Nomenclature

$\rho$

Radius ratio or torque ratio

$\tau _i$

Torque produced by the tendon at the $i^{th}$ joint

$\theta _i$

Joint angle at the $i^{th}$ joint

$b$

Base length or palm width

$b_r$

Non-dimensionalized base distance ( $=b/r_1$ )

$c_i$

Contact length at the $i^{th}$ contact

$c_{ir}$

Non-dimensionalized contact length $c_{ir}=c_i/r_1$

$d_x$ , $d_y$

Displacement of the object in the x and y direction, respectively

$f_i$

Contact force at the $i^{th}$ contact

$f_{a/b/c/d}$

Non-dimensionalized contact forces ( $f_a=f_1/T$ ) and so on

$K_g$

Grasp stiffness matrix

$k_i$

Joint stiffness at the $i^{th}$ joint

$k_{a/b/c/d}$

Non-dimensionalized joint stiffness ( $k_a=k_1/(Tr_1)$ ) and so on

$l_i$

Link length of the $i^{th}$ link

$l_{ir}$

Non-dimensionalized link length ( $l_{ir}=l_i/r_1$ )

$O_r$

Non-dimensionalized object radius ( $=r_o/r_1$ )

$Q_k$

Grasp stiffness measure

$r_i$

Radius of the $i^{th}$ joint pulley

$r_o$

Radius of the object

$T$

Tendon tension

$t$

Distance between line joining joints and contact point (link thickness)

2. System description

In this work, we study an underactuated, tendon-driven, two-fingered hand shown in Fig. 1(a). For our analysis, the hand grasps an object in the horizontal plane. In this case, the gravity does not act in the plane of the grasp. In the condition when the hand is facing up or down, an external (gravitational) force should be considered on the object. Single tendon runs through both the fingers. Both the fingers are, thus, actuated with tendons having equal tension as shown in Fig. 1(a). Tendons are routed over joint pulleys as shown. The design of hand is similar to the design of SDM hand [Reference Dollar and Howe3]. The mechanism is a constant torque ratio mechanism. The ratio of the torque created by the tendon at the two joints of a finger is constant (independent of the joint angles).

Figure 1. Grasp geometry.

We assume frictionless contact between the tendon and the joint pulleys or between the fingers and the object. The interaction forces between the finger and object ( $f_1, f_2, f_3, f_4$ ) are therefore perpendicular to the finger surface. The fingers have an active close mechanism – they flex when the tendon is pulled and extend when the tendon is released (due to action of the joint spring). When the fingers are not interacting with an object, the free movement of the finger is governed by the pulley radii and the stiffness of the springs. While Fig. 1(a) shows the two-fingered hand without joint limits, Fig. 5 shows the hand design with joint limits at the proximal joints of the fingers.

A schematic of the hand grasping an object is shown in Fig. 1(b). In this study, we have used a circular object. A circular object simplifies the analysis as it can be specified with one parameter (radius). At the same time, a circular object provides useful insight into the nature of the grasp for objects of different sizes. Thus, we found circular object to be an adequate assumption.

Grasp of the object takes place as follows: hand is positioned near the object with fingers open, fingers start closing and the proximal phalanx makes contact with the object, the finger continues to close and distal phalanx makes contact with the object, the hand pushes the object within the grasp to a position of equilibrium for both the hand and the object.

In certain situations, the grasp may fail due to the fingers slipping on the object. We detect such cases by checking for a feasible equilibrium of hand and object. To analyze this, we develop the grasp model to find the interaction forces between the fingers and the object. Grasp modeling is presented in the following section.

3. Grasp modeling

This section presents the procedure for computing the contact forces and the net force on the object. We consider the case of frictionless contact between the fingers and the object. With the frictionless assumption, we perform a worst-case analysis of the underactuated hand. The frictionless case also provides insights into the design features that lead to an unstable grasp.

To compute the contact forces, we consider the object fixed in space at a known position and find the configuration of the hand in contact with the object. Using the configuration of the hand, obtained from geometry, we compute the contact forces required for equilibrium of the fingers. Using the value of the contact forces, applied to the object, we compute the net force on the object.

The reference frame ( $x-y$ ) attached to the palm of the hand is shown in Fig. 1(b). For the position of the object shown, we can use geometry to determine the contact locations ( $c_1$ , $c_2$ , $c_3$ , and $c_4$ ) and the joint angles ( $\theta _{1}$ , $\theta _{2}$ , $\theta _{3}$ , $\theta_4$ ). For brevity, we do not show the full expressions here.

The expressions for equilibrium of the fingers are Eqs. (1) to (4):

(1) \begin{align} \tau _{1} =Tr_1 =f_{1}c_{1}+f_{2}(c_{2}+l_{1}cos\theta _{2})+k_{1}\theta _{1} \end{align}

(2) \begin{align} \tau _{2} = Tr_2 =f_{2}c_{2}+k_{2}\theta _{2} \end{align}

(3) \begin{align} \tau _{3} = Tr_3 =f_{3}c_{3}+f_{4}(c_{4}+l_{3}cos\theta _{4})+k_{3}\theta _{3} \end{align}

(4) \begin{align} \tau _{4} =Tr_4 =f_{4}c_{4}+k_{4}\theta _{4} \end{align}

In the equations above, $f_1$ , $f_2$ , $f_3$ , and $f_4$ are the contact forces on the links 1, 2, 3, and 4 respectively. The torques at the joint ( $\tau _1$ , $\tau _2$ , $\tau _3$ , and $\tau _4$ ) are produced due to the tension in the tendon ( $T$ ). Torque at the $i^{th}$ joint produced due to tendon tension is given by: $\tau _i=Tr_i$ . Here, $r_i$ is the radius of the $i^{th}$ joint pulley. From the preceding expression, we find that the joint torque produced due to tendon tension is independent of the configuration of the finger. The angular stiffness of the springs at the joints are $k_1$ , $k_2$ , $k_3$ , and $k_4$ .

The next step in modeling of the system is to non-dimensionalize the preceding expressions. We divide Eqs. (1), (2), (3), and (4) by the term $Tr_1$ to obtain the following expressions (Eqs. (5) to (8)):

(5) \begin{align} 1 =\frac{f_{1}}{T}\frac{c_{1}}{r_{1}}+\frac{f_2}{T}\left ( \frac{c_2}{c_1} + \frac{l_1}{c_1}cos\theta _{2} + \frac{k_1}{Tr_1}\theta _{1} \right ) \end{align}

(6) \begin{align}\frac{r_2}{r_1} =\frac{f_2}{T}\frac{c_2}{r_1}+\frac{k_2}{Tr_1}\theta _{2} \end{align}

(7) \begin{align} 1 =f_ac_{1r}+f_b(c_{2r}+l_{1r}cos\theta _{2})+k_a\theta _{1} \end{align}

(8) \begin{align} \rho =f_bc_{2r}+k_b\theta _{2} \end{align}

where

\begin{align*} \rho = \frac{r_2}{r_1}, \; f_a=\frac{f_1}{T}, \; f_b=\frac{f_2}{T}, \; c_{1r}=\frac{c_1}{r_1}, \; c_{2r}=\frac{c_2}{r_1},\\ \; l_{1r}=\frac{l_1}{r_1}, \; k_a=\frac{k_1}{Tr_1}, \; k_b=\frac{k_2}{Tr_1} \end{align*}

In the expressions above, forces are non-dimensionalized by division of the tendon tension ( $T$ ), all lengths are non-dimensionalized by the proximal joint radius $r_1$ , and the stiffnesses are non-dimensionalized by the product $Tr_1$ .

Rearranging to find the non-dimensionalized forces $f_a$ (Eq. (9)) and $f_b$ (Eq. (10)).

(9) \begin{equation} f_a=\frac{1}{c_{1r}}\left [ 1-k_a\theta _1 - (\rho -k_b\theta _2)\left (1+\frac{l_{1r}}{c_{2r}}cos\theta _2 \right ) \right ] \end{equation}

And,

(10) \begin{equation} f_b=\frac{1}{c_{2r}}(\rho -k_b\theta _{2}) \end{equation}

Similarly, the expressions of the contact forces for the other finger, $f_c$ (Eq. (11)) and $f_d$ (Eq. (12)) can be calculated as:

(11) \begin{equation} f_c=\frac{1}{c_{3r}}\left [ 1-k_c\theta _3 - (\rho -k_d\theta _4)\left (1+\frac{l_{3r}}{c_{4r}}cos\theta _4 \right ) \right ] \end{equation}

And,

(12) \begin{equation} f_d=\frac{1}{c_{4r}}(\rho -k_d\theta _{4}) \end{equation}

The non-dimensionalized object radius is given by $O_r=r_o/r_1$ , and the non-dimensionalized base distance is given by $b_r=b/r_1$ . Radius ratio ( $\rho =r_2/r_1$ ) is an important design parameter which is equal to the inter-phalangeal torque ratio (Eq. (13)).

(13) \begin{equation} \frac{\tau _{2}}{\tau _{1}}=\frac{r_2}{r_1} \end{equation}

In our analysis, we take the radius ratio/ torque ratio of the left and right finger to be equal. Hence, the radius ratio is given by Eq. (14)

(14) \begin{equation} \rho =\frac{r_2}{r_1}=\frac{r_4}{r_3} \end{equation}

The analysis presented in this paper is for the non-dimensionalized parameters $O_r$ , $\rho$ , and $b_r$ . For example, we have presented the result for $O_r=10.0$ , $\rho =1.0$ , and $b_r=6.0$ . Since these parameters are non-dimensionalized with respect to the radius of the proximal joint pulley $r_1$ , we can obtain the actual dimensions of the parameters by first choosing $r_1$ . The choice of $r_1$ will be based on physical constraint on its dimensions. For example, we might choose $r_1=5.0\; \textrm{mm}$ ; then, the base distance we are working with becomes $b=b_r r_1=30.0 \; \textrm{mm}$ , the object size becomes $r_o=O_r r_1=50.0 \; \textrm{mm}$ , and the radius of the distal joint pulley becomes $r_2=r_1 \rho =5.0 \; \textrm{mm}$ .

If we have a tendon tension $T=5.0 \; \textrm{N}$ , then the torque at both the joints is $\tau _i=Tr_1=25.0 \; \textrm{Nmm}$ , $i=1,2,3,4$ . The joint angles for an equilibrium position of hand and object are $\theta _1=0.39 \; \textrm{rad}$ , $\theta _2=1.61\; \textrm{rad}$ , $\theta _3=0.39\; \textrm{rad}$ , and $\theta _4=1.61 \; \textrm{rad}$ . And the corresponding contact forces are $f_1=f_3=0.206 \; \textrm{N}$ and $f_2=f_4=0.457 \; \textrm{N}$ . And the non-dimensionalized forces are $f_a=f_c=f_1/T=0.0412$ and $f_b=f_d=f_2/T=0.0914$ .

Since the goal of this study is to evaluate the success of a hand design to grasp objects, detection of successful and defective grasps through simulation is critical. We do this by searching for a feasible equilibrium of the hand and object. If a feasible equilibrium of both hand and object is found, the grasp is successful else it is defective. Defective grasp physically means that the fingers will slip on the object and grasp will fail. The process of detecting grasp success and failure cases is called grasp analysis.

As explained earlier, to ascertain whether a grasp is successful, we “search” for a feasible equilibrium position of both the hand and the object. For this, we fix the object at different locations relative to the palm and determine the net force on the object applied by the hand. If the net force is zero, we have found the equilibrium position. If not, we iterate the process with a new candidate location of the object. This process when applied to hand designs with joint limits has to first eliminate the location of the object where the joint limits of the fingers will be violated.

In the following section, we describe, in detail, the various conditions that has been taken into account to detect grasp success and failure.

4. Grasp analysis

In grasp analysis, we computationally determine the success and failure of grasps. We also determine the strength of the grasp using the grasp stiffness measure. We consider four- and five-point grasps. A five-point grasp is one in which the object touches the palm of the hand. In a four-point grasp, contact takes place only with the fingers. The steps involved in determining the success and failure of the grasp are as follows:

1. 1. Locate the object at a position relative to the palm. The object is considered fixed at this location, and we determine the force exerted by the underactuated hand on the object.

2. 2. From geometry, determine the contact locations and joint angles for the fingers to contact the object (Fig. 1(b)).

3. 3. Determine the forces required for the equilibrium of the fingers in the computed configuration.

4. 4. Check if all the contact forces are positive. If any contact force is negative, the position is not a feasible position of the object and a new position needs to be considered.

5. 5. If all the contact forces are positive, compute the net force on the object applied by the fingers.

6. 6. If the net force is zero, an equilibrium has been found. If the net force is not zero, then the present position is not the equilibrium position of the hand-object system and a new position of the object should be considered.

7. 7. Iterate the process till an equilibrium is found. If equilibrium is not found for bounded candidate positions of the object, the object is not graspable by the hand.

Using the steps given above, we can determine whether an object is graspable by hand. The algorithm for determining success and failure of grasp was implemented in MATLAB. Figure 2 shows two cases of successful grasps with the object having contact only with the fingers (Fig. 2(a)) and having contact with the fingers and the palm (Fig. 2(b)). The range of object sizes that the hand can grasp is one of the performance measures we consider in this work. If the object is graspable, we can define the stiffness of the grasp as detailed in the following section.

Figure 2. Two cases of successful grasp by the hand (a) having four contacts only with the fingers (b) having five contacts, four with the fingers and one with the palm.

4.1. Grasp stiffness

When a grasped object is perturbed from its position, the hand passively applies a restoring force on the object to bring it back to its equilibrium position. The matrix $K_g$ maps the perturbation to the restoring force (Eq. (15)). The form of the grasp stiffness matrix is shown in Eq. (16). We define the grasp stiffness performance measure ( $Q_k$ ) as the determinant of the grasp stiffness matrix (Eq. (17).

(15) \begin{equation} \begin{bmatrix} \Delta F_x\\ \Delta F_y \end{bmatrix}=K_g \begin{bmatrix} \Delta x\\ \Delta y \end{bmatrix} \end{equation}

(16) \begin{equation} K_g = \begin{bmatrix} \dfrac{\Delta F_x}{\Delta x} & \dfrac{\Delta F_x}{\Delta y}\\[9pt] \dfrac{\Delta F_y}{\Delta x} & \dfrac{\Delta F_y}{\Delta y} \end{bmatrix} \end{equation}

(17) \begin{equation} Q_k = \left | K_g \right | \end{equation}

6. Experimental validation

In this section, we perform experimental validation for immobilizing grasps. We compare the immobilizing grasps found in simulation to that in the experiment. For experiments, we have made the two-fingered underactuated hand with 3D printing. Table III shows the comparison between simulation (MATLAB Simscape Multibody) and experimental results. The comparison is performed for two object sizes and two radius ratios ( $\rho =1.0$ and $\rho =1.5$ ). Absolute values of the parameters used for verifying the results is shown in Table IV.

Table III. Comparison of simulation and experimental results ( $l_{1r} = 11.25 \; (45 \; \textrm{mm})$ ). I: Immobilized, NI: Not immobilized.

Table IV. Values of the parameters used for experimental results.

In the results presented in Table III, we validate our work with the match between the experimental results and simulation results. In the experimental results, immobilizing grasp is one in which joint limits are reached. In such a grasp, tendon tension can be increased to resist larger external forces on the object. Figures 12(a) and (b) show the grasps with hands made with the parameters from Table IV.

Figure 12. Experimental verification of results.

The experimental result also shows the same trend presented in Fig. 7. The major trends are as follows:

• Larger objects are immobilized: We see this in case of both $\rho =1.0$ and $\rho =1.5$ . The larger object ( $r_o=40 \; \textrm{mm}$ or $O_r=10.0$ ) is immobilized while the smaller object ( $r_o=25 \; \textrm{mm}$ or $O_r=6.25$ ) is not immobilized.

• Using larger torque ratio increases the chances of achieving an immobilizing grasp: If we compare the grasps by hand design with $\rho =1.5$ and $\rho =1.0$ on the smaller object ( $r_o=25 \; \textrm{mm}$ or $O_r=6.25$ ) (Table III number 2 and 4), we observe that the equilibrium angle for $\rho =1.5$ is smaller. This implies that we can create an immobilizing grasp with smaller joint limit angle for a hand having a larger torque ratio.

• Larger joint limits lead to larger range of objects immobilized: This is evident from the result for smaller objects. Consider $O_r = 6.25$ and $\rho = 1.5$ (Table III number 4). Since the equilibrium of the object is reached at $\theta _{1}=\theta _{2}=0.87 \; \textrm{rad}$ , a joint limit angle larger than this will create immobilizing grasp on the object. However, we should also note that the size of the largest object grasp also decreases with increase in the joint limit angle.