2.1. Design of SAU-RFC finger

Figure 1 illustrates the representative examples of self-adaptive underactuated finger mechanism. Fig. 1(a) is a simplified illustration of the self-adaptive finger described in ref. [Reference Laliberté, Birglen and Gosselin9], which is composed of multiple four-bar linkage with two DOFs and one actuation. The structure conforms the object and generates discrete contact forces as the actuation link rotates. This finger is suitable for grasping large payload, but its point contacts apply uneven pressure and may limit range of motion. Another approach is to take advantage of the material’s flexibility, as shown in Fig. 1(b), a well-known soft gripper from SRT Co. [24]. This gripper is actuated by air pressure and continuously wraps around the object, distributing the contact force evenly, resulting in a more stable grasp. However, it comes at a cost in terms of grasping position precision and payload capacity. The ideal goal of compliant grasping is to apply uniform force regardless of the size or shape of the object. The concept of typical self-adaptive fingers yields useful inspiration for the development of SAU-RFC finger.

Figure 1. Conceptual examples of self-adaptive underactuated finger mechanism. (a) A self-adaptive underactuated finger made of multiple four-bar linkages and (b) a soft self-adaptive finger made of silicone.

SAU-RFC finger with two DOFs and rigid-flexible coupling structure is shown in Fig. 2. The flexion of finger is achieved by pulling of flexible belt, while the finger is extended by the reverse torque of the proximal and middle springs. In particular, the elastic coefficient of the middle spring is greater than that of the proximal spring to ensure that the relative position of the three phalanxes remains unchanged before the proximal phalanx touches the object. As the key part of the finger, the first end of flexible belt is connected to the auxiliary link, and the second end crosses the inside of the proximal phalanx and passes through the belt guide path to attach to the external traction mechanism. Note that the flexible belt not only acts as the power transmission carrier to achieve the motion of finger but also acts as a contact surface of object to form a large area envelope contact. Furthermore, it is inevitable that the flexible belt will always suffer from a relaxation problem over time because of its material characteristic. However, the possibility of slack or even breakage of a belt is far less than that of a thread/rope because it is made of interwoven threads. We finally select polyethylene fiber material, which not only has low elongation rate with 5% but also has the features of high wear resistance and high strength.

Figure 2. The mechanical structure of SAU-RFC finger. 1,2,3 – the proximal, middle, and distal phalanx; 4 – the flexible belt; 5 – the limit baffle; 6 – the auxiliary link; 7 – the object; 8 – the middle spring; 9 – the proximal spring; 10 – the finger base; 11 – the bearings; 12 – the soft cover; 13 – the upper base; 14 – the threaded lead screw pair; 15 – the fixer; 16 – the servomotor-I; and 17 – the gear pair.

In addition, the surface of the distal phalanx is also covered with soft material. The traction mechanism is mainly composed of a threaded lead screw pair, a fixer, and a servomotor-I, in which the lead screw is fitted together with the output shaft of servomotor-I, the nut attaches to the first end of fixer, and the second end of fixer is connected to the flexible belt. Based on the connection relationship, the rotary motion of servomotor-I is converted to the pulling motion of belt. Benefiting from the underactuated structure and the actuator is far away from the finger, SAU-RFC finger has a greater advantage in compact design. Due to the flexibility of belt, SAU-RFC finger has high resistance to external shocks and grasping stability.

2.2. Design of SAU-RFC hand

Basing on SAU-RFC Finger, SAU-RFC hand is designed, as shown in Fig. 3. It has three fingers with same structure, a palm base, and four actuators. The palm base has one DOF and is equipped with a side-turning (ST) mechanism. Furthermore, these three fingers can be divided into two ST fingers and one non-side-turning (NST) finger, and the ST fingers can perform synchronous reverse rotation laterally with each other. It is realized by the ST mechanism installed on the palm base, which is composed of a gear pair and a servomotor-II. The gear pair contains an input shaft and two output shafts, and the input shaft is fitted together with the output shaft of servomotor-II, while the output shafts are respectively attached to the corresponding ST fingers. In addition, the rotation angle range of the proximal, the middle, and the distal joint are $0^{\circ } \sim 100^{\circ }$ , $0^{\circ } \sim 75^{\circ }$ , and $0^{\circ } \sim 90^{\circ }$ , respectively. The lateral rotation angle range of ST finger is $0^{\circ } \sim 60^{\circ }$ . In the interest of matching to the outline of most objects, structural parameters are selected and modified based on the size of human finger and the extreme position of phalanxes, which are shown in Table I.

Table I. The structure parameters of SAU-RFC hand.

Figure 3. The SAU-RFC hand. (a) Front view and (b) side view.

2.3. Analysis of grasping process

Figure 4 shows the interaction between a single SAU-RFC finger and the object over the grasping process. Fig. 4(a) is the initial state of the finger, when the grasping action is started, and the flexible belt starts to pull the finger under the action of the traction mechanism. On the one hand, due to the restriction of the middle spring, the middle phalanx will not rotate around the middle joint, and under the influence of the four-bar linkage, the distal phalanx also will not rotate around the distal joint. On the other hand, the elastic coefficient of the middle spring is greater than that of the proximal spring, causing the proximal phalanx rotates around the joint shaft and other phalanxes keep the relative position unchanged.

Figure 4. The grasping process of SAU-RFC finger.

Therefore, before the finger or the flexible belt touches the object, the entire finger will rotate around the proximal joint to form a larger grasping range, as shown in Fig. 4(b). After the flexible belt contacts the object for the first time, it will automatically deform according to the shape of the object to form an envelope contact with the object. At the same time, due to the obstacle of the object, the proximal phalanx no longer rotates around the proximal joint, but the flexible belt will continue to pull the finger and gradually overcome the resistance of the middle spring. Then, under the effect of the four-bar linkage, the middle and the distal phalanxes are driven to rotate around their joint shaft, respectively, as shown in Fig. 4(c). When the flexible belt continues to pull, each phalanx will automatically adjust its grasping angle according to the shape of the object until the finger reaches a more stable grasping posture, and finally completing the grasping action, as shown in Fig. 4(d), the automatic adjustment of grasping angle also happens to reflect the SAU-RFC finger’s self-adaptive effect.

In particular, Fig. 5 presents the coupled four-bar linkage between the middle and distal joints, which has the following relationship:

(1) \begin{equation} \begin{cases} \begin{aligned} L&=\sqrt{l_{1}^{2}+l_{2}^{2}-2l_{1}l_{2}\textrm{cos}\alpha _{1}} \\ \\[-9pt]A&=\textrm{arccos}\left (\frac{L^{2}+l_{4}^{2}-l_{3}^{2}}{2Ll_{4}}\right ) \\ \\[-9pt] B&=\textrm{arccos}\left (\frac{L^{2}+l_{2}^{2}-l_{1}^{2}}{2Ll_{2}}\right ) \\ \\[-9pt] \alpha _{2}&=\pi -\alpha _{3}-A+B.\\ \end{aligned} \end{cases} \end{equation}

where, $l_{1}$ , $l_{2}$ , $l_{3}$ , and $l_{4}$ are the length of each bar in the four-bar linkage, respectively, $\alpha _{1}$ is the rotation angle of the middle joint, $\alpha _{2}$ is the rotation angle of the distal joint, and $\alpha _{3}$ is the fixed angle determined by the mechanical structure of the distal phalanx.

Figure 5. The four-bar linkage of SAU-RFC finger. (a) 2D view and (b) 3D view.

The motion ratio of $\alpha _{2}$ to $\alpha _{1}$ and the angle $\alpha _{3}$ of the distal phalanx are designed according to the average grasping posture of the human hand. In this way, the change curve for the rotation angle of the distal joint with respect to that of the middle joint is drawn, which have an approximate linear relationship, as shown in Fig. 6. For the convenience of the calculation of inverse kinematics, linear fitting is performed on them (with $R^2 = 0.993$ ), which is expressed as:

(2) \begin{equation} \alpha _{2}=0.99\alpha _{1}+6.5. \end{equation}

In order to simplify the calculation, the relationship between and can be approximately replaced by Eq. (2).

Figure 6. The grasping process of SAU-RFC finger.

Moreover, note that the flexible belt forms a large area of contact with the object and has more contact points, which promotes uniform contact force distribution. Under the combined action of the joint spring and the flexible belt, the SAU-RFC finger produces a good self-adaptive grasping effect, which means that it can achieve the enveloping grasp for objects of different shapes and sizes.

3.1. Kinematics

As depicted in Fig. 7, the reference frames of each phalanx is established by the Denavit−Hartenberg method. $x_{0}y_{0}z_{0}$ is the fixed coordinate $\{0\}$ and the others are moving coordinates system. The homogeneous transformation matrix $^{1}A_{i}^{i-1}(i=1,2,3)$ of the phalanxes for NST finger can be obtained:

(3) \begin{equation} ^{1}A_{i}^{i-1}= \begin{bmatrix}\textrm{cos}\theta _{i} &\quad -\textrm{sin}\theta _{i} &\quad 0 &\quad a_{i}\textrm{cos}\theta _{i} \\ \\[-8pt]\textrm{sin}\theta _{i} &\quad\textrm{cos}\theta _{i} &\quad 0 &\quad a_{i}\textrm{sin}\theta _{i} \\ \\[-8pt]0 &\quad 0 &\quad 1 &\quad 0 \\ \\[-8pt]0 &\quad 0 &\quad 0 &\quad 1 \\ \end{bmatrix}. \end{equation}

Figure 7. D-H reference frames of SAU-RFC finger. (a) NST finger and (b) ST finger.

And the transformation matrix $^{2}A_{i}^{i-1}(i=1,2,3,4)$ of the phalanxes for ST finger are determined by:

(4a) \begin{equation} ^{2}A_{1}^{0}= \begin{bmatrix} -\textrm{sin}\theta _{1}&\quad 0 &\quad\textrm{cos}\theta _{1} &\quad -a_{1}\textrm{sin}\theta _{1} \\ \\[-8pt] \textrm{cos}\theta _{1} &\quad 0 &\quad \textrm{sin}\theta _{1} &\quad a_{1}\textrm{cos}\theta _{1} \\ \\[-8pt] 0 &\quad 1 &\quad 0 &\quad 0 \\ \\[-8pt] 0 &\quad 0 &\quad 0 &\quad 1 \\ \end{bmatrix} \end{equation}

and

(4b) \begin{equation} ^{2}A_{i}^{i-1}= \begin{bmatrix} \textrm{cos}\theta _{i} &\quad -\textrm{sin}\theta _{i} &\quad 0 &\quad a_{i}\textrm{cos}\theta _{i} \\ \\[-8pt]\textrm{sin}\theta _{i} &\quad \textrm{cos}\theta _{i} &\quad 0 &\quad a_{i}\textrm{sin}\theta _{i} \\ \\[-8pt]0 &\quad 0 &\quad 1 &\quad 0 \\ \\[-8pt]0 &\quad 0 &\quad 0 &\quad 1 \\ \end{bmatrix}\;i=2,3,4. \end{equation}

Then, the transformation matrix $^{1}T_{3}^{0}$ of NST finger’s fingertip relative to the coordinate $\{0\}$ is expressed by:

(5) \begin{align} ^{1}T_{3}^{0} & ={^{1}A_{1}^{0}}\;{^{1}A_{2}^{1}}\;{^{1}A_{3}^{2}} \nonumber\\[3pt]& = \begin{bmatrix} \textrm{cos}\sum \limits _{i=1}^{3}{\theta _{i}} &\quad -\textrm{sin}\sum \limits _{i=1}^{3}{\theta _{i}} &\quad 0 &\quad \sum \limits _{i=1}^{3}{\left (a_{i}\textrm{cos}\sum \limits _{j=1}^{i}{\theta _{j}}\right )}\\ \\[-7pt]\textrm{sin}\sum \limits _{i=1}^{3}{\theta _{i}} &\quad \textrm{cos}\sum \limits _{i=1}^{3}{\theta _{i}} &\quad 0 &\quad \sum \limits _{i=1}^{3}{\left (a_{i}\textrm{sin}\sum \limits _{j=1}^{i}{\theta _{j}}\right )}\\ \\[-7pt]0 &\quad 0 &\quad 1 &\quad 0 \\ \\[-7pt]0 &\quad 0 &\quad 0 &\quad 1 \\ \end{bmatrix}. \end{align}

Moreover, the transformation matrix $^{2}T_{4}^{0}$ of the ST finger’s fingertip relative to the coordinate $\{0\}$ is calculated as:

(6) \begin{equation} \begin{aligned} ^{2}T_{4}^{0} &={^{2}A_{1}^{0}}\;{^{2}A_{2}^{1}}\;{^{2}A_{3}^{2}}\;{^{2}A_{4}^{3}} \\ &= \begin{bmatrix} -\textrm{sin}{\theta _{1}}\textrm{cos}\sum \limits _{i=2}^{4}{\theta _{i}} &\quad \textrm{sin}{\theta _{1}}\textrm{sin}\sum \limits _{i=2}^{4}{\theta _{i}} &\quad \textrm{cos}{\theta _{1}} &\quad -\textrm{sin}{\theta _{1}}{\left (\sum \limits _{i=2}^{4}{\left (a_{i}\textrm{cos}\sum \limits _{j=2}^{i}{\theta _{j}}\right )}+a_{1}\right )}\\ \\[-7pt]\textrm{cos}{\theta _{1}}\textrm{cos}\sum \limits _{i=2}^{4}{\theta _{i}} &\quad -\textrm{cos}{\theta _{1}}\textrm{sin}{\sum \limits _{i=2}^{4}{\theta _{i}}} &\quad \textrm{sin}{\theta _{1}} &\quad \textrm{cos}{\theta _{1}}{\left (\sum \limits _{i=2}^{4}{\left (a_{i}\textrm{cos}\sum \limits _{j=2}^{i}{\theta _{j}}\right )}+a_{1}\right )}\\ \\[-7pt]\textrm{sin}\sum \limits _{i=2}^{4}{\theta _{i}} &\quad \textrm{cos}\sum \limits _{i=2}^{4}{\theta _{i}}&\quad 0 &\quad \sum \limits _{i=2}^{4}{\left (a_{i}\textrm{sin}\sum \limits _{j=2}^{i}{\theta _{j}}\right )}\\ \\[-7pt]0 &\quad 0 &\quad 0 &\quad 1 \\ \end{bmatrix}.\\ \end{aligned} \end{equation}

Specifically, for the NST finger, the matrix $^{1}T_{3}^{0}$ is the function of joint variables $\theta _{i}(i=1,2,3)$ (Eq. 5), and the pose of fingertip in the coordinate $\{0\}$ is expressed as the first three values in the last column of $^{1}T_{3}^{0}$ . Hence, the pose of fingertip is solved by measuring the value of variables $\theta _{i}(i=1,2,3)$ , as mentioned in Section 2, $\theta _{2}$ and $\theta _{3}$ have a linear relationship (Eq. 2), so it is determined by the variables $\theta _{i}(i=1,2)$ . Similar analysis can be done for ST finger. Figure 8 presents the workspaces of NST and ST fingers’ fingertip, and it is observed that the SAU-RFC hand has a large workspace.

Figure 8. The workspace of SAU-RFC fingers’ fingertip. (a) NST finger and (b) ST finger.

3.2. Kinetics

Once the flexible belt of SAU-RFC finger touches the object, it will cause several contact forces in different orientations and sizes. It is complicated to analyze these forces one by one. Therefore, the composition of forces is adopted to make all the component forces of object to the flexible belt synthesize an equivalent force at the midpoint of the flexible belt. For the convenience of analysis, assuming that the outline of the object being grasped is cylindrical. The grasping force diagram of SAU-RFC finger is shown in Fig. 9, where,

Figure 9. The grasping force diagram of SAU-RFC finger.

$f_{1}$ – the equivalent external force on the proximal phalanx (or the flexible belt), $\textrm{N}$ ;

$f_{2},f_{3}$ – the external force on the middle and distal phalanxes, $\textrm{N}$ ;

$\theta _{1},\theta _{2},\theta _{3}$ – the rotation angle of the proximal, middle, and distal phalanxes, $^{\circ }$ ; note that $\theta _{2}$ and $\theta _{3}$ are associated with $\alpha _{1}$ and $\alpha _{2}$ in Eq. 2: $\alpha _{2}=\theta _{3}-\theta _{2},\alpha _{1}=\theta _{2}-\sigma$ .

$a_1,a_2,a_3$ – the length of the proximal, middle, and distal phalanxes, $\textrm{mm}$ ;

$d_1,d_2,d_3$ – the distance between the joint shaft and the force contact point of the proximal, middle, and distal phalanxes, respectively, $\textrm{mm}$ ;

$T_m$ – the output torque of servomotor, $\textrm{N}\cdot \textrm{mm}$ ;

$\beta$ – the rotation angle of servomotor, $^{\circ }$ ;

$\alpha$ – the rotation angle of $l_{3}$ -bar in the four-bar linkage relative to the proximal phalanx, $^{\circ }$ ;

$r_1,r_2$ – the distance between the first end of the flexible belt and the proximal phalanx joint shaft, and the $l_{3}$ -bar shaft, respectively, $\textrm{mm}$ ;

$M_1,M_2$ – the initial torque of the proximal and middle spring, $\textrm{N}\cdot \textrm{mm}$ ;

$k_1,k_2$ – the stiffness coefficient of the proximal and middle spring, $\textrm{N}\cdot \textrm{mm}/^{\circ }$ .

The middle and the distal phalanxes are coupling connected by the four-bar linkage, then $f_{2}$ and $f_{3}$ are determined by:

(7) \begin{equation} f_{2}d_{2}=f_{3}\left (d_{3}+a_{2}\textrm{cos}\theta _{3}\right ). \end{equation}

Then the expression between the force and position is calculated from the principle of virtual work:

(8) \begin{equation} \begin{bmatrix} f_{1}& f_{2} \\ \end{bmatrix} \begin{bmatrix} -d_{1}& 0 \\ \\[-9pt] K & N \end{bmatrix} \begin{bmatrix} \delta \theta _{1} \\ \\[-9pt] \delta \theta _{2} \\ \end{bmatrix}= \begin{bmatrix} k_{1}\theta _{1}+M_1& k_{2}\theta _{2}+M_2 & -T_m\\ \end{bmatrix} \begin{bmatrix} \delta \theta _{1} \\ \\[-9pt] \delta \theta _{2} \\ \\[-9pt] \delta \beta \\ \end{bmatrix}. \end{equation}

where, $K=-a_{1}\textrm{cos}{\left (\theta _{2}-\theta _1\right )}-\dfrac{d_{2}a_{1}\textrm{cos}{\left (\theta _{3}-\theta _{2}-\theta _1\right )}-d_{2}d_3}{d_{3}+a_{2}\textrm{cos}\theta _3}$ and $N=-d_{2}-\dfrac{d_{2}a_{2}\textrm{cos}{\left (\theta _{3}-\theta _{2}\right )}+2d_{2}d_3}{d_{3}+a_{2}\textrm{cos}\theta _3}$ .

In addition, $\delta \beta$ , $\delta \theta _{1}$ , and $\delta \theta _{2}$ have the following conversion relationship:

(9) \begin{equation} \begin{cases} \begin{aligned} {} {} {}\delta\beta/360&=\delta x/s \\ \\[-10pt] {} {} {}\delta x&\approx r_{1}\delta\theta_{1}+r_{2}\delta\alpha \\ \\[-10pt] {} {} {}\alpha-\theta_{1}&=-0.007\left(\theta_{2}-\theta_{1}\right)^2+1.7\left(\theta_{2}-\theta_{1}\right)+21 \\ \\[-10pt] {} {} {}\delta\alpha&=\left(-0.014\left(\theta_{2}-\theta_{1}\right)+1.7\right)\delta\theta_{2}+\left(0.014\left(\theta_{2}-\theta_{1}\right)-1.7\right)\delta\theta_{1}.\end{aligned} \end{cases} \end{equation}

where $\delta x$ is the tiny reduction of flexible belt and $s$ is the lead of the lead screw pair. Function $\alpha (\theta _2)$ is obtained by quadratic fitting based on the geometric parameters, as shown in Fig. 10. Then the right half of Eq. (8) is expressed as:

(10) \begin{equation} \begin{bmatrix} k_{1}\theta _{1}+M_1&\quad k_{2}\theta _{2}+M_2 &\quad -T_m\\ \end{bmatrix} \begin{bmatrix} 1& \quad 0 \\ \\[-8pt]0&\quad 1 \\ \\[-8pt]P&\quad Q \\ \end{bmatrix} \begin{bmatrix} \delta \theta _{1} \\ \\[-8pt] \delta \theta _{2} \\ \end{bmatrix}. \end{equation}

where $\left. P=360\left (r_{1}+r_{2}\left (0.014(\theta _{2}-\theta _1)-0.7\right )\right ) \middle/ s \right .$ and $\left. Q=360r_{2}\left (-0.014(\theta _{2}-\theta _1)+1.7\right )\middle/s \right .$ . Substitu-ting Eq. (10) into Eq. (8) yields

(11) \begin{equation} \begin{bmatrix} f_{1}&\quad f_{2} \\ \end{bmatrix}= \begin{bmatrix} k_{1}\theta _{1}+M_1&\quad k_{2}\theta _{2}+M_2 &\quad -T_m\\ \end{bmatrix} \begin{bmatrix} -\dfrac{1}{d_1} &\quad 0 \\ \\[-7pt] \dfrac{K}{d_{1}N} &\quad \dfrac{1}{N} \\ \\[-7pt] \dfrac{KQ}{d_{1}N}-\dfrac{P}{d_{1}}&\quad \dfrac{Q}{N} \\ \end{bmatrix}. \end{equation}

Finally, assuming the necessary parameters: $k_{1}=0.05\,\textrm{N}\cdot \textrm{mm}/^{\circ }$ , $k_{2}=0.14\,\textrm{N}\cdot \,\textrm{mm}/^{\circ }$ , $a_{1}=51 \,\textrm{mm}$ , $a_{2}=36 \,\textrm{mm}$ , $d_{1}=25 \,\textrm{mm}$ , $d_{2}=22 \,\textrm{mm}$ , $d_{3}=14 \,\textrm{mm}$ , $T_{m}=1 \,\textrm{N}\cdot \textrm{mm}$ , $M_{1}=0.2 \,\textrm{N}\cdot \textrm{mm}$ , $M_{1}=0.54 \,\textrm{N}\cdot \textrm{mm}$ , $r_{1}=59 \,\textrm{mm}$ , and $r_{2}=16 \,\textrm{mm}$ . The relationship between the grasping force and the rotation angle of the proximal, middle, and distal phalanxes is illustrated in Fig. 11, in which the left column is the 3D view and the right column is the corresponding 2D view. In this figure, the first row is the 3D view of the grasping force, while the second row is the corresponding 2D view. Figure 11 indicates that when $\theta _1$ increases, the grasping force of $f_1$ the proximal phalanx decreases, while the grasping forces $f_2$ of the middle phalanx and $f_3$ of the distal phalanx increase. When $\theta _2$ increases, the grasping force $f_1$ gradually increases, while the grasping forces $f_2$ and $f_3$ gradually decrease. This is because the elastic force of the spring will increase with the increase of joint angle, and the three phalanxes share the power of the servomotor-I, while its output torque is constant.

Figure 10. The relationship between $\alpha$ and $\theta _{2}$ .

Figure 11. The relationship between the grasping force and the rotation angle of joints.

3.3. Contact force distribution model of flexible belt

In order to further study the change in the contact force distribution caused by replacing the rigid proximal phalanx with the flexible belt, a contact force distribution model is established. As shown in Fig. 12, the flexible belt is regarded as a rigid serial multilink with $n$ joints, which is cut and subdivided into $n+1$ phalanx units. The joints are numbered in the order from 1 to $n$ , and the phalanx units are numbered in the order from 0 to $n$ . The joint $i$ connects the unit $i-1$ and the unit $i$ and drives the unit $i$ . The phalanx unit 0 is fixed, which is equivalent to the “base” of the flexible belt, while each of the remaining phalanx units forms a contact force with the object, that is, $F_1,F_2,F_3,\cdots,F_n$ .

Figure 12. The model of contact force distribution.

To simplify the calculation, let each force form a torque at point $O$ and define some parameters of the finger as follows:

$T_i$ – the moment caused by the contact force $F_i$ on phalanx unit $i$ , $\textrm{N}\cdot \textrm{mm}$ ;

$T_f$ – the equivalent moment caused by the tension $f$ of flexible belt, $\textrm{N}\cdot \textrm{mm}$ ;

$T_{md}$ – the equivalent moment caused by the contact force on middle and distal phalanxes, $\textrm{N}\cdot \textrm{mm}$ ;

$T_{st}$ – the equivalent moment caused by the elastic force of proximal and middle springs, $\textrm{N}\cdot \textrm{mm}$ ;

$l_i$ – the length of phalanx unit $i$ , $\textrm{mm}$ ;

$e_i$ – the distance between acting point of contact force $F_i$ and joint $i$ , $\textrm{mm}$ ;

$\lambda _i$ – the rotation angle of joint $i$ relative to joint $i-1$ , $^{\circ }$ .

Then, establish the moment equilibrium of flexible belt:

(12) \begin{equation} \begin{aligned} T_{f}&=T_{md}+T_{st}+F_{1}e_{1}+F_{2}\left (e_2+l_{1}\textrm{cos}\lambda _2\right )+F_{3}\left (e_{3}+l_{2}\textrm{cos}\lambda _{3}+l_{1}\textrm{cos}\left (\lambda _{3}+\lambda _2\right )\right )\\[3pt] &\cdots +F_{n}\left (e_{n}+l_{n-1}\textrm{cos}\lambda _{n}+l_{n-2}\textrm{cos}\left (\lambda _{n}+\lambda _{n-1}\right )+\cdots +l_{1}\textrm{cos}\left (\lambda _{n}+\lambda _{n-1}+\cdots +\lambda _{2}\right )\right ).\\ \end{aligned} \end{equation}

By introducing $T_i$ , the above equation is further expressed as:

(13) \begin{equation} \begin{cases} \begin{aligned} T_f&=\sum \limits _{i=1}^{n}{T_i}+T_{md}+T_{st} \\ \\[-8pt] T_i&= F_{i}\left (e_{i}+l_{i-1}\textrm{cos}\lambda _{i}+l_{i-2}\textrm{cos}\left (\lambda _{i}+\lambda _{i-1}\right )+\cdots +l_{1}\textrm{cos}\left (\lambda _{i}+\lambda _{i-1}+\cdots +\lambda _{2}\right )\right ).\\ \end{aligned} \end{cases} \end{equation}

The value of $n$ is close to infinity and the material of flexible belt is uniform, then the length of each phalanx unit can be regarded as the same. Meanwhile, assuming the contact point of each phalanx unit $i$ to the object is at the joint $i+1$ , and the flexible belt takes an arc-shaped grasping posture, that is,

(14) \begin{equation} \begin{cases} \begin{aligned} l_1&=l_2=\cdots =l_n \\ e_i&=l_i\\ \lambda _1&=\lambda _2 = \cdots =\lambda _{n}. \end{aligned} \end{cases} \end{equation}

Thus, Eq. (13) is calculated as:

(15) \begin{equation} T_i=F_{i}\left (e_{i}+l_{i}\textrm{cos}\lambda _{i}+l_{i}\textrm{cos}(2\lambda _i)+\cdots +l_{i}\textrm{cos}\left (\left (i-1\right )\lambda _{i}\right )\right ). \end{equation}

Aiming at the four cases where $l_i$ is $0.01$ , $0.05$ , $0.10$ , and $0.15 \textrm{mm}$ , respectively, the relationship between $T_i$ and the parameter $F_i,\lambda _i$ is plotted in Fig. 13. For the length parameter $l_i$ of phalanx unit, Fig. 13(a)−(d) show that $T_i$ increases when $l_i$ increases from $0.01$ to $0.15\,\textrm{mm}$ , and its maximum value increases from about $0.5$ to $8\,\textrm{N} \cdot \textrm{mm}$ . Therefore, when $l_i$ is small enough, the contact moment of flexible belt can be lower than $0.5\,\textrm{N} \cdot \textrm{mm}$ , and the difference in contact force among phalanx units is also smaller, which means that the flexible belt has a more uniform force distribution than the traditional rigid finger structure.

Figure 13. The relationship between the parameter $T_i$ and $F_i,\lambda _i$ . (a) $l_i=0.01\,\textrm{mm}$ ; (b) $l_i=0.05\,\textrm{mm}$ ; (c) $l_i=0.10\,\textrm{mm}$ ; and (d) $l_i=0.15\,\textrm{mm}$ .

Meanwhile, every subgraph in Fig. 13 shows that $T_i$ increases with the increase of $F_i$ , while $T_i$ decreases with the increase of $\lambda _i$ . It indicates that a higher flexible belt contact moment is required when grasping the large-mass and large-volume object than grasping the small-mass and small-volume object, which means that enveloping grasp (power grasp) requires a higher driving torque than pinching grasp (precise grasp).

4.1. Experiments of grasping mode

In order to illustrate the rationality of the mechanical structure, a series of grasping experiments on SAU-RFC hand are performed. The prototype of SAU-RFC hand and its various grasping configurations are shown in Fig. 14. SAU-RFC hand is made of mixed materials such as resin (white part) and nylon (black part). Remark that it can automatically select the grasping mode according to the distance of objects from the palm: enveloping grasp or pinching grasp.

Figure 14. The prototype and grasping configuration of SAU-RFC hand. (a) configuration-I; (b) configuration-II; and (c) configuration-III.

When the object is close to the palm, enveloping grasp is performed, which is shown in Fig. 15. As is to see, before touching the object, the three phalanxes rotate around the proximal joint at a constant relative angle, after the proximal phalanx touches the object first, the middle and distal phalanxes start to rotate under the action of the self-adaptive mechanism until they contact the object. Note that each phalanx automatically adjusts its angle according to the contour of object until the fingers reach a more stable grasping posture, which just reflects the self-adaptivity.

Figure 15. The enveloping grasp process of SAU-RFC hand.

When the object is far from the palm, consecutive snapshots of pinching grasp are given in Fig. 16. The middle spring makes the middle and distal phalanxes remain upright, so that SAU-RFC hand can pinch the object tightly. Meanwhile, the soft cover on the surface of the distal phalanx increases the friction.

Figure 16. The pinching grasp process of SAU-RFC hand.

4.2. Experiments of grasping objects

The grasping versatility of SAU-RFC hand is verified on set of objects to include the flexible belt affects grasping performance, considering the grasping benchmark from other studies. It is mainly based on the Yale-CMU-Berkeley (YCB) object set [Reference Calli, Walsman, Singh, Srinivasa and Abbeel25], which comprises 60 objects in 5 categories, including food items and tools.

Often, there is variation added to the object position and orientation in the realistic pick-and-place environment, which is more representative of practical applications. Therefore, experiment is conducted with YCB benchmark involving 60 unique objects grasped 10 times in random object orientation, as shown in Fig. 17. This figure also presents how the rotation of ST fingers and the grasping mode (pinching grasp/enveloping grasp) is employed to help grasp objects depending on their geometry. During the 10 times of grasping, the random placement of objects will cause some unsuitable orientations, resulting in the grasping failure, and the unsuccessful results are summarized in Table II.

Table II. Statistics of unsuccessful grasping results of SAU-RFC hand.

Figure 17. Examples of SAU-RFC hand grasping various objects from the YCB object set.

SAU-RFC hand successfully grasped the majority of the objects only failing on the smaller flat objects and the heavier ones, and the successful/ unsuccessful objects are shown in Fig. 18. The effectiveness of pinching grasp is proved on small objects, and enveloping grasp can hold the object tightly for large objects. Overall, SAU-RFC hand has the ability to grasp objects of various shapes self-adaptively benefiting from the compliance of belt. But the flat objects prove a significant challenge for this hand, because it is no suitable grasping posture to match the narrow edge of flat objects. Similarly, the flexibility of the flexible belt results in power grasp being more subject to failure when facing sharp or heavy (more than $2.8\,\textrm{kg}$ ) objects. Nevertheless, it does show the physical properties of objects that the SAU-RFC hand is likely to struggle with, indicating limitations in the design.

Figure 18. (a) Items from the YCB object set that the SAU-RFC hand was able to grasp in 10 times and (b) items from the YCB object set that the SAU-RFC hand was unable to grasp in 10 times.

4.3. Validation of grasping stability

Stability is an important indicator to measure the grasping capability of the robot hand; thus, disturbance experiments are carried out after grasping by introducing random perturbations in the hand pose and external collision. The hand pose is perturbed by shaking the object to the same degree from random directions. Figure 19 shows the snapshot of perturbation in the hand pose after grasping different objects. The left column is the selected four objects, which are cylinder bottle, electric drill, stapler, and multimeter from top to bottom, while the right column is the video sequence of the object state. The tests show that object weighing below $2.8\,\textrm{kg}$ can be grasped without sliding even when disturbed by random hand pose, which demonstrates good stability.

Figure 19. Disturbance experiments after grasping different objects with SAU-RFC hand.

The external collision is caused by a bottle falling freely at different heights and hitting the fixed position of the hand in a positive direction. Specifically, SAU-RFC hand holding the object is placed horizontally, a $400\,\textrm{g}$ bottle is freely fallen within a height of $20-80\,\textrm{cm}$ above it, and the initial height is recorded when the object slips off. Experiment presents that the fingers are severely deformed and the object slips off when the bottle falls freely about $45\,\textrm{cm}$ above the hand and calculate the impulse value as $1.19\,\textrm{kg} \cdot \textrm{m/s}$ by momentum theorem, indicating SAU-RFC finger has high resistance to external shocks. If the fingers are fabricated by stronger material, the impulse will be higher. Overall, the experiments demonstrate that the novel mechanical design of the SAU-RFC hand has a good adaptability and stability of grasping.