This paper presents the design and optimization of four versions of self-adaptive, a.k.a. underactuated, fingers based on four-bar linkages. These fingers are designed to be attached to and used with the same standard translational grippers as one finds in the manufacturing and packaging industries. This paper aims at showing self-adaptive fingers as simply as possible and analysing the resulting trade-off between complexity and performance. To achieve this objective, the simplest closed-loop 1 degree-of-freedom (DOF) linkage, namely the four-bar linkage, is used to build these fingers. However, it should be pointed out that if this work does consider a single four-bar linkage as the basic building block of the fingers, four variations of this four-bar linkage are actually discussed, including some with a prismatic joint. The ultimate purpose of this work is to evaluate whether the simplest linkages for adaptive fingers can produce the same level of performance in terms of grasp forces as more complex designs. To this end, a kinetostatic analysis of the four fingers is first presented. Then, the fingers are all numerically optimized considering various force-based metrics, and results are presented. Finally, these results are analysed and prototypes shown.

Self-adaptive, also known as underactuated (Birglen et al., 2007), hands and fingers have been used in the last decade by both the research community and the industry as a compromise between complex anthropomorphic robotic hands and classical industrial grippers. Complex dexterous hands could require more than three fingers and 9 actuated degrees of freedom (DOF) to become dexterous and provide sufficient motion capability for object manipulation, while classical industrial grippers are made for simpler tasks that only require a motion produced by a 1 DOF mechanism. Underactuated hands and fingers offer a simplicity in control not accessible to fully actuated designs since the number of actuators is smaller than the number of DOF. Often in an underactuated hand, a single actuator drives the whole hand. Additionally, underactuated hands are made not to depend on sensors for their operation while still having shape adaptation capabilities, i.e. being able to conform to a vast range of shapes of objects to grasp. Underactuated hands are also referred to as self-adaptive because of this property and to differentiate themselves from other applications of underactuation in robotics such as passive walkers. The first self-adaptive hand reported in the scientific literature was probably the Soft Gripper, introduced by Shigeo Hirose (Shigeo and Umetani, 1978), which was actuated by two wires and had two 10-phalanx fingers. Newer designs have been demonstrated since, commonly showing anthropomorphic inspiration but not exclusively (Begoc et al., 2007; Catalano et al., 2012; Dollar and Howe, 2006). Other robotic devices close to self-adaptive fingers have also been reported which use structural compliance to achieve conformal grasps. They are referred to as soft hands and grippers. Amongst these soft grippers, many can be found built using a particular design of bio-inspired fingers based on the Fin Ray Effect (FRE; see Fig. 1) (Crooks et al., 2017; Shan and Birglen, 2020) for which commercial products exist and are marketed by the company Festo. Underactuation and soft grasping are two common techniques to provide robotic systems with tools for securing and manipulating arbitrarily shaped objects.

Fin Ray Effect fingers grasping a fruit.

Commercial products exist for grasping hands, either relying on underactuated mechanisms or soft robotics techniques. For instance, Robotiq's 2F-85 and 2F-140 Adaptive Grippers, RightHand Robotics RightPick, Soft Robotics Inc. mGrip, or the Gripper Company fingers are all commercially available devices and are widely used in many markets. However, they are all new products, while the manufacturing and packaging industry has been using pneumatic parallel grippers in their operations for a very long time, and replacing them with any of these new designs is costly and time-consuming. The only commercial product not requiring the replacement of the end effectors in existing workcells is the FRE-based fingers from Festo since they can be simply attached to the gripper in place. Another similar solution avoiding the complete replacement of existing hardware was introduced in Carpenter et al. (2014) in which an adaptive jaw was proposed that can be secured to and driven by a parallel gripper. This adaptive jaw consists of three parallel hydraulic cylinders that are connected to a common local reservoir. By providing only an addition to a standard gripper, this solution eliminates the need to engineer a complex cable or linkage system to provide finger adaptability. Yet another solution, proposed by the second author and inspired by the FRE, consists in designing a passively adaptive linkage to be attached to standard industrial grippers; see Fig. 2 for an example. When in contact with an object, this linkage can deform in such a way that it provides an enveloping motion around the object it is in contact with. Actuation is thus provided by the motion of the gripper moving the base of the adaptive linkage but the latter embeds no actuation or sensing element. In Birglen (2015), such a linkage named the PaCoMe finger was introduced and consists in a three-phalanx self-adaptive mechanical finger. The proposed design is inspired by the FRE fingers but using rigid links instead of compliant ones and discards unnecessary crossbeams connecting the front and back of the fingers. The PaCoMe finger has 3 DOF and constitutes a closed-loop six-bar linkage in which specific joints embed a spring and a joint stopper. This finger was shown to produce stable enveloping and precision grasps while being attached to an off-the-shelf translational pneumatic gripper. An extension of this work was then demonstrated in Birglen (2019), based on a slightly simpler linkage and specifically designed to match the requirements of collaborative robotics applications. This second design was based on the idea that a transmission linkage producing full mobility to the phalanges might not be mandatory to achieve a successful grasp. The simpler mechanism presented in Birglen (2019) was shown to be a valid alternative to the original six-bar linkage version, based both on theoretical and experimental results shown in the paper. The same idea is also shown in Kok (2018) and Abdeetedal et al. (2018), where one 2 DOF finger and then one 1 DOF self-adaptive finger are introduced to be used with industrial parallel grippers. Finally, another example of a self-adaptive finger with 1 DOF is presented in Zheng and Zhang (2019), where a four-bar mechanism along with an eccentric cam is used.

PaCoMe adaptive finger prototypes attached to Schunk gripper.

This paper aims at continuing the discussion on this line of thought, namely
producing self-adaptive fingers as simply as possible, by analysing the
performance of designs with only a single DOF. To this end, four novel
variations of self-adaptive designs are presented, all based on four-bar
mechanisms. Two of these designs have two phalanges, while the other two have
three. An uncommon feature of these designs is that two of them (one in each
category: two- and three-phalanx fingers) use a prismatic joint. Prismatic
joints in robotic fingers are not unheard of, but they are uncommon as they
significantly depart from anthropomorphic inspiration. However, for
industrial grasping, the main market of the previously mentioned commercial
products, anthropomorphic designs are not necessarily relevant. The main
contributions of this work are as follows:

the introduction of new, simple 1 DOF designs of linkage-based adaptive fingers, potentially including a prismatic joint,

the comparison of the performances produced by these fingers with other more complex designs, as previously reported in the literature.

Prototypes of the 3 DOF PaCoMe fingers based on six-bar linkages described in Birglen (2015) are shown in Fig. 2 where they are attached to a standard Schunk KGG 100-80 gripper. As mentioned before, these fingers are based on a single-loop linkage with six revolute joints only, and they have three phalanges. These phalanges are constituted by the three consecutive binary links central to the hand and connected by three revolute joints. Two additional links and three revolute joints are used to form the transmission linkage of the finger at the outer side of each finger. This transmission linkage aims at avoiding constraint of the DOF of the finger and ensuring that the allowed motion of the phalanges produces the desired shape-adaptation property for the mechanism. The revolute joints in the transmission linkage are equipped with springs with stoppers to fully constrain and preload the linkage when it is not subjected to external contacts at the phalanges. The last remaining link, i.e. the base of the linkage, is connected to the gripper movable jaw, whose motion is one of a prismatic joint. Thus, the gripper acts a linear actuator for the fingers. A simplified design of this design was shown in Birglen (2019) in which the finger only had 2 DOF, and there are kinematic couplings between the rotations of the phalanges.

Following this trend in decreasing the number of DOF in self-adaptive fingers to simplify manufacturing and decrease cost, this paper proposes to reduce the DOF one step further to a single one. While there are a multitude of 1 DOF grippers proposed in the literature, it should be made clear that the mechanisms proposed here are fundamentally different from these grippers. What is proposed here is a 1 DOF mechanical finger to attach to a separate 1 DOF gripper. Hence, the complete mechanism has 2 DOF: one for the finger and one for the gripper. Contact forces are provided at the finger only, and the gripper is creating the motion, closing the distance between the object and the mechanical finger. A 0 DOF finger could actually be made and would be a single rigid part. However, a 0 DOF finger would not be able to provide any shape adaptation. Hence, 1 DOF for an adaptive finger is the absolute minimal number of DOF one can use. Further highlighting the difference between the solution proposed here and simple articulated grippers, the mechanisms proposed here use two or three phalanges to envelop objects, while articulated 1 DOF grippers only create pinch grasps (a.k.a. precision grasps).

The four 1 DOF self-adaptive fingers proposed in this paper.

All the fingers proposed in this work are illustrated in Fig. 3. Points

Summary of the main parameters of the fingers.

Fingers no. 1 and no. 2 as defined in this paper have two phalanges, and
their geometry as well as their associated parameters are shown in Fig. 3a and b. The transmission linkage of finger no. 1 is constituted
by two revolute joints in points

Figure 3c and d similarly illustrate the parameters of fingers no. 3 and no. 4. This time both mechanisms have three phalanges, and, thus, only a single joint remains for the transmission linkage (which in that sense is not “linkage” per se, being reduced to a joint). This remaining joint is either revolute for finger no. 3 or prismatic for finger no. 4. In the latter case, the distal phalanx is again rigidly connected to the prismatic joint. In both mechanisms, a spring is added to this last joint to statically constrain the finger in the absence of a contact, similarly to fingers no. 1 and no. 2. In all cases, when the translational gripper modelled by the prismatic joint at the base of the four mechanisms is driven, the finger is brought into contact with the object to be grasped, and this contact causes a motion of the linkages in reaction to the force arising at the connection, assuming the force of the gripper is sufficient to overcome friction at the contact point, as illustrated in Fig. 4. Similarly to fingers no. 3 and no. 4, the quadrilateral at the top of the fingers depicted in Fig. 3c–d is the distal phalanges of these fingers, which are rigid bodies.

Simulation of a closing sequence of a geometry of finger no. 1 mounted on a translational gripper.

In order to be able to compare between the four fingers modelled in this
paper and the ones previously reported in the literature, two steps are
required. As the performances of self-adaptive fingers are usually
quantified by the magnitudes of the contact forces they produce, one must
first establish these forces. Then, in a second step and to obtain a
meaningful comparison, the fingers must be optimized with respect to these
forces in a similar fashion as reported in the literature so that the best
fingers possible are used for the comparisons, and these comparisons are
meaningful. Assuming that dynamic forces are negligible, a common hypothesis
in underactuated grasping since the masses and inertias of the links are
usually relatively small, one can use the virtual work principle to
calculate the generated contact forces. The total virtual work for each of
the four fingers is

It should be noted that the expressions of

All the fingers discussed in this paper have 1 DOF and can therefore be
constrained by a single contact. Thus, two cases or contact scenarios are
possible for fingers no. 1 and no. 2 since they have two phalanges, and
three contact scenarios are possible for fingers no. 3 and no. 4 since these
have three phalanges. A contact scenario is defined as a situation with one
contact point at one specific phalanx. For a contact scenario

By choosing the 2 DOF of the mechanisms (one for the finger and one for
the actuation) as the translation

Again, fingers no. 1 and no. 2 have two phalanges, which means that there are
only two contact scenarios possible, while fingers no. 3 and no. 4 have
three contact scenarios since they have three phalanges. The Jacobian
matrices can be calculated for each mechanism for each contact scenario by
finding an expression of

The last case, namely contact scenario 3, for fingers no. 3 and no. 4, yields

In Eqs. (9)–(11),

These equations can be used to express the Jacobian matrix in Eq. (8), and the components of the latter depend on the geometry of each four-bar mechanism, which is very different for each finger since they all have a unique combination of number of phalanges and type of joints.

The transmission matrix

For fingers no. 1 and no. 2, the transmission linkage has two joints since
it is formed by two rotational joints for finger no. 1 and one rotational
and one prismatic for finger no. 2, so

For finger no. 2 these equations become

With fingers no. 3 and no. 4, their transmission “linkage” only has a
single joint, revolute for finger no. 3 and prismatic for finger no. 4, and
therefore, the expressions are a bit simpler. In these cases,

and for finger no. 4 one obtains

Substituting Eqs. (8)–(21) into Eq. (7), one can compute the generated contact forces at the phalanges for all fingers and all contact scenarios, as well as the gripper actuation force required for static equilibrium.

Once the contact force generated by a finger can be calculated as described
in the Sect. 2, one can start the optimization process of these forces.
The same fitness function needs to be used for the four mechanisms in order
to have a standardized evaluation of their performances, and this function
must be as close as possible to the ones used in the literature. In general,
many different optimization criteria are used in the literature for
underactuated robotic grippers (Kragten, 2010), but in the context of this
work, three fitness functions were used:

the percentage of the positive contact forces generated over the workspace,

the average value of the coefficient of variation of the contact forces,

and the mechanical advantage of the mechanisms.

It is known that underactuated and self-adaptive fingers do not always
generate positive contact forces at all phalanges in all configurations and
this is one of their main drawbacks. Generating a negative contact force
means that the finger has to pull on the surface of the object to be
grasped, which is impossible in most cases. When negative contact forces
occur, the finger can eventually lose contact with the object after sliding
along its surface, a phenomenon known as ejection. It is desirable to
minimize the occurrence of these negative forces, and this is the reason why
this first fitness function, evaluating how often over the finger's
workspace positive forces are encountered, is used both here and in the
literature. The associated metric of performance can be written as

The second optimization criterion used in this paper is the average value of
the coefficient of variation of the contact forces. The contact forces
generated by the mechanism could be unbalanced even if they are always
positive, and this is detrimental to the objective of securing objects of
vastly different shapes and sizes. It is usually more desirable to even out
the generated forces in the workspace of the fingers in order to be able to
apply a relatively constant pressure on objects whatever their geometries.
The average value of the coefficient of variation can be defined
mathematically as

where

Finally, the third optimization criterion introduced here is the mechanical
advantage of the mechanism. The forces generated by the fingers (while being
positive and even with optimal coefficient of variation, i.e. of close
magnitudes) could still be significantly weaker than the actuation force
provided by the gripper. This decrease of the contact forces compared to the
gripper's is detrimental if fine force control is required and also strongly
weakens the grasp as one potentially “loses” a significant part of the
actuation effort. This phenomenon was identified as one of the major
weaknesses of FRE fingers in Carpenter et al. (2014)
and can cause slippage of the object from the hand during high-speed pick
and place operations. Quantifying the mechanical advantage of a
self-adaptive finger gives an idea of the general efficiency of the fingers
since it is in a sense the ratio between the gripper and the finger forces.
Ideally, the contact force generated by the finger onto an object should be
equal to the closing force delivered by the translational actuator at the
base to maximize the efficiency. The mathematical equation for the
mechanical advantage can be written as

To integrate this equation into the optimization process, the first step is
to normalize the values of this mechanical advantage to get values ranging
from 0 to 1. The second step is to calculate its average for each contact
scenario

Geometrical parameters of the fingers to be optimized.

Optimization results.

Table 3 and Fig. 5 present and illustrate the values of the optimized
parameters and the values of the optimization functions for all the fingers.
All these optimized fingers generate positive contact forces exclusively
throughout their workspaces since the value of

Geometry of the optimized fingers.

Plots of the generated forces as a function of the input angle and the gripper force for the four fingers.

As for the forces, plotted in Fig. 6a for finger no. 1,

CAD models for the optimal designs of the fingers; the gripper is shown
in the closed

For finger no. 2, it can be seen in Fig. 6c that the generated contact
force on the proximal phalanx

For finger no. 3, one can divide the workspace into roughly two parts based
on the plot shown in Fig. 6e: for

Finally, for finger no. 4 Fig. 6g illustrates that all the phalanges
generate contact forces that have very close values when

Finally, for the comparison with more complex linkages previously shown in the literature, if one looks at Birglen (2019), the best percentage of the workspace corresponding to fully positive contact forces is reported to reach 9.7 %, and the average coefficient of variations of the contact forces is 0.73. Considering the second reference design proposed in Kok (2018), a value of 7 % is reported for the positive contact force workspace. With all the fingers presented in this paper, the achieved percentage of positive contact forces in the workspace is 100 %. Therefore, from this metric, there seems to be a clear advantage of using simpler linkages. The coefficient of variation of the optimized fingers of this paper with the same number of phalanges as in Birglen (2019) is of similar magnitude, but the ones with fewer phalanges are 2 to 3 times better.

This paper presented four different designs of self-adaptive mechanical fingers that can be actuated by the standard translational grippers used in the industry to transform these grippers into complete underactuated hands. The four designs are simple in terms of kinematics when compared to previous prototypes from the literature since all of them are based on variations of four-bar mechanisms and, thus, have only 1 DOF. Prismatic joints were also considered in two of these four fingers, which is uncommon in artificial fingers, even industrial ones. A general kinetostatic analysis was first presented in which a Jacobian and transmission matrix were calculated for each finger and for each of the defined contact scenarios in order to compute the contact forces generated by the different fingers. Then, optimization criteria were discussed for the evaluation of the performances of the fingers. The three optimization functions used in this paper, namely the percentage of the workspace where positive contact forces are generated by the mechanisms, the average value of the coefficient of variation of the contact forces, and the mechanical advantage of the mechanisms, gave a better understanding of the magnitude and variations of the generated contact forces between different phalanges. Although the mechanisms are simple in terms of geometry, their performances can be considered at least comparable with other prototypes with greater mobility (more DOF) or even better. However, it should be noted that the optimization was focused exclusively on the generated forces, and the enveloping capability of the fingers was not taken into consideration, and simulations show that fingers with rotational joints tend to have better enveloping grasps than the ones using a translational joint. This does not mean that using a prismatic joint in underactuated fingers will definitely lead to poor enveloping grasps, but the same finger optimization routine could give very different results if the ability to generate enveloping grasps were explicitly considered. Future works include manufacturing and experimental validation of the effectiveness of the optimal designs presented in this work and shown in Fig. 7.

Code can be freely accessed at

The data are available upon request from the corresponding author.

FN conducted the mathematical modelling, programmed all MATLAB functions used for the paper including those needed the optimizations, produced Figs. 3 and 6, analysed the data, and prepared the manuscript. LB provided the original idea for the work, the four kinematic designs of the fingers, and the methodology, produced Figs. 1, 2, 4, 5, and 7, designed the CAD models of the prototypes, and revised the manuscript.

Some authors are members of the editorial board of MS. The peer-review process was guided by an independent editor, and the authors have also no other competing interests to declare.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This paper was edited by Mohamed Amine Laribi and reviewed by Araceli Zapatero and one anonymous referee.

^{®}Effect, International Journal of Advanced Robotic Systems, 14.4, 1729881417721155, 2017.