Table 1. Therefore, these hands are hard to be utilized in human daily life and mostly stay in experimental stage. For a reliable and compact mechanical hand system, the number of driven units must be limited in a reasonable range to simplify the whole system. The general methods are as follows:

using flexible joints instead of rotation joints;

coupling the minor joints and fingers; and

using continuum differential mechanism⁸ and under actuated driven system.

Figure 1.

Definition of human hand’s structure.²¹

Table 1.

Compare of the complexity for different hands.^a

Hand	DoFs	Motors
The ACT hand⁵	23	20
The Awiwi hand²¹	19	42
The Robonaut hand²⁴	12	16
An open source hand⁹	6	6
The RIC hand¹²	6	3
A multigrasp hand prosthesis¹⁰	6	4

DoF: degree of freedom; ACT: anatomically correct test bed; RIC: Rehabilitation Institute of Chicago.

^aThe motors of the ACT hand are only for the thumb, index and middle finger.

However, if too many DoFs are removed especially those of the thumb joints, the mechanical hand loses the dextrous manipulation ability and turns to be a hand gripper as the last three hands shown in Table 1.

According to the discussion above, this article proposed a novel light-duty and compact hand which has enough DoFs to realize common dextrous manipulation. The complexity of the system is decreased by coupling minor DoFs through hand posture analysis. Meanwhile, the geometry parameters of the hand are optimized through grasp evaluation.

The general structure configuration of the hand is shown in Figure 2. Since the ring finger and little finger play the minor part in human hand’s postures, this arrangement has enough ability to complete most daily manipulations and could balance between grasp and dextrous manipulation. Moreover, in order to further simplify the mechanical structure to realize light scale, several minor joints are cancelled. The distal interphalangeal (DIP) joints of fingers except the thumb are replaced with four-bar mechanism, the metacarpophalangeal (MC) joint of the middle finger is fixed and the MC joints of the index and little finger are coupled. In this distribution, the thumb arrangement is extremely crucial. The opposite configuration of the thumb and the other fingers is the key factor of the power grasp. The position and the rotation range of the trapezometacarpal (TMC) joint would largely affect the stability and capacity of precision grasp. In addition, all of the thumb joints need to be fully active controlled to form optimal posture of power and precision grasp cooperated with the other fingers.

Figure 2.

Structure of the light-duty four-finger hand. Thumb is obviously different from the other fingers and all joints are active driven. DIP joints of other fingers are formed by four-bar mechanism; MC joints of index finger and little finger are coupled. DIP: distal interphalangeal; MC: . metacarpophalangeal.

The global coordinate system $\bar{X} - \bar{Y} - \bar{Z}$ of the hand is set at the root of the middle finger as shown in Figure 2. The coordinates of the fingers and joints are named as {I}, {M}, {L}, {T} and {T₁}, {T₂}, {T₃}, {T₄} (the four joint frames of thumb), respectively. The global origin coincides with the origin of {M}. The local coordinates of each phalanx are built following the Denavit and Hartenberg (D–H) method. And geometry parameters are named as follows: phalanx length and joint angles of little finger are defined as L_l1, L_l2, L_l3, L_l4 and θ_l1, θ_l2, θ_l3, θ_l4.

Joint coupling relationship

To build a hand with fewer active driven DoFs, the first problem need to be considered is the priority of each DoF. According to the PCA taxonomy method, all the DoFs including in the first principal component (PC) and adduction of all MC, metacarpophalangeal (MP), interphalangeal (IP) and TMC joints (including lateral motion and flexion) are independently controlled. The DIP and proximal interphalangeal (PIP) joints are coupled following the PC relationship. As some typical samples shown in Figure 3, groups of common human hand’s postures are observed by data glove and series of information about joint angles are obtained. According to the experiment data, motion ranges of every joint are uncovered, as shown in Table 2. It indicates that the TMC joint (θ_t1 = 0, θ_t2 = 0) is in the zero point when the metacarpal bone of the thumb stays in the palm plane and is perpendicular to the middle finger.

Figure 3.

Several postures of daily capture manipulations.

Table 2.

Preset motion range of each joint.

Joint	Range (angle)
TMC	−30° − 110° Side rotation (θ_t1)
	−3° − 30° Flexion (θ_t2)
MP (θ_t3)	0° − 90°
IP (θ_t4)	0° − 90°
MC	0° − 30° Side rotation (θ₁)
	0° − 90° Flexion (θ₂)
PIP (θ₃)	0° − 100°
DIP (θ₄)	0° − 50°

TMC: trapezometacarpal; MP: metacarpophalangeal; IP: interphalangeal; MC: metacarpophalangeal; PIP: proximal interphalangeal; DIP: distal interphalangeal.

The experiment data of the joints are divided into two groups, PIP–DIP joint group and MC–PIP–DIP joint group, and are, respectively, analysed by PCA. The results are shown in Figure 4. In the PIP–DIP joint group, the weight of the first PC is high up to 0.93. According to the eigenvector of the first PCs, the joints can be coupled following the relationship $θ_{3} : θ_{4} = 3.87$ (as defined in Figure 2). Similarly in the MC–PIP–DIP joint group, the sum weight of the first and second PCs is 0.96, so the joints can be coupled following the PC vector $V_{m p d} = W_{m p d 1} \times V_{m p d 1} + W_{m p d 2} \times V_{m p d 2}$ , $θ_{2} : θ_{3} : θ_{4} = 3.8 : 4.45 : 1$ . Meanwhile, the couple relationship of MC–PIP joint group is $θ_{2} : θ_{3} = 1 : 1.15$ . Analysis results above totally depend on the experimental data and do not change with different initial geometry parameters of the hand. So these coupling relationship could guide the whole system design process.

Figure 4.

PCs of joint groups. (a) is the first two PCs of PIP–DIP joint group. The eigenvectors of the first two PCs are $V_{p d 1} = [0.97; 0.25]$ and $V_{p d 2} = [- 0.25; 0.97]$ , respectively, and their weights are $W_{p d 1} = 0.93$ and $W_{p d 2} = 0.07$ . (b) The first three PCs of MC–PIP–DIP joint group. The eigenvectors of the first three PCs of are $V_{m p d 1} = [0.42; 0.88; 0.22]$ , $V_{m p d 2} = [0.91; - 0.39; - 0.17]$ and $V_{m p d 3} = [0.06; - 0.27; 0.96]$ , respectively, and their weights are $W_{m p d 1} = 0.75$ , $W_{m p d 2} = 0.21$ and $W_{m p d 3} = 0.05$ . PC: principal component; PIP: proximal interphalangeal; DIP: distal interphalangeal; MC: metacarpophalangeal.

Initial geometry parameters

In general, mechanical hands share the same size scale as human hands. Considering the difficulty of manufacture, the length of phalanxes is designed about 1.3 times the length of human fingers. Their initial values are taken from numerical ranges, $L_{1} \in [10, 20]$ mm, $L_{2} \in [30, 50]$ mm, $L_{3} \in [25, 35]$ mm and $L_{4} \in [25, 35]$ mm. The distances between fingers are also set approximate to human hands. Thus, the origin coordinates of index, middle and little fingers are $P_{I 0} = [0; 30; 0]$ , $P_{M 0} = [0; 0; 0]$ and $P_{L 0} = [0; - 30; 0]$ . The thumb is an exception. Maxime Chalon³ summarized several guidelines from different aspects of thumb designing. The two key points adopted in this research are the following:

The rotation axis, Z-axis, of {T₂}, {T₃} and {T₄} (local coordinates of joints defined in Figure 2) are not parallel to each other but a little bit inclined. This character increases the contact area between the thumb tip and the target. Therefore, there is a $θ_{twist} = - 5 °$ twist angle in this investigation.

The origin position of the thumb coordinates could be found out from two postures shown in Figure 5. When all fingers are extended like Figure 5(a), the limiting position of the thumb tip in $\bar{X}$ direction is around the PIP joint of index finger. Meanwhile, when all the fingers gather together as Figure 5(b), the tips of all fingers should be able to contact each other. Based on these two general recognitions, the coordinates origin of the thumb is settled at $P_{T 0} = [- 73, 30, 0]$ .

Figure 5.

Position of the thumb origin.

Another problem is the direction of {T₁}, {T₂} coordinates. In the view of anatomy, the TMC joint is inclined to be described as saddle shape¹. The Z-axes of {T₁}, {T₂} are neither intersected nor perpendicular with each other, and they both form a certain intersection angle with $\bar{X}$ -axis. Adopting this structure, hands could pass the Kapandji test and reach 10 scores, which indicates the complete opposite structure and preferable dexterity. But this structure has limited influence when utilized for four-finger hand, especially hands with coupled MC joints, in which the index and little fingers are kept symmetrical about $\bar{X}$ -axis. On the other hand, when the Z-axes of {T₁}, {T₂} are arranged along $\bar{X}$ -axis and $\bar{Y}$ -axis, the power grasp planning process will be largely simplified with little loss of dexterity. In conclusion, this simpler distribution is more preferable.

Optimization of phalanxes length

Based on the above initial values set according to several qualitative analysis, the length parameters of phalanxes are optimized through dexterity analysis for the single finger (except the thumb). Furthermore, the dexterity is measured by the condition number C_δ of the Jacobian matrix²⁵ equation (1). The details of deducing the Jacobian matrix and the condition number are given in Appendix 1.

condition number: C_{δ} = δ_{min} / δ_{max}

where δ is the singular value of J.

In this research, the dexterity of the whole system is not discussed because it is affected by too many parameters, such as the geometry parameters of the structure and the relative positions between different fingers. Aiming at single finger, the Jacobian matrix $J (l_{1}, l_{2}, l_{3}, l_{4}, θ_{1}, θ_{2}, θ_{3})$ is built following the D–H parameters. Among the seven variables of Jacobian matrix, θ₁, θ₂ and θ₃ are dynamic parameters that change during manipulations. When each parameter of phalanx varies in the value range, the variation tendencies of C_δ in different joint angle conditions are calculated, as shown in Figure 6. The influence of each parameter can be summarized as follows:

C_δ rises with the increase in θ₃. Meanwhile, when θ₁ > 0, C_δ rises with the increase in θ₁, and conversely when θ₁ < 0.

C_δ has a leap around θ₂ = 80° and stays stable in other stages.

C_δ does not change with the L₁. The decrease in L₄ will increase C_δ.

L₂ and L₃ will have reverse effects on C_δ when the values of θ₁, θ₂ and θ₃ vary. In different conditions of θ₁ and θ₂, C_δ rises as L₂ declines. When θ₃ < 90°, C_δ increases with the increase in L₂ and the decrease in L₃, but conversely when θ₃ > 90°.

Figure 6.

Dexterity of single finger. Unit of l₂, l₃, l₄ and l₅ is millimetre, and unit of θ₁, θ₂ and θ₃ is degree.

Above all, although the Jacobian matrix has as many as seven variables, most of these variables have specific influence on C_δ. Obviously, $Θ = {[θ_{1}; θ_{2}; θ_{3}] | | θ_{1} |_{max}, θ_{2} > 80 °, (θ_{3})_{max}}$ is the joint angles range when the value of C_δ is great. And the optimal value of L₄ is $min (L_{4}) = 25 mm$ . The length of L₁ has no influence on C_δ with the change of Θ. However, when L₃ and L₂ differ, C_δ decreases with the rise of L₁ (shown in Figure 7). Therefore, the optimal value of L₁ is also $min (L_{1}) = 10 mm$ .

Figure 7.

Change of condition numbers with different joint couples. (a) L₂ = 30 mm, L₄ = 25 mm; θ₁ = 2.9, θ₂ = 1.5, θ₃ = 15.7 (b) L₃ = 25 mm, L₄ = 25 mm; θ₁ = 2.9, θ₂ = 1.5, θ₃ = 15.7.

On the other hand, according to the conclusion above, L₂ and L₃ have a metabolic and more complex influence. Especially, θ₃ changes the influences of L₂ and L₃ on the system. Therefore, the maximum and average value of C_δ with different θ₃ changing from 0° to 120° and different value combinations of L₂ and L₃ are computed and compared, as shown in Figure 8. Considering manufacture problems that the proximal phalanx should have enough space to accommodate tendons, sensors and electronic components. In addition, to insure a higher average value of C_δ at the same time, L₂ = 50 mm and L₃ = 35 mm are settled.

Figure 8.

Average and maximum value of C_δ in different conditions.

Above all, the length of the phalanxes are L₁ = 10 mm, L₂ = 50 mm, L₃ = 35 mm and L₄ = 25 mm. The length of the thumb phalanxes are set as the same with other fingers for the coherence of the whole hand.

Parameters of four-bar linkage

As discussed above, the distal and medial phalanxes of fingers except for the thumb are replaced by a four-bar linkage as shown in Figure 9, of which L₃ and the coupling relationship θ₃ : θ₄ are settled. The length L_A = 8 mm is determined according to the structural limit, and L_B = 31.44 mm and L_C = 4.6 mm are computed through graphical method. It needs to be emphasized that the coupling relationship between θ₃ and θ₄ can be formulated by a three-order polynomial as equation (2). With the increase in θ₃, the value of θ₃ : θ₄ falls down. This feature is determined by the four-bar linkage. So the ratio of θ₃ and θ₄ satisfies the results from PCA only at the initial phase of flexion . Meanwhile, DIP can reach a larger rotation angle at the end of flexion, which ensures a complete handhold posture.

\begin{array}{l} θ_{4} = - 0.0645 θ_{3}^{3} + 0.2812 θ_{3}^{2} + 0.1913 θ_{3} - 0.0052 \end{array}

Figure 9.

Four-bar linkage of DIP and PIP joints. DIP: distal interphalangeal; PIP: proximal interphalangeal.

where unit θ of is radian.

Evaluation of power grasp ability

Most researchers evaluate the grasp ability after the prototypes are built up. However, in this research, evaluation is accomplished during designing to optimize the global parameters, including the relative position of four fingers and limits of θ_t1 and θ_t2.

According to the classical Cutkosy taxonomy¹⁷, power grasp have two subclassifications, prismatic and circular grasp. These two types of postures are distinguished by the lateral rotation of the MC joints of index, middle and little fingers. Prismatic grasp postures contain no lateral rotation and are mostly used for carrying cylindrical targets. Circular grasp postures have lateral rotation and are used for grasping spherical targets. In order to get simple and clear numerical indices of grasp ability, these two types of postures are abstracted into the $\bar{X} - \bar{Z}$ and $\bar{X} - \bar{Y}$ plane separately and quantitatively evaluated by the envelope.

Discussion in $\bar{X} - \bar{Z}$ plane

Since the four-finger hand in this article has many DoFs, several constrains should be set in the discussion.

The hand keeps the completely opposite posture so the lateral rotation angle of the fingers are constants, $θ_{t 1} = 90 °$ , $θ_{l 1} = θ_{m 1} = θ_{i 1} = 0 °$ .

Except for the thumb, θ₂ and θ₃ of the other fingers change following the PCA rules ( $θ_{2} : θ_{3} =$ constant) and these fingers are defined as PCA finger.

The posture of the thumb is determined by two principles. First, four fingers form force-closure grasp in $\bar{X} - \bar{Z}$ space. Second, the area of the grasp posture envelope is maximized.

In order to present explicit and feasible evaluation method, constraints above are abstracted to simple geometry limitations. Therefore, the evaluation can be easily applied to other different hands. As the mechanical hand presented in this research, the grasp postures are evaluated and divided into four stages as shown in Figure 10. For each stage, the status of PCA finger is regarded as a given condition.

Grasp stage 1: Because the flexions of PCA fingers are very slight, force closure cannot be formed in $\bar{X} - \bar{Z}$ space. Therefore, this stage is defined as ‘Bad Power Grasp’. However, these postures are generally utilized for grasping flat targets (shown in Figure 10(a)). Because force equilibrium can be realized only in one direction in this situation, force criterion is replaced by position criterion that ensures the normal vector of the finger tips being parallel to each other, in order to obtain valid posture solutions. Frictions between the fingers and the target help to balance the target’s gravity. The joint angles of the thumb can be solved from equation (3), but the solution is not unique. To acquire the unique solution while ensuring a relatively large grasp envelope, θ_t2 and θ_t3 are limited to the extrema.

{\begin{array}{l} \frac{π}{2} + θ_{t 2} + θ_{t 3} + θ_{t 4} = θ_{m 2} + θ_{m 3} + θ_{m 4} \\ θ_{t 2} = min (θ_{t 2}), θ_{t 3} = max (θ_{t 3}) \end{array}

Grasp stage 2: This stage is a transition from bad power grasp to normal power grasp and force-closure grasp is still unattainable. However, this series of postures is efficient to grasp targets with large size such as basketballs, since friction acts a really important part in grasping. The evaluation equation of this stage is the same as equation (3), except for θ_t3 = 0°.

Grasp stage 3: From this stage on, all the postures can fulfil the force criteria. Since the statement of the PCA fingers are definite, the normal vector of thumb distal phalanx ${\vec{n}}_{t 4}$ must point to the $- \bar{Z}$ direction to maintain the force-closure posture. Meanwhile, the distance between the thumb tip and the PCA-finger tips can be used as the criteria of the envelope, which is much simpler than calculating the cover area. The evaluation equation of this stage can be formulated as equation (4). Besides, the $\bar{Z}$ coordinates of all fingers are limited to be equal to standardize the evaluation criterion, which will be discussed as follows.

\begin{array}{l} | (P_{i t i p})_{x} - (P_{t t i p})_{x} |_{max} \\ constraints : {\begin{array}{l} {(n_{t 4})}_{z} < 0 \\ {(P_{t t i p})}_{z} = (P_{i t i p})_{z} \end{array} \end{array}

Grasp stage 4: This stage is similar to stage 3 and shares the same criteria equation. Since the hand postures during these two stages meet the force criteria strictly, they are considered as the most effective stages for power grasp. The only difference between these two stages is that θ_t3 in the fourth stage is not equal to zero, which means contact area between the fingers and the target increases. Thus, hand postures in this period are more suitable for grasping small size targets and could afford larger contact force.

Figure 10.

Grasp stages in $\bar{X} - \bar{Z}$ space. These graphics show the change process of each stage. Dot lines stand for the initial postures and solid lines for the terminal postures. The blocks of colour stand for the grasp envelopes. Only initial and terminal states of each stage are indicated and the other states are just showed by the trajectories of joints.

Corresponding to the grasp stages discussed above, two assessed indices, the acreage and the size ratio of grasp envelope, are proposed to quantitatively evaluate the grasp ability. The acreage of the grasp envelope shows the sizes of the affordable targets. The size ratio is defined as the largest size of grasp envelope in $\bar{X}$ and $\bar{Z}$ direction, which can not only show the scale of affordable targets, but also partly show the shape properties of targets. During the whole posture changing period, the envelope area increases first and then decreases as shown in Figure 11(a). Meanwhile, the size ratio is always increasing as shown in Figure 11(b). This is coincident with the discussion above that the postures of the first and second stages are suitable for targets with large dimension in particular direction and the postures of the third and fourth stages are suitable for targets with balanced size.

Figure 11.

Quantitative evaluation results of grasp postures. Points on the line are the demarcations of two different stages. (a) Acreage of different capture states and (b) size ratio of grasping envelope.

The evaluation results can also be used for grasp planning because the grasp envelope calculated above stands for the envelope of grasp contact points in actual. According to this consideration, the grasp postures can be chosen based on the coherence of planned contact points and known grasp envelope.

Discussion in $\bar{X} - \bar{Y}$ space

Postures of circular grasp have actions both in the $\bar{X} - \bar{Y}$ and $\bar{X} - \bar{Z}$ planes. The grasp ability in the $\bar{X} - \bar{Z}$ plane has been explicitly discussed above. In this section, based on the circular grasp, the grasp ability in the $\bar{X} - \bar{Y}$ plane is discussed further. The constraints are as follows:

PCA finger: In order to keep the grasp envelope in the $\bar{X} - \bar{Y}$ plane to remain the gravity of the target on $\bar{Z}$ -axis, the tips of PCA fingers must be perpendicular to the $\bar{X} - \bar{Y}$ plane. In addition, to increase the force-closure feasible region of the thumb in the $\bar{X} - \bar{Y}$ plane, the lateral rotations of index and little fingers are in the extreme position, and their angles satisfy a fixed relationship $θ_{i 1} = - θ_{l 1}$ .

{\begin{array}{l} θ_{2} increases by certain step value \\ θ_{i 1} = max (θ_{i 1}) \\ θ_{2} + θ_{3} + θ_{4} = π / 2 \end{array}

Thumb: The normal vector of the thumb tip ${\vec{n}}_{t 4}$ is limited in the same plane with ${\vec{n}}_{i 4}$ , ${\vec{n}}_{m 4}$ and ${\vec{n}}_{l 4}$ of PCA fingers. Under this premise, force-closure criteria can be guaranteed by keeping $\vec{-} n_{t 4}$ stay between ${\vec{n}}_{i 4}$ and ${\vec{n}}_{l 4}$ . In this research, ${\vec{n}}_{4}$ is the same as the Y-axis of {4} frame of each finger.

\begin{array}{l} | (P_{t t i p})_{x} |_{min} \\ θ_{t 1} increases by certain step value \\ constraints : {\begin{array}{l} π / 3 \leq ∠ Y_{t 4} \bar{Y} \leq 2 π / 3 \\ Y_{t 4} • \bar{Z} \leq 0 \\ {(P_{t t i p})}_{z} = (P_{m t i p})_{z} \end{array} \end{array}

Since the postures of the thumb that meet the above conditions are limited, all the feasible solutions can be calculated. Figure 12 shows several grasp postures in the $\bar{X} - \bar{Y}$ plane. As shown in Figure 12(a), when θ₂ is small, the postures are suitable for grasping disc targets, and as θ₂ gradually grows up, the postures will be capable for spherical and cylindrical targets and then turns into precision grasp.

Figure 12.

Grasp in $\bar{X} - \bar{Y}$ plane. Units of axes are millimetre. (a) θ_i2 = 0.192, (b) θ_i2 = 0.593, (c) θ_i2 = 0.995 and (d) θ_i2 = 1.4.

During the changing process, the main influence factors of the grasp envelope are the position and the rotation range of TMC joint. As shown in Figure 13(a), when θ₂ is small, the thumb postures have many invalid solutions since the range of θ_t2 is not large enough. At this moment, the thumb tip is in the introflexion status and cannot implement grasping in fact. In order to enhance the grasp ability for disc targets, the inferior limit of θ_t2 should be less than a certain value, which depends on the moving mode of PCA fingers. Figure 13 shows the grasp envelopes when θ_t2 has different inferior limits. When the limit decreases to −60°, the thumb gains more valid postures and the valid grasp range is amplified greatly. However, they do not keep changing as the inferior limit of θ_t2 further decreases.

Figure 13.

Grasp in $\bar{X} - \bar{Y}$ plane with lower θ₂. Units of axes are millimetre.

Figure 13(b) and (c) compares the results of different TMC joint positions. In Figure 13(c), the TMC joint is set in the same $\bar{X}$ position as the middle finger instead of the index finger. This improvement optimizes the force distributions of the finger tips and increases the grasp stability by reinforcing the antagonistic structure of postures. However, it contributes nothing to the grasp envelope and reduces the space in the palm for tendons, which means this arrangement is suitable for hands driven directly by motors.

Since there exist several thumb postures corresponding to each PCA-finger posture, it’s hard to present a certain envelope index to evaluate the grasp. Therefore, the angle value of θ_t1, which could indirectly reflect the capture range, is taken as the evaluation index instead. Furthermore, effective rate δ_t1 of θ_t1 is defined in equation (7). The fitting curves of the three indices are shown in Figure 14. It indicates that there are few, even no valid solutions at the beginning, if the inferior limit of θ_t2 is too large. In addition, with the bending of PCA fingers, the effective range of θ_t1 decreases step by step.

δ_{t 1} = \frac{max (θ_{t 1}) - min (θ_{t 1})}{range of θ_{t 1}}

Figure 14.

δ_t1 Changing process. Lines in the graphics fit the variation tendency of the datum. If there exists no valid solution, the maximum, minimum and effective rate are all set to zero. The unit of θ_i2 is radian. (a) TMC = [2π/9,−π/6], (b) TMC = [2π/9,−π/3] and (c) different thumb origin_y.

The discussion above puts forward several very simple indices to preliminarily evaluate the grasp ability of the four-finger hand. On this basis, the range of θ_t2 is revised. Meanwhile, the evaluation results can also be used for future planning process. In addition, this analysis can be easily carried out upon other hands and helpful to determine the key structure parameters.

Structure of the four-finger tendon-driven hand

General structure

Based on the analysis above, the four-finger tendon-driven hand is designed using the geometry parameters given in Table 3. Its prototype is shown in Figure 15. This mechanical hand contains four fingers including the PCA fingers and the thumb. All the DIP joints of the PCA fingers are coupled with the corresponding PIP joints and the lateral rotations of index and little finger are coupled. Flexion of all joints is controlled by tendons and extension is passively controlled by springs. The linear motion of all the tendons is realized by ball screws. The lateral rotation of the TMC joint is driven directly by a motor, in order to reduce the complexity of the thumb tendon arrangement. The palm includes a tendon guiding system, a 2-DoFs wrist and driven motors of MC and TMC joints. The other driven motors are installed on the forearm. The whole structure of the hand is manufactured by a three-dimensional (3D) print and the total mass is about 0.5 kg.

Table 3.

Geometry parameters of the four-finger tendon-driven hand.

Length (mm)	L ₁	L ₂	L ₃	L ₄
PCA fingers	10	50	35	25
Thumb	0	50	35	25
Joint angle (°)	θ ₁	θ ₂	θ ₃	θ ₄
Index finger	[0,30]	[0,85]	[0,105]
Middle finger		[0,85]	[0,105]
Little finger	[−30,0]	[0,85]	[0,105]
Thumb	[−30,110]	[−60,40]	[0,90]	[0,90]
Twist of IP and MP joints, θ_twist = −5°

PCA: principal components analysis.

Figure 15.

The prototype of the four-finger tendon-driven hand.

The control system is inherited from the last generation of our mechanical hand²⁶, as shown in Figure 16, which uses EC13 DC brushless motors from Maxon Company (Jiangsu, China), Gold Solo Whistle drivers and Gold Maestro (GMAS) motion controllers from Elmo Company (Shanghai, China). Moreover, all the joints are equipped with position sensor in order to supervise the change of the joint angles.

Figure 16.

The driven system of the hand.

Discussion about the tendon system

While tendon driven is a kind of indirect driven type, improper arrangement will enlarge the uncertainty of the control system, such as the inability to identify the relationship between the tendon length and the joint angle. Therefore, the design of the tendon system is quiet crucial for the mechanical hand. The primary purpose is to build up complete guiding path for each tendon to ensure the length of tendons being calculable in every moment.

Guiding path for a single finger

Figure 17 shows the arrangement of PCA fingers in detail. The tendon and the spring of the same joint are arranged on the opposite side of the finger. When the tendon passes an axle, it twines around the axle for a certain angle in order to keep the tendon length countable and to avoid the separation of the tendon and the axle. For each tendon, except for the joint driven by itself, the pressure inflicted by the tendon on the axles within its path always points to the axis and generates no rotational torque. However, the frictions between the tendon and the axle cause torques, so bearings are introduced into the hand to lead the tendons, in order to reduce the frictions as much as possible.

Figure 17.

The tendon path of the single finger. The dash line is the path of the tendons.

At this stage, the change of the tendon length can be calculated precisely by equation (8). ΔL_i stands for the length change of the tendon mounted on the i th joint. Δθ_i is the angle variate of the i th joint and r_i is the joint radius of the i th joint. Δθ_j, r_j are the parameters of other joints on the guiding path.

Δ L_{i} = Δ θ_{i} r_{i} + \sum Δ θ_{j} r_{j}

Guiding path in the palm

The guiding path in the palm is a 3D path. It should ensure the length of all the tendons countable under the influence of lateral rotation of the first joint of fingers and the two rotation DoFs of the wrist. At the same time the whole system should be compact. Therefore, two basic routing methods are proposed as shown in Figure 18. Two bearings in the same plane and with paralleled axes are used to lead the tendons to pass through the rotation axle. Meanwhile, two bearings with perpendicular axes lead the tendons to change the routing plane. With different combination of these two basic methods, it is possible to guide the tendons strictly in the 3D space. At this stage, the relationship between the length and the angle is no longer a multiple relationship as above but can be presented as a monadic equation determined only by the first lateral rotation joint of each finger.

Figure 18.

Two methods of tendon orienting. (a) The tendon orienting method of rotation axis. (b) The method of changing the tendons path plane.

According to the discussion above, equation (8) is rewritten as equation (9). $Δ θ_{w 1}, r_{w 1}, Δ θ_{w 2} and r_{w 2}$ stand for the variation of the wrist joint and $f_{i} (θ_{1})$ is the function of the lateral rotation angle.

\begin{array}{l} Δ L_{i} = Δ θ_{i} r_{i} + \sum Δ θ_{j} r_{j} + f_{i} (θ_{1}) + Δ θ_{w 1} r_{w 1} + Δ θ_{w 2} r_{w 2} \end{array}

Conclusion

Aiming to imitate the abilities of human hand with simple and manufacturable structures, this article presents a novel light-duty tendon-driven hand with four fingers. The DoFs of the system are reduced based on the PCA theory. And optimization of structure parameters is carried out based on the analysis of dexterity and grasp posture. Through these discussions, the whole system is able to achieve low mass and low complexity, while a majority of daily gestures can still be implemented. In addition, considering about further reducing the mass, most of the active joints are driven by tendons. As to the control difficulties increased by the tendon system, a routing scheme is introduced. Therefore, the tendon length can be decoupled with the joint angle through mathematical methods to keep the tendons countable during manipulations.

In the future work, we will further study the relationship among the grasp posture, the tendon tension and the grasp force. The grasp force is a key index for the grasp ability and stability, which mainly depends on the driven motors and the grasp postures. On this basis, combining the evaluations discussed in this article, the grasp posture planning is also going to be studied further.

Footnotes

Acknowledgements

The authors express their sincere thanks to all the editors and the reviewers for their critical and constructive review of this article.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by the National Natural Science Foundation of China (no. 51675264) and Funding of Jiangsu Innovation Program for Graduate Education (no. KYLX16_0388).

Appendix 1

The detailed calculating process of the condition number C_δ discussed in the ‘Optimization of phalanxes length’ section is based on the single finger and divided into several steps as follows. The index finger is chosen as an example.

The frames for this finger are built following the D–H convention as shown in Figure 1A. {I₀} is the fixed base frame and attached to the first joint of the index finger. {I₅} is attached to the finger tip and has the same orientation as {I₄}. Meanwhile, the transformation 5 1 Z , 5 2 Z , 5 3 Z , 5 4 Z , 5 5 Z and rotation matrices 5 0 R are presented by the D–H parameters. Each transformation can be written as follows:

First, the Jacobian matrix I 5 J reference to {I₅} frame is formulated. The j th column of I 5 J is written as follows:

where (.)_z stands for the z-component of a vector. The Jacobian matrix I 0 J reference to the base frame {I₀} is formulated as equation (1C).

The relationship between the finger-tip velocity and the joint velocity is formulated as equation (1D). v_ft stands for the linear velocity and the angular velocity of the finger tip in the Cartesian space and is denoted as v f t = [ v x v y v z ω x ω y ω z ] T . θ ˙ stands for the joint velocity in the joint space and is denoted as θ ˙ = [ θ ˙ i 1 θ ˙ i 2 θ ˙ i 3 θ ˙ i 4 ] T

Generally, θ ˙ can be calculated as θ ˙ = I 0 J − 1 v f t . However, I 0 J 6 × 4 has no left inverse matrix. Hence, I 0 J is transformed through elementary row operation at first. The relationship between the finger-tip velocity and the joint velocity is rewritten as E r v f t = E r I 0 J θ ˙ , in which E_r stands for the elementary transformation matrix. The new Jacobian matrix is defined as J ′ = E r I 0 J and has two zero rows, i.e. J ′ = [ J ′ 4 0 0 ] T . Equation (1D) can be expanded as follows:

where c 1 , s 1 , x c are parameters during the derivation. Equation (1E) shows that w_y and w_z are linearly related to v ′ f t = [ v x v y v z w x ] T and can be simplified as equation (1G).

As the definition of the condition number²⁵ C_δ, we obtain

On the other hand, while

we obtain

Equation (1I) is rewritten as

As shown in equation (1M), δ_i is the singular value of I 0 J 1 . Therefore, the condition number C_δ can be obtained from original Jacobian matrix I 0 J . The Jacobian matrix obtained in this article for the four-finger hand is formulated as equation (1N).

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