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 [1–3] 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 [5–7]. 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 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. [14–16]. 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 [19–22]. 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.

*J*can be represented by the joint twist

_{i}**t**

*. The planar joint twist for a revolute joint and for a prismatic joint can be expressed in Plücker axis coordinates as*

_{i}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.

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 **t*** _{r}* and a prismatic joint along each prismatic twist

**t**

*(with corresponding assigned joint compliance*

_{p}*c*for each joint), then the compliance behavior is realized with the mechanism. Thus, a decomposition of

_{i}**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.

**t**

*, a unique point, the instantaneous center of rotation for the twist motion, is calculated using*

_{r}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.

where **n** is a unit 2-vector indicating the direction of the spring axis, $d=(r\u0303\xd7n)\xb7k$ is a scalar indicating the distance of the spring axis from the coordinate frame, $r\u0303$ 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.

**w**

*and stiffness*

_{i}*k*> 0, then the Cartesian stiffness associated with the mechanism is

_{t}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 **w*** _{i}* and spring stiffness

*k*so that the given stiffness is achieved.

_{i}**w**

*is given, the perpendicular position*

_{l}**r**to the wrench axis can be calculated using

where $\Omega $ is the matrix defined in Eq. (4).

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

**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

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 **t*** _{p}* 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.

*J*,

_{i}*i*= 1, 2, 3, 4. The location of each

*J*is uniquely specified by its twist

_{i}**t**

*. A wrench*

_{i}**w**

*that is reciprocal to two joint twists*

_{ij}**t**

*and*

_{i}**t**

*must satisfy both*

_{j}**w**

*must pass through both joints*

_{ij}*J*and

_{i}*J*(the locations of the two twist centers) and thus is uniquely determined by

_{j}**t**

*and*

_{i}**t**

*. Mathematically, wrench*

_{j}**w**

*can be calculated by*

_{ij}**w**reciprocal to

**t**

*and*

_{i}**t**

*has the same line of action of*

_{j}**w**

*and can be expressed as*

_{ij}where *α* is a scalar.

**C**, and two wrenches

**w**

_{12}and

**w**

_{34}that pass through joints (

*J*

_{1},

*J*

_{2}) and joints (

*J*

_{3},

*J*

_{4}), respectively. For a four-joint realization,

**C**can be expressed in the form of

*J*

_{3}and

*J*

_{4}[using Eq. (9)] yields

*J*

_{1}and

*J*

_{2}(using Eq. (9) again) yields

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

**C**can be expressed in the form of Eq. (11), then for any permutation of {1, 2, 3,4}

**C**is symmetric $(wijTCwrs=wrsTCwij)$ and that

**w**

*and*

_{ij}**w**

*have the same line of action (*

_{ji}**w**

*=*

_{ij}*α*

**w**

*). Thus, condition (13) for all permutations of {1, 2, 3, 4} is equivalent to the set of three equations*

_{ji}The sufficiency of this set of conditions is evaluated next.

**w**

*and*

_{ij}**w**

*yields the following four equations:*

_{ir}Note that the four Eqs. (20) and (21) are satisfied due to the definition of $C\u0303$ in Eq. (15) and are independent from each other if any three twists **t*** _{i}* 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 $C\u0303=C$. 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).

**C**can be expressed in the form of Eq. (11), then for all permutations {

*i*,

*j*,

*r*,

*s*} of {1, 2, 3, 4}

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).

For two permutations of {1, 2, 3, 4} with different values of *s*, Eq. (28) gives two independent equations. Since $(C\u0303\u2212C)$ must satisfy the four Eqs. (20) and (21), the 3 × 3 symmetric matrix $(C\u2212C\u0303)$ satisfies six independent homogeneous equations. Thus, $C\u2212C\u0303=0$, which means that $C=C\u0303$ 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 matrix**C**can 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*$wijTCwir(wijTts)(wirTts)=wijTCwrj(wijTts)(wrjTts)\u2003\u25fb$(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 **t*** _{i}* if there are no constraints on the coefficients

*c*'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.

_{i}*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)]

Proposition 2. *A compliance matrix**C**can be passively realized with a four-joint serial mechanism having joint twists**t*_{i} 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*$(wijTCwir)(wijTts)(wirTts)\u22650\u2003\u25fb$(32)

### Parallel Mechanisms With Four Springs.

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

**w**

*. If*

_{i}**t**

*is the twist that is reciprocal to both*

_{ij}**w**

*and*

_{i}**w**

*, then*

_{j}**t**

*is located at the intersection of the two lines along the spring wrenches*

_{ij}**w**

*and*

_{i}**w**

*. Similar to Eq. (10),*

_{j}**t**

*can be determined by*

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

Proposition 3. *A stiffness matrix**K**can be passively realized with a four-spring parallel mechanism having spring wrenches**w*_{i} 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*$tijTKtir(tijTws)(tirTws)\u22650\u2003\u25fb$(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}.

**K**can be realized with the mechanism, the stiffness for each spring can be determined by

### Realization Uniqueness.

*at a given configuration*of a redundant four-joint serial mechanism, the joint compliances

*c*typically are unique. If any three

_{i}**t**

*'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*

_{i}**t**

*'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*

_{i}*c*can be calculated using any permutation of {

_{i}*j*,

*r*,

*s*}

Although *c _{i}* can be obtained with different permutations, Proposition 1b guarantees that the obtained compliance

*c*is the same for the given joint twist

_{i}**t**

*.*

_{i}## 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.

**w**

*is a force along line*

_{ij}*l*passing through both joints

_{ij}*J*and

_{i}*J*, then the twist resulting from

_{j}**w**

*acting on the compliant mechanism is*

_{ij}**t**

*must be on the line of action of*

_{ij}**w**

*, i.e., on line*

_{rs}*l*passing through joints

_{rs}*J*and

_{r}*J*. Thus, any force along line

_{s}*l*results in a twist on line

_{ij}*l*.

_{rs}Below, we show that the inequality conditions (32) for passive realization require that the locus of **t*** _{ij}* be on a segment along line

*l*bounded by joints

_{rs}*J*and

_{r}*J*. Two cases are considered: either line

_{s}*l*intersects

_{ij}*l*between

_{rs}*J*and

_{s}*J*or it does not.

_{r}First, consider the case in which line *l _{ij}* does not intersect

*l*between

_{rs}*J*and

_{r}*J*. For the four-joint locations illustrated in Fig. 2, consider, without loss of generality,

_{s}**w**

_{12}and

**w**

_{34}.

**C**is passively realized with the mechanism, then

**t**

_{12}=

**Cw**

_{12}must be located on the segment

*J*

_{3}

*J*

_{4}. Since

**C**can be expressed in the form of Eq. (11) with

*c*≥ 0

_{i}Since *J*_{3} and *J*_{4} are on the same side of *l*_{12}, $t3Tw12$ and $t4Tw12$ have the same sign. Since *c*_{3} and *c*_{4} are positive, $c3(t3Tw12)$ and $c4(t4Tw12)$ have the same sign. Thus, the positive (or negative) combination of **t**_{3} and **t**_{4} in Eq. (44) indicates that **t**_{12} must be located on the line segment between *J*_{3} and *J*_{4}.

Conversely, if **t**_{12} is located on the finite segment between *J*_{3} and *J*_{4}, then the coefficients of **t**_{3} and **t**_{4} in Eq. (44) must have the same sign, i.e., $c3(t3Tw12)$ and $c4(t4Tw12)$ have the same sign. Therefore, *c*_{3} and *c*_{4} must also have the same sign. Below, we show that *c*_{3} and *c*_{4} must be non-negative.

**C**is PSD

Therefore, *c*_{3} and *c*_{4} must be non-negative. The geometric interpretation of the realization conditions for this case is illustrated in Fig. 2.

*l*intersects

_{ij}*l*at the segment between

_{rs}*J*and

_{r}*J*. Here, we suppose, without loss of generality, that

_{s}*l*

_{13}intersects line

*l*

_{24}between

*J*

_{2}and

*J*

_{4}. 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

*c*≥ 0. Then

_{i}Since *J*_{2} and *J*_{4} are on the opposite sides of line *l*_{13}, $t2Tw13$ and $t4Tw13$ have opposite signs. Thus, **t**_{13} must be on line *l*_{24} outside the finite segment bounded by *J*_{2} and *J*_{4}. Conversely, if **t**_{13} is located outside the line segment *J*_{2}*J*_{4}, the coefficients of **t**_{2} and **t**_{3} in Eq. (45), $c2(t2Tw13)$ and $c4(t4Tw13)$, have opposite signs. Since $(t2Tw13)$ and $(t4Tw13)$ have opposite signs, *c*_{1} and *c*_{2} have the same sign. Since **C** is PSD, *c*_{1} and *c*_{2} must be non-negative. Figure 3 shows the geometric interpretation of the realization conditions for this case.

For a serial mechanism with four joints *J _{i}* 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 compliance**C**can 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 l*_{12}*results in a twist located on line segment J*_{3}*J*_{4}*; and a force along l*_{34}*results in a twist located on line segment J*_{1}*J*_{2}. - (b)
*A force along l*_{23}*results in a twist located on line segment J*_{1}*J*_{4}*; and a force along l*_{14}*results in a twist located on line segment J*_{2}*J*_{3}*. ◻*

Note that in Proposition 4, condition (a) implies one equality condition $w12TCw34=0$ and four inequality conditions *c _{i}* ≥ 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.

**w**

*. A stiffness matrix*

_{i}**K**can be realized with the mechanism if and only if

where *k _{i}* ≥ 0.

**w**

*and*

_{i}**w**

*at*

_{j}*T*uniquely identifies a unit twist

_{ij}**t**

*reciprocal to both wrenches. The equality realization condition requires that any twist located at the intersection of*

_{ij}**w**

*and*

_{i}**w**

*results in a wrench that passes through the intersection of the axes of the other two wrenches,*

_{j}**w**

*and*

_{r}**w**

*. Below, we show that the wrench must be inside a portion of the pencil bounded by the lines along the axes of*

_{s}**w**

*and*

_{r}**w**

*. Since the wrench*

_{s}**w**

*=*

_{ij}**Kt**

*passes through the intersection of*

_{ij}**w**

*and*

_{r}**w**

*,*

_{s}**w**

*can be expressed as*

_{ij}*α*and

*β*are scalars. By inequality condition (35)

*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

**w**

*must be inside an area bounded by the two axes of*

_{ij}**w**

*and*

_{r}**w**

*. It is readily shown that the area does not contain the intersection of the axes of*

_{s}**w**

*and*

_{i}**w**

*,*

_{j}*T*. In fact, if a wrench

_{ij}**w**has axis passing through both locations of

*T*and

_{ij}*T*(the center of twist

_{rs}**t**

*),*

_{rs}**w**can be expressed as

If $\alpha \u0303$ and $(tijTwr)$ have the same sign, $\beta \u0303$ and $(tijTws)$ must have the opposite sign, and vice versa. Therefore, the wrench resulting from twist **t*** _{ij}* 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

**w**

*and*

_{i}**w**

*as shown in Fig. 4.*

_{j}**w**

*is in the area defined above (the shaded area in Fig. 4), the following equality conditions are satisfied:*

_{ij}In summary, for a parallel mechanism having four springs **w*** _{i}*, we have

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

- (a)
*A twist at the intersection of**w*_{1}*and**w*_{2}*, T*_{12}*, results in a wrench that passes through the intersection of**w*_{3}*and**w*_{4}*, T*_{34}*, and lies in the area bounded by the axes of**w*_{3}*and**w*_{4}*that does not contain T*_{12}*; and a twist at T*_{34}*results in a wrench that passes through the intersection of**w*_{1}*and**w*_{2}*, T*_{12}*and lies in the area bounded by the axes of**w*_{1}*and**w*_{2}*that does not contain T*_{34}. - (b)
*A twist at the intersection of**w*_{1}*and**w*_{3}*, T*_{13}*, results in a wrench that passes through the intersection of**w*_{2}*and**w*_{4}*and lies in the area bounded by the axes of**w*_{2}*and**w*_{4}*that does not contain T*_{13}*; and a twist at the intersection of**w*_{2}*and**w*_{4}*, T*_{24}*, results in a wrench that passes through the intersection of**w*_{1}*and**w*_{3}*and lies in the area bounded by the axes of**w*_{1}*and**w*_{3}*that does not contain T*_{24}. ◻

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 *k _{i}* ≥ 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.

**C**and two wrenches

**w**

_{1}and

**w**

_{2}satisfying

Then, the twist **t**_{1} = **Cw**_{1} must lie on the line of action of **w**_{2}, and the twist **t**_{2} = **Cw**_{2} must lie on the line of action of **w**_{1} as shown in Fig. 6. The following statements hold true:

- (a)
Any wrench

**w**passing through the intersection point of the two wrenches,*T*_{12}, when multiplied by**C**, results in a twist**t**located on line*l*_{12}passing through the centers of**t**_{1}and**t**_{2}as shown in Fig. 6. - (b)
Any wrench along line

*l*_{12}passing through the centers of**t**_{1}and**t**_{2}, when multiplied by**C**, results in a twist**t**_{12}located at*T*_{12}(where wrenches**w**_{1}and**w**_{2}intersect). - (c)
The center of compliance must be inside the triangle formed by the centers of twists

**t**_{1},**t**_{2,}and**t**_{12}(the shaded area shown in Fig. 6).

**w**

_{1}and

**w**

_{2}satisfy condition (51) and wrench

**w**passes through point

*T*

_{12}, the intersection of

**w**

_{1}and

**w**

_{2}, then

**w**can be expressed as

*α*and

*β*are scalars. The twist resulting from

**w**is

Thus, **t** must be on the line passing through the centers of twists **t**_{1} and **t**_{2}.

**w**

_{12}is a wrench passing through

**t**

_{1}and

**t**

_{2}

Thus, the twist resulting from **C** multiplying **w**_{12} is reciprocal to both **w**_{1} and **w**_{2}, which indicates that the twist center is located at the intersection of **w**_{1} and **w**_{2}, i.e., at *T*_{12}.

Since the three wrenches **w**_{1}, **w**_{2,} and **w**_{12} 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 **t**_{1}, **t**_{2,} and **t**_{12} 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 *l _{c}* that passes through the center of compliance

*C*.

_{c}**C**, i.e.,

It is shown below that the centers of twists in $Tw$ form a straight line *l _{t}* passing through the center of compliance

*C*.

_{c}**w**in $Wp$ has the same direction $n,\u2009Wp$ can be expressed in the form

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

**C**can be represented in block form as

**w**can then be written as

**t**

*is independent of the value of*

_{n}*d*. Using Eq. (3), the center of twist

**t**

*,*

_{c}*T*, is calculated as

_{c}**t**associated with

**w**lies on the straight line

*l*passing through the centers of

_{t}**t**

*and*

_{n}**t**

*.*

_{c}Since **t*** _{c}* is the third column of the compliance matrix, Eq. (59) indicates that it is located at the center of compliance

*C*(

_{c}*T*=

_{c}*C*). Thus, the

_{c}**t**associated with

**w**must lie on the straight line

*l*passing through the compliance center

_{t}*C*as illustrated in Fig. 8(a). Since

_{c}**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

*l*as shown in Fig. 8(b). When

_{c}**w**approaches the line passing through the center of compliance,

*l*, then

_{c}**w**

^{T}

**t**

*→ 0, and the center if the corresponding*

_{c}**t**goes to infinity along the line

*l*as shown in Fig. 8(c). When

_{t}**w**is far away from the compliance center,

*d*→ ±

*∞*, then the center of

**t**approaches

*C*, the center of the compliance, as shown in Fig. 8(d).

_{c}In summary, we have

Proposition 6. *Consider a family of parallel wrenches*$Wp$*and the corresponding set of twists*$Tw$*defined in Eq. (54)*.

- (a)
*The locus of*$t\u2208Tw$*is a straight line l*_{t}passing through the center of compliance. - (b)
*For any*$w\u2208Wp$*, the corresponding twist*$t\u2208Tw$*is separated by line l*_{c}, the line passing through the center of compliance C_{c}. - (c)
*When the axis of*$w\u2208Wp$*approaches the line l*_{c}passing through compliance center, the location of*t**goes to infinity along line l*_{t}. - (d)
*When the axis of*$w\u2208Wp$*is far away from the compliance center, d → ±∞ and the location of**t**approaches the center of compliance C*_{c}along l_{t}. ◻

*l*that passes through the center of stiffness

_{t}*C*, and denote $Wt$ as the collection of wrenches that result from multiplying the stiffness matrix

_{k}**K**by the twists in $Tl$, i.e.,

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

**t**

*be a twist at the stiffness center and*

_{c}**t**

_{0}be an arbitrary point on

*l*. Any twist on the line can be expressed as

_{t}*α*and

*β*are scalars. The corresponding resulting wrench is

**t**

*is at the center of stiffness,*

_{c}**Kt**

*is a pure couple having the form*

_{c}Thus, the wrench **w** = **Kt** has the direction of wrench **w**_{0} = **Kt**_{0} regardless of the values of *α* and *β* in Eq. (61). Therefore, all wrenches in $Wt$ are parallel to the wrench **w**_{0}, which means that $Wt$ forms a family of parallel wrenches.

Suppose that *l _{c}* is the line of action of the wrench in $Wt$ passing through the stiffness center

*C*. Similar to Proposition 6, we have:

_{k}Proposition 7. *Consider a family of twists*$Tl$*located on a straight line l _{t} passing through the stiffness center C_{k} and the set of corresponding wrenches*$Wt$

*defined in Eq. (60).*

- (a)
$Wt$

*forms a family of parallel wrenches.* - (b)
*For any twist*$t\u2208Tl$*, the line of action of the corresponding wrench**w**is on the opposite side of line l*_{c}. - (c)
*When the center of twist**t**→ C*_{k}along line l_{t}, the line of action of the corresponding*w**goes to infinity.* - (d)
*When**t**→ ±∞ along line l*_{t}, the line of action of the corresponding*w**approaches line l*_{c}. ◻

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 *C _{c}*. The geometry associated with the sequence of operations in the synthesis procedure is illustrated in Fig. 10.

- (1)Choose a line
*l*_{1}arbitrarily relative to*C*and obtain the unit wrench_{c}**w**_{1}associated with*l*_{1}. Two revolute joints of the serial mechanism will be located on this line. Calculate the twist resulting from that wrench acting on**C**The center of$t1=Cw1$(62)**t**_{1},*T*_{1}, is determined using Eq. (3). - (2)Choose a line
*l*_{2}passing through the point*T*_{1}and obtain the unit wrench**w**_{2}associated with*l*_{2}. The other two joints of the mechanism will be on this line. Calculate the twist resulting from**w**_{2}The center of$t2=Cw2$(63)**t**_{2},*T*_{2}, will be on line*l*_{1}(the axis of**w**_{1}), i.e.,The line passing through points$w1TCw2=0$(64)*T*_{1}and*T*_{2}is denoted as*l*. - (3)
Choose a line

*l*_{3}on one side of line*l*such that the center of the corresponding twist**t**_{3}=**Cw**_{3}is on the opposite side of*l*.Line

*l*_{3}can be selected using the properties of Proposition 6 presented in Sec. 4.1.- (a)
First, choose a candidate line

*l*that does not intersect line segment_{p}*T*_{1}*T*_{2}. Obtain the unit wrench $w\u0303p$ associated with*l*and calculate the corresponding twist $tp=Cw\u0303p$. Then, by the property presented in Sec. 4.1, the center of_{p}**t**,_{p}*T*, and the compliance center_{p}*C*must be on the same side of_{c}*l*._{p} - (b)
Translate

*l*toward_{p}*C*until_{c}*T*is on the opposite side of_{p}*l*. This line is selected as*l*_{3}and the center of the corresponding twist**t**_{3}is*T*_{3}.This can always be accomplished due to the fact that when

*l*→_{p}*C*,_{c}*T*→_{p}*∞*.

- (a)
- (4)
Choose a line

*l*_{4}that passes through point*T*_{3}and does not intersect line segment*T*_{1}*T*_{2}. The intersections of these four lines (shown in Fig. 10) are the locations of the four joints (*J*_{1},*J*_{2},*J*_{3},*J*_{4}) of the mechanism that realizes the specified compliance. - (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 *C _{k}*. The geometry associated with the sequence of operations in the synthesis procedure is illustrated in Fig. 11.

- (1)Choose an arbitrary location
*T*_{1}relative to the stiffness center*C*and obtain the unit twist_{k}**t**_{1}at*T*_{1}using the formula for**t**in Eq. (1). Two spring axes will intersect here in the parallel mechanism. The wrench resulting from_{r}**t**_{1}isThe line of action of $w\u03031$,$w\u03031=Kt1$(65)*l*_{1}, is obtained. - (2)Choose an arbitrary point
*T*_{2}on line*l*_{1}, the unit twist**t**_{2}centered at*T*_{2}is calculated using Eq. (1). Point*T*_{2}is the intersection of the other two spring axes. Then, wrench $w\u03032$ corresponding to**t**_{2}is obtainedThe line of action of $w\u03032$,$w\u03032=Kt2$(66)*l*_{2}is obtained. By the reciprocal condition, line*l*_{2}must pass through*T*_{1}, and satisfyThe line passing through points$t1TKt2=0$(67)*T*_{1}and*T*_{2}is denoted as*l*. - (3)
Choose a line

*l*_{3}such that the corresponding wrench $w\u03033$ results in a twist $t3=Cw\u03033$ centered at the opposite side of line*l*.Line

*l*_{3}can be selected using the properties of Proposition 7 presented in Sec. 4.1.- (a)
First choose a candidate line

*l*that does not intersect line segment_{p}*T*_{1}*T*_{2}. Using the unit wrench along*l*, $w\u0303p$, the corresponding twist $tp=Cwp$ is calculated. Then, by the property presented in Sec. 4.1, the center of_{p}**t**,_{p}*T*, and the stiffness center_{p}*C*are both located on the same side of_{k}*l*._{p} - (b)
Translate

*l*toward_{p}*C*until_{k}*T*is on the opposite side of_{p}*l*. This location*T*is selected as_{p}*T*_{3}which will be the intersection of two spring axes in the mechanism.This can always be accomplished due to the fact that when

*l*→_{p}*C*,_{k}*T*→_{p}*∞*. Line*l*_{3}intersects*l*_{1}at*T*_{13}and intersects*l*_{2}at*T*_{23}.

- (a)
- (4)
Choose an arbitrary point

*T*_{4}on line*l*_{3}between*T*_{13}and*T*_{23}, the four lines passing through points (*T*_{1},*T*_{3}), (*T*_{1},*T*_{4}), (*T*_{2},*T*_{4}), and (*T*_{2},*T*_{3}) as shown in Fig. 11 are identified as the four spring axes (**w**_{1},**w**_{2},**w**_{3},**w**_{4}) for the parallel mechanism. - (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

The location of the center of stiffness/compliance for this behavior is calculated to be at $((17/14),\u2212(16/14))$. 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.

*J*

_{1}and

*J*

_{2}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

*l*

_{1}(unit wrench) is

where the first two components of **w**_{1} indicate the direction and the third component indicates the perpendicular distance of the line to the origin (according to the right-hand rule).

**t**

_{1}associated with

**w**

_{1}is calculated to be

and using Eq. (3), the center of **t**_{1}, *T*_{1}, is calculated to be located at (1.6842, −1.8947).

*J*

_{3}and

*J*

_{4}, must lie on a line passing through point

*T*

_{1}. Since

*T*

_{1}is determined, line

*l*

_{2}can be determined by selecting the direction or slope arbitrarily. Here, line

*l*

_{2}is selected to be parallel to the

*y*-axis as shown in Fig. 12. It can be seen that the distance from

*l*

_{2}to the origin is

*d*

_{2}= 1.6842. The line vector of

*l*

_{2}is

**t**

_{2}associated with

**w**

_{2}is calculated to be

The center of **t**_{2}, *T*_{2}, is calculated using Eq. (3) to be (0.4, −0.6). Note that *T*_{2} lies on line *l*_{1}.

*l*

_{3}determines the location of joints

*J*

_{2}and

*J*

_{3}. Line

*l*

_{3}must be selected such that the location of the corresponding twist,

*T*

_{3}, is on the opposite side of line

*l*connecting points

*T*

_{1}and

*T*

_{2}. Here, line

*l*

_{3}is selected to have slope −1 and distance 0.4 from the origin as shown in Fig. 12. The line vector associated with

*l*

_{3}is

**w**

_{3}is calculated to be

The center of **t**_{3}, *T*_{3}, is located at (0.9562, −2.1751).

*T*

_{3}is selected. The selection of this line

*l*

_{4}determines the location of joints

*J*

_{1}and

*J*

_{4}. It must be selected such that it does not intersect line segment

*T*

_{1}

*T*

_{2}. Here, line

*l*

_{4}is selected to be parallel to the

*x*-axis as shown in Fig. 13. The line vector associated with

*l*

_{4}is

Then, the locations of the four joints *J*_{1}, *J*_{2}, *J*_{3}, and *J*_{4} are determined by the intersections of two lines (*l*_{1}, *l*_{4}), (*l*_{1}, *l*_{3}), (*l*_{2}, *l*_{3}), and (*l*_{3}, *l*_{4}), respectively, as shown in Fig. 12.

*l*

_{1}passes through

*J*

_{1}and

*J*

_{2}, wrench

**w**

_{1}is reciprocal to

**t**

_{1}and

**t**

_{2}. Thus

*J*are calculated as

_{i}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.

*T*

_{1}can be selected arbitrarily. Here,

*T*

_{1}is where spring wrenches

**w**

_{1}and

**w**

_{2}will intersect and is selected to be (2, −1.5). The unit twist at

*T*

_{1}is

**t**

_{1}is calculated to be

*l*

_{1}, is obtained as

and is shown in Fig. 13.

*T*

_{2}can be arbitrarily selected from line

*l*

_{1}. The other two spring wrenches

**w**

_{3}and

**w**

_{4}intersect at this point. Here, point

*T*

_{2}is selected such that line

*T*

_{1}

*T*

_{2}is parallel to the

*x*-axis. Using Eq. (68), the coordinates of

*T*

_{2}are calculated to be (0.8125, −1.5). The unit twist at

*T*

_{2}is

**t**

_{2}is

*l*

_{2}, is obtained as

and is shown in Fig. 13. Note that *l*_{2} passes through *T*_{1}.

*l*

_{3}is selected. Here, the line is selected to be parallel to

*x*-axis again and to have the distance

*d*

_{3}= 0.5 to the origin of the coordinate frame as shown in Fig. 13. The line vector associated with

*l*

_{3}is

The center of **t**_{3}, *T*_{3}, is located at (1.6111, −2.3333).

*T*

_{4}can be selected from any point of the line segment

*T*

_{13}

*T*

_{23}. 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

*T*

_{1}

*T*

_{3},

*T*

_{1}

*T*

_{4},

*T*

_{2}

*T*

_{4}, and

*T*

_{2}

*T*

_{3}. The four spring wrenches are

**t**

*reciprocal to wrenches*

_{ij}**w**

*and*

_{i}**w**

*are needed. Since*

_{j}**t**

_{12},

**t**

_{34},

**t**

_{14}, and

**t**

_{23}are located at

*T*

_{1},

*T*

_{2},

*T*

_{3}, and

*T*

_{4}, respectively, they can be easily calculated using Eq. (1). The four twists needed are

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).