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.
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 ) 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. , 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. , 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 Ji and Ji+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.
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.
These properties will be used in the synthesis of compliance with a serial mechanism.
2.1.2 Compliance Realization With a Serial Mechanism.
2.2 Compliance Center Relative Position.
The relationship between the location of the compliance center and the configuration of a mechanism capable of realizing the behavior is presented in Ref.  for the general spatial case. For the planar case, the relationship can be expressed in a simpler form.
Thus, the center of compliance is the joint compliance ci weighted average of the joint locations ri. 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.
Suppose a complianceCwith principal compliances (λx, λy, λτ) is realized by ann-joint serial mechanism. Then,
- the distances of the joints to the principal axes must satisfy:(16)(17)
- the distances of the joints to the compliance centerCcmust satisfy:(18)
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.
2.5 Screw Representation of Link Length Constraints.
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 Ji and Ji+1 is constrained, i.e., ‖JiJi+1‖ = li. 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.
As proved in Ref. , to realize a given compliance C with a 5R mechanism, any joint Js in the mechanism must be located on a quadratic curve determined by C and the locations of the other four joints (Ji, Jj, Jp, Jq). This curve is characterized by a 3 × 3 symmetric matrix Aijpq constructed below.
It is proved  that the curve defined in Eq. (45) passes through the four joints (Ji, Jj, Jp, Jq) 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 ci 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.
Equation (46) can be viewed as a mapping from a line represented by wij into a point represented by tij through the compliance. As proved in Ref. , to ensure that all coefficients ci in Eq. (42) are nonnegative, tij must be located inside the triangle formed by the other three joints Jp, Jq, and Js. 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 ci in Eq. (42) must be positive. Thus, we have:
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, twisttijis located within triangleJpJqJsand twisttpqis located within triangleJiJjJs.
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 ci ≥ 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 Jn 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 Cc; (2) the three principal compliances λx, λy, and λτ; and (3) the directions of the principal axes. By using these values, the circle Γc 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.
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 of lines at J1. If we denote the collection of all twists obtained by the compliance mapping:then the centers of all twists in 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 Γw 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 associated with circle Γ2:where G2 is the 3 × 3 matrix defined in Eq. (37).(48)
Here, t34 is selected to be close to line segment and not enclosed by .
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:(49)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:(50)Since w3 is tangent to curve , by the results obtained in Sec. 2.5, joint twist t3, and therefore joint J3, must be located on circle Γ2 (Fig. 5(b)).(51)
- Select the location of J4. The locus of J4 is a circle Γ3 of radius l3 centered at J3. Determine the line defined by:It can be proved that if t34 is close to line w12, twist t12 is close to line w34.(52)
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).
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.
In Ref. , it was shown that two joint compliances ci and cj 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 ci 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 Jis 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).
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 (Ji, Jj) and their corresponding joint compliances (ci, cj) 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 andThe location of t16 is obtained using Eq. (2).(62)
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).
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.
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.
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.
- Select the location of J1. To satisfy inequality (31), the distance between joint J1 and the compliance center Cc must be less than 4 m. Here, J1 is located at position (3, 3) (inside the rectangle enclosed by the four lines and 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:
- 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 . The joint twist of J2 is expressed as follows:The wrench passing through J1 and J2 is expressed as follows:The twist associated with w12 is expressed as follows: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:The associated quadratic curve is expressed as follows:and is illustrated in Fig. 9.Choose a point (not enclosed by curve ) close to line segment . This point is selected to be (3.2, 3.2), which is the center of t34. Then, twist t34 is expressed as follows:
- Select the location of J3. Consider a line that passes through the center of t34 (3.2, 3.2) and is tangent to curve . 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):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:The center of twist , (3.9050, 4.4255), is the location of J3. The (unit) joint twist of J3 is expressed as follows:
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 beThe 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: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:
- 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):where twist t is defined in Eq. (34) and the symmetric matrix A1234 calculated from Eqs. (43)–(44) is expressed as follows:The intersection of the quadratic curve and circle Γ4 occurs outside of Γc 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: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 ci. By using Eq. (47):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:The calculated compliance is expressed as follows:which verifies the realization.
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).
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.
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.
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.
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 , then point t12 must be on the line passing through J3 and J4 selected by the procedure. Thus, if t34 is close to , t12 is close to the line passing through J3 and J4. However, if t34 is selected on , 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 .
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.
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.