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The underactuated fingers used in numerous robotic systems are evaluated by grasping force, configuration space, actuation method, precision of operation, compactness and weight. In consideration of all such factors a novel linkage based underactuated finger with a self-adaptive actuation mechanism is proposed to be used in prosthetics hands, where the finger can accomplish flexion and extension. Notably, the proposed mechanism can be characterized as a combination of parallel and series links. The mobility of the system has been analyzed according to the Chebychev-Grübler-Kutzbach criterion for a planar mechanism. With the intention of verifying the effectiveness of the mechanism, kinematics analysis has been carried out, by means of the geometric representation and Denavit-Hartenberg (D-H) parameter approach. The presented two-step analysis followed by a numerical study, eliminates the limitations of the D-H conversion method to analyze the robotics systems with both series and parallel links. In addition, the trajectories and configuration space of the proposed finger mechanism ha ve been determined by the motion simulations. A prototype of the proposed finger mechanism has been fabricated using 3D printing and it has been experimentally tested to validate its functionality. The kinematic analysis, motion simulations, experimental investigations and finite element analysis have demonstrated the effectiveness of the proposed mechanism to gain the expected motions.

Progressively, the researchers have developed different robotic fingers with various functionalities and mechanisms. Such robotic fingers are functional in various applications, for instance prosthetic hands, industrial grippers, surgical robots and scape robot arms. The effectiveness of an artificial finger will depend on its ability to apply an extensive range of grasping forces, generation of precise motion patterns and establishment of a comprehensive configuration space [

Nearly, in the past three decades of period, numerous prosthetic finger mechanisms have been developed by the use of tendon-based mechanisms [

However, in consideration of power grasping applications with a high workload and finger contact force, such prosthetic hands have a need of powerful actuators to be placed inside the fingers, which will create an excessive weight, which should bear by the amputee. This research proposes a novel linkage based finger mechanism, which is mostly applicable for the grasping applications with a high workload and finger contact force. Besides, the proposed mechanism cause to move by the cables, can be made with less in weight since there is no need to place any actuator inside the finger. Subsequently, a self-adaptive actuation method is presented.

In the last 150 years, several approaches have been proposed for the calculation of the mobility of the mechanisms. In the second half of the 19th century and the beginning of the 20th century the Chebychev-Grübler-Kutzbach criterion for multi-loop mechanisms were set up. Different versions of these formula were proposed all along the 20th century by Dobrovolski (1949-1951), Artobolevskii (1953) Kolchin (1960) Rössner (1961), Boden (1962), Ozol (1963), Manolescu and Manafu (1963), Bagci (1971), Hunt (1978), Tsai (1999) [

The novel linkage mechanism presented in this paper can be identified as a combination of parallel and series links. Consequently, the mobility of the system has been analyzed according to Chebychev-Grübler-Kutzbach criterion for a planar mechanism. A combination of geometric relations and Denavit-Hartenberg conversion have been deliberated to analyze the forward kinematics of the proposed mechanism. In addition, the trajectories and configuration space of the proposed finger mechanism are presented. The fabricated prototype has been experimentally tested and the comparison between the experimental and simulation results are demonstrated. Furthermore, finite element analysis has been carried out to ensure the structural performance. As presented in a previous publication of ours, this particular finger design has been used to develop a prosthetic hand for power grasping applications [

The linkage mechanism explicated in the previous paragraph has been progressively applied to develop a prototype finger.

with higher workloads. Elastic rubber loops are placed between PPT1 and PPT2, IPT1 and IPT2. Those rubber loops act as springs to obtain the reverse actuation and bring the finger the initial position.

Furthermore, a simple hypothesis is proposed, in order to achieve self-adaptive grasping.

The kinematic analysis is essential to yield the improved performance of the mechanical design and to develop appropriate control algorithms. The full kinematic structure of the proposed prosthesis is shown in

representation of the finger mechanism has been simplified by assuming the middle links act as the bones, where the outer links act as muscles.

F = 3 ( n − j − 1 ) + ∑ i = 1 j f i (1)

where n is the number of links, j is the number of kinematic pairs and f_{i} is DoF of the i^{th} pair. For the simplified configuration without length varying links n equals 3, j equals 2 and Σf_{i} equals 2. Therefore, the effective DoF of the mechanism is equal to 2. Therefore, without considering the motions of the length varying links which represent muscles, the mechanism is capable of generating only two-motion configuration with a fixed set of joint angles. Furthermore, kinematic analysis of the mechanism has been carried out in two steps. In the first phase, the geometric representation has been depicted with the intention to initiate the relationships between the joint angles and the link

lengths. Subsequently, in the second phase, the forward kinematic analysis has been carried out by means of Denavit-Hartenberg (D-H) approach, to determine the positions of the critical points on the finger with respect to the different CD and DG distances.

_{1} and DIP joint angle is 180 − θ_{2}. Rotation angles of the links CD, EF, DG and FI are represented by α_{1}, β_{1}, α_{2} and β_{2} respectively.

According to the geometry, following distances are equal.

A C = A E = B D = B F = G H = H I (2)

Let us consider the links of the proximal phalanx. According the geometry,

E F cos β 1 − A B = B F sin θ 1 (3)

A B − C D cos α 1 = B D sin θ 1 (4)

By considering, Equation (2), (3) and (4) it can be derived that,

2 A B = E F cos β 1 + C D cos α 1 (5)

Moreover, according to the geometry it can be established that,

A C = B D cos θ 1 + C D sin α 1 (6)

A E = B F cos θ 1 + E F sin β 1 (7)

By considering, Equation (2), (6) and (7) it can be derived that,

C D sin α 1 = E F sin β 1 (8)

According to the Pythagorean theorem,

C D = ( A B − B D sin θ 1 ) 2 + ( A C − B D cos θ 1 ) 2 (9)

By considering the general principles of trigonometry and algebra, Equation (9) can be simplified as,

C D 2 − A B 2 − A C 2 − B D 2 − 2 B D = A B sin θ 1 + A C cos θ 1 (10)

According to trigonometry it can be derived that,

( A B ) sin θ 1 + ( A C ) cos θ 1 = C 1 sin ( θ 1 + δ 1 ) (11)

where;

C 1 = ± A B 2 + A C 2 (12)

δ 1 = tan − 1 ( A C A B ) (13)

By substituting to the right hand side of the Equation (10), from Equation (11), (12) and (13),

C D 2 − A B 2 − A C 2 − B D 2 − 2 B D = ± A B 2 + A C 2 ( sin ( θ 1 + ( tan − 1 ( A C A B ) ) ) ) (14)

Equation (14) can simplified as below to establish a relationship between the angle θ_{1} and the link lengths.

θ 1 = sin − 1 ( C D 2 − A B 2 − A C 2 − B D 2 − 2 B D ( ± A B 2 + A C 2 ) ) − ( tan − 1 ( A C A B ) ) (15)

θ 1 = sin − 1 ( A 1 ) − δ 1 (16)

where;

A 1 = C D 2 − A B 2 − A C 2 − B D 2 − 2 B D ( ± A B 2 + A C 2 ) (17)

δ 1 = tan − 1 ( A C A B ) (18)

According to the Pythagorean theorem it can be derived that,

E F = ( A B + B F sin θ 1 ) 2 + ( A E − B F cos θ 1 ) 2 (19)

By substituting the θ_{1} from Equation (16), the Equation (19) can be written as,

E F = ( A B + B F sin ( sin − 1 ( A 1 ) − δ 1 ) ) 2 + ( A E − B F cos ( sin − 1 ( A 1 ) − δ 1 ) ) 2 (20)

By substituting the θ_{1} from Equation (16), the Equation (6) can be written as,

A C = B D cos ( sin − 1 ( A 1 ) − δ 1 ) + C D sin α 1 (21)

Equation (21) can simplified as below to establish a relationship between the angle α_{1} and the link lengths.

α 1 = sin − 1 A C − B D cos ( sin − 1 ( A 1 ) − δ 1 ) C D (22)

By substituting the θ_{1} from Equation (16), the Equation (7) can be written as,

A E = B F cos ( sin − 1 ( A 1 ) − δ 1 ) + E F sin β 1 (23)

By substituting EF from Equation (20), the Equation (23) can simplified as below to establish a relationship between the angle β_{1} and the link lengths.

β 1 = sin − 1 A E − B F cos ( sin − 1 ( A 1 ) − δ 1 ) ( A B + B F sin ( sin − 1 ( A 1 ) − δ 1 ) ) 2 + ( A E − B F cos ( sin − 1 ( A 1 ) − δ 1 ) ) 2 (24)

In the same way by considering the links in the intermediate phalanx, it can be proved that the relationship between the angle θ_{2} and the link lengths are given by,

θ 2 = sin − 1 ( D G 2 − B H 2 − B D 2 − G H 2 − 2 G H ( ± B H 2 + B D 2 ) ) − ( tan − 1 ( B D B H ) ) (25)

θ 2 = sin − 1 ( A 2 ) − δ 2 (26)

where;

A 2 = D G 2 − B H 2 − B D 2 − G H 2 − 2 G H ( ± B H 2 + B D 2 ) (27)

δ 2 = tan − 1 ( B D B H ) (28)

Furthermore, the relationship between the angle α_{2} and the link lengths are given by,

α 2 = sin − 1 B D − G H cos ( sin − 1 ( A 2 ) − δ 2 ) D G (29)

The relationship between the angle β_{2} and the link lengths are given by,

β 2 = sin − 1 B F − H I cos ( sin − 1 ( A 2 ) − δ 2 ) ( B H + H I sin ( sin − 1 ( A 2 ) − δ 2 ) ) 2 + ( B F − H I cos ( sin − 1 ( A 2 ) − δ 2 ) ) 2 (30)

The forward kinematic analysis has been carried out by means of the Denavit-Hartenberg (DH) parameter approach. According to the D-H procedure described by the Rocha et al. [_{i} is the distance between z_{i} − 1 and z_{i}. d_{i} is the distance between x_{i} − 1 and x_{i}. α_{i} is the angle between z_{i} − 1 and z_{i} measured along x_{i}, while θ_{i} is the angle between x_{i} − 1 and x_{i}, measured along z_{i}. Then the homogeneous transformation matrices for each joint should be determined and finally the overall homogeneous transformation matrix by premultiplication of

Link No. | D-H Parameter | |||
---|---|---|---|---|

α_{(i-1)} (degrees) | a_{(i-1)} (mm) | d_{i} (mm) | θ_{i} (degrees) | |

0 | 0 | 0 | 0 | 0 |

1 | 0 | AB | 0 | θ_{1} |

2 | 0 | BH | 0 | θ_{2} |

the individual joint transformation matrices should be determined. Accordingly,

By referring to the link frame assignment and D-H link parameters. The rotation around the Z_{0} axis can be denoted by,

T 0 = [ cos 0 − sin 0 0 0 ( sin 0 ) ( cos 0 ) ( cos 0 ) ( cos 0 ) − sin 0 ( − sin 0 ) ( 0 ) ( sin 0 ) ( sin 0 ) ( cos 0 ) ( sin 0 ) cos 0 ( cos 0 ) ( 0 ) 0 0 0 1 ] (31)

T 0 = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] (32)

Subsequently, the translation by AB, followed by a rotation around the Z_{1} axis, can be denoted by,

T 1 0 = [ cos θ 1 − sin θ 1 0 A B ( sin θ 1 ) ( cos 0 ) ( cos θ 1 ) ( cos 0 ) − sin 0 ( − sin 0 ) ( 0 ) ( sin θ 1 ) ( sin 0 ) ( cos θ 1 ) ( sin 0 ) cos 0 ( cos 0 ) ( 0 ) 0 0 0 1 ] (33)

T 1 0 = [ cos θ 1 − sin θ 1 0 A B sin θ 1 cos θ 1 0 0 0 0 1 0 0 0 0 1 ] (34)

Successively, the translation by BH, followed by a rotation around the Z_{2} axis, can be denoted by,

T 2 1 = [ cos θ 2 − sin θ 2 0 B H ( sin θ 2 ) ( cos 0 ) ( cos θ 2 ) ( cos 0 ) − sin 0 ( − sin 0 ) ( 0 ) ( sin θ 2 ) ( sin 0 ) ( cos θ 2 ) ( sin 0 ) cos 0 ( cos 0 ) ( 0 ) 0 0 0 1 ] (35)

T 2 1 = [ cos θ 2 − sin θ 2 0 B H sin θ 2 cos θ 2 0 0 0 0 1 0 0 0 0 1 ] (36)

According to the Denavit-Hartenberg convention,

T 2 0 = ( T 0 ) ( T 1 0 ) ( T 2 1 ) (37)

By substituting from the Equation (32), (34) and (36), the Equation (37) can be written as,

T 2 0 = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] [ cos θ 1 − sin θ 1 0 A B sin θ 1 cos θ 1 0 0 0 0 1 0 0 0 0 1 ] [ cos θ 2 − sin θ 2 0 B H sin θ 2 cos θ 2 0 0 0 0 1 0 0 0 0 1 ] (38)

Subsequently, the Equation (38) can be simplified as below,

T 2 0 = [ cos ( θ 1 + θ 2 ) − sin ( θ 1 + θ 2 ) 0 B H cos θ 1 + A B sin ( θ 1 + θ 2 ) cos ( θ 1 + θ 2 ) 0 B H sin θ 1 0 0 1 0 0 0 0 1 ] (39)

The position of the point D with respect to the origin can be defined as,

D x 0 y 0 z 0 = T 1 0 [ D x 1 D y 1 D z 1 1 ] (40)

where,

[ D x 1 D y 1 D z 1 1 ] = [ 0 B D 0 1 ] (41)

Substituting from Equation (34) and (41), the Equation (40) can be written as,

D x 0 y 0 z 0 = [ cos θ 1 − sin θ 1 0 A B sin θ 1 cos θ 1 0 0 0 0 1 0 0 0 0 1 ] [ 0 B D 0 1 ] (42)

Furthermore, the Equation (42) can be simplified as below to describe the position of the point D with respect to the origin.

D x 0 y 0 z 0 = [ − B D sin θ 1 + A B B D cos θ 1 0 1 ] (43)

Likewise, the position of the point G with respect to the origin can be defined as,

G x 0 y 0 z 0 = T 2 0 [ G x 2 G y 2 G z 2 1 ] (44)

where,

[ G x 2 G y 2 G z 2 1 ] = [ 0 G H 0 1 ] (45)

Substituting from Equation (36) and (45), the Equation (44) can be written as,

G x 0 y 0 z 0 = [ cos ( θ 1 + θ 2 ) − sin ( θ 1 + θ 2 ) 0 B H cos θ 1 + A B sin ( θ 1 + θ 2 ) cos ( θ 1 + θ 2 ) 0 B H sin θ 1 0 0 1 0 0 0 0 1 ] [ 0 G H 0 1 ] (46)

Afterwards the Equation (46) can be simplified as below to describe the position of the point G with respect to the origin.

G x 0 y 0 z 0 = [ − G H sin ( θ 1 + θ 2 ) + B H cos θ 1 + A B G H cos ( θ 1 + θ 2 ) + B H sin θ 1 0 1 ] (47)

Successively, the position of the point K with respect to the origin can be defined as,

K x 0 y 0 z 0 = T 2 0 [ K x 2 K y 2 K z 2 1 ] (48)

where,

[ K x 2 K y 2 K z 2 1 ] = [ H J J K 0 1 ] (49)

Substituting from Equation (39) and (49), the Equation (48) can be written as,

K x 0 y 0 z 0 = [ cos ( θ 1 + θ 2 ) − sin ( θ 1 + θ 2 ) 0 B H cos θ 1 + A B sin ( θ 1 + θ 2 ) cos ( θ 1 + θ 2 ) 0 B H sin θ 1 0 0 1 0 0 0 0 1 ] [ H J J K 0 1 ] (50)

Furthermore, the Equation (50) can be simplified as below to describe the position of the point K with respect to the origin.

K x 0 y 0 z 0 = [ H J cos ( θ 1 + θ 2 ) − J K sin ( θ 1 + θ 2 ) ) + B H cos θ 1 + A B H J sin ( θ 1 + θ 2 ) + J K cos ( θ 1 + θ 2 ) + B H sin θ 1 0 1 ] (51)

The prototype of the linkage finger mechanism has been designed with the link parameters demonstrated in _{1} and θ_{2} from Equation (16) and (26) to Equation (43), (47) and (51) the X and Y coordinates of the points D, G and K have been plotted as illustrated in

Furthermore, experimental investigations have been carried out to confirm the motions of the proposed finger mechanism. Accordingly, the sequence of the finger motions throughout the actuation of the prototype were captured by using a digital camera, as presented in

Parameter | Constant | Variable | ||||
---|---|---|---|---|---|---|

AC, AE, BD, BF, GH, HI | AB, BH | HJ | LJ, JK | CD, DG | EF, FI | |

Length (mm) | 10 | 40 | 30 | 9 | 33 to 40 | 40 to 47 |

angles of the prototype were measured with respect to different CD and DG distances. As presented in

The finite element analysis (FEA) by Solidworks simulations proved that the finger is sturdy to withstand the standard finger forces. Designed prosthetic finger mechanism is expected to carry 20 N payload at distal phalanx. In addition to the play load it is assumed that 5 N is applied by the elastics rubber loops as a spring effect. Further the tension of nylon strings has been is taken as 10 N where it applies a pull on PPB2 and IPB2. Von misses stress, resultant displacement and equivalent strain of the finger have been determined by FEA. According to the results, the designed finger has a maximum von Mises Stress of 3.36e+007 N/m^{2}, maximum resultant displacement of 1.55 mm and maximum equivalent strain of 0.008 where the Yield strength, Tensile strength and Elastic modulus of the material (PLA) are 7e+007 N/m^{2}, 7.3e+007 N/m^{2}, 3.5e+009 N/m^{2} respectively. The minimum factor of safety has been determined as 2. The convergence analysis has been carried out for randomly chosen mesh sizes and the results are presented in

This paper proposes a novel underactuated linkage finger mechanism, which is a combination of series and parallel links. Although the Chebychev-Grübler-Kutzbach criterion has limitations to analyze parallel robots, the simplification of the kinematic representation has been applicable to analyze the mobility of the proposed linkage mechanism. However, it was only able to demonstrate the effective DoF for finger grasps and was unable to demonstrate a clear understanding on friction of the mechanism. Even though the mobility analysis proposed that the mechanism has two DoF, the finger can be actuated by a single linear actuator. Hence, the presented linkage finger mechanism is competent as an underactuated mechanism. Furthermore, the proposed mechanism can be developed, without placing any actuators inside the finger, thus lightweight prosthetic devices can be introduced to accomplish grasps with high workload and finger contact forces. The explicated two-step kinematic analysis, first generating the relationships between the link lengths and the joint angles, then D-H conversion towards the forward kinematics is an impeccable approach to analyze the systems, which are developed as a hybrid of series and parallel links. Furthermore, the experimental investigation has verified the functionality of the finger and the exactitude of the analysis carried out. The outcomes of the kinematic analysis, establishment of the trajectories and the configuration space for the proposed mechanism, will be beneficial for future research towards developing control algorithms for the finger actuation.

The authors are indebted to the Department of Mechanical and Manufacturing Engineering, University of Ruhuna, Sri Lanka, for providing the fabrication facilities.

Herath, H.M.C.M., Gopura, R.A.R.C. and Lalitharatne, T.D. (2018) An Underactuated Linkage Finger Mechanism for Hand Prostheses. Modern Mechanical Engineering, 8, 121-139. https://doi.org/10.4236/mme.2018.82009