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Anthropomorphic finger with optimized geometric parameters for pinching and grasping tasks
Mechanism and Machine Theory 49 (2012) 52–66
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Anthropomorphic finger with optimized geometric parameters for pinching and grasping tasks Norsinnira Zainul Azlan ⁎, Hiroshi Yamaura Department of Mechanical and Control, Graduate School of Science and Engineering, Tokyo Institute of Technology, Ookayama 2-12-1-I3-1, Meguro-ku, Tokyo 152-8552, Japan
a r t i c l e
i n f o
Article history: Received 20 January 2011 Received in revised form 17 October 2011 Accepted 1 November 2011 Available online 7 December 2011 Keywords: Robotic finger Anthropomorphic design Optimization Grasping Pinching
a b s t r a c t This paper presents the parameter optimization procedure for a new anthropomorphic finger mechanism in performing pinching and self-adaptive grasping operation. The mechanism is made based on the observation of human finger during grasping and pinching. It is constructed from seven bar linkages and its upper middle phalanx is incorporated with a curved guiding slot and lead screw mechanism to adjust the link's effective length in accomplishing these two tasks. The finger's geometrical parameters are optimized based on four design criteria, which are the anthropomorphic pinching kinematics and force; and grasping kinematics and force to enhance its performance. The numerical result obtained gives an improved parameter set compared to the initial guess. A prototype with small size and light weight actuators and sensors has been implemented based on the optimum solution to achieve a compact anthropomorphic design with grasping and pinching capabilities. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Artificial hands are the mechanisms developed to either or both mimic the physical appearance and serve the functions of a real hand. Some of the applications of this type of hand include research purpose, industry, space exploration, prosthetics and humanoid robots. The hands developed for prosthetics and humanoid robots applications require a special characteristic, which is “anthropomorphism”. The term “anthropomorphism” refers to the capability of the robotic end effectors to resemble the human hand in terms of both functionality and physical appearance including its shape, size, and weight. A humanoid robot with anthropomorphic hand looks more natural and the replacement of this kind of hand for the amputees helps them in carrying out their daily tasks. Over the last decades, numerous researches have been conducted to investigate the practical solution of this hand. The main challenge in developing an anthropomorphic hand is to design a mechanism that can both realize the functions and motions, and at the same time have similar shape, size and weight to a human hand. However, there are not many components including the motors and reduction gears that are small and light enough to realize practical hand movements [1]. Some of the previous robotic hands including the UTAH/ MIT Hand [2] and Stanford/ JPL Hand [3] have achieved good performances in replicating human capabilities, but they are bulky and heavy due to the large size and number of actuators and sensors incorporated to realize the complex motions. An alternative to the design problem is to reduce the number of degree of freedom (DOF) in the hand by decreasing the number of actuators utilized such as in the TBM Hand [4]. The fingers developed in Refs. [5–7] and LARM Hand [8] are constructed from various arrangement of bar linkages with only one DOF. Although this technique approaches the human hands’ weight and size due to the low number of motors used, it also affects the versatility of the hand.
⁎ Corresponding author. Tel.: + 81 80 3022 0843; fax: + 81 3 5734 2419. E-mail address: [email protected] (N. Zainul Azlan). 0094-114X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2011.11.005
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Another approach in the hand development is to adopt the underactuated mechanism such as in the tendon based SPRING Hand [9] and RTRII Hand [10], and the bar linkage based SARAH HAND [11–13]. This type of mechanism utilizes low number of actuators without sacrificing the hand's DOF. The number of DOF is made higher than the number of active motors by replacing the traditional actuators such as electro-magnetic motors with small passive elements such as springs and mechanical limits. This mechanism also results in the self-adaptive grasp capability of the finger to the object to be grasped [14] and avoids the need of sensors and complex feedback control strategies. The index finger in HIT/DLR Prosthetic hand [15] is developed based on the underactuated linkage mechanism and actuated by a single motor. However its middle and distal phalanges are actuated through coupling linkages, in which these phalanges rotate at the same time with the angle ratio 1:1. The Iowa hand [16] exhibits a significantly lighter weight compared to the available commercial prosthetic hands by constructing its fingers from a number of springs, compression links, cables and conduits. In this hand, each of the springs acts as a joint. However, all the compact fingers in Refs. [5–16] are limited to grasping function only. In reality, human fingers can also pinch thin and small objects with the fingertips, such as moving a pencil while writing or holding a piece of paper. A piece of paper cannot be grasped, as this will crumple it. Therefore, this study proposes a new anthropomorphic finger mechanism that is able perform both pinching and self-adaptive grasping operations. The basic structure of the mechanism is based on bar linkage SARAH hand [11–13], in which the upper link of the middle phalange is designed with a curved guiding slot and a lead screw mechanism so that its effective length can be adjusted to accomplish these two operations. To enhance its efficiency and performance, the linkages’ length is optimized based on the anthropomorphic finger pinching and grasping kinematics and also the force optimality criteria. A number of constraints are set up to avoid any mechanical interference in the design and to fulfill the anthropomorphic, as well as the mathematical requirements. A finger prototype is then built based on the optimum length obtained. 2. Finger mechanism The mechanism consists of three phalanges as in human fingers, which are the distal, middle and proximal phalanges as illustrated in Fig. 1. Its basic structure is made of seven bar linkages based on the SARAH finger mechanism [11–13]. It possesses the light weight and unbulky design advantages of the underactuated finger and overcomes its limitations, in which in this study, the robotic finger is designed to be able to perform both fingertip pinching and self-adaptive grasping tasks. Link 1 of the finger is fixed while other linkages are able to rotate. The mechanism is proposed based on the observation of human finger during grasping and pinching. All of human finger joints turn in the same direction while grasping, but during pinching, only the first two joints rotate in the same direction whereas the last joint turns in opposite direction. For example, during grasping, all of the three joints of a human's left hand index finger turn in clockwise direction as shown in Fig. 2(a). In contrast, during pinching operation, only the first two joints turn in clockwise direction, while the last or distal interphalangeal (DIP) joint rotates in counterclockwise direction as shown in Fig. 2(b). To fulfill this requirement, the effective length of the upper link of the artificial finger's middle phalanx has to be relatively long while grasping, but short during pinching. Therefore, this upper link is designed with a curved guiding slot and lead screw mechanism comprising a screw, a nut and a micro-motor as shown in Fig. 1. The micro-motor moves the distal phalanx's pin which is attached to the nut along the slot to adjust the middle phalanx's length, depending whether the finger is in grasping or pinching mode. In pinching operation, the input torque provided on link 2 turns the whole finger to pinch the object. The micro-motor moves the pin of the distal phalanx through the lead screw mechanism, along the guiding slot from its rest position at point C to point B of the upper link of the middle phalanx. This shortens the effective length between the pin and point A until the finger reaches the final pinching configuration as shown in Fig. 3. During grasping operation, the distal phalanx’ pin remains at point C or is moved to point C by the micro-motor. This lengthens the middle phalanx's length or the effective distance between the pin and point A so that the final grasping configuration can be realized as in Fig. 4. The finger behaves like a linkage based underactuated finger mechanism with self adaptable capability during this operation. In the beginning, the finger acts as a single body and when the input torque is applied on link 2, the finger rotates point A
micro-motor
screw
guiding slot nut distal phalanx’s pin point C
link 2 point B link 1
proximal phalanx
middle phalanx
Fig. 1. Finger mechanism design.
distal phalanx
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(a)
(b)
Fig. 2. Human finger's configuration during (a) grasping and (b) pinching.
point A point B
link 2
distal input torque
phalanx’s pin
guiding slot point C
(b)
(a)
Fig. 3. Final finger mechanism configuration in pinching operation (a) front view and (b) isometric view.
around joint 1 until the middle phalanx touches the object. The middle phalanx will stop to move after being in contact with the object. Next, further input torque will cause the distal phalanx to rotate until the finger has wrapped the object as illustrated in Fig. 5. In this mechanism, the force exerted by the finger depends mainly on the torque provided at link 2. A geared motor with high torque can be chosen to serve this purpose. The micro-motor in the middle phalanx can be a low torque motor, responsible to adjust the phalanx's length only. With the combination of these two motors in this configuration, no gear transmission is needed within the phalanges. Thus, more space is available to place electronic devices within the finger and the mechanism's weight and size can be reduced while at the same time, having sufficient force necessary to accomplish grasping and pinching tasks. Besides
link 2
point A
input torque
point B guiding slot
point C
(a)
distal phalanx’s pin
(b)
Fig. 4. Final finger mechanism configuration in grasping operation (a) front view and (b) isometric view.
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input torque
input torque
joint 1 input torque
object
object
object
55
Fig. 5. Finger mechanism grasping sequence.
that, the number of force sensors incorporated can be decreased as the mechanism possesses the self-adaptive grasping capability. 3. Geometric optimization The length of the links in a linkage based finger plays a crucial role in its behaviour [17], including its kinematics and also the force it exerts. Not any combination of the links’ length in bar type fingers can realize both grasping and pinching configuration and operation. An unsuitable combination of the linkages’ length may limit the finger's motion and also cause undesirable mechanical interferences. Therefore, this study focuses on the optimization of the finger's linkage geometric parameters to enhance its capability in performing the two tasks similar to a human finger. Considering this requirement, the problem is formulated as a multi-objective optimization subjected to • • • •
pinching kinematics objective function pinching force objective function grasping kinematics objective function grasping force objective function
In this optimization problem, an optimum solution of the design variables which include the length of the linkages, l1, l2, l3, l4, l5, l6p, l6g, l7, and l8 as shown in Fig. 6 is investigated. The lengths l6p and l6g represent the effective length of the upper link of the middle phalanx during pinching and grasping respectively. The length l9 is not included in the optimization process but it is calculated based on the optimum solution obtained as will be described in Section 5. 3.1. Objective function 1: pinching kinematics In pinching kinematics objective function, the finger has to resemble the human finger final configuration in pinching thin or small objects. The artificial finger is considered as two units of four bar linkage mechanism, with the first four bar linkage is made of links with lengths l1, l2, l3 and l4; and the second four bar linkage is built from the bars with lengths l4, l5, l7 and l6p or l6g, as shown in Fig. 7. The output of the first four bar linkage, θ14 is equal to the input of the second, θ22 and the value of angle θ11 is fixed to be 0 ∘. At the desired angle θ21which is measured from a human's middle phalanx, θ21hp the output angles made by the robotic finger, θ23p and θ14can be calculated using the closure equation of the second four bar linkage as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 −Bp − Bp 2 −C p 2 þ Ap 2 5; C p −Ap
−1 4
θ23p ¼ 2 tan
θ14 ¼ tan
−1
ð1Þ
l7 sin θ24ð0Þ þ l6p sin θ23p −l5 sin θ21 l7 cos θ24ð0Þ þ l6p cos θ23p −l5 cos θ21
! ð2Þ
l3 l2
l6g l6p
l4
l1
l9 l7
l5 Fig. 6. Design variables.
l8
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θ 23p
l 6g θ23g l 7
l3 l2 input torque
l4
θ 13
θ12
l 6p θ14 = θ 22 l5
θ 31
θ 21
l1 θ11
θ 24 l 8
l9
four bar linkage 2
four bar linkage1 Fig. 7. Two four-bar linkages.
where Ap ¼ 2l6p l7 cos θ24ð0Þ −2l5 l6 cos θ21 Bp ¼ 2l6p l7 sin θ24ð0Þ −2l5 l6 sin θ21 C p ¼ l25 þ l26p þ l27 −l24 −2l5 l7 cos θ24ð0Þ −θ21
ð3Þ
θ14(0) and θ24(0)are the values of θ14and θ24 while the finger is in the rest position respectively. These values can be calculated using the four bar linkage closure equation, with the angles θ21 and θ31 are equal to zero and the angle θ12as the initial input angle to link 2, θ12(0) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 −B0i − B20i −C 20i þ A20i 5; C 0i −A0i
−1 4
θi4ð0Þ ¼ −2 tan
i ¼ 1; 2
ð4Þ
where A01 ¼ 2l1 l4 −2l2 l4 cos θ12ð0Þ B01 ¼ −2l2 l4 sin θ12ð0Þ
C 01 ¼ l21 þ l22 þ l24 −l23 −2l1 l2 cosð−θ12 Þ
A02 ¼ 2l5 l7 −2l4 l7 cos θ14ð0Þ B02 ¼ −2l4 l7 sin θ14ð0Þ
C 02 ¼ l24 þ l25 þ l27 −l26g −2l4 l5 cos −θ14ð0Þ
ð5Þ
During pinching operation, the middle phalanx is not in contact with any object. The angle between links l4 and l5 is constant during this motion. Therefore the position of link l4 after the same angular displacement as link l5 from its rest position should be equal to the value of θ14, calculated from the second four bar linkage as described in Eq. (2). This necessary position of link l4 in making the finger to reach the desired θ21hpandθ31hp is denoted by θ14i and it can be calculated using θ14(0) which is obtained from the first four bar linkage closure equation in Eq. (4). θ14i can be mathematically expressed as θ14i ¼ θ14ð0Þ − 2π−θ21hp
ð6Þ
The objective function due to pinching kinematics, f1 has to minimize the difference made by the angleθ23p and its counterpart by human finger, θ23hp when θ21 is equal to the human middle phalanx's angle,θ21hp. The angles θ21hp and θ23hp are measured from a photo of a human finger pinching a thin object in horizontal position. In addition, the objective function also has to minimize the difference between the resulting θ14and θ14i which are obtained from the second and first four bar linkage closure equations respectively so that the finger can produce the desired θ21hpandθ31hp. Mathematically, f1 can be described as
f1 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2 θ23hp −θ23p þ ðθ14i −θ14 Þ2 =2
ð7Þ
Only the angles in the final configuration instead of all values during the whole pinching motion are considered in the objective function. This is because only the last set of angles rather than the finger's movement behaviour is more crucial in ensuring a successful pinching kinematics as in human finger.
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3.2. Objective function I1: pinching force The pinching force of the artificial finger can be calculated by performing static analysis on its free body diagram during pinching as shown in Fig. 8. Since the summation of force in the vertical direction is zero, Fp ¼ N
ð8Þ
where Fp and N are the pinching and normal forces acting on the distal phalanx respectively. With the assumption that the pinching force is exerted at the middle of the distal phalanx, Fp can be calculated from the moment equation around point A as F p ¼ T i =ðla −f r lb Þ
ð9Þ
where Ti fr d
: input torque on link 2 : coefficient of static friction : distance between line l8 and the edge of the link, considering the linkage's thickness la ¼ l1 cos θ11 þ l5 cos θ21 þ 0:5l8 cos θ31 lb ¼ −l1 sin θ11 −l5 sin θ21 −0:5l8 sin θ31 −d sinðθ31 −π=2Þ
ð10Þ
The criterion for achieving optimum pinching force is formulated by taking into account the difference between the desired pinching force made by the human finger, Fph and the artificial finger, Fp with the provided input torque. Therefore, the objective function f2 can be represented as
f2 ¼
F ph −F p f r ¼0:5
ð11Þ
F ph
3.3. Objective function III: grasping kinematics In grasping kinematics optimization, the finger is also considered as two sets of four bar linkage as shown in Fig. 7. The link with length l1 and the angle θ12 is considered as the base linkage and input angle of the first four bar linkage respectively. On the other hand, the link with length l5 and the angle θ21act as the base link and base angle in the second four bar linkage respectively. The value of θ14while the upper linkage of the middle phalanx is at its maximum length can be calculated as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 −B1 − B1 2 −C 1 2 þ A1 2 5; C 1 −A1
−1 4
θ14 ¼ 2 tan
ð12Þ
where A1 ¼ 2l1 l4 cos θ11 −2l2 l4 cos θ12 B1 ¼ 2l1 l4 sin θ11 −2l2 l4 sin θ12 2 2 2 2 C 1 ¼ l1 þ l2 þ l4 −l3 −2l1 l2 cosðθ11 −θ12 Þ
ð13Þ
From Eq. (12), the base angle of the second four bar linkage, θ21before the middle phalanx is in contact with the object can be obtained as θ21 ¼ θ14 −θ22ð0Þ
ð14Þ
Ti
θ21
l6p
A lb
l6g distal phalanx’s edge
l5 l8 la
d
N Objec t
fr N
Fig. 8. Finger free body diagram during pinching.
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whereθ22(0) = θ14(0). Therefore the output of the second four bar linkage can be obtained as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 −B2 − B2 2 −C 2 2 þ A2 2 5; C 2 −A2
−1 4
θ24 ¼ 2 tan
θ23g ¼ tan
−1
l5 sin θ21 þ l7 sin θ24 −l4 sin θ22 l5 cos θ21 þ l7 cos θ24 −l4 cos θ22
ð15Þ ð16Þ
where A2 ¼ 2l5 l7 cos θ21 −2l4 l7 cos θ22 B2 ¼ 2l5 l7 sin θ21 −2l4 l7 sin θ22 C 2 ¼ l24 þ l25 þ l27 −l26g −2l4 l5 cosðθ21 −θ22 Þ
ð17Þ
Finally, the value of θ31 after the middle phalanx has touched the object can be determined as θ31 ¼ θ24 −θ24ð0Þ
ð18Þ
The corresponding angles that the human's middle phalanx, θ21hg and distal phalanx, θ31hg make while grasping small (1 cm radius), medium (2 cm radius) and large (3 cm radius) size cylindrical objects are measured from the photos taken. These two angles are indicated in Fig. 9. Then, these values are compared to angles θ21and θ31 made by the robotic finger with the upper link of the middle phalanx is at its maximum length. The objective function in this optimality criterion, f3 has to minimize the difference between these angles and this can be mathematically expressed as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3 2 2 uX f3 ¼ t θ21hg r¼ri −θ21 r¼ri þ θ31hg r¼ri −θ31 r¼ri =6;
ð19Þ
i¼1
where r is the object's radius, r1, r2 and r3 equal to 1 cm, 2 cm and 3 cm respectively. 3.4. Objective function IV: grasping force The grasping force calculation is performed based on the work done in Ref. [18]. Referring to Fig. 10, the grasping force exerted by middle, fg1 and distal phalanges, fg2 can be described by f g1 ¼
h2 l5 U ðk3 −h3 cos θ3 ÞT i ; k2 k3 ðh2 þ l1 Þðh3 þ l5 Þ
ð20Þ
f g2 ¼
h2 h3 T i ; k3 ðh2 þ l1 Þðh3 þ l5 Þ
ð21Þ
where Ti : input torque on link 1 hi : signed distance between point Oi and intersection of lines (Oi − 1Oi) and (Pi − 1Pi) k1, k2 and k3 contact points of the proximal, middle and distal phalanges with the object respectively U ¼ k2 k3 h3 þ k2 k3 l2 −h2 k3 l2 cosθ2 þ h2 h3 l2 cosθ2 cosθ3 −h2 h3 k2 cosðθ2 þ θ3 Þ
θ21hg
θ 31hg
Fig. 9. θ21hg and θ31hg measured from photo.
ð22Þ
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P1
59
P2 θ1
θ2
O1 O2
P3
k2
O3 fg1
k3 fg2
θ3
Fig. 10. Geometric details for grasping force calculation.
If both grasping forces fg1 and fg2 in Eqs. (20) and (21) are positive, it means that a successful grasping is achieved, but if one or more of the forces are negative, ejection has occurred. Therefore, the main criterion of the fourth objective function, f4 is the grasping force exerted by all of the phalanges on cylindrical objects with 1 cm radius must be greater than zero. The force while grasping cylindrical objects with 2 cm and 3 cm radius are not taken into consideration because human hands usually need the palm or thumb to grasp large objects. If a successful grasping is achieved, the finger is also required to provide sufficient grasp force, fg to support the desired object weight. This value can be determined as f g ¼ f 1g sinðθ21 −π=2Þ þ f 2g sinðθ31 −π=2Þ
ð23Þ
The value of f4 is set to 1 as a penalty if any of the phalanx forces while grasping 1 cm radius cylindrical object are negative. However, if no ejection occurs, the objective function has to minimize the difference between fg and the supported load, Wd. Mathematically, f4 can be expressed by 8 1 > < f 4 ¼ f g −W d > : Wd
∃f gi jr¼1cm b0;
i ¼ 1; 2
∀f gi jr¼1cm > 0;
i ¼ 1; 2
ð24Þ
3.5. Constraints The multi-objective optimization problem is subjected to multiple constraints which are due to interferences during both pinching and grasping, finger size and four bar linkage closure equations. The vertical component of the link with length l6g during pinching configuration is set to be at least 1 mm shorter than the total vertical length between point P2, as in Fig. 10 and the surface of the distal phalanx with length l8. This constraint is necessary to avoid any mechanical interference between the link at length l6g and the object during pinching. Hence, the constraint c1 due to this factor can be formulated as c1 : l6g sin θ23g þ 1 b l6p sin θ23p þ l7 j sin θ24 j
ð25Þ
where θ23g θ23p and θ24are the values during final pinching configuration. To avoid any linkage interference between the link of length l5 and the guiding slot for l6p while the finger is grasping the 1 cm radius cylindrical object, the distance between these two points is set to be at least 3 mm horizontally from each other as linkage’s edge
l6p linkage’s thickness
Object (r= 1cm)
l5
distance between l6p slot and end of l5 Fig. 11. Finger configuration in grasping object.
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hinge pin, diameter=2mm
li
2mm
thickness=4mm Fig. 12. Basic structure of linkages with length li, i = 1 to 5.
illustrated in Fig. 11. The distance is selected taking into consideration the thickness of the linkage and the diameter of the distal phalanx pin. This can be mathematically represented as c2 : l6p cos θ23p þ 3bj−l4 cos θ14 þ l5 cos θ21 j
ð26Þ
where θ14, θ21 and θ23p are the final angles while grasping a cylindrical object of 1 cm radius. In terms of the finger size, the total finger length is constrained to be less than 75 mm, so that it is close to the size of an average human's index finger. This constraint can be represented by c3 : l1 þ l5 þ l8 b 75
ð27Þ
Considering the kinematics equations of the four bar linkages in Eqs. (1), (4), (12) and (15), the optimization problem is also subjected to the constraints c4 and c5 below. These constraints ensure that the solutions to these equations are real and therefore the mechanism can be assembled [19]. 2
2
2
c4 : Bp −C p þ Ap > 0 2
2
2
c5 : Bi −C i þ Ai > 0;
ð28Þ i ¼ 0; 1; 2
ð29Þ
3.6. Optimization formulation Taking all the four objective functions in Eqs. (7), (11), (19) and (24) into consideration, the optimization problem has to minimize the multi-objective function F ¼ w1 f 1 þ w2 f 2 þ w3 f 3 þ w4 f 4
ð30Þ
where wi, i = 1, 2, 3 and 4 are the weight coefficients, indicating the importance of each objective function. At the same time, the solution to this problem must also satisfy all the constraints governed by Eqs. (25)–(29) simultaneously. The optimization algorithm has been performed by the nonlinear continuous optimization algorithm, Sequential Quadratic Programming (SQP) using the ‘fmincon’ command available in Matlab Optimization Toolbox software [20]. Although the command l6g
l 6p guiding slot Fig. 13. Middle phalanx's upper linkage.
l9
l7
d =2mm l8 Fig. 14. Distal phalanx linkage.
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Table 1 Hardware components. Components
Dimension (mm)
Weight (g)
Motor 1 and encoder Motor 2 Potentiometer Force sensor
ϕ13.0 × 52.0 ϕ10.0 × 8.0 × 11.7 ϕ8.0 × 5.0 14.0 × 102.0 × 0.189
40.1 11.0 0.7 0.4
provides the solution by the convergence of local optimum, this technique has been chosen because it is easy to use, does not require tedious computational effort and does not consume long calculation time. The steps implemented by ‘fmincon’ function in finding the constrained minimum of a function of several variables can be referred in [8]. 4. Prototype development The hardware components of the finger are chosen to achieve a compact and light design. The basic structure of the finger prototype is made from aluminum since the material possesses high strength but low weight. All of the hinge pins’ diameter and the horizontal distance between its center and the edge of the link are set to be 2 mm. The bar linkages’ length is developed based on the optimized solution obtained and their thickness is fixed to be 4 mm. Figs. 12, 13 and 14 illustrate the basic structure of the linkages with length li where i =1 to 5, upper linkage of the middle phalanx and distal phalanx respectively. A high torque geared motor with rated torque 0.475 Nm which is manufactured by Citizen Micro Co. Ltd. is selected to provide the input torque on link 1. On the other hand, a micro-motor from Mabuchi Motor Co. Ltd. with the weight of 11 g and nominal torque 0.00053 Nm is chosen to drive the lead screw mechanism in adjusting the length of the middle phalanx. An encoder made by Citizen Micro Co. Ltd. and a rotary potentiometer are used to measure the joint positions, and the Tekscan Inc. Flexi force sensor is utilized in measuring the force exerted by the finger. To facilitate future works, the electrical commands are provided using xPC Target, Matlab software. The dimension and weight of the sensors and actuators used in the mechanism are summarized in Table 1. 5. Results 5.1. Optimization numerical results For the numerical optimization, a set of initial guess, upper and lower boundaries have been chosen as in Table 2. It is a good practice to provide the parameters’ boundaries because this may promote a fast convergence to the solution since the search space is restricted. The corresponding angles made by a human finger during pinching,θ21hp, θ23hp, θ21hg andθ31hg are measured Table 2 Initial guess, lower and upper boundaries of the design variables. Length (mm)
Initial guess (mm)
Lower boundary (mm)
Upper boundary (mm)
l1 l2 l3 l4 l5 l6p l6g l7 l8
26.0 28.0 30.0 24.0 25.0 26.0 40.0 20.0 24.0
20.0 20.0 20.0 20.0 24.0 25.0 20.0 15.0 20.0
30.0 30.0 50.0 30.0 30.0 30.0 50.0 30.0 30.0
Table 3 Corresponding angles made by human finger during pinching and grasping. Angle
Measured value (°)
θ21hp θ23hp θ31hp θ21hg|r = 1cm θ31hg|r = 1cm θ21hg|r = 2cm θ31hg|r = 2cm θ21hg|r = 3cm θ31hg|r = 3cm
290 270 0 270 180 300 245 325 280
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Table 4 Optimized parameter solution and initial guess. Length
Initial guess (mm)
Optimized solution (mm)
l1 l2 l3 l4 l5 l6p l6g l7 l8 l9
26.0 28.0 30.0 24.0 25.0 26.0 40.0 20.0 24.0 34.8
27.3 30.0 36.2 27.0 24.0 25.4 46.7 20.9 20.0 24.3
Table 5 Objective functions values. Objective function
Initial guess
Optimized solution
f1 f2 f3 f4 F
19.2134 0.0139 13.3779 1.0000 33.6052
0.0001 0.0041 0.4120 0.1309 0.5471
Object
Object
(a)
(b)
Object Object
(c)
(d)
Fig. 15. Optimized artificial finger in (a) pinching, grasping a (b) 1 cm (c) 2 cm and (d) 3 cm cylindrical object.
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from the photos taken and these values are summarized in Table 3. The initial input angle θ12(0) and the input torque, Ti have been set to be 147∘ and 0.1 Nm respectively. It has been desired for the finger to exert a pinching force, Fph of 3.0 N with the provided input torque and the coefficient of static friction equals to 0.5. The finger is also required to be able to support a 1 cm radius cylindrical object with weight, Wd 1.5 N. The grasping forces have been assumed to act at the middle of each phalanx. During the optimization process, all the angles have been calculated and measured in degrees. The optimized solution that gives the minimum anthropomorphic multi-objective function and satisfies all the constraints has been obtained as in Table 4. The resulting errors or objective functions are summarized in Table 5. In this case, all of the weight coefficients have been set to 1, implying equal importance for all the four objective functions. The resulting value of l9 has been obtained based on the optimum solution using the cosine law l9 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l27 þ l28 −2l7 l8 cos θ24ð0Þ
ð31Þ
The result shows that compared to the initial guess, the optimized parameters produce less error in fulfilling both the kinematics configuration and force exerted for the pinching as well as grasping task. The mechanism with initial guess parameters gives high errors in the pinching and grasping kinematics optimality criteria. Besides that, it is also unable to grasp a 1 cm object, where ejection has occurred. Fig. 15 illustrates the finger with the optimized design variables during pinching and grasping cylindrical objects of 1 cm, 2 cm and 3 cm radius. The dashed lines in the figure represent the outer edges of the linkages that are actually in contact with the objects. It can also be seen from the figure that the optimal solution also complies with the constraints imposed by Eqs. (25) and (26), in which no mechanical interference occur. The SQP technique implemented in this study gives the optimization solution based on the convergence to local optimum. In this method, the result depends greatly on the initial guess selected. Different sets of initial guess will result in distinct optimum design variables and multiple-objective function value, F. Therefore in this work, the initial guess is first chosen intuitively as the values close to a human finger. Then, a few tests are performed using different sets of initial guess, which are varied slightly from the one chosen in the beginning. Finally, the initial guess set that gives the lowest multiple-objective function, F as in Table 2 has been selected in the optimization procedure. Although the SQP algorithm requires few trials to select the initial guess set, the technique is chosen because it is easy to implement and does not require high computational effort and long calculation time. Table 6 shows the numerical optimization results for different combinations of the optimality criteria under the same initial guess, lower boundary and upper boundary sets as in Table 2. Referring to Table 6, if only the pinching kinematics and force objective functions are considered in the optimization problem by changing the coefficient weights to w1 = 1, w2 = 1, w3 = 0 and w4 = 0, the finger is able to reach the pinching configuration and exert the desired pinching force successfully. Although this solution gives a lower multi-objective function, F compared to the optimal solution, the higher error of objective function f3 indicates that it is less efficient in making the finger to grasp objects, as illustrated in Fig. 16(a). Table 6 Weight coefficient, solution and objective function of the optimization problem considering different combination of optimality criteria. Pinching kinematics and force
Grasping kinematics and force
Pinching and grasping kinematics
Pinching and grasping kinematics and force
Weight w1 w2 w3 w4
1 1 0 0
0 0 1 1
1 0 1 0
1 1 1 1
Solution (mm) l1 l2 l3 l4 l5 l6p l6g l7 l8 l9
24.8 28.5 32.4 25.6 24.1 25.9 45.1 19.5 25.3 28.1
29.7 30.0 38.2 27.0 24.0 25.0 41.5 19.0 20.0 27.2
27.2 30.0 36.2 27.0 24.0 25.4 46.7 21.0 23.8 26.7
27.3 30.0 36.2 27.0 24.0 25.4 46.7 20.9 20.0 24.3
Objective function f1 f2 f3 f4 F Total (f1+ f2 + f3+ f4)
0.0002 0.0000 4.4806 0.0800 0.0002 4.5608
20.0240 0.0648 0.8871 0.0528 0.9399 21.0287
0.0003 0.0484 0.4113 0.2668 0.4116 0.7268
0.0001 0.0041 0.4120 0.1309 0.5471 0.5471
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θ14i
object
θ14
object
(a)
(b)
Fig. 16. Robotic finger with (a) pinching kinematics and force optimality criteria only in grasping configuration (b) grasping kinematics and force optimality criteria only in pinching configuration.
On the other hand, if the values of the weight coefficient are assigned to w1 = 0, w2 = 0, w3 = 1 and w4 = 1 to consider only the grasping kinematics and force optimality criteria, the solution obtained satisfies the grasping requirement, but the link with length l6p is unable to achieve the desired pinching configuration, θ23hpunder the provided θ21hpandθ31hp, as shown in Fig. 16 (b). It can also be observed from the figure that the resulted angular displacement of link l4, denoted byθ14, does not reach the angular position, θ14i that is necessary to achieve the desired θ21hpandθ31hp configuration. In short, the value of θ14i is unequal toθ14. The solution in considering only the pinching and grasping kinematics objective functions has also been determined by setting w1 = 1, w2 = 0, w3 = 1 and w4 = 0. Even though it gives a smaller total error (f1 + f2 + f3 + f4) compared to the previous two cases, it does not result in an optimum solution for all the four optimality criteria. The total error value is higher compared to the solution provided by the optimum design variables. The results in these three cases show that it is important to consider all the four factors which are due to both pinching and grasping kinematics and force in the multi-objective optimization formulation. The linkage based robotic finger's capability in completing both of the functions as in a human finger can be improved by considering all these four factors. 5.2. Prototype A prototype has been built based on the optimum solution obtained and is shown in Fig. 17. It can be seen from Fig. 18 that the prototype has successfully achieved the final pinching and grasping configurations. Therefore, the optimization procedure improves the finger performance compared to implementing the finger with just any guessed linkage lengths. 6. Conclusions A new finger mechanism and its linkages’ length optimization to resemble a human finger in performing pinching and grasping with self-adaptive grasp capability are presented in this paper. The mechanism is made of seven bar linkages with the upper link of the middle phalanx designed with a curved guiding slot and lead screw mechanism to adjust its effective length in accomplishing pinching and grasping tasks. Its geometrical parameter has been formulated as a multi-objective optimization design procedure based on four anthropomorphic design goals, which are the pinching kinematics and force; and grasping kinematics and force. The numerical result demonstrates that the optimal solution has greatly reduced the error compared to the one resulted from the initial
Fig. 17. Finger prototype.
N. Zainul Azlan, H. Yamaura / Mechanism and Machine Theory 49 (2012) 52–66
(a)
(c)
65
(b)
(d)
Fig. 18. Finger prototype in (a) pinching, grasping a (b) 1 cm (c) 2 cm and (d) 3 cm cylindrical object.
guess. The weight coefficients have been varied in the numerical analysis to consider different combination of optimality criteria, and the results have shown that it is vital to take all the four objective functions into account in improving the finger performance. A finger prototype has been constructed based on the optimum solution obtained and it has been seen that the finger conforms to the desired configurations. Implementing the optimum linkages length solution together with the choice of light weight and small size actuators and sensors in the design contribute to the realization and enhancement of a compact anthropomorphic finger mechanism in carrying out both pinching and grasping operations. Future works involve the development of a control algorithm in controlling the finger while manipulating objects and the construction of the whole hand consisting of a thumb and other fingers. References [1] K. Hoshino, I. Kawabuchi, Dexterous robot hand with pinching function at fingertips, IEEE/RAS-EMBS International Conference on Biomedical Robotics and Biomechatronics, Italy, 2006, pp. 1113–1118. [2] S.G. Jacobsen, E.K. Iversen, D.F. Knutti, R.T. Johnson, K.B. Biggers, Design of the Utah/M.I.T. dextrous hand, IEEE International Conference on Robotics and Automation, U.S.A, 1986, pp. 1520–1532. [3] M.T. Mason, J.K. Salisbury, Robot Hands and the Mechanics of Manipulation, MIT Press, 1985. [4] N. Dechev, W.L. Cleghorn, S. Naumann, Multiple finger, passive adaptive grasp prosthetic hand, Mechanism and Machine Theory 41 (2006) 897–911. [5] E.N. Haulin, R. Vinet, Multiobjective optimization of hand prosthesis mechanisms, Mechanism and Machine Theory 38 (2003) 3–26. [6] Y. Liu, H. Wang, B. Li, W. Zhou, The underactuation and motion-coupling in robotic fingers and two new 1-DOF motion-coupling anthropomorphic fingers, IEEE International Conference on Robotics and Biomimetics, Thailand, 2009, pp. 1573–1578. [7] G. Guo, T.T. Lee, W.A. Gruver, J. Zhang, Design of a planar multijointed prosthetic finger mechanism, ASME Mechanisms Conference, U.S.A, 1990, pp. 165–170. [8] N.E.N. Rodriguez, G. Carbone, M. Ceccarelli, Optimal design of driving mechanism in a 1-DOF anthropomorphic finger, Mechanism and Machine Theory 41 (2006) 897–911. [9] M.C. Carrozza, C. Suppo, F. Sebastiani, B. Massa, F. Vecchi, R. Lazzarini, M.R. Cutkosky, P. Dario, The SPRING hand: development of a self-adaptive prosthesis for restoring natural grasping, Autonomous Robots 16 (2004) 125–141.
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[10] B. Massa, S. Rocella, M.C. Carozza, P. Dario, Design and development of an underactuated prosthetic hand, IEEE International Conference on Robotics and Automation, 2002, pp. 3374–3379. [11] T. Laliberte, C.M. Gosselin, Underactuation in space robotic hands, International Symposium on Artificial Intelligence and Robotics & Automation in Space, Canada, 2001, pp. 1–8. [12] T. Laliberte, L. Birglen, C.M. Gosselin, Underactuation in robotic grasping hands, Machine Intelligence & Robotic Control 4 (3) (2002) 1–11. [13] SARAH HAND. Available from: bhttp://robot.gmc.ulaval.ca/en/research/theme304.html>. [14] L. Birglen, C.M. Gosselin, Grasp-state plane analysis of two-phalanx underactuated fingers, Mechanism and Machine Theory 41 (2006) 807–822. [15] D.W. Zhao, L. Jiang, H. Huang, M.H. Jin, H.G. Cai, H. Liu, Development of multi-DOF anthropomorphic prosthetic hand, IEEE International Conference on Robotics and Biomimetics, China, 2006, pp. 878–883. [16] J. Yang, E.P. Pitarch, K.A. Malek, A. Patrick, L. Lindkvist, A multi-fingered hand prosthesis, Mechanism and Machine Theory 39 (2004) 555–581. [17] T. Laliberte, C.M. Gosselin, Simulation and design of underactuated mechanical hands, Mechanism and Machine Theory 33 (1) (1998) 39–57. [18] L. Birglen, C.M. Gosselin, Geometric design of three-phalanx underactuated fingers, Transaction of the ASME 128 (2006) 356–364. [19] K.J. Waldron, G.L. Kinzel, Kinematics, Dynamics, and Design of Machinery, Wiley, 1999. [20] Optimization Toolbox For Use with MATLAB. Available from: bhttp://www.mathworks.com/help/pdf_doc/optim/optim_tb.pdf > last accessed: 13 January 2011.
Contents lists available at SciVerse ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Anthropomorphic finger with optimized geometric parameters for pinching and grasping tasks Norsinnira Zainul Azlan ⁎, Hiroshi Yamaura Department of Mechanical and Control, Graduate School of Science and Engineering, Tokyo Institute of Technology, Ookayama 2-12-1-I3-1, Meguro-ku, Tokyo 152-8552, Japan
a r t i c l e
i n f o
Article history: Received 20 January 2011 Received in revised form 17 October 2011 Accepted 1 November 2011 Available online 7 December 2011 Keywords: Robotic finger Anthropomorphic design Optimization Grasping Pinching
a b s t r a c t This paper presents the parameter optimization procedure for a new anthropomorphic finger mechanism in performing pinching and self-adaptive grasping operation. The mechanism is made based on the observation of human finger during grasping and pinching. It is constructed from seven bar linkages and its upper middle phalanx is incorporated with a curved guiding slot and lead screw mechanism to adjust the link's effective length in accomplishing these two tasks. The finger's geometrical parameters are optimized based on four design criteria, which are the anthropomorphic pinching kinematics and force; and grasping kinematics and force to enhance its performance. The numerical result obtained gives an improved parameter set compared to the initial guess. A prototype with small size and light weight actuators and sensors has been implemented based on the optimum solution to achieve a compact anthropomorphic design with grasping and pinching capabilities. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Artificial hands are the mechanisms developed to either or both mimic the physical appearance and serve the functions of a real hand. Some of the applications of this type of hand include research purpose, industry, space exploration, prosthetics and humanoid robots. The hands developed for prosthetics and humanoid robots applications require a special characteristic, which is “anthropomorphism”. The term “anthropomorphism” refers to the capability of the robotic end effectors to resemble the human hand in terms of both functionality and physical appearance including its shape, size, and weight. A humanoid robot with anthropomorphic hand looks more natural and the replacement of this kind of hand for the amputees helps them in carrying out their daily tasks. Over the last decades, numerous researches have been conducted to investigate the practical solution of this hand. The main challenge in developing an anthropomorphic hand is to design a mechanism that can both realize the functions and motions, and at the same time have similar shape, size and weight to a human hand. However, there are not many components including the motors and reduction gears that are small and light enough to realize practical hand movements [1]. Some of the previous robotic hands including the UTAH/ MIT Hand [2] and Stanford/ JPL Hand [3] have achieved good performances in replicating human capabilities, but they are bulky and heavy due to the large size and number of actuators and sensors incorporated to realize the complex motions. An alternative to the design problem is to reduce the number of degree of freedom (DOF) in the hand by decreasing the number of actuators utilized such as in the TBM Hand [4]. The fingers developed in Refs. [5–7] and LARM Hand [8] are constructed from various arrangement of bar linkages with only one DOF. Although this technique approaches the human hands’ weight and size due to the low number of motors used, it also affects the versatility of the hand.
⁎ Corresponding author. Tel.: + 81 80 3022 0843; fax: + 81 3 5734 2419. E-mail address: [email protected] (N. Zainul Azlan). 0094-114X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2011.11.005
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53
Another approach in the hand development is to adopt the underactuated mechanism such as in the tendon based SPRING Hand [9] and RTRII Hand [10], and the bar linkage based SARAH HAND [11–13]. This type of mechanism utilizes low number of actuators without sacrificing the hand's DOF. The number of DOF is made higher than the number of active motors by replacing the traditional actuators such as electro-magnetic motors with small passive elements such as springs and mechanical limits. This mechanism also results in the self-adaptive grasp capability of the finger to the object to be grasped [14] and avoids the need of sensors and complex feedback control strategies. The index finger in HIT/DLR Prosthetic hand [15] is developed based on the underactuated linkage mechanism and actuated by a single motor. However its middle and distal phalanges are actuated through coupling linkages, in which these phalanges rotate at the same time with the angle ratio 1:1. The Iowa hand [16] exhibits a significantly lighter weight compared to the available commercial prosthetic hands by constructing its fingers from a number of springs, compression links, cables and conduits. In this hand, each of the springs acts as a joint. However, all the compact fingers in Refs. [5–16] are limited to grasping function only. In reality, human fingers can also pinch thin and small objects with the fingertips, such as moving a pencil while writing or holding a piece of paper. A piece of paper cannot be grasped, as this will crumple it. Therefore, this study proposes a new anthropomorphic finger mechanism that is able perform both pinching and self-adaptive grasping operations. The basic structure of the mechanism is based on bar linkage SARAH hand [11–13], in which the upper link of the middle phalange is designed with a curved guiding slot and a lead screw mechanism so that its effective length can be adjusted to accomplish these two operations. To enhance its efficiency and performance, the linkages’ length is optimized based on the anthropomorphic finger pinching and grasping kinematics and also the force optimality criteria. A number of constraints are set up to avoid any mechanical interference in the design and to fulfill the anthropomorphic, as well as the mathematical requirements. A finger prototype is then built based on the optimum length obtained. 2. Finger mechanism The mechanism consists of three phalanges as in human fingers, which are the distal, middle and proximal phalanges as illustrated in Fig. 1. Its basic structure is made of seven bar linkages based on the SARAH finger mechanism [11–13]. It possesses the light weight and unbulky design advantages of the underactuated finger and overcomes its limitations, in which in this study, the robotic finger is designed to be able to perform both fingertip pinching and self-adaptive grasping tasks. Link 1 of the finger is fixed while other linkages are able to rotate. The mechanism is proposed based on the observation of human finger during grasping and pinching. All of human finger joints turn in the same direction while grasping, but during pinching, only the first two joints rotate in the same direction whereas the last joint turns in opposite direction. For example, during grasping, all of the three joints of a human's left hand index finger turn in clockwise direction as shown in Fig. 2(a). In contrast, during pinching operation, only the first two joints turn in clockwise direction, while the last or distal interphalangeal (DIP) joint rotates in counterclockwise direction as shown in Fig. 2(b). To fulfill this requirement, the effective length of the upper link of the artificial finger's middle phalanx has to be relatively long while grasping, but short during pinching. Therefore, this upper link is designed with a curved guiding slot and lead screw mechanism comprising a screw, a nut and a micro-motor as shown in Fig. 1. The micro-motor moves the distal phalanx's pin which is attached to the nut along the slot to adjust the middle phalanx's length, depending whether the finger is in grasping or pinching mode. In pinching operation, the input torque provided on link 2 turns the whole finger to pinch the object. The micro-motor moves the pin of the distal phalanx through the lead screw mechanism, along the guiding slot from its rest position at point C to point B of the upper link of the middle phalanx. This shortens the effective length between the pin and point A until the finger reaches the final pinching configuration as shown in Fig. 3. During grasping operation, the distal phalanx’ pin remains at point C or is moved to point C by the micro-motor. This lengthens the middle phalanx's length or the effective distance between the pin and point A so that the final grasping configuration can be realized as in Fig. 4. The finger behaves like a linkage based underactuated finger mechanism with self adaptable capability during this operation. In the beginning, the finger acts as a single body and when the input torque is applied on link 2, the finger rotates point A
micro-motor
screw
guiding slot nut distal phalanx’s pin point C
link 2 point B link 1
proximal phalanx
middle phalanx
Fig. 1. Finger mechanism design.
distal phalanx
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(a)
(b)
Fig. 2. Human finger's configuration during (a) grasping and (b) pinching.
point A point B
link 2
distal input torque
phalanx’s pin
guiding slot point C
(b)
(a)
Fig. 3. Final finger mechanism configuration in pinching operation (a) front view and (b) isometric view.
around joint 1 until the middle phalanx touches the object. The middle phalanx will stop to move after being in contact with the object. Next, further input torque will cause the distal phalanx to rotate until the finger has wrapped the object as illustrated in Fig. 5. In this mechanism, the force exerted by the finger depends mainly on the torque provided at link 2. A geared motor with high torque can be chosen to serve this purpose. The micro-motor in the middle phalanx can be a low torque motor, responsible to adjust the phalanx's length only. With the combination of these two motors in this configuration, no gear transmission is needed within the phalanges. Thus, more space is available to place electronic devices within the finger and the mechanism's weight and size can be reduced while at the same time, having sufficient force necessary to accomplish grasping and pinching tasks. Besides
link 2
point A
input torque
point B guiding slot
point C
(a)
distal phalanx’s pin
(b)
Fig. 4. Final finger mechanism configuration in grasping operation (a) front view and (b) isometric view.
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input torque
input torque
joint 1 input torque
object
object
object
55
Fig. 5. Finger mechanism grasping sequence.
that, the number of force sensors incorporated can be decreased as the mechanism possesses the self-adaptive grasping capability. 3. Geometric optimization The length of the links in a linkage based finger plays a crucial role in its behaviour [17], including its kinematics and also the force it exerts. Not any combination of the links’ length in bar type fingers can realize both grasping and pinching configuration and operation. An unsuitable combination of the linkages’ length may limit the finger's motion and also cause undesirable mechanical interferences. Therefore, this study focuses on the optimization of the finger's linkage geometric parameters to enhance its capability in performing the two tasks similar to a human finger. Considering this requirement, the problem is formulated as a multi-objective optimization subjected to • • • •
pinching kinematics objective function pinching force objective function grasping kinematics objective function grasping force objective function
In this optimization problem, an optimum solution of the design variables which include the length of the linkages, l1, l2, l3, l4, l5, l6p, l6g, l7, and l8 as shown in Fig. 6 is investigated. The lengths l6p and l6g represent the effective length of the upper link of the middle phalanx during pinching and grasping respectively. The length l9 is not included in the optimization process but it is calculated based on the optimum solution obtained as will be described in Section 5. 3.1. Objective function 1: pinching kinematics In pinching kinematics objective function, the finger has to resemble the human finger final configuration in pinching thin or small objects. The artificial finger is considered as two units of four bar linkage mechanism, with the first four bar linkage is made of links with lengths l1, l2, l3 and l4; and the second four bar linkage is built from the bars with lengths l4, l5, l7 and l6p or l6g, as shown in Fig. 7. The output of the first four bar linkage, θ14 is equal to the input of the second, θ22 and the value of angle θ11 is fixed to be 0 ∘. At the desired angle θ21which is measured from a human's middle phalanx, θ21hp the output angles made by the robotic finger, θ23p and θ14can be calculated using the closure equation of the second four bar linkage as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 −Bp − Bp 2 −C p 2 þ Ap 2 5; C p −Ap
−1 4
θ23p ¼ 2 tan
θ14 ¼ tan
−1
ð1Þ
l7 sin θ24ð0Þ þ l6p sin θ23p −l5 sin θ21 l7 cos θ24ð0Þ þ l6p cos θ23p −l5 cos θ21
! ð2Þ
l3 l2
l6g l6p
l4
l1
l9 l7
l5 Fig. 6. Design variables.
l8
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θ 23p
l 6g θ23g l 7
l3 l2 input torque
l4
θ 13
θ12
l 6p θ14 = θ 22 l5
θ 31
θ 21
l1 θ11
θ 24 l 8
l9
four bar linkage 2
four bar linkage1 Fig. 7. Two four-bar linkages.
where Ap ¼ 2l6p l7 cos θ24ð0Þ −2l5 l6 cos θ21 Bp ¼ 2l6p l7 sin θ24ð0Þ −2l5 l6 sin θ21 C p ¼ l25 þ l26p þ l27 −l24 −2l5 l7 cos θ24ð0Þ −θ21
ð3Þ
θ14(0) and θ24(0)are the values of θ14and θ24 while the finger is in the rest position respectively. These values can be calculated using the four bar linkage closure equation, with the angles θ21 and θ31 are equal to zero and the angle θ12as the initial input angle to link 2, θ12(0) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 −B0i − B20i −C 20i þ A20i 5; C 0i −A0i
−1 4
θi4ð0Þ ¼ −2 tan
i ¼ 1; 2
ð4Þ
where A01 ¼ 2l1 l4 −2l2 l4 cos θ12ð0Þ B01 ¼ −2l2 l4 sin θ12ð0Þ
C 01 ¼ l21 þ l22 þ l24 −l23 −2l1 l2 cosð−θ12 Þ
A02 ¼ 2l5 l7 −2l4 l7 cos θ14ð0Þ B02 ¼ −2l4 l7 sin θ14ð0Þ
C 02 ¼ l24 þ l25 þ l27 −l26g −2l4 l5 cos −θ14ð0Þ
ð5Þ
During pinching operation, the middle phalanx is not in contact with any object. The angle between links l4 and l5 is constant during this motion. Therefore the position of link l4 after the same angular displacement as link l5 from its rest position should be equal to the value of θ14, calculated from the second four bar linkage as described in Eq. (2). This necessary position of link l4 in making the finger to reach the desired θ21hpandθ31hp is denoted by θ14i and it can be calculated using θ14(0) which is obtained from the first four bar linkage closure equation in Eq. (4). θ14i can be mathematically expressed as θ14i ¼ θ14ð0Þ − 2π−θ21hp
ð6Þ
The objective function due to pinching kinematics, f1 has to minimize the difference made by the angleθ23p and its counterpart by human finger, θ23hp when θ21 is equal to the human middle phalanx's angle,θ21hp. The angles θ21hp and θ23hp are measured from a photo of a human finger pinching a thin object in horizontal position. In addition, the objective function also has to minimize the difference between the resulting θ14and θ14i which are obtained from the second and first four bar linkage closure equations respectively so that the finger can produce the desired θ21hpandθ31hp. Mathematically, f1 can be described as
f1 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2 θ23hp −θ23p þ ðθ14i −θ14 Þ2 =2
ð7Þ
Only the angles in the final configuration instead of all values during the whole pinching motion are considered in the objective function. This is because only the last set of angles rather than the finger's movement behaviour is more crucial in ensuring a successful pinching kinematics as in human finger.
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3.2. Objective function I1: pinching force The pinching force of the artificial finger can be calculated by performing static analysis on its free body diagram during pinching as shown in Fig. 8. Since the summation of force in the vertical direction is zero, Fp ¼ N
ð8Þ
where Fp and N are the pinching and normal forces acting on the distal phalanx respectively. With the assumption that the pinching force is exerted at the middle of the distal phalanx, Fp can be calculated from the moment equation around point A as F p ¼ T i =ðla −f r lb Þ
ð9Þ
where Ti fr d
: input torque on link 2 : coefficient of static friction : distance between line l8 and the edge of the link, considering the linkage's thickness la ¼ l1 cos θ11 þ l5 cos θ21 þ 0:5l8 cos θ31 lb ¼ −l1 sin θ11 −l5 sin θ21 −0:5l8 sin θ31 −d sinðθ31 −π=2Þ
ð10Þ
The criterion for achieving optimum pinching force is formulated by taking into account the difference between the desired pinching force made by the human finger, Fph and the artificial finger, Fp with the provided input torque. Therefore, the objective function f2 can be represented as
f2 ¼
F ph −F p f r ¼0:5
ð11Þ
F ph
3.3. Objective function III: grasping kinematics In grasping kinematics optimization, the finger is also considered as two sets of four bar linkage as shown in Fig. 7. The link with length l1 and the angle θ12 is considered as the base linkage and input angle of the first four bar linkage respectively. On the other hand, the link with length l5 and the angle θ21act as the base link and base angle in the second four bar linkage respectively. The value of θ14while the upper linkage of the middle phalanx is at its maximum length can be calculated as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 −B1 − B1 2 −C 1 2 þ A1 2 5; C 1 −A1
−1 4
θ14 ¼ 2 tan
ð12Þ
where A1 ¼ 2l1 l4 cos θ11 −2l2 l4 cos θ12 B1 ¼ 2l1 l4 sin θ11 −2l2 l4 sin θ12 2 2 2 2 C 1 ¼ l1 þ l2 þ l4 −l3 −2l1 l2 cosðθ11 −θ12 Þ
ð13Þ
From Eq. (12), the base angle of the second four bar linkage, θ21before the middle phalanx is in contact with the object can be obtained as θ21 ¼ θ14 −θ22ð0Þ
ð14Þ
Ti
θ21
l6p
A lb
l6g distal phalanx’s edge
l5 l8 la
d
N Objec t
fr N
Fig. 8. Finger free body diagram during pinching.
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whereθ22(0) = θ14(0). Therefore the output of the second four bar linkage can be obtained as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 −B2 − B2 2 −C 2 2 þ A2 2 5; C 2 −A2
−1 4
θ24 ¼ 2 tan
θ23g ¼ tan
−1
l5 sin θ21 þ l7 sin θ24 −l4 sin θ22 l5 cos θ21 þ l7 cos θ24 −l4 cos θ22
ð15Þ ð16Þ
where A2 ¼ 2l5 l7 cos θ21 −2l4 l7 cos θ22 B2 ¼ 2l5 l7 sin θ21 −2l4 l7 sin θ22 C 2 ¼ l24 þ l25 þ l27 −l26g −2l4 l5 cosðθ21 −θ22 Þ
ð17Þ
Finally, the value of θ31 after the middle phalanx has touched the object can be determined as θ31 ¼ θ24 −θ24ð0Þ
ð18Þ
The corresponding angles that the human's middle phalanx, θ21hg and distal phalanx, θ31hg make while grasping small (1 cm radius), medium (2 cm radius) and large (3 cm radius) size cylindrical objects are measured from the photos taken. These two angles are indicated in Fig. 9. Then, these values are compared to angles θ21and θ31 made by the robotic finger with the upper link of the middle phalanx is at its maximum length. The objective function in this optimality criterion, f3 has to minimize the difference between these angles and this can be mathematically expressed as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3 2 2 uX f3 ¼ t θ21hg r¼ri −θ21 r¼ri þ θ31hg r¼ri −θ31 r¼ri =6;
ð19Þ
i¼1
where r is the object's radius, r1, r2 and r3 equal to 1 cm, 2 cm and 3 cm respectively. 3.4. Objective function IV: grasping force The grasping force calculation is performed based on the work done in Ref. [18]. Referring to Fig. 10, the grasping force exerted by middle, fg1 and distal phalanges, fg2 can be described by f g1 ¼
h2 l5 U ðk3 −h3 cos θ3 ÞT i ; k2 k3 ðh2 þ l1 Þðh3 þ l5 Þ
ð20Þ
f g2 ¼
h2 h3 T i ; k3 ðh2 þ l1 Þðh3 þ l5 Þ
ð21Þ
where Ti : input torque on link 1 hi : signed distance between point Oi and intersection of lines (Oi − 1Oi) and (Pi − 1Pi) k1, k2 and k3 contact points of the proximal, middle and distal phalanges with the object respectively U ¼ k2 k3 h3 þ k2 k3 l2 −h2 k3 l2 cosθ2 þ h2 h3 l2 cosθ2 cosθ3 −h2 h3 k2 cosðθ2 þ θ3 Þ
θ21hg
θ 31hg
Fig. 9. θ21hg and θ31hg measured from photo.
ð22Þ
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P1
59
P2 θ1
θ2
O1 O2
P3
k2
O3 fg1
k3 fg2
θ3
Fig. 10. Geometric details for grasping force calculation.
If both grasping forces fg1 and fg2 in Eqs. (20) and (21) are positive, it means that a successful grasping is achieved, but if one or more of the forces are negative, ejection has occurred. Therefore, the main criterion of the fourth objective function, f4 is the grasping force exerted by all of the phalanges on cylindrical objects with 1 cm radius must be greater than zero. The force while grasping cylindrical objects with 2 cm and 3 cm radius are not taken into consideration because human hands usually need the palm or thumb to grasp large objects. If a successful grasping is achieved, the finger is also required to provide sufficient grasp force, fg to support the desired object weight. This value can be determined as f g ¼ f 1g sinðθ21 −π=2Þ þ f 2g sinðθ31 −π=2Þ
ð23Þ
The value of f4 is set to 1 as a penalty if any of the phalanx forces while grasping 1 cm radius cylindrical object are negative. However, if no ejection occurs, the objective function has to minimize the difference between fg and the supported load, Wd. Mathematically, f4 can be expressed by 8 1 > < f 4 ¼ f g −W d > : Wd
∃f gi jr¼1cm b0;
i ¼ 1; 2
∀f gi jr¼1cm > 0;
i ¼ 1; 2
ð24Þ
3.5. Constraints The multi-objective optimization problem is subjected to multiple constraints which are due to interferences during both pinching and grasping, finger size and four bar linkage closure equations. The vertical component of the link with length l6g during pinching configuration is set to be at least 1 mm shorter than the total vertical length between point P2, as in Fig. 10 and the surface of the distal phalanx with length l8. This constraint is necessary to avoid any mechanical interference between the link at length l6g and the object during pinching. Hence, the constraint c1 due to this factor can be formulated as c1 : l6g sin θ23g þ 1 b l6p sin θ23p þ l7 j sin θ24 j
ð25Þ
where θ23g θ23p and θ24are the values during final pinching configuration. To avoid any linkage interference between the link of length l5 and the guiding slot for l6p while the finger is grasping the 1 cm radius cylindrical object, the distance between these two points is set to be at least 3 mm horizontally from each other as linkage’s edge
l6p linkage’s thickness
Object (r= 1cm)
l5
distance between l6p slot and end of l5 Fig. 11. Finger configuration in grasping object.
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hinge pin, diameter=2mm
li
2mm
thickness=4mm Fig. 12. Basic structure of linkages with length li, i = 1 to 5.
illustrated in Fig. 11. The distance is selected taking into consideration the thickness of the linkage and the diameter of the distal phalanx pin. This can be mathematically represented as c2 : l6p cos θ23p þ 3bj−l4 cos θ14 þ l5 cos θ21 j
ð26Þ
where θ14, θ21 and θ23p are the final angles while grasping a cylindrical object of 1 cm radius. In terms of the finger size, the total finger length is constrained to be less than 75 mm, so that it is close to the size of an average human's index finger. This constraint can be represented by c3 : l1 þ l5 þ l8 b 75
ð27Þ
Considering the kinematics equations of the four bar linkages in Eqs. (1), (4), (12) and (15), the optimization problem is also subjected to the constraints c4 and c5 below. These constraints ensure that the solutions to these equations are real and therefore the mechanism can be assembled [19]. 2
2
2
c4 : Bp −C p þ Ap > 0 2
2
2
c5 : Bi −C i þ Ai > 0;
ð28Þ i ¼ 0; 1; 2
ð29Þ
3.6. Optimization formulation Taking all the four objective functions in Eqs. (7), (11), (19) and (24) into consideration, the optimization problem has to minimize the multi-objective function F ¼ w1 f 1 þ w2 f 2 þ w3 f 3 þ w4 f 4
ð30Þ
where wi, i = 1, 2, 3 and 4 are the weight coefficients, indicating the importance of each objective function. At the same time, the solution to this problem must also satisfy all the constraints governed by Eqs. (25)–(29) simultaneously. The optimization algorithm has been performed by the nonlinear continuous optimization algorithm, Sequential Quadratic Programming (SQP) using the ‘fmincon’ command available in Matlab Optimization Toolbox software [20]. Although the command l6g
l 6p guiding slot Fig. 13. Middle phalanx's upper linkage.
l9
l7
d =2mm l8 Fig. 14. Distal phalanx linkage.
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Table 1 Hardware components. Components
Dimension (mm)
Weight (g)
Motor 1 and encoder Motor 2 Potentiometer Force sensor
ϕ13.0 × 52.0 ϕ10.0 × 8.0 × 11.7 ϕ8.0 × 5.0 14.0 × 102.0 × 0.189
40.1 11.0 0.7 0.4
provides the solution by the convergence of local optimum, this technique has been chosen because it is easy to use, does not require tedious computational effort and does not consume long calculation time. The steps implemented by ‘fmincon’ function in finding the constrained minimum of a function of several variables can be referred in [8]. 4. Prototype development The hardware components of the finger are chosen to achieve a compact and light design. The basic structure of the finger prototype is made from aluminum since the material possesses high strength but low weight. All of the hinge pins’ diameter and the horizontal distance between its center and the edge of the link are set to be 2 mm. The bar linkages’ length is developed based on the optimized solution obtained and their thickness is fixed to be 4 mm. Figs. 12, 13 and 14 illustrate the basic structure of the linkages with length li where i =1 to 5, upper linkage of the middle phalanx and distal phalanx respectively. A high torque geared motor with rated torque 0.475 Nm which is manufactured by Citizen Micro Co. Ltd. is selected to provide the input torque on link 1. On the other hand, a micro-motor from Mabuchi Motor Co. Ltd. with the weight of 11 g and nominal torque 0.00053 Nm is chosen to drive the lead screw mechanism in adjusting the length of the middle phalanx. An encoder made by Citizen Micro Co. Ltd. and a rotary potentiometer are used to measure the joint positions, and the Tekscan Inc. Flexi force sensor is utilized in measuring the force exerted by the finger. To facilitate future works, the electrical commands are provided using xPC Target, Matlab software. The dimension and weight of the sensors and actuators used in the mechanism are summarized in Table 1. 5. Results 5.1. Optimization numerical results For the numerical optimization, a set of initial guess, upper and lower boundaries have been chosen as in Table 2. It is a good practice to provide the parameters’ boundaries because this may promote a fast convergence to the solution since the search space is restricted. The corresponding angles made by a human finger during pinching,θ21hp, θ23hp, θ21hg andθ31hg are measured Table 2 Initial guess, lower and upper boundaries of the design variables. Length (mm)
Initial guess (mm)
Lower boundary (mm)
Upper boundary (mm)
l1 l2 l3 l4 l5 l6p l6g l7 l8
26.0 28.0 30.0 24.0 25.0 26.0 40.0 20.0 24.0
20.0 20.0 20.0 20.0 24.0 25.0 20.0 15.0 20.0
30.0 30.0 50.0 30.0 30.0 30.0 50.0 30.0 30.0
Table 3 Corresponding angles made by human finger during pinching and grasping. Angle
Measured value (°)
θ21hp θ23hp θ31hp θ21hg|r = 1cm θ31hg|r = 1cm θ21hg|r = 2cm θ31hg|r = 2cm θ21hg|r = 3cm θ31hg|r = 3cm
290 270 0 270 180 300 245 325 280
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Table 4 Optimized parameter solution and initial guess. Length
Initial guess (mm)
Optimized solution (mm)
l1 l2 l3 l4 l5 l6p l6g l7 l8 l9
26.0 28.0 30.0 24.0 25.0 26.0 40.0 20.0 24.0 34.8
27.3 30.0 36.2 27.0 24.0 25.4 46.7 20.9 20.0 24.3
Table 5 Objective functions values. Objective function
Initial guess
Optimized solution
f1 f2 f3 f4 F
19.2134 0.0139 13.3779 1.0000 33.6052
0.0001 0.0041 0.4120 0.1309 0.5471
Object
Object
(a)
(b)
Object Object
(c)
(d)
Fig. 15. Optimized artificial finger in (a) pinching, grasping a (b) 1 cm (c) 2 cm and (d) 3 cm cylindrical object.
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63
from the photos taken and these values are summarized in Table 3. The initial input angle θ12(0) and the input torque, Ti have been set to be 147∘ and 0.1 Nm respectively. It has been desired for the finger to exert a pinching force, Fph of 3.0 N with the provided input torque and the coefficient of static friction equals to 0.5. The finger is also required to be able to support a 1 cm radius cylindrical object with weight, Wd 1.5 N. The grasping forces have been assumed to act at the middle of each phalanx. During the optimization process, all the angles have been calculated and measured in degrees. The optimized solution that gives the minimum anthropomorphic multi-objective function and satisfies all the constraints has been obtained as in Table 4. The resulting errors or objective functions are summarized in Table 5. In this case, all of the weight coefficients have been set to 1, implying equal importance for all the four objective functions. The resulting value of l9 has been obtained based on the optimum solution using the cosine law l9 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l27 þ l28 −2l7 l8 cos θ24ð0Þ
ð31Þ
The result shows that compared to the initial guess, the optimized parameters produce less error in fulfilling both the kinematics configuration and force exerted for the pinching as well as grasping task. The mechanism with initial guess parameters gives high errors in the pinching and grasping kinematics optimality criteria. Besides that, it is also unable to grasp a 1 cm object, where ejection has occurred. Fig. 15 illustrates the finger with the optimized design variables during pinching and grasping cylindrical objects of 1 cm, 2 cm and 3 cm radius. The dashed lines in the figure represent the outer edges of the linkages that are actually in contact with the objects. It can also be seen from the figure that the optimal solution also complies with the constraints imposed by Eqs. (25) and (26), in which no mechanical interference occur. The SQP technique implemented in this study gives the optimization solution based on the convergence to local optimum. In this method, the result depends greatly on the initial guess selected. Different sets of initial guess will result in distinct optimum design variables and multiple-objective function value, F. Therefore in this work, the initial guess is first chosen intuitively as the values close to a human finger. Then, a few tests are performed using different sets of initial guess, which are varied slightly from the one chosen in the beginning. Finally, the initial guess set that gives the lowest multiple-objective function, F as in Table 2 has been selected in the optimization procedure. Although the SQP algorithm requires few trials to select the initial guess set, the technique is chosen because it is easy to implement and does not require high computational effort and long calculation time. Table 6 shows the numerical optimization results for different combinations of the optimality criteria under the same initial guess, lower boundary and upper boundary sets as in Table 2. Referring to Table 6, if only the pinching kinematics and force objective functions are considered in the optimization problem by changing the coefficient weights to w1 = 1, w2 = 1, w3 = 0 and w4 = 0, the finger is able to reach the pinching configuration and exert the desired pinching force successfully. Although this solution gives a lower multi-objective function, F compared to the optimal solution, the higher error of objective function f3 indicates that it is less efficient in making the finger to grasp objects, as illustrated in Fig. 16(a). Table 6 Weight coefficient, solution and objective function of the optimization problem considering different combination of optimality criteria. Pinching kinematics and force
Grasping kinematics and force
Pinching and grasping kinematics
Pinching and grasping kinematics and force
Weight w1 w2 w3 w4
1 1 0 0
0 0 1 1
1 0 1 0
1 1 1 1
Solution (mm) l1 l2 l3 l4 l5 l6p l6g l7 l8 l9
24.8 28.5 32.4 25.6 24.1 25.9 45.1 19.5 25.3 28.1
29.7 30.0 38.2 27.0 24.0 25.0 41.5 19.0 20.0 27.2
27.2 30.0 36.2 27.0 24.0 25.4 46.7 21.0 23.8 26.7
27.3 30.0 36.2 27.0 24.0 25.4 46.7 20.9 20.0 24.3
Objective function f1 f2 f3 f4 F Total (f1+ f2 + f3+ f4)
0.0002 0.0000 4.4806 0.0800 0.0002 4.5608
20.0240 0.0648 0.8871 0.0528 0.9399 21.0287
0.0003 0.0484 0.4113 0.2668 0.4116 0.7268
0.0001 0.0041 0.4120 0.1309 0.5471 0.5471
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θ14i
object
θ14
object
(a)
(b)
Fig. 16. Robotic finger with (a) pinching kinematics and force optimality criteria only in grasping configuration (b) grasping kinematics and force optimality criteria only in pinching configuration.
On the other hand, if the values of the weight coefficient are assigned to w1 = 0, w2 = 0, w3 = 1 and w4 = 1 to consider only the grasping kinematics and force optimality criteria, the solution obtained satisfies the grasping requirement, but the link with length l6p is unable to achieve the desired pinching configuration, θ23hpunder the provided θ21hpandθ31hp, as shown in Fig. 16 (b). It can also be observed from the figure that the resulted angular displacement of link l4, denoted byθ14, does not reach the angular position, θ14i that is necessary to achieve the desired θ21hpandθ31hp configuration. In short, the value of θ14i is unequal toθ14. The solution in considering only the pinching and grasping kinematics objective functions has also been determined by setting w1 = 1, w2 = 0, w3 = 1 and w4 = 0. Even though it gives a smaller total error (f1 + f2 + f3 + f4) compared to the previous two cases, it does not result in an optimum solution for all the four optimality criteria. The total error value is higher compared to the solution provided by the optimum design variables. The results in these three cases show that it is important to consider all the four factors which are due to both pinching and grasping kinematics and force in the multi-objective optimization formulation. The linkage based robotic finger's capability in completing both of the functions as in a human finger can be improved by considering all these four factors. 5.2. Prototype A prototype has been built based on the optimum solution obtained and is shown in Fig. 17. It can be seen from Fig. 18 that the prototype has successfully achieved the final pinching and grasping configurations. Therefore, the optimization procedure improves the finger performance compared to implementing the finger with just any guessed linkage lengths. 6. Conclusions A new finger mechanism and its linkages’ length optimization to resemble a human finger in performing pinching and grasping with self-adaptive grasp capability are presented in this paper. The mechanism is made of seven bar linkages with the upper link of the middle phalanx designed with a curved guiding slot and lead screw mechanism to adjust its effective length in accomplishing pinching and grasping tasks. Its geometrical parameter has been formulated as a multi-objective optimization design procedure based on four anthropomorphic design goals, which are the pinching kinematics and force; and grasping kinematics and force. The numerical result demonstrates that the optimal solution has greatly reduced the error compared to the one resulted from the initial
Fig. 17. Finger prototype.
N. Zainul Azlan, H. Yamaura / Mechanism and Machine Theory 49 (2012) 52–66
(a)
(c)
65
(b)
(d)
Fig. 18. Finger prototype in (a) pinching, grasping a (b) 1 cm (c) 2 cm and (d) 3 cm cylindrical object.
guess. The weight coefficients have been varied in the numerical analysis to consider different combination of optimality criteria, and the results have shown that it is vital to take all the four objective functions into account in improving the finger performance. A finger prototype has been constructed based on the optimum solution obtained and it has been seen that the finger conforms to the desired configurations. Implementing the optimum linkages length solution together with the choice of light weight and small size actuators and sensors in the design contribute to the realization and enhancement of a compact anthropomorphic finger mechanism in carrying out both pinching and grasping operations. Future works involve the development of a control algorithm in controlling the finger while manipulating objects and the construction of the whole hand consisting of a thumb and other fingers. References [1] K. Hoshino, I. Kawabuchi, Dexterous robot hand with pinching function at fingertips, IEEE/RAS-EMBS International Conference on Biomedical Robotics and Biomechatronics, Italy, 2006, pp. 1113–1118. [2] S.G. Jacobsen, E.K. Iversen, D.F. Knutti, R.T. Johnson, K.B. Biggers, Design of the Utah/M.I.T. dextrous hand, IEEE International Conference on Robotics and Automation, U.S.A, 1986, pp. 1520–1532. [3] M.T. Mason, J.K. Salisbury, Robot Hands and the Mechanics of Manipulation, MIT Press, 1985. [4] N. Dechev, W.L. Cleghorn, S. Naumann, Multiple finger, passive adaptive grasp prosthetic hand, Mechanism and Machine Theory 41 (2006) 897–911. [5] E.N. Haulin, R. Vinet, Multiobjective optimization of hand prosthesis mechanisms, Mechanism and Machine Theory 38 (2003) 3–26. [6] Y. Liu, H. Wang, B. Li, W. Zhou, The underactuation and motion-coupling in robotic fingers and two new 1-DOF motion-coupling anthropomorphic fingers, IEEE International Conference on Robotics and Biomimetics, Thailand, 2009, pp. 1573–1578. [7] G. Guo, T.T. Lee, W.A. Gruver, J. Zhang, Design of a planar multijointed prosthetic finger mechanism, ASME Mechanisms Conference, U.S.A, 1990, pp. 165–170. [8] N.E.N. Rodriguez, G. Carbone, M. Ceccarelli, Optimal design of driving mechanism in a 1-DOF anthropomorphic finger, Mechanism and Machine Theory 41 (2006) 897–911. [9] M.C. Carrozza, C. Suppo, F. Sebastiani, B. Massa, F. Vecchi, R. Lazzarini, M.R. Cutkosky, P. Dario, The SPRING hand: development of a self-adaptive prosthesis for restoring natural grasping, Autonomous Robots 16 (2004) 125–141.
66
N. Zainul Azlan, H. Yamaura / Mechanism and Machine Theory 49 (2012) 52–66
[10] B. Massa, S. Rocella, M.C. Carozza, P. Dario, Design and development of an underactuated prosthetic hand, IEEE International Conference on Robotics and Automation, 2002, pp. 3374–3379. [11] T. Laliberte, C.M. Gosselin, Underactuation in space robotic hands, International Symposium on Artificial Intelligence and Robotics & Automation in Space, Canada, 2001, pp. 1–8. [12] T. Laliberte, L. Birglen, C.M. Gosselin, Underactuation in robotic grasping hands, Machine Intelligence & Robotic Control 4 (3) (2002) 1–11. [13] SARAH HAND. Available from: bhttp://robot.gmc.ulaval.ca/en/research/theme304.html>. [14] L. Birglen, C.M. Gosselin, Grasp-state plane analysis of two-phalanx underactuated fingers, Mechanism and Machine Theory 41 (2006) 807–822. [15] D.W. Zhao, L. Jiang, H. Huang, M.H. Jin, H.G. Cai, H. Liu, Development of multi-DOF anthropomorphic prosthetic hand, IEEE International Conference on Robotics and Biomimetics, China, 2006, pp. 878–883. [16] J. Yang, E.P. Pitarch, K.A. Malek, A. Patrick, L. Lindkvist, A multi-fingered hand prosthesis, Mechanism and Machine Theory 39 (2004) 555–581. [17] T. Laliberte, C.M. Gosselin, Simulation and design of underactuated mechanical hands, Mechanism and Machine Theory 33 (1) (1998) 39–57. [18] L. Birglen, C.M. Gosselin, Geometric design of three-phalanx underactuated fingers, Transaction of the ASME 128 (2006) 356–364. [19] K.J. Waldron, G.L. Kinzel, Kinematics, Dynamics, and Design of Machinery, Wiley, 1999. [20] Optimization Toolbox For Use with MATLAB. Available from: bhttp://www.mathworks.com/help/pdf_doc/optim/optim_tb.pdf > last accessed: 13 January 2011.