Singularity Analysis of an Extensible Kinematic Architecture: Assur Class N, Order N 1

Transcription

1 David H. Myszka Andrew P. Murray University of Dayton, Dayton, OH James P. Schmiedeler The Ohio State University, Columbus, OH Singularity Analysis of an Extensible Kinematic Architecture: Assur Class N, Order N 1 This paper presents an analysis to create a general singularity condition for a mechanism that contains a deformable closed contour. This kinematic architecture is particularly relevant to rigid-link, shape-changing mechanisms. Closed contour shape-changing mechanisms will be shown to belong to the Assur classification because of the pattern of interconnections among the links. The general singularity equations are reduced to a condensed form, which allows geometric relationships to be readily detected. The analysis is repeated for alternative input links. A method for formulating the singularity condition for an Assur Class N, knowing the condition for a Class N 1 mechanism, is given. This approach is illustrated with several examples. DOI: / Introduction Contributed by the Mechanisms and Robotics Committee for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received April 15, 2008; final manuscript received June 26, 2008; published online August 5, Review conducted by Jorge Angeles. Paper presented at the ASME 2008 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference IDETC2008, Brooklyn, New York, Aug. 3 6, During the operation of a mechanism, a configuration can arise such that the input link is no longer able to move the mechanism. The mechanism becomes stationary, and the mechanical advantage reduces to zero 1. This stationary position of a mechanism is termed a singular configuration and must be avoided when attempting to operate the mechanism with a single input crank. Mathematically, a singular position occurs when the matrix that relates the input speeds with the output speeds becomes rank deficient, as discussed by Gosselin and Angeles 2. The singularity analysis of a particular mechanism is the study of the conditions that lead to the stationary configuration. A recent promising application of deformable closed contour linkages is shape-changing rigid-body mechanisms 3. Planar linkages, comprising a closed chain of N rigid links connected with revolute joints, can be designed to approximate a shape change defined by a set of morphing closed curves. The value N is selected by balancing the approximation to the desired curves, which is typically better with a larger N, and the complexity of the mechanism, which is generally reduced with a smaller N 4. These rigid links may be constrained with additional links to yield a single-degree-of-freedom shape-changing mechanism. In such a case, of the N links forming the closed contour, one remains binary, while N 1 are ternary links. The other revolute joints on the ternary links connect constraining dyads. A mechanism with six links N=6n the deformable closed contour is shown in Fig. 1. Also, the mechanism contains five ternary links and five constraining dyads N 1=5. A classical approach for describing a planar mechanism composition is with Assur groups. An Assur kinematic chain AKCs defined as the minimum arrangement of n links that creates a rigid structure 5,6. That is, another rigid structure cannot be created by suppressing one or more links of an AKC. The structure shown in Fig. 1 as an Assur kinematic chain. An AKC can be converted into a single-degree-of-freedom mechanism by inserting an additional binary link. The mechanism shown in Fig. 1 bs an Assur mechanism. Replacing a link of an existing mechanism with an AKC does not alter the number of degrees of freedom of the mechanism. Thus, mechanisms of great complexity can be constructed by the sequential addition of AKCs to simpler chains 7. The usefulness of AKCs arises in position analysis. Since an AKC is a structure, constraint equations can be readily generated to identify the positions of all links in the chain. This strategy allows a modularized approach to position analysis in which complex linkages can be analyzed by dividing them into a sequence of groups 8. The class of an AKC commonly refers to the number of links contained in a deformable closed contour 9, and the accepted nomenclature is to specify the class with a Roman numeral. The order of an AKC, sometimes known as the series, refers to the number of external kinematic joints or the number of binary links in the closed chain that are replaced with ternary or quaternary links 10. For example, if an Assur Class IV mechanism has two of the four links that comprise the closed chain as ternary, it is referred to as order 2, denoted as IV/2. A general model of the Assur VI/5 is illustrated in Fig. 1 b. The deformable closed contour mechanisms described earlier are Assur Class N, order N 1. The class arises because N links are incorporated in the shape-changing closed chain, and the order refers to the N 1 constraining dyads used to constrain the mechanism. The order is also identified with N 1 ternary links included in the deformable closed contour. The formulation of singularities for a shape-changing mechanism is necessary to ensure that the linkage can be driven between a set of morphing curves, but the singular configurations in these complex linkages are seldom intuitive. For example, the Assur V/4 shown in Fig. 2 is in a singular configuration in which no links are parallel or perpendicular. Furthermore, lines along the links are extended to illustrate that no other special relationships, such as three lines intersecting at a point, exist. This paper presents a method to create a concise singularity equation for an Assur mechanism that contains a deformable closed contour. In designing shape-changing mechanisms of Assur Class N, order N 1, the authors have experienced significant difficulty in locating an input link that rotates monotonically to drive among the desired shapes because the singular configurations are not intuitively identifiable. Generating the most concise statement of the singularities enables the designer to identify, and thus Journal of Mechanisms and Robotics Copyright 2008 by ASME FEBRUARY 2009, Vol. 1 /

2 Fig. 2 A singularity configuration of an Assur V/4 g 2 and a 2 can be subsequently suppressed to form another closed kinematic chain of mobility 0. Still, the mechanism is analogous to an Assur III/2. It is noted that the mechanism in Fig. 3 is merely a four-bar linkage since the three links in the closed contour form a rigid structure. This trivial case is not a shape-changing mechanism but is explored to develop singularity equations that provide a starting point for further generalizations. The mechanism has 11 physical parameters a i, b i, d i, e i, i =1,2, f, g 2x, and g 2y and 7 joint variables 1, 1, 1, 2, 2, 2, and. The loop closure equations for the mechanism are a 1 cos 1 + e 1 cos 1 g 2x a 2 cos 2 b 2 cos 2 =0 a 1 sin 1 + e 1 sin 1 g 2y a 2 sin 2 b 2 sin 2 =0 d 1 cos 1 + d 2 cos 2 + f cos =0 Fig. 1 Assur kinematic chain that becomes an Assur mechanism d 1 sin 1 + d 2 sin 2 + f sin =0 avoid, the special geometrical relationships that lead to singularities. This avoidance dramatically reduces the design space to be searched, speeding the synthesis process and enabling design optimization based on more advanced criteria than simply singularity avoidance, such as force transmission. The remainder of the paper is organized as follows. The singularity condition is generated for a mechanism with three links in the closed contour in Sec. 2. Section 3 expands the analysis to an Assur IV/3 mechanism, while Sec. 4 deals with an Assur V/4. Section 5 utilizes these results to generalize the singularity condition for an Assur N/ N 1. Lastly, several illustrative examples are given in Sec Singularity Analysis of a Mechanism With Three Links in the Closed Contour A mechanism with only three links defining the closed contour portion is shown in Fig. 3. The mechanism is composed of six moving links, two of which are ternary links, and seven revolute joints, and it has a single degree of freedom. A closed kinematic chain of mobility 0 is obtained by suppressing one dyad, a 1. However, the resulting structure is not an AKC since additional links Fig. 3 Mechanism with three links in the closed contour / Vol. 1, FEBRUARY 2009 Transactions of the ASME

3 1 1 C 1 =0, 2 2 C 2 =0 1 where C 1 and C 2 are constant interior angles within the rigid ternary links, noting that 2b 1 e 1 cos C 1 =b 2 1 +e d 1 and 2b 2 d 2 cos C 2 =e 2 2 b 2 2 d 2 2. The time derivatives of the loop equations relate the velocities of the joint parameters, which is important in identifying the singular conditions. Being linear, the velocity equations can be written as where K M 3 v 3 =0 K 3 = A 1s,A 1c,0,0,0,0 T 2 3 and E1s A2s B2s E 1c A 2c B 2c D 1s D 2s F s M 3 = D 1c D 2c F c v 3 = 1, 2, 2, 1, 2, T 5 The nomenclature used in Eqs. 3 and 4 and continued through the remainder of the paper is A is = a i sin, A ic = a i cos, B is = b i sin, B ic = b i cos 6 D is = d i sin, D ic = d i cos, E is = e i sin, E ic = e i cos for 1 i N 1, and F s = f sin, F c = f cos 7 When driving the mechanism with a 1, a singularity is encountered when the matrix M 3 in Eq. 2 loses rank, det M 3 =0 8 Taking the determinant generates the singularity equation A 2s B 2c D 1c F s A 2c B 2s D 1c F s A 2s B 2c D 1s F c + A 2c B s D 1s F s + A 2c D 2s E 1s F c A 2c D 2c E 1s F s A 2s D 2s E 1c F c + A 2s D 2c E 1c F s =0 9 Fully expanded, Eq. 9 contains eight terms. Experience with trigonometric substitutions leads the authors to simplify the singularity equation to a concise form with two terms, d 1 b 2 sin 2 2 sin 1 + e 1 d 2 sin 1 2 sin 2 =0 10 Equation 10 presents the general singularity condition for a mechanism with three links in its closed contour, two of which are ternary links. That is, the mechanism driven by 1 will become stationary if Eq. 10 holds. Written in the form of Eq. 10, special geometrical relationships that create the singularity are readily identified. For example, a singularity exists if links e 1, b 2, and a 2 are parallel, giving 1 = 2 = 2. Note, however, that this creates two additional constraint equations on a single-degree-of-freedom mechanism. Therefore, this case will not arbitrarily exist unless an additional relationship between the physical parameters in Eq. 1 is enforced. This situation is further explored in Sec. 6. Starting from the determinant, the authors were not able to produce Eq. 10 using MATLAB s standard algebraic and trigonometric simplification tools without knowing the form of the equation a priori. Obtaining the most concise statement of the singularities enables the designer to identify, and thus avoid, the special geometrical relationships that lead to singularities. Without the benefit of mathematical software, another tool that generates the most concise statement is desired and presented in Sec Singularity Analysis of a Mechanism With a Four- Link Deformable Closed Contour: Assur IV/3 A mechanism with four links defining the deformable closed contour is shown in Fig. 4. This single-degree-of-freedom mechanism is categorized as an Assur IV/3, which contains seven moving links, three being ternary, and ten revolute joints. It has 17 physical parameters a i, b i, d i, e i, i=1,2,3, f, g 2x, g 2y, g 3x, and g 3y and 11 joint variables 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, and. The loop closure equations are a 1 cos 1 + e 1 cos 1 g 2x a 2 cos 2 b 2 cos 2 =0 a 1 sin 1 + e 1 sin 1 g 2y a 2 sin 2 b 2 sin 2 =0 a 2 cos 2 + e 2 cos 2 g 3x a 3 cos 3 b 3 cos 3 =0 a 2 sin 2 + e 2 sin 2 g 3y a 3 sin 3 b 3 sin 3 =0 d 1 cos 1 + d 2 cos 2 + d 3 cos 3 + f cos =0 d 1 sin 1 + d 2 sin 2 + d 3 sin 3 + f sin =0 1 1 C 1 =0, 2 2 C 2 =0 2 2 C 3 =0, 3 3 C 4 =0 11 In addition to C 1 and C 2 defined in the previous case, constants C 3 and C 4 are used in the loop equations to ensure that the ternary links remain rigid. Again, the constants can be expressed in terms of the physical parameters, 2d 2 e 2 cos C 3 =d 2 2 +e b 2 and 2b 3 d 3 cos C 4 =e 2 3 b 2 3 d 2 3. Taking the time derivative of the loop equations and arranging into matrix form gives where Fig. 4 Assur IV/3 mechanism K M 4 v 4 =0 12 Journal of Mechanisms and Robotics FEBRUARY 2009, Vol. 1 /

4 K 4 = A 1s,A 1c,0,0,0,0,0,0,0,0 T 13 E1s A2s B2s E 1c A 2c B 2c A 2s 0 E 2s A 3s B 3s A 2c 0 E 2c A 3c B 3c D 1s D 2s D 3s f s M 4 = D 1c D 2c D 3c f c and v 4 = 1, 2, 2, 2, 3, 3, 1, 2, 3, T 15 When driving the mechanism with a 1, a singularity will be encountered when det M 4 =0 16 Taking the determinant generates the singularity equation A 2c A 3s F c E 1s D 2s B 3s + A 2c A 3s F s E 1s D 2c B 3c + A 2s A 3s F s E 1c D 2s B 3c + A 2s A 3c F s E 1c D 2c B 3s A 2c A 3c F s E 1s D 2c B 3s A 2c A 3s F c E 1s D 2s B 3c A 2s A 3c F c E 1c D 2s B 3s A 2s A 3s F s E 1c D 2c B 3c + A 2c A 3c F c E 1s E 2s D 3s + A 2c A 3s F s E 1s E 2c D 3c + A 2s A 3c F s E 1c E 2s D 3c + A 2s A 3s F c E 1c E 2c D 3s A 2s A 3s F s E 1c E 2c D 3c A 2c A 3c F s E 1s E 2s D 3c A 2c A 3s F c E 1s E 2c D 3s A 2s A 3c F c E 1c E 2s D 3s + A 2s A 3c F s E 1s B 2c D 3c + A 2c A 3s F c E 1s B 2c D 3s + A 2c A 3s F s E 1c B 2s D 3c + A 2s A 3c F c E 1c B 2s D 3s A 2c A 3s F s E 1s B 2c D 3c A 2s A 3c F c E 1s B 2c D 3s A 2s A 3c F s E 1c B 2s D 3c A 2c A 3s F c E 1c B 2s D 3s + A 2s A 3s F c D 1s B 2c B 3c + A 2c A 3s F c D 1c B 2c B 3s + A 2c A 3s F s D 1c B 2s B 3c + A 2c A 3c F c D 1s B 2s B 3s A 2s A 3c F c D 1s B 2c B 3s A 2c A 3s F c D 1s B 2s B 3c A 2s A 3s F s D 1c B 2c B 3c A 2c A 3c F s D 1c B 2s B 3s =0 17 After factoring and substituting the appropriate trigonometric relationships, the singularity equation for an Assur IV/3 mechanism, driven with a 1, reduces to d 1 b 2 b 3 sin 2 2 sin 3 3 sin 1 + e 1 d 2 b 3 sin 1 2 sin 3 3 sin 2 + e 1 b 2 d 3 sin 1 2 sin 3 2 sin 3 + e 1 e 2 d 3 sin 1 2 sin 2 3 sin 3 =0 18 The 32 terms in Eq. 17 reduce to four terms in Eq. 18, and, again, specialized geometrical relationships can be more readily detected with the reduced version. For example, a singularity exists if links e 1, a 2, a 3, and b 3 are parallel, giving 1 = 2 = 3 = 3. Note, however, that this creates three additional constraint equations on a single-degree-offreedom mechanism. Therefore, this case will not arbitrarily exist unless two additional relationships between the physical parameters in Eq. 11 are enforced. With a judicious change to the definition of the variables, these results can be utilized for mechanisms driven with the other outer dyad, a 3. Note that Eq. 12 can be regrouped with the middle dyad a 2 as the driving link resulting in a different K and M. Following the process outlined above, a singularity equation with 24 terms is generated. A reduced singularity condition with only three terms is d 1 b 2 b 3 sin 2 1 sin 3 3 sin 1 + e 1 d 2 b 3 sin 1 1 sin 3 3 sin 2 + e 1 e 2 d 3 sin 1 1 sin 2 3 sin 3 = Singularity Analysis of a Mechanism With a Five- Link Deformable Closed Contour: Assur V/4 A mechanism with five links defining the deformable closed contour is shown in Fig. 5. This single-degree-of-freedom mechanism is categorized as an Assur V/4 with 9 moving links, 4 of which are ternary, and 13 revolute joints. Completing an analysis identical to those leading to Eqs. 10 and 18, the following singularity condition applies to Assur V/4: d 1 b 2 b 3 b 4 sin 2 2 sin 3 3 sin 4 4 sin 1 + e 1 d 2 b 3 b 4 sin 1 2 sin 3 3 sin 4 4 sin 2 + e 1 e 2 d 3 b 4 sin 1 2 sin 2 3 sin 4 4 sin 3 + e 1 b 2 d 3 b 4 sin 1 2 sin 3 2 sin 4 4 sin 3 + e 1 e 2 b 3 d 4 sin 1 2 sin 2 3 sin 4 3 sin 4 + e 1 b 2 b 3 d 4 sin 1 2 sin 3 2 sin 4 3 sin 4 + e 1 e 2 e 3 d 4 sin 1 2 sin 2 3 sin 3 4 sin 4 + e 1 b 2 e 3 d 4 sin 1 2 sin 3 2 sin 3 4 sin 4 = / Vol. 1, FEBRUARY 2009 Transactions of the ASME

5 Fig. 5 Assur V/4 mechanism The general equation of 64 terms reduces to the 8 terms of Eq. 20. By judiciously redefining the parameters, the condition can be utilized for mechanisms driven with the other outer dyad, a 4. As in the Assur IV/3 case, the singularity analysis can be completed with a 2 as the driving link. The reduced singularity condition is d 1 b 2 b 3 b 4 sin 2 1 sin 3 3 sin 4 4 sin 1 + e 1 d 2 b 3 b 4 sin 1 1 sin 3 3 sin 4 4 sin 2 + e 1 e 2 d 3 b 4 sin 1 1 sin 2 3 sin 4 4 sin 3 + e 1 e 2 b 3 d 4 sin 1 1 sin 2 3 sin 4 3 sin 4 + e 1 e 2 e 3 d 4 sin 1 1 sin 2 3 sin 3 4 sin 4 =0. 21 The general equation of 40 terms reduces to the 5 terms of Eq. 21, which more clearly reveals geometric relationships. As before, by redefining parameters, the condition can be utilized for mechanisms driven with a 3. 5 Singularity Analysis of a Mechanism With an N-Link Deformable Closed Contour: Assur NÕ N 1 When extending the Assur class of the mechanism, it is apparent that the new singularity condition builds on the condition associated with the lower class mechanism. A series of rules that allow construction of the higher class mechanism can be formulated. A general model of the Assur N/ N 1s illustrated in Fig. 6. The singularity condition for Class N 1, when driving with 1, can be used to generate the Class N singularity condition by making the following modifications: 1. Each of the initial 2 N 3 terms is multiplied by b N 1 sin N 1 N 1. Fig. 6 Assur N/ N 1 mechanism 2. 2 N 3 new terms are created, each including the product e 1 d N 1 sin N Each of the 2 N 3 terms identified in step 2 are multiplied by the product of b 2 or e 2 and b 3 or e 3 up to b n 2 or e n 2. For example, the four new terms generated when N=5 are multiplied by b 2 b 3, b 2 e 3, e 2 b 3, and e 2 e Each new term from step 3 that contains a b 2 is multiplied by sin Each new term from step 3 that contains an e 2 is multiplied by sin Each new term from step 3 that contains a b N 2 is multiplied by sin N 1 N Each new term from step 3 that contains an e N 2 is multiplied by sin N 2 N If N 4, the physical parameters of the 2 N 3 new terms created in step 3 are reviewed in pairs. Each new term that contains b i b i+1 is multiplied by sin +1. Each new term that contains b i e i+1 is multiplied by sin +1. Each new term that contains e i b i+1 is multiplied by sin +1. Each new term that contains e i e i+1 is multiplied by sin +1. The final equation contains 2 N 2 terms. The singularity condition for Class N, when driving with a 2, can be generated from the condition when driving with a 1 by making the following modifications: 1. Each term that begins with e 1 b 2 is eliminated. 2. In all instances, 1 is substituted for 2. The new equation will have 2 N 3 +1 terms. Using these proce- Table 1 Mechanism parameters for Example 1 i a i b i d i e i g ix g iy f Journal of Mechanisms and Robotics FEBRUARY 2009, Vol. 1 /

6 Fig. 7 Singularity discussed in Example 1 dures, singularity equations for higher Assur class mechanisms have been generated when driving with a 1 and a 2 and are available from the authors upon request. 6 Examples 6.1 Example 1: Combination of Physical and Joint Parameters Resulting in a Singularity. A singular configuration for an Assur IV/3 mechanism, with the properties identified in Table 1, is shown in Fig. 7. No two links are parallel. Additionally, lines along the links have been extended to assist an inspection for any special relationships such as three lines intersecting at a single point, and none exist. The singularity is attributed to a combination of physical and joint parameters that satisfy Eq Example 2: Singularity With Adjacent Parallel Links. By inspecting Eq. 18, it is clear that an Assur IV/3 mechanism will encounter a singularity if the adjacent links e 1, b 2, and a 2 are parallel. As noted earlier, this singularity condition presents two additional constraints, 2 = = 1 23 Since the mechanism has a single degree of freedom, one physical parameter is dependent on the others. Consider a mechanism with the arbitrary physical parameters given in the first section of Table 2. For that mechanism to reach a singularity with links e 1, b 2, and a 2 being parallel, the final physical parameter f can be determined by substituting Eqs. 22 and 23nto the Assur IV/3 loop equations in Eq. 11. Solving, f = The resulting mechanism with these physical parameters is shown in an arbitrary configuration in Fig. 8 a. The joint Fig. 8 Singularity discussed in Example 2 values consistent with the singularity are given in the second section of Table 2, and the resulting mechanism at the singularity is shown in Fig. 8 b. 6.3 Example 3: Singularity With Isolated Parallel Links. By inspecting Eq. 18, an additional relationship that will induce a singularity in an Assur IV/3 mechanism is when the isolated links e 1, a 2, a 3, and b 3 are parallel. This singularity condition presents three additional constraints, 2 = 1 3 = = 1 26 Since the mechanism has a single degree of freedom, two physical parameters will be dependent on the others. Consider a mechanism with the arbitrary physical parameters given in the first section of Table 3. For that mechanism to reach a singularity with links e 1, a 2, a 3, and b 3 being parallel, the two remaining physical parameters, a 1 and f, can be determined by substituting Eqs nto the Assur IV/3 loop equations Table 2 Mechanism parameters for Example 2 i a i b i d i e i g ix g iy / Vol. 1, FEBRUARY 2009 Transactions of the ASME

7 Table 3 Mechanism parameters for Example 3 i a i b i d i e i g ix g iy given in Eq. 11. Solving, a 1 = and f = The resulting mechanism with these physical parameters is shown in an arbitrary configuration in Fig. 9 a. The joint values consistent with the singularity are given in the second section of Table 3 and the resulting mechanism at the singularity is shown in Fig. 9 b. 7 Conclusions This paper presents an analysis to create the singularity condition of a closed-loop Assur mechanism. The general singularity equation was reduced to a condensed form, which allows geometric relationships to be readily detected. The identification of singularities of this extensible kinematic architecture is particularly relevant to rigid-link, shape-changing mechanisms. A procedure for generating the reduced form of the singularity condition for an Assur Class N mechanism, knowing the condition for Class N 1, is presented. Additionally, a procedure for producing the singularity condition with an alternate driving dyad is presented. This approach was illustrated with several examples. Fig. 9 Singularity discussed in Example 3 References 1 Litvin, F., and Tan, J., 1989, Singularities in Motion and Displacement Functions of Constrained Mechanical Systems, Int. J. Robot. Res., 8 2, pp Gosselin, C., and Angeles, J., 1990, Singularity Analysis of Closed-Loop Kinematic Chains, IEEE Trans. Rob. Autom., 6 3, pp Persinger, J., Schmiedeler, J., and Murray, A., 2007, Synthesis of Planar, Shape-Changing Rigid Body Mechanisms Approximating Closed Curves, Proceedings of the ASME Design Engineering Technical Conference. 4 Murray, A., Schmiedeler, J., and Korte, B., 2008, Synthesis of Planar, Shape- Changing Rigid Body Mechanisms, ASME J. Mech. Des., 130, p Galletti, C. U., 1979, On the Position Analysis of Assur s Groups of High Class, Meccanica, 14 1, pp Galletti, C. U., 1986, A Note on Modular Approaches to Planar Linkage Kinematic Analysis, Mech. Mach. Theory, 21 5, pp Mruthyunjaya, T. S., 2003, Kinematic Structure of Mechanisms Revisited, Mech. Mach. Theory, 38 4, pp Peisach, E., 2005, Displacement Analysis of Linkages: Evolution of Investigations in a Historical Aspect, Proceedings of the Third International Workshop on History of Machines and Mechanisms, pp Manolescu, N. I., 1968, For a United Point of View in the Study of Structural Analysis of Kinematic Chains and Mechanisms, J. Mech., 3, pp Romaniak, K., 2007, Methodology of the Assur Group Creation, Proceedings of the 12th IFToMM World Congress. Journal of Mechanisms and Robotics FEBRUARY 2009, Vol. 1 /