Abstract

Physiological studies of the human finger indicate that friction in the tendon-pulley system accounts for a considerable fraction of the total output force (9–12%) in a high-load static posteccentric configuration. Such a phenomenon can be exploited for robotic and prosthetic applications, as it can result in (1) an increase of output force or (2) a reduction of energy consumption and actuator weight. In this study, a simple frictional, two-link, one-degree-of-freedom model of a human finger was created. The model is validated against in vitro human finger data, and its behavior is examined with respect to select physiological parameters. The results point to clear benefits of incorporating friction in tendon-driven robotic fingers for actuator mass and output force. If it is indeed the case that the majority of high-load hand grasps are posteccentric, there is a clear benefit of incorporating friction in tendon-driven prosthetic hand replacements.

1 Introduction

The field of robotic prosthetics research is continuously marking progress toward functional hand replacements for amputees, with prototypes close to attaining functionalities similar to those of the human hand [4]. Despite the advance attained, however, approximately 20–40% of users of such robotic (usually myoelectric) prostheses reject them [6, 28, 44]. This is due to inherent real-world requirements that are usually not taken into consideration [10], such as (1) low power consumption, (2) low device weight, (3) high dexterity, and (4) ease of use. We argue that by studying the morphology and the biomechanics of the human hand, we can identify and exploit principles [34] that can be used to create devices that can be of use in the real world. Toward such an aim, we investigate the presence of friction in the human finger tendon-pulley system and its importance with regard to the force capabilities of the finger itself.

Tendon-driven robotic systems are usually designed and constructed so that the influence of mechanical friction is minimized. In particular, tendon-driven robotic and prosthetic hands utilize Teflon- or nylon-coated cables and pulleys, and aim toward frictionless tendon transmission. This is due to friction introducing a number of unwanted side effects, such as (1) nonlinearities [45], (2) hysteresis effects, and (3) energy and power losses [18, 22, 51]. Designers of robotic hands such as the Shadow hand [43], the smart hand [53], and other tendon-driven prosthetic hands [13, 15, 16] either intentionally try to minimize friction in their systems (especially in the tendon-pulley system), or consider friction to have no importance in their performance.

In the case of the tendon-pulley system in the human finger, however, friction has been found to be beneficial. It has been experimentally shown that during high-load flexion of the interphalangeal joints, eccentric and concentric forces differ by 9–12%, a difference that can be directly attributed to tendon-sheath friction [41]. In addition, the fact that tendons compress when tensioned, and also the orientation of tendon and sheath fibers, further contribute to the appearance of frictional forces [50]. An extreme version of such utilization is present in chiropterans and specifically bats, which employ frictional forces to dangle on their fingers without the application of muscular force [35, 37, 39]. The tendon locking mechanism (TLM) provides bats with the ability to lock their digits in place when the tendon is flexed. This is due to surface properties of tendon and pulley. This locking prevents the digit from extending, and consequently it maintains contraction of the digital flexor muscles without energy consumption while hanging.

It is relevant to note the kind of activities in the context of eccentric and concentric configurations. Yoshio Matsumoto et al. [27] recorded a life log of a healthy person and analyzed it based on the ICF.1 Even though the ICF classification is too vague for some of the activities, a number of them appear to be eccentric in nature with respect to the fingers, namely, (1) lifting, (2) putting down objects, (3) carrying in the hands, and (4) pulling. These activities compose the majority of performed actions during the day of a healthy person, summing to over 60%.

This frictional phenomenon allows the human finger to produce a higher output force at the fingertips during an extension movement. Considering this fact, we question the ongoing principle of minimizing friction in robotic and prosthetic tendon-driven hands. If we can characterize and exploit this phenomenon in robotic and prosthetic systems, it can result in (1) the increase of output force or (2) the reduction of energy consumption and actuator weight. In this study, a simple two-link, one-degree-of-freedom (DOF) model of a human finger was created to identify the influence of friction on concentric and eccentric postmovements in a tendon-driven segment. As we are modeling a biological system, the design choices are primarily influenced by the biomechanical properties of the human finger.

In Section 2 we start with describing the physiology of the human finger system, including bones, tendons, and pulleys. We continue by presenting work related to biological modeling in Section 3. In Section 4 we describe the model and the design choices behind it. In Section 5 we validate the model against in vitro human data. We then present the model's behavior in Section 6, ending with a discussion of our findings in Section 7.

2 Physiology of the Human Finger

Overall, the human hand has 27 bones divided into three groups: 8 carpal bones in the wrist, 5 metacarpal bones, and 14 phalanges of the fingers. The phalanges are three for each digit and two for the thumb. They are named proximal (PP), middle (MP), and distal phalanges (DP) (with the middle phalanx missing in the thumb), according to their position. The joints of the human finger are in sequence from proximal to distal: metacarpophalangeal (MCP), proximal interphalangeal (PIP), and distal interphalangeal (DIP) joints. Due to the complexity of the muscle actuation of the human hand, here we will merely mention the two main tendons responsible for finger actuation (involved in most repetitive daily work): the flexor digitorum profundus (FDP) and flexor digitorum superficialis (FDS).

The FDP tendon passes along the finger through a series of pulleys, which maintain a reasonably constant moment arm2 [7] for flexing or extending the finger. Before inserting into the distal phalanx, the FDP passes through a split in the FDS tendon (Figures 1 and 2). The FDP is the only tendon with the ability to flex all three joints of the finger; in doing so, it provides most of the finger's strength [7]. Although the FDP is the primary finger flexor, the FDS becomes increasingly active as more force is needed. Due to the complex interaction between the two tendons, we will only consider the FDP tendon for the upcoming models, as the aim of this work is to highlight the frictional aspects and not to capture the exact mechanics of the human finger tendons.

Figure 1. 

Human finger and its flexor tendons (superficialis (FDS) and profundus (FDP)). Provided by A. Schweizer.

Figure 1. 

Human finger and its flexor tendons (superficialis (FDS) and profundus (FDP)). Provided by A. Schweizer.

Figure 2. 

Abstracted human finger and its flexor tendons (superficialis and profundus).

Figure 2. 

Abstracted human finger and its flexor tendons (superficialis and profundus).

2.1 Flexor Tendon-Pulley System

The tendon sheath3 is a double-walled tube, surrounding the tendons and containing synovial fluid. The sheath provides a low-friction gliding (not to be confused with the friction in the tendon-pulley system) as well as a nutritional environment for the flexor tendon [20]. The sheath begins at the neck of the metacarpal phalanx, ends at the distal interphalangeal joint, and is held against the phalanges by pulleys. These pulleys primarily act to prevent tendon bowstringing4 across the joints during flexion or extension of the finger.

The pulleys themselves can be divided into three types based on their location: a palmar aponeurosis pulley, five annular (ring-shaped) pulleys (A1, A2, A3, A4, and A5), and three cruciate (crosslike) pulleys (C1, C2, and C3). The A2 and A4 pulleys are located on the proximal and middle phalanges, while the A1, A3, and A5 pulleys are located at the palmar surface of the MCP, PIP, and DIP joints (Figure 3). The most important pulleys for normal function are the A2 and A4 pulleys, with the A3 and other pulleys coming into play when A2 and A4 have been damaged [20]. Although the A3 pulley is weaker and closer to the PIP joint, it is more flexible and stretches, thereby transferring the load to the A2 and A4 pulleys, which may fail first during high loads [26].

Figure 3. 

The different pulleys in the human finger. Reproduced from [38].

Figure 3. 

The different pulleys in the human finger. Reproduced from [38].

2.2 Eccentric and Concentric Contractions

Although the term contraction implies shortening, when referring to the muscular system it means muscle fibers generating tension with the help of motor neurons. There are two major categories of contractions, (1) isometric and (2) isotonic. Isometric contractions occur when the muscle contracts but there is no resultant limb movement. Muscle contractions that result in limb movement are known as isotonic contractions. There are two types of isotonic muscle contractions: (1) eccentric and (2) concentric. Eccentric muscle contractions occur when the muscle contracts but increases in length. This type of contraction usually occurs in the direction of gravity to control a movement. Concentric muscle contractions occur when the muscle shortens in length in order to make the limb move.

Because we will be modeling the tendon-pulley system of the human finger statically, these two terms will be used a little differently. The term eccentric configuration will be used here for a static configuration from which an eccentric contraction (resulting in an extension movement) would follow if the fingertip force Ftip were increased by a very small amount. In contrast, the term concentric configuration will be used here for a static configuration from which a concentric contraction (flexion movement) would follow if the fingertip force Ftip were decreased by a very small amount. Why exactly this is the case we will discuss in Section 4.1.2.

3 Related Work

In 1995, Walbeehm and McGrouther [50] investigated the mechanical interaction of tendon loading and motion between tendons and the A2 pulley in cadaveric hands. Using scanning electron microscopy, they found transverse ridges5 on the inner surface of the A2 pulley and on the palmar surface of the FDP tendon. The direction of the fibers of the gliding pair demonstrated a preferential direction for friction because the shape of the tendon and the direction of the fibers changed when the tendon was under tension. When flexing, the friction would be less, but as soon as the system became static, or eccentric, the directional angle of the fibers changed to favor friction (Figure 4).

Figure 4. 

SEM performed on the inner surface of the A2 pulley. The fibers shown are perpendicular to the direction of movement of the tendons. Reproduced from [50].

Figure 4. 

SEM performed on the inner surface of the A2 pulley. The fibers shown are perpendicular to the direction of movement of the tendons. Reproduced from [50].

In 1995, Uchiyama et al. presented an experimental setup [48, 49] that allowed direct in vitro measurement of friction at the tendon-pulley interface [12]. Though their results did not perfectly match their theoretical model, they showed that friction favors the eccentric configuration and measured the friction in this system (the friction coefficient was 0.040 ± 0.014).

In a sport climbing study [40], A. Schweizer investigated in vivo friction between flexor tendons and pulleys by comparing the eccentric and concentric maximum strengths of flexion, both in the PIP joint and in the wrist joint, with an isokinetic device. The strength deficit (difference between the maximum eccentric and concentric strengths) of these two movements was found and used to determine the friction between flexor tendon and pulleys. Under maximal load, friction was responsible for approximately 9% of the holding force (static friction coefficient μ = 0.075) during the crimp grip. Friction showed a clear correlation with the degree of flexion of the PIP joint (being maximal at a flexion angle of approximately 85°).

In 2009, Schweizer investigated the influence of a preceding flexion or extension movement on the static configuration of human finger flexor tendons and pulleys [42]. Using freshly frozen human cadaver fingers, his experiments showed that while the tendons were statically loaded with 40 N, the generated flexion torque in the PIP joint was 11% greater if there was a preceding extension movement than if there was a preceding flexion movement in that joint. This difference has been directly attributed to the tendon-pulley friction.

Of the many existing models that try to capture the moving behavior of the human finger, most [1, 8, 23] fail to include friction. Furthermore, the only model that incorporates friction has too many limitations, making it difficult to examine extensively [41]. It is worth mentioning that some studies on an abstract frictional two-link tendon-sheath system do exist; however, they are directed toward creating a controller that compensates for the friction present [32, 33].

4 Model Definition

Due to the complexity of the tendon-pulley interaction in the human finger, several simplifications and assumptions have to be made. The finger is statically modeled using Coulomb friction. A dynamic representation was not favored, as the difference between a static and dynamic model would not be very large using the Coulomb approximation. Due to the small masses and dimensions of the bones, the dynamics (velocity, acceleration, moments of inertia, etc.) become almost negligible, which again would lead to a static model. Furthermore, for dynamics, the Coulomb approximation uses the dynamical friction coefficient μd, on which very little information can be found in literature. We only model the A2 pulley, as it has been shown to be the main contributor of friction in the human finger [36, 40]. Additionally, the PIP joint is considered to be friction-free; [41] and [3] state that friction in this joint is present but very small. Finally, the DIP joint is considered to be stiff and has therefore no influence on the system.

Figure 5 shows the basic abstraction and the parameters used in this work. A frictionless tendon-driven finger system with an actuated PIP joint can be modeled as a rigid two-link system, defined by the free link length (L), the PIP joint angle (φ), the free segment mass (m), and the two pulleys a2 and a4, which define the angle α. The model parameters used are shown in Table 1 and were obtained from physiological studies of the human finger tendon-pulley system [25, 26]. The segment mass was arbitrarily set at 10 g, as no appropriate physiological data was found. The model orientation having the PP perpendicular to gravity was chosen, because it corresponds to the majority of physiological studies available.

Figure 5. 

Human finger and the abstraction used in our model. The distal interphalangeal joint (DIP) is fixed, while the proximal interphalangeal joint (PIP) is not. The model consists of a two-link, one-joint frictional tendon-driven system. The metacarpal bone and joint are not modeled.

Figure 5. 

Human finger and the abstraction used in our model. The distal interphalangeal joint (DIP) is fixed, while the proximal interphalangeal joint (PIP) is not. The model consists of a two-link, one-joint frictional tendon-driven system. The metacarpal bone and joint are not modeled.

Table 1. 

Model parameters.

L = 49.4 mm 
m = 10 g 
μ = 0.075 
g = 9.81 m/s2 
f = 20 mm 
p = 16.6 mm 
e2 = 5.2 mm 
e4 = 6.5 mm 
s = 12.5 mm 
L = 49.4 mm 
m = 10 g 
μ = 0.075 
g = 9.81 m/s2 
f = 20 mm 
p = 16.6 mm 
e2 = 5.2 mm 
e4 = 6.5 mm 
s = 12.5 mm 

Friction in the tendon-sheath system is modeled using the Coulomb frictional model and the capstan friction equation [24]. As a result, the force required to actuate the now frictional segment is in addition a function of the friction coefficient of the tendon-sheath system (μ) and of the arc angle of the pulley the tendon is in contact with (α). Before we describe the model implementation in detail, we will briefly present the friction implementation used for our model.

4.1 Coulomb Friction

The Coulomb friction approximation [31] is used in our model. Though in general the relationship between normal forces and frictional forces is not exactly linear, even its simplest expression encapsulates the fundamental effects of sticking and sliding. There are several friction models whose approximations incorporate additional effects [30], such as the Stribeck effect, viscous friction, stiction, rate dependence, and so on. Two prominent examples are Armstrong's model [2] and the LuGre model [17, 29].

Considering a biological system (such as the tendon-pulley system of a human finger), the parameters these models require are not easily determined. This is one of the main reasons why the Coulomb approximation is used in our model. This choice does not greatly amplify the imprecisions of the model, as the human finger does not have many known constant parameters (compared to, e.g., a mechanical pendulum). Furthermore, the Coulomb approximation applied on a circle segment (which we are going to see later on) maintains its simple formulation.

The classical Coulomb friction law is a typical example that can be considered as a set-valued force law. The Coulomb law states that the sliding friction is proportional to the normal force of a contact. The magnitude of the static friction force is less than or equal to the maximum static friction force, which is also proportional to the normal contact force.

4.1.1 Sliding

The friction force λT has a magnitude μλN and acts in the direction opposed to the relative tangential velocity, that is,
formula
where is the velocity of the block, λT is the frictional force, μ is the friction coefficient, and λN is the normal force acting on the block. The Sgn function is defined as
formula

It is important to highlight that, while the classical sgn function is defined with sgn(0) = 0, the Sgn multifunction is set-valued at = 0 [19].

4.1.2 Sticking

Continuing from the above example, if the block sticks, then the friction force must lie in the interval −μλN ≤ λT ≤ μλN. Going back to the human finger, normally an eccentric or concentric configuration refers to a movement. For simplicity, we define an eccentric configuration as a static configuration for which a posteccentric movement would follow (and similarly for concentric), meaning the following: Looking at Figure 6, during an eccentric configuration the friction force λT is at its maximum. A slight push in the right direction would lead to a movement (posteccentric movement), and a slight push in the other direction would lead to no movement: The friction force λT would be increased or decreased (depending on which maximum λT has attained), but the velocity is still at zero.

Figure 6. 

A simple Coulomb example with a block.

Figure 6. 

A simple Coulomb example with a block.

4.2 Capstan Friction

The capstan equation, also known as Eytelwein's formula [24], relates the holding force to the loading force of a flexible rope wound around a pulley. If there is friction between the rope and the pulley surface (and the forces on the two sides are not equal), a frictional force Ff will act on the rope and the pulley so as to oppose its motion.

As the rope bends over a small segment of the pulley, the tension in the rope will increase from T to T + dT in an angle of dϕ. The normal force is the differential dN. The frictional force is μdN and acts to oppose slippage (Figure 7). Integrating over the total contact angle gives the ratio of the tension forces in terms of the coefficient of friction μ and the contact angle α, yielding the capstan equation:
formula
Figure 7. 

Capstan derivation from a circle segment.

Figure 7. 

Capstan derivation from a circle segment.

4.3 Model Implementation

A simple model which describes the tendon-pulley system in the finger is shown in Figure 8. The pulleys themselves are not modeled per se, but instead they define where the tendon force is attached (point 3) and in which direction the tendon force is pointing (point 2). The angle α, used instead for the calculation of the frictional force λT, is defined by points 2 and 3, and can be formulated as
formula
Furthermore, the moment arm of F1 w.r.t. the PIP joint is defined as the cross product of the force and its position vector :
formula
Taking the moment and the force equilibrium of the system from Figure 8 gives us
formula
formula
formula
However, as we are not interested in x0x and y0y (the reaction forces in the PIP joint), it is possible to use only Equation 4, the moment equilibrium equation.
Figure 8. 

The abstracted simple model, indicating the parameters used for calculations.

Figure 8. 

The abstracted simple model, indicating the parameters used for calculations.

As a next step, we are interested in the interaction between the pulley and the tendon (see Figure 9). Equations 7 and 8 describe the system with capstan friction occurring in the A2 pulley. There are twopossible options we can consider: having either Ftendon or Ftip fixed. Fixing Ftendon with a constant force will give us better insight into the output force at the fingertip, whereas fixing Ftip will give us better insight into the tendon force as influenced by friction (note that the tendon force in a tendon-driven robotic or prosthetic hand has to be sustained by an actuator, so it can be looked on as an actuator force in this situation).

Figure 9. 

The pulley is modeled as a circle segment to fit the capstan approximation.

Figure 9. 

The pulley is modeled as a circle segment to fit the capstan approximation.

4.4 Ftip Fixed

In a scenario in which the Ftip is fixed, the system equation and the capstan equations evolve as follows:
formula
formula
where F1 is given by
formula
Note that F1 is the force acting on the tendon in the segmental system, but the friction force has not yet been considered.

4.5 Ftendon Fixed

Because in the model friction is only present in the A2 pulley, the fixed tendon force Ftendon is directly modified by the capstan equation:
formula
formula
Furthermore, the equations for what we are now interested in, Ftip (with a given Ftendon), are
formula
formula

5 Model Validation

Raw physiological data for an experiment involving the static response to eccentric and concentric loading of a human finger was obtained from [42]. For a detailed description of the experiment, we direct the reader to Section 2 of the above reference. The raw data was processed to produce a graph of Ftip versus the joint angle ϕ. As the mean physiological parameters of the model presented in Table 1 did not result in a direct match to the in vitro data (Figure 10), we modified the model by hand to generate a close match to the physiological data. Since the model is based on physiological parameters, we can make an informed decision on their dependence to best match the physiological data.

Figure 10. 

In vitro data plotted together with the unmodified model. The model parameters, which are mean values from the literature and presented in Table 1, do not result in an acceptable fit to the human data.

Figure 10. 

In vitro data plotted together with the unmodified model. The model parameters, which are mean values from the literature and presented in Table 1, do not result in an acceptable fit to the human data.

A first fit of the data can be seen in Figure 11. In order to obtain the fit, the following steps were taken: (1) a force offset of Foffset = 0.7 N was added to the model, (2) the friction coefficient was modified by a factor factorμ = 2.3, and (3) the parameters s, e2, and f were modified by factors factors = 1.063, factore2 = 0.9, and factorf = 0.99, correspondingly.

Figure 11. 

In vitro data plotted together with the model, with its parameters modified by hand to best fit the human data.

Figure 11. 

In vitro data plotted together with the model, with its parameters modified by hand to best fit the human data.

The added force offset required could be due to measurement error in the physiological data. We are not certain why the friction coefficient had to be increased by more than two times. A possible explanation is that the change is due to additional phenomena that are not accounted for with the current model. One such phenomenon, modeling tendon bowstringing, is discussed in Section 7.1. Furthermore, the friction coefficient used originally is based on studies on rock-climbing subjects [41] and might be a nonrepresentative mean. Finally, could be the case that the A4 pulley significantly contributes to friction and needs to be incorporated in the model. The modifications of the remaining physiological parameters are minimal, and we can assume they are due to the natural deviation of this particular finger from the mean values found in the literature.

A similar fit was obtained by the following modifications: (1) a force offset Foffset = 0.75 N was added to the model, (2) the friction coefficient was modified by a factor factorμ = 2.2, and (3) the parameters s and f were modified by factors factors = 1.063 and factorf = 0.9, correspondingly. Going a step further, we can attempt to obtain a better fit by separately modifying the parameter e2 for concentric and eccentric contractions. The physiological reasoning is twofold: (1) the A2 pulley is not a rigid structure [36], and (2) there are differences between eccentric and concentric finger loading. When the finger is eccentrically loaded, the direction of structural deformation in the A2 pulley will be toward the joint. As the tendon is bowstringing, this will result in a natural increase in e2. When the finger is concentrically loaded, structural deformation will be in the opposite direction, resulting in a decrease of e2. Of course, these would be minuscule changes; they would, however lead to a significant increase in the output force, as they are directly affecting the tendon's moment arm. As seen in Figure 12, e2 changes from 7.02 mm for the eccentric condition to 4.888 mm for the concentric condition, a change of only 2.132 mm, but with a significant impact on the Ftip force profile for each respective case.

Figure 12. 

In vitro data plotted together with the model, with its parameters modified to best fit the human data. The parameter e2 was modified separately for concentric and eccentric configurations of the model, while all other parameters were constant between configurations.

Figure 12. 

In vitro data plotted together with the model, with its parameters modified to best fit the human data. The parameter e2 was modified separately for concentric and eccentric configurations of the model, while all other parameters were constant between configurations.

6 Model Behavior

In this section, we examine the behavior of the model with respect to: (1) different tendon-pulley friction coefficients μ, (2) resultant external force-bearing capabilities for a given actuator force, (3) the resultant actuator force required to hold a given external force, (4) the magnitude of friction force λT over a range of external forces Fload, and (5) the mechanical advantage of the system, MA = Ftip/Ftendon, given an external force. These behaviors were examined with the PIP joint angle in the range φ = {0, 110}°. This range was chosen as a good approximation to the range of motion of the human index finger PIP joint [5]. The base parameters used in all calculations are based on the literature and are as presented in Table 1 unless otherwise noted.

6.1 Friction Coefficient Variation

The friction coefficient of the tendon-pulley system was examined in the range μf = μ × {0, 5}, for μ as given in Table 1. This choice offers a reasonable selection of frictional coefficients of materials that could potentially be used for the development of a robotic prosthetic hand with tendon-pulley friction. Even though μmax = 5μ = 0.3750 is much smaller than the frictional coefficient of steel on steel (μsteel = 0.8) [47], it clearly demonstrates the effect of the friction coefficient on the behavior of the system.

Considering a mechanical finger system in which the friction is set to zero (μ = 0), the actuator force is shown in Figure 13, plotted in black. In this situation, the concentric and eccentric curves are merged to one single curve, and their difference vanishes. Increasing μ leads to a decrease in required eccentric tendon force Ftendonecc and simultaneously to an increase in required concentric tendon force Ftendonconc, with a constant fingertip force of Ftip = 40 N.

Figure 13. 

All variations of the frictional coefficient μ. Here Ftip is fixed at 40 N. For greater distinction, the frictionless curve (μ = 0 · μ0) has been plotted in black.

Figure 13. 

All variations of the frictional coefficient μ. Here Ftip is fixed at 40 N. For greater distinction, the frictionless curve (μ = 0 · μ0) has been plotted in black.

The effect of varying the frictional coefficient μf on the external load that can be sustained, given a constant tendon force Ftendon = 40 N, can be seen in Figure 14. Increasing μf leads to an increase in the eccentrically sustained external force Ftipecc and simultaneously to a decrease in the concentrically sustained external force Ftipconc.

Figure 14. 

All variations of the frictional coefficient μ. Here Ftendon is fixed at 40 N. For greater distinction, the frictionless curve (μ = 0 · μ0) has been plotted in black.

Figure 14. 

All variations of the frictional coefficient μ. Here Ftendon is fixed at 40 N. For greater distinction, the frictionless curve (μ = 0 · μ0) has been plotted in black.

6.2 Resultant External Load

Figure 15 shows the resulting sustainable external loading force Ftip of the system with a constant tendon force Ftendon = 100 N over the range of motion of the PIP joint (angle φ). The maximum difference FtipeccFtipconc occurs at φ = 91° and is 2.1 N. This is a direct consequence of the friction model used, as the angle α and consequently the arc p of the tendon in contact with the sheath increase with flexion. Over all flexion angles φi, the frictional model favors posteccentric over postconcentric configurations with regard to the external force that can be sustained with a fixed actuator force.

Figure 15. 

Dependence of external force Ftip for a 100-N tendon force Ftendon on the joint's flexion angle φ.

Figure 15. 

Dependence of external force Ftip for a 100-N tendon force Ftendon on the joint's flexion angle φ.

6.3 Resultant Actuator Force

The resultant actuator force capabilities of the model in eccentric and concentric configurations was examined at the external force extrema. Ftipmin = 0 is used to describe the no-load configuration, with only the mass of the MP segment used to calculate Ftendon. Correspondingly, Ftipmax is used to describe Ftendon under a maximum load configuration of 100 N, which has been found to be the mean boundary failure load force of the PIP joint in the human finger [26].

Figure 16 shows the actuator force (Ftendon) that has to be applied under no external force. As only the mass of the segment has to be sustained at a particular PIP angle φ, the required actuator force is very small. In addition, the differences between eccentric and concentric configurations are minimal, with Fecc-conmax = 0.0077 N occurring at a flexion angle of φ = {30, 40}°. In other words, under low loads, the influence of friction in the tendon-pulley system is very low and can be neglected.

Figure 16. 

Dependence of actuator force Ftendon for a zero external fingertip load Fload on the joint's flexion angle φ. The influence of friction in the system is minimal, due to the low mass of the finger.

Figure 16. 

Dependence of actuator force Ftendon for a zero external fingertip load Fload on the joint's flexion angle φ. The influence of friction in the system is minimal, due to the low mass of the finger.

Figure 17 shows the resulting actuator force (Ftendon) that has to be applied for a 100-N loading force. The difference between the concentric and eccentric forces is increasing in the flexion angle φ. The maximum difference Fecc-conmax = 25.82 N occurs at a flexion angle φ = 91°. Again, as is the case with the resultant actuator force, this is a direct consequence of our model.

Figure 17. 

Dependence of actuator force Ftendon for a 100-N external fingertip load Fload on the joint's flexion angle φ.

Figure 17. 

Dependence of actuator force Ftendon for a 100-N external fingertip load Fload on the joint's flexion angle φ.

6.4 Frictional Force Magnitude

Figure 18 shows the frictional force λT over the entire flexion range in a static eccentric configuration when the load at the fingertip (Ftip) is varied between Ftipmin and Ftipmax. The force λT is the difference between Ftendon and F1 of the pulley model, given a friction coefficient μ from Table 1. The frictional influence increases both with angle and with external load. Further, for loads close to Ftipmax the region of φ where frictional force is high appears to be larger than for lower loads. This indicates a clear benefit of friction for posteccentric movements under large loading, while at the same time free-air movements, where no external load is present, are virtually unaffected over the entire range of motion of the PIP joint.

Figure 18. 

Frictional force λT over a range of fingertip forces (Ftip) and joint angles (φ). The frictional force is maximal under high external forces and for configurations around PIP angle φ = 90°.

Figure 18. 

Frictional force λT over a range of fingertip forces (Ftip) and joint angles (φ). The frictional force is maximal under high external forces and for configurations around PIP angle φ = 90°.

7 Discussion

7.1 Modeling Bowstringing

The model presented above uses α for the frictional pulley angle. The reason behind this choice is merely to obtain some representative results, as the angle α does slightly increase during flexion of the finger; however, it does not have any physiological meaning. To assign a physiological meaning to this angle [25], we could define a circle that best fits the four points 1–4 in Figure 19. As the pulley length p2 at A2 (which can be interpreted as the circle segment on which friction occurs) does not change, the pulley angle can be defined as
formula
Figure 19. 

The drawing on the left shows approximately what the tendon's path looks like, and the one on the right shows how a circle can be used to represent this path.

Figure 19. 

The drawing on the left shows approximately what the tendon's path looks like, and the one on the right shows how a circle can be used to represent this path.

Theoretically, three points are enough to define a circle. However, preliminary results indicate that using only points 1–3, the fitted circle does not accurately represent the curvature at the A2 pulley. Therefore, a fourth point (4) has to be included. This leads to an overdetermined system that has to be solved with a least-squares method, using either an algebraic or a geometric fit.

7.2 Characterization of High-Load Configurations and Adaptation to Robotics

There is clearly an advantage of frictional over nonfrictional tendon-driven systems for posteccentric configurations. However, it is still questionable whether friction in a tendon-driven system is beneficial. Currently no such postmovement categorization exists in the literature. To create such a categorization, a task characterization for the given system is necessary. In the case of robotic hands, such a task space can be defined as the grasp taxonomy of the human hand. There are a number of such taxonomies, with Cutkosky's being one of the most prominent in robotics [14]. In addition, a different categorization exists for upper limb prostheses, corresponding to activities of daily living (ADLs), with the Wolf motor test being widely used [52]. In this case, ADLs would need to be decomposed into an appropriate grasp class where we could then look for postmovement configurations.

The results of Matsumoto et al. [27] definitely represent a step in a good direction, as they indicate that the majority of daily life activities are of an eccentric nature. Here we further hypothesize that the majority of high-load configurations involve posteccentric movements. If this is indeed the case, we are safe to conclude that friction in a tendon-driven hand system is of great benefit.

7.3 Adaptation to Robotics

Assuming our above hypothesis is correct, two mutually exclusive beneficial optimizations can be performed on a robotic hand system: (1) reducing actuator size or (2) increasing force output.

In the first case, we can reduce the size and weight of the actuators compared to a nonfrictional system, while being certain that low- to medium-load configurations will not be affected by such a reduction. Assuming a frictionless model of a robotic hand, the tendon force Ftendon required to maintain an external load Ftip can be calculated. An appropriate choice of an actuator can then be made that will satisfy the requirement of maintaining the external load. By simply modifying the transmission system to a frictional tendon-pulley system, as Ftendonecc > Ftendonμ=0, the actuator force capabilities and thus the size can be reduced. At the same time, we still satisfy the requirement of maintaining the original external force Ftip.

Alternatively, by maintaining the size and correspondingly the force capabilities of a frictionless system, we can increase the maximum eccentric holding load by incorporating a frictional tendon-driven system at no expense. Given an actuator with Ftendon able to maintain Ftip for a near-frictionless system, by incorporating friction in this system, as Ftendonecc > Ftendonμ=0, the Ftip that can now be maintained is higher.

We can then identify optimal model parameters and their influence on the characteristic behavior of the system. Potential optimization areas include the size, location, and morphology of the A2 pulley and tendon material with an appropriate friction coefficient. Going a step further, and noting that during a low-loaded configuration friction might be undesirable for performance reasons (e.g., positional precision) while friction under high loads does have a large positive influence in eccentric configurations, we can aim for designing an adaptive tendon-pulley system in such a way that friction is not present in unfavorable situations. Such a system would be able to exploit friction under beneficial circumstances and minimize it otherwise.

7.4 Force Output Metric for Frictional Tendon-Driven Systems

The standard metric for characterizing the force capabilities of robotic prosthetic hands involves measuring either the grasp force of the entire hand [9, 16, 21] or the pinch force between the thumb and (usually) the index finger [11, 46]. As has been described in this article, however, assuming a tendon-driven robotic hand, what those measurements display is in fact Ftendonconc, as the friction between the tendons and the pulleys is not zero. If it is indeed the case that high-load configurations mostly involve eccentric movements, friction will have a positive effect in such systems. Furthermore, assuming the above assumptions hold, the actual force capability of such a system for high-load configurations should in fact be defined by Ftipecc, where Ftipecc = Ftipμ=0 + λT. The model presented in this study can be used to identify the frictional influence of a particular robotic system.

8 Conclusion

The tremendous gap still existing between the natural biomechatronic system of the human hand and the designs of artificial anthropomorphic hands will occupy the research effort in this field for a long while. This work provides a frictional model that can be used in the design of tendon-driven robotic hands, with the main aim of increasing the holding force of such hands for eccentric configurations with no related actuator cost (i.e., additional weight or force output). It should further serve to acquaint the reader with what it means to have friction in a tendon-driven robotic system and what it means to actually take advantage of this phenomenon.

We presented a two-link, 1-DOF model of a tendon-driven finger that includes friction in the tendon-pulley system and was validated against in vitro human data. Assuming that most hand activities are eccentric and furthermore that these activities can be highly loaded (compared to concentric activities), we showed that for these cases friction can be of great benefit. Without increasing the overall weight of the robotic hand by using stronger and heavier actuators, and only taking advantage of friction, a robotic hand is capable of increasing its output force.

It is essential to realize a hand activity classification into high-load and low-load activities. Furthermore, and especially concerning this work, it is of great interest to evaluate the ratio between eccentric and concentric configurations in daily hand activities. This would give us better insights into the exact ratio between eccentric and concentric movements regarding high-load activities.

Acknowledgments

The authors would like to thank Prof. Rolf Pfeifer, Juan Pablo Carbajal from the AI Lab, UZH, and Prof. Fumiya Iida from the Bio-Inspired Robotics Lab, for their valuable input. This research was supported by the Swiss National Science Foundation through the National Centre of Competence in Research Robotics (NCCR Robotics).

Notes

1 

International Classification of Functioning, Disability and Health, also known as ICF, is a classification of the health components of functioning and disability.

2 

The moment arm is defined as the perpendicular distance from the point of rotation (the joint) to the line of action of the force (the tendon's path).

3 

The sheath in the human finger is often misnamed in literature or merely confounded with the pulley: The sheath is a tube providing low friction and will not be addressed here, whereas the pulleys (annular and cruciate) are the main friction contributors to the tendon.

4 

The system supplies mechanical advantage by maintaining the flexor tendons close to the joint's axis of motion. In doing so, the pulleys prevent bowstringing. Bowstringing is therefore increasing the distance from the tendon to the joint's axis.

5 

Rough linear elevations.

References

1. 
An
,
K. N.
,
Chao
,
E. Y.
,
Cooney
,
W. P.
, &
Linscheid
,
R. L.
(
1979
).
Normative model of human hand for biomechanical analysis.
Journal of Biomechanics
,
12
,
775
788
.
2. 
Armstrong-Helouvry
,
B.
,
Dupont
,
P.
, &
de Wit
,
C. C.
(
1994
).
A survey of models, analysis tools and compensation methods for the control of machines with friction.
Automatica
,
30
,
1083
1138
.
3. 
Ateshian
,
G. A.
, &
Hung
,
C. T.
(
2005
).
Patellofemoral joint biomechanics and tissue engineering.
Clinical Orthopaedics and Related Research
,
81
,
81
90
.
4. 
Balasubramanian
,
R.
, &
Matsuoka
,
Y.
(
2008
).
Biological stiffness control strategies for the anatomically correct testbed (ACT) hand.
In
IEEE International Conference on Robotics and Automation
(pp.
737
742
).
5. 
Becker
,
J.
, &
Thakor
,
N.
(
1988
).
A study of the range of motion of human fingers with application to anthropomorphic designs.
IEEE Transactions on Biomedical Engineering
,
35
(
2
),
110
117
.
6. 
Biddiss
,
E.
, &
Chau
,
T.
(
2007
).
Upper-limb prosthetics: Critical factors in device abandonment.
American Journal of Physical Medicine & Rehabilitation/Association of Academic Physiatrists
,
86
(
12
),
977
987
.
7. 
Brand
,
P. W.
, &
Hollister
,
A.
(
1993
).
Clinical mechanics of the hand.
Burlington, MA
:
Mosby Year Book
.
8. 
Buchholz
,
B.
(
1992
).
A kinematic model of the human hand to evaluate its prehensile capabilities.
Journal of Biomechanics
,
25
,
149
162
.
9. 
Carrozza
,
M.
,
Cappiello
,
G.
,
Beccai
,
L.
,
Zaccone
,
F.
,
Micera
,
S.
, &
Dario
,
P.
(
2004
).
Design methods for innovative hand prostheses.
In
Engineering in Medicine and Biology Society, 2004. IEMBS '04. 26th Annual International Conference of the IEEE
,
Vol. 2
(pp.
4345
4348
).
10. 
Carrozza
,
M.
,
Cappiello
,
G.
,
Micera
,
S.
,
Edin
,
B.
,
Beccai
,
L.
, &
Cipriani
,
C.
(
2006
).
Design of a cybernetic hand for perception and action.
Biological Cybernetics
,
95
(
6
),
629
644
.
11. 
Lee
,
W.-c.
, &
Wu
,
C.-W.
(
2010
).
A novel design of a prosthetic hand.
In
2010 IEEE International Conference on Systems Man and Cybernetics (SMC)
(pp.
1821
1824
).
12. 
Coert
,
J.
,
Uchiyama
,
S.
,
Amadio
,
P.
,
Berglund
,
L.
, &
An
,
K.
(
1995
).
Flexor tendon-pulley interaction after tendon repair: A biomechanical study.
The Journal of Hand Surgery: Journal of the British Society for Surgery of the Hand
,
20
(
5
),
573
577
.
13. 
Controzzi
,
M.
,
Cipriani
,
C.
,
Jehenne
,
B.
,
Donati
,
M.
, &
Carrozza
,
M.
(
2010
).
Bioinspired mechanical design of a tendon-driven dexterous prosthetic hand.
In
2010 Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC)
(pp.
499
502
).
14. 
Cutkosky
,
M.
(
1989
).
On grasp choice, grasp models, and the design of hands for manufacturing tasks.
IEEE Transactions on Robotics and Automation
,
5
(
3
),
269
279
.
15. 
Dalley
,
S.
,
Wiste
,
T.
,
Varol
,
H.
, &
Goldfarb
,
M.
(
2010
).
A multigrasp hand prosthesis for transradial amputees.
In
2010 Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC)
(pp.
5062
5065
.
16. 
Dalley
,
S.
,
Wiste
,
T.
,
Withrow
,
T.
, &
Goldfarb
,
M.
(
2009
).
Design of a multifunctional anthropomorphic prosthetic hand with extrinsic actuation.
IEEE/ASME Transactions on Mechatronics
,
14
(
6
),
699
706
.
17. 
de Wit
,
C. C.
,
Olsson
,
H.
,
Astrom
,
K. J.
, &
Lischinsky
,
P.
(
1995
).
A new model for control of systems with friction.
IEEE Transactions on Automatic Control
,
40
(
3
),
419
425
.
18. 
Fite
,
K.
,
Withrow
,
T.
,
Shen
,
X.
,
Wait
,
K.
,
Mitchell
,
J.
, &
Goldfarb
,
M.
(
2008
).
A gas-actuated anthropomorphic prosthesis for transhumeral amputees.
IEEE Transactions on Robotics
,
24
(
1
),
159
169
.
19. 
Flores
,
P.
,
Leine
,
R.
, &
Glocker
,
C.
(
2010
).
Modeling and analysis of planar rigid multibody systems with translational clearance joints based on the non-smooth dynamics approach.
Multibody Systems Dynamics
,
23
,
165
190
.
20. 
Freivalds
,
A.
(
2004
).
Biomechanics of the upper limbs: Mechanics, modeling, and musculoskeletal injuries.
Boca Raton, FL
:
CRC Press
.
21. 
Gosselin
,
C.
,
Pelletier
,
F.
, &
Laliberté
,
T.
(
2008
).
An anthropomorphic underactuated robotic hand with 15 DOFs and a single actuator.
In
IEEE International Conference on Robotics and Automation 2008 (ICRA 2008)
(pp.
749
754
).
22. 
Laliberté
,
T.
,
Baril
,
M.
,
Guay
,
F.
, &
Gosselin
,
C.
(
2010
).
Towards the design of a prosthetic underactuated hand.
Mechanical Sciences Journal
,
2
,
19
26
.
23. 
Landsmeer
,
J. M. F.
(
1955
).
Anatomical and functional investigation of the articulation of human fingers.
Acta Anatomica (Basel)
,
25
(
Suppl. 24
),
1
69
.
24. 
Levin
,
E.
(
1991
).
Friction experiments with a capstan.
American Journal of Physics
,
59
,
80
84
.
25. 
Lin
,
G.-T.
,
Amadio
,
P. C.
,
An
,
K.-N.
, &
Cooney
,
W. P.
(
1989
).
Functional anatomy of the human digital flexor pulley system.
The Journal of Hand Surgery
,
14
(
6
),
949
956
.
26. 
Marco
,
R. A. W.
,
Sharkey
,
N. A.
,
Smith
,
T. S.
, &
Zissimos
,
A. G.
(
1998
).
Pathomechanics of closed rupture of the flexor tendon pulleys in rock climbers. Journal of Bone and Joint Surgery.
American Volume
,
80
(
7
),
1012
1019
.
27. 
Matsumoto
,
Y.
,
Nishida
,
Y.
,
Motomura
,
Y.
, &
Okawa
,
Y.
(
2011
).
A concept of needs-oriented design and evaluation of assistive robots based on ICF.
In
IEEE International Conference on Rehabilitation Robotics.
.
28. 
Millstein
,
S. G.
,
Heger
,
H.
, &
Hunter
,
G. A.
(
1986
).
Prosthetic use in adult upper limb amputees: A comparison of the body powered and electrically powered prostheses.
Prosthetics and Orthotics International
,
10
(
1
),
27
34
.
29. 
Olsson
,
H.
(
1996
).
Control systems with friction.
Ph.D. thesis, Lund University, Sweden
.
30. 
Olsson
,
H.
,
Astrom
,
K. J.
,
de Wit
,
C. C.
,
Gafvert
,
M.
, &
Lischinsky
,
P.
(
1998
).
Friction models and friction compensation.
European Journal of Control
,
4
(
3
),
176
195
.
31. 
Palaci
,
I.
(
2007
).
Atomic force microscopy studies of nanotribology and nanomechanics.
Ph.D. thesis, EPFLausanne, France
.
32. 
Palli
,
G.
, &
Melchiorri
,
C.
(
2006
).
Model and control of tendon-sheath transmission systems.
In
Proceedings of the 2006 IEEE International Conference on Robotics and Automation, 2006. ICRA 2006
(pp.
988
993
).
33. 
Palli
,
G.
, &
Melchiorri
,
C.
(
2006
).
Optimal control of tendon-sheath transmission systems.
In
Proceedings of the IFAC Symposium on Robot Control
,
Vol. 8
(pp.
73
78
).
34. 
Pfeifer
,
R.
, &
Bongard
,
J.
(
2006
).
How the body shapes the way we think: A new view of intelligence.
Cambridge, MA
:
MIT Press
.
35. 
Quinn
,
T. H.
, &
Baumel
,
J. J.
(
1993
).
Chiropteran tendon locking mechanism.
Journal of Morphology
,
216
(
2
),
197
208
.
36. 
Roloff
,
I.
,
Schoffl
,
V. R.
,
Vigouroux
,
L.
, &
Quaine
,
F.
(
2006
).
Biomechanical model for the determination of the forces acting on the finger pulley system.
Journal of Biomechanics
,
39
(
5
),
915
923
.
37. 
Schaffer
,
J.
(
1905
).
Anatomisch histologische Untersuchung über den Bau der Zehen bei Flederma¨usen und einigen kletternden Sa¨uge-Tieren.
Zeitschrift für wissenschaftliche Zoologie
,
83
,
231
284
.
38. 
Schmitz
. (
n.d.
).
Flexor pulley system.
Online. Available at: http://www.orthobullets.com/hand/6004/flexor-pulley-system (accessed July 2012).
39. 
Schutt
,
J.
(
1993
).
Digital morphology in the Chiroptera: The passive digital lock.
Acta Anatomica (Basel)
,
148
,
219
227
.
40. 
Schweizer
,
A.
(
2008
).
Biomechanics of the interaction of finger flexor tendons and pulleys in rock climbing.
Sports Technology
,
1
(
6
),
249
256
.
41. 
Schweizer
,
A.
,
Frank
,
O.
,
Ochsner
,
P. E.
, &
Jacob
,
H. A. C.
(
2003
).
Friction between human finger flexor tendons and pulleys at high loads.
Journal of Biomechanics
,
36
(
1
),
63
71
.
42. 
Schweizer
,
A.
,
Moor
,
B. K.
,
Nagy
,
L.
, &
Snedecker
,
J. G.
(
2009
).
Static and dynamic human flexor tendon-pulley interaction.
Journal of Biomechanics
,
42
(
12
),
1856
1861
.
43. 
Shadow Robot Company
.
The Shadow dextrous hand.
Online. Available at: http://www.shadow.org.uk/products/newhand.shtml (accessed July 2012).
44. 
Silcox
,
D. H.
,
Rooks
,
M. D.
,
Vogel
,
R. R.
, &
Fleming
,
L. L.
(
1993
).
Myoelectric prostheses. A long-term follow-up and a study of the use of alternate prostheses.
The Journal of Bone and Joint Surgery
,
75
(
12
),
1781
1789
.
45. 
Smagt
,
P. v. d.
,
Grebenstein
,
M.
,
Urbanek
,
H.
,
Fligge
,
N.
,
Strohmayr
,
M.
,
Stillfried
,
G.
,
Parrish
,
J.
, &
Gustus
,
A.
(
2009
).
Robotics of human movements.
Journal of Physiology, Paris
,
103
(
3–5
),
119
132
.
46. 
Takaki
,
T.
, &
Omata
,
T.
(
2009
).
High performance anthropomorphic robot hand with grasp force magnification mechanism.
In IEEE International Conference on Robotics and Automation, 2009. ICRA '09.
(pp.
1697
1703
).
47. 
The engineering toolbox
.
Static steel on steel friction coefficient.
Online. Available at: http://www.engineeringtoolbox.com/friction-coefficients-d_778.html (accessed July 2012).
48. 
Uchiyama
,
S.
,
Amadio
,
P. C.
, &
An
,
K.
(
1997
).
Gliding resistance of extrasynovial and intrasynovial tendons through the A2 pulley.
The Journal of Bone and Joint Surgery
,
79-A
,
219
224
.
49. 
Uchiyama
,
S.
,
Coert
,
J. H.
,
Berglund
,
L.
,
Amadio
,
P. C.
, &
An
,
K.-N.
(
1995
).
Method for the measurement of friction between tendon and pulley.
Journal of Orthopaedic Research
,
13
(
1
),
83
89
.
50. 
Walbeehm
,
E.
, &
McGrouther
,
D.
(
1995
).
An anatomical study of the mechanical interactions of flexor digitorum superficialis and profundus and the flexor tendon sheath in zone 2.
The Journal of Hand Surgery: Journal of the British Society for Surgery of the Hand
,
20
(
3
),
269
280
.
51. 
Wiste
,
T.
,
Dalley
,
S.
,
Withrow
,
T.
, &
Goldfarb
,
M.
(
2009
).
Design of a multifunctional anthropomorphic prosthetic hand with extrinsic actuation.
In
IEEE International Conference on Rehabilitation Robotics, 2009. ICORR 2009
(pp.
675
681
.
52. 
Wolf
,
S. L.
,
Lecraw
,
D. E.
,
Barton
,
L. A.
, &
Jann
,
B. B.
(
1989
).
Forced use of hemiplegic upper extremities to reverse the effect of learned nonuse among chronic stroke and head-injured patients.
Experimental Neurology
,
104
(
2
),
125
132
.
53. 
Zollo
,
L.
,
Roccella
,
S.
,
Guglielmelli
,
E.
,
Carrozza
,
M. C.
, &
Dario
,
P.
(
2007
).
Biomechatronic design and control of an anthropomorphic artificial hand for prosthetic and robotic applications.
IEEE/ASME Transactions on Mechatronics
,
12
(
4
),
418
429
.

Author notes

Contact author.

∗∗

Artificial Intelligence Lab, UZH, Andreasstrasse 15, CH-8050, Zurich, Switzerland. E-mail: dermitza@ifi.uzh.ch

Bio-Inspired Robotics Lab, ETHZ, LEO D 9.2, Leonhardstrasse 27, CH-8092, Zurich, Switzerland. E-mail: mmorales@student.ethz.ch

Uniklinik Balgrist, UZH, Forchstrasse 340, CH-8008, Zurich, Switzerland. E-mail: andreas.schweizer@balgrist.ch