2.1. Model of a compliant grasp

In a hand, each finger/object combination can be viewed as a serial mechanism. A compliant grasp then can be modeled as a system of serial mechanisms connected in parallel to the held object. Unlike the fully serial mechanisms studied in previous work [Reference Huang and Schimmels15, Reference Huang and Schimmels16, Reference Huang and Schimmels17], in which the connection between links are all elastic revolute joints, the fingertip makes a hard point contact connection to the held body. As such, at the point of contact, only force (no torque) can be transmitted to the body as illustrated in Fig. 2(a).

Figure 2. Model of a compliant robotic hand with multiple 3-joint fingers. The fingertip of each finger only provides pure force (no torque) at the contact point and is modeled as a revolute joint without elastic behavior.

It can be seen that, with the above assumptions, each finger can provide only point compliance to the body (yielding a rank deficient planar stiffness matrix at the contact point) regardless of number of the compliant joints in the finger. Since the fingertip does not transmit torque, the connection between the finger tip and the object it contacts is equivalent to a free revolute joint at the contact point as shown in Fig. 2(b). Thus, when in contact with the held object, each 3-joint finger can be viewed as a $4R$ serial mechanism with the last joint having no elastic behavior.

2.2. Screw representation of mechanism geometry

Consider a fully parallel mechanism consisting of $n$ line springs. The position of each spring can be represented by a unit wrench defined as the spring wrench. In Plücker ray coordinates, a planar line spring wrench has the form:

(3) \begin{eqnarray}{\bf w}= \left [ \begin{array}{r}{\bf n} \\[4pt] d \end{array} \right ], \end{eqnarray}

where the unit 2-vector $\bf n$ indicates the direction of the wrench (spring axis) and where

(4) \begin{eqnarray} d = ({\bf r}\times{\bf n}) \cdot{\bf k}, \end{eqnarray}

where $\bf r$ is the position vector from the origin to any point on the spring axis, and $\bf k$ is the unit vector orthogonal to the plane. Conversely, a unit wrench $\bf w$ in the form of Eq. (3) can be used to represent a line in the plane.

If a stiffness matrix $\bf K$ is passively realized with a parallel mechanism having $n$ springs ${\bf w}_i$ $(1, \ldots, n)$ , then [Reference Huang and Schimmels32]

(5) \begin{eqnarray}{\bf K}= k_1{\bf w}_1{\bf w}_1^T + k_2{\bf w}_2{\bf w}_2^T + \cdots + k_n{\bf w}_n{\bf w}_n^T, \end{eqnarray}

where $k_i \geq 0$ is the spring stiffness associated with spring wrench ${\bf w}_i$ .

Now consider a fully serial mechanism consisting of $n$ elastic revolute joints. For a joint $J$ , the unit twist $\bf t$ centered at the joint is defined as its joint twist. In Plücker axis coordinates, joint twist $\bf t$ has the form:

(6) \begin{eqnarray}{\bf t} = \left [ \begin{array}{r}{\bf v} \\[4pt] 1 \end{array} \right ], \end{eqnarray}

where ${\bf v}={\bf r}\times{\bf k}$ , $\bf k$ is the unit vector orthogonal to the plane, and $\bf r$ is the position vector of the revolute joint with respect to the coordinate frame. The 2-vectors $\bf r$ and $\bf v$ (the translational component of $\bf t$ ) are related by:

(7) \begin{eqnarray}{\bf r} ={\boldsymbol{\Omega }}{\bf v}, \end{eqnarray}

where $\boldsymbol{\Omega }$ is the $2\times 2$ skew-symmetric matrix defined as:

(8) \begin{eqnarray}{\boldsymbol{\Omega }} =\left [ \begin{array}{c@{\quad}c} 0 & -1 \\[4pt] 1 & 0 \end{array} \right ]. \end{eqnarray}

Conversely, a unit twist $\bf t$ in the form of Eq. (6) can be used to represent the location of a point in the plane.

If a compliance matrix $\bf C$ is passively realized with a serial mechanism having $n$ joints described by joint twists ${\bf t}_i$ , $(1, \ldots, n)$ , then [Reference Huang and Schimmels33]

(9) \begin{eqnarray}{\bf C}= c_1{\bf t}_1{\bf t}_1^T + c_2{\bf t}_2{\bf t}_2^T + \cdots + c_n{\bf t}_n{\bf t}_n^T, \end{eqnarray}

where $c_i \geq 0$ is the joint compliance at the joint associated with ${\bf t}_i$ .

2.3. Compliance of multi-serial mechanisms

As shown in Fig. 2, a compliant hand can be viewed as parallel system in which each finger is a serial mechanism. For an $m$ -serial mechanism, if each ${\bf C}_i$ (calculated using Eq. (9)) is the compliance matrix attained at the elastically suspended body by the $i$ th serial mechanism, then the overall Cartesian stiffness of the multi-serial system is

(10) \begin{eqnarray}{\bf K}={\bf C}_1^{-1} +{\bf C}_2^{-1} + \cdots +{\bf C}_m^{-1}. \end{eqnarray}

However, for the compliant hand illustrated in Fig. 2(b), because the fingertip in contact with the object surface is modeled as a free joint (each finger only exerts pure force (no torque) to the body), the planar compliance provided by each finger is rank deficient. In other words, each finger provides only point planar compliance (a $2\times 2$ matrix) rather than general planar compliance (a $3\times 3$ matrix). As such, Eq. (10) cannot be directly applied to a hand as modeled in Fig. 2(b). A different approach to determine the Cartesian stiffness of a hand needs to be developed, as described in the next section.

3.1. Properties of point stiffness and compliance

Consider a 2-spring parallel mechanism having spring wrenches $({\bf w}_1,{\bf w}_2)$ and spring rates $(k_1, k_2)$ . The stiffness of the 2-spring system is

(11) \begin{eqnarray}{\bf K}= k_1{\bf w}_1{\bf w}_1^T + k_2{\bf w}_2{\bf w}_2^T = \left [ \begin{array}{c@{\quad}c}{\bf K}_{t} &{\bf b} \\[5pt]{\bf b}^T & k_r \end{array} \right ], \end{eqnarray}

which is a $3\times 3$ rank-2 PSD matrix.

Suppose ${\bf w}_1$ and ${\bf w}_2$ intersect at point $P$ (Fig. 3), then $P$ must be the stiffness center of $\bf K$ . If the coordinate frame is located at $P$ , the stiffness matrix has the form:

(12) \begin{eqnarray}{\bf K}= \left [ \begin{array}{c@{\quad}c}{\bf K}_{P} &{\bf 0} \\[4pt]{\bf 0} & 0 \end{array} \right ], \end{eqnarray}

where ${\bf K}_{P}$ is a $2\times 2$ PD matrix. Thus, $\bf K$ is a point stiffness at $P$ . Since the leading $2\times 2$ diagonal block in the stiffness matrix $\bf K$ is invariant under coordinate translation, ${\bf K}_t ={\bf K}_P$ . As a consequence, the $2\times 2$ translational matrix ${\bf K}_P$ can be obtained from the leading block of the $3\times 3$ stiffness $\bf K$ in any coordinate frame. Conversely, it can be proved that any point stiffness in the form of Eq. (12) is readily realized with 2 springs intersecting at the coordinate frame origin. Therefore, to achieve a point stiffness at a location, only the geometry and the elastic properties of two springs intersecting at that location need to be determined. Also, note that not every rank-2 planar $3\times 3$ stiffness matrix can be expressed in the form of Eq. (12), that is, not every rank-2 matrix represents a point stiffness. For example, the stiffness matrix

(13) \begin{eqnarray}{\bf K} = \left [ \begin{array}{c@{\quad}c@{\quad}c} 1& 0 & 0 \\[4pt] 0 & 0 & 0\\[4pt] 0 & 0 & 1 \end{array} \right ] \end{eqnarray}

does not represent a point planar matrix. Although $\bf K$ in Eq. (13) is a rank-2 PSD matrix, the leading $2\times 2$ block is not full rank and the matrix cannot be realized by two springs intersecting at a point.

Figure 3. Any point planar stiffness at point $P$ can be achieved by 2 springs intersecting at $P$ .

Figure 4. A point stiffness achieved by a serial mechanism. A free revolute joint is located at the point of interest.

In summary, for the planar case, we have

Now consider a serial mechanism (Fig. 4) having $n$ elastic revolute joints described by joint twists $({\bf t}_1,{\bf t}_2, \ldots,{\bf t}_n)$ with $n\geq 3$ . Then, the $3\times 3$ compliance matrix associated with the serial mechanism using Eq. (9) is

(14) \begin{eqnarray}{\bf C}_e = c_1{\bf t}_1{\bf t}_1^T + c_2{\bf t}_2{\bf t}_2^T + \cdots + c_n{\bf t}_n{\bf t}_n^T. \end{eqnarray}

If another joint $J_P$ with compliance $c_P$ is added at the distal link endpoint $P$ , then

(15) \begin{eqnarray}{\bf C} ={\bf C}_e + c_P{\bf t}_P{\bf t}_P^T, \end{eqnarray}

where ${\bf t}_P$ is the joint twist associated with $J_P$ . When the coordinate frame is located at point $P$ , ${\bf t}_P$ is

(16) \begin{eqnarray}{\bf t}_P = \left [ \begin{array}{c}{\bf 0} \\[4pt] 1 \end{array} \right ]. \end{eqnarray}

Any full-rank compliance ${\bf C}_e$ described at the same coordinate frame has the form

(17) \begin{eqnarray}{\bf C}_e= \left [ \begin{array}{c@{\quad}c}{\bf C}_P &{\bf g} \\[4pt]{\bf g}^T & c_r \end{array} \right ]. \end{eqnarray}

Using Eqs. (15)-(17), the total compliance at $P$ is

(18) \begin{eqnarray}{\bf C} = \left [ \begin{array}{c@{\quad}c}{\bf C}_P &{\bf g} \\[4pt]{\bf g}^T & c_r + c_p \end{array} \right ]. \end{eqnarray}

The location of its compliance center relative to endpoint $P$ is calculated to be:

(19) \begin{eqnarray}{\bf r}_c = -\frac{1}{(c_r + c_p)}{\boldsymbol{\Omega }}{\bf g}, \end{eqnarray}

where $\boldsymbol{\Omega }$ is the skew-symmetric matrix defined in Eq. (8). The inverse of $\bf C$ , the $3\times 3$ stiffness matrix $\bf K$ , is

(20) \begin{eqnarray}{\bf K} ={\bf C}^{-1} = \left [ \begin{array}{c@{\quad}c} \left ({\bf C}_P -\dfrac{1}{(c_r+c_p)}{\bf g}{\bf g}^T \right )^{-1} & -\dfrac{1}{h}{\bf C}_P^{-1}{\bf g} \\[15pt] -\dfrac{1}{h}{\bf g}^T{\bf C}_P^{-1} & \dfrac{1}{h} \end{array} \right ], \end{eqnarray}

where $h = (c_r + c_p) -{\bf g}^T{\bf C}_P^{-1}{\bf g}$ .

If $c_p \rightarrow +\infty$ , then ${\bf r}_c \rightarrow{\bf 0}$ , meaning that the compliance center is located at $P$ , and the associated stiffness matrix in SE(2) is

(21) \begin{eqnarray}{\bf K} = \left [ \begin{array}{c@{\quad}c}{\bf C}_P^{-1} &{\bf 0} \\[4pt]{\bf 0} & 0 \end{array} \right ] = \left [ \begin{array}{c@{\quad}c}{\bf K}_P &{\bf 0} \\[4pt]{\bf 0} & 0 \end{array} \right ]. \end{eqnarray}

Thus, a point compliance is achieved by placing a free revolute joint at the end of the serial chain at the point of interest. The $2\times 2$ point compliance is the leading $2\times 2$ block in the compliance matrix of the fully elastic serial mechanism. Note that, unlike the stiffness matrix, the leading $2\times 2$ block matrix in the compliance matrix $\bf C$ associated with an elastic serial mechanism is not invariant under coordinate translation. The point compliance ${\bf C}_P$ can be obtained from the $3\times 3$ planar compliance matrix $\bf C$ only if it is expressed at a coordinate frame located at $P$ . Also, to obtain a full-rank point planar compliance, the elastic joints cannot be collinear. This is because, if all joints are located on the same line, any two of the translational components ${\bf v}_i$ and ${\bf v}_j$ in joint twists ${\bf t}_i$ and ${\bf t}_j$ must be linearly dependent, making ${\bf C}_P$ in Eq. (18) rank deficient (rank 1). In summary,

3.2. Realization of a point compliance

For a point stiffness at a given location, since only force and translation are considered, the force-deflection relation can be expressed as

(22) \begin{eqnarray}{\bf f} ={\bf K}_{P}{\bf v}, \end{eqnarray}

where the force $\bf f$ and the displacement $\bf v$ (the translational component of the twist) are 2-vectors. This relationship can be alternatively expressed as:

(23) \begin{eqnarray}{\bf v} ={\bf C}_{P}{\bf f}, \end{eqnarray}

where ${\bf C}_{P} ={\bf K}_{P}^{-1}$ is the point compliance at $P$ .

Consider a serial mechanism having $n$ elastic joints, and the point compliance ${\bf C}_P$ is at the most distal link tip $P$ . If ${\bf r}_i$ is the 2-vector indicating the location of joint $J_i$ relative to the coordinate frame $Pxy$ , then, using Eq. (7), the translational component of the joint twist ${\bf t}_i$ is

(24) \begin{eqnarray}{\bf v}_i =\boldsymbol{\Omega }^T{\bf r}_i. \end{eqnarray}

If $c_i$ is the joint compliance at $J_i$ , then the $2 \times 2$ point compliance at $P$ associated with the $n$ -joint mechanism configuration is

(25) \begin{eqnarray}{\bf C}_P = c_1{\bf v}_1{\bf v}_1^T + c_2{\bf v}_2{\bf v}_2^T + \cdot + c_n{\bf v}_n{\bf v}_n^T. \end{eqnarray}

From Proposition 2, the point compliance can be achieved by a fully serial mechanism having $n\geq 2$ elastic joints. To realize a full-rank PD $2\times 2$ point compliance at point $P$ , at least 2 elastic joints are needed. It can be proved that, if for any two joints, a given passive point compliance is expressed as

(26) \begin{eqnarray}{\bf C}_P = c_1{\bf v}_1{\bf v}_1^T + c_2{\bf v}_2{\bf v}_2^T, \end{eqnarray}

then both the coefficients $c_1$ and $c_2$ must be positive. In fact, if ${\bf r}_i$ is the position of joint $J_i$ with respect to $P$ , then using Eq. (24), ${\bf r}_i^T{\bf v}_i = 0$ . Since ${\bf C}_P$ is PD,

(27) \begin{eqnarray}{\bf r}_1^T{\bf C}_P{\bf r}_1 = c_2 \left({\bf r}_1^T{\bf v}_2\right)^2 \gt 0. \end{eqnarray}

Thus, $c_2 \gt 0$ . Similarly, $c_1\gt 0$ .

This result for 2 joints is not true for 3 joints. As shown in ref. [Reference Huang and Schimmels34], an arbitrary point compliance matrix ${\bf C}_P$ can always be expressed in the form of Eq. (25) at any configuration of a $3R$ serial mechanism, and the 3 coefficients $(c_1, c_2, c_3)$ are uniquely determined by the locations of the three elastic joints. These coefficients, however, are not guaranteed to all be positive. Based on the result for 2 joints, it is readily shown that among the 3 coefficients $c_i$ s, at most one is negative. Conditions on the configuration of a $3R$ serial mechanism that ensure passive realization of a given point compliance are presented in ref. [Reference Huang and Schimmels34].

When synthesizing a point compliance with a finger for which the locations of the base joint and the fingertip contact are determined, an equivalent and easier to use condition is derived below.

Consider a given $2\times 2$ point compliance matrix ${\bf C}_P$ at point $P$ . Suppose that a $3R$ serial mechanism has joints $J_1$ , $J_2$ , and $J_3$ and suppose that ${\bf r}_i$ is the position of $J_i$ relative to point $P$ . Denote $\rho _i$ as the line collinear with position vector ${\bf r}_i$ and denote $l_i$ as the line of action of force ${\bf f}_i ={\bf K}_P{\bf v}_i$ . At a given configuration, the compliance is expressed as:

(28) \begin{eqnarray}{\bf C}_P = c_1{\bf v}_1{\bf v}_1^T + c_2{\bf v}_2{\bf v}_2^T + + c_3{\bf v}_3{\bf v}_3^T. \end{eqnarray}

For an arbitrary permutation $(i, j, s)$ of $\{1, 2, 3\}$ , since ${\bf r}_s^T{\bf v}_s = 0$ and ${\bf C}_P{\bf f}_s ={\bf v}_s$ ,

(29) \begin{eqnarray}{\bf r}_s^T{\bf C}_P{\bf f}_s = c_i \left({\bf r}_s^T{\bf v}_i\right) \left({\bf v}_i^T{\bf f}_s\right) + c_j \left({\bf r}_s^T{\bf v}_j\right) \left({\bf v}_j^T{\bf f}_s\right) = 0. \end{eqnarray}

Thus, $c_i$ and $c_j$ have the same sign if and only if $[\left({\bf r}_s^T{\bf v}_i\right) \left({\bf v}_i^T{\bf f}_s\right)]$ and $[\left({\bf r}_s^T{\bf v}_j\right) \left({\bf v}_j^T{\bf f}_s\right)]$ have different signs. Consider the following two cases.

1. Case 1: $\left({\bf r}_s^T{\bf v}_i\right)$ and $\left({\bf r}_s^T{\bf v}_j\right)$ have the same sign. In this case, joints $J_i$ and $J_j$ are on the same side of line $\rho _s$ . Then $\left({\bf v}_i^T{\bf f}_s\right)$ and $\left({\bf v}_j^T{\bf f}_s\right)$ must have different signs. Thus, $J_i$ and $J_j$ must be separated by $l_s$ .

2. Case 2: $\left({\bf r}_s^T{\bf v}_i\right)$ and $\left({\bf r}_s^T{\bf v}_j\right)$ have different signs. In this case, joints $J_i$ and $J_j$ are separated by $\rho _s$ . Then $\left({\bf v}_i^T{\bf f}_s\right)$ and $\left({\bf v}_j^T{\bf f}_s\right)$ must have the same sign. Thus, $J_i$ and $J_j$ must be on the same side of $l_s$ .

Thus, $c_i$ and $c_j$ have the same sign if and only if joints $J_i$ and $J_j$ are separated by either line $l_s$ or line $\rho _s$ (but not both). Since, of the 3 coefficients $(c_1, c_2, c_3)$ , at most one is negative; therefore, if $c_i$ and $c_j$ have the same sign, they must both be positive. In summary,

This general result is simplified when the 3-joint locations ${\bf r}_i$ are not collinear and the first joint (base joint) location is specified. For a given point compliance ${\bf C}_P$ expressed in the form of Eq. (28), the force ${\bf f}_1$ associated with translational twist ${\bf v}_1$ at joint $J_1$ is

(30) \begin{eqnarray}{\bf f} _1 ={\bf K}_P{\bf v}_1. \end{eqnarray}

Then,

(31) \begin{eqnarray}{\bf C}_P{\bf f}_1 = c_1\left({\bf v}_1^T{\bf f}_1\right){\bf v}_1+ c_2\left({\bf v}_2^T{\bf f}_1\right){\bf v}_2 + c_3\left({\bf v}_3^T{\bf f}_1\right){\bf v}_3. \end{eqnarray}

Since ${\bf C}_P{\bf f}_1 ={\bf v}_1$ , if $J_3$ is located on line $l_1$ (the line of action of ${\bf f}_1$ ), then ${\bf v}_3^T{\bf f}_1 = 0$ , and Eq. (31) becomes

(32) \begin{eqnarray}{\bf v}_1 = c_1\left({\bf v}_1^T{\bf f}_1\right){\bf v}_1+ c_2\left({\bf v}_2^T{\bf f}_1\right){\bf v}_2. \end{eqnarray}

Since ${\bf v}_1$ and ${\bf v}_2$ are not collinear, they are linearly independent, and $c_2 = 0$ . Thus, if $J_3$ is located along $l_1$ ,

(33) \begin{eqnarray}{\bf C}_P = c_1{\bf v}_1{\bf v}_1^T + c_3{\bf v}_3{\bf v}_3^T. \end{eqnarray}

As proved in Eqs. (26)-(27), both $c_1$ and $c_3$ must be positive regardless of the location of $J_2$ .

Figure 5. Necessary and sufficient condition on the joint locations of a $3R$ mechanism for the realization of a point compliance at a given point $P$ . Joints $J_i$ and $J_j$ must satisfy either: (a) $J_i$ and $J_j$ are separated by line $l_s$ and not by line $\rho _s$ or (b) $J_i$ and $J_j$ are separated by line $\rho _s$ and not by line $l_s$ .

Since $c_2$ changes its sign when $J_3$ moves across the line of action of ${\bf f}_1$ , rotation of link $L_3$ about the fixed point $P$ from a location along ${\bf f}_1$ will result in a sign change in $c_2$ . If a rotation yields a location of $J_3$ such that $J_2$ and $J_3$ are separated by either $l_1$ or $\rho _1$ (not both), then the 3 coefficients $c_1$ , $c_2$ , and $c_3$ must all be positive.

In the realization of an arbitrary point compliance using a $3R$ finger with the locations of base joint $J_1$ and fingertip $P$ specified, only an evaluation of the locations of $J_2$ and $J_3$ relative to lines $\rho _1$ and $l_1$ is needed. This approach will be used in the synthesis of compliance with multi-finger hands.

4.1. Approach

Suppose that a specific $3\times 3$ symmetric PD matrix $\bf K$ is desired for an object grasped by an $m$ -finger hand. To do this, the first step in the synthesis process is to identify a 2m -spring fully parallel mechanism for which pairs of springs intersect on the held object surface at locations $P_i$ . In doing this, the given stiffness is strategically decomposed into the sum of rank-2 PSD matrices (each ultimately corresponding to a point planar stiffness provided by a finger). The desired stiffness would then be described by:

(34) \begin{eqnarray}{\bf K} ={\bf K}_1 +{\bf K}_2 +\ldots +{\bf K}_m, \end{eqnarray}

where $m$ is the number of fingers in the hand.

In this point stiffness decomposition approach, each ${\bf K}_i$ is then realized using the results of Proposition 2 by a $3R$ finger with the fingertip contacting the object surface at point $P_i$ . The $m$ fingers together realize the desired $\bf K$ .

Specifically, the approach uses previously developed geometric construction-based synthesis procedures [Reference Huang and Schimmels15, Reference Huang and Schimmels17] for parallel mechanisms. As such, the decomposition of Eq. (34) is not arbitrary, but strategic. In the process, $2m$ springs are selected with each pair $({\bf w}_q,{\bf w}_s)$ intersecting at point $P_i$ on the surface of the object as illustrated in Fig. 6(a). By Proposition 1, the stiffness ${\bf K}_i$ associated with $({\bf w}_q,{\bf w}_s)$ must be a point stiffness at $P_i$ .

Figure 6. Realization of an elastic behavior with a multi-finger hand. (a) Realization of the stiffness by a $2m$ -spring mechanism with each pair of springs intersecting at the object surface. (b) The point compliance ${\bf C}_{Pi} ={\bf K}_{Pi}^{-1}$ associated with each pair of springs is realized by a 3-joint finger.

Next, each rank-2 stiffness is converted to a finger compliance matrix at point $P_i$ . Suppose that the $2\times 2$ leading block of ${\bf K}_i$ is ${\bf K}_{Pi}$ , then

(35) \begin{eqnarray}{\bf C}_{Pi} ={\bf K}_{Pi}^{-1} \end{eqnarray}

is the $2\times 2$ point compliance matrix at $P_i$ to be realized by a finger. Note that ${\bf K}_{Pi}$ is expressed in the global coordinate frame $Oxy$ and does not change when translated to the frame at $P_i$ , whereas ${\bf C}_{Pi}$ obtained from Eq. (35) is the associated point compliance expressed only in the coordinate frame at $P_i$ .

For each point compliance ${\bf C}_{Pi}$ , one 3-joint finger (a $3R$ serial mechanism) is configured such that the last link contacts the object surface at $P_i$ as shown in Fig. 6(b). As such, the desired compliant behavior is achieved by the grasp.

4.2. Spring distribution constraints

As shown in ref. [Reference Huang and Schimmels20] in order for a parallel mechanism to realize a specified elastic behavior, the locations and orientations of the springs in the mechanism must satisfy a set of necessary conditions. Below, the necessary conditions on the spring distribution of a parallel mechanism are reviewed.

For a given stiffness matrix $\bf K$ , the distribution conditions [Reference Huang and Schimmels20] on the springs in a parallel mechanism for the realization of $\bf K$ are described by a circle $\Gamma _k$ centered at the stiffness center $C_c$ and a pair of lines $\gamma ^+$ and $\gamma ^-$ intersecting at $C_c$ . The circle and the line pair are determined by the principal stiffnesses of the stiffness matrix.

Suppose that $k_{\chi }$ and $k_{\eta }$ are the two translational principal stiffnesses along the principal $\chi$ - and $\eta$ -axis, respectively, and $k_{\tau }$ is the rotational principal stiffness of stiffness matrix $\bf K$ . The radius of $\Gamma _k$ is

(36) \begin{eqnarray} r_k = \sqrt{\frac{ k_{\tau }}{ k_{\chi }+ k_{\eta } }}. \end{eqnarray}

The angles between line $\gamma ^+$ ( $\gamma ^-$ ) and the principal $\chi$ -axis are

(37) \begin{eqnarray} \theta = \pm \sin ^{-1}\sqrt{ \frac{k_{\chi }}{ k_{\chi }+ k_{\eta } }}. \end{eqnarray}

If ${\bf r}_i$ is the perpendicular vector from compliance center $C_c$ to the $i$ th spring axis, then the following conditions must be satisfied by the set of springs in the parallel mechanism:

1. (i) At least one spring line of action intersects $\Gamma _k$ , and at least one spring line of action does not intersect $\Gamma _k$ ;

2. (ii) All ${\bf r}_i$ s span more than a half plane, and at least 2-vectors ${\bf r}_i$ and ${\bf r}_j$ are separated by either $\gamma ^+$ or $\gamma ^-$ but not both.

Since the synthesis approach is based on the realization of the compliance with a $2m$ -spring parallel mechanism, these necessary conditions should be considered before and while performing the synthesis procedure.

4.3. Synthesis procedure

The more detailed procedure presented below is used to synthesize a given compliant behavior with a multi-finger hand. Each finger has 3 elastic joints and fixed link lengths. In a coordinate frame $Oxy$ , the desired stiffness for the held body is $\bf K$ .

Step 1: Parallel mechanism synthesis

Realize $\bf K$ with a fully parallel mechanism to obtain the corresponding point compliance matrices at the surface of the object. The desired stiffness is first realized by a $2m$ -spring parallel mechanism in which pairs of springs intersect at the object surface. Here, cases for which the number of fingers $m= 2$ , $m= 3$ , and $m \gt 3$ are considered.

(i) $m = 2$ . Using the process developed in ref. [Reference Huang and Schimmels29], 4-spring wrenches $({\bf w}_1,{\bf w}_2,{\bf w}_3,{\bf w}_4)$ and the corresponding spring rates $(k_1, k_2, k_3, k_4)$ are obtained. In the geometry-based spring selection procedure [Reference Huang and Schimmels29], the intersections of $({\bf w}_1,{\bf w}_2)$ and $({\bf w}_3,{\bf w}_4)$ , $P_1$ and $P_2$ , are located on the surface of the held object as shown in Fig. 7(a).

Figure 7. Realization of the compliant behavior with a $2m$ -spring parallel mechanism. (a) A 4-spring mechanism. (b) A 6-spring mechanism. For both cases, the intersection of each pair of springs is located at the surface of the object.

The two rank-2 stiffness matrices associated with $({\bf w}_1,{\bf w}_2)$ and $({\bf w}_3,{\bf w}_4)$ are

(38) \begin{equation}{\bf K}_1 = k_1{\bf w}_1{\bf w}_1^T + k_2{\bf w}_2{\bf w}_2^T, \end{equation}

(39) \begin{equation}{\bf K}_2 = k_3{\bf w}_3{\bf w}_3^T + k_4{\bf w}_4{\bf w}_4^T. \end{equation}

The $3\times 3$ planar stiffness matrices ${\bf K}_1$ and ${\bf K}_2$ are converted to $2\times 2$ point stiffness matrices at $P_1$ and $P_2$ , ${\bf K}_{P1}$ and ${\bf K}_{P2}$ , by extracting the leading $2\times 2$ blocks of ${\bf K}_1$ and ${\bf K}_2$ , respectively.

The twists associated with fingertip contact points $P_1$ and $P_2$ are calculated as

(40) \begin{equation}{{\hat{\textbf{t}}}}_{P1} ={\bf w}_1\times{\bf w}_2, \end{equation}

(41) \begin{equation}{{\hat{\textbf{t}}}}_{P2} ={\bf w}_3\times{\bf w}_4, \end{equation}

which are each converted to unit twists, ${\bf t}_{P1}$ and ${\bf t}_{P2}$ . The position vectors of $P_1$ and $P_2$ in $Oxy$ , ${{\tilde{\textbf{r}}}}_{P1}$ and ${{\tilde{\textbf{r}}}}_{P2}$ , are then obtained using Eq. (7):

(42) \begin{equation}{{\tilde{\textbf{r}}}}_{P1} ={\boldsymbol{\Omega }}{\bf v}_{P1}, \end{equation}

(43) \begin{equation}{{\tilde{\textbf{r}}}}_{P2} ={\boldsymbol{\Omega }}{\bf v}_{P2}, \end{equation}

where ${\bf v}_{P1}$ and ${\bf v}_{P2}$ are the first two components of the unit twists ${\bf t}_{P1}$ and ${\bf t}_{P2}$ , respectively, and $\boldsymbol \Omega$ is the matrix defined in Eq. (8).

The two $2\times 2$ point compliance matrices at $P_1$ and $P_2$ are then obtained:

(44) \begin{eqnarray}{\bf C}_{P1} ={\bf K}_{P1}^{-1},\quad {\bf C}_{P2} ={\bf K}_{P2}^{-1}. \end{eqnarray}

(ii) $m = 3$ . Similarly, using the process presented in ref. [Reference Huang and Schimmels29], 6-spring wrenches $({\bf w}_1,{\bf w}_2, \ldots,{\bf w}_6)$ and corresponding spring rates $(k_1, k_2, k_3, k_4, k_5, k_6)$ are selected such that pairs of springs wrenches intersect on the object surface. The intersections of $({\bf w}_1,{\bf w}_2)$ , $({\bf w}_3,{\bf w}_4)$ , and $({\bf w}_5,{\bf w}_6)$ are illustrated in Fig. 7(b).

Following an equivalent process to that described in Eqs. (40)-(44), the $2\times 2$ point compliance matrices ${\bf C}_{Pi}$ at $P_i$ $(i = 1, 2, 3)$ and the position vectors ${\bf r}_{Pi}$ are obtained.

(iii) $m \gt 3$ . The 3-finger process can be extended to the case $m \gt 3$ . For example, for $m = 4$ , first obtain three rank-2 point stiffness matrices ${\bf K}_{P1}$ , ${\bf K}_{P2}$ , ${\bf K}_{P3}$ on the object surface using the above process for $m = 3$ . Then ${\bf C}_{P1} ={\bf K}_{P1}^{-1}$ can be realized with a 3-joint finger. Let

\begin{eqnarray*}{{\hat{\textbf{K}}}}_{2} ={\bf K}_{P2} +{\bf K}_{P3}. \end{eqnarray*}

Then, ${{\hat{\textbf{K}}}}_{2}$ can be realized with three fingers using the 3-finger synthesis process. As such, the desired stiffness is achieved with a hand having a total of 4 fingers by successive use of the 3-finger process.

Figure 8. Realization of point compliance ${\bf C}_{Pi}$ by a 3-joint finger having fixed link lengths. The locations of joints $J_{i2}$ and $J_{i3}$ must be separated by line $l_{i1}$ (or $\rho _{i1}$ ) and be on the same side of line $\rho _{i1}$ (or $l_{i1}$ ).

Step 3: Determination of hand configuration

Obtain each joint location in the global coordinate frame. For finger $i$ , each joint location is calculated using

(49) \begin{eqnarray}{{\tilde{\textbf{r}}}}_{ij} ={{\tilde{\textbf{r}}}}_{Pi} +{\bf r}_{ij},\quad j = 1, 2, 3. \end{eqnarray}

With the final step, all finger configurations and elastic properties are identified.

4.4. Discussion

The synthesis of a grasp stiffness depends on the locations where fingers contact the object, which directly relate to the selection of spring intersections on the object surface. In selecting a set of springs, the spring distribution necessary conditions [Reference Huang and Schimmels20] (reviewed in Section 4.2) need to be considered. It should be noted that the realizability of a stiffness with a given hand depends not only on the given stiffness but also on the shape of the object and the geometry of the hand. Not every stiffness matrix can be realized by a compliant grasp due to the conditions associated with point contact on the object surface. If the compliant behavior cannot be realized by a $2m$ -spring parallel mechanism with springs intersecting at the object surface using the results of refs. [Reference Huang and Schimmels15] or [Reference Huang and Schimmels17], then the behavior cannot be achieved by any $m$ -finger compliant hand modeled as shown in Fig. 2(b) regardless of the number of elastic joints or the locations of fingertip contact with the object.

When the desired locations of fingertip contact are identified, each contact point $P_i$ must be in the workspace of a finger so that all contact points can be reached by a given hand. This requirement should be considered in selecting the location of the hand base and in selecting the intersection locations of spring pairs in the first step.

As stated previously, if the contact point and the base joint locations are determined, a 3-joint finger is equivalent to a 4-bar mechanism. The ability of a finger to achieve an arbitrary point compliance depends on its relative link lengths. For example, if the mechanism is Grashof with $L_3$ being the shortest link, then any point compliance matrix can be realized [Reference Huang and Schimmels34]. Thus, in designing finger geometry, the last link (the one in contact with the object) should be the shortest one. The contact point should not be close to the boundary of the finger workspace so that it has sufficient mobility to vary the locations of the other joints in order to achieve the desired point compliance.

Hard fingertip contact with the surface of a held object is modeled as a free revolute joint. This model is based on the assumption that the fingertip does not slip on the object surface. This requires that friction between the fingertip and the object surface is sufficient to prevent slipping. If slipping does occur, force closure from all fingertips of a hand cannot be established, and a stable grasp cannot be accomplished. The theory presented in this paper is valid only for non-slip cases. In the synthesis process, the locations of fingertip contact points should be judiciously selected so that force closure and a stable grasp can be achieved for a desired full-rank compliance matrix.

5.1. Distribution condition evaluation

The two necessary conditions on the parallel spring distribution described in Section 4.2 are evaluated for the given stiffness matrix $\bf K$ (in Eq. (51)) and the object.

The three principal stiffnesses of $\bf K$ are

\begin{eqnarray*} [k_{\chi }, k_{\eta }, k_{\tau }] = [33.3333, 0.9259, 12.5000], \end{eqnarray*}

and the principal axes are parallel to the coordinate frame axes. The radius of circle $\Gamma _k$ is calculated using Eq. (36) to be:

\begin{eqnarray*} r_k = 0.6040. \end{eqnarray*}

The angles between line $\gamma ^+$ is calculated using Eq. (37) as:

\begin{eqnarray*} \theta = 80.5377^{\circ }. \end{eqnarray*}

Circle $\Gamma _k$ and lines $\gamma ^{\pm }$ are illustrated in Fig. 9. Since $\Gamma _k$ does not enclose the peg, a set of $2m$ springs can be selected to meet the two distribution conditions, where $m$ is the number of fingers eventually used in the realization.

5.2. Compliance synthesis with a 2-finger hand

The desired compliance is first realized with a 4-spring parallel mechanism with pairs of springs intersecting at locations within the acceptable area on the body surface ( $x\lt 3$ shown in Fig. 9). Then, each point compliance associated with the intersecting spring pairs is synthesized with a 3-joint finger.

Figure 10. Compliance synthesis with a 4-spring parallel mechanism. The intersections of spring pairs $({\bf w}_1,{\bf w}_2)$ and $({\bf w}_3,{\bf w}_4)$ will be the contact locations of the two fingertips.

Step 2: Compliance synthesis for each finger

The desired compliance is realized with the 2-finger hand by identifying the joint stiffnesses and the configuration of each finger. In the following, only the synthesis for one finger ( ${\bf C}_{P1}$ at $P_1$ ) is presented.

Since the base joint $J_{11}$ location (given in the base location and geometry) and the finger tip $P_1$ location (calculated in Step 1) are determined, only the locations of the second and third joints $J_{12}$ and $J_{13}$ need to be identified.

In the local coordinate frame $P_1xy$ , the position vector (using Eq. (45)) and the corresponding translational twist of base joint $J_{11}$ are

\begin{eqnarray*}{\bf r}_{11} = [-5.2566, 0.6]^T,\quad {\bf v}_{11} = [0.6, 5.2566]^T. \end{eqnarray*}

The locations of $J_{12}$ and $J_{13}$ can be identified by the following procedure. The geometry associated with each step is illustrated in Fig. 11(a).

1. 1. Calculate force ${\bf f}_{11}$ associated with joint twist ${\bf v}_{11}$ :

   \begin{eqnarray*}{\bf f}_{11} ={\bf K}_{P1}{\bf v}_{11} = [-1.5436, 1.1160]^T. \end{eqnarray*}
   The line of action of ${\bf f}_{11}$ , $l_{11}$ , obtained is shown in Fig. 11(a).

2. 2. Select the location of $J_{13}$ . Joint $J_{13}$ must be on circle $\Gamma _3$ of radius $L_3$ centered at $P_1$ . Here, in the local frame $P_1xy$ , ${\bf r}_{13}$ is arbitrarily selected to be:

   \begin{eqnarray*}{\bf r}_{13} = L_3[ -\cos 25^{\circ }, \sin 25^{\circ }]^T = [ -1.8126, 0.8452]^T. \end{eqnarray*}

3. 3. Determine the location of $J_{12}$ . Once $J_{13}$ is selected, joint $J_{12}$ must be located at the intersection of circles $\Gamma _{11}$ and $\Gamma _{12}$ with radii $L_1$ and $L_2$ and centered at $J_{11}$ and $J_{13}$ , respectively. There are two intersection points $J_{12}$ and $J^{\prime}_{12}$ . Both are acceptable in that each is separated from $J_{13}$ by either $\rho _{11}$ or $l_{11}$ , but not both. The one that is separated from $J_{13}$ by $l_{11}$ is chosen. This joint location is determined to be:

   \begin{eqnarray*}{\bf r}_{12} = [ -2.9353, 2.5004]^T. \end{eqnarray*}
   The translational components of the joint twists associated with the selected joints are
   \begin{eqnarray*} [{\bf v}_{11},{\bf v}_{12},{\bf v}_{13} ] = \left [ \begin{array}{r@{\quad}r@{\quad}r} -0.6000 & -2.5004 & -0.8452 \\[4pt] -5.2566 & -2.9353 & -1.8126 \end{array} \right ]. \end{eqnarray*}

4. 4. Calculate the joint compliances using Eqs. (46)-(48):

   \begin{eqnarray*} [c_{11}, c_{12,} c_{13}] = [0.2017, 0.0106, 0.0293]. \end{eqnarray*}

With this final step, a finger to achieve the point compliance ${\bf C}_{P1}$ is obtained. The joint locations of $J_{12}$ and $J_{13}$ in the original coordinate frame $Oxy$ are determined using Eq. (49) to be:

\begin{eqnarray*}{{\tilde{\textbf{r}}}}_{12} & = & [-0.6787, 3.5004 ]^T, \\[4pt]{{\tilde{\textbf{r}}}}_{13} & = & [0.4440, 1.8452]^T. \end{eqnarray*}

The 3-joint locations in the original coordinate frame $Oxy$ and the values of corresponding joint compliances are listed in Table I.

Table I. Finger joint locations and joint compliances for the obtained hand configuration.

Figure 11. Synthesis of point compliance at $P_1$ with a finger of a given hand. Each of the following two approaches yields an acceptable joint configuration. (a) Rotate $L_3$ about $P_1$ (counterclockwise) such that $J_{13}$ and $J_{12}$ are separated by line $l_{11}$ . (b) Rotate $L_3$ about $P_1$ (clockwise) such that $J_{13}$ and $J_{12}$ are separated by line $l_{11}$ .

Alternatively, joint $J_{13}$ could have been selected to be on the other side of line $l_{11}$ by rotating $L_3$ clockwise (as shown in Fig. 11(b)). If $J_{13}$ is arbitrarily selected as:

\begin{eqnarray*}{\bf r}_{13} = L_3[ -\cos 70^{\circ }, \sin 70^{\circ } ]^T = [ -0.6840, 1.8794]^T, \end{eqnarray*}

then, the two intersection points of circles $\Gamma _1$ and $\Gamma _2$ are located at $J_{12}$ and $J^{\prime}_{12}$ as illustrated in Fig. 11(b).

Only $J_{12}$ is separated by $l_{11}$ from $J_{13}$ which is located at:

\begin{eqnarray*}{\bf r}_{12} = [ -2.2569, 0.6440]^T. \end{eqnarray*}

Then, the 3 alternative joint locations are determined. The 3-joint twists are

\begin{eqnarray*} [{\bf v}_{11},{\bf v}_{12},{\bf v}_{13}] = \left [ \begin{array}{r@{\quad}r@{\quad}r} -0.6000 & -0.6440& -1.8794 \\[4pt] -5.2566 & -2.2569 & -0.6840 \end{array} \right ]. \end{eqnarray*}

The 3-joint compliances are calculated using Eqs. (46)-(48):

\begin{eqnarray*} [c_{11}, c_{12}, c_{13}] = [0.1906, 0.0957, 0.0146]. \end{eqnarray*}

The alternative locations of $J_{12}$ and $J_{13}$ in the original coordinate frame $Oxy$ are

\begin{eqnarray*}{{\tilde{\textbf{r}}}}_{12} & = & [-0.0003, 1.6440 ]^T, \\[4pt]{{\tilde{\textbf{r}}}}_{13} & = & [1.5725, 2.8794]^T. \end{eqnarray*}

The 3-joint locations in the original coordinate frame $Oxy$ and the values of corresponding joint compliances for the alternative hand configuration are listed in Table II.

Table II. Finger joint locations and joint compliances for the alternative hand configuration.

Using the same process, point compliance ${\bf C}_{P2}$ at point $P_2$ can be achieved by the second finger. Since the compliant behavior is symmetric about the $x$ -axis along the peg’s symmetric axis, one can choose the finger configuration symmetric about the axis with the same joint compliances. Figure 12(a) and (b) illustrates the two selected symmetric hand configurations that realize the desired compliance. Note that, due to kinematic redundancy and the ability to select each joint compliance using variable stiffness actuation, there are infinitely many other possible configurations that realize the desired elastic behavior.

Figure 12. Compliance synthesis with a 2-finger hand. The desired compliance can be achieved by the same hand at different symmetric configurations (a) and (b) with different joint stiffnesses.

5.3. Result verification

For the hand configuration shown in Fig. 12(a), the $2\times 2$ point compliance matrices at the fingertip $P_1$ are calculated using the joint compliances and joint twists obtained in the procedure.

\begin{eqnarray*}{\bf C}_{P1} = c_{11}{\bf v}_{11}{\bf v}_{11}^T + c_{12}{\bf v}_{12}{\bf v}_{12}^T + c_{13}{\bf v}_{13}{\bf v}_{13}^T = \left [ \begin{array}{r@{\quad}r} 0.1600 & 0.7589 \\[4pt] 0.7589 & 5.7600 \end{array} \right ]. \end{eqnarray*}

The joint translational twists associated with the symmetric finger configuration at $P_2$ in Fig. 12(a) are

\begin{eqnarray*} [{\bf v}_{11},{\bf v}_{12},{\bf v}_{13} ] = \left [ \begin{array}{r@{\quad}r@{\quad}r} 0.6000 & 2.5004 & 0.8452 \\[4pt] -5.2566 & -2.9353 & -1.8126 \end{array} \right ]. \end{eqnarray*}

The associated point compliance is

\begin{eqnarray*}{\bf C}_{P2} = c_{21}{\bf v}_{21}{\bf v}_{21}^T + c_{22}{\bf v}_{22}{\bf v}_{22}^T + c_{23}{\bf v}_{23}{\bf v}_{23}^T = \left [ \begin{array}{r@{\quad}r} 0.1600 & -0.7589 \\[4pt] -0.7589 & 5.7600 \end{array} \right ]. \end{eqnarray*}

The $3\times 3$ stiffness matrices in the local coordinate frames at $P_1$ and $P_2$ are

\begin{eqnarray*}{{\tilde{\textbf{K}}}}_{P1} & = & \left [ \begin{array}{c@{\quad}c}{\bf C}_{P1}^{-1} &{\bf 0} \\[4pt]{\bf 0}& 0 \end{array} \right ] = \left [ \begin{array}{r@{\quad}r@{\quad}r} 16.6667 & -2.1960 & 0 \\[4pt] -2.1960 & 0.4630 & 0 \\[4pt] 0 & 0& 0 \end{array} \right ],\\[4pt]{{\tilde{\textbf{K}}}}_{P2} & = & \left [ \begin{array}{c@{\quad}c}{\bf C}_{P2}^{-1} &{\bf 0} \\[4pt]{\bf 0} & 0 \end{array} \right ] = \left [ \begin{array}{r@{\quad}r@{\quad}r} 16.6667 & 2.1960 & 0 \\[4pt] 2.1960 & 0.4630 & 0 \\[4pt] 0 & 0& 0 \end{array} \right ]. \end{eqnarray*}

The transformation (translation) matrices from local coordinate frames $P_1xy$ and $P_2xy$ to the original frame $Oxy$ , respectively, are

\begin{eqnarray*}{\bf T}_{P1} = \left [ \begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & 1 \\[4pt] 0 & 1 & -2.2566 \\[4pt] 0 & 0 & 1 \end{array} \right ],{\bf T}_{P2} = \left [ \begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & -1 \\[4pt] 0 & 1 & -2.2566 \\[4pt] 0 & 0 & 1 \end{array} \right ]. \end{eqnarray*}

In the original coordinate frame, the grasp stiffness matrix of the 2-finger hand is

\begin{eqnarray*}{\bf K}_h ={\bf T}_{P1}^T{{\tilde{\textbf{K}}}}_{P1}{\bf T}_{P1} +{\bf T}_{P2}^T{{\tilde{\textbf{K}}}}_{P2}{\bf T}_{P2} = \left [ \begin{array}{c@{\quad}c@{\quad}c} 33.3333 & 0 & 0 \\[4pt] 0 & 0.9259 & 6.4815 \\[4pt] 0 & 6.4815 & 57.8704 \end{array} \right ], \end{eqnarray*}

which confirms that the desired stiffness matrix in Eq. (51) is achieved.

Using the same process, it is also verified that the hand configuration in Fig. 12(b) realizes the desired elastic behavior.

5.4. Compliance synthesis with a hand having 3 or more fingers

The desired compliant behavior can be also realized with a hand having 3 or more joints. Similar to the process used for a 2-finger hand, the desired behavior is first realized with a 6-spring parallel mechanism with spring pairs intersecting at the surface of the peg and the corresponding point compliance matrix at each contact point is obtained. Then, each point compliance is realized with a finger of the hand.

If use of more fingers is desired, as discussed in Section 4.3, the synthesis procedures for 2-finger and 3-finger hands can be used to decompose the given stiffness matrix into the sum of $m$ rank-2 point stiffness matrices ${\bf K}_{Pi}$ . Then, using the process (Step 2) described in Section 4.3, each ${\bf K}_{Pi}$ is achieved by a 3-joint finger.