Compliance Realization With Planar Serial Mechanisms Having Fixed Link Lengths

Huang, Shuguang; Schimmels, Joseph M.

doi:10.1115/1.4053819

Abstract

In this article, the synthesis of any specified planar compliance with a serial elastic mechanism having previously determined link lengths is addressed. For a general n-joint serial mechanism, easily assessed necessary conditions on joint locations for the realization of a given compliance are identified. Geometric construction-based synthesis procedures for five-joint and six-joint serial mechanisms having kinematically redundant fixed link lengths are developed. By using these procedures, a given serial manipulator can achieve a large set of different compliant behaviors by using variable stiffness actuation and by adjusting the mechanism configuration.

1 Introduction

To regulate contact forces and ensure accurate relative positioning, passive compliance is needed in constrained robotic manipulation. A general model for compliance is a rigid body supported by an elastic suspension. A compliant behavior is characterized by the relationship between a force (wrench) applied to the body and the resulting displacement (twist) of the body. If small displacements are considered, the wrench–twist relationship can be represented by a symmetric positive definite matrix, the compliance matrix C, or the stiffness matrix K, the inverse of C.

In practice, an elastic suspension can be achieved by elastic components connected in parallel or in series. Realization of a given compliance involves identifying the geometric and elastic properties of each component such that the desired compliance is attained. This article focuses on serial mechanisms with revolute joints, each having some type of passive compliance. The previous work in this area addressed the problem of finding any mechanism (one with unspecified geometry) to realize a selected compliance. Here, we address the issue of assessing whether a given mechanism is capable of realizing a selected compliance and, if so, how it must be configured to do so. A serial manipulator having fixed link lengths can achieve a large set of different compliant behaviors by adjusting the joint compliance (e.g., using a cobot with variable stiffness actuation [1]) and by adjusting the mechanism configuration (using kinematic redundancy).

1.1 Related Work.

Many researchers investigated general compliant behaviors. In the analysis of spatial compliance, screw theory and Jacobian analysis [2–9], and Lie groups [10,11] have been widely used. In the recent work on the synthesis of compliance, mechanisms are designed to realize any specified compliance. Most previous synthesis approaches were based on an algebraic rank-1 decomposition of the stiffness/compliance matrix [12–15]. In Refs. [16,17], some geometric considerations on the mechanism were included in the synthesis procedures. In Ref. [18], a completely geometry-based approach to the realization of an arbitrary spatial stiffness was presented.

In Refs. [19,20], the synthesis of planar stiffness with parallel mechanisms having specific topologies was presented. In Refs. [21–26], compliant behaviors associated with mechanisms composed of distributed elastic components were investigated.

In closely related work in the realization of planar compliances [27–30], geometry-based approaches were developed for the design of fully parallel or fully serial mechanisms having n (3 ≤ n ≤ 6) elastic components. Necessary and sufficient conditions on the elastic component locations of corresponding mechanisms of a given topology were identified for the realization of any specified planar compliant behavior. The link lengths in these mechanisms were not considered in the synthesis procedures [27–30].

In Ref. [31], conditions required to achieve a special isotropic compliance in a 2D Euclidean space with a serial mechanism with specified link lengths was presented.

1.2 Contribution of the Paper.

Previously developed necessary and sufficient conditions [27–30] on mechanism geometry for the realization of a given compliance must be satisfied for any n-component (3 ≤ n ≤ 6) mechanism. These conditions are the foundation for the development of general planar compliance synthesis procedures. The main limitations of this prior work are as follows:

Each n-joint serial mechanism had no constraints imposed on its link lengths. Thus, the serial mechanism obtained from the synthesis procedure to realize one selected compliance is very unlikely to be able to realize a different compliance.
The issue of whether or how a specified compliance can be realized by a given mechanism was not addressed.

These restrictions limit the use of the existing theories in practical application and are the motivation of this work. When link lengths are considered, the distance between two adjacent joints J_i and J_i+1 is constant. In selecting a configuration of an n-joint serial mechanism for the realization of a compliance, (n − 1) nonlinear constraints on the n-joint locations must be satisfied. The main contributions of this article are as follows:

Identification of a set of necessary conditions on a general n-joint serial mechanism. These conditions provide greater insight into the distribution of joint locations of a serial mechanism in the realization of a compliance.
Development of new synthesis procedures that take into account the known link lengths of any specific serial manipulator. By using these procedures, a large and continuous, but constrained, space of compliances can be realized with a single mechanism by identifying its configuration and joint compliances.

1.3 Overview.

This article addresses the passive realization of an arbitrary planar (3 × 3) compliance with a serial compliant mechanism having fixed link lengths and variable stiffness actuators.

This article is outlined as follows. In Sec. 2, screw representation of planar mechanism configuration is first reviewed. A set of necessary conditions on the geometry of a general n-joint serial mechanism for the realization of a given compliance is then identified. Necessary and sufficient conditions for the realization of a compliance with 5R and 6R mechanisms with prescribed link lengths are presented in Secs. 3 and 4, respectively. Geometry-based synthesis procedures for these mechanisms to realize a given compliance are developed. In Sec. 5, a numerical example is provided to demonstrate the synthesis procedures for both 5R and 6R mechanisms. Finally, a brief discussion and summary are presented in Secs. 6 and 7.

2 Technical Background

In this section, the technical background needed for planar compliance realization with an n-joint serial mechanism is presented. First, the use of screw representation to describe mechanism configuration is reviewed. Next, a requirement on the compliance center location expressed in terms of mechanism joint locations is derived, and a requirement on the distribution of joint locations relative to the compliance center is identified. Then, screw representation of link length constraints and the associated geometric restrictions are presented.

2.1 Elastic Behavior Realized With a Serial Mechanism.

First, screw representations of a point on a plane and a line on a plane are reviewed. The realization of a planar compliance at a mechanism configuration represented by a set of screws is then summarized.

2.1.1 Screw Representation of Points/Lines in a Plane.

It is known that in a plane, a point can be represented by a unit twist t and a line can be represented by a unit wrench w [32]. In Plücker axis coordinates, a planar unit twist t has the following form:

t = [\begin{matrix} u \\ 1 \end{matrix}]

(1)

where

u = r \times \hat{k}

⁠, r is the position vector of the point (instantaneous center of the twist) with respect to the coordinate frame, and

\hat{k}

is the unit vector perpendicular to the plane. Thus, for any unit twist t, the location of its instantaneous center is expressed as follows:

r = Ω u

(2)

where

Ω

is the 2 × 2 antisymmetric matrix defined as follows:

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

(3)

Thus, t is uniquely described by the location r of a point J as shown in Fig. 1(a).

Fig. 1

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Screw representation of a point and a line. (a) A point J is uniquely represented by a unit twist t in Eq. (1). The point location r identifies the instantaneous center of the twist. (b) A line l is uniquely represented by a unit wrench w in Eq. (4). The line is the line-of-action (axis) of the wrench.

In Plücker ray coordinates, a unit wrench w has the following form:

w = [\begin{matrix} n \\ d \end{matrix}]

(4)

where n is a unit 2-vector indicating the direction of the wrench and where

d = (r_{p} \times n) \cdot \hat{k}

(5)

where r_p is the position vector from the origin to any point on the wrench axis. The axis of w is uniquely defined as the line l having direction n with perpendicular distance d to the origin (as shown in Fig. 1(b)). Thus, any line in the plane can be represented by a unit wrench.

If a line l represented by wrench w passes through a point J represented by twist t, then the two screws must be reciprocal:

t^{T} w = 0

(6)

These properties will be used in the synthesis of compliance with a serial mechanism.

2.1.2 Compliance Realization With a Serial Mechanism.

Consider a serial mechanism with n revolute joints J_i (i = 1, 2, …, n). If each joint location J_i is described by joint twist t_i, the mechanism Cartesian compliance C is expressed as follows [15]:

C = c_{1} t_{1} t_{1}^{T} + c_{2} t_{2} t_{2}^{T} + \dots + c_{n} t_{n} t_{n}^{T}

(7)

where c_i ≥ 0 is the joint compliance at joint J_i, i = 1, 2, …, n. Thus, to passively realize a compliance C with an n-joint serial mechanism, a set of n-joint twists t_i and corresponding joint compliances c_i that satisfy Eq. need to be identified. For a specified joint twist, the location of the associated joint is determined by Eq. .

An n-joint planar serial mechanism with fixed link lengths has n degrees-of-freedom. If each joint has modulated passive compliance c_i, there are n additional independent variables. If the end-effector position and orientation with respect to the base joint are specified, the total number of independent variables associated with a compliant mechanism is (2n − 3). Since a planar 3 × 3 passive compliance matrix is symmetric, it has six independent parameters. To realize an arbitrary compliance at an end-effector pose relative to the robot base, the number of joints, n, must satisfy

2 n - 3 \geq 6 ⟹ n \geq 4.5

Since n is an integer,

n \geq 5

Therefore, to achieve an arbitrary compliant behavior with a serial mechanism having fixed link lengths, a given position of the base, and a given position and orientation of the end-effector, the mechanism must have at least five joints.

2.2 Compliance Center Relative Position.

For every planar fully elastic behavior, there is a unique point at which the compliance matrix can be diagonalized. This point is called the center of compliance. For the planar case, the center of compliance and the center of stiffness are coincident. A general planar compliance matrix has the following form:

C = [\begin{matrix} C_{11} & v \\ v^{T} & c_{22} \end{matrix}]

(8)

where

C_{11} \in R^{2 \times 2}

⁠,

v \in R^{2}

and c₂₂ > 0. The location r_c of the compliance center C_c is determined by

r_{c} = \frac{Ω v}{c_{22}}

(9)

where the 2 × 2 matrix

Ω

is defined in Eq. .

The relationship between the location of the compliance center and the configuration of a mechanism capable of realizing the behavior is presented in Ref. [33] for the general spatial case. For the planar case, the relationship can be expressed in a simpler form.

Proposition 1

Suppose a planar compliance C is realized at a particular configuration of an n-joint mechanism. If r_i is the position vector of each joint in an arbitrary coordinate frame, and c_i is the corresponding joint compliance, then the location of the center of compliance is expressed as follows:

r_{c} = \frac{c_{1} r_{1} + c_{2} r_{2} + \dots + c_{n} r_{n}}{c_{1} + c_{2} + \dots + c_{n}}

(10)

Thus, the center of compliance is the joint compliance c_i weighted average of the joint locations r_i. It can be seen that the location of the compliance center in Eq. (10) takes the same form as the location of the mass center for particle masses, which indicates the analogy between the two types of centers. Therefore, the compliance center must be within the convex hull formed by the n-joint locations.

2.3 Joint Location Distribution Conditions.

The condition that the compliance center must be inside the area determined by the joint locations is only a necessary condition to realize the behavior. Most compliant behaviors cannot be achieved by a serial mechanism even if the compliance center is located within the corresponding area associated with the mechanism geometry. Necessary and sufficient conditions for mechanisms having 3, 4, 5, and 6 joints are identified in Refs. [27–30].

Below, an additional set of easily assessed necessary conditions on the distribution of elastic components is identified.

At the center of compliance, a compliance matrix can be expressed in diagonal form in the principal frame by screw transformation:

C = diag [λ_{x}, λ_{y}, λ_{τ}]

(11)

where λ_x and λ_y are the two translational principal compliances and λ_τ is the rotational principal compliance.

For a given C realized by an n-joint mechanism in which d_i is the distance of joint J_i from the compliance center, denote:

d_{min} = min {d_{1}, d_{2}, \dots, d_{n}}

(12)

d_{max} = max {d_{1}, d_{2}, \dots, d_{n}}

(13)

Suppose that, in the principal frame, each joint J_i has coordinates (x_i, y_i) and

d_{min}^{x} = min {| x_{1} |, \dots, | x_{n} |}, d_{max}^{x} = max {| x_{1} |, \dots, | x_{n} |}

(14)

d_{min}^{y} = min {| y_{1} |, \dots, | y_{n} |}, d_{max}^{y} = max {| y_{1} |, \dots, | y_{n} |}

(15)

Then,

d_{min}^{x}

and

d_{max}^{x}

indicate the minimum and maximum distances from joints to the principal y-axis, and

d_{min}^{y}

and

d_{max}^{y}

indicate the minimum and maximum distances from the joints to the principal x-axis. The distribution of joint locations relative to the compliance center must satisfy geometric constraints determined by the three principal compliances.

Proposition 2

Suppose a complianceCwith principal compliances (λ_x, λ_y, λ_τ) is realized by ann-joint serial mechanism. Then,

the distances of the joints to the principal axes must satisfy:
$d_{min}^{x} \leq \sqrt{\frac{λ_{y}}{λ_{τ}}} \leq d_{max}^{x}$
(16)
$d_{min}^{y} \leq \sqrt{\frac{λ_{x}}{λ_{τ}}} \leq d_{max}^{y}$
(17)
the distances of the joints to the compliance centerC_cmust satisfy:
$d_{min} \leq \sqrt{\frac{λ_{x} + λ_{y}}{λ_{τ}}} \leq d_{max}$
(18)

Proof

Consider the principal frame at the center of compliance in which the C matrix is in the diagonal form of Eq. . Suppose r_i = [x_i, y_i]^T is the position vector of joint J_i, then using Eq. , the corresponding joint twist t_i is expressed as follows:

t_{i} = [y_{i}, - x_{i}, 1]^{T}

(19)

The distance from the frame origin (the compliance center) to joint J_i is expressed as follows:

d_{i} = \sqrt{x_{i}^{2} + y_{i}^{2}}

(20)

If C is realized by a mechanism at the configuration described by joint twists (t₁, t₂, …, t_n), then

C = \sum_{i = 1}^{n} c_{i} t_{i} t_{i}^{T} = \sum_{i = 1}^{n} c_{i} [\begin{matrix} y_{i} \\ - x_{i} \\ 1 \end{matrix}] [y_{i}, - x_{i}, 1]

(21)

where each c_i > 0 is the joint compliance of J_i.

Thus,

λ_{x} = c_{1} y_{1}^{2} + c_{2} y_{2}^{2} + \dots + c_{n} y_{n}^{2}

(22)

λ_{y} = c_{1} x_{1}^{2} + c_{2} x_{2}^{2} + \dots + c_{n} x_{n}^{2}

(23)

λ_{τ} = c_{1} + c_{2} + \dots + c_{n}

(24)

From Eqs. and ,

λ_{y} \leq (c_{1} + c_{2} + + \dots + c_{n}) (d_{max}^{x})^{2} = λ_{τ} (d_{max}^{x})^{2}

Hence,

\frac{λ_{y}}{λ_{τ}} \leq (d_{max}^{x})^{2} ⟺ \sqrt{\frac{λ_{y}}{λ_{τ}}} \leq d_{max}^{x}

(25)

Similarly,

d_{min}^{x} \leq \sqrt{\frac{λ_{y}}{λ_{τ}}}

(26)

Thus, inequality is proved. By using the same reasoning, inequality is proved.

To prove inequality , adding Eqs. and yields

\begin{aligned} λ_{x} + λ_{y} & = c_{1} (x_{1}^{2} + y_{1}^{2}) + c_{2} (x_{2}^{2} + y_{2}^{2}) + \dots + c_{n} (x_{n}^{2} + y_{n}^{2}) \\ = c_{1} d_{1}^{2} + c_{2} d_{2}^{2} + \dots + c_{n} d_{n}^{2} \\ \leq (c_{1} + c_{2} + \dots + c_{n}) d_{max}^{2} \\ = λ_{τ} d_{max}^{2} \end{aligned}

Similarly,

λ_{τ} d_{min}^{2} \leq λ_{x} + λ_{y}

Thus,

λ_{τ} d_{min}^{2} \leq λ_{x} + λ_{y} \leq λ_{τ} d_{max}^{2}

(27)

which leads to inequality .

The inequalities in Proposition 2 have geometric significance. The two equations

x = \sqrt{\frac{λ_{y}}{λ_{τ}}} and x = - \sqrt{\frac{λ_{y}}{λ_{τ}}}

(28)

define two lines

l_{x}^{+}

and

l_{x}^{-}

parallel to and symmetric about the principal y-axis as illustrated in Fig. 2. The two equations

y = \sqrt{\frac{λ_{x}}{λ_{τ}}} and y = - \sqrt{\frac{λ_{x}}{λ_{τ}}}

(29)

define two lines

l_{y}^{+}

and

l_{y}^{-}

parallel to and symmetric about the principal x-axis (Fig. 2). Proposition 2(i) states that to realize a given compliance with a configuration of a serial mechanism, the joint locations cannot be either all inside or all outside area Λ_x between

l_{x}^{-}

and

l_{x}^{+}

⁠; i.e., at least two joint locations must be separated by only one line of

l_{x}^{-}

and

l_{x}^{+}

⁠. The same statement holds for area Λ_y between the other two lines

l_{y}^{-}

and

l_{y}^{+}

⁠.

Fig. 2

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Joint location restrictions relative to the compliance center and principal axes. At least one joint must be located inside circle Γ_c and at least one joint must be outside Γ_c. At least one joint must be located within area Λ_x between lines $l_{x}^{-}$ and $l_{x}^{+}$ ⁠; and at least one joint must be located outside Λ_x. At least one joint must be located inside the area Λ_y between lines $l_{y}^{-}$ and $l_{y}^{+}$ ⁠, and at least one joint must be located outside Λ_y.

Inequality imposes restriction on the distance from the joint locations to the compliance center. To realize a given compliance with a serial mechanism, the joint locations must surround the compliance center C_c and cannot be either all inside or all outside circle Γ_c centered at C_c having radius:

r_{λ} = \sqrt{\frac{λ_{x} + λ_{y}}{λ_{τ}}}

(30)

as illustrated in Fig. 2. Therefore, to realize a specified compliance with a serial mechanism, the space reachable by the mechanism joints must include the compliance center C_c and the joint locations cannot be enclosed by circle Γ_c.

2.4 Implications of Joint Location Restrictions.

Proposition 1 requires that the mechanism joints surround the compliance center. Proposition 2 places requirement on how the joints surround the compliance center. Although there is some overlap in conditions (i) and (ii) of Proposition 2, the two sets of inequalities are independent.

For an n-joint mechanism with each link length l_i, the boundary of the space reachable by the last (most distal) joint J_n is a circle Γ_w of radius r_w centered at the base joint J₁. Propositions 1 and 2 also impose restrictions on the distance d_c between the mechanism base J₁ and compliance center C_c, and on the radius r_w of Γ_w. Since the joints must surround the compliance center and circle Γ_w cannot be contained by circle Γ_c, the following conditions must be satisfied:

r_{λ} - r_{w} < d_{c} < r_{w}

(31)

which also requires

r_{w} > \frac{1}{2} r_{λ} = \frac{1}{2} \sqrt{\frac{λ_{x} + λ_{y}}{λ_{τ}}}

(32)

If any condition in inequalities – is not satisfied, the given compliance cannot be realized by the mechanism.

Note that the conditions in Propositions 1 and 2 and their implications in Eqs. (31)–(32) are only necessary conditions to achieve a given compliance. To ensure passive realization of a compliance, additional conditions are needed [27–30].

2.5 Screw Representation of Link Length Constraints.

If the location of joint J_i is specified, then the locus of possible joint locations of J_i+1 is a circle Γ_i of radius l_i centered at J_i. Suppose joint J_i is located at a given position (x_i, y_i), then joint J_i+1 must be located at a point (x, y), which satisfies:

(x - x_{i})^{2} + (y - y_{i})^{2} = l_{i}^{2}

(33)

Using screw representation, point (x, y) is associated with a unit twist t given by

t = [y, - x, 1]^{T}

(34)

As such, Eq. can be written in a homogeneous form as follows:

t^{T} (T_{i}^{T} E_{i} T_{i}) t = 0

(35)

where

E_{i} = [\begin{matrix} I & 0 \\ 0^{T} & - l_{i}^{2} \end{matrix}], T_{i} = [\begin{matrix} 1 & 0 & - y_{i} \\ 0 & 1 & x_{i} \\ 0 & 0 & 1 \end{matrix}]

and I is the 2 × 2 identity matrix.

Suppose t is a unit twist located on circle Γ_i of Eq. and suppose w is the corresponding wrench w = Kt. Then,

t = C w

Substituting the aforementioned equation into Eq. yields

w^{T} (C T_{i}^{T} E_{i} T_{i} C) w = 0

(36)

Let

G_{i} = C (T_{i}^{T} E_{i} T_{i}) C

(37)

then G_i is a 3 × 3 symmetric matrix that relates acceptable joint locations to acceptable line locations. The collection of all wrenches corresponding to the twists on circle Γ_i as mapped through K is expressed as follows:

W_{i} = {w : w^{T} G_{i} w = 0}

(38)

Consider a different mapping from wrenches (lines) to twists (points) defined by:

t = G_{i} w

(39)

Then,

w^{T} G_{i} w = 0 ⟺ t^{T} G_{i}^{- 1} t = 0

(40)

If we denote

T_{i} = {t : t^{T} G_{i}^{- 1} t = 0}

(41)

then Eq. defines a 1-to-1 mapping from

W_{i}

to

T_{i}

⁠. It can be proved that the set

T_{i}

defined in Eq. is a quadratic curve in the plane. All wrenches

w \in W_{i}

corresponding to the twists on circle Γ_i (through the stiffness mapping w = Kt) must be tangent to curve

T_{i}

(as illustrated in Fig. 3). Conversely, if a wrench w is tangent to the quadratic curve

T_{i}

⁠, then the twist t = Cw must be located on circle Γ_i. For any given point P in the plane not enclosed by curve

T_{i}

⁠, there are two wrenches with axes passing through P and tangent to

T_{i}

⁠. The two twists corresponding to the two wrenches obtained by the compliance mapping must both be located on circle Γ_i. This property will be used in the synthesis procedure presented below for a serial mechanism with fixed link lengths. The use of lines rather than points allows the placement of two points (joint locations) to be considered simultaneously.

Fig. 3

Twists t on circle Γi and the corresponding wrenches w∈Ti through the stiffness mapping. Wrench w must be tangent to the quadratic curve Ti. For point P not enclosed by Ti, there are two wrenches wp and w′p with axes passing through P and tangent to Ti. The twist t corresponding to any wrench tangent to Ti must be located on circle Γi.

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Twists t on circle Γ_i and the corresponding wrenches $w \in T_{i}$ through the stiffness mapping. Wrench w must be tangent to the quadratic curve $T_{i}$ ⁠. For point P not enclosed by $T_{i}$ ⁠, there are two wrenches w_p and w′_p with axes passing through P and tangent to $T_{i}$ ⁠. The twist t corresponding to any wrench tangent to $T_{i}$ must be located on circle Γ_i.

3 Compliance Realization With a 5R Mechanism

In this section, the realization of an arbitrary compliance with a five-joint serial mechanism having specified link lengths is addressed. Since each link length is fixed, the distance between two adjacent joints J_i and J_i+1 is constrained, i.e., ‖J_iJ_i+1‖ = l_i. To impose this constraint, a new set of realization conditions is identified first. Then, a geometry-based synthesis procedure for the realization of compliance with a 5R serial mechanism with specified link lengths is developed.

3.1 Realization Condition.

Consider a 5R serial mechanism having specified link lengths. A given compliance C can be passively realized with the mechanism at a configuration if and only if C can be expressed as follows:

C = c_{1} t_{1} t_{1}^{T} + c_{2} t_{2} t_{2}^{T} + \dots + c_{5} t_{5} t_{5}^{T}

(42)

with c_i ≥ 0. A set of necessary and sufficient conditions on the mechanism configuration for the realization of C without considering the link length restrictions was presented in Ref. [29]. Below, a different set of conditions is presented.

As proved in Ref. [29], to realize a given compliance C with a 5R mechanism, any joint J_s in the mechanism must be located on a quadratic curve determined by C and the locations of the other four joints (J_i, J_j, J_p, J_q). This curve is characterized by a 3 × 3 symmetric matrix A_ijpq constructed below.

Consider a 3 × 3 matrix H_ijpq defined as follows:

H_{i j p q} = (w_{i j}^{T} C w_{p q}) (w_{i p} w_{j q}^{T}) - (w_{i p}^{T} C w_{j q}) (w_{i j} w_{p q}^{T})

(43)

where w_ij is the unit wrench passing through joints J_i and J_j. The symmetric matrix associated with H_ijpq is expressed as follows:

A_{i j p q} = H_{i j p q} + H_{i j p q}^{T}

(44)

Consider an arbitrary unit twist t located at (x, y) expressed in the form of Eq. . The equation

f_{i j p q} (x, y) = t^{T} A_{i j p q} t = 0

(45)

defines a quadratic (conic) curve in the xy-plane.

It is proved [30] that the curve defined in Eq. (45) passes through the four joints (J_i, J_j, J_p, J_q) and that the compliance matrix C can be expressed in the form of Eq. (42), if and only if the one remaining joint is located on the curve. However, this condition alone does not ensure a passive realization of the compliance, since Eq. (45) does not require that the coefficients c_i in Eq. (42) are all nonnegative. A set of necessary and sufficient conditions for passive realization of a compliance with a 5R serial mechanism is described in this section.

Suppose that five-joint twists t_i satisfy Eq. for a selected set of c_is. Consider the unit wrench w_ij passing through two joints J_i and J_j, and consider the corresponding twist t_ij defined by

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

(46)

Equation (46) can be viewed as a mapping from a line represented by w_ij into a point represented by t_ij through the compliance. As proved in Ref. [29], to ensure that all coefficients c_i in Eq. (42) are nonnegative, t_ij must be located inside the triangle formed by the other three joints J_p, J_q, and J_s. For example, if t₁₂ is located within the triangle formed by joints J₃, J₄, and J₅ as shown in Fig. 4, then the coefficients c₃, c₄, and c₅ in Eq. (42) must be positive. If the equivalent condition also holds for twist t₃₄ (or t₄₅, t₃₅), then all five coefficients c_i in Eq. (42) must be positive. Thus, we have:

Fig. 4

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Realization condition on mechanism configuration. One joint must be located on the quadratic curve determined by the other four joint locations; twists t₁₂ and t₃₄ are located within triangles J₃J₄J₅ and J₁J₂J₅, respectively.

Proposition 3

A5Jserial mechanism realizes a given compliance C at a configuration in which the joint twists are (t₁, t₂, …, t₅) if and only if

each joint is located on the quadratic curve of Eq.(45)determined by four of the five joints, and
for any permutation (i, j, p, q, s) from ${1, 2, 3, 4, 5}$ ⁠, twistt_ijis located within triangleJ_pJ_qJ_sand twistt_pqis located within triangleJ_iJ_jJ_s.

Note that as shown in Ref. [29], when the five-joint locations are identified, the joint compliance of joint J_s can be uniquely determined with (i, j, p, q) being any permutation of

{1, 2, 3, 4, 5}

excluding s, i.e.,

(i, j, p, q) = {1, 2, 3, 4, 5} ∖ s

⁠:

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

(47)

3.2 Construction-Based Synthesis Procedure.

The synthesis of a compliance with a given mechanism (one having specified link lengths) is primarily based on the conditions presented in Proposition 3 with additional guidance provided by Propositions 1 and 2. The synthesis procedure identifies a configuration of a 5R mechanism by determining the location of each joint. The joint compliance c_i ≥ 0 at each joint is also determined in the procedure.

As stated in Sec. 2.1.2, n = 5 is the minimum number of joints in a serial mechanism needed to achieve an arbitrary compliance if the link lengths and the locations of the base joint J₁ and the endpoint joint J_n are specified. In the geometry-based synthesis process, for the system to have sufficient degrees-of-freedom, only one joint location can be specified (e.g., for n = 5, either only base location J₁ or only distal joint location J₅) to reliably obtain the specified compliance.

For a given compliance matrix C, first calculate (1) the location of the compliance center C_c; (2) the three principal compliances λ_x, λ_y, and λ_τ; and (3) the directions of the principal axes. By using these values, the circle Γ_c defined in Eq. (30), and the four lines parallel to the principal axes defined in Eqs. (28)–(29) are constructed to provide guidance in the selection of joint locations.

In the synthesis procedure described here, two twists t₁₂ and t₃₄ must be located in triangles J₃J₄J₅ and J₁J₂J₅, respectively. The locations of these twists are selected first to satisfy condition (ii) of Proposition 3 before determining the locations of J₂ and J₃ in the subsequent steps.

A more detailed description of the synthesis procedure is presented below. The geometry corresponding to each step in the procedure is illustrated in Figs. 5(a)–5(d).

Identify the location of one joint, typically the base joint J₁, arbitrarily.
Choose the location of J₂. Since the location of J₁ (with joint twist t₁) is specified, the locus of J₂ locations is a circle of radius l₁, Γ₁. The collection of lines passing through J₁ and J₂ is a pencil $P_{12}$ of lines at J₁. If we denote the collection of all twists obtained by the compliance mapping:
$T_{12} = {t = C w : w \in P_{12}}$
then the centers of all twists in $T_{12}$ form a straight line represented by wrench w₁ = Kt₁, which is the locus of twist t₁₂ locations. Since twist t₁₂ must be in the triangle formed by J₃, J₄, and J₅, this line must intersect circle Γ_w defined in Sec. 2.4 for the compliance to be realized by the mechanism. Judiciously select point t₁₂ on the line such that conditions in Propositions 2 and 3 are easier to satisfy. The line associated with wrench w₁₂ = Kt₁₂ will pass through J₁ with a slope determined by t₁₂. The intersection of line w₁₂ and circle Γ₁ determines the location of J₂ as shown in Fig. 5(a).
Select the location of twist t₃₄ such that it lies within the triangle formed by the locations of J₁, J₂, and J₅. Since the location of J₅ is not yet determined, the location of t₃₄ is selected before selecting J₃ and J₄ separately so that the triangle condition will be satisfied for virtually all possible locations of J₅. The location of t₃₄ is selected based on the selected two joints (J₁, J₂) and quadratic curve $T_{2}$ associated with circle Γ₂:
$t^{T} G_{2}^{- 1} t = 0$
(48)
where G₂ is the 3 × 3 matrix defined in Eq. (37).
Here, t₃₄ is selected to be close to line segment $\bar{J_{1} J_{2}}$ and not enclosed by $T_{2}$ ⁠.
Select the location of J₃. The locus of J₃ locations is a circle Γ₂ of radius l₂ centered at J₂ (x₂, y₂).
Consider the wrench w₃ that passes through t₃₄ and is tangent to the curve defined in Eq. (48). Mathematically, w₃ satisfies the following two equations:
$t_{34}^{T} w_{3} = 0$
(49)
$w_{3}^{T} G_{2} w_{3} = 0$
(50)
Solving these two equations yields two lines (or unit wrenches). Choose one w₃ from the two solutions, then the location of J₃ is determined by twist:
$t_{3} = C w_{3}$
(51)
Since w₃ is tangent to curve $T_{2}$ ⁠, by the results obtained in Sec. 2.5, joint twist t₃, and therefore joint J₃, must be located on circle Γ₂ (Fig. 5(b)).
Select the location of J₄. The locus of J₄ is a circle Γ₃ of radius l₃ centered at J₃. Determine the line defined by:
$w_{34} = K t_{34}$
(52)
It can be proved that if t₃₄ is close to line w₁₂, twist t₁₂ is close to line w₃₄.
Since w₃ passes through point t₃₄ as selected in step 4, w₃₄ must pass through J₃. Line w₃₄ and circle Γ₃ intersect at two points. Select J₄ to be the one closer to point t₁₂ (Fig. 5(c)) to ensure that t₁₂ is inside triangle J₃J₄J₅.
Select the location of J₅. The locus of J₅ locations is a circle Γ₄ of radius l₄ centered at J₄. A quadratic curve f₁₂₃₄ passing through the four joints (J₁, J₂, J₃, J₄) is determined using Eq. (45). This curve intersects circle Γ₄ at two points. Select J₅ so that t₁₂ is inside triangle J₃J₄J₅ and t₃₄ is inside triangle J₁J₂J₅ (Fig. 5(d)).
Determine the joint compliances. The five-joint compliances at the joint locations are each calculated using Eq. (47).

Fig. 5

Synthesis of a compliance with a 5R serial mechanism: (a) select the location of joint J2 on circle Γ1, (b) select the location of joint J3 so that it is on circle Γ2, (c) select the location of joint J4 on circle Γ3, and (d) locate joint J5 at the intersection of circle Γ4 and curve f1234

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Synthesis of a compliance with a 5R serial mechanism: (a) select the location of joint J₂ on circle Γ₁, (b) select the location of joint J₃ so that it is on circle Γ₂, (c) select the location of joint J₄ on circle Γ₃, and (d) locate joint J₅ at the intersection of circle Γ₄ and curve f₁₂₃₄

The process described earlier enforces link length constraints. For the selected five joints, the conditions in Proposition 3 are satisfied, which guarantees that each joint compliance calculated in step 6 is positive. Therefore, the compliance is passively achieved by the mechanism in the selected configuration.

4 Compliance Realization With a 6R Mechanism

In this section, the synthesis of a planar compliance with a 6R mechanism having given link lengths is addressed. As the number of joints increases, the mechanism degrees-of-freedom are increased. As such, more constraints can be considered in the synthesis process. First, new compliance realization conditions on a general 6R serial mechanism are presented. Then, a synthesis procedure for the realization of compliance with a given 6R mechanism with a set of constraints is developed.

4.1 Realization Condition.

Consider a 6R serial mechanism with each joint J_i represented by joint twist t_i (i = 1, 2, …, 6). Any given compliance C can be expressed in the following form:

C = c_{1} t_{1} t_{1}^{T} + c_{2} t_{2} t_{2}^{T} + \dots + c_{6} t_{6} t_{6}^{T}

(53)

For any given configuration, the coefficients c_is in Eq. can be uniquely determined using the following procedure.

Since C is symmetric, Eq. can be expressed in the vector form as follows:

\tilde{c} = c_{1} {\tilde{t}}_{1} + c_{2} {\tilde{t}}_{2} + \dots + c_{6} {\tilde{t}}_{6}

(54)

where

\tilde{c} = [c_{11}, c_{12}, c_{13}, c_{22}, c_{23}, c_{33}]^{T}

⁠, and

{\tilde{t}}_{i}

is the six-vector representation of

t_{i} t_{i}^{T}

⁠. If we denote:

c = [c_{1}, c_{2}, \dots, c_{6}]^{T} \in R^{6}, \tilde{T} = [{\tilde{t}}_{1}, {\tilde{t}}_{2}, \dots, {\tilde{t}}_{6}] \in R^{6 \times 6}

then, Eq. is expressed as follows:

\tilde{c} = \tilde{T} c

(55)

The joint compliance variables are obtained using:

c = {\tilde{T}}^{- 1} \tilde{c}

(56)

Thus, for any given compliance C, the coefficients c_i are uniquely determined by Eq. for a given mechanism configuration.

Note that the coefficients c_is from Eq. may be positive or negative. A necessary and sufficient condition for the passive realization of a given compliance, however, is that each c_i in Eq. is nonnegative:

{\tilde{T}}^{- 1} \tilde{c} \geq 0

(57)

The six inequalities in Eq. impose constraints on the mechanism configuration. However, due to the matrix inverse operation of

\tilde{T}

⁠, the geometric significance of these inequalities is not evident. Thus, the conditions in Eq. cannot be used directly in a geometric construction-based synthesis procedure to achieve a given compliance.

In Ref. [30], it was shown that two joint compliances c_i and c_j have the same sign if and only if the two joints are separated by the quadratic curve of Eq. (45) determined by the other four joints, and that all six c_i are positive if and only if every two joints are separated by the quadratic curve of Eq. (45) determined by the other four joints. Here, a property on any two joint locations and their corresponding joint compliances is identified.

Consider a set of six joints J_is for the realization of a given compliance C. If joint J₅ is located on the quadratic curve f₁₂₃₄ determined by four other joints (J₁, J₂, J₃, J₄), then c₆ = 0 in Eq. (53) regardless of the location of J₆. Now consider varying the location of J₅ while keeping all other joint locations unchanged (Fig. 6).

Fig. 6

Synthesis of a compliance with six elastic joints. If J5 and J6 are separated by curve f1234, c5 and c6 must have the same sign. If J5 moves without crossing curve g5, c5 maintains the same sign.

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Synthesis of a compliance with six elastic joints. If J₅ and J₆ are separated by curve f₁₂₃₄, c₅ and c₆ must have the same sign. If J₅ moves without crossing curve g₅, c₅ maintains the same sign.

As shown in Ref. [30], the value of c₅ can be calculated as follows:

c_{5} = \frac{(w_{13}^{T} t_{6}) (w_{24}^{T} t_{6}) (w_{12}^{T} C w_{34}) - (w_{12}^{T} t_{6}) (w_{34}^{T} t_{6}) (w_{13}^{T} C w_{24})}{D_{56}}

(58)

where w_ij is the unit wrench passing through J_i and J_j, and where the denominator D₅₆ is expressed as follows:

\begin{aligned} D_{56} & = (w_{12}^{T} t_{5}) (w_{34}^{T} t_{5}) (w_{13}^{T} t_{6}) (w_{24}^{T} t_{6}) \\ - (w_{12}^{T} t_{6}) (w_{34}^{T} t_{6}) (w_{13}^{T} t_{5}) (w_{24}^{T} t_{5}) \end{aligned}

(59)

Since all joint locations except J₅ are unchanged, the numerator of c₅ in Eq. does not change. A sign change of c₅ depends only on the denominator D₅₆ in Eq. . Let t = [y, − x, 1]^T and consider the function g₅(x, y) defined by

\begin{aligned} g_{5} (x, y) & = (w_{12}^{T} t) (w_{34}^{T} t) (w_{13}^{T} t_{6}) (w_{24}^{T} t_{6}) \\ - (w_{12}^{T} t_{6}) (w_{34}^{T} t_{6}) (w_{13}^{T} t) (w_{24}^{T} t) \end{aligned}

(60)

Then,

g_{5} (x, y) = 0

(61)

defines a quadratic curve in the plane. It can be seen that this curve passes through the five joints (J₁, J₂, J₃, J₄, J₆), and thus, it is uniquely determined by the locations of these five joints.

If c₅ changes its sign, joint J₅ must cross curve g₅(x, y) = 0. Thus, if J₅ moves without crossing curve g₅(x, y) such that J₅ and J₆ are separated by curve f₁₂₃₄, then c₅ and c₆ are either both positive or both negative. If J₅ crosses curve g₅(x, y) with J₅ and J₆ being separated by curve f₁₂₃₄, then both c₅ and c₆ change their sign. Note that this property is also true for any two joints (J_i, J_j) and their corresponding joint compliances (c_i, c_j) and will be used for the synthesis procedure for 6R mechanisms having fixed link lengths.

4.2 Construction-Based Synthesis Procedures.

In this section, synthesis procedures used to realize a given compliance with a 6R serial mechanism are presented. In the process, the first joint J₁ (base joint) and the most distal joint J₆ (connected to the end-effector) are specified. First, a procedure that uses five elastic joints in a 6R mechanism is presented, Then, a procedure for which all six joints are elastic is presented.

4.2.1 Compliance Synthesis With Five Elastic Joints.

This synthesis procedure identifies the locations of the six joints and the corresponding joint compliances (with one joint compliance equal to zero). This procedure is based on the 5R synthesis procedure presented in Sec. 3.2. The geometry corresponding to each step in the procedure is illustrated in Fig. 7.

Select the locations of two joints arbitrarily, typically, the base joint J₁ and the distal joint J₆.
Calculate the twist t₁₆ associated with wrench w₁₆. Since J₁ and J₆ are specified, the wrench w₁₆ passing through J₁ and J₆ is determined and
$t_{16} = C w_{16}$
(62)
The location of t₁₆ is obtained using Eq. (2).
Select J₂. Follow step 2 in the procedure for 5R mechanisms presented in Sec. 3.2.
Select J₃. Follow steps 3 and 4 in the procedure for 5R mechanisms presented in Sec. 3.2.
Select J₄. Determine the quadratic curve f₁₂₃₆ associated with the four selected joints (J₁, J₂, J₃, J₆) using Eq. (45) and determine circle Γ₃ of radius l₃ centered at J₃. Joint J₄ is at the intersection of circle Γ₃ and curve f₁₂₃₆. In selecting the location of J₄, the distance between J₄ and J₆ must be less than (l₄ + l₅) and t₂₃ must be inside triangle J₄J₆J₁.
Determine the location of J₅. Since J₄ and J₆ are specified, J₅ is determined by the intersection of the two circles Γ₄ and Γ₅ centered at J₄ and J₆ with radius l₄ and l₅, respectively. There are two possible locations for J₅ as shown in Fig. 7. Choose either one.
Determine the joint compliances. Since the compliance is effectively realized by five elastic joints (J₁, J₂, J₃, J₄, J₆) in the 6R mechanism, c₅ = 0. The other five-joint compliances can be calculated using either Eq. (47) or (56).

Fig. 7

Synthesis of a compliance with a 6R serial mechanism. The locations of J1 and J6 are specified.

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Synthesis of a compliance with a 6R serial mechanism. The locations of J₁ and J₆ are specified.

In this procedure, although only five joints are used to provide joint compliance, six joints are needed for the kinematic mobility necessary to satisfy the geometric constraints (specified locations of J₁ and J₆).

4.2.2 Compliance Synthesis With Six Elastic Joints.

Synthesis of a compliance with all six nonzero compliance joints is outlined as follows:

Identify a configuration of the 6R mechanism that realizes the given compliance with five joints as described in the procedure of Sec. 4.2.1. Suppose that in the realization, the given compliance is realized with five elastic joints (J₁, J₂, J₃, J₄, J₆) as illustrated in Fig. 7. The procedure ensures that all five-joint compliances c₁, c₂, c₃, c₄, and c₆ are positive.
Move J₄ away from the quadratic curve f₁₂₃₆ such that J₄ and J₅ are separated by f₁₂₃₆. Since joints J₃ and J₆ are selected, the mechanism is equivalent to a four-bar mechanism with J₃ and J₆ fixed. By rotating link J₃J₄ (or J₆J₅) about J₃ (or J₆), J₄ is moved away from the curve. Two example configurations are shown in Fig. 8. Since joints J₄ and J₅ are separated by curve f₁₂₃₆ for each configuration, the joint compliances c₄ and c₅ at these two joints must have the same sign. Since the two configurations are only slightly varied from a configuration at which c₄ > 0, both c₄ and c₅ are positive at one of these two configurations.
Calculate the joint compliances using Eq. (56) at the two configurations selected in step 2, and choose the one that has all positive joint compliances.

Fig. 8

Synthesis of a compliance with a 6R serial mechanism. There are two configurations (a) and (b) at which joints J4 and J5 are separated by curve f1236. The joint compliances c4 and c5 have the same sign at these two configurations.

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Synthesis of a compliance with a 6R serial mechanism. There are two configurations (a) and (b) at which joints J₄ and J₅ are separated by curve f₁₂₃₆. The joint compliances c₄ and c₅ have the same sign at these two configurations.

With the final step, the configuration of the mechanism and all joint compliances are determined and the compliance is passively achieved with the 6R mechanism.

5 Example

In this section, a numerical example is provided to illustrate the synthesis procedures. In a global frame, the compliance matrix to be realized is expressed as follows:

C = [\begin{matrix} 12.50 m/N & - 13.54 m/N & 3.41 N^{- 1} \\ - 13.54 m/N & 15.91 m/N & - 3.74 N^{- 1} \\ 3.41 N^{- 1} & - 3.74 N^{- 1} & 0.95 (N \cdot m)^{- 1} \end{matrix}]

By using Eq. , the compliance center is calculated to be

r_{c} = [3.9368, 3.5895]^{T}

The two principal axes and the corresponding two translational principal compliances are expressed as follows:

[e_{1}, e_{2}] = [\begin{matrix} - 0.9926 & 0.1218 \\ - 0.1218 & - 0.9926 \end{matrix}], [λ_{x}, λ_{y}] = [0.2457, 1.2004]

and the rotational principal compliance is λ_τ = 0.95. The circle Γ_c and the two pairs of lines in the principal frame at the compliance center C_c are calculated as follows:

Γ_{c} : r_{λ} = 1.2338, l_{x}^{\pm} : x^{'} = \pm 1.1241, l_{y}^{\pm} : y^{'} = \pm 0.5086

and are illustrated in Fig. 9.

Fig. 9

Compliance synthesis process. Selection of joint locations for joints J1, J2, J3, and J4.

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Compliance synthesis process. Selection of joint locations for joints J₁, J₂, J₃, and J₄.

In the following, the synthesis of C with a 5R serial mechanism having given link lengths is first performed. Then, the synthesis of C with a 6R serial mechanism is presented.

5.1 5R Mechanism Synthesis.

Consider a 5R serial mechanism in which each link has the same length:

l_{1} = l_{2} = l_{3} = l_{4} = 1 m

The space reachable by the last joint J₅ is a circle Γ_w of radius r_w centered at J₁. The radius r_w is expressed as follows:

r_{w} = \sum l_{i} = 4 m

Select the location of J₁. To satisfy inequality (31), the distance between joint J₁ and the compliance center C_c must be less than 4 m. Here, J₁ is located at position (3, 3) (inside the rectangle enclosed by the four lines $l_{x}^{\pm}$ and $l_{y}^{\pm}$ as illustrated in Fig. 9), which satisfies the two lower bound inequalities of (16) and (17). The joint twist of J₁ is expressed as follows:
$t_{1} = [3, - 3, 1]^{T}$
Select the location of J₂. Based on the selected location of J₁ and Proposition 2, position (3, 4) is selected to be the location of J₂, which is separated from J₁ by $l_{y}^{+}$ ⁠. The joint twist of J₂ is expressed as follows:
$t_{2} = [4, - 3, 1]^{T}$
The wrench passing through J₁ and J₂ is expressed as follows:
$w_{12} = [0, 1, 3]^{T}$
The twist associated with w₁₂ is expressed as follows:
$t_{12} = C w_{12} = [- 3.31, 4.69, - 0.89]^{T}$
which is located at (5.2697, 3.7191).
Select the center of twist t₃₄. By using Eq. (37), the matrix G₂ is expressed as follows:
$G_{2} = [\begin{matrix} 0.6276 & - 4.3893 & 0.1510 \\ - 4.3893 & 10.0249 & - 1.1749 \\ 0.1510 & - 1.1749 & 0.0417 \end{matrix}]$
The associated quadratic curve $T_{2}$ is expressed as follows:
$T_{2} : t^{T} G_{2}^{- 1} t = 0$
and is illustrated in Fig. 9.
Choose a point (not enclosed by curve $T_{2}$ ⁠) close to line segment $\bar{J_{1} J_{2}}$ ⁠. This point is selected to be (3.2, 3.2), which is the center of t₃₄. Then, twist t₃₄ is expressed as follows:
$t_{34} = [3.2, - 3.2, 1]^{T}$
Select the location of J₃. Consider a line that passes through the center of t₃₄ (3.2, 3.2) and is tangent to curve $T_{2}$ ⁠. There are two lines represented by wrenches w₃ and w′₃ satisfying these conditions as shown in Fig. 9. Here, unit wrench w₃ is selected, using Eqs. (49)–(50):
$w_{3} = [0.9945, 0.1046, - 2.8476]^{T}$
By the results presented in Sec. 3.2, the twist associated with w₃ through the compliance mapping must be located on circle Γ₂. This twist is calculated to be:
${\hat{t}}_{3} = C w_{3} = [1.3043, - 1.1508, 0.2947]^{T}$
The center of twist ${\hat{t}}_{3}$ ⁠, (3.9050, 4.4255), is the location of J₃. The (unit) joint twist of J₃ is expressed as follows:
$t_{3} = [4.4255, - 3.9050, 1]^{T}$
Select the location of J₄. Joint J₄ must be on circle Γ₃ with radius l₃ = 1 centered at joint J₃.
The wrench associated with twist t₃₄ is calculated to be
$w_{34} = K t_{34} = [- 1.2780, 0.4969, 7.5962]^{T}$
The intersection of line w₃₄ and circle Γ₃ will be the location of J₄. If (x, y) are the coordinates of J₄, then the corresponding joint twist is t₄ = [y, − x, 1]^T. The location of J₄ is obtained by solving the following equations:
$\begin{aligned} (x - x_{3})^{2} + (y - y_{3})^{2} & = l_{3} = 1 \\ t_{4}^{T} w_{34} & = 0 \end{aligned}$
For the two sets of solutions to the equations, select the one that better causes the set of joints to surround the compliance center. Here, the solution (x, y) = (4.8370, 4.0632) is selected to be the location of J₄. The joint twist of J₄ is expressed as follows:
$t_{4} = [4.0632, - 4.8370, 1]^{T}$
Determine the location of J₅. For the selected four joints (J₁, J₂, J₃, J₄), a quadratic curve passing through these four joints is obtained using Eq. (45):
$f_{1234} : t^{T} A_{1234} t = 0$
where twist t is defined in Eq. (34) and the symmetric matrix A₁₂₃₄ calculated from Eqs. (43)–(44) is expressed as follows:
$A_{1234} = [\begin{matrix} 0.1429 & 0.1184 & 0.8554 \\ 0.1184 & - 0.1892 & - 1.2253 \\ 0.8554 & - 1.2253 & - 7.3636 \end{matrix}]$
The intersection of the quadratic curve and circle Γ₄ occurs outside of Γ_c at (5.6552, 3.4882), which is the location of J₅. Thus, all five-joint locations are identified and shown in Fig. 10. The joint twist of J₅ is expressed as follows:
$t_{5} = [3.4882, - 5.6552, 1]^{T}$
Thus, all five-joint locations are identified and shown in Fig. 10.
Since all conditions in Proposition 3 are satisfied for the selected five-joint locations, the given compliance is passively realized by the serial mechanism.
Determine the joint compliances c_i. By using Eq. (47):
$c = [0.3352, 0.1643, 0.1344, 0.0867, 0.2294] (N \cdot m)^{- 1}$
With this final step, the mechanism configuration and joint compliances are identified. By using the obtained results for joint compliances identified earlier and the joint twists:
$[t_{1}, t_{2}, t_{3}, t_{4}, t_{5}] = [\begin{matrix} 3 & 4 & 4.4255 & 4.0632 & 3.4882 \\ - 3 & - 3 & - 3.9050 & - 4.8370 & - 5.6552 \\ 1 & 1 & 1 & 1 & 1 \end{matrix}]$
The calculated compliance is expressed as follows:
$\sum_{i = 1}^{5} c_{i} t_{i} t_{i}^{T} = [\begin{matrix} 12.50 & - 13.54 & 3.41 \\ - 13.54 & 15.91 & - 3.74 \\ 3.41 & - 3.74 & 0.95 \end{matrix}]$
which verifies the realization.

Fig. 10

Selection of the location of joint J5. The joint must be located on quadratic curve f1234 determined by the other four joints.

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Selection of the location of joint J₅. The joint must be located on quadratic curve f₁₂₃₄ determined by the other four joints.

5.2 6R Mechanism Synthesis.

If a 6R serial mechanism is considered, due to the increase in the degrees-of-freedom, the locations of joints J₁ and J₆ can be specified arbitrarily based on the conditions of Proposition 2. As stated in Sec. 4.2.1, a given compliance can be realized by a 6R mechanism with only five effective elastic joints. In the following, the synthesis of C with five joints is performed. Then, using the procedure described in Sec. 4.2.2, the synthesis of C with all six joints having nonzero compliance is performed.

5.2.1 Realization With Five Elastic Joints.

Select joints J₁ and J₆ to be located at positions (3,3) and (3,6), respectively. The locations of J₂ and J₃ can be selected using the process of 5R mechanism synthesis described in Sec. 4.2.1. Here, J₂ and J₃ locations are selected to be the same as that in Sec. 5.1: J₂ is located at (3, 4), and J₃ is located at (3.9050, 4.4255). Since joint J₆ is specified, for five-joint synthesis, we only need to select one joint, either J₄ or J₅, to be located on the quadratic curve f₁₂₃₆ determined by the four joint locations (J₁, J₂, J₃, J₆).

By using Eq. , curve f₁₂₃₆ is calculated and illustrated in Fig. 11. The wrench associated with the line passing through J₁ and J₆ is determined as follows:

w_{16} = [1, 0, - 3]^{T}

The twist associated with w₁₆ is calculated to be:

t_{16} = C w_{16} = [2.2700, - 2.3200, 0.56]^{T}

which is located at (4.1429, 4.0536) as shown in Fig. 11.

Fig. 11

Synthesis of C with a 6R serial mechanism with the base joint location J1 and last joint location J6 specified. The synthesis procedure yields a mechanism for which only five of the six joints are elastic.

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Synthesis of C with a 6R serial mechanism with the base joint location J₁ and last joint location J₆ specified. The synthesis procedure yields a mechanism for which only five of the six joints are elastic.

Similarly, unit wrench w₂₃ and the corresponding twist t₂₃ are calculated to be

\begin{aligned} w_{23} & = [0.9050, 0.4255, - 2.3435]^{T} \\ t_{23} & = [- 2.4401, 3.2807, - 0.7316]^{T} \end{aligned}

Twist t₂₃ is located at (4.4840, 3.3351) and is illustrated in Fig. 11.

First, consider the case that J₄ is on curve f₁₂₃₆. The location of J₄ is the intersection of circle Γ₃ and curve f₁₂₃₆, which is calculated to be J′₄ (4.8546, 4.1120). Since t₁₆ is outside triangle J₂J₃J′₄, the realization condition (ii) of Proposition 3 is not satisfied, and the given compliance cannot be passively realized at this configuration.

Now consider the case that J₅ is located on curve f₁₂₃₆. The location of J₅ can be determined by the intersection of circle Γ₅ and curve f₁₂₃₆, which is point (5.3498, 3.7598). Since (J₁, J₂, J₃, J₅, J₆) are all located on curve f₁₂₃₆, the location of J₄ is irrelevant (c₄ = 0). From Fig. 11, it can be seen that t₁₆ is located inside triangle J₂J₃J₅ and t₂₃ is located inside triangle J₁J₅J₆. Thus, all conditions in Proposition 3 are satisfied, and the given compliance can be passively realized at this configuration.

The five elastic joint twists selected are given as follows:

[t_{1}, t_{2}, t_{3}, t_{5}, t_{6}] = [\begin{matrix} 3 & 4 & 4.4255 & 3.7598 & 3 \\ - 3 & - 3 & - 3.9050 & - 5.3498 & - 6 \\ 1 & 1 & 1 & 1 & 1 \end{matrix}]

By using Eq. , the five-joint compliances are calculated as follows:

[c_{1}, c_{2}, c_{3}, c_{5}, c_{6}] = [0.3313, 0.1744, 0.1292, 0.2651, 0.05]

with units (N · m)⁻¹. It is readily verified that:

\sum c_{i} t_{i} t_{i}^{T} = [\begin{matrix} 12.50 & - 13.54 & 3.41 \\ - 13.54 & 15.91 & - 3.74 \\ 3.41 & - 3.74 & 0.95 \end{matrix}]

5.2.2 Realization With Six Joints.

Consider a configuration close to that obtained in Sec. 5.2.1. When all six joints have nonzero passive compliances, J₄ and J₅ must be separated by curve f₁₂₃₆. There are infinitely many possible locations of J₄ and J₅ on the opposite sides of curve f₁₂₃₆ that satisfy link length restrictions. Figure 12 illustrates two cases (J₄, J₅) and (J′₄, J′₅) in which the two joints are separated by curve f₁₂₃₆.

Fig. 12

Synthesis of C with a 6R serial mechanism with the base joint location J1 and last joint location J6 specified. In the synthesis, all six joints are elastic. Note that joints J4 and J5 must be separated by curve f1236.

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Synthesis of C with a 6R serial mechanism with the base joint location J₁ and last joint location J₆ specified. In the synthesis, all six joints are elastic. Note that joints J₄ and J₅ must be separated by curve f₁₂₃₆.

First, consider the case that J₅ is outside of the curve. Choose the location of J₅ at (5.6, 3.9156) on Γ₅, then the location of J₄ can be obtained by solving the equations:

\begin{aligned} (x - 5.6)^{2} + (y - 3.9156)^{2} & = l_{4}^{2} = 1 \\ (x - 3.9050)^{2} + (y - 4.4255)^{2} & = l_{3}^{2} = 1 \end{aligned}

which yields (4.6185, 3.7249) as the location of J₄.

For the selected six joints (J₁, J₂, J₃, J₄, J₅, J₆) shown in Fig. 12, the joint twist matrix is expressed as follows:

[T] = [\begin{matrix} 3 & 4 & 4.4255 & 3.7249 & 3.9165 & 3 \\ - 3 & - 3 & - 3.9050 & - 4.6185 & - 5.6000 & - 6 \\ 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}]

By using Eq. , the six-joint compliances are obtained:

c = [0.3214, 0.1758, 0.1104, 0.1297, 0.1450, 0.0678]

with units (N · m)⁻¹.

The obtained results can be verified using Eq. :

\sum_{i = 1}^{6} c_{i} t_{i} t_{i}^{T} = [\begin{matrix} 12.50 & - 13.54 & 3.41 \\ - 13.54 & 15.91 & - 3.74 \\ 3.41 & - 3.74 & 0.95 \end{matrix}]

which confirms that passive realization of the given compliance is achieved.

If the location of the fifth joint is chosen to be inside the curve, for example, at J₅′ (5.1, 3.4359) as shown in Fig. 12, the location of the fourth joint is determined to be at J₄′ (4.9049, 4.4167). At this configuration (J₁, J₂, J₃, J₄′, J₅′, J₆), the values of the joint compliances are calculated to be:

c = [0.2833, 0.2905, - 0.0775, 0.2097, 0.1902, 0.0537]

Since c₃ is negative, the given compliance C cannot be passively achieved at this configuration.

If a different compliance is desired at the same end-effector location, the joints J₁ and J₆ can be selected at the same locations for the mechanism as specified in Sec. 5.2.1. Following the steps as described earlier, the synthesis procedure will yield a different mechanism configuration and a different set of joint compliances that realize the desired compliant behavior.

6 Discussion

It is known that any planar compliance can be realized with an n-joint (n ≥ 3) serial mechanism if there are no constraints on the link lengths. To increase the space of realizable compliant behaviors when link lengths are specified, more joints are needed. For a mechanism having the number of joints n ≤ 5, increasing n by 1 increases the dimension of realizable space by 1. When n ≥ 6, increasing the number of joints does not significantly enlarge the realizable space.

If link lengths are specified, a minimum of five joints are needed to realize an arbitrary compliance (within a space bounded by inequalities) at given base and end-effector positions. Thus, for a given compliance, a given 5R or given 6R serial mechanism can be configured to achieve the behavior.

In each step of the processes presented in Secs. 3 and 4, the selection of joint locations is not unique. Since the procedure is geometric construction based, graphics tools can be used to select a better mechanism configuration in the realization of compliance.

In the second step of the procedure presented in Sec. 3, it is suggested that point t₃₄ be selected close to the line segment formed by the two selected joints J₁ and J₂. It can be proved that, if t₃₄ is on segment $\bar{J_{1} J_{2}}$ ⁠, then point t₁₂ must be on the line passing through J₃ and J₄ selected by the procedure. Thus, if t₃₄ is close to $\bar{J_{1} J_{2}}$ ⁠, t₁₂ is close to the line passing through J₃ and J₄. However, if t₃₄ is selected on $\bar{J_{1} J_{2}}$ ⁠, the matrix A₁₂₃₄ in Eq. (44) will be rank-deficient, and the associated quadratic curve f₁₂₃₄ in Eq. (45) will be degenerate. For this case, the conditions of Proposition 3 cannot be applied. As such, t₃₄ should be selected close to but not on segment $\bar{J_{1} J_{2}}$ ⁠.

7 Summary

In this article, the realization of planar compliance with a serial mechanism having specified link lengths is addressed. Insight into the joint distribution of a general serial mechanism in the realization of a given compliance is provided. Synthesis procedures to achieve a compliance with a serial mechanism having fixed link lengths and either five or six elastic joints are developed. The theories presented in this article enable one to assess the ability of a given compliant serial mechanism to realize any given compliance and to select a configuration of a given mechanism that achieves a realizable elastic behavior. Since the developed synthesis procedures are completely geometry based, computer graphics tools can be used in the process to obtain a mechanism configuration that attains the desired compliance.

Acknowledgment

This research was supported in part by the National Science Foundation under Grant CMMI-2024554.

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The authors attest that all data for this study are included in the paper. Data provided by a third party listed in Acknowledgment.

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Compliance Realization With Planar Serial Mechanisms Having Fixed Link Lengths

Abstract

1 Introduction

1.1 Related Work.

1.2 Contribution of the Paper.

1.3 Overview.

2 Technical Background

2.1 Elastic Behavior Realized With a Serial Mechanism.

2.1.1 Screw Representation of Points/Lines in a Plane.

2.1.2 Compliance Realization With a Serial Mechanism.

2.2 Compliance Center Relative Position.

2.3 Joint Location Distribution Conditions.

2.4 Implications of Joint Location Restrictions.

2.5 Screw Representation of Link Length Constraints.

3 Compliance Realization With a 5R Mechanism

3.1 Realization Condition.

3.2 Construction-Based Synthesis Procedure.

4 Compliance Realization With a 6R Mechanism

4.1 Realization Condition.

4.2 Construction-Based Synthesis Procedures.

4.2.1 Compliance Synthesis With Five Elastic Joints.

4.2.2 Compliance Synthesis With Six Elastic Joints.

5 Example

5.1 5R Mechanism Synthesis.

5.2 6R Mechanism Synthesis.

5.2.1 Realization With Five Elastic Joints.

5.2.2 Realization With Six Joints.

6 Discussion

7 Summary

Acknowledgment

Conflict of Interest

Data Availability Statement

References

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Compliance Realization With Planar Serial Mechanisms Having Fixed Link Lengths

Abstract

1 Introduction

1.1 Related Work.

1.2 Contribution of the Paper.

1.3 Overview.

2 Technical Background

2.1 Elastic Behavior Realized With a Serial Mechanism.

2.1.1 Screw Representation of Points/Lines in a Plane.

2.1.2 Compliance Realization With a Serial Mechanism.

2.2 Compliance Center Relative Position.

2.3 Joint Location Distribution Conditions.

2.4 Implications of Joint Location Restrictions.

2.5 Screw Representation of Link Length Constraints.

3 Compliance Realization With a 5R Mechanism

3.1 Realization Condition.

3.2 Construction-Based Synthesis Procedure.

4 Compliance Realization With a 6R Mechanism

4.1 Realization Condition.

4.2 Construction-Based Synthesis Procedures.

4.2.1 Compliance Synthesis With Five Elastic Joints.

4.2.2 Compliance Synthesis With Six Elastic Joints.

5 Example

5.1 5R Mechanism Synthesis.

5.2 6R Mechanism Synthesis.

5.2.1 Realization With Five Elastic Joints.

5.2.2 Realization With Six Joints.

6 Discussion

7 Summary

Acknowledgment

Conflict of Interest

Data Availability Statement

References

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