Geometric Approach to the Realization of Planar Elastic Behaviors With Mechanisms Having Four Elastic Components

Huang, Shuguang; Schimmels, Joseph M.

doi:10.1115/1.4039399

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.

Fig. 1

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Simple planar compliant mechanisms: (a) serial mechanism. Each joint has joint compliance c_i (b) parallel mechanism. Each spring has stiffness k_i.

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 spatial compliance analysis, screw theory [8–11] and Lie groups [12] have been used.

In previous work in the realization of spatial compliances, synthesis of elastic behaviors with simple mechanisms (i.e., parallel and serial mechanisms without helical joints) was addressed [1,2,13,14]. The realization of an arbitrary spatial stiffness matrix with a parallel mechanism having both simple springs and screw springs was presented in Refs. [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.

Consider a serial mechanism having four joints as shown in Fig. 1(a). The type and location of joint J_i can be represented by the joint twist t_i. The planar joint twist for a revolute joint and for a prismatic joint can be expressed in Plücker axis coordinates as

t_{r} = [\begin{matrix} v \\ 1 \end{matrix}], t_{p} = [\begin{matrix} n \\ 0 \end{matrix}]

(1)

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

If a mechanism has four joints with joint twists t_i and joint compliance c_i > 0, then the Cartesian compliance matrix associated with the mechanism [2,3] is

C = c_{1} t_{1} t_{1}^{T} + c_{2} t_{2} t_{2}^{T} + c_{3} t_{3} t_{3}^{T} + c_{4} t_{4} t_{4}^{T}

(2)

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

Given the value of a twist t_r, a unique point, the instantaneous center of rotation for the twist motion, is calculated using

r = Ω v

(3)

where

Ω

is the 2 × 2 antisymmetric matrix associated with a cross product

Ω = [\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}]

(4)

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

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

w_{l} = [\begin{matrix} n \\ d \end{matrix}], w_{t} = [\begin{matrix} 0 \\ 1 \end{matrix}]

(5)

where n is a unit 2-vector indicating the direction of the spring axis, $d = (\tilde{r} \times n) \cdot k$ is a scalar indicating the distance of the spring axis from the coordinate frame, $\tilde{r}$ is a position vector from the coordinate frame to any point along the spring axis, and k is the unit vector perpendicular to the plane of the mechanism.

If a parallel mechanism has four springs with spring wrench w_i and stiffness k_t > 0, then the Cartesian stiffness associated with the mechanism is

K = k_{1} w_{1} w_{1}^{T} + k_{2} w_{2} w_{2}^{T} + k_{3} w_{3} w_{3}^{T} + k_{4} w_{4} w_{4}^{T}

(6)

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

When the value of a wrench w_l is given, the perpendicular position r to the wrench axis can be calculated using

r = - d Ω n

(7)

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

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

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

w^{T} t = t^{T} w = 0

(8)

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

Since a translational twist 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.

First, consider a serial mechanism with four elastic joints J_i, i = 1, 2, 3, 4. The location of each J_i is uniquely specified by its twist t_i. A wrench w_ij that is reciprocal to two joint twists t_i and t_j must satisfy both

w_{i j}^{T} t_{i} = 0 and w_{i j}^{T} t_{j} = 0

(9)

The line of action of w_ij must pass through both joints J_i and J_j (the locations of the two twist centers) and thus is uniquely determined by t_i and t_j. Mathematically, wrench w_ij can be calculated by

w_{i j} = t_{i} \times t_{j}, \forall i \neq j

(10)

Any wrench w reciprocal to t_i and t_j has the same line of action of w_ij and can be expressed as

w = α w_{i j}, \forall i \neq j

where α is a scalar.

Consider a symmetric matrix C, and two wrenches w₁₂ and w₃₄ that pass through joints (J₁, J₂) and joints (J₃, J₄), respectively. For a four-joint realization, C can be expressed in the form of

C = c_{1} t_{1} t_{1}^{T} + c_{2} t_{2} t_{2}^{T} + c_{3} t_{3} t_{3}^{T} + c_{4} t_{4} t_{4}^{T}

(11)

then, multiplying Eq. by a wrench passing through two joints J₃ and J₄ [using Eq. ] yields

\begin{matrix} C w_{34} = (c_{1} t_{1} t_{1}^{T} + c_{2} t_{2} t_{2}^{T} + c_{3} t_{3} t_{3}^{T} + c_{4} t_{4} t_{4}^{T}) w_{34} \\ = c_{1} (t_{1}^{T} w_{34}) t_{1} + c_{2} (t_{2}^{T} w_{34}) t_{2} \end{matrix}

Then, multiplying the result from left by a wrench passing through the other two joints J₁ and J₂ (using Eq. again) yields

w_{12}^{T} C w_{34} = 0

(12)

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

Similarly, it is readily shown that if C can be expressed in the form of Eq. , then for any permutation of {1, 2, 3,4}

w_{i j}^{T} C w_{r s} = 0

(13)

Note that C is symmetric

(w_{i j}^{T} C w_{r s} = w_{r s}^{T} C w_{i j})

and that w_ij and w_ji have the same line of action (w_ij = αw_ji). Thus, condition for all permutations of {1, 2, 3, 4} is equivalent to the set of three equations

w_{12}^{T} C w_{34} = 0, w_{13}^{T} C w_{24} = 0, w_{14}^{T} C w_{23} = 0

(14)

The sufficiency of this set of conditions is evaluated next.

Below, we prove that if any two equations in Eq. are satisfied, then C can be expressed in the form of Eq. . To prove this, we define a symmetric matrix

\tilde{C}

as

\tilde{C} = {\tilde{c}}_{1} t_{1} t_{1}^{T} + {\tilde{c}}_{2} t_{2} t_{2}^{T} + {\tilde{c}}_{3} t_{3} t_{3}^{T} + {\tilde{c}}_{4} t_{4} t_{4}^{T}

(15)

where

{\tilde{c}}_{1} = \frac{w_{23}^{T} C w_{24}}{(w_{23}^{T} t_{1}) (w_{24}^{T} t_{1})}

(16)

{\tilde{c}}_{2} = \frac{w_{34}^{T} C w_{14}}{(w_{34}^{T} t_{2}) (w_{14}^{T} t_{2})}

(17)

{\tilde{c}}_{3} = \frac{w_{14}^{T} C w_{12}}{(w_{14}^{T} t_{3}) (w_{12}^{T} t_{3})}

(18)

{\tilde{c}}_{4} = \frac{w_{12}^{T} C w_{23}}{(w_{12}^{T} t_{4}) (w_{23}^{T} t_{4})}

(19)

We show that any two equations in condition ensure

\tilde{C} = C

⁠. To illustrate this, let i = 2 and j = 4 in Eq. , then

t_{2}^{T} w_{24} = 0 and t_{4}^{T} w_{24} = 0

Thus

\begin{matrix} \tilde{C} w_{24} = ({\tilde{c}}_{1} t_{1} t_{1}^{T} + {\tilde{c}}_{2} t_{2} t_{2}^{T} + {\tilde{c}}_{3} t_{3} t_{3}^{T} + {\tilde{c}}_{4} t_{4} t_{4}^{T}) w_{24} \\ = {\tilde{c}}_{1} (t_{1}^{T} w_{24}) t_{1} + {\tilde{c}}_{3} (t_{3}^{T} w_{24}) t_{3} \end{matrix}

Since

w_{23}^{T} t_{3} = 0

w_{23}^{T} \tilde{C} w_{24} = {\tilde{c}}_{1} (t_{1}^{T} w_{24}) (w_{23}^{T} t_{1})

Substituting Eq. into the above equation yields

w_{23}^{T} \tilde{C} w_{24} = w_{23}^{T} C w_{24}

which is equivalent to

w_{23}^{T} (\tilde{C} - C) w_{24} = 0

Applying the same process to all w_ij and w_ir yields the following four equations:

w_{23}^{T} (\tilde{C} - C) w_{24} = 0, w_{14}^{T} (\tilde{C} - C) w_{34} = 0

(20)

w_{12}^{T} (\tilde{C} - C) w_{14} = 0, w_{12}^{T} (\tilde{C} - C) w_{23} = 0

(21)

If two equations in condition are satisfied, for example

w_{12}^{T} C w_{34} = 0, w_{13}^{T} C w_{24} = 0

(22)

then two additional equations similar to Eqs. and can be constructed.

By the definition of

\tilde{C}

in Eq. , it is easy to verify that

w_{12}^{T} \tilde{C} w_{34} = 0, and w_{13}^{T} \tilde{C} w_{24} = 0

(23)

Combining Eqs. and yields

w_{12}^{T} (\tilde{C} - C) w_{34} = 0, w_{13}^{T} (\tilde{C} - C) w_{24} = 0

(24)

Thus, the matrix

(\tilde{C} - C)

satisfies six equations in Eqs. , , and . Since a 3 × 3 symmetric matrix has six independent entries, the six independent equations require that

\tilde{C} - C = 0 \Leftrightarrow \tilde{C} = C

(25)

Note that the four Eqs. (20) and (21) are satisfied due to the definition of $\tilde{C}$ 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 $\tilde{C} = C$ ⁠. It can be seen that if any two equations in Eq. (14) are satisfied, then condition (12) must be satisfied for any permutation of {1, 2, 3, 4}. As such, condition (14) is a necessary and sufficient condition for C to be expressed in the form of Eq. (11).

Also, note that if C can be expressed in the form of Eq. , then for all permutations {i, j, r, s} of {1, 2, 3, 4}

c_{s} = \frac{w_{i j}^{T} C w_{i r}}{(w_{i j}^{T} t_{s}) (w_{i r}^{T} t_{s})} = \frac{w_{i j}^{T} C w_{r j}}{(w_{i j}^{T} t_{s}) (w_{r j}^{T} t_{s})}

(26)

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

In fact, if condition holds for any two different permutations, then, by the construction of

\tilde{C}

in Eq. ,

\tilde{C}

must satisfy the same condition

\frac{w_{i j}^{T} \tilde{C} w_{i r}}{(w_{i j}^{T} t_{s}) (w_{i r}^{T} t_{s})} = \frac{w_{i j}^{T} \tilde{C} w_{r j}}{(w_{i j}^{T} t_{s}) (w_{r j}^{T} t_{s})}

(27)

Combining Eqs. and yields

\frac{w_{i j}^{T} (\tilde{C} - C) w_{i r}}{(w_{i j}^{T} t_{s}) (w_{i r}^{T} t_{s})} = \frac{w_{i j}^{T} (\tilde{C} - C) w_{r j}}{(w_{i j}^{T} t_{s}) (w_{r j}^{T} t_{s})}

(28)

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

In summary, we have

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

(a)
For any two sets of (i, j, r, s) from the three permutations (1, 2, 3, 4), (1, 3, 2, 4), and (1, 4, 2, 3)
$w_{i j}^{T} C w_{r s} = 0$
(29)
(b)
For any two permutations (i, j, r, s) of {1, 2, 3, 4} with different values of s
$\frac{w_{i j}^{T} C w_{i r}}{(w_{i j}^{T} t_{s}) (w_{i r}^{T} t_{s})} = \frac{w_{i j}^{T} C w_{r j}}{(w_{i j}^{T} t_{s}) (w_{r j}^{T} t_{s})} ◻$
(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_i's in Eq. (11). The mechanism, however, must be capable of having either positive or negative joint compliances. If the conditions in Proposition 1 are not satisfied, the compliant behavior cannot be realized with the mechanism at the configuration even if negative (active) compliance is allowed for each joint.

If variable stiffness actuation is used for a passive realization, each coefficient in Eq. must be non-negative, i.e., the coefficients defined in Eqs. – must be non-negative. It can be proved that, if for any four different permutations with s = 1, 2, 3, 4 [i.e., Eqs. –]

\frac{(w_{i j}^{T} C w_{i r})}{(w_{i j}^{T} t_{s}) (w_{i r}^{T} t_{s})} \geq 0

then, the conditions must hold for all permutations. In summary, we have:

Proposition 2. A compliance matrixCcan be passively realized with a four-joint serial mechanism having joint twistst_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)
$w_{i j}^{T} C w_{r s} = 0$
(31)
(b)
For any four permutations (i, j, r, s) of {1, 2, 3, 4} with s = 1, 2, 3, 4
$\frac{(w_{i j}^{T} C w_{i r})}{(w_{i j}^{T} t_{s}) (w_{i r}^{T} t_{s})} \geq 0 ◻$
(32)

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

Parallel Mechanisms With Four Springs.

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

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

t_{i j} = w_{i} \times w_{j}

(33)

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

Proposition 3. A stiffness matrixKcan be passively realized with a four-spring parallel mechanism having spring wrenchesw_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)
$t_{i j}^{T} K t_{r s} = 0$
(34)
(b)
For any four permutations (i, j, r, s) of {1, 2, 3, 4} with s = 1, 2, 3, 4
$\frac{t_{i j}^{T} K t_{i r}}{(t_{i j}^{T} w_{s}) (t_{i r}^{T} w_{s})} \geq 0 ◻$
(35)

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

It can be seen that, in order to passively realize a given stiffness matrix with a four-spring parallel mechanism, two equality conditions and four inequality conditions must be satisfied. If a given K can be realized with the mechanism, the stiffness for each spring can be determined by

k_{1} = \frac{t_{23}^{T} K t_{24}}{(t_{23}^{T} w_{1}) (t_{24}^{T} w_{1})}

(36)

k_{2} = \frac{t_{34}^{T} K t_{14}}{(t_{34}^{T} w_{2}) (t_{14}^{T} w_{2})}

(37)

k_{3} = \frac{t_{14}^{T} K t_{12}}{(t_{14}^{T} w_{3}) (t_{12}^{T} w_{3})}

(38)

k_{4} = \frac{t_{12}^{T} K t_{23}}{(t_{12}^{T} w_{4}) (t_{23}^{T} w_{4})}

(39)

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

Realization Uniqueness.

When a compliance is realized at a given configuration of a redundant four-joint serial mechanism, the joint compliances c_i typically are unique. If any three t_i'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 t_i's requires that these three joints are not located on a single line. The joint compliances can be determined by Eqs. –. Note that for each joint, the unique corresponding joint compliance c_i can be calculated using any permutation of {j, r, s}

c_{i} = \frac{w_{j r}^{T} C w_{j s}}{(w_{j r}^{T} t_{i}) (w_{j s}^{T} t_{i})}

(40)

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

Similarly, if in a parallel mechanism, any three springs are not parallel to each other, the realization of the given stiffness with the mechanism is unique, and the spring stiffness can be determined using Eqs. –, or using any permutation of {j, r, s}

k_{i} = \frac{t_{j r}^{T} K t_{j s}}{(t_{j r}^{T} w_{i}) (t_{j s}^{T} w_{i})}

(41)

Geometric Analysis of Four-Component Elastic Behavior Realization

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

Geometric Interpretation of Realization Conditions for Serial Mechanisms.

The realization conditions can be interpreted in terms of requirements on the geometry of the mechanism. If w_ij is a force along line l_ij passing through both joints J_i and J_j, then the twist resulting from w_ij acting on the compliant mechanism is

t_{i j} = C w_{i j}

(42)

The realization equality conditions require

w_{r s}^{T} C w_{i j} = 0 \Leftrightarrow w_{r s}^{T} t_{i j} = 0

(43)

which indicates that t_ij must be on the line of action of w_rs, i.e., on line l_rs passing through joints J_r and J_s. Thus, any force along line l_ij results in a twist on line 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_rs bounded by joints J_r and J_s. Two cases are considered: either line l_ij intersects l_rs between J_s and J_r or it does not.

First, consider the case in which line l_ij does not intersect l_rs between J_r and J_s. For the four-joint locations illustrated in Fig. 2, consider, without loss of generality, w₁₂ and w₃₄.

Fig. 2

Realization conditions for the case that l12 does not cross line l34 between J3 and J4: (a) A force w12 applied to the mechanism along line l12 must result in a twist t12 located on line l34 between joints J3 and J4 and (b) a force w34 applied to the mechanism along line l34 must result in a twist t34 located on line l12 between joints J1 and J2

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Realization conditions for the case that l₁₂ does not cross line l₃₄ between J₃ and J₄: (a) A force w₁₂ applied to the mechanism along line l₁₂ must result in a twist t₁₂ located on line l₃₄ between joints J₃ and J₄ and (b) a force w₃₄ applied to the mechanism along line l₃₄ must result in a twist t₃₄ located on line l₁₂ between joints J₁ and J₂

For this case, we first show that if C is passively realized with the mechanism, then t₁₂ = Cw₁₂ must be located on the segment J₃J₄. Since C can be expressed in the form of Eq. with c_i ≥ 0

t_{12} = C w_{12} = c_{3} (t_{3}^{T} w_{12}) t_{3} + c_{4} (t_{4}^{T} w_{12}) t_{4}

(44)

Since J₃ and J₄ are on the same side of l₁₂, $t_{3}^{T} w_{12}$ and $t_{4}^{T} w_{12}$ have the same sign. Since c₃ and c₄ are positive, $c_{3} (t_{3}^{T} w_{12})$ and $c_{4} (t_{4}^{T} w_{12})$ have the same sign. Thus, the positive (or negative) combination of t₃ and t₄ in Eq. (44) indicates that t₁₂ must be located on the line segment between J₃ and J₄.

Conversely, if t₁₂ is located on the finite segment between J₃ and J₄, then the coefficients of t₃ and t₄ in Eq. (44) must have the same sign, i.e., $c_{3} (t_{3}^{T} w_{12})$ and $c_{4} (t_{4}^{T} w_{12})$ have the same sign. Therefore, c₃ and c₄ must also have the same sign. Below, we show that c₃ and c₄ must be non-negative.

Since C is PSD

\begin{matrix} w_{12}^{T} C w_{12} = c_{3} (t_{3}^{T} w_{12}) (t_{3}^{T} w_{12}) + c_{4} (t_{4}^{T} w_{12}) (t_{4}^{T} w_{12}) \\ = c_{3} {(t_{3}^{T} w_{12})}^{2} + c_{4} {(t_{4}^{T} w_{12})}^{2} \geq 0 \end{matrix}

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

Next, consider the case in which line l_ij intersects l_rs at the segment between J_r and J_s. Here, we suppose, without loss of generality, that l₁₃ intersects line l₂₄ between J₂ and J₄. If C is realized with the mechanism at the configuration illustrated, then C can be expressed in the form of Eq. with all coefficients c_i ≥ 0. Then

t_{13} = C w_{13} = c_{2} (t_{2}^{T} w_{13}) t_{2} + c_{4} (t_{4}^{T} w_{13}) t_{4}

(45)

Since J₂ and J₄ are on the opposite sides of line l₁₃, $t_{2}^{T} w_{13}$ and $t_{4}^{T} w_{13}$ have opposite signs. Thus, t₁₃ must be on line l₂₄ outside the finite segment bounded by J₂ and J₄. Conversely, if t₁₃ is located outside the line segment J₂J₄, the coefficients of t₂ and t₃ in Eq. (45), $c_{2} (t_{2}^{T} w_{13})$ and $c_{4} (t_{4}^{T} w_{13})$ ⁠, have opposite signs. Since $(t_{2}^{T} w_{13})$ and $(t_{4}^{T} w_{13})$ have opposite signs, c₁ and c₂ have the same sign. Since C is PSD, c₁ and c₂ must be non-negative. Figure 3 shows the geometric interpretation of the realization conditions for this case.

Fig. 3

Realization conditions for the case in which l13 crosses line l24 between J2 and J4: (a) A force w13 applied on the mechanism along line l13 must result in a twist t13 located on line l24 outside the segment between J2 and J4 and (b) a force w24 applied to the mechanism along line l24 results in a twist t24 located on line l13 outside the segment between J1 and J3

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Realization conditions for the case in which l₁₃ crosses line l₂₄ between J₂ and J₄: (a) A force w₁₃ applied on the mechanism along line l₁₃ must result in a twist t₁₃ located on line l₂₄ outside the segment between J₂ and J₄ and (b) a force w₂₄ applied to the mechanism along line l₂₄ results in a twist t₂₄ located on line l₁₃ outside the segment between J₁ and J₃

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

(a)
A force along l₁₂results in a twist located on line segment J₃J₄; and a force along l₃₄results in a twist located on line segment J₁J₂.
(b)
A force along l₂₃results in a twist located on line segment J₁J₄; and a force along l₁₄results in a twist located on line segment J₂J₃. ◻

Note that in Proposition 4, condition (a) implies one equality condition $w_{12}^{T} C w_{34} = 0$ 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.

Consider a parallel mechanism having four springs w_i. A stiffness matrix K can be realized with the mechanism if and only if

K = k_{1} w_{1} w_{1}^{T} + k_{2} w_{2} w_{2}^{T} + k_{3} w_{3} w_{3}^{T} + k_{4} w_{4} w_{4}^{T}

(46)

where k_i ≥ 0.

Suppose that among the four springs, no three are parallel as illustrated in Fig. 4. The intersection of two wrench axes w_i and w_j at T_ij uniquely identifies a unit twist t_ij reciprocal to both wrenches. The equality realization condition requires that any twist located at the intersection of w_i and w_j results in a wrench that passes through the intersection of the axes of the other two wrenches, w_r and w_s. Below, we show that the wrench must be inside a portion of the pencil bounded by the lines along the axes of w_r and w_s. Since the wrench w_ij = Kt_ij passes through the intersection of w_r and w_s, w_ij can be expressed as

w_{i j} = K t_{i j} = α w_{r} + β w_{s}

where α and β are scalars. By inequality condition

\frac{t_{i s}^{T} K t_{i j}}{(t_{i j}^{T} w_{r}) (t_{i s}^{T} w_{r})} = \frac{t_{i s}^{T} (α w_{r} + β w_{s})}{(t_{i j}^{T} w_{r}) (t_{i s}^{T} w_{r})} = \frac{α}{t_{i j}^{T} w_{r}} \geq 0

(47)

Fig. 4

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Realization conditions for four-spring parallel mechanisms. The wrench resulting from a twist t_ij whose center T_ij is located at the intersection of two wrench axes w_i and w_j, must pass through the intersection of the other two wrench axes, T_rs, and must lie in the interior of the shaded area that does not contain point T_ij.

and

\frac{t_{i r}^{T} K t_{i j}}{(t_{i j}^{T} w_{s}) (t_{i r}^{T} w_{s})} = \frac{t_{i r}^{T} (α w_{r} + β w_{s})}{(t_{i j}^{T} w_{s}) (t_{i r}^{T} w_{s})} = \frac{β}{t_{i j}^{T} w_{s}} \geq 0

(48)

Thus, α and

t_{i j}^{T} w_{r}

have the same sign, and β and

t_{i j}^{T} w_{s}

have the same sign. Since the scalars α and β have fixed signs, wrench w_ij must be inside an area bounded by the two axes of w_r and w_s. It is readily shown that the area does not contain the intersection of the axes of w_i and w_j, T_ij. In fact, if a wrench w has axis passing through both locations of T_ij and T_rs (the center of twist t_rs), w can be expressed as

w = \tilde{α} w_{r} + \tilde{β} w_{s}

(49)

and satisfies

t_{i j}^{T} w = \tilde{α} (t_{i j}^{T} w_{r}) + \tilde{β} (t_{i j}^{T} w_{s}) = 0

(50)

If $\tilde{α}$ and $(t_{i j}^{T} w_{r})$ have the same sign, $\tilde{β}$ and $(t_{i j}^{T} w_{s})$ 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_i and w_j as shown in Fig. 4.

Also, it can be proved that if w_ij is in the area defined above (the shaded area in Fig. 4), the following equality conditions are satisfied:

\frac{t_{i s}^{T} K t_{i j}}{(t_{i j}^{T} w_{s}) (t_{i r}^{T} w_{s})} \geq 0, \frac{t_{i r}^{T} K t_{i j}}{(t_{i j}^{T} w_{s}) (t_{i r}^{T} w_{s})} \geq 0

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

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

(a)
A twist at the intersection ofw₁andw₂, T₁₂, results in a wrench that passes through the intersection ofw₃andw₄, T₃₄, and lies in the area bounded by the axes ofw₃andw₄that does not contain T₁₂; and a twist at T₃₄results in a wrench that passes through the intersection ofw₁andw₂, T₁₂and lies in the area bounded by the axes ofw₁andw₂that does not contain T₃₄.
(b)
A twist at the intersection ofw₁andw₃, T₁₃, results in a wrench that passes through the intersection ofw₂andw₄and lies in the area bounded by the axes ofw₂andw₄that does not contain T₁₃; and a twist at the intersection ofw₂andw₄, T₂₄, results in a wrench that passes through the intersection ofw₁andw₃and lies in the area bounded by the axes ofw₁andw₃that does not contain T₂₄. ◻

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

Fig. 5

Realization conditions for four-spring parallel mechanisms: (a) The wrench caused by a twist at T12 must be in the shaded area bounded by the two wrench axes w3 and w4 and (b) The wrench caused by a twist at T34 must be in the shaded area bounded by the two wrench axes w1 and w2

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Realization conditions for four-spring parallel mechanisms: (a) The wrench caused by a twist at T₁₂ must be in the shaded area bounded by the two wrench axes w₃ and w₄ and (b) The wrench caused by a twist at T₃₄ must be in the shaded area bounded by the two wrench axes w₁ and w₂

Note that, similar to the serial mechanism, in Proposition 5, condition (a) implies one equality condition $t_{12}^{T} K t_{34} = 0$ 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.

Consider a compliance matrix C and two wrenches w₁ and w₂ satisfying

w_{1}^{T} C w_{2} = 0

(51)

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

Fig. 6

Two wrenches satisfying reciprocal condition (51). Any wrench passing through T12 results in a twist located on line l12. Any wrench along line l12 results in a twist located at point T12. The center of the compliance must be in the interior of the shaded triangle.

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Two wrenches satisfying reciprocal condition (51). Any wrench passing through T₁₂ results in a twist located on line l₁₂. Any wrench along line l₁₂ results in a twist located at point T₁₂. The center of the compliance must be in the interior of the shaded triangle.

(a)
Any wrench w passing through the intersection point of the two wrenches, T₁₂, when multiplied by C, results in a twist t located on line l₁₂ passing through the centers of t₁ and t₂ as shown in Fig. 6.
(b)
Any wrench along line l₁₂ passing through the centers of t₁ and t₂, when multiplied by C, results in a twist t₁₂ located at T₁₂ (where wrenches w₁ and w₂ intersect).
(c)
The center of compliance must be inside the triangle formed by the centers of twists t₁, t_2, and t₁₂ (the shaded area shown in Fig. 6).

In fact, if wrenches w₁ and w₂ satisfy condition and wrench w passes through point T₁₂, the intersection of w₁ and w₂, then w can be expressed as

w = α w_{1} + β w_{2}

where α and β are scalars. The twist resulting from w is

t = C w = α C w_{1} + β C w_{2} = α t_{1} + β t_{2}

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

If w₁₂ is a wrench passing through t₁ and t₂

t_{1}^{T} w_{12} = 0, t_{2}^{T} w_{12} = 0

(52)

then

w_{1}^{T} C w_{12} = 0, w_{2}^{T} C w_{12} = 0

(53)

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

Since the three wrenches w₁, w_2, and w₁₂ are reciprocal about C, analysis of three-spring parallel mechanisms [7] shows that the center of compliance must be in the interior of the triangle formed by the axes of the three wrenches. It also can be seen that the three twists t₁, t_2, and t₁₂ are reciprocal about the stiffness K = C⁻¹. The space of realizable elastic centers is bounded by the joint locations (the joint twist centers) in serial mechanisms and by the spring axes in parallel mechanisms.

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

Fig. 7

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The locus of compliance centers of elastic behaviors realized with a four-joint serial and four-spring parallel mechanism. (a) For a four-joint serial mechanism, the center is inside the polygon formed by the union of all triangles formed by any three joints. (b) For a four-spring parallel mechanism, the center is inside the polygon formed by the union of all triangles formed by any three spring wrenches.

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 $W_{p}$ ⁠. In the plane, each w in $W_{p}$ can be represented by its line of action l. Thus, $W$ is represented by a family of parallel lines $L_{p}$ in the plane. In $L_{p}$ ⁠, there is a unique line l_c that passes through the center of compliance C_c.

Suppose

T_{w}

is the collection of all twists resulting from multiplying the wrenches of

W_{p}

by C, i.e.,

T_{w} = {t = C w, w \in W_{p}}

(54)

It is shown below that the centers of twists in $T_{w}$ form a straight line l_t passing through the center of compliance C_c.

Since every w in

W_{p}

has the same direction

n, W_{p}

can be expressed in the form

W_{p} = {w = [\begin{matrix} n \\ d \end{matrix}], d \in (- \infty, \infty)}

(55)

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

The compliance matrix C can be represented in block form as

C = [\begin{matrix} D & g \\ g^{T} & c_{33} \end{matrix}]

(56)

The twist caused by w can then be written as

\begin{matrix} t = C w = [\begin{matrix} D & g \\ g^{T} & c_{33} \end{matrix}] [\begin{matrix} n \\ d \end{matrix}] = [\begin{matrix} D n + d g \\ g^{T} n + d c_{33} \end{matrix}] \\ = [\begin{matrix} D n \\ g^{T} n \end{matrix}] + d [\begin{matrix} g \\ c_{33} \end{matrix}] \end{matrix}

(57)

The two components of t in Eq. are denoted as

t_{n} = [\begin{matrix} D n \\ g^{T} n \end{matrix}], t_{c} = [\begin{matrix} g \\ c_{33} \end{matrix}]

(58)

It can be seen that t_n is independent of the value of d. Using Eq. , the center of twist t_c, T_c, is calculated as

r_{c} = \frac{1}{c_{33}} Ω g

(59)

Thus,

T_{w}

defined in Eq. can be expressed as

T_{w} = {t = t_{n} + d t_{c}, d \in (- \infty, \infty)}

which means that every t associated with w lies on the straight line l_t passing through the centers of t_n and 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_c). Thus, the t associated with w must lie on the straight line l_t passing through the compliance center C_c as illustrated in Fig. 8(a). Since C is positive definite, $w^{T} t = w^{T} C w > 0$ ⁠, the axis of w never intersects the corresponding center of t. Therefore, w and t are always separated by line l_c as shown in Fig. 8(b). When w approaches the line passing through the center of compliance, l_c, then w^Tt_c → 0, and the center if the corresponding t goes to infinity along the line l_t as shown in Fig. 8(c). When w is far away from the compliance center, d → ±∞, then the center of t approaches C_c, the center of the compliance, as shown in Fig. 8(d).

Fig. 8

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A family of parallel wrenches $W_{p}$ and the corresponding twists $T_{w}$ ⁠: (a) the locus of t = Cw is a line l_t passing through the compliance center C_c, (b) the line of action of wrench w and the center of twist t = Cw are separated by line l_c, (c) when w → l_c from two sides, the center of t → ±∞ along line l_t, and (d) When w → ±∞, the center of t approaches C_c along line l_t

In summary, we have

Proposition 6. Consider a family of parallel wrenches $W_{p}$ and the corresponding set of twists $T_{w}$ defined in Eq. (54).

(a)
The locus of $t \in T_{w}$ is a straight line l_t passing through the center of compliance.
(b)
For any $w \in W_{p}$ , the corresponding twist $t \in T_{w}$ is separated by line l_c, the line passing through the center of compliance C_c.
(c)
When the axis of $w \in W_{p}$ approaches the line l_c passing through compliance center, the location oftgoes to infinity along line l_t.
(d)
When the axis of $w \in W_{p}$ is far away from the compliance center, d → ±∞ and the location oftapproaches the center of compliance C_c along l_t. ◻

Conversely, consider the collection of twists,

T_{l}

⁠, located on a straight line l_t that passes through the center of stiffness C_k, and denote

W_{t}

as the collection of wrenches that result from multiplying the stiffness matrix K by the twists in

T_{l}

⁠, i.e.,

W_{t} = {w = K t, t \in T_{l}}

(60)

Then, as shown below, $W_{t}$ is a family of parallel wrenches.

Let t_c be a twist at the stiffness center and t₀ be an arbitrary point on l_t. Any twist on the line can be expressed as

t = α t_{c} + β t_{0}

where α and β are scalars. The corresponding resulting wrench is

w = K t = α K t_{c} + β K t_{0}

(61)

Since t_c is at the center of stiffness, Kt_c is a pure couple having the form

w_{c} = K t_{c} = {[0, 0, m]}^{T}

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

Suppose that l_c is the line of action of the wrench in $W_{t}$ passing through the stiffness center C_k. Similar to Proposition 6, we have:

Proposition 7. Consider a family of twists $T_{l}$ located on a straight line l_t passing through the stiffness center C_k and the set of corresponding wrenches $W_{t}$ defined in Eq. (60).

(a)
$W_{t}$ forms a family of parallel wrenches.
(b)
For any twist $t \in T_{l}$ , the line of action of the corresponding wrenchwis on the opposite side of line l_c.
(c)
When the center of twistt→ C_k along line l_t, the line of action of the correspondingwgoes to infinity.
(d)
Whent→ ±∞ along line l_t, the line of action of the correspondingwapproaches line l_c. ◻

The properties of $W_{t}$ are illustrated in Fig. 9.

Fig. 9

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A family of twists $T_{l}$ centered on a line l_t passing through the stiffness center C_k and the corresponding wrenches $W_{t}$ ⁠: (a) the lines of action of wrenches resulting from w = Kt form a family of parallel lines, (b) the line of action of w and the center of t are on the opposite sides of line l_c, (c) when the center of t → C_k, the line of action of w → ±∞, and (d) when the center of t → ±∞, the line of action of w approaches line l_c

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.

Fig. 10

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Synthesis of a compliance with a four-joint serial mechanism based on geometry. The realization is achieved by selecting the locations of the four joints in the mechanism.

(1)
Choose a line l₁ arbitrarily relative to C_c and obtain the unit wrench w₁ associated with l₁. Two revolute joints of the serial mechanism will be located on this line. Calculate the twist resulting from that wrench acting on C
$t_{1} = C w_{1}$
(62)
The center of t₁, T₁, is determined using Eq. (3).
(2)
Choose a line l₂ passing through the point T₁ and obtain the unit wrench w₂ associated with l₂. The other two joints of the mechanism will be on this line. Calculate the twist resulting from w₂
$t_{2} = C w_{2}$
(63)
The center of t₂, T₂, will be on line l₁ (the axis of w₁), i.e.,
$w_{1}^{T} C w_{2} = 0$
(64)
The line passing through points T₁ and T₂ is denoted as l.
(3)
Choose a line l₃ on one side of line l such that the center of the corresponding twist t₃ = Cw₃ is on the opposite side of l.
Line l₃ can be selected using the properties of Proposition 6 presented in Sec. 4.1.
- (a)
  First, choose a candidate line l_p that does not intersect line segment T₁T₂. Obtain the unit wrench ${\tilde{w}}_{p}$ associated with l_p and calculate the corresponding twist $t_{p} = C {\tilde{w}}_{p}$ ⁠. Then, by the property presented in Sec. 4.1, the center of t_p, T_p, and the compliance center C_c must be on the same side of l_p.
- (b)
  Translate l_p toward C_c until T_p is on the opposite side of l. This line is selected as l₃ and the center of the corresponding twist t₃ is T₃.
  This can always be accomplished due to the fact that when l_p → C_c, T_p → ∞.
(4)
Choose a line l₄ that passes through point T₃ and does not intersect line segment T₁T₂. The intersections of these four lines (shown in Fig. 10) are the locations of the four joints (J₁, J₂, J₃, J₄) of the mechanism that realizes the specified compliance.
(5)
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.

Fig. 11

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Synthesis of a compliance with a four-spring parallel mechanism based on geometry. The realization is achieved by selecting the spring axes of the four springs in the mechanism.

(1)
Choose an arbitrary location T₁ relative to the stiffness center C_k and obtain the unit twist t₁ at T₁ using the formula for t_r in Eq. (1). Two spring axes will intersect here in the parallel mechanism. The wrench resulting from t₁ is
${\tilde{w}}_{1} = K t_{1}$
(65)
The line of action of ${\tilde{w}}_{1}$ ⁠, l₁, is obtained.
(2)
Choose an arbitrary point T₂ on line l₁, the unit twist t₂ centered at T₂ is calculated using Eq. (1). Point T₂ is the intersection of the other two spring axes. Then, wrench ${\tilde{w}}_{2}$ corresponding to t₂ is obtained
${\tilde{w}}_{2} = K t_{2}$
(66)
The line of action of ${\tilde{w}}_{2}$ ⁠, l₂ is obtained. By the reciprocal condition, line l₂ must pass through T₁, and satisfy
$t_{1}^{T} K t_{2} = 0$
(67)
The line passing through points T₁ and T₂ is denoted as l.
(3)
Choose a line l₃ such that the corresponding wrench ${\tilde{w}}_{3}$ results in a twist $t_{3} = C {\tilde{w}}_{3}$ centered at the opposite side of line l.
Line l₃ can be selected using the properties of Proposition 7 presented in Sec. 4.1.
- (a)
  First choose a candidate line l_p that does not intersect line segment T₁T₂. Using the unit wrench along l_p, ${\tilde{w}}_{p}$ ⁠, the corresponding twist $t_{p} = C w_{p}$ is calculated. Then, by the property presented in Sec. 4.1, the center of t_p, T_p, and the stiffness center C_k are both located on the same side of l_p.
- (b)
  Translate l_p toward C_k until T_p is on the opposite side of l. This location T_p is selected as T₃ which will be the intersection of two spring axes in the mechanism.
  This can always be accomplished due to the fact that when l_p → C_k, T_p → ∞. Line l₃ intersects l₁ at T₁₃ and intersects l₂ at T₂₃.
(4)
Choose an arbitrary point T₄ on line l₃ between T₁₃ and T₂₃, the four lines passing through points (T₁, T₃), (T₁, T₄), (T₂, T₄), and (T₂, T₃) as shown in Fig. 11 are identified as the four spring axes (w₁, w₂, w₃, w₄) for the parallel mechanism.
(5)
Calculate the value of stiffness for each joint using Eqs. (36)–(39).

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

Examples

Examples are provided to illustrate the synthesis procedures. The elastic behavior to be realized in a given coordinate frame is given by

K = [\begin{matrix} 3 & - 2 & 1 \\ - 2 & 6 & 5 \\ 1 & 5 & 9 \end{matrix}], C = K^{- 1} = [\begin{matrix} 1.16 & 0.92 & - 0.64 \\ 0.92 & 1.04 & - 0.68 \\ - 0.64 & - 0.68 & 0.56 \end{matrix}]

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

Serial Mechanism Synthesis.

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

The first line on which two joints J₁ and J₂ lie can be chosen arbitrarily. Here, the line is chosen to make a 45 deg angle with the x-axis and the distance from the line to the origin of the coordinate frame is

d_{1} = 0.5 \sqrt{2}

as shown in Fig. 12. The line vector of l₁ (unit wrench) is

w_{1} = {[cos 45 °, sin 45 °, 0.5 \sqrt{2}]}^{T}

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

Fig. 12

Realization of a given compliance with a four-joint serial mechanism. In the process, the locations of the four joints (J1, J2, J3, J4) in the mechanism and the corresponding joint compliances (c1, c2, c3, c4) are identified.

View large Download slide

Realization of a given compliance with a four-joint serial mechanism. In the process, the locations of the four joints (J₁, J₂, J₃, J₄) in the mechanism and the corresponding joint compliances (c₁, c₂, c₃, c₄) are identified.

The twist t₁ associated with w₁ is calculated to be

t_{1} = C w_{1} = {[1.4400, 1.2800, - 0.7600]}^{T}

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

The other two joints, J₃ and J₄, must lie on a line passing through point T₁. Since T₁ is determined, line l₂ can be determined by selecting the direction or slope arbitrarily. Here, line l₂ is selected to be parallel to the y-axis as shown in Fig. 12. It can be seen that the distance from l₂ to the origin is d₂ = 1.6842. The line vector of l₂ is

w_{2} = {[0, 1, 1.6842]}^{T}

The twist t₂ associated with w₂ is calculated to be

t_{2} = C w_{2} = {[- 0.1579, - 0.1053, 0.2632]}^{T}

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

The selection of line l₃ determines the location of joints J₂ and J₃. Line l₃ must be selected such that the location of the corresponding twist, T₃, is on the opposite side of line l connecting points T₁ and T₂. Here, line l₃ is selected to have slope −1 and distance 0.4 from the origin as shown in Fig. 12. The line vector associated with l₃ is

w_{3} = {[- cos 45 °, sin 45 °, 0.4]}^{T}

The twist associated with w₃ is calculated to be

t_{3} = C w_{3} = {[- 0.4257, - 0.1871, 0.1957]}^{T}

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

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

w_{4} = {[1, 0, 2.1751]}^{T}

Fig. 13

Realization of a given stiffness with a four-spring parallel mechanism. In the process, the four axes of the springs (w1, w2, w3, w4) in the mechanism and the corresponding spring stiffnesses (k1, k2, k3, k4) are identified.

View large Download slide

Realization of a given stiffness with a four-spring parallel mechanism. In the process, the four axes of the springs (w₁, w₂, w₃, w₄) in the mechanism and the corresponding spring stiffnesses (k₁, k₂, k₃, k₄) are identified.

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

The locations of the four joints are

\begin{matrix} r_{1} = {[- 1.1751, - 2.1751]}^{T}, r_{2} = {[0.7828, - 0.2172]}^{T}, \\ r_{3} = {[1.6842, - 1.1185]}^{T}, r_{4} = {[1.6842, - 2.1751]}^{T} \end{matrix}

Since l₁ passes through J₁ and J₂, wrench w₁ is reciprocal to t₁ and t₂. Thus

w_{12} = w_{1} = \frac{\sqrt{2}}{2} {[1, 1, 1]}^{T}

Similarly

\begin{matrix} w_{23} = w_{3} = {[- cos 45 °, sin 45 °, 0.4]}^{T} \\ w_{34} = w_{2} = {[0, 1, 1.6842]}^{T} \\ w_{14} = w_{4} = {[1, 0, 2.1751]}^{T} \end{matrix}

The joint compliance constants are determined using Eqs. –

c_{1} = 0.0180, c_{2} = 0.2349, c_{3} = 0.1119, c_{4} = 0.1953

With this procedure, the mechanism configuration (illustrated in Fig. 12) and the corresponding joint compliances are determined. To verify the result, the joint twists associated with joint J_i are calculated as

[t_{1}, t_{2}, t_{3}, t_{4}] = [\begin{matrix} - 2.1751 & - 0.2172 & - 1.1185 & - 2.1751 \\ 1.1751 & - 0.7828 & - 1.6842 & - 1.6842 \\ 1 & 1 & 1 & 1 \end{matrix}]

Summing of the elastic components using Eq. yields

\sum_{i = 1}^{4} c_{i} t_{i} t_{i}^{T} = [\begin{matrix} 1.16 & 0.92 & - 0.64 \\ 0.92 & 1.04 & - 0.68 \\ - 0.64 & - 0.68 & 0.56 \end{matrix}]

which confirms the realization of the compliance with the mechanism.

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

Parallel Mechanism Synthesis.

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

First, point T₁ can be selected arbitrarily. Here, T₁ is where spring wrenches w₁ and w₂ will intersect and is selected to be (2, −1.5). The unit twist at T₁ is

t_{1} = {[- 1.5, - 2, 1]}^{T}

The wrench associated with t₁ is calculated to be

{\tilde{w}}_{1} = K t_{1} = {[0.5, - 4, - 2.5]}^{T}

The line of action of

{\tilde{w}}_{1}

⁠, l₁, is obtained as

y = - 8 x + 5

(68)

and is shown in Fig. 13.

The second point T₂ can be arbitrarily selected from line l₁. The other two spring wrenches w₃ and w₄ intersect at this point. Here, point T₂ is selected such that line T₁T₂ is parallel to the x-axis. Using Eq. , the coordinates of T₂ are calculated to be (0.8125, −1.5). The unit twist at T₂ is

t_{2} = {[- 1.5, - 0.8125, 1]}^{T}

The wrench associated with t₂ is

{\tilde{w}}_{2} = K t_{2} = {[- 1.875, 3.125, 3.4375]}^{T}

The line of action of

{\tilde{w}}_{2}

⁠, l₂, is obtained as

y = - 1.6667 x + 1.8333

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

Next, line l₃ is selected. Here, the line is selected to be parallel to x-axis again and to have the distance d₃ = 0.5 to the origin of the coordinate frame as shown in Fig. 13. The line vector associated with l₃ is

{\tilde{w}}_{3} = {[1, 0, 0.5]}^{T}

The twist associated with

{\tilde{w}}_{3}

is calculated to be

t_{3} = C {\tilde{w}}_{3} = {[0.84, 0.58, - 0.36]}^{T}

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

The final point T₄ can be selected from any point of the line segment T₁₃T₂₃. Here, this point is selected to be located at (1, −0.5) (illustrated in Fig. 13). The four-spring wrench axes are identified by the four lines T₁T₃, T₁T₄, T₂T₄, and T₂T₃. The four spring wrenches are

[w_{1}, w_{2}, w_{3}, w_{4}] = [\begin{matrix} 0.4229 & 0.7071 & 0.1843 & 0.6919 \\ 0.9062 & - 0.7071 & 0.9829 & - 0.7220 \\ 2.4467 & - 0.3536 & 1.0750 & 0.4512 \end{matrix}]

In order to calculate the spring constants using Eqs. –, four twists t_ij reciprocal to wrenches w_i and w_j are needed. Since t₁₂, t₃₄, t₁₄, and t₂₃ are located at T₁, T₂, T₃, and T₄, respectively, they can be easily calculated using Eq. . The four twists needed are

[t_{12}, t_{34}, t_{14}, t_{23}] = [\begin{matrix} - 1.5 & - 1.5000 & - 2.3333 & - 0.5 \\ - 2.0 & - 0.8125 & - 1.6111 & - 1.0 \\ 1 & 1 & 1 & 1 \end{matrix}]

The spring constants calculated using Eqs. – are

k_{1} = 0.8740, k_{2} = 3.8278, k_{3} = 2.5359, k_{4} = 1.7624

With this procedure, the mechanism configuration (illustrated in Fig. 13) and the corresponding spring stiffnesses are determined. To verify the result, summing the elastic components using Eq. yields

\sum_{i = 1}^{4} k_{i} w_{i} w_{i}^{T} = [\begin{matrix} 3 & - 2 & 1 \\ - 2 & 6 & 5 \\ 1 & 5 & 9 \end{matrix}]

which confirms that the stiffness is realized with the mechanism.

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

Summary

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

Funding Data

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

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Geometric Approach to the Realization of Planar Elastic Behaviors With Mechanisms Having Four Elastic Components

Introduction

Related Work.

Overview.

Planar Compliance Realization Conditions

Technical Background.

Realization Conditions for Mechanisms Having Four Elastic Components.

Parallel Mechanisms With Four Springs.

Realization Uniqueness.

Geometric Analysis of Four-Component Elastic Behavior Realization

Geometric Interpretation of Realization Conditions for Serial Mechanisms.

Geometric Interpretation of Realization Conditions for Parallel Mechanisms.

Center of Planar Compliant Behaviors.

Synthesis Procedures

Mathematical Preliminaries.

Synthesis Procedures.

Serial Mechanism Synthesis.

Parallel Mechanism Synthesis.

Examples

Serial Mechanism Synthesis.

Parallel Mechanism Synthesis.

Summary

Funding Data

References

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Geometric Approach to the Realization of Planar Elastic Behaviors With Mechanisms Having Four Elastic Components

Introduction

Related Work.

Overview.

Planar Compliance Realization Conditions

Technical Background.

Realization Conditions for Mechanisms Having Four Elastic Components.

Parallel Mechanisms With Four Springs.

Realization Uniqueness.

Geometric Analysis of Four-Component Elastic Behavior Realization

Geometric Interpretation of Realization Conditions for Serial Mechanisms.

Geometric Interpretation of Realization Conditions for Parallel Mechanisms.

Center of Planar Compliant Behaviors.

Synthesis Procedures

Mathematical Preliminaries.

Synthesis Procedures.

Serial Mechanism Synthesis.

Parallel Mechanism Synthesis.

Examples

Serial Mechanism Synthesis.

Parallel Mechanism Synthesis.

Summary

Funding Data

References

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