Abstract
Purpose
This paper presents a method to elaborate the selections of these parameters to achieve stable grasps. The performance of a prosthetic hand is mainly determined by its mechanical design. However, the effects of the geometric parameters of the hand configuration and the object sizes on the grasp stability are unknown.
Design/methodology/approach
First, the thumb functions of human hands are analyzed based on the anatomical model, and the configuration characteristics of the thumbs for typical prosthetic hands are summarized. Then a method of optimizing the thumb configuration is proposed by measuring the kinematic transmission performance of robotics. On the basis of the thumb configuration analysis, a design method of the prosthetic hand configuration is proposed based on form closure theory. The discriminant function of form closure is used to analyze and determine the hand configuration parameters.
Findings
An application of this method – the newly developed HIT V prosthetic hand – elaborates the optimization of the thumb configuration and the hand configuration, where the relation between the key hand configuration parameters and the discriminant function on condition of satisfying form closure, sustained by analytical equations and graphs, is revealed and visualized. An experimental verification shows that it is an effective method to design the prosthetic hand configuration available for grasping typical objects in our daily life.
Originality/value
The paper shows how to easily determine the geometric dimensions of the palm, phalanges and hand configuration, so that the desired range of object sizes can be obtained.
Keywords
Citation
Jiang, L., Zeng, B. and Fan, S. (2015), "A design approach to the configuration of a prosthetic hand", Industrial Robot, Vol. 42 No. 4, pp. 359-370. https://doi.org/10.1108/IR-02-2015-0029
Publisher
:Emerald Group Publishing Limited
Copyright © 2015, Emerald Group Publishing Limited
1. Introduction
As the special role of the thumb in grasping and manipulating is widely recognized, the thumb design of anthropopathic hands has been one of the most important interests. Kapandji (1971) considered that the thumb could move opposite to the other fingers to synergistically accomplish most of functions of human hands. In recent years, the dexterous hand designers have made great efforts to replicate the complex motions of the thumb. Each of the thumbs of DLR-I hand (Hirzinger et al., 1999), DLR-II hand (Butterfass et al., 2001), HIT-DLR II hand (Liu et al., 2008), Gifu-II hand (Kawasaki et al., 2002), Gifu-III hand (Mouri et al., 2002), Dexhand (Chalon et al., 2011) and Utah/MIT hand (Jacobsen et al., 1984) has two degrees of freedom (DOFs) (flection/extension and abduction/adduction) in trapezium–metacarpus (TM) joints. The thumb of HIT-DLR I hand (Liu et al., 2008) has an additional DOF to realize the motion relative to the palm, which improves grasping performance greatly. Similarly to the dexterous hands, many prosthetic hands such as Smarthand (Cipriani et al., 2010), RTR II hand (Zollo et al., 2007), extrinsic actuation (EA) hand (Dalley et al., 2009) are capable of abducting/adducting, although they have fewer DOFs than dexterous hands due to the limited size and weight. Furthermore, considering the practicability, the abduction/adduction motions of the thumbs of i-limb hand (Connolly, 2008) and bebionic hand (Medynski and Rattray, 2011) are manually adjusted, which is beneficial for reducing weight and structure complexity on the basis of ensuring the thumb functions. Although various forms of thumb designs exist, most of them are dependent on designers’ experience and inspirations on mimicking human beings. To replicate the thumb functions of human hands more objectively, it is important to know well the anatomical structure and characteristics of human thumbs, which will helps to simplify the thumb model. Then, some proper quantitative measures can be used to determine the key model parameters.
The measures of a robot performance can be generally divided into two categories: local measures and global measures. The former includes manipulability, the minimum singular value of Jacobian matrix, the condition number of Jacobian matrix, joint range availability and so on. Manipulability (Yoshikawa, 1985) is used to describe singularity based on the determinant of Jacobian matrix, but cannot be used as a practical measure of the ill-conditioning degree. The minimum singular value of Jacobian matrix (Klein and Blaho, 1987) can be used as a measure of controlling an upper bound on required joint velocities. The end-effector responds faster as it increases. The condition number of Jacobian matrix (Salisbury and Craig, 1982) is described to evaluate the kinematic isotropy and the stability of forward/inverse kinematics solution. Joint range availability (Liegeois, 1977) is used to judge the evenness of a joint angle distribution and is easily computed. Global measures include workspace and global performance index (GPI). The latter is the mean of local measures in workspace (Gosselin and Angeles, 1991). For a dexterous hand, the dexterity of the thumb can be evaluated synthetically by both the two categories of measures. However, only the local measures can be used to evaluate the dexterity of the thumb of a prosthetic hand because the workspace of a prosthetic hand is usually a curve or a surface.
In order to acquire great grasping performance, it is not comprehensive only to optimize the thumb configuration. There are many design parameters about the topology of the hand such as the numbers of fingers and phalanges per finger; the parameters about the geometry of the palm and the relative length of the phalanges (Kragten et al., 2011); and the parameters about the thumb position on the palm and the relative position of all the fingers (included in so-called the hand configuration). It is unknown about the difference of the importance degree of these configuration parameters, and how they are synthetically selected when the condition of stable grasps is satisfied. Conventional methods can be summarized as follows:
The configuration design is mainly dependent on the designers’ experience together with the grasping simulations. The simulations are used for verifying major grasping patterns and further optimizing the hand configuration parameters. This method has the obvious limitations on conducting the hand configuration design, which results from not only a large amount of simulation work coordinated with the mechanism design but also a lack of theoretical basis such as HIT-DLR prosthetic hand (Huang et al., 2006).
Some important fingers are selected to be optimized specially. In a study by Wang et al (2012), a configuration method of the thumb position on the palm was proposed through optimizing two typical postures based on the Euler Theorem about Finite Rotation of Rigid Body. However, the method is not suitable for a prosthetic hand configuration design because the optimization is related with workspace volume, which a general prosthetic hand cannot form. In a study by Kragten et al. (2011), the effect of the dimensions of the phalanges and palm for a two-finger underactuated hand on the range of objects that can be successfully grasped was analyzed based on the planar geometric theory. Then optimal performance was considered as the ability to grasp the largest range of objects normalized by the size of the hand. However, this method was mainly applied to design a planar underactuated hand with two fingers and is not suitable for the configuration design of a multi-finger hand. It is worth emphasizing that the Kragten’s elaboration about a design method is taken as a reference in this paper.
The human hand model is considered as a standard to conduct the design of prosthetic hand configurations to accurately replicate the human hand. In a study by Shin et al. (2011), a human hand Denavit-Hartenberg (DH) model was constructed and verified by using a visual tracker. Then, a robot hand was developed based on the analysis data of human hands. In a study by Berceanu et al. (2010), a new experimental approach was proposed regarding the kinematics of the human finger movement, and the laws of variation for the main joints in the human finger were determined by using video capture of a particular finger movement. In a study by Tarnita and Marghitu (2013), an anthropomorphic hand arm system was developed and a study based on the tools of nonlinear dynamics to visualize the steady state of kinematics of human wrist joint movement was presented. Based on analyzing the kinematic data of the flexion-extension angles for an artificial system and for human hand motion, the periodic movement of the wrist joint of the artificial hand system was concluded. All the papers above inspire us to analyze a prosthetic hand and conduct its design considering more human data. However, there are many differences between prosthetic hands and human hands such as grasping dexterity. It is a long way and hard to achieve the entire equality due to the limitations such as actuators. In addition, a highly dexterous hand is hardly available for amputees.
To overcome the disadvantages of the methods applied to the prosthetic hand configuration design, a method is detailed to determine the geometric design parameters including the thumb configuration and the hand configuration. The innovation of this method is that the effect of these geometric parameters and object sizes on the grasp stability is addressed based on the form closure theory and visualized in a design chart by an application. The method is expected to serve to easily determine the geometric dimensions of the palm, phalanges and hand configuration, so that the desired range of object sizes can be obtained.
The structure of the paper is as follows. Section 2 presents a method to optimize the thumb configuration by the measures of evaluating motion and force transmission performance of robotics. Section 3 presents a method to optimize the hand configuration based on the form closure theory. A case and the relevant results of the optimization are demonstrated in detail in Section 4, followed by conclusions in Section 5.
2. Design of thumb configuration
2.1 Functionality analysis of thumb
From an anatomical point of view, the thumb is composed of five bones and four joints. The five bones are, respectively, Scaphoideum, Trapezium, Metacarpus, Proximal Phalanx and Distal Phalanx. The four joints are, respectively, scaphoideum–trapezium (ST) joint, TM joint, metacarpus–phalanx (MP) joint, distal–inter–phalanx (DIP) joint, where ST joint has little impact on the thumb function because its movement is only in a small range, thus not considered in this paper.
The three joints of the thumb have five DOFs in total, and the characteristics are summarized (Bullock et al., 2012; Rezzoug and Gorce, 2008; Santos and Valero-Cuevas, 2006; Valero-Cuevas et al., 2003; Wu et al., 2009). The TM joint is a saddle joint with two DOFs (abduction/adduction and flection/extension). The two axes are close but not intersecting each other. The distance between them is unknown, and the angle is about 40 degree. The MP joint is a condyloid joint with two DOFs. The complex movement in the two directions leads to the rotation of the proximal phalanx. The DIP joint is a hinge joint with a flection/extension DOF. In general, DIP joint axis and the thumb axis are considered to present a small declination angle rather than perpendicular to each other, which makes the distal phalanx pronate during flection. The thumb movement is determined by the synergy of TM, MP and DIP joints, where TM joint plays the most important role.
2.2 Configuration analysis of thumb
It is almost impossible to replicate exactly a human thumb due to the limitations on current technologies such as actuators. Consequently, it is vital for prosthetic hands to realize the basic functions of human thumbs by simplifying a thumb model. Because the thumb of a prosthetic hand has relatively few DOFs, the configurations of typical prosthetic hands can be categorized into two types: “211” and “111”.
A “211” configuration is the type where DIP joint and MP joint, respectively, have a DOF, while TM joint has two DOFs, as shown in Figure 1(a). It can be seen from the figure that the two axes A1 and A2 of TM joint have a distance. According to the difference of their spatial angle, they are further divided into orthogonal and non-orthogonal axes. This type of thumb is easy to construct and is capable of grasping most of the objects in the daily life. Therefore, this configuration is applied to quite a number of thumbs of prosthetic hands, such as Smarthand, Cyberhand (Carrozza et al., 2006), RTR II hand, Vanderbilt hand and NAIST II hand (Kurita et al., 2011). Figure 1(b) denotes a variant of “211” configuration, namely, “21” configuration. A “21” configuration is the type where DIP joint and MP joint are designed as a whole. That is because a DIP joint able to move independently is not a necessary part in the practical application. Fluid hand (Gaiser et al., 2009), Vincent hand (Schulz et al., 2011) (Configuration B) and i-limb hand (Configuration C) belong to this configuration. In addition, although the thumb of i-limb hand has a passive abduction/adduction DOF, it can be also classified as this configuration.
A “111” configuration is the type where DIP joint, MP joint and TM joint, respectively, have a DOF (flection/extension). This configuration is the easiest to realize, which is applied the most extensively to the thumbs of prosthetic hands such as Southampton hand (Crowder et al., 1999), UB III hand (Lotti et al., 2005) and Tsing Hua series hands (Che and Zhang, 2010; Li et al., 2012). It can be seen from Figure 1(d) that the three axes A1 , A2 and A3 are parallel with each other. In this case, there is no difference between the thumb and the other fingers. Compared with the thumbs with “211” configuration, it has fewer grasping patterns, and the grasping performance is worse. With regard to the weakness, the thumb of HIT IV hand (Wang et al., 2010) adopts a spatial four-bar linkage combined with a planar four-bar linkage to accomplish the spatially conical movement, which makes its posture more anthropopathic, as shown in Figure 1(e). The two axes A1 and A2 of TM and MP joints are non-orthogonal with a fixed angle, which is specially optimized. In addition, it is a variant of “111” configuration. To visualize the analysis above, a table with the typical prosthetic hands and their thumb configurations is shown as follows.
After choosing the thumb configuration, some proper quantitative measures can be used to analyze the thumb performance and determine the key parameters. In this paper, the thumb performance is analyzed only from a kinematic point of view and relative measures are introduced as follows (Table I).
2.3 Kinematic performance of thumb
A thumb of a prosthetic hand can be considered as a serial robot. Therefore, the relative theories of serial robots can be used to analyze the mechanism of a thumb. Assuming a robotic manipulator with n DOFs, the joint angle vector is Θ = [Θ 1, Θ 2, ..., Θ n ]Tand the joint velocity vector is Θ˙ = [Θ˙ 1, Θ˙ 2, ..., Θ˙ n ]T. The manipulation velocity vector is v , which is related to the joint velocity vector Θ˙ by: Equation 1
where v ∈ R m(m-dimensional Euclidean space), Inline Equation 1 , J ( Θ ) is the Jacobian matrix and J ( Θ ) ∈ R m × n.
If rank J(Θ) < m for some Θ , the manipulator is in a singular state. As it is well known, for any J ( Θ ), there exist orthogonal matrices U ∈ R m × nand V ∈ R m × nyielding: Equation 2
whereΣ = [Σ 1 O] ∈ R m × n,Σ 1 = d i a g (σ 1, σ 2, ..., σ m ) withσ 1≥σ 2≥...≥σ m ≥0. Equation (2) is the singular value decomposition of J ( Θ ) matrix, andσ 1, σ 2..., σ m are the singular values of J ( Θ ).
Salisbury and Craig (1982) used the condition number of J ( Θ ) matrix as a measure of evaluating the kinematic transmission ability. The condition number is defined by: Equation 3
where ‖ ⋅ ‖denotes Euclide norm; J ( Θ ) +represents the generalized inverse matrix of J ( Θ ).
It is proved that the relation between the condition number and the singular value is: Equation 4
Obviously, its range is1≤K≤∞. Because the minimum value may be zero, the reciprocal I of K is chosen instead [equation (5)]. Equation 5
It denotes a ratio of the minimum principal axis to the maximum principal axis of a velocity ellipsoid mapping from the joint space to the task space (Yoshikawa, 1985), and describes the velocity isotropy. Isotropy is the property that the robot has the same kinematic transmission ability along all directions when the manipulability ellipsoid is a sphere, while maintaining gradual kinematic performance in the solution space. On the other hand, it describes the force transmission ability because of the dual relationship between velocity mapping and statics mapping in both the joint space and the task space.
To evaluate the performance of a robot in the workspace, a global measure based on kinematic isotropy can be defined in accordance with the concept of GPI presented by Gosselin as follows, which is a grid-based numerical method. Each value of the index WI represents a mean of condition numbers in the workspace for a robot. Equation 6
where N is the number of random samples.
In summary, equation (6) can be considered as one of the measures of optimizing the thumb configuration to obtain the variation of the index for different geometric parameters (e.g. the lengths of phalanxes) of the thumb. Then, the selections of these geometric parameters can be determined by a better value of the index. Obviously, WI is better as the average condition number becomes closer to 1. Maybe there are other measures for the optimization which are not only related to the kinematic performance. These measures are not discussed in this paper, but this optimization method is also effective.
3. Design of hand configuration
3.1 Form closure criterion
For a given reference frame, a grasp G can be defined as a set of point contacts: (Xiong, 1994) Equation 7
where p is the line vector of action for the contact point (briefly line vector) described by the unit outer normal u at the contact point and a position vector of the contact r . Define: Equation 8
A discriminant function of form closure for a grasp is defined as the minimum value of the following optimization problem: Equation 9
where the feasible set Ω 0is the solution set of the following system of line inequalities. Equation 10
wherey = [y 1, y 2, ..., y t ]Tandy t + 1are artificial variables. S is the number of p , while t is the dimension of p . When t equals to six, form closure will be determined. Otherwise, t is smaller than six, relative form closure will be determined. Once all the line vectors of contact points p are determined, J0 (G) can be calculated and, as a result, the closure of a grasp can be determined.
3.2 Modeling and hypothesis
To advantageously elaborate the method of the hand configuration, it is initially assumed that the hand has four identical fingers that consist of two phalanxes and a thumb with three phalanxes (identical proximal and distal phalanxes and an extra metacarpus), as visualized in Figure 2. It is of generality for this simplification because the following calculation method is the same, although the number of contact points increases for the fingers with three phalanxes. The parameters comprise the length of the proximal phalanx l1 , distal phalanx l2 and metacarpus lt . The other design parameters are some relative angles and will be introduced in the next part.
The shape or the sectional shape of the reference rigid objects is circular, which is continuous and characterized by the radius r. Stiffness or friction in the joints or at the contact points is not taken into account. The index finger and ring finger are assumed a symmetrical distribution relative to the middle finger. It is also assumed that the phalanges of a planar finger are straight, and the contact points are on the connecting line between the joints. Each phalanx can have only one contact point with the object. The phalanges are thus always tangent to the object surface.
In addition, the prosthetic hands are designed based on the underactuated or coupled theory. Therefore, the line vectors of contact points will be, respectively, discussed, of which the solutions are given.
3.3 Form closure analysis of power grasp
3.3.1 Relative form closure analysis of a cylinder grasp
Relative form closure is a particular case of form closure in a two-dimension space, which can be applied to analyze a cylinder grasp. In this case, the dimension of line vectors is three, and the previous criterion is also available. While grasping a cylinder object, all the sections are circular, as shown in Figure 3. When the criterion of form closure is satisfied in the plane of thumb-palm-middle finger, the grasp is considered as stable.
For an underactuated hand, the analysis is shown in detail as follows:
The parameters to be optimized should be firstly determined in terms of the configuration characteristics of the thumb and fingers, and all the initial ranges of these parameters should be given reasonably.
In the plane of thumb-palm-middle finger, all the contact points will be calculated. When the thumb adducts opposite to the middle finger and begins a power grasp, the angleΦ t1between the palm and the metacarpus is considered fixed. By taking the metacarpus and the palm as the location data, the contact points between the cylinder and the thumb, the palm and the middle finger are, respectively, calculated.Considering the contact point calculations of the thumb as a calculation example, the contact point H with the metacarpus is geometrically described as follows: Equation 11where u H is the unit outer normal vector of the contact point A (similarly hereinafter).The angleΦ t2between the proximal phalanx of the thumb and the palm is calculated as follows: Equation 12The contact point G with the proximal phalanx of the thumb is described as the following equations set: Equation 13The angleΦ t3between the distal phalanx of the thumb and the palm is calculated as follows: Equation 14The contact point I with the distal phalanx of the thumb is described as the following equations set: Equation 15The solving method of the contact points of middle finger is the same as that of thumb, which will not be elaborated.
The contact points mentioned above are automatically discriminated through J0 (G) function whether to satisfy form closure criterion and then the ranges of the parameters satisfying the criterion will be revealed.
For a hand with coupled joints (assuming the ratio as 1), the analysis is shown in detail as follows:
Due to the application of the coupled design, there is only one phalanx contacting with the cylinder for a finger. In this case, the number of contact points is four. Based on Step 1 in the underactuated analysis, the actual contact points have to be determined by the coupled motion relation, which means by the comparison betweenΦ t1−Φ t2andΦ t2−Φ t3, the second contact point is discriminated whether on the proximal phalanx or on the distal phalanx. Taking the actual contact point calculations of the thumb as an example, the calculation method is described as follows:IfΦ t1−Φ t2≤Φ t3−Φ t2, the proximal phalanx contacts with the cylinder and the contact points are H and G, otherwise the distal phalanx contacts and the points are H and I.To calculate contact point I, letΦ t1−Φ t2 = Φ t3−Φ t2. The equation of the distal phalanx is: Equation 16The distance fromO s to this line is made equal to r and thenΦ t2,Φ t3and the coordinate of C will be solved.
In general, the length of metacarpus is designed differently. When the metacarpus is short and the relation among the metacarpus AB, the object radius r andΦ t1satisfies the equation (17), the thumb hardly rotates in the previous location data, as shown in Figure 4(a). It was found through quantities of grasping simulations that on condition of equation (18), the thumb could continually rotate and then separate from the metacarpus until the proximal and distal phalanxes, respectively, contacted with the cylinder. Therefore, another stable grasp state was obtained, as visualized in Figure 4(b). Equation 17 Equation 18The contact points F and G and the angleΦ 2can be calculated by the equations (19), (20) and (21). Equation 19 Equation 20 Equation 21where Φ 2 represents the rotation angle between the proximal phalanx and the metacarpus.
The four contact points are automatically discriminated whether to satisfy form closure criterion through J0 (G) function and then the ranges of the parameters satisfying the criterion will be revealed.
3.3.2 Form closure analysis of a sphere grasp
A sphere grasp can be also analyzed based on the form closure theory, which is an application of form closure theory in a three-dimension space. In this case, the dimension of line vectors is six.
For an underactuated hand, the analysis is shown in detail as follows:
First, the parameters to be optimized should be determined in terms of the configuration characteristics of the thumb and fingers, and all the initial ranges of these parameters should be given reasonably. While grasping, the maximum section of a sphere is made to coincide with the plane of thumb-palm-middle finger. Then, the contact points in this plane are calculated. In the plane of index finger-palm, the sectional radius is different due to the separation between this circular section and the palm, as visualized in Figure 5. Consequently, the center of this section has to be determined. Furthermore, all the contact points will be calculated, which are automatically discriminated whether to satisfy form closure criterion by J0 (G) function. Last the ranges of the parameters satisfying the criterion will be revealed.
In Figure 5, the angle Φ f1 between the proximal phalanx of the index finger and the palm is calculated as follows: Equation 22
where d f represents the deviation of the MP joint of the index finger from that of the middle finger in the vertical direction of the palm; d w represents the distance of adjacent fingers; d t f represents the deviation of the MP joint of the index finger from that of the middle finger along the direction of the palm.
The contact point M with the proximal phalanx of the index finger is described as the following equations set: Equation 23
The angle Φ f2 between the distal phalanx of the index finger and the palm is calculated as follows: Equation 24
The contact point N with the distal phalanx of the index finger is geometrically described as follows: Equation 25
The solving method of the contact points of the ring and little finger is the same as that of the index finger, which will not be elaborated.
For a hand with coupled joints, it is necessary to determine the actual contact points. The other analysis is similar. It is worthy of noting that the little finger cannot contact with the sphere if the radius is smaller than twice of the width between the adjacent fingers. In this case, the number of contact points is six, otherwise the number is seven. The grasping simulations are visualized in Figure 6. The whole flow chart of form closure discriminance is visualized in Figure 7.
4. Application and results
The method is applied to a design case to illustrate its practical application. The design of the HIT V prosthetic hand is taken as a reference (Jiang et al., 2014). This hand was designed and constructed in 2014 with the intention of application in rehabilitation, as shown in Figure 8(a). It is a multisensory and integrated five-finger prosthetic hand with six active DOFs. To achieve a high degree of modularity, it was decided to construct five equal fingers each consisting of two joints and a flection/extension DOF, while the thumb has a metacarpus and an extra DOF of adduction/abduction.
4.1 Analysis of thumb configuration
Figure 8(b) shows that the thumb of HIT V hand has a typical “21” configuration (Configuration B), and its two axes with some distance in the TM joint are non-orthogonal. Due to the modular design, the proximal phalanx and the fingertip of the thumb are identical to the ones of the other fingers. Because the actuator is integrated in the proximal phalanx and multiple sensors are integrated in the distal phalanx, the lengths of the phalanxes are dependent on the actuator and the configuration of these sensors, which means that the lengths of lt 2 and lt 3 are determined by the special structure of the finger. The metacarpus lt 1 and the angle α between the two axes are difficult to choose, thus they are taken as unknown parameters. In consideration of the anthropomorphic appearance and structure, lt 1 should be equal to or greater than 15 and equal to or smaller than 60, andα should be equal to or greater than −90° and equal to or smaller than 0. WI in equation (6) is taken as the optimal index, of which each value represents a mean of condition numbers in the workspace of the thumb for a given group of geometric parameters (lt 1 and α). Then, this paper showed the variation of this index for different groups of lt 1 and α. The selections are determined by a better value of WI (the average condition number closer to 1).
The values of I are visualized in Figure 9 when lt 1 and α change, of which the distribution turns out to be a saddle. The values of I are the greatest in area A and B, where the performance of the thumb is best. On the contrary, the values of I are smallest in area C, where the performance of the thumb is worst. Thus, the parameters lt 1 and α should be chosen in in area A and B in terms of the special structure. When lt 2 and lt 3 are long, the values of lt 1 in area A should be chosen to avoid an overlong thumb to influence the appearance. Otherwise, the values in area B should be chosen for short phalanxes. Without loss of generality, the ratio lt 1/L can be used as a parameter, where L = l t 1 + l t 2 + l t 3. In addition, the selection of α in the range −90°- −55° should be considered synthetically together with grasping patterns and the thumb layout on the palm.
4.2 Analysis of hand configuration
The hand configuration design mainly includes the following parameters: the phalanx length of each finger, the width between adjacent fingers, the relative positions of fingers in the paralleled and vertical directions of the palm and the sizes of objects. However, it is impossible to optimize all the parameters simultaneously. A feasible method is to determine the important extent of these parameters and analyze them, respectively. For HIT V prosthetic hand (each finger with two joints, coupled), it is necessary to judge the numbers of actual contact points and their positions before calculating the line vectors at contact points. Due to the limited sizes of the hand and its special structure, the width d w between adjacent fingers is fixed, which will not be discussed in detail.
First, the parameters to be optimized should be determined. The analysis steps are shown as follows:
In the previous thumb configuration analysis, lt 1 has been determined. Considering the object size to be grasped in our daily life, the maximum radius is enough when it is 35. The parameters in the vertical direction of the palm are ignored temporarily (let d f = d r = d t = 0). The relative positions between the MP joints of the fingers (TM joint of thumb) and that of the middle finger along the direction of the palm are specially analyzed. The ranges of the following three parameters are estimated by the adults’ hand models: Equation 26
where Δ x 1 represents the deviation of the MP joints of the index and ring finger from the one of the middle finger along the direction of the palm, and Δ x 2 represents the deviation between the little finger and the middle finger.
The calculation results show that in the ranges of the above-mentioned parameters, the condition of relative form closure is not satisfied, while the condition of form closure is satisfied. Consequently, it is incomplete to judge the form closure only by changing this group of parameters when lt 1 and r are fixed:
Several groups of parameters in Step 1are selected. Then, only lt 1 is changed, while the others are invariable. Compared with the results in Step 1, it can be concluded that lt 1 can make great influence over relative form closure, which will be changed as lt 1 changes in a small range.
Several groups of parameters in Step 1are selected. Only r is changed, while the others are invariable. Compared with the results in Step 1, it can be concluded that r can make great influence over relative form closure, which will be changed as r changes in a small range.
Several groups of parameters in Steps 2 and 3 are, respectively, selected. df , dr and dl are changed, while the others are invariable. Compared with the results in Steps 2 and 3, the changes are not obvious.
In summary, d t m , lt 1 and r are considered as the most important parameters for the hand configuration design. To clarify the discussions above, Table II is given as follows.
The distributions of form closure function J0 (G) are visualized in Figures 10 and 11 when d t m , lt 1 and r change. It is hard to describe the variations of function values as the changes of the three variables. Therefore, we divided the object size r into a group of values in the range (25, 40) with a step size two. Then a group of pseudo-color maps are used to depict the distributions of form closure function J0 (G) for different r when d t m and lt 1 change. The functions J01 (G) and J02 (G) for any r are shown in one graph.
We can discuss two states from the following figures according to the different values of r and lt 1.
First, for grasping an object with the radius smaller than or near to 30, the value range can be divided into several parts by the two lines close to l t 1 = 25 and l t 1 = 50, as shown in Figure 10, and the areas satisfying closure condition have the trends of rightward and upward move as r increases:
When 15≤l t 1≤25 and 50≤d t m ≤70, the form closure criterion is satisfied.
When 25≤l t 1≤50, the form closure criterion will be satisfied if d t m ≥75 for r < 30. In addition, it will be also satisfied if 50≤d t m ≤55 for the radius near to 30.
When l t 1≥50, the form closure criterion will be satisfied for any d t m in its range.
Second for grasping an object with the radius over 30, the areas satisfying closure condition also have the trends of rightward and upward move as r increases, so that l t 1 = 30 and l t 1 = 55 are considered as the dividing lines, as shown in Figure 11:
When 15≤l t 1≤30 and 60≤d t m ≤75, the form closure criterion is satisfied.
When 30≤l t 1≤55, the form closure criterion will be satisfied only if d t m ≥80. Another area satisfying the criterion exists when 35≤l t 1≤50 and 50≤d t m ≤60 for r ≤ 35. Furthermore, this area will continually move rightward as r increases. The criterion will be satisfied when 40≤l t 1≤55 and 50≤d t m ≤60 for r ≥ 35.
When l t 1≥55, the form closure criterion will be satisfied if d t m ≥55.
In summary, considering the special structure and the appearance of HIT V hand, l t 1 and d t m are selected according to the Step 1 of the first state. Without loss of generality, the ratio lt1 /dtm can be used as an unknown parameter. In addition, the determined l t 1 here accord with the one determined by the thumb configuration analysis, which proves the validity of the thumb configuration analysis from another point of view. Other configuration parameters mainly involved with relative positions of the fingers can be selected or properly changed according to the previous analysis and the anthropopathic requirement, which will not influence the results of form closure judgment.
4.3 Grasping experiments
Theoretically, a multi-pattern prosthetic hand should have enabled upper-amputees to perform all the activities of daily life. However, it is hard work. A feasible method is that the grasp patterns used more frequently should be considered preferentially. Studies have been conducted to categorize grasp gestures of human hands and to analyze the using frequency of different grasp types in activities of daily living (ADLs). Taylor and Schwarz (1955) divided the grasping patterns into six sorts based on object shapes; Napier (1956) classified them into precision manipulation and power grasp based on different demands of accuracy and grasping force, while Sollerman and Ejeskär (1995) assorted the grasping gestures to eight sorts according to the relative positions between the thumb and other fingers and specified their using frequency; Zheng et al. (2011) made a statistic analysis of using frequency of 16 grasping patterns based on Cutkosky’s (1989) method. As can be seen from Taylor’s method and Zheng’s data, primary gestures including cylindrical, spherical, tip, lateral, palmar and hook, which need to be achieved, account for 77.6 per cent of daily grasping patterns. Additionally, pointing, clenching fist and platform also need to be taken into account as a supplement. Figure 12 shows that our newly developed HIT V hand can accomplish all of them successfully.
5. Conclusion
In this paper, the thumb function is analyzed based on the anatomical structure of human thumbs, the configuration and characteristics of typical prosthetic hands are summarized, and a method is introduced to optimize thumb configuration parameters of prosthetic hands in terms of the measures of kinematic transmission ability. Furthermore, a design method of the prosthetic hand configuration is put forward based on form closure theory, in which almost all the configuration parameters are taken into account. A discriminant function of form closure is used to analyze the hand configuration to determine the important parameters satisfying the closure criterion. Our newly developed HIT V prosthetic hand is taken as an example in this paper to illustrate in detail the configuration optimization of the thumb with a special structure, and summarize the variation relation between the important hand configuration parameters and the discriminant function when a stable or relatively stable grasp is achieved. Both of the two parts accomplish the optimization. It is concluded that the parameters ranges satisfying the closure criterion can be divided into several subsections and the optimal choice is determined by different structure characteristics. It is concluded that the design steps – sustained by analytical equations, graphs and grasping experiments – constitute an effective method to design a prosthetic hand. In this paper, only a kinematic performance measure – the reciprocal of the condition number – is applied to optimize the thumb configuration. However, it is more suitable for a prosthetic hand to use an index about grasping force, which may be a future interest. In addition, it is necessary to extend the method to apply to a hand with three-phalanx fingers and make a deeper study on stable grasps.
Corresponding author
Shaowei Fan can be contacted at: [email protected]
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Acknowledgements
This work is in part supported by the National Program on Key Basic Research Project (973 Program, No. 2011CB013306, 2011CB013305), the National Natural Science Foundation of China (No. 51175106, 61203346) and the China Postdoctoral Science Foundation Funded Project (No. 2013M540276, 2014T70316).