This paper addresses the passive realization of any selected planar elastic behavior with redundant elastic manipulators. The class of manipulators considered are either serial mechanisms having four compliant joints or parallel mechanisms having four springs. Sets of necessary and sufficient conditions for mechanisms in this class to passively realize an elastic behavior are presented. The conditions are interpreted in terms of mechanism geometry. Similar conditions for nonredundant cases are highly restrictive. Redundancy yields a significantly larger space of realizable elastic behaviors. Construction-based synthesis procedures for planar elastic behaviors are also developed. In each, the selection of the mechanism geometry and the selection of joint/spring stiffnesses are completely decoupled. The procedures require that the geometry of each elastic component be selected from a restricted space of acceptable candidates.

Introduction

Compliant behavior is widely used in robotic manipulation to provide force regulation to avoid excessive forces when in contact with a physical constraint. If displacements from equilibrium are small, a compliant behavior can be described by the linear relationship between the applied force (wrench) and displacement (twist). The linear mapping is characterized by a symmetric positive semi-definite (PSD) matrix, the stiffness matrix K, or its inverse, the compliance matrix C.

A general compliance can be modeled as an elastically suspended rigid body. The elastic suspension can be attained by using a compliant mechanism having elastic components connected in series or in parallel (as illustrated in Fig. 1). To realize a compliance, both the mechanism configuration (i.e., the location of the elastic joints of a serial mechanism or the location of the springs in a parallel mechanism) and the elastic behavior of each joint/spring must be identified. It is known [13] that, for a given compliant behavior, there are infinitely many mechanisms and sets of joint/spring elastic properties that achieve the desired behavior. In application, each joint/spring property can be attained using a conventional (constant) torsional/line spring. Time-varying elastic behaviors can be obtained using variable stiffness actuation (VSA) [4]. Identification of the mechanism geometry required to realize a given compliance (provided that each joint/spring elastic property is selectable) is the primary motivation for this work.

Although the use of VSAs increases the realizable space of compliances for a mechanism, a significant amount of compliances cannot be attained due to the mechanism kinematic mobility [57]. To increase the size of the space of realizable elastic behaviors further, redundant mechanisms can be used. A redundant mechanism allows the mechanism configuration to vary without changing the end-effector pose. A redundant mechanism is particularly useful when the length of each link in a mechanism is limited or the work space is constrained by obstacles.

This paper addresses the passive realization of any selected planar elastic behavior with simple redundant elastic manipulators having four elastic components. Serial mechanisms considered (Fig. 1(a)) consist of rigid links connected by revolute joints, each loaded with a spring (joint compliance). The parallel mechanisms considered (Fig. 1(b)) consist of springs, each independently connected to the compliantly suspended body. In the realization, only passive springs, i.e., elastic behaviors that can be achieved without closed-loop control, are considered. Thus, each joint compliance/stiffness must be positive. The physical significance of the realization conditions for simple elastic mechanisms provides a foundation for the design of more complicated elastic mechanisms.

Related Work.

In spatial compliance analysis, screw theory [811] and Lie groups [12] have been used.

In previous work in the realization of spatial compliances, synthesis of elastic behaviors with simple mechanisms (i.e., parallel and serial mechanisms without helical joints) was addressed [1,2,13,14]. The realization of an arbitrary spatial stiffness matrix with a parallel mechanism having both simple springs and screw springs was presented in Refs. [1416]. A decomposition with invariant properties [17,18] for the purpose of realization with simple and screw springs was identified. The duality between the stiffness matrix realized with a parallel mechanism and the compliance matrix realized with a serial mechanism was identified in Ref. [3]. In these approaches, the realization of a compliant behavior was based on an algebraic decomposition of the stiffness/compliance matrix into rank-1 components. These approaches did not consider mechanism geometry in the decomposition.

More recently, realizations of spatial compliant behaviors with some considerations for the mechanism geometry were presented [1922]. In Ref. [23], a procedure to synthesize an arbitrary planar stiffness with a symmetric four-spring parallel mechanism was developed.

In recent work, the realization of translational compliances with 3R serial mechanisms having specified link lengths has been addressed [24]. In Ref. [5], conditions on mechanism link lengths to achieve all planar translational compliances were identified and synthesis procedures to realize an arbitrary 2 × 2 compliance matrix were developed. The results obtained in Ref. [5] for 3R mechanisms were then extended to general planar serial mechanisms with three (revolute and/or prismatic) joints [6]. In Refs. [25] and [26], synthesis of isotropic compliance in E(2) and E(3) were addressed.

In our most recent work [7], a geometric construction-based approach to the design of three-component simple elastic mechanisms that realize general planar elastic behavior was described. The three-component cases correspond to nonredundant mechanisms. In a three-joint serial mechanism, the locations of the three joints form a triangle. Necessary and sufficient conditions for the realization of a given compliance require that a force acting along one side of the triangle results in a twist having its instantaneous center located at the vertex opposite to that side of the triangle. The same realization condition applies to a three-spring parallel mechanism in which the three wrench axes form a triangle. As such, the space of compliant behaviors realized at a given configuration is very limited even if each joint compliance/stiffness can vary infinitely.

This paper addresses compliance realization with redundant mechanisms having four elastic components. Due to the mechanism's increase in the degree-of-freedom, a much larger space of elastic behaviors can be realized. The space is increased for two reasons. First, as previously stated, redundancy allows the mechanism configuration to vary when the end-effector pose is specified. Second, even when the mechanism configuration is not allowed to vary, the restrictions on elastic behavior are less conservative than the three-component cases.

Overview.

This paper addresses the passive realization of an arbitrary planar (3 × 3) elastic behavior with a redundant compliant mechanism. The mechanisms considered are four-joint serial mechanisms and four-spring parallel mechanisms for which each joint compliance or spring stiffness is selectable. Requirements on mechanism geometry to realize a given elastic behavior are identified. Geometric construction-based synthesis procedures are developed. These procedures enable one to select the geometry of each elastic component from the infinite, but restricted set of options available.

The paper is outlined as follows: In Sec. 2, the theoretical background for planar compliance realization with a serial or parallel mechanism is presented. Necessary and sufficient conditions on mechanism configurations to realize an elastic behavior are identified. In Sec. 3, the physical implications of the realization conditions are presented for serial and parallel mechanisms. Using these conditions, the bounds on the realizable space of elastic behaviors for a given mechanism are interpreted in terms of the locus of elastic behavior centers. In Sec. 4, geometric construction-based synthesis procedures for the realization of an arbitrary planar elastic behavior (using either a parallel or serial mechanism) are presented. Section 5 provides numerical examples to illustrate the synthesis procedures for both serial and parallel mechanisms. Finally, a brief summary is presented in Sec. 6.

Planar Compliance Realization Conditions

In this section, the technical background for planar compliance realization with a serial or parallel mechanism having four compliant components is presented. Necessary and sufficient conditions to realize an elastic behavior are then derived for both a serial and a parallel mechanism.

Technical Background.

Consider a serial mechanism having four joints as shown in Fig. 1(a). The type and location of joint Ji can be represented by the joint twist ti. The planar joint twist for a revolute joint and for a prismatic joint can be expressed in Plücker axis coordinates as 
tr=[v1],tp=[n0]
(1)

where v = r × k, k is the unit vector perpendicular to the plane, r is the 2-vector indicating the location of the revolute joint relative to the coordinate frame used to describe the compliance C, and n is the unit 2-vector indicating the direction of the prismatic joint axis.

If a mechanism has four joints with joint twists ti and joint compliance ci > 0, then the Cartesian compliance matrix associated with the mechanism [2,3] is 
C=c1t1t1T+c2t2t2T+c3t3t3T+c4t4t4T
(2)

It is known that any positive definite compliance matrix C can be decomposed into the form of Eq. (2). Thus, if a mechanism is designed to have a revolute joint located at the instantaneous center of each revolute joint twist tr and a prismatic joint along each prismatic twist tp (with corresponding assigned joint compliance ci for each joint), then the compliance behavior is realized with the mechanism. Thus, a decomposition of C with four rank-1 components yields the design of a four-joint serial mechanism configuration that realizes C. The rank-1 decomposition of C in Eq. (2) is not unique. There are infinitely many mechanisms that realize a given elastic behavior.

Given the value of a twist tr, a unique point, the instantaneous center of rotation for the twist motion, is calculated using 
r=Ωv
(3)
where Ω is the 2 × 2 antisymmetric matrix associated with a cross product 
Ω=[0110]
(4)

Although the twist associated with a revolute joint in a serial mechanism has a unique location in the plane, a twist associated with a prismatic joint has an arbitrary location in the plane.

For a parallel mechanism (illustrated in Fig. 1(b)), the type and location of a spring are represented by a unit wrench defined as a spring wrench [1]. The planar spring wrench for a line spring and for a torsional spring can be expressed in Plücker ray coordinates as 
wl=[nd],wt=[01]
(5)

where n is a unit 2-vector indicating the direction of the spring axis, d=(r̃×n)·k is a scalar indicating the distance of the spring axis from the coordinate frame, r̃ is a position vector from the coordinate frame to any point along the spring axis, and k is the unit vector perpendicular to the plane of the mechanism.

If a parallel mechanism has four springs with spring wrench wi and stiffness kt > 0, then the Cartesian stiffness associated with the mechanism is 
K=k1w1w1T+k2w2w2T+k3w3w3T+k4w4w4T
(6)

If a given positive definite stiffness matrix K is decomposed into the form of Eq. (6), a parallel mechanism can be designed to have the spring wrench wi and spring stiffness ki so that the given stiffness is achieved.

When the value of a wrench wl is given, the perpendicular position r to the wrench axis can be calculated using 
r=dΩn
(7)

where Ω is the matrix defined in Eq. (4).

For a torsional spring, since the spring wrench is a free vector, its location is arbitrary.

A wrench w and twist t are called reciprocal [8] if w performs no work along t. If wrench w and twist t are expressed in Plücker ray and axis coordinates, respectively, then w and t are reciprocal if and only if 
wTt=tTw=0
(8)

For planar cases, a twist and a wrench are reciprocal when the instantaneous center of the twist is on the line of action of the wrench.

Since a translational twist tp in Eq. (1) is a free vector in the plane, a serial mechanism consisting of only prismatic joints cannot achieve a full-rank compliant behavior. Thus, in order to realize an arbitrary compliance with a serial mechanism, revolute joints must be used. Similarly, to realize an arbitrary stiffness with a parallel mechanism, simple line springs must be used. Realization conditions impose requirements on the revolute joint locations in a serial mechanism, or on the axes of line springs in a parallel mechanism. These conditions are derived next.

Realization Conditions for Mechanisms Having Four Elastic Components.

This section identifies necessary and sufficient conditions that a four-joint simple elastic mechanism must satisfy to realize a specified elastic behavior.

First, consider a serial mechanism with four elastic joints Ji, i = 1, 2, 3, 4. The location of each Ji is uniquely specified by its twist ti. A wrench wij that is reciprocal to two joint twists ti and tj must satisfy both 
wijTti=0andwijTtj=0
(9)
The line of action of wij must pass through both joints Ji and Jj (the locations of the two twist centers) and thus is uniquely determined by ti and tj. Mathematically, wrench wij can be calculated by 
wij=ti×tj,ij
(10)
Any wrench w reciprocal to ti and tj has the same line of action of wij and can be expressed as 
w=αwij,ij

where α is a scalar.

Consider a symmetric matrix C, and two wrenches w12 and w34 that pass through joints (J1, J2) and joints (J3, J4), respectively. For a four-joint realization, C can be expressed in the form of 
C=c1t1t1T+c2t2t2T+c3t3t3T+c4t4t4T
(11)
then, multiplying Eq. (11) by a wrench passing through two joints J3 and J4 [using Eq. (9)] yields 
Cw34=(c1t1t1T+c2t2t2T+c3t3t3T+c4t4t4T)w34=c1(t1Tw34)t1+c2(t2Tw34)t2
Then, multiplying the result from left by a wrench passing through the other two joints J1 and J2 (using Eq. (9) again) yields 
w12TCw34=0
(12)

The geometric arrangement of the joints satisfying Eq. (12) is a necessary condition for the realization of specified compliance C.

Similarly, it is readily shown that if C can be expressed in the form of Eq. (11), then for any permutation of {1, 2, 3,4} 
wijTCwrs=0
(13)
Note that C is symmetric (wijTCwrs=wrsTCwij) and that wij and wji have the same line of action (wij = αwji). Thus, condition (13) for all permutations of {1, 2, 3, 4} is equivalent to the set of three equations 
w12TCw34=0,w13TCw24=0,w14TCw23=0
(14)

The sufficiency of this set of conditions is evaluated next.

Below, we prove that if any two equations in Eq. (14) are satisfied, then C can be expressed in the form of Eq. (11). To prove this, we define a symmetric matrix C̃ as 
C̃=c̃1t1t1T+c̃2t2t2T+c̃3t3t3T+c̃4t4t4T
(15)
where 
c̃1=w23TCw24(w23Tt1)(w24Tt1)
(16)
 
c̃2=w34TCw14(w34Tt2)(w14Tt2)
(17)
 
c̃3=w14TCw12(w14Tt3)(w12Tt3)
(18)
 
c̃4=w12TCw23(w12Tt4)(w23Tt4)
(19)
We show that any two equations in condition (14) ensure C̃=C. To illustrate this, let i = 2 and j = 4 in Eq. (9), then 
t2Tw24=0andt4Tw24=0
Thus 
C̃w24=(c̃1t1t1T+c̃2t2t2T+c̃3t3t3T+c̃4t4t4T)w24=c̃1(t1Tw24)t1+c̃3(t3Tw24)t3
Since w23Tt3=0 
w23TC̃w24=c̃1(t1Tw24)(w23Tt1)
Substituting Eq. (16) into the above equation yields 
w23TC̃w24=w23TCw24
which is equivalent to 
w23T(C̃C)w24=0
Applying the same process to all wij and wir yields the following four equations: 
w23T(C̃C)w24=0,w14T(C̃C)w34=0
(20)
 
w12T(C̃C)w14=0,w12T(C̃C)w23=0
(21)
If two equations in condition (14) are satisfied, for example 
w12TCw34=0,w13TCw24=0
(22)
then two additional equations similar to Eqs. (20) and (21) can be constructed.
By the definition of C̃ in Eq. (15), it is easy to verify that 
w12TC̃w34=0,andw13TC̃w24=0
(23)
Combining Eqs. (22) and (23) yields 
w12T(C̃C)w34=0,w13T(C̃C)w24=0
(24)
Thus, the matrix (C̃C) satisfies six equations in Eqs. (20), (21), and (24). Since a 3 × 3 symmetric matrix has six independent entries, the six independent equations require that 
C̃C=0C̃=C
(25)

Note that the four Eqs. (20) and (21) are satisfied due to the definition of C̃ in Eq. (15) and are independent from each other if any three twists ti are linearly independent. These four equations are also independent of the equations in Eq. (14). Satisfying two equations in Eq. (14) provides a sufficient number of independent equations to ensure that C̃=C. It can be seen that if any two equations in Eq. (14) are satisfied, then condition (12) must be satisfied for any permutation of {1, 2, 3, 4}. As such, condition (14) is a necessary and sufficient condition for C to be expressed in the form of Eq. (11).

Also, note that if C can be expressed in the form of Eq. (11), then for all permutations {i, j, r, s} of {1, 2, 3, 4} 
cs=wijTCwir(wijTts)(wirTts)=wijTCwrj(wijTts)(wrjTts)
(26)

Thus, condition (26) for any {i, j, r, s} is a necessary condition for C to be expressed in the form of Eq. (11). It can be proved that condition (26) for any two permutations with different values of s is also sufficient for C to be expressed in the form of Eq. (11).

In fact, if condition (26) holds for any two different permutations, then, by the construction of C̃ in Eq. (15), C̃ must satisfy the same condition 
wijTC̃wir(wijTts)(wirTts)=wijTC̃wrj(wijTts)(wrjTts)
(27)
Combining Eqs. (26) and (27) yields 
wijT(C̃C)wir(wijTts)(wirTts)=wijT(C̃C)wrj(wijTts)(wrjTts)
(28)

For two permutations of {1, 2, 3, 4} with different values of s, Eq. (28) gives two independent equations. Since (C̃C) must satisfy the four Eqs. (20) and (21), the 3 × 3 symmetric matrix (CC̃) satisfies six independent homogeneous equations. Thus, CC̃=0, which means that C=C̃ is expressed in the form of Eq. (11). Therefore, condition (26) for any two different permutations is also necessary and sufficient for C to be expressed in the form of Eq. (11).

In summary, we have

Proposition 1. A symmetric matrixCcan be expressed in the form of Eq. (11) if and only if one of the following statements holds:

  • (a)
    For any two sets of (i, j, r, s) from the three permutations (1, 2, 3, 4), (1, 3, 2, 4), and (1, 4, 2, 3) 
    wijTCwrs=0
    (29)
  • (b)
    For any two permutations (i, j, r, s) of {1, 2, 3, 4} with different values of s 
    wijTCwir(wijTts)(wirTts)=wijTCwrj(wijTts)(wrjTts)
    (30)

Note that if condition (29) or (30) holds for two permutations described in Propositions (1a) and (1b), the condition must hold for all permutations of {1, 2, 3, 4}.

The conditions presented in Proposition 1 are necessary and sufficient conditions for a given C to be realized with a mechanism with joint locations determined by ti if there are no constraints on the coefficients ci's in Eq. (11). The mechanism, however, must be capable of having either positive or negative joint compliances. If the conditions in Proposition 1 are not satisfied, the compliant behavior cannot be realized with the mechanism at the configuration even if negative (active) compliance is allowed for each joint.

If variable stiffness actuation is used for a passive realization, each coefficient in Eq. (11) must be non-negative, i.e., the coefficients defined in Eqs. (16)(19) must be non-negative. It can be proved that, if for any four different permutations with s = 1, 2, 3, 4 [i.e., Eqs. (16)(19)] 
(wijTCwir)(wijTts)(wirTts)0
then, the conditions must hold for all permutations. In summary, we have:

Proposition 2. A compliance matrixCcan be passively realized with a four-joint serial mechanism having joint twiststi if and only if the following two conditions are both satisfied:

  • (a)
    For any two sets of (i, j, r, s) from the three permutations (1, 2, 3, 4), (1, 3, 2, 4), and (1, 4, 2, 3) 
    wijTCwrs=0
    (31)
  • (b)
    For any four permutations (i, j, r, s) of {1, 2, 3, 4} with s = 1, 2, 3, 4 
    (wijTCwir)(wijTts)(wirTts)0
    (32)

It can be seen that, in order to passively realize a given compliance matrix with a 4-joint serial mechanism at a given configuration, two equality conditions (31) and four inequality conditions (32) must be satisfied.

Parallel Mechanisms With Four Springs.

By duality, the realization of a specified stiffness with a parallel mechanism having four springs can be obtained.

Consider a parallel mechanism having four simple springs. The line of action (the spring axis) of each spring is identified with the spring wrench wi. If tij is the twist that is reciprocal to both wi and wj, then tij is located at the intersection of the two lines along the spring wrenches wi and wj. Similar to Eq. (10), tij can be determined by 
tij=wi×wj
(33)

The results presented in Proposition 2 can be modified for the realization of stiffness with a parallel mechanism.

Proposition 3. A stiffness matrixKcan be passively realized with a four-spring parallel mechanism having spring wrencheswi if and only if the following two conditions are both satisfied:

  • (a)
    For any two sets of (i, j, r, s) from the three permutations (1, 2, 3, 4), (1, 3, 2, 4), and (1, 4, 2, 3) 
    tijTKtrs=0
    (34)
  • (b)
    For any four permutations (i, j, r, s) of {1, 2, 3, 4} with s = 1, 2, 3, 4 
    tijTKtir(tijTws)(tirTws)0
    (35)

Similar to the serial case, if condition (34) is satisfied for any two permutations described in Proposition (3a), it must be satisfied for all permutations. If condition (34) is not satisfied, the compliant behavior cannot be realized with the mechanism even if negative stiffness (active, but independently controlled behavior) is allowed for each spring. If condition (35) is satisfied for any four permutations with s being each of the springs, then the condition must be satisfied for all permutations of {1, 2, 3, 4}.

It can be seen that, in order to passively realize a given stiffness matrix with a four-spring parallel mechanism, two equality conditions (34) and four inequality conditions (35) must be satisfied. If a given K can be realized with the mechanism, the stiffness for each spring can be determined by 
k1=t23TKt24(t23Tw1)(t24Tw1)
(36)
 
k2=t34TKt14(t34Tw2)(t14Tw2)
(37)
 
k3=t14TKt12(t14Tw3)(t12Tw3)
(38)
 
k4=t12TKt23(t12Tw4)(t23Tw4)
(39)

To be passively realized, each ki in Eqs. (36)(39) must be non-negative (which is equivalent to Proposition 3b).

Realization Uniqueness.

When a compliance is realized at a given configuration of a redundant four-joint serial mechanism, the joint compliances ci typically are unique. If any three ti's are linearly independent (the generic case), the realization of the given compliance with the mechanism at the configuration must be unique. The linear independence of three ti's requires that these three joints are not located on a single line. The joint compliances can be determined by Eqs. (16)(19). Note that for each joint, the unique corresponding joint compliance ci can be calculated using any permutation of {j, r, s} 
ci=wjrTCwjs(wjrTti)(wjsTti)
(40)

Although ci can be obtained with different permutations, Proposition 1b guarantees that the obtained compliance ci is the same for the given joint twist ti.

Similarly, if in a parallel mechanism, any three springs are not parallel to each other, the realization of the given stiffness with the mechanism is unique, and the spring stiffness can be determined using Eqs. (36)(39), or using any permutation of {j, r, s} 
ki=tjrTKtjs(tjrTwi)(tjsTwi)
(41)

Geometric Analysis of Four-Component Elastic Behavior Realization

The implications of the realization conditions can be understood in the geometry of the mechanism. First, the physical interpretations of the realization conditions for serial mechanisms and for parallel mechanisms are presented. Then, using these conditions, the bounds on the realizable space of elastic behaviors for a given mechanism are interpreted in terms of the locus of elastic behavior centers.

Geometric Interpretation of Realization Conditions for Serial Mechanisms.

The realization conditions can be interpreted in terms of requirements on the geometry of the mechanism. If wij is a force along line lij passing through both joints Ji and Jj, then the twist resulting from wij acting on the compliant mechanism is 
tij=Cwij
(42)
The realization equality conditions (31) require 
wrsTCwij=0wrsTtij=0
(43)
which indicates that tij must be on the line of action of wrs, i.e., on line lrs passing through joints Jr and Js. Thus, any force along line lij results in a twist on line lrs.

Below, we show that the inequality conditions (32) for passive realization require that the locus of tij be on a segment along line lrs bounded by joints Jr and Js. Two cases are considered: either line lij intersects lrs between Js and Jr or it does not.

First, consider the case in which line lij does not intersect lrs between Jr and Js. For the four-joint locations illustrated in Fig. 2, consider, without loss of generality, w12 and w34.

For this case, we first show that if C is passively realized with the mechanism, then t12 = Cw12 must be located on the segment J3J4. Since C can be expressed in the form of Eq. (11) with ci ≥ 0 
t12=Cw12=c3(t3Tw12)t3+c4(t4Tw12)t4
(44)

Since J3 and J4 are on the same side of l12, t3Tw12 and t4Tw12 have the same sign. Since c3 and c4 are positive, c3(t3Tw12) and c4(t4Tw12) have the same sign. Thus, the positive (or negative) combination of t3 and t4 in Eq. (44) indicates that t12 must be located on the line segment between J3 and J4.

Conversely, if t12 is located on the finite segment between J3 and J4, then the coefficients of t3 and t4 in Eq. (44) must have the same sign, i.e., c3(t3Tw12) and c4(t4Tw12) have the same sign. Therefore, c3 and c4 must also have the same sign. Below, we show that c3 and c4 must be non-negative.

Since C is PSD 
w12TCw12=c3(t3Tw12)(t3Tw12)+c4(t4Tw12)(t4Tw12)=c3(t3Tw12)2+c4(t4Tw12)20

Therefore, c3 and c4 must be non-negative. The geometric interpretation of the realization conditions for this case is illustrated in Fig. 2.

Next, consider the case in which line lij intersects lrs at the segment between Jr and Js. Here, we suppose, without loss of generality, that l13 intersects line l24 between J2 and J4. If C is realized with the mechanism at the configuration illustrated, then C can be expressed in the form of Eq. (2) with all coefficients ci ≥ 0. Then 
t13=Cw13=c2(t2Tw13)t2+c4(t4Tw13)t4
(45)

Since J2 and J4 are on the opposite sides of line l13, t2Tw13 and t4Tw13 have opposite signs. Thus, t13 must be on line l24 outside the finite segment bounded by J2 and J4. Conversely, if t13 is located outside the line segment J2J4, the coefficients of t2 and t3 in Eq. (45), c2(t2Tw13) and c4(t4Tw13), have opposite signs. Since (t2Tw13) and (t4Tw13) have opposite signs, c1 and c2 have the same sign. Since C is PSD, c1 and c2 must be non-negative. Figure 3 shows the geometric interpretation of the realization conditions for this case.

For a serial mechanism with four joints Ji arranged such that no three are collinear as illustrated in Fig. 2, the necessary and sufficient condition for realization can be expressed geometrically as:

Proposition 4. A complianceCcan be realized with a four-joint serial mechanism at a given configuration if and only if the following two conditions are satisfied:

  • (a)

    A force along l12results in a twist located on line segment J3J4; and a force along l34results in a twist located on line segment J1J2.

  • (b)

    A force along l23results in a twist located on line segment J1J4; and a force along l14results in a twist located on line segment J2J3. ◻

Note that in Proposition 4, condition (a) implies one equality condition w12TCw34=0 and four inequality conditions ci ≥ 0 [if C can be expressed in the form of Eq. (11)]. Thus, if condition (a) [or condition (b)] is satisfied, one only needs to check an equality condition (29) for any different permutation to ensure the realization of the behavior.

Geometric Interpretation of Realization Conditions for Parallel Mechanisms.

By duality, the geometric interpretation of the realization conditions can be obtained for parallel mechanisms and stiffness matrices.

Consider a parallel mechanism having four springs wi. A stiffness matrix K can be realized with the mechanism if and only if 
K=k1w1w1T+k2w2w2T+k3w3w3T+k4w4w4T
(46)

where ki ≥ 0.

Suppose that among the four springs, no three are parallel as illustrated in Fig. 4. The intersection of two wrench axes wi and wj at Tij uniquely identifies a unit twist tij reciprocal to both wrenches. The equality realization condition requires that any twist located at the intersection of wi and wj results in a wrench that passes through the intersection of the axes of the other two wrenches, wr and ws. Below, we show that the wrench must be inside a portion of the pencil bounded by the lines along the axes of wr and ws. Since the wrench wij = Ktij passes through the intersection of wr and ws, wij can be expressed as 
wij=Ktij=αwr+βws
where α and β are scalars. By inequality condition (35) 
tisTKtij(tijTwr)(tisTwr)=tisT(αwr+βws)(tijTwr)(tisTwr)=αtijTwr0
(47)
and 
tirTKtij(tijTws)(tirTws)=tirT(αwr+βws)(tijTws)(tirTws)=βtijTws0
(48)
Thus, α and tijTwr have the same sign, and β and tijTws have the same sign. Since the scalars α and β have fixed signs, wrench wij must be inside an area bounded by the two axes of wr and ws. It is readily shown that the area does not contain the intersection of the axes of wi and wj, Tij. In fact, if a wrench w has axis passing through both locations of Tij and Trs (the center of twist trs), w can be expressed as 
w=α̃wr+β̃ws
(49)
and satisfies 
tijTw=α̃(tijTwr)+β̃(tijTws)=0
(50)

If α̃ and (tijTwr) have the same sign, β̃ and (tijTws) must have the opposite sign, and vice versa. Therefore, the wrench resulting from twist tij must pass through the intersection of the other two wrench axes and must be in the interior of the area bounded by the two axes that does not contain the intersection of wrench axes wi and wj as shown in Fig. 4.

Also, it can be proved that if wij is in the area defined above (the shaded area in Fig. 4), the following equality conditions are satisfied: 
tisTKtij(tijTws)(tirTws)0,tirTKtij(tijTws)(tirTws)0

In summary, for a parallel mechanism having four springs wi, we have

Proposition 5. A stiffnessKcan be realized with a four-spring parallel mechanism if and only if the following two conditions are satisfied:

  • (a)

    A twist at the intersection ofw1andw2, T12, results in a wrench that passes through the intersection ofw3andw4, T34, and lies in the area bounded by the axes ofw3andw4that does not contain T12; and a twist at T34results in a wrench that passes through the intersection ofw1andw2, T12and lies in the area bounded by the axes ofw1andw2that does not contain T34.

  • (b)

    A twist at the intersection ofw1andw3, T13, results in a wrench that passes through the intersection ofw2andw4and lies in the area bounded by the axes ofw2andw4that does not contain T13; and a twist at the intersection ofw2andw4, T24, results in a wrench that passes through the intersection ofw1andw3and lies in the area bounded by the axes ofw1andw3that does not contain T24. ◻

The results of Proposition (5a) are illustrated in Fig. 5.

Note that, similar to the serial mechanism, in Proposition 5, condition (a) implies one equality condition t12TKt34=0 and four inequality conditions ki ≥ 0 [if K can be expressed in the form of Eq. (46)]. Thus, if condition (a) [or condition (b)] of Proposition 5 is satisfied, one only needs to check an equality condition (34) for any different permutation to ensure the realization of the behavior.

Center of Planar Compliant Behaviors.

For any planar compliant behavior, the unique point at which the behavior can be described by a diagonal compliance (stiffness) matrix is defined as the center of compliance (stiffness). For the planar case, the centers of stiffness and compliance are coincident. In Ref. [7], it was shown that if a compliance (stiffness) is realized with a three-joint (three-spring) serial (parallel) mechanism, the center must be inside the triangle formed by the three joints (springs) as vertices (sides). Below, these results are extended to mechanisms having four compliant components.

Consider a compliance matrix C and two wrenches w1 and w2 satisfying 
w1TCw2=0
(51)

Then, the twist t1 = Cw1 must lie on the line of action of w2, and the twist t2 = Cw2 must lie on the line of action of w1 as shown in Fig. 6. The following statements hold true:

  • (a)

    Any wrench w passing through the intersection point of the two wrenches, T12, when multiplied by C, results in a twist t located on line l12 passing through the centers of t1 and t2 as shown in Fig. 6.

  • (b)

    Any wrench along line l12 passing through the centers of t1 and t2, when multiplied by C, results in a twist t12 located at T12 (where wrenches w1 and w2 intersect).

  • (c)

    The center of compliance must be inside the triangle formed by the centers of twists t1, t2, and t12 (the shaded area shown in Fig. 6).

In fact, if wrenches w1 and w2 satisfy condition (51) and wrench w passes through point T12, the intersection of w1 and w2, then w can be expressed as 
w=αw1+βw2
where α and β are scalars. The twist resulting from w is 
t=Cw=αCw1+βCw2=αt1+βt2

Thus, t must be on the line passing through the centers of twists t1 and t2.

If w12 is a wrench passing through t1 and t2 
t1Tw12=0,t2Tw12=0
(52)
then 
w1TCw12=0,w2TCw12=0
(53)

Thus, the twist resulting from C multiplying w12 is reciprocal to both w1 and w2, which indicates that the twist center is located at the intersection of w1 and w2, i.e., at T12.

Since the three wrenches w1, w2, and w12 are reciprocal about C, analysis of three-spring parallel mechanisms [7] shows that the center of compliance must be in the interior of the triangle formed by the axes of the three wrenches. It also can be seen that the three twists t1, t2, and t12 are reciprocal about the stiffness K = C−1. The space of realizable elastic centers is bounded by the joint locations (the joint twist centers) in serial mechanisms and by the spring axes in parallel mechanisms.

Using the realization conditions for a serial and parallel mechanism, it is evident that if a compliant behavior is realized with a serial mechanism, the location of the center must be within the polygon formed by the convex hull of the four joints as shown in Fig. 7(a). For a parallel mechanism of four springs, the location of the center must be inside the polygon formed by the union of all triangles formed by any three spring wrenches as shown in Fig. 7(b).

Synthesis Procedures

In this section, synthesis procedures for the realization of an elastic behavior with a four-joint serial mechanism and with a four-spring parallel mechanism are presented. The procedures are based on the geometric interpretations of the realization conditions obtained in Sec. 3. First, some geometric properties of wrenches and twists associated with an elastic behavior are derived. These properties are used in the synthesis procedures that follow.

Mathematical Preliminaries.

Consider a family of parallel unit wrenches Wp. In the plane, each w in Wp can be represented by its line of action l. Thus, W is represented by a family of parallel lines Lp in the plane. In Lp, there is a unique line lc that passes through the center of compliance Cc.

Suppose Tw is the collection of all twists resulting from multiplying the wrenches of Wp by C, i.e., 
Tw={t=Cw,wWp}
(54)

It is shown below that the centers of twists in Tw form a straight line lt passing through the center of compliance Cc.

Since every w in Wp has the same direction n,Wp can be expressed in the form 
Wp={w=[nd],d(,)}
(55)

where d indicates the perpendicular distance from the line of action to the origin of the coordinate frame used to describe the compliance.

The compliance matrix C can be represented in block form as 
C=[DggTc33]
(56)
The twist caused by w can then be written as 
t=Cw=[DggTc33][nd]=[Dn+dggTn+dc33]=[DngTn]+d[gc33]
(57)
The two components of t in Eq. (57) are denoted as 
tn=[DngTn],tc=[gc33]
(58)
It can be seen that tn is independent of the value of d. Using Eq. (3), the center of twist tc, Tc, is calculated as 
rc=1c33Ωg
(59)
Thus, Tw defined in Eq. (54) can be expressed as 
Tw={t=tn+dtc,d(,)}
which means that every t associated with w lies on the straight line lt passing through the centers of tn and tc.

Since tc is the third column of the compliance matrix, Eq. (59) indicates that it is located at the center of compliance Cc (Tc = Cc). Thus, the t associated with w must lie on the straight line lt passing through the compliance center Cc as illustrated in Fig. 8(a). Since C is positive definite, wTt=wTCw>0, the axis of w never intersects the corresponding center of t. Therefore, w and t are always separated by line lc as shown in Fig. 8(b). When w approaches the line passing through the center of compliance, lc, then wTtc → 0, and the center if the corresponding t goes to infinity along the line lt as shown in Fig. 8(c). When w is far away from the compliance center, d → ±, then the center of t approaches Cc, the center of the compliance, as shown in Fig. 8(d).

In summary, we have

Proposition 6. Consider a family of parallel wrenchesWpand the corresponding set of twistsTwdefined in Eq. (54).

  • (a)

    The locus oftTwis a straight line lt passing through the center of compliance.

  • (b)

    For anywWp, the corresponding twisttTwis separated by line lc, the line passing through the center of compliance Cc.

  • (c)

    When the axis ofwWpapproaches the line lc passing through compliance center, the location oftgoes to infinity along line lt.

  • (d)

    When the axis ofwWpis far away from the compliance center, d → ±∞ and the location oftapproaches the center of compliance Cc along lt. ◻

Conversely, consider the collection of twists, Tl, located on a straight line lt that passes through the center of stiffness Ck, and denote Wt as the collection of wrenches that result from multiplying the stiffness matrix K by the twists in Tl, i.e., 
Wt={w=Kt,tTl}
(60)

Then, as shown below, Wt is a family of parallel wrenches.

Let tc be a twist at the stiffness center and t0 be an arbitrary point on lt. Any twist on the line can be expressed as 
t=αtc+βt0
where α and β are scalars. The corresponding resulting wrench is 
w=Kt=αKtc+βKt0
(61)
Since tc is at the center of stiffness, Ktc is a pure couple having the form 
wc=Ktc=[0,0,m]T

Thus, the wrench w = Kt has the direction of wrench w0 = Kt0 regardless of the values of α and β in Eq. (61). Therefore, all wrenches in Wt are parallel to the wrench w0, which means that Wt forms a family of parallel wrenches.

Suppose that lc is the line of action of the wrench in Wt passing through the stiffness center Ck. Similar to Proposition 6, we have:

Proposition 7. Consider a family of twistsTllocated on a straight line lt passing through the stiffness center Ck and the set of corresponding wrenchesWtdefined in Eq. (60).

  • (a)

    Wtforms a family of parallel wrenches.

  • (b)

    For any twisttTl, the line of action of the corresponding wrenchwis on the opposite side of line lc.

  • (c)

    When the center of twistt→ Ck along line lt, the line of action of the correspondingwgoes to infinity.

  • (d)

    Whent→ ±∞ along line lt, the line of action of the correspondingwapproaches line lc. ◻

The properties of Wt are illustrated in Fig. 9.

Synthesis Procedures.

In this subsection, the results of Propositions 6 and 7 are used in synthesis procedures to realize an elastic behavior with a four-joint serial mechanism or a four-spring parallel mechanism.

Serial Mechanism Synthesis.

The procedure for a four-joint serial mechanism is presented below. This procedure identifies the locations of four joints in a serial mechanism and the corresponding compliance values that realize a specified compliance matrix C having compliance center Cc. The geometry associated with the sequence of operations in the synthesis procedure is illustrated in Fig. 10.

  1. (1)
    Choose a line l1 arbitrarily relative to Cc and obtain the unit wrench w1 associated with l1. Two revolute joints of the serial mechanism will be located on this line. Calculate the twist resulting from that wrench acting on C 
    t1=Cw1
    (62)
    The center of t1, T1, is determined using Eq. (3).
  2. (2)
    Choose a line l2 passing through the point T1 and obtain the unit wrench w2 associated with l2. The other two joints of the mechanism will be on this line. Calculate the twist resulting from w2 
    t2=Cw2
    (63)
    The center of t2, T2, will be on line l1 (the axis of w1), i.e., 
    w1TCw2=0
    (64)
    The line passing through points T1 and T2 is denoted as l.
  3. (3)

    Choose a line l3 on one side of line l such that the center of the corresponding twist t3 = Cw3 is on the opposite side of l.

    Line l3 can be selected using the properties of Proposition 6 presented in Sec. 4.1.

    • (a)

      First, choose a candidate line lp that does not intersect line segment T1T2. Obtain the unit wrench w̃p associated with lp and calculate the corresponding twist tp=Cw̃p. Then, by the property presented in Sec. 4.1, the center of tp, Tp, and the compliance center Cc must be on the same side of lp.

    • (b)

      Translate lp toward Cc until Tp is on the opposite side of l. This line is selected as l3 and the center of the corresponding twist t3 is T3.

      This can always be accomplished due to the fact that when lpCc, Tp.

  4. (4)

    Choose a line l4 that passes through point T3 and does not intersect line segment T1T2. The intersections of these four lines (shown in Fig. 10) are the locations of the four joints (J1, J2, J3, J4) of the mechanism that realizes the specified compliance.

  5. (5)

    Calculate the value of joint compliance for each joint using Eqs. (16)(19).

It can be seen that when the four joint locations are selected by this process, the realization conditions in Proposition 4 are satisfied. Therefore, the compliance C is realized with a serial mechanism having the obtained configuration.

Parallel Mechanism Synthesis.

The procedure for a four-spring parallel mechanism is presented below. This procedure identifies the locations of four line springs and their stiffness values in a parallel mechanism that realizes a specified stiffness matrix K having stiffness center Ck. The geometry associated with the sequence of operations in the synthesis procedure is illustrated in Fig. 11.

  1. (1)
    Choose an arbitrary location T1 relative to the stiffness center Ck and obtain the unit twist t1 at T1 using the formula for tr in Eq. (1). Two spring axes will intersect here in the parallel mechanism. The wrench resulting from t1 is 
    w̃1=Kt1
    (65)
    The line of action of w̃1, l1, is obtained.
  2. (2)
    Choose an arbitrary point T2 on line l1, the unit twist t2 centered at T2 is calculated using Eq. (1). Point T2 is the intersection of the other two spring axes. Then, wrench w̃2 corresponding to t2 is obtained 
    w̃2=Kt2
    (66)
    The line of action of w̃2, l2 is obtained. By the reciprocal condition, line l2 must pass through T1, and satisfy 
    t1TKt2=0
    (67)
    The line passing through points T1 and T2 is denoted as l.
  3. (3)

    Choose a line l3 such that the corresponding wrench w̃3 results in a twist t3=Cw̃3 centered at the opposite side of line l.

    Line l3 can be selected using the properties of Proposition 7 presented in Sec. 4.1.

    • (a)

      First choose a candidate line lp that does not intersect line segment T1T2. Using the unit wrench along lp, w̃p, the corresponding twist tp=Cwp is calculated. Then, by the property presented in Sec. 4.1, the center of tp, Tp, and the stiffness center Ck are both located on the same side of lp.

    • (b)

      Translate lp toward Ck until Tp is on the opposite side of l. This location Tp is selected as T3 which will be the intersection of two spring axes in the mechanism.

      This can always be accomplished due to the fact that when lpCk, Tp. Line l3 intersects l1 at T13 and intersects l2 at T23.

  4. (4)

    Choose an arbitrary point T4 on line l3 between T13 and T23, the four lines passing through points (T1, T3), (T1, T4), (T2, T4), and (T2, T3) as shown in Fig. 11 are identified as the four spring axes (w1, w2, w3, w4) for the parallel mechanism.

  5. (5)

    Calculate the value of stiffness for each joint using Eqs. (36)(39).

It can be seen that when the four spring axes are selected by this process, the realization conditions in Proposition 5 are satisfied. Therefore, the stiffness K is realized with the parallel mechanism.

Examples

Examples are provided to illustrate the synthesis procedures. The elastic behavior to be realized in a given coordinate frame is given by 
K=[321265159],C=K1=[1.160.920.640.921.040.680.640.680.56]

The location of the center of stiffness/compliance for this behavior is calculated to be at ((17/14),(16/14)). Since the center must be inside the polygon formed by the locations of the four elastic joints in a serial mechanism (or the four spring components in a parallel mechanism), this point is used as a reference in selecting each elastic component.

Serial Mechanism Synthesis.

Following the procedure provided in Sec. 4.2.1, the locations of joints in a serial mechanism and their joint compliances can be obtained.

The first line on which two joints J1 and J2 lie can be chosen arbitrarily. Here, the line is chosen to make a 45 deg angle with the x-axis and the distance from the line to the origin of the coordinate frame is d1=0.52 as shown in Fig. 12. The line vector of l1 (unit wrench) is 
w1=[cos45°,sin45°,0.52]T

where the first two components of w1 indicate the direction and the third component indicates the perpendicular distance of the line to the origin (according to the right-hand rule).

The twist t1 associated with w1 is calculated to be 
t1=Cw1=[1.4400,1.2800,0.7600]T

and using Eq. (3), the center of t1, T1, is calculated to be located at (1.6842, −1.8947).

The other two joints, J3 and J4, must lie on a line passing through point T1. Since T1 is determined, line l2 can be determined by selecting the direction or slope arbitrarily. Here, line l2 is selected to be parallel to the y-axis as shown in Fig. 12. It can be seen that the distance from l2 to the origin is d2 = 1.6842. The line vector of l2 is 
w2=[0,1,1.6842]T
The twist t2 associated with w2 is calculated to be 
t2=Cw2=[0.1579,0.1053,0.2632]T

The center of t2, T2, is calculated using Eq. (3) to be (0.4, −0.6). Note that T2 lies on line l1.

The selection of line l3 determines the location of joints J2 and J3. Line l3 must be selected such that the location of the corresponding twist, T3, is on the opposite side of line l connecting points T1 and T2. Here, line l3 is selected to have slope −1 and distance 0.4 from the origin as shown in Fig. 12. The line vector associated with l3 is 
w3=[cos45°,sin45°,0.4]T
The twist associated with w3 is calculated to be 
t3=Cw3=[0.4257,0.1871,0.1957]T

The center of t3, T3, is located at (0.9562, −2.1751).

Finally, a line passing through T3 is selected. The selection of this line l4 determines the location of joints J1 and J4. It must be selected such that it does not intersect line segment T1T2. Here, line l4 is selected to be parallel to the x-axis as shown in Fig. 13. The line vector associated with l4 is 
w4=[1,0,2.1751]T

Then, the locations of the four joints J1, J2, J3, and J4 are determined by the intersections of two lines (l1, l4), (l1, l3), (l2, l3), and (l3, l4), respectively, as shown in Fig. 12.

The locations of the four joints are 
r1=[1.1751,2.1751]T,r2=[0.7828,0.2172]T,r3=[1.6842,1.1185]T,r4=[1.6842,2.1751]T
Since l1 passes through J1 and J2, wrench w1 is reciprocal to t1 and t2. Thus 
w12=w1=22[1,1,1]T
Similarly 
w23=w3=[cos45°,sin45°,0.4]Tw34=w2=[0,1,1.6842]Tw14=w4=[1,0,2.1751]T
The joint compliance constants are determined using Eqs. (16)(19) 
c1=0.0180,c2=0.2349,c3=0.1119,c4=0.1953
With this procedure, the mechanism configuration (illustrated in Fig. 12) and the corresponding joint compliances are determined. To verify the result, the joint twists associated with joint Ji are calculated as 
[t1,t2,t3,t4]=[2.17510.21721.11852.17511.17510.78281.68421.68421111]
Summing of the elastic components using Eq. (2) yields 
i=14cititiT=[1.160.920.640.921.040.680.640.680.56]
which confirms the realization of the compliance with the mechanism.

Note that by the synthesis procedure, the mechanism geometry is identified by selecting the locations of the four joints. In the construction of a serial mechanism, the connection order of these four joints does not influence the compliance achieved with the mechanism.

Parallel Mechanism Synthesis.

Similar to the serial case, the stiffness is synthesized with a parallel mechanism using the procedure described in Sec. 4.2.2. The mechanism geometry associated with the sequence of operations in the synthesis procedure is illustrated in Fig. 13.

First, point T1 can be selected arbitrarily. Here, T1 is where spring wrenches w1 and w2 will intersect and is selected to be (2, −1.5). The unit twist at T1 is 
t1=[1.5,2,1]T
The wrench associated with t1 is calculated to be 
w̃1=Kt1=[0.5,4,2.5]T
The line of action of w̃1, l1, is obtained as 
y=8x+5
(68)

and is shown in Fig. 13.

The second point T2 can be arbitrarily selected from line l1. The other two spring wrenches w3 and w4 intersect at this point. Here, point T2 is selected such that line T1T2 is parallel to the x-axis. Using Eq. (68), the coordinates of T2 are calculated to be (0.8125, −1.5). The unit twist at T2 is 
t2=[1.5,0.8125,1]T
The wrench associated with t2 is 
w̃2=Kt2=[1.875,3.125,3.4375]T
The line of action of w̃2, l2, is obtained as 
y=1.6667x+1.8333

and is shown in Fig. 13. Note that l2 passes through T1.

Next, line l3 is selected. Here, the line is selected to be parallel to x-axis again and to have the distance d3 = 0.5 to the origin of the coordinate frame as shown in Fig. 13. The line vector associated with l3 is 
w̃3=[1,0,0.5]T
The twist associated with w̃3 is calculated to be 
t3=Cw̃3=[0.84,0.58,0.36]T

The center of t3, T3, is located at (1.6111, −2.3333).

The final point T4 can be selected from any point of the line segment T13T23. Here, this point is selected to be located at (1, −0.5) (illustrated in Fig. 13). The four-spring wrench axes are identified by the four lines T1T3, T1T4, T2T4, and T2T3. The four spring wrenches are 
[w1,w2,w3,w4]=[0.42290.70710.18430.69190.90620.70710.98290.72202.44670.35361.07500.4512]
In order to calculate the spring constants using Eqs. (36)(39), four twists tij reciprocal to wrenches wi and wj are needed. Since t12, t34, t14, and t23 are located at T1, T2, T3, and T4, respectively, they can be easily calculated using Eq. (1). The four twists needed are 
[t12,t34,t14,t23]=[1.51.50002.33330.52.00.81251.61111.01111]
The spring constants calculated using Eqs. (36)(39) are 
k1=0.8740,k2=3.8278,k3=2.5359,k4=1.7624
With this procedure, the mechanism configuration (illustrated in Fig. 13) and the corresponding spring stiffnesses are determined. To verify the result, summing the elastic components using Eq. (46) yields 
i=14kiwiwiT=[321265159]
which confirms that the stiffness is realized with the mechanism.

Note that by the synthesis procedure, the mechanism geometry is identified by selecting the line of action (axis) for each spring. In the construction of a parallel mechanism, each spring can be anywhere along its axis.

Summary

In this paper, the realization of an arbitrary planar elastic behavior using redundant serial and parallel mechanisms having four elastic components is addressed. A set of necessary and sufficient conditions for a mechanism to realize a given planar compliance is presented and the physical interpretations of the realization conditions in terms of the mechanism geometry are provided. Since the conditions on the mechanism geometry and joint compliances are completely decoupled, the methods identified can be used for mechanisms having VSAs to realize a given compliance by changing the mechanism configuration and joint stiffnesses. In application, one can use the method to design a compliant mechanism with better geometry from the infinite, but restricted, set of candidates available. Since those restrictions are explicit in the mechanism geometry, the method makes it possible to use graphic tools to readily design a compliant mechanism for the realization of any specified planar compliance.

Funding Data

  • National Science Foundation (Grant No. IIS-1427329).

References

References
1.
Huang
,
S.
, and
Schimmels
,
J. M.
,
1998
, “
The Bounds and Realization of Spatial Stiffnesses Achieved With Simple Springs Connected in Parallel
,”
IEEE Trans. Rob. Autom.
,
14
(
3
), pp.
466
475
.
2.
Huang
,
S.
, and
Schimmels
,
J. M.
,
2000
, “
The Bounds and Realization of Spatial Compliances Achieved With Simple Serial Elastic Mechanisms
,”
IEEE Trans. Rob. Autom.
,
16
(
1
), pp.
99
103
.
3.
Huang
,
S.
, and
Schimmels
,
J. M.
,
2002
, “
The Duality in Spatial Stiffness and Compliance as Realized in Parallel and Serial Elastic Mechanisms
,”
ASME J. Dyn. Syst., Meas., Control
,
124
(
1
), pp.
76
84
.
4.
Ham
,
R. V.
,
Sugar
,
T. G.
,
Vanderborght
,
B.
,
Hollander
,
K. W.
, and
Lefeber
,
D.
,
2009
, “
Compliant Actuator Designs: Review of Actuators With Passive Adjustable Compliance/Controllable Stiffness for Robotic Applications
,”
IEEE Rob. Autom. Mag.
,
16
(
3
), pp.
81
94
.
5.
Huang
,
S.
, and
Schimmels
,
J. M.
,
2016
, “
Realization of Point Planar Elastic Behaviors Using Revolute Joint Serial Mechanisms Having Specified Link Lengths
,”
Mech. Mach. Theory
,
103
, pp.
1
20
.
6.
Huang
,
S.
, and
Schimmels
,
J. M.
,
2016
, “
Synthesis of Point Planar Elastic Behaviors Using 3-Joint Serial Mechanisms of Specified Construction
,”
ASME J. Mech. Rob.
,
9
(
1
), p.
011005
.
7.
Huang
,
S.
, and
Schimmels
,
J. M.
,
2017
, “
Geometric Construction-Based Realization of Planar Elastic Behaviors With Parallel and Serial Manipulators
,”
ASME J. Mech. Rob.
,
9
(
5
), p.
051006
.
8.
Ball
,
R. S.
,
1900
,
A Treatise on the Theory of Screws
,
Cambridge University Press
,
Cambridge, UK
.
9.
Dimentberg
,
F. M.
,
1965
,
The Screw Calculus and Its Applications in Mechanics
,
Foreign Technology Division, Wright-Patterson Air Force Base
,
Dayton, OH
.
10.
Griffis
,
M.
, and
Duffy
,
J.
,
1991
, “
Kinestatic Control: A Novel Theory for Simultaneously Regulating Force and Displacement
,”
ASME J. Mech. Des.
,
113
(
4
), pp.
508
515
.
11.
Patterson
,
T.
, and
Lipkin
,
H.
,
1993
, “
Structure of Robot Compliance
,”
ASME J. Mech. Des.
,
115
(
3
), pp.
576
580
.
12.
Loncaric
,
J.
,
1987
, “
Normal Forms of Stiffness and Compliance Matrices
,”
IEEE J. Rob. Autom.
,
3
(
6
), pp.
567
572
.
13.
Roberts
,
R. G.
,
1999
, “
Minimal Realization of a Spatial Stiffness Matrix With Simple Springs Connected in Parallel
,”
IEEE Trans. Rob. Autom.
,
15
(
5
), pp.
953
958
.
14.
Ciblak
,
N.
, and
Lipkin
,
H.
,
1999
, “
Synthesis of Cartesian Stiffness for Robotic Applications
,”
IEEE
International Conference on Robotics and Automation
, Detroit, MI, May 10–15, pp.
2147
2152
.
15.
Huang
,
S.
, and
Schimmels
,
J. M.
,
1998
, “
Achieving an Arbitrary Spatial Stiffness With Springs Connected in Parallel
,”
ASME J. Mech. Des.
,
120
(
4
), pp.
520
526
.
16.
Roberts
,
R. G.
,
2000
, “
Minimal Realization of an Arbitrary Spatial Stiffness Matrix With a Parallel Connection of Simple Springs and Complex Springs
,”
IEEE Trans. Rob. Autom.
,
16
(
5
), pp.
603
608
.
17.
Huang
,
S.
, and
Schimmels
,
J. M.
,
2000
, “
The Eigenscrew Decomposition of Spatial Stiffness Matrices
,”
IEEE Trans. Rob. Autom.
,
16
(
2
), pp.
146
156
.
18.
Huang
,
S.
, and
Schimmels
,
J. M.
,
2001
, “
A Classification of Spatial Stiffness Based on the Degree of Translational–Rotational Coupling
,”
ASME J. Mech. Des.
,
123
(
3
), pp.
353
358
.
19.
Huang
,
S.
, and
Schimmels
,
J. M.
,
2001
, “
Minimal Realizations of Spatial Stiffnesses With Parallel or Serial Mechanisms Having Concurrent Axes
,”
J. Rob. Syst.
,
18
(
3
), pp.
135
246
.
20.
Huang
,
S.
, and
Schimmels
,
J. M.
,
2002
, “
Realization of Those Elastic Behaviors That Have Compliant Axes in Compact Elastic Mechanisms
,”
J. Rob. Syst.
,
19
(
3
), pp.
143
154
.
21.
Choi
,
K.
,
Jiang
,
S.
, and
Li
,
Z.
,
2002
, “
Spatial Stiffness Realization With Parallel Springs Using Geometric Parameters
,”
IEEE Trans. Rob. Autom.
,
18
(
3
), pp.
274
284
.
22.
Hong
,
M. B.
, and
Choi
,
Y. J.
,
2009
, “
Screw System Approach to Physical Realization of Stiffness Matrix With Arbitrary Rank
,”
ASME J. Mech. Rob.
,
1
(
2
), p.
021007
.
23.
Huang
,
S.
, and
Schimmels
,
J. M.
,
2011
, “
Realization of an Arbitrary Planar Stiffness With a Simple Symmetric Parallel Mechanism
,”
ASME J. Mech. Rob.
,
3
(
4
), p.
041006
.
24.
Petit
,
F. P.
,
2014
, “Analysis and Control of Variable Stiffness Robots,”
Ph.D. thesis
, ETH Zurich, Zurich, Switzerland.
25.
Verotti
,
M.
, and
Belfiore
,
N. P.
,
2016
, “
Isotropic Compliance in E(3): Feasibility and Workspace Mapping
,”
ASME J. Mech. Rob.
,
8
(
6
), p.
061005
.
26.
Verotti
,
M.
,
Masarati
,
P.
,
Morandini
,
M.
, and
Belfiore
,
N.
,
2016
, “
Isotropic Compliance in the Special Euclidean Group SE(3)
,”
Mech. Mach. Theory
,
98
, pp.
263
281
.