Control-Oriented Finger Kinematic Model: Geometry-Based Approach

The development of human finger/hand modeling has attracted much interest, and its research has widely been done since they promise a variety of practical applications in medicine and rehabilitation, prosthetics, artificial hands, robotics, man–machine interface device, human–computer interaction, and so on. The success of human hand modeling in these areas relies strongly on an understanding of anatomical/functional aspects of the hand, kinematic/dynamic properties of the hand, and the purposes of the use of models. A variety of research on finger/hand modeling in various points of view has been made in human fingers simulation [1], finger motion coordination [2,3], virtual hand modeling and simulation [4–7], articulated human hand [8,9], and so on.

One of the main functions of the hand models is to simulate object grasping which is realized by the hand coordination which is the combination of finger movements encompassing the interjoint coordination of each finger and the interfinger coordination of the hand. The implementation of the multifinger coordination [10] in models can be a pivotal direction of modeling. It is noted that although the advantages and their modeling directions of the previous researches are best understood, this work focuses on the establishment of a novel control-oriented finger kinematic model to be used in future combined design–control strategies. The work aims to provide a finger kinematic model capable of describing the natural finger joint coordination by a single parameter. The proposed finger kinematic model is based on a hand shape in an open configuration with each finger being in some degrees of flexion. It is assumed that the form of the hand under consideration can be observed when it is power/in-hand grasping a cylindrical object. In this case, the object physically exists. However, the existence of a physical object will not be required in the process of precision grasping, which is characterized by grasping and holding an object by the fingertips. In this case, the fingers encompass a virtual cylinder. In both cases, the fingers are in some degree of flexion. The resulting model is applicable to control systems design for robotic and artificial hands with a potential to replicate multifinger coordination of the human hand. The degrees of flexion of a finger can be determined by the joint rotation angles in each finger, which angles can be determined by the radius of the real/virtual cylindrical object being assumed to be grasped. With the finger model, further discussion on multifinger coordination is made.

Here, we would like to note that the main contribution of the work is to provide a kinematic finger model that formularizes the naturalistic hand gesture during flexion or extension as closely as possible. Furthermore, the proposed model can determine finger joint rotation configuration by a single control parameter even though three settings are required to define its finger gesture. A single parameterized control would benefit in the field of under-actuated artificial robotic hand design and controls [11]. In addition, the ability to predict the most probable joint angle trajectories in order to obtain a suitable wearable mechanism for limb rehabilitation for grasping, walking, or reaching motions is very important. Existing methods require the derivation of specific parameters, such as segmental motions, forces, velocities, cycle periods, etc., or using initial Cartesian trajectories, which may be difficult to find for individuals with limited limb mobility. Therefore, there is a need for developing novel models that can predict the joint trajectories from the subjects’ anthropometric parameters.

This paper is organized as follows. In Sec.  2, a geometry-based finger kinematic model is presented that can determine the joint angle configuration of the index finger. It is shown that the proposed model is a one degree-of-freedom (DOF) model that calculates rotation angles of three joints each with a single control parameter. Section 3 describes the multifinger coordination function using the proposed model. Section 4 summarizes the work. Appendix  A extends the proposed model, which can be directly applied to the middle finger, to the ring and little finger. Appendix  B investigates a kinematic finger model using an elliptical object.

One possible shape of a human hand is in an open configuration with each of the fingers being in some degree of flexion subconsciously around its corresponding joints. This can be assumed to be a “naturalistic” shape observed in the human hand. Human long fingers (index, middle, ring, and pinky fingers) have three joints each, i.e., metacarpophalangeal (MCP) joint, proximal interphalangeal (PIP) joint, and distal interphalangeal (DIP) joint. Each finger has four DOF (flexion/extension in DIP, flexion/extension in PIP, and flexion/extension and abduction/adduction in the MCP joint) [12]. For simplicity, in the present work, it is assumed that only the flexion/extension motion in the MCP joint is considered, which leads to a planar finger model with three degrees of freedom. In what follows, a mathematical model for depicting joint rotation coordination of a finger is proposed for naturalistic shape representation, followed by a multifinger coordination scheme for object manipulation, discussed in Sec. 4.

In order to develop a mathematical model of the finger joint motion for the naturalistic shape of a hand, two types of the real/virtual object being grasped are considered: cylinder and cylindroid (a cylinder with an elliptical cross section). A joint configuration model using a cylinder is discussed in this section (for the index finger) and in Appendix  A (for the third to fifth fingers), and a model using a cylindroid (for the index finger) can be seen in Appendix  B.

As observed in Eqs. (3), (6), and (8), the joint rotation configuration can be determined by only a single parameter R given the fixed dimension of semi-thickness (i.e., t_0I, t_1I, t_2I, and t_3I) and length of phalanges (L₁, L₂, and L₃). It is worth to note that the set of Eqs. (3), (6), and (8) represent a 1-DOF joint rotation configuration model of the human fingers. Furthermore, it is noted that these angles are not coupled. For the different profiles of R to time, both flexion and extension motions of a finger can be materialized.

Multiple works, mentioned above, control independent angular values to generate the grasping motion. Others utilize optimization methods to create such values through recursive calculation, which requires significant computational efforts. Some of the proposed solutions cannot coordinate the movement in time. Therefore, exploring new modeling methods is needed. The proposed model reduces the computational strain by utilizing a single variable which governs three angular values of each phalange to harmoniously coordinate each finger for generating the grasping motion. The merit of the proposed model is in its simplicity when used in the application to robotic hand design, assessment and control systems, such as articulated prosthetic devices, visual animation, and virtual reality, among others.

In order to investigate the changes in joint angles with the change of R over time, several sets of R profile have been considered. Figures 3–7 show the flexion motion of an index finger utilizing the proposed model with different profiles of R. The following values of the semi-thickness and phalangers’ length of the index finger under consideration are used for simulation: t_0I = 1.5, t_1I = 1.31, t_2I = 0.94, and t_3I = 0.49 cm. L₁ = 4.54, L₂ = 2.29, and L₃ = 1.76 cm. In Fig. 3, R = R(t) is chosen to vary linearly with respect to time. As expected, the smaller the radius R, the larger the joint angles. However, it is observed that the calculated profile (behavior) of each joint angle seems not to be similar with that of a human finger [2,3,10,14,15]. Following this, different profiles of R are examined to see which of them is best suited for simulating the human finger motion. The candidate profiles under investigation are (1) parabola (Fig. 4), (2) hyperbolic tangent (Fig. 5), (3) cosine (Fig. 6), and (4) spline (Fig. 7) functions. For a hyperbolic tangent function, following the formula adopted from Ref. [3], the coefficient of the function is set as c₁ = (R_max + R_min)/2, c₂ = (R_max − R_min)/2, c₃ = 0.8, and c₄ = 0.4 for the calculation example. The spline curve was shaped with the points of (0, R_max), (0.5, 0.9 R_max), (1, 0.8(R_max + R_min)/2), (1.8, 1.2 R_min), and (2, R_min). From the calculation results, it can be noted that the configuration parameter R with the profile of hyperbolic tangent function seems to fit for describing the natural flexion motion of a finger, as reported in Refs. [3,10]. This can be easily justified from the plots of stepwise finger motion (shown in Figs. 3–7 with the time span of 0.1 s). Note that in Fig. 5, a broad span between consecutive movements is observed in the middle of the excursion of flexion, which means that the finger moves fast in the range. This notion will be discussed in more detail in Sec. 4. Joint configuration models for the third to fifth fingers and the thumb are given in Appendix  A.

For the evaluation of the proposed finger model, a set of motion capture experiments with two subjects has been performed. Optical markers were attached on the index finger of each subject, and the movement of the finger was recorded, as shown in Fig. 8(a). Through the motion capture process, finger movements have been made with several consecutive flexion and extension motions, and the planar coordinates for joints and the fingertip have been acquired. From the coordinates, the fingertip trajectory and joint angle data of each subject were calculated.

A comparison of the joint angle configurations in joint space between the experiment and the model is shown in Table 1. Joint angles of the model for each point were calculated via Eqs. (3), (6), and (8) with the value of R corresponding to the coordinates of the fingertip point under consideration. It can be seen that the experiment and model tend to be very similar in the configurations of the joint angles for the similar fingertip points taken from the experiment and the model. In comparing both the trajectory and the joint angle configuration between the model and the experiments, it can be seen that the proposed model closely simulates the naturalistic motion of the human finger in Cartesian, as well as joint space.

The joint configuration model of an index finger using cylindroid real/virtual object grasping is given in Appendix  B. The proposed model can be treated as a one-DoF model: however, it is not suitable to be directly used in control systems design, since the joint angle configurations are obtained via the numerical approach of solving systems of nonlinear equations. Refer to Appendix  B for more details.

In many hand motions, several fingers work together to perform specific tasks. The notion of “working together” can be explained in terms of the multifinger-coordination function of the hand. The hand coordination is a combination of finger movements encompassing (1) interjoint coordination in each finger and (2) interfinger coordination in the hand.

From the survey of research works on multifinger coordination [2,8,10], it is observed that the interjoint coordination is related to the sequence of movement in the phalanges of a finger. For a voluntary extension movement (opening), a proximal-to-distal sequence is evident in the long fingers and distal-to-proximal sequence in the thumb. For the flexion movement (closing), reversed sequences are held in the long fingers and the thumb, respectively. Furthermore, in the motion analysis of interfinger coordination, the following results are given: the MCP joint of each finger moves together and has its peak velocity at approximately 50% movement in its excursion. Similar trends are observed in interphalangeal (IP) joints with their peak velocity at about 57% of the whole movement (see Ref. [10] for details).

In this work, the joint configuration model proposed in Sec. 2 is used to implement the concept of multifinger coordination. The notion of coordinated motions endows the model with the capability to simulate the natural characteristics of the human hand. The joint configuration model has one degree of freedom that is capable of determining each of the joint rotation angles (at the MCP, PIP, and DIP joints) in the finger simultaneously by only a single parameter R and representing the most elemental finger movement, i.e., flexion and extension in a quite simple way. Since the model has no constraints on the angles between joints and each joint angle is independently determined by just one single parameter, it can be easily used to implement interjoint coordination.

Figure 9 shows a schematic to explain the concept of the proposed coordinated motion of a hand. The respective solid-line blocks in each of the long fingers represent the proposed joint configuration kinematic model for the rotation angles of the MCP, PIP, and DIP joints, respectively. As described in Eqs. (3), (6), and (8), each joint angle is a function of the joint configuration parameter R_ik which is the output from the interjoint motion coordination block assigned to the finger (represented as a broken-line block in the figure). The interjoint motion coordination block plays a role to set the profile of R_ik and to provide each joint model with its value. The establishment of the profile R_ik can be made based on the following need (1) to meet the sequence of movement in phalanges during flexion or extension movement (discussed in Ref. [10], for example) or (2) to make any specific shape of fingers.

The coordination in the multifinger level is realized in the interfinger motion coordination block. It aims at determining human hand configurations that are shapes of hand which need to be made for performing specific tasks, such as grasp, or object manipulation [16–18]. As mentioned earlier, the realization of the interjoint coordination using the proposed kinematic model is directly related to the temporal aspects of a single finger movement that considers the order of rotation initiation of each joint.

On the other hand, the interfinger coordination is thought to be related to the spatial aspects of multifinger movements that consider the arrangement of virtual cylinders with different radii R_i, i = I, M, R, P, respectively, following the outputs from grasping algorithms [19,20], control systems [21–23], and a library/database of hand gesture ([24], for instance) to make a specific shape, as shown in Fig. 11.

A novel finger kinematic model for the human hand configuration was derived based on the geometry of a long finger which is assumed to be softly enclosing the surface of a (virtual) cylindrical object. The proposed model (1) provides the joint rotation configuration (represented by three degrees of rotation motion) of the long finger with a single variable/control parameter which is the radius of the (virtual) cylindrical object, (2) calculates the values of the three joint rotation angles independently, and (3) describes the interjoint coordination function which can be combined with the interfinger coordination function to materialize the multifinger coordination function. The proposed index finger kinematic model can be easily applied to the model of other fingers (shown in Appendix  A). A kinematic finger model encompassing an elliptical virtual object was also taken into consideration in Appendix  B. It is shown that the model requires to solve nonlinear simultaneous equations numerically. Experimental validation of the model shows that it can simulate closely naturalistic human finger movements with applications in mimicking human hand shapes, as well as robotic hands design and control, among others.

• NSF (CAREER Award Id # 1751770; Funder ID: 10.13039/100000001).

• 2018 Research-Year Grant of Jeonju University, South Korea.

For the third to fifth digits, it is assumed that each MCP joint of each finger has deviated from the origin J_1I being the center of rotation of the MCP joint of the index finger. Let its deviation be s_xij and s_yij where i = I and j = M, R and P. Note that the subscripts I, M, R, and P represent an index, middle, ring, and pinky (little) fingers, respectively.

With the notation in Fig. 1, a planar thumb model for joint rotation configuration can be derived in a similar manner as in Sec. 2.2. For the purpose of simplicity, it is assumed that β_0T and L_0T related to the anthropometric data of the human hand are known.

In this section, the second joint rotation configuration model is proposed by considering another virtual object being encompassed by fingers. In the work, a naturalistic shape of fingers of the human hand is again assumed as one that is observed when a human hand is grasping a (virtual) cylindroid object with an elliptical cross section as shown in Fig. 13. By maintaining numerical eccentricity (defined as e = c/a, where $c=(a2−b2$⁠)) of the ellipse constant, one can obtain a joint rotation configuration kinematic model of the index finger which can be manipulated by a single control variable, either a or b.

The joint angle configuration of the index finger can be obtained by formulating and solving geometric constraints which need to be met to describe the posture of the fingers when they encompass the virtual elliptic object under consideration. These constraints are described with the coordinates of the points of interest. Let us begin with the geometric aspect of the proximal phalanx of the index finger shown in Fig. 13.

The joint rotation angle of the PIP joint in the index finger is obtained by the procedure described above.

Using the procedure above, the joint rotation configuration of the DIP joint in the index finger is obtained.

Figure 14 shows how the joint angle of each joints, MCP, PIP, and DIP, respectively, are calculated in consecutive order. Two examples of the calculation with different sets of a and b are shown in Fig. 15. It is noted that the model should be solved numerically to get the joint angle configuration of a finger.

Note that when a = b, the object has a circular cross section. The joint rotation configuration can be obtained numerically by solving the constraints (with a = b = R) discussed here as well as the formulas in Sec. 2.2.