Kinematic Analysis and Dimensional Synthesis of a Meso-Gripper

Grippers and robotic hands are essential and important end-effectors of robotic manipulators. The development of grippers or robotic hands able to pick up objects for different applications has attracted much attention within the research community over the last four decades. Grippers vary from simple configurations, such as two-jaw parallel designs, to complex, handlike shapes with advanced sensors, control units, and servomotors. The initial type of gripper application used in industry was for picking up raw and finished parts [1]. The majority of grippers are variations of three fundamental designs, namely: parallel, three-fingered, and angled grippers. A two-finger gripper has the minimum number of fingers and the minimum complexity of a hand. According to typical types of industrial gripper applications, the minimum gripping force of a parallel gripper is difficult to control especially for small loads. Angled grippers can be adjusted to various angles according to work piece and space limitations. Also, robotic grippers that mimic anthropomorphic hands have also been investigated and applied in industry. Within such systems, fingers are the most important components of a precision grasping “hand” [2]. These can mainly be classified into three types: low-cost grippers for industrial production, precision grasping, and underactuated grasping [3]. Recent research on universal high flexibility, multifunctional robotic grippers has been carried out, and some systems have been developed [4]. However, it is still difficult to grip very small, fragile, and light objects with such grippers. With the development of microscale technologies in electronics, optics, and biology, microgrippers are required for microrobot and micro-assembly. Microgrippers also require to grip and handle the micro-objects securely and accurately with the range below 100 μm, preferably with no gripper changes. Some specifically designed microgrippers handle objects ranging from nanometers up to 500 μm. In a variety of circumferences, a robot has to switch hands for gripping objects with different sizes, shapes, and ranges of gripping force that requires. A general purpose hand that can accomplish all of these tasks will make gripping much more efficient and reduce changeover times.

To address this gap, in this research a meso-gripper which has metamorphic characteristics and two gripping modes shown in Fig. 1 is proposed, developed, and tested.

This paper is organized as follows. In Sec. 2, a mesoscale for gripping technology is defined according to human grasping behaviors and processes. The gripper is then designed and developed by integrating a remote center of motion (RCM) mechanism with a cross four-bar (CFB) linkage. In Sec. 3, the dimensional synthesis of the gripper is outlined for a specified gripping task followed by analysis of the synthesizing mechanism. In Sec. 4, the design is modified via 3D model by considering stiffness and layout of the assembly and a differential mechanism included to increase the gripper flexibility. In Sec. 5, from the model, a physical prototype is fabricated using 3D printing and manually tested to evaluate and verify the feasibility of the gripper design. The paper then concludes with the further consideration of gripping forces during the design process and comments on the outcomes and future direction of research.

Generally speaking, the mesoscopic scale comprises different length scales in different research areas. A comparison of the relevant length scales in material science and molecular biology shows that the mesoscopic length is 10⁶–10⁸nm (1–100 mm) [5]. However, some specifically designed microgrippers handle objects up to 500 μm; therefore, the minimum gripping size of a meso-gripper has been defined in this work as 500 μm. A meso-gripper bridges the gap between microgrippers and macrogrippers by extending the macrogripping range, preferably without switching grippers for industrial assembly. Therefore, in Fig. 2, this work defines the working gripping dimension range of meso-grippers as from 0.5 to 100 mm, based on typical requirements for the assembly of multimedia products.

Two manual gripping examples were considered, and various configurations of fingers observed to help define the functionality of the proposed meso-gripper. These examples comprise two parts, a very thin hex socket screw with a diameter of 1.5 mm, and a plastic cuboid with dimensions 28 mm × 24 mm × 17 mm. Both were observed to be processed in four steps, namely: searching, reaching, gripping, and moving [6] (see Fig. 3). The objective for each step in assembling these two parts is the same but the configurations are different due to size and shapes of the parts.

The gripping process of hex socket is as follows:

• Distal segments of thumb and index fingers contact each other, then move to the hex socket.

• The contacted fingers reach above the hex socket.

• The hex socket is gripped by fingertips.

• The hex socket is moved.

The gripping process for plastic cuboid is

• The thumb and index fingers move to the plastic cuboid.

• Distal segments of thumb and index fingers reach and contact two edges of the cuboid.

• Pads of thumb and index fingertips grip the sides of the part for clamping.

• The cuboid is moved.

Comparing these two processes, gripping the hex socket screw requires only one degree-of-freedom (DOF), while gripping the cuboid requires 2DOFs (see Fig. 3). This analysis greatly simplifies the finger design for gripping these objects. Two degrees-of-freedom, one for positioning and another for clamping, are the minimum number of DOF required in these gripping processes.

Gripping using a human hand is activated because the neural and visual systems work as sensors with the brain operating as a control system. According to the gripping analysis in Sec. 2.2, 2DOF fingers could be adopted to grip mesoscale parts in two modes. The key challenge is to build a 2DOF finger with two gripping modes. Therefore, a passive adjusting concept is proposed in this section for gripping mesoscale parts.

Progress in research and development for grippers has been made in underactuated grasping; for example, Birglen and Gosselin have used a five-bar linkage and a four-bar linkage connected in series to design three-phalanx underactuated fingers [7]. Ceccarelli et al. have designed three-phalanx underactuated fingers with CFB linkage in series [8]. To create stable, encompassing grasps with subsets of fingers, soft fingertips that deform during contact and apply a larger special spread of frictional forces and moments than their rigid counterparts have been studied [9]. CFB linkage is found to be good candidates to achieve passive adjusting motion and has similar characteristics as to soft fingertip. Passive adjusting mechanism has characteristics such as adaptivity, under-actuativity, efficiency, and multifunction whereby such a system can adjust itself automatically depending on the locations and shapes of objects.

A CFB linkage is shown in Fig. 4(a). When crank link AD rotates from initial angle γ with respect to vertical line to angle γ′, the coupler link CD reaches to position C′D′. In turn, the mechanism can adjust passively when one point of link CD contacts an object. If the positions of two points of the coupler link CD are determined, the configuration of the whole mechanism will be fixed. A soft-fingertip model composing with CFB-damper-spring component is shown in Fig. 4(b). Friction is represented as f at the fingers’ contact surfaces. The normal force applied on the each fingertip is F_n. The mass of object and mass of CFB linkage is G_m and G_l, respectively. K_s and R_d represent the stiffness of the spring and damper ratio, respectively. Analysis of contact positions should be conducted before applying it as a fingertip for gripping. Furthermore, if the motion of one endpoint of link CD contacts with the symmetrical side, then these two CFB mechanisms will be angled fingertips.

RCM mechanisms are widely used as a wrist for a minimally invasive surgery (MIS) robot to provide a fixed point moving around the surgical incision. An RCM manipulator was first developed by Taylor et al. [10], then the concept and mechanism were used in MIS for precision operations as a steady hand [11]. Bai expanded the mechanism to multiple RCMs for complicated applications by investigating the relationship between RCM mechanism and deployable mechanism. A multiple RCM mechanism [12] was proposed, and some applications, such as foldable stages and a surgical helmet for ophthalmology, were demonstrated in Ref. [13]. The kinematic analysis of the RCMs shows that the end link of the mechanism has the same rotation angle as the drive link no matter how many links are connected between them. As shown in Fig. 5, links HJ and GE rotate around two remote centers O1 and O2, respectively, and have the same rotational angles as the drive link AC [13]. This characteristic makes the mechanism useful and easily controllable. In this paper, an RCM mechanism will be used as a middle and proximal phalanx to provide gripping force and positioning motion.

The overall gripper model was obtained by integrating the RCM, CFB mechanisms, damper, and spring. The gripping process mimics the human hand, as shown in Fig. 6. The process shows that the integrated mechanism passively adapts to parallel sides of a cuboid.

Considerable development in theoretical research on metamorphic mechanisms [14] has been made in the past 15 years such as the metamorphic hand [15] and walking machines. The meso-gripper during passive adjusting mode shown in Fig. 7(a) has 2DOFs when considering one side of the gripper. The DOF of the mechanism will reduce to one if one point of CFB mechanism touches the object. If two points of the coupler link of the CFB linkage touch the object, the DOF degrades to −1, i.e., it is over-constrained. Therefore, the meso-gripper is an example of metamorphic mechanism. A detailed analysis of this characteristic is outlined below.

The process of cuboid gripping is similar to the four steps shown in Fig. 3. The object M is supposed to be fixed to the frame, while point I of the CFB mechanism slides on the surface of the object until point H contacts it as shown in Fig. 7(a). The solid line diagram shows the point at which the mechanism reaches the cuboid and the dashed diagram the final configuration. The equivalent mechanism during the gripping process is shown in Fig. 7(b). The DOF (F) analysis of this mechanism during this process is calculated as follows using Gruebler’s equation, where n = number of moving links, p_l = number of lower pairs, and p_h = number of higher pairs:

• Searching step: F = 3 n − 2p_l − 2 p_h = 3 × 8 − 2 × 11 = 24 − 22 = 2

• Reaching step: F = 3 n − 2p_l − 2 p_h = 3 × 9 − 2 × 13 = 27 − 26 = 1

• Gripping step: F = 3 n − 2p_l − 2 p_h = 3 × 7 − 2 × 11 = 21 − 22 = − 1

The number of movable links of the mechanism changes during the gripping process, causing a subsequent change in its topology.

The angled gripping mode of the gripper has a similar motion to the passive adjusting mode. The DOFs and topology of the mechanism vary at different gripping steps.

Geometric constraint programming was proposed to solve general kinematic synthesis problem, such as planar four-bar linkages for motion generation, path generation, and function generation [16]. Detailed dimensional synthesis and kinematic analysis of the multi-RCM mechanism can be found in Refs. [12,13]. The design process of the meso-gripper using this approach will be presented in this section.

The objective of dimensional synthesis is to determine the value of each geometric parameter of a mechanism by taking account of its desired performance. Geometric parameters vary with the design criteria. Gripping range is one of the most important criteria which differentiate the meso-gripper from other designs. The dimensions of the mechanism will be presented based on the task requirements.

A geometrical constraint approach is presented to help design mechanisms over a specific gripping range. The approach is closely related to the synthesis of multi-RCM mechanisms [12]. The differences mainly lie in the prototype design and manufacturing considerations.

The task requirements’ specification is as follows:

1. The gripping range of the meso-gripper should be 0–55 mm.

2. The range of angled gripping mode is 0–6 mm.

3. The range of passive adjusting mode is 6–55 mm.

4. The mechanism of gripper should be compact.

The synthesis process incorporating the geometrical constraint approach is outlined as follows:

• Step 1. Determination of initial positions according to the task requirements: Fig. 8 gives an enlarged range of 0–60 mm for the optimization. The RCM point and the first frame point are determined by considering the compactness of the whole gripper.

• Step 2. Design of the CFB linkage: The dimensions of CFB linkage are shown in Fig. 9. The length of follower link should be longer than crank link to build angled configuration for angled gripping. The connecting link will connect to the RCM mechanism.

• Step 3. Determination of position at angled gripping mode: The CFB linkage is incorporated into the graph obtained from step 1. The length of drive link is set at 40 mm, i.e., larger than 30 mm which is at the borderline limit. A circle of 40 mm radius is drawn with the RCM point its center. The two endpoints of the coupler link are positioned on the symmetrical line and 3 mm borderline, respectively. One endpoint of connecting link is placed on the circle and a line drawn from the RCM point to the end of the circle, making the angle between it and connecting link of the CFB linkage 150 deg. The position of CFB linkage at the angled gripping mode is then determined, as shown in Fig. 10 .

• Step 4. Determination of position at initial passive adjusting mode: As is shown in Fig. 11 , one end of the coupler link is positioned at the 30 mm borderline, connecting one endpoint of the connecting link at the circle. This makes the angle between the connecting link and the RCM line 150 deg.

• Step 5. Synthesis of the RCM mechanism: By using synthesis method of the RCM mechanism [12], the dimensions of the RCM mechanism can be determined. Based on the first frame point, the second frame point with a dimension of 10 mm and a 135 deg angle is drawn about the horizontal line. Parallelogram 1 is constructed according to the two-point frame line and drive link. Parallelogram 2 is constructed using the connecting link and the RCM position line. Based on these two parallelograms, parallelogram 3 is then determined as shown in Fig. 12 .

• Step 6. Verification: Three angles should be verified to determine the feasibility of the synthesized mechanism. If angle δ is larger than θ, then the synthesized mechanism can reach the preset position as shown in Fig. 13 . Angle Ψ should be no greater than 90 deg to make the 55 mm object gripping successful. In Fig. 13, angle Ψ is greater than 90 deg, so a modification should be made.

• Step 7. Optimization: The initial position of the RCM mechanism is changed in order to reduce the verification angle Ψ to 90 deg by rotating the crank link of CFB mechanism, as shown in Fig. 14 . As a consequence, the final design meets the maximum gripping size requirement of 55 mm.

In order to verify the parameters of the synthesized mechanism, kinematic analysis of the mechanism is carried out. The most important geometric parameters are the drive angle and the motion between contact surface and the object. Two different modes of gripping: passive adjusting and angled gripping are analyzed in this section. The rotating angle (RA) of driving link and the distance moved by point I on the contacting surface are the basic parameters required for further design of the drive method and the subsequent gripping force analysis.

As is shown in Fig. 15, the solid line diagram shows the final position for gripping a 55 mm object, and the dashed line diagram shows the initial position of the mechanism. Point I slides on the object surface using a slide link as shown. It is determined that the rotating angle (RA) of the drive link is 6.42 deg, and slide distance of point I about y axis (Uy) is 4.3 mm.

As shown in Fig. 16, the gripping process for an object of 6 mm requires a maximum RA of drive link AC of around 2.9 deg and contacting point I sliding around 1.28 mm.

In Sec. 3, a geometrical constraint method was presented to design a gripper for a required gripping range. Kinematic analysis of the mechanism helps an understanding of the gripping process and determines the drive position. In this section, modified design is presented by considering stiffness of the mechanism and layout of the assembly for prototyping and manufacturing.

As shown in Fig. 17(b), the integrated mechanism has four layers at the position of cross four-bar mechanism because of its multiple joints. This layout leads to the tip of the gripper being less stiff for accelerated gripping. So it is more advantageous to reduce the layers of the mechanism to provide a smaller gripping torque at the tip linkages.

The CFB mechanism should be modified by changing the multiple joint of connecting link into a single joint with the coupler link of the mechanism reduced in size to provide adequate space for tip contact surface (Fig. 18).

Figures 19 and 20 show the kinematics of the meso-gripper at passive adjusting and angled gripping modes. The rotating angles of drive link at these two modes are around 8.65 deg and 2.91 deg, and sliding distances of contacting point I are 2.94 mm and 1.24 mm, respectively.

A typical RCM mechanism comprises a six-bar over-constraint mechanism containing three-parallelogram loops, as shown in Fig. 21(a). The mechanism transfers to 1DOF by removing one connecting link of the parallelogram loop, as shown in Fig. 21(b). However, this simplification reduces the stiffness of the mechanism because one supporting link is removed. At passive adjusting mode, the weight of the object and associated acceleration may result in a very large torque on some links; meanwhile, the meso-gripper must be designed to be compact with each component being very small. This requires that the stiffness of the whole mechanism must be considered. Jensen [17] provided a design with pulley coaxial with the pivot by fixing connecting links on the pulley wheel, as shown in Fig. 22(a). Tendon- or belt-actuated mechanisms are limited to small gripping forces and lead to increased friction and elasticity. In this section, an alternative approach to increase the stiffness of RCM mechanism is proposed by using redundant links, as detailed in Fig. 22(b).

The modified RCM mechanism is developed in Fig. 23(a). Due to its geometric characteristics, the dimensions of the mechanism are determined by simplifying the angulated link, as shown in Fig. 23(b).

A CFB linkage is used for passive adjustment due to the complicated contacting surfaces. References [18] and [19] propose a method to exactly duplicate the kinematic characteristics of a rigid-link four-bar mechanism by using the centrodes of the four-bar linkage. A mechanism that is overconstrained may be the best choice in problems of machine design when larger and variable loads must be sustained by means of mass and compliance, especially when the maintenance of mechanical accuracy is important [20]. The continuous trajectories of fixed and moving centrodes are the contact-aided surfaces for exactly duplicating the kinematic characteristics of rigid four-bar linkage.

According to kinematic analysis shown in Fig. 24, the drive angles of crank links are calculated as 18.06 deg and 5.65 deg for adjusting and angled modes individually. The larger angle of 18.06 deg should be selected to generate trajectories of fixed and moving centrodes of modified CFB mechanism.

Initial and final positions of the modified CFB mechanism are obtained, while trajectories of fixed and moving centrodes of the mechanism are generated. By copying corresponding files of the trajectories to 3D modeling software, the over-constrained mechanism considering dimensions of crank and connecting links is generated, as shown in Fig. 25(c).

The concept of underactuation [3] in robotic gipping with fewer actuators than DOFs allows the two fingers to adjust to irregular shapes without the need for complex control strategies and sensors. Differential mechanisms are used in robotic hands to provide underactuation, such as a movable pulley, seesaw mechanism, fluidic T-pipe, and planetary and bevel gear differentials [21]. In this paper, two fingers are driven by a movable pulley for irregular shapes, as shown in Fig. 26(a). This differential system is located at the palm of the gripper with the two ends of the tendon fixed symmetrically to the two pulley wheels, as shown in Fig. 26(b). The actuated power is distributed to the two fingers to facilitate gripping of noncentered or irregularly shaped objects.

The final 3D model of the meso-gripper with passive adjusting and angled gripping modes is shown in Fig. 27. The whole design was scaled up to 200% for better 3D printing. The gripping range for passive adjusting mode is 12–110 mm and for angled gripping mode 0–12 mm.

Most of the components of the gripper were manufactured using 3D printing. The material used for gripper body is polylactide thermoplastic (PLA). For the coupler link of CFB linkage, a silicone elastomer was used. The prototype of the gripper can grip different objects manually using both modes.

As shown in Fig. 28, objects used for testing the gripper’s gripping capabilities include regular shapes (cylinder, cone, and hexagon) and irregular shapes (flat, sharp, and pinecone). These objects varied from 0.5 to 105 mm in size and from 0.5 to 1000 g in mass. For different types of objects, the gripping approach was different, e.g., vertical gripping, horizontal gripping, passive adjusting mode gripping, or angled gripping. In all cases, the meso-gripper performed successfully.

This paper has introduced the concept and defined the range of the mesoscale for robotic gripper design as well as formalization of a methodology for such a gripping system. It has also demonstrated and validated this methodology through the design, analysis, and testing of a meso-gripper combining two integrated operational modes, passive adjusting and angled gripping.

Based on the gripping process of the human hand, the gripper was proposed by integrating the RCM mechanism and CFB linkage. A geometrical constraint method was used for dimensional synthesis of the mechanism. The kinematics of the synthesized mechanism was analyzed for model designed. The modified design considering stiffness and layout of the mechanism generated a new model. A 3D-printed manually operated prototype was tested for gripping different types of objects. The result shows the gripper with passive adjusting and angled gripping modes can achieve universal gripping within the mesoscale as small as 0.5 mm with a gripping load of as little as 0.5 g. The general purpose meso-gripper successfully addresses the gap identified in the introduction.

Future research will consider the gripping force during dimensional synthesis. Sensors attached to the fingertip will be used to measure the gripping force imposed on objects in the gripper control.

The authors would like to thank the Engineering and Physical Sciences Research Council (EPSRC), United Kingdom, for the support under Grant No. EP/K018345/1, and the first author would also like to thank the international Doctoral Training Partnership (DTP) from the EPSRC.

CFB =

cross four-bar

DOF =

degree-of-freedom

f =

friction

F =

degree-of-freedom

F_n =

force applied to finger tip

G_i =

mass of mechanism

G_m =

mass of object m

K_s =

stiffness of the spring

meso-gripper =

mesoscopic scale gripper

MIS =

minimally invasive surgery

n =

number of moving links

p_h =

number of higher pairs

p_l =

number of lower pairs

PLA =

polylactide thermoplastic

R_d =

damper ratio

RA =

rotating angle

RCM =

remote center of motion

Uy =

slide distance of point I about y axis

δ =

angle between two links in parallelogram

θ =

angle RCM link at initial and final position

Ψ =

angle between couple link in CFB mechanism and vertical line