This paper addresses the passive realization of any selected planar elastic behavior with a parallel or a serial manipulator. Sets of necessary and sufficient conditions for a mechanism to passively realize an elastic behavior are presented. These conditions completely decouple the requirements on component elastic properties from the requirements on mechanism kinematics. The restrictions on the set of elastic behaviors that can be realized with a mechanism are described in terms of acceptable locations of realizable elastic behavior centers. Parallel–serial mechanism pairs that realize identical elastic behaviors (dual elastic mechanisms) are described. New construction-based synthesis procedures for planar elastic behaviors are developed. Using these procedures, one can select the geometry of each elastic component from a restricted space of kinematically allowable candidates. With each selection, the space is further restricted until the desired elastic behavior is achieved.

Introduction

Compliant behavior in manipulation is an important topic in robotics research and industrial application. General compliant behavior can be modeled as a body suspended by an elastic parallel or serial mechanism. For small displacements from equilibrium, elastic behavior can be described by a symmetric positive semidefinite (PSD) matrix, the stiffness matrix K, which maps the applied force to displacement, or its inverse, the compliance matrix C, which maps the displacement to force.

A desired compliance can be obtained using mechanisms containing multiple passive elastic components, with each providing compliant constraint along or about a single axis. In robotic applications, a desired elastic behavior may be achieved using an elastic mechanism mounted on the manipulator end effector or the desired behavior may be designed into the robot manipulator itself. In the design of the behavior using either approach, the geometric construction of the mechanism/manipulator is an important consideration.

In some manipulation tasks, a time-varying compliant behavior is needed. For this purpose, variable stiffness actuators (VSAs) [1] that allow joint compliance to be changed in real time are used. Although the use of VSA's significantly enlarges the space of realizable compliant behaviors, varying the joint stiffness values alone, however, may not be adequate to achieve a desired behavior. Identification of the mechanism geometry required to realize a given compliance (as well as the joint compliances) is the primary motivation for this work.

This work is also motivated by the desire for a better understanding of compliant behavior achieved with a parallel or serial mechanism. In the planar case, since the elastic components in a mechanism are easy to illustrate, the physical significance of realization conditions can be readily understood in terms of the mechanism geometry.

Related Work.

Screw theory [2] has been widely used in elastic behavior analysis [36], while Lie groups [7] have also been used.

In previous work in the realization of spatial compliances, the bounds of elastic behaviors achieved with simple mechanisms (i.e., parallel and serial mechanisms without helical joints) were identified [8,9]. Synthesis procedures to achieve a simple-mechanism realizable stiffness or compliance matrix were developed [8,9] and later refined [10,11]. The synthesis of an arbitrary spatial stiffness matrix with a parallel system with both screw and simple springs was presented in Ref. [12] and the process further refined in Ref. [13]. Stiffness matrix decompositions for the purpose of realization with screw and simple springs revealed inherent elastic behavior properties [14,15].

Each of these approaches to spatial elastic behavior realization involved a decomposition of the stiffness matrix without regard to mechanism geometry. More recent work has included some geometric considerations in the realization of spatial elastic behaviors [1619].

In recent work [20] on planar elastic mechanism realization, a procedure to synthesize an arbitrary planar stiffness was developed for a restricted class of mechanism. As part of the procedure, the geometric parameters of a symmetric four-spring parallel mechanism were selected.

Most recently, the realization of a specified point planar elastic behavior (compliance in Euclidian space E(2)) using 3R serial mechanisms with specified link lengths has been addressed. In Ref. [21], optimization was used to identify the combination of mechanism configuration and joint stiffnesses that achieve an approximation of the desired elastic behavior. In Refs. [22] and [23], the synthesis of isotropic compliance in E(2) and E(3) with serial mechanisms has been addressed. In Ref. [24], conditions on mechanism geometry to achieve all compliances in E(2) were identified and synthesis procedures for the realization of an arbitrary 2 × 2 compliance were presented. The results obtained for 3R mechanisms [24] were then extended to general serial mechanisms having three (revolute and/or prismatic) joints [25].

Overview.

This paper addresses the passive realization of an arbitrary planar (3 × 3) elastic behavior with either a parallel or a serial mechanism. Unlike most previous synthesis procedures that involved mathematically decomposing the stiffness matrix in one step without regard to mechanism geometry, the synthesis procedures presented here are completely geometry based (no matrix decomposition needed). This allows one to select the geometry of each elastic component from a restricted space of kinematically allowable candidates. Then, with each selection, the space is further restricted until the desired elastic behavior is achieved.

The paper is outlined as follows: Section 2 presents the theoretical background for planar compliance realization with a parallel or serial mechanism. Necessary and sufficient conditions for an elastic behavior to be realized with a mechanism are obtained. In Sec. 3, the physical implications of the realization conditions are presented. The restrictions on the set of elastic behaviors that can be realized (described in terms of the locations of realizable elastic behavior centers) with a mechanism are identified and the concept of dual elastic mechanisms is introduced. In Sec. 4, geometric construction-based syntheses of a planar compliant behavior using either a parallel or a serial mechanism are presented. In Sec. 5, a numerical example is provided to illustrate the synthesis procedures. A brief summary is presented in Sec. 6.

Planar Compliance Realization Conditions

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

Technical Background.

It is known that any rank-m 6 × 6 PSD matrix K can be decomposed into a sum of m rank-1 PSD matrices, i.e., 
K=k1w1w1T+k2w2w2T++kmwmwmT
(1)
where ki>0 is a constant and wi6 is a unit wrench defined as the spring wrench [8]. Each rank-1 PSD stiffness Ki=kiwiwiT can be uniquely realized with a simple spring or a screw spring [12] having a line of action along wi and spring constant ki. The decomposition (1) can be written as 
K=WKJW
(2)

where W=[w1,w2,,wm] is the wrench matrix and KJ=diag(k1,k2,,km) is the joint-space stiffness matrix.

If stiffness is decomposed into the form of Eq. (1), the elastic behavior can be realized with a set of springs connected in parallel in which each spring provides a single axis of compliant constraint. In general, the rank-1 decomposition of K is not unique. There are infinitely many sets of springs that realize a given elastic behavior.

By duality [26], a decomposition of a compliance matrix C (the inverse of stiffness matrix K) yields a set of compliant joint twists associated with a serial mechanism. Using a similar process, a compliance matrix C can be realized with a serial mechanism in which each joint twist provides a rank-1 PSD component.

For the planar case, an elastic behavior is characterized by a 3 × 3 PSD stiffness matrix K or its inverse, the compliance matrix C. The spring wrenches in a parallel mechanism and the joint twists in a serial mechanism are 3-vectors. To realize an arbitrary elastic behavior, only simple mechanisms (zero or infinite pitch spring wrenches or joint twists) are needed. For a parallel mechanism, only line springs and torsional springs are needed. For a serial mechanism, only revolute and prismatic joints are needed.

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]
(3)

where n is a unit 2-vector indicating the direction of the spring axis and d=(r×n)·k is a scalar indicating the distance of the spring axis from the coordinate frame used to describe the stiffness K, r is the perpendicular position vector from the coordinate frame to the spring axis, and k is the unit vector perpendicular to the plane of the mechanism.

When the value of a wrench wl is given, the perpendicular position r to the wrench axis can be calculated using 
r=dΩn
(4)
where Ω is the 2 × 2 anti-symmetric matrix associated with a cross product 
Ω=[0110]
(5)

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

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]
(6)

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

Given the value of a twist tr, a unique point, the instantaneous center of rotation for the twist motion is calculated using 
r=Ωv
(7)

For a twist associated with a prismatic joint, since tp is a free vector in twist space, the location of the joint is arbitrary in the mechanism chain.

A wrench w and twist t are called reciprocal [2] if w performs no work along t. If wrench w and twist t are expressed in Plücker ray and axis coordinates (as in Eqs. (3) and (6)), respectively, then w and t are reciprocal if and only if 
wTt=tTw=0
(8)

Realization Conditions.

The space of stiffness matrices that can be realized with a given mechanism by adjusting the spring constant of each spring in the mechanism is determined by the mechanism kinematics.

Consider a parallel mechanism having three spring wrenches (w1,w2,w3). Below, we prove that a necessary and sufficient condition for a stiffness matrix K to be realized with the mechanism is 
wi×K(wj×wk)=0,{i,j,k}={1,2,3}
(9)
To prove the condition is necessary, we suppose that K is realized with the mechanism. Then, by Eq. (2), K can be expressed as 
K=WKJWT
(10)
where W=[w1,w2,w3] is the wrench matrix and KJ=diag(k1,k2,k3) (with ki0) is the joint stiffness matrix. For the 3 × 3 matrix W, its inverse can be expressed as 
W1=1λ[w2×w3,w3×w1,w1×w2]T
(11)
where λ is the triple product of (w1,w2,w3) 
λ=(w1×w2)·w3
(12)
Multiplying Eq. (10) by WT from the right yields 
K[w2×w3,w3×w1,w1×w2]=λ[k1w1,k2w2,k3w3]
Thus, for {i,j,k}={1,2,3} 
K(wj×wk)=λkiwi
(13)

Thus, wi×K(wj×wk)=0, which proves that condition (9) is necessary.

In the evaluation of the condition sufficiency, consider that condition (9) is satisfied, then, there exist scalars αi such that 
K[w2×w3,w3×w1,w1×w2]=[α1w1,α2w2,α3w3]
Using Eq. (11) 
W1KWT=diag(α1λ,α2λ,α3λ)=KJ
Since K is PSD, (αi/λ)0. Thus 
K=WKJWT
which proves that the stiffness K is realized with the mechanism. The three joint stiffness constants can be calculated using 
ki=αiλ
where λ is the scalar defined in Eq. (12) and 
α1=w1TK(w2×w3)w1Tw1
 
α2=w2TK(w3×w1)w2Tw2
 
α3=w3TK(w1×w2)w3Tw3

By duality, condition (9) applies to a serial mechanism having joint twists (t1,t2,t3) and a compliance matrix C. In Eq. (9), one can simply replace the stiffness matrix K with the compliance matrix C and replace the spring wrenches wi with the joint twists ti to obtain the condition for a serial mechanism. The joint compliances can also be obtained accordingly.

In summary, we have:

Proposition 1. Consider a parallel mechanism having spring wrenches(w1,w2,w3)and a serial mechanism having joint twists(t1,t2,t3). Then, if the elastic constants (ki or ci) are selectable,

  • (a)
    A stiffness matrixKcan be realized with the parallel mechanism if and only if 
    wi×K(wj×wk)=0,{i,j,k}={1,2,3}
    (14)
  • (b)
    A compliance matrixCcan be realized with the serial mechanism if and only if 
    ti×C(tj×tk)=0,{i,j,k}={1,2,3}
    (15)

The realization conditions (14) and (15) are mathematical requirements for a parallel and a serial mechanism, respectively, to achieve an arbitrary given compliance behavior. In these conditions, each spring wrench wi or joint twist ti is treated as a vector in 3 and the cross product is an operation between these 3-vectors. The physical significance of these conditions is provided in Sec. 2.3.

Below, for any full-rank compliance behavior, an equivalent set of conditions is derived from Eqs. (14) and (15). These conditions do not use the cross product operation and have clear physical significance.

For a full-rank K,αi0. Multiplying Eq. (13) from the left by C=K1 yields 
(wj×wk)=αiCwi
For any ij 
αiwjTCwi=wjT(wj×wk)=0
(16)
Since αi0, Eq. (16) can be expressed as 
wiTCwj=0,ij
To determine the spring constants ki, consider 
K=k1w1w1T+k2w2w2T+k3w3w3T
(17)
Multiplying K in Eq. (17) from the right by Cwi yields 
KCwi=kiwi(wiTCwi)wi=ki(wiTCwi)wi
Thus 
ki=1wiTCwi

The result obtained for stiffness matrix K for a parallel mechanism applies to its dual involving the compliance matrix C for a serial mechanism. Thus, we have:

Proposition 2. Consider a parallel mechanism having spring wrenches(w1,w2,w3)and a serial mechanism having joint twists(t1,t2,t3). Then, if the elastic constants (ki or ci) are selectable

  • (a)
    A full-rank elastic behaviorK(C) can be realized with a parallel mechanism if and only if 
    wiTCwj=0,ij
    (18)
    The spring constant associated withwi is determined using 
    ki=1wiTCwi,i=1,2,3
    (19)
  • (b)
    A full-rank elastic behaviorC(K) can be realized with a serial mechanism if and only if 
    tiTKtj=0,ij
    (20)
    The joint compliance associated withti is determined using 
    ci=1tiTKti,i=1,2,3
    (21)

Note that the conditions in Eq. (14) or Eq. (18) for parallel mechanism and the conditions in Eq. (15) or Eq. (20) for serial mechanisms can be used to determine whether a given elastic behavior can be realized based on the mechanism kinematics alone. If these conditions are satisfied, the realization of the specified behavior is ensured if the non-negative spring coefficients in Eq. (19) or joint compliances in Eq. (21) can be physically attained.

Also, note that realization condition (14) for a parallel mechanism applies for all stiffness matrices (including those nonfull-rank elastic behaviors) while condition (18) applies only to full-rank elastic behaviors. Since full-rank elastic behaviors are of most interest, in the rest of this paper, only full-rank stiffness and compliance matrices are considered and the conditions presented in Proposition 2 are used.

Physical Significance of Realization Conditions.

Since the cross product operation is normally not used on screws, the physical significance of conditions (14) and (15) is not evident. However, if the three vectors in Eq. (14) associated with the cross product are interpreted as planar twists (i.e., t=wi×wj), the physical meaning of the realization conditions can be obtained.

For a parallel mechanism having three spring wrenches w1,w2, and w3, consider a twist t reciprocal to the two spring wrenches wi and wj. By Eq. (8), the planar twist can be expressed as 
t=γ(wi×wj)
where γ is a scalar and the twist t is located at the intersection of the two spring wrench axes. Realization condition (14) requires 
Kt=αwk

where α is a scalar. Thus, if a parallel mechanism realizes the stiffness, a twist located at the intersection of any two spring axes (wi and wj) yields a wrench along the axis of the third spring wk.

Similarly, for a serial mechanism having three joint twists t1,t2, and t3, realization condition (15) implies that a wrench passing through any two joints (ti and tj) results in a twist motion about the third joint tk.

The physical significance of conditions (18) and (20) is evident. For an elastic behavior realized with a parallel mechanism, the twist resulting from a force along one spring wrench must be reciprocal to the other two spring wrenches. For an elastic behavior realized with a serial mechanism, the wrench resulting from a motion along one joint twist must be reciprocal to the other two joint twists. The realization conditions for parallel and serial mechanisms provide the relationship between the mechanism geometry and the elastic behavior to be realized. If the conditions are not satisfied, then a specified planar elastic behavior cannot be obtained no matter how the joint stiffnesses vary.

Planar Elastic Behaviors and Mechanisms

The implications of the realization conditions can be understood in the geometry of the mechanism. First, additional physical interpretations of conditions (18) and (20) are presented. Next, the bounds on the realizable space of elastic behaviors for a given mechanism are interpreted in terms of the locus of elastic behavior centers using these conditions. Then, conditions on the parallel and serial mechanisms that can achieve the same subspace of elastic behaviors are identified. The two mechanisms are defined to be dual elastic mechanisms. The geometric properties of a pair of dual elastic mechanisms are presented.

Implications of the Realization Conditions.

Consider a parallel mechanism consisting of three line springs. Spring wrench behavior is independent of the location along the spring axis and, if none of the springs is parallel to another, the three spring axes form a triangle ABC as shown in Fig. 1.

If a force f1 is applied to the elastically constrained body along spring wrench w1, the force can be expressed as f1=αw1 where α is a scalar. The twist motion t1 resulting from f1 is 
t1=Cf1=αCw1

By condition (18), twist t1 is reciprocal to the other two spring wrenches w2 and w3. Thus, the instantaneous center associated with t1 must be located at the intersection of these two spring axes, point C. The resulting motion is a rotation about vertex C as illustrated in Fig. 2(a). Therefore, if an applied force is along one spring axis, the resulting motion is a rotation about the opposite vertex of the triangle formed by the three spring axes.

Also, it can be proved that if an applied force passes through a vertex of the triangle, the resulting twist corresponds to an instantaneous center located on the spring axis opposite to the vertex. To prove this, consider a force f passing through vertex A as shown in Fig. 2(b). Then, f can be expressed as a linear combination of w1 and w2 
f=αw1+βw2
where α and β are arbitrary scalars. Condition (18) requires 
w3TCf=w3TC(αw1+βw2)=0

which means that the twist resulting from f acting on the body must be reciprocal to w3. Thus, the twist instantaneous center must be on spring axis w3.

In summary, for a planar parallel mechanism having three line springs,

  • (i)

    A force along one spring axis results in a twist having an instantaneous center of rotation located at the opposite vertex of the triangle formed by the three spring axes.

  • (ii)

    A force passing through a vertex of the triangle results in a twist center on the line along the opposite side of the triangle.

By duality and condition (20), for a planar serial mechanism having three revolute compliant joints:

  • (i)

    A rotation about one joint results in a wrench passing through the other two joints;

  • (ii)

    A rotation about an arbitrary point on the line passing through two joints results in a wrench passing through the third joint.

Center of Planar Elastic Behavior.

For any planar elastic behavior, there is a unique point at which the behavior can be described by a diagonal stiffness (compliance) matrix. This point is defined as the center of stiffness (compliance). For a given planar elastic behavior, the centers of stiffness and compliance are coincident. Any force passing through the center results in a pure translation (i.e., a twist with infinite pitch or with instantaneous center at infinity), and any twist at the center results in a pure couple (i.e., a wrench with zero pitch). It can be seen that, if a force f results in a pure translation, the line of action of the force must pass through the center of stiffness.

If, in a given coordinate frame, a 3 × 3 stiffness K and compliance C are expressed in a partitioned form as 
K=[AbbTk33],C=[EggTc33]
(22)
where A and E are 2 × 2 symmetric matrices, and b and g are 2-vectors, the locations of stiffness center and compliance center can be determined using [20] 
rk=ΩA1bandrc=Ωg/c33
(23)

where Ω2×2 is the anti-symmetric matrix defined in Eq. (5). For the same planar elastic behavior (C=K1), rk=rc.

Conditions (18) and (20) constrain the possible location of the elastic centers associated with a parallel or serial mechanism. Below, we show that, for any stiffness behavior realized with a parallel mechanism of three line springs, the center of the behavior must be inside the triangle formed by the three spring axes.

Consider a force f that passes through vertex A between w1 and w2 which can be expressed as 
f(q)=(1q)w1+qw2,q[0,1]
(24)

As q varies from 01, the direction of f varies from w1 to w2, and as previously shown, since f passes through vertex A, the instantaneous center T(q) of the resulting twist must vary from point C to point B along the w3 axis. We show that T(q) cannot move from CB along finite line CB as q varies from 01.

Suppose that for all q[0,1] the center of the twist is finite. Then the location of the twist instantaneous center, T(q) (which moves along w3), is a continuous function of q. Note that T(0) = C and T(1) = B. With the finite path from CB along the axis of w3, there is a point q̂(0,1) such that f(q̂) passes through the instantaneous center T(q̂) (Fig. 3(a)), which means f(q̂) is reciprocal to the motion caused by itself. Thus, at q=q̂

 
fTCf=0

This conflicts the fact that C is positive definite and f0. Therefore, the finite path from CB is not valid. The path of T(q) as q increases from 0 to 1 must be opposite to that illustrated in Fig. 3(a), and there must be q(0,1) such that f(q) in Eq. (24) results in a twist at infinity, which is a pure translation. Thus, the line of action of f(q) must pass through the center of stiffness. Since the force f(q) is a positive combination of w1 and w2, the line of action of f(q) is within the area bounded by spring axes w1 and w2 (the shaded area of Fig. 3(b)). Therefore, the center of stiffness must be in this area.

Applying the same reasoning to vertices B and C, it is proved that the location of the stiffness center is within the triangle ABC formed by the three spring axes (Fig. 4(a)).

By duality, for a serial mechanism having three revolute joints, no matter how the joint compliances are selected, the center of compliance must be within the triangle formed by the locations of the three joints J1J2J3 as shown in Fig. 4(b).

Since the locus of stiffness centers is determined for a given mechanism with fixed geometry, it is easy to assess whether a specified elastic behavior can be attained by evaluating the location of the behavior center. If the center is not in the region bounded by the mechanism geometry, then the behavior cannot be realized with the mechanism regardless of the value of each joint stiffness/compliance. Also, the location of the center can be used to: (1) help determine the placement of the elastic components in the design of a new mechanism or (2) determine the location within the manipulator workspace that a specified compliance can be achieved in an existing mechanism.

Dual Mechanisms in Parallel and Serial Constructions.

Suppose a parallel mechanism has three line spring wrenches w1,w2, and w3. Consider the following three 3-vectors (s1,s2,s3) defined by 
s1=w2×w3,s2=w3×w1,s3=w1×w2
(25)

If the three 3-vectors si are viewed as planar twists in Plücker axis coordinates, and ti is the unit twist associated with si, then the three unit twists (t1,t2,t3) are uniquely determined by the three wrenches (w1,w2,w3) and each ti is reciprocal to two wrenches (wj,wk).

Consider the serial mechanism composed of three joints J1, J2, and J3 having joint twists t1,t2, and t3, respectively. Since twist t1 is reciprocal to wrenches w2 and w3, J1 must be at the intersection of the two wrenches. Similarly, joint J2 must be at the intersection of wrenches w1 and w3, and joint J3 must be at the intersection of wrenches w1 and w2. Therefore, as shown in Fig. 5, the triangle formed by the three line spring axes in the parallel mechanism is coincident with the triangle formed by the three revolute joints in the serial mechanism. We define such a pair of parallel and serial mechanisms as dual elastic mechanisms.

Given a pair of dual elastic mechanisms with spring wrenches (w1,w2,w3) and joint twists (t1,t2,t3), it can be proved that for an elastic behavior described in stiffness matrix K or compliance matrix C=K1 
tiTKtj=0wiTCwj=0,ij
(26)
Thus, an arbitrary elastic behavior can be realized with one mechanism if and only if it can be realized with its dual elastic mechanism. The realizable spaces of elastic behaviors for the two mechanisms are exactly the same. Also, it can be proved that, if ki is the spring constant associated with spring wrench wi in the parallel mechanism and ci is the joint compliance associated with joint twist ti, then ki and ci satisfy 
kici=1(wiTti)2,i=1,2,3
(27)

Note that, since t1 is reciprocal to two wrenches w2 and w3,t1 cannot be reciprocal to wrench w1, unless the three wrenches are linearly dependent or t1=0. Thus, w1Tt10. Similarly, wiTti0 for i = 2, 3.

It can be seen that, for the generic case, a three-spring parallel mechanism and a three-joint serial mechanism are a pair of dual elastic mechanisms if and only if the two triangles formed by the three springs and formed by the three joints are coincident. For some (nongeneric) cases, the triangle for a parallel or serial mechanism does not exist. The dual mechanisms have different geometry. Below, two cases are considered:

  • (a)
    Two springs are parallel in a parallel mechanism. Suppose a parallel mechanism has three springs w1,w2, and w3 with w1w2. The dual elastic serial mechanism has two revolute joints and one prismatic joint. The two revolute joints are located at the two intersection points of springs w1 and w3 and w2 and w3. The prismatic joint is perpendicular to the two parallel spring axes. The geometry of the two mechanism's wrench and twist axes is illustrated in Fig. 6(a) 
    Fig. 6

    Dual elastic mechanisms in nongeneric cases. (a) A parallel mechanism with two parallel springs. The dual elastic serial mechanism has two revolute joints each located at the intersection of nonparallel springs and one prismatic joint perpendicular to the two parallel springs. (b) A parallel mechanism with one torsional spring. The dual elastic serial mechanism has two prismatic joints perpendicular to the two line springs and one revolute joint located at the intersection point of the two line springs.

    Fig. 6

    Dual elastic mechanisms in nongeneric cases. (a) A parallel mechanism with two parallel springs. The dual elastic serial mechanism has two revolute joints each located at the intersection of nonparallel springs and one prismatic joint perpendicular to the two parallel springs. (b) A parallel mechanism with one torsional spring. The dual elastic serial mechanism has two prismatic joints perpendicular to the two line springs and one revolute joint located at the intersection point of the two line springs.

    .
  • (b)

    A parallel mechanism has one torsional spring. Suppose w3 is the torsional spring in a parallel mechanism. The dual elastic serial mechanism has two prismatic joints J1 and J2 and one revolute joint J3. The directions of the two prismatic joints are perpendicular to the two line springs w1 and w2, respectively, and the revolute joint is located at the intersection of the two line springs w1 and w2. The geometry of the two mechanism's wrench and twist axes is illustrated in Fig. 6(b).

Elastic Behavior Synthesis Procedures

In this section, procedures for the realization of planar elastic behavior using geometric construction-based methods are presented. First, a synthesis procedure for a parallel mechanism having three springs is provided. Next, a synthesis procedure for a serial mechanism having three joints is presented. These two types of mechanisms are the most general in that all full-rank planar stiffness/compliance matrices can be realized with these two types. Then, synthesis procedures for a parallel mechanism having a torsional spring and for a serial mechanism having prismatic joints are discussed.

Parallel Elastic Mechanism.

Suppose a stiffness K described in a body-based frame is to be realized. The following synthesis procedure identifies a set of spring axes and their corresponding spring constants that realize the given K. The location of the stiffness center of the behavior, Ck, can be calculated using Eq. (23). The geometry associated with the sequence of operations in the synthesis procedure is illustrated in Fig. 7.

  1. (1)

    Select the first spring w1: The spring axis can be chosen arbitrarily relative to the stiffness center Ck.

  2. (2)
    Calculate the twist t1 resulting from wrench w1 
    t1=Cw1
    The location of the instantaneous center of rotation associated with t1, T1, is calculated using Eq. (7).
  3. (3)

    Select the second spring w2: Due to the reciprocal condition (18), all candidate wrenches are from the pencil of lines passing through point T1. Choose a direction for a wrench passing through T1, then w2 is determined.

  4. (4)
    Calculate the twist t2 resulting from w2 
    t2=Cw2
    The location of the instantaneous center associated with t2, T2, is determined using Eq. (7). Since t2 satisfies the reciprocal condition (18), T2 must be on the axis of the first spring w1.
  5. (5)

    Identify the third spring w3: The axis of w3 is uniquely determined by the line passing through points T1 and T2.

With the final step, all three spring wrenches are determined. The stiffness coefficient for each spring can be calculated using Eq. (19).

Note that in the generic case, the three wrenches (w1,w2,w3) generated from the procedure are associated with line springs and form a triangle. If, in the process of realizing a given elastic behavior, one or more spring axes are selected to pass through the center of stiffness, then the three spring axes will not form a triangle. Consider the following two cases:

  • (a)
    If the first spring is selected to pass through the stiffness center Ck in step 1, the twist t1=Cw1 is a pure translation. Due to the reciprocal condition (18), both the second and third spring axes, w2 and w3, must be perpendicular to the direction of translation t1. The location of the second spring axis w2 can be arbitrarily selected. The location of w3 can be determined by passing through the instantaneous center of the twist t2=Cw2 (as shown in Fig. 8(a) 
    Fig. 8

    Nongeneric parallel mechanism stiffness realization cases. (a) If the first spring axis is selected to pass through Ck, then the spring axes w2 and w3 must be perpendicular to the translation t1 resulting from w1. The location of w2 can be selected arbitrarily and the spring axis w3 passes through the instantaneous center of twist t2=Cw2, T2. (b) If the second spring axis w2 passes through the stiffness center Ck, the third spring axis w3 must be parallel to spring axis w1 and pass through T1.

    Fig. 8

    Nongeneric parallel mechanism stiffness realization cases. (a) If the first spring axis is selected to pass through Ck, then the spring axes w2 and w3 must be perpendicular to the translation t1 resulting from w1. The location of w2 can be selected arbitrarily and the spring axis w3 passes through the instantaneous center of twist t2=Cw2, T2. (b) If the second spring axis w2 passes through the stiffness center Ck, the third spring axis w3 must be parallel to spring axis w1 and pass through T1.

    ). If both the first and second springs are chosen to pass through the stiffness center, then the third spring must be a torsional spring.
  • (b)

    If in step 3, the second spring wrench w2 (passing through point T1, the instantaneous center of t1=Cw1) is chosen to pass through the stiffness center Ck (illustrated in Fig. 8(b)), then the twist t2=Cw2 is a pure translation (twist instantaneous center at infinity). The third spring wrench w3 must also pass through T1 and be parallel to w1. Thus, the behavior is realized with three line springs w1,w2, and w3 as shown in Fig. 8(b).

Serial Elastic Mechanism.

Similar to the parallel mechanism case, the synthesis procedure identifies the set of joint locations (configuration of the mechanism) and corresponding joint compliance constants that realize the given C. The location of the compliance center, Cc, can be calculated using Eq. (23) and K=C1. The geometry associated with the sequence of operations in the synthesis procedure is illustrated in Fig. 9.

  1. (1)

    Select the first joint location for t1: The location of the joint, J1, can be arbitrarily chosen relative to the center Cc.

  2. (2)
    Calculate the wrench w1 resulting from the twist 
    w1=Kt1
    The perpendicular position to the line of action of w1 is determined using Eq. (4).
  3. (3)

    Select the second joint location for t2: Due to the reciprocal condition (20), all candidate joints are located on the line of action of wrench w1. Choose a joint location on the line along w1, then the joint twist t2 is determined.

  4. (4)
    Calculate the wrench w2 resulting from t2 
    w2=Kt2
    Since w2 satisfies the reciprocal condition, the line of action of w2 passes through the first joint J1.
  5. (5)

    Identify the third joint twist t3: the joint is uniquely determined by the intersection of the two lines along wrenches w1 and w2.

With the final step, all three joint locations are determined. The joint compliance coefficient for each elastic joint can be calculated using Eq. (21).

Note that in the generic case, the three twists (t1,t2,t3) generated from the procedure are joint twists of revolute joints. If in the process of realizing a given elastic behavior, the location of one joint is selected to be at the center of compliance, then one or two prismatic joints must be used. Consider the following two cases.

  • (a)
    If the location of the first joint is selected at the compliance center Cc in step 1 (illustrated in Fig. 10(a) 
    Fig. 10

    Nongeneric serial mechanism compliance realization cases. (a) If the first joint is selected to pass through the center of compliance Cc, the other two joints must be prismatic. The direction of the second prismatic axis can be arbitrarily chosen. The direction of the third prismatic axis is perpendicular to the line of action of the wrench w2=Kt2. (b) If the second joint J2 is on the line passing through J1 and the compliance center Cc, the resulting wrench w2=Kt2 must be parallel to w1. The third joint is prismatic and is perpendicular to w1.

    Fig. 10

    Nongeneric serial mechanism compliance realization cases. (a) If the first joint is selected to pass through the center of compliance Cc, the other two joints must be prismatic. The direction of the second prismatic axis can be arbitrarily chosen. The direction of the third prismatic axis is perpendicular to the line of action of the wrench w2=Kt2. (b) If the second joint J2 is on the line passing through J1 and the compliance center Cc, the resulting wrench w2=Kt2 must be parallel to w1. The third joint is prismatic and is perpendicular to w1.

    ), the wrench w1=Kt1 is a pure couple. Due to the reciprocal condition (20), both the second and third joint twists, t2 and t3, must be pure translation and the corresponding joints must be prismatic. The direction of the prismatic joint axis for J2, n2, can be selected arbitrarily. Given the selection of n2, the direction of the prismatic joint axis for J3 must be perpendicular to the wrench axis determined by 
    w2=Ktp2
    where tp2=[n2T,0]T is the joint twist of J2.
  • (b)

    If in step 3, the location of the second joint J2 is selected such that line J1J2 passes through the center of compliance Cc (illustrated in Fig. 10(b)), then the wrench w2=Kt2 is parallel to w1. The third joint must be prismatic and perpendicular to w1. Thus, the compliance is realized with a serial mechanism having two revolute joints and one prismatic joint. Since translation is a free vector, the location of the prismatic joint on the serial chain is arbitrary.

Discussion.

In the generic case, the synthesis procedure presented in Sec. 4.1 yields three line springs in a parallel mechanism, and the synthesis procedure presented in Sec. 4.2 yields three revolute joints in a serial mechanism. In the case that a torsional spring is desired in a parallel mechanism or a prismatic joint is desired in a serial mechanism, the synthesis procedures can be modified.

A three-spring parallel mechanism can have, at most, one torsional spring to realize a full-rank stiffness matrix. Since the spring wrench associated with a torsional spring has the form of wt in Eq. (3), it must be selected as the first spring in the synthesis procedure presented in Sec. 4.1 (or the last spring if the other two springs pass through the center of stiffness).

A three-joint serial mechanism can have, at most, two prismatic joints to realize a full-rank compliance matrix. The twist associated with a prismatic joint has the form of tp in Eq. (6). Since the third joint is uniquely determined by the first two joints, the revolute joint should be assigned last in the synthesis procedure.

Examples

Examples are provided to demonstrate the geometry-based synthesis procedures for the realization of a specified elastic behavior. First, the realization of a stiffness matrix with a parallel mechanism is demonstrated. Then, the realization of the same elastic behavior with a serial mechanism is presented.

The elastic behavior to be realized in a known coordinate frame is given by 
K=[321265159],C=125[292316232617161714]

The location of the center of stiffness/compliance for this behavior is calculated using Eq. (23) to be at ((17/14),(16/14)). Since the center must be inside the triangle formed by the spring components (or location of the elastic joints), this point is used as a reference in selecting each component.

Parallel Mechanism Realization.

The geometry associated with the sequence of operations in the synthesis procedure is illustrated in Fig. 11.

The first spring can be arbitrarily chosen. Here, a horizontal spring passing through the origin is selected. The spring wrench is 
w1=[1,0,0]T
The twist associated with w1 is calculated to be 
t1=Cw1=1625[2916,2316,1]T

Using Eq. (7), the location the instantaneous center T1 associated with twist t1 is found to be (1.4375,1.8125).

Given the selection of the first spring, the second spring wrench is any one that passes through point T1. Here, the slope of the line is chosen to be 4. The unit wrench passing through T1 with this slope is 
w2=[0.2425,0.9701,1.8342]T
The twist associated with the second spring wrench is 
t2=Cw2=0.2122[0,0.0716,1]T

Using Eq. (7), the location of the instantaneous center T2 associated with t2 is calculated as (0.0716, 0).

The third spring wrench is along the line passing through T1 and T2. The spring wrench is 
w3=[0.6019,0.7986,0.0570]T
The three spring constants calculated using Eq. (19) are 
k1=0.8621,k2=2.6699,k3=5.4680
The process is verified by summing the stiffness components using Eq. (1) yielding 
K=i=13kiwiwiT=[321265159]

Note that the synthesis procedure identifies the line of action and stiffness constant for each spring. In the construction of a parallel mechanism, each spring can be anywhere along its line of action.

Serial Mechanism Realization.

For the parallel mechanism obtained in Sec. 5.1, the dual elastic serial mechanism is readily determined using the results of Sec. 3.3. The three joints of the serial mechanism are located at the three vertices T1, T2, and T3 of the triangle formed by the three spring axes (shown in Fig. 11). The joint compliances calculated using Eq. (27) are 
c1=0.3531,c2=0.1202,c3=0.0867

An alternative design using the synthesis procedure of Sec. 4.2 is derived below. The geometry associated with the sequence of operations in the synthesis procedure is illustrated in Fig. 12.

The first joint location can be chosen arbitrarily. Here, the location of J1 is selected at (1, 0). The unit joint twist associated with J1 is calculated using Eq. (6) 
t1=[0,1,1]T
The wrench w1 associated with t1 is calculated to be 
w1=Kt1=[3,1,4]T
Using Eq. (4), the equation of the line of action of w1 is determined to be 
y=13x43
Given the selection of the first joint, the second joint J2 is located at any point on the line of action of w1. Here, the location of J2 is selected to be r2=[0,(4/3)]T. Then, joint twist associated with the second joint, J2, is calculated using Eq. (6) to be 
t2=[43,0,1]T
The wrench associated with t2 is calculated to be 
w2=Kt2=[3,233,233]T
The line of action of w2 is 
y=239x+239
The intersection of the two action lines w1 and w3 is ((7/4),(23/12)), which is the location of J3. The unit twist associated with this location is 
t3=[2312,74,1]T
The three joint compliances are calculated using Eq. (21) 
c1=0.2,c2=0.0857,c3=0.2743
The process is verified by summing the compliance components, similar to Eq. (1) yielding 
C=i=13cititiT=125[292316232617161714]

Note that the synthesis procedure identifies the location and joint compliance coefficient for each elastic joint. In the construction of a serial mechanism, the connection order of these joints does not influence the elastic behavior achieved with the mechanism.

Summary

In this paper, the realization of an arbitrary planar elastic behavior using parallel and serial mechanisms 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 are provided. The methods presented in this paper allow one to synthesize any compliant behavior by selecting each elastic component in a parallel or serial mechanism based on its geometry without decomposition of the compliance/stiffness matrix. Each selected component restricts the space of allowable candidates in subsequent selection. Since the conditions on the mechanism geometry and joint compliances are decoupled, the methods identified can be used for mechanisms having VSAs to realize a desired compliant behavior by changing the mechanism configuration and joint stiffnesses. In application, one can use the method to judiciously select a better mechanism geometry for a specified compliance from the infinite, but restricted, set of options available. This method makes those restrictions to the mechanism geometry explicit.

Funding Data

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

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