Limits of Extramobile and Intramobile Motion of Cylindrical Developable Mechanisms

Jared Butler1 School of Engineering Design, Technology, and Professional Programs, The Pennsylvania State University, State College, PA 16802 e-mail: [email protected] Limits of Extramobile and Jacob Greenwood Intramobile Motion of Cylindrical Compliant Mechanisms Research Group, Department of Mechanical Engineering, Brigham Young University, Developable Mechanisms Provo, UT 84602 Mechanisms that can both deploy and provide motions to perform desired tasks offer a mul- e-mail: [email protected] tifunctional advantage over traditional mechanisms. Developable mechanisms (DMs) are Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/13/1/011024/6611384/jmr_13_1_011024.pdf by guest on 20 November 2021 devices capable of conforming to a predetermined developable surface and deploying Larry L. Howell from that surface to achieve specific motions. This paper builds on the previously identified Compliant Mechanisms Research Group, behaviors of extramobility and intramobility by introducing the terminology of extramobile Department of Mechanical Engineering, and intramobile motions, which define the motion of developable mechanisms while interior Brigham Young University, and exterior to a developable surface. The limits of motion are identified using defined con- Provo, UT 84602 ditions. It is shown that the more difficult of these conditions to kinematically predict may be e-mail: [email protected] treated as a non-factor during the design of cylindrical developable mechanisms given certain assumptions. The impact of toggle positions for each case is discussed. Physical Spencer Magleby prototypes demonstrate the results. [DOI: 10.1115/1.4048833] Compliant Mechanisms Research Group, Department of Mechanical Engineering, Keywords: folding and origami, mechanism design, mechanism synthesis Brigham Young University, Provo, UT 84602 e-mail: [email protected] 1 Introduction possibility would be mechanisms that lie on the outside of a rocket body where penetrating the pressure vessel would lead to cat- Mechanisms that can create customized paths, positions, and astrophic failure. Past work investigated whether a given mecha- force outputs are important, and combining these behaviors with nism is capable of these behaviors of moving into or away from a other functionalities is an area of increasing interest. In particular, developable surface (referred to as intramobility and extramobility mechanisms capable of both deploying and moving to perform [13]). These behaviors allows a DM to (1) lie on a pre-existing desired tasks offer multifunctional benefits over traditional mecha- surface and (2) exhibit some amount of motion without penetrating nisms. Examples include deployable straight-line linkages [1,2], the surface. Mechanisms that exhibit extramobility and intramobi- deployable mechanisms with intentionally shaped parts [3], shape- lity can move without interfering with existing subsystems, morphing structures [4], and mechanisms that conform to or approximate predetermined shapes [5–7]. Other means of obtaining desired shapes or behaviors have been shown through the use of compliant parts [8,9] and harmonic linkages [10]. Developable mechanisms (DMs) are devices that are able to conform to a predetermined developable surface (such as a cylinder or cone [11]) and deploying from that surface to achieve specific motions. An example is shown in Fig. 1. Their ability to lie within a surface makes them compact, and if embedded into or fab- ricated from part of the surface (such as if a compliant mechanism were made from part of the surface), they can occupy no additional volume, becoming hyper-compact. Because of the prevalence of developable surfaces in many engineering applications, DMs provide a way to integrate multifunctionality into previously under- utilized surfaces. Foundational work in this field has defined DMs [12], described behaviors unique to cylindrical [13] and conical [14] DMs, and demonstrated their usefulness in certain applications, such as minimally invasive surgical devices [15]. Because DMs are designed within the context of a developable surface, the movement of the mechanism relative to that surface becomes of particular importance. For example, a device may be made to exist on the interior of a pressurized pipe and requires all parts to remain interior to the pipe during actuation. Another 1 Corresponding author. (a) (b) Contributed by Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received June 1, 2020; final Fig. 1 This four-bar linkage embedded within a cylinder illus- manuscript received September 21, 2020; published online November 19, 2020. trates a developable mechanism that (a) conforms to and Assoc. Editor: Stephen Canfield. (b) emerges from a cylinder Journal of Mechanisms and Robotics Copyright © 2020 by ASME FEBRUARY 2021, Vol. 13 / 011024-1 thereby providing multifunctionality in pre-existing systems with extramobile motion is motion where all moving parts of a mechanism minimal impact. However, the ability to move into or away from remain exterior to the reference surface throughout their motion. a developable surface does not define the possible motions of a Because DMs on cylinders are planar mechanisms, it is often con- mechanism while interior or exterior to that surface. In this paper, venient to model them when viewed along the cylinder centerline, as we advance the understanding of these mechanisms by showing shown in Fig. 2. While cylindrical DMs physically are created using how to determine the range of motion that can be achieved by a curved links, it can be advantageous in the kinematic modeling of DM while interior or exterior to a developable surface. DMs (such as determining the Grashof condition for a mechanism) Another challenge that exists in the design of mechanisms is the to view each linkage as a straight line. Figure 2 shows a (a) develop- existence of toggle positions (where three pin joints are collinear) able mechanism and its (b) straight-linkage equivalent. The two and change-points (where all joints are collinear). Early identifica- mechanisms are kinematically equivalent since the distance tion of these positions can aid in maintaining a mechanism’s between pivots is identical. This paper uses both methods to repre- desired characteristics throughout its motion [16–18]. Change-points sent the links within a cylindrical DM. Note that for all figures in in DMs have not been previously investigated, meaning that the plau- this paper, the black link is ground, the purple link is the coupler sibility of extramobile and intramobile motion of a cylindrical DM is link, and the orange and blue links are links 2 and 4, respectively. unclear when the linkage is a change-point mechanism. Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/13/1/011024/6611384/jmr_13_1_011024.pdf by guest on 20 November 2021 This paper presents nomenclature and methods to determine the limits of extramobile and intramobile motions. These limits of motion are identified using three defined conditions, and it is 3.1 Conditions for Intramobile and Extramobile Motions. shown that the more difficult of these conditions to model kinemat- Based on the possible motions of all links, the requirement that ically may have negligible influence on the design of cylindrical all parts of a mechanism must remain interior or exterior to the refer- DMs given certain assumptions. It is shown that these conditions ence surface can be decomposed into the following conditions: can be further reduced under certain mechanism configurations. • Condition 1: No grounded link may rotate from the conformed Possibilities of change-point mechanisms existing in intramobile position far enough to again intersect the reference surface. and extramobile DMs are investigated. A discussion is then pro- • Condition 2: No grounded link may rotate interior to (exterior vided on the implications of intramobile and extramobile motions. to) the reference surface for extramobile (intramobile) motion. • Condition 3: No portion of the coupler may cross the reference surface. 2 Developable Mechanism Background These conditions define the limits of intramobile and extramobile When modeled with zero thickness, DMs are constrained to have motions. The motion of a cylindrical DM will remain extramobile at least one position in which their joints are all coincident to and or intramobile if none of these three conditions are violated. It is aligned with the ruling lines of a developable surface [12]. (This therefore useful to accurately identify the limits of each of these position is referred to as the conformed position.) This constraint conditions. Predicting the motion limits of grounded links (Condi- creates unique conditions that influence the possible outcomes of tions 1 and 2) is straightforward. In contrast, predicting the location mechanism synthesis. Constraining kinematic linkages to conform of a coupler relative to the reference surface (Condition 3) can be to predetermined developable surfaces can result in the definition much more complex. However, if the first two conditions were to of mechanism behaviors within the context of their motion relative always occur prior to Condition 3, Condition 3 may be ignored to those surfaces, such as the ability to be extramobile, intramobile, during the design process, making its complexity a non-factor and transmobile [13]. when designing for extramobile and intramobile motions. Sections In a zero-thickness model, the surface to which the mechanism 3 and 4 demonstrate that Conditions 1 and 2 will always be violated conforms is called the “reference” surface. When conformed, all prior to Condition 3 for any four-bar mechanism exhibiting intra- of the mechanism’s joint axes must intersect and be aligned to the mobile or extramobile motion given the following assumptions: ruling lines on the reference surface. It should be noted that the reference surface does not need to represent a physical surface. (A) All links have an arc length ≤ πR. As such, the reference surface is merely a representation of where (B) All links have the same curvature as the reference surface. the joint axes must align in space. A cylindrical DM is a mechanism (C) All links are modeled with zero thickness. that has at least one position where all parts of the mechanism (D) All grounded links only extend in one direction past their conform to a cylindrical reference surface. grounded pivot. (E) The coupler does not extend beyond either of the moving pivots. 3 Intramobile and Extramobile Motion A grounded link is defined as any moving link pinned to ground and a coupler is a moving link attached to two grounded links. We define intramobile motion as motion where all moving parts of Assumptions A and D are necessary for mechanisms to exhibit a mechanism remain interior to the reference surface. Similarly, intramobility or extramobility [13]. Assumption B is a requirement for cylindrical developable mechanisms. Assumptions C and E (a) (b) Fig. 2 (a) A developable mechanism with curved links and (b) its Fig. 3 The maximum amount of rotation outside the reference straight-linkage kinematic equivalent (Color version online.) surface for a grounded link 011024-2 / Vol. 13, FEBRUARY 2021 Transactions of the ASME  (a) (b) (c) Fig. 4 (a) The convex side of a coupler may intersect the reference surface before the endpoints intersect the surface. Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/13/1/011024/6611384/jmr_13_1_011024.pdf by guest on 20 November 2021 γ is used to determine if the coupler (b) has not or (c) has violated Condition 3 prior to Condition 1 or 2. Solid and dashed lines correlate to the initial and rotated positions, respectively. (Color version online.) build on previous work in this area [12,13] and provide a foundation Second, the convex side of the coupler must intersect the refer- for mechanisms with thickness and more complex geometries. ence surface prior to Condition 1 or 2 being violated. This can be evaluated by analyzing the rotation of the coupler, γ, when Condi- tion 2 is violated. If γ < π, the coupler has not already crossed into 3.2 Conditions for Extramobile Motion. This section will the reference surface when Condition 2 is met, as shown in detail Conditions 1–3 for both Grashof and non-Grashof extramo- Fig. 4(b). If γ > π, the coupler has crossed into the reference bile cylindrical developable mechanisms. Note that special-case surface prior to Condition 2 being met, as shown in Fig. 4(c). Grashof mechanisms (change-point mechanisms) will be discussed Greenwood [13] put extramobile and intramobile mechanisms in a later section. into three classes and demonstrated how only specific types of four-bar mechanisms [19,20] can be created within each class. 3.2.1 Conditions 1 and 2. For grounded links (i.e., links 2 and These classes are shown in Fig. 5. (These three classes are decom- 4 in traditional four-bar mechanisms), the maximum exterior rota- posed into subclass A and B mechanisms due to symmetry, allow- tion for the link (the point at which Condition 1 is violated) can ing our discussion to be simplified to the analysis of only subclass A be calculated as described in the equation below and shown in mechanisms.) To demonstrate that Condition 3 is not violated prior Fig. 3, where S is the arc length of the link. to Conditions 1 and 2, we will look at the possible motions of the δextramobile,max = π for (0 < S ≤ πR) (1) coupler in each of these three classes. Without loss of generality, we will assign θ1 (angle of the ground link) in each class to equal 0. Class 1. mechanisms are conformed in their open configuration To violate Condition 2, a grounded link would need to move (Fig. 5(a)). Under Class 1, and using Barker’s classification for exterior to the reference surface, then return to its initial position planar four-bar linkages [21], it is possible to obtain a GCCC on the reference surface. At this point, a continuation of motion (double crank), GCRR/GRRC (crank rocker), GRCR (double would move the link interior to the reference surface. Hence, the rocker), and RRR2/RRR4 (triple rocker) (excluding change-point limits of extramobile motion for grounded links are represented mechanisms) [13]. Note that change-points will be discussed in a by the conformed position of the link and Eq. (1). later section. Only GRCR and RRR2/RRR4 are capable of reaching both their open and crossed configurations, resulting in the convex 3.2.2 Condition 3. To violate Condition 3 prior to Condition 1 side of the coupler facing the reference surface as the coupler or 2, the convex side of the curved coupler would need to intersect moves toward the surface. The other mechanism types (GCCC and the reference surface prior to the endpoints crossing the surface, as GCRR/GRRC) cannot invert the coupler and the convex side of shown in Fig. 4(a). Hence, there are two scenarios that must be nec- the coupler can therefore not contact the reference surface prior to essary for this to happen. Conditions 1 and 2. First, the coupler must invert orientation (convex side of arc For mechanism types GRCR and RRR2/RRR4, the mechanism can facing the reference surface), as shown in Fig. 4(a). This is only deploy off the surface, toggle to its crossed configuration, and then possible if the mechanism can reach both its open and crossed con- link 2 can re-conform to the surface, as shown in Fig. 6. We will figurations in the same circuit (as is the case with double rockers and show that when link 2 comes back to the conformed position, the all non-Grashof mechanisms). convex side of the coupler has not penetrated the reference surface. (a) (b) (c) Fig. 5 There are three classes of extra/intramobile mechanisms. The black line represents the ground link: (a) Class 1 mechanism, (b) Class 2 mechanism, and (c) Class 3 mechanism. Journal of Mechanisms and Robotics FEBRUARY 2021, Vol. 13 / 011024-3  Fig. 8 The maximum amount of rotation inside the reference Fig. 6 Class 1 mechanism in its open (solid, conformed) and surface for a grounded link crossed (dashed) configuration. Straight lines represent curved links, as discussed previously in Fig. 2. type cannot reach both the open and crossed circuits, meaning it Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/13/1/011024/6611384/jmr_13_1_011024.pdf by guest on 20 November 2021 cannot toggle the coupler to place the convex side adjacent to the Under Class 1, all joints must reside on the same half of a circle reference surface. Therefore, for Class 3, Conditions 1 and 2 will while in the conformed position. This constrains the longest link l to always occur before the convex side of the coupler comes into be the link closest to the center of the circle, as shown in Fig. 6. It contact with the reference surface (Condition 3). can be seen that the angles adjacent to l, α, and ρ must be less than π/2. Furthermore, when link 2 re-conforms to the surface (reaches the crossed configuration for the same value of θ2 at the conformed 3.3 Conditions for Intramobile Motion. For grounded links position), a symmetric polygon is formed by links 3 and 4 in the (usually links 2 and 4 in traditional four-bar mechanisms), the open and crossed positions. maximum interior rotation for the link (the maximum rotation before Because ρ must always be less than π/2, and because the mirrored violating Condition 1) can be calculated as described in the equation polygon is symmetric, the angle opposite ρ is equivalent to ρ and below and shown in Fig. 8, where S is the arc length of the link. must always be less than π/2. The angle γ must therefore always S be less than π, preventing the coupler curve from moving past its δintramobile, max = π − for (0 < S ≤ πR) (2) tangent position and into the reference surface. It is then concluded R that the defining limits to extramobility for Class 1, given the assumptions above, are set by Conditions 1 and 2. Condition 2 can be violated if any grounded link moves away Class 2. mechanisms are conformed in their crossed configura- from its initial position on the reference surface then moves back tion (Fig. 5(b)). Under Class 2, it is possible to obtain GCCC and to its initial position. At this point, a continuation of motion will GRCR mechanisms (excluding change-point mechanisms). move that link exterior to the reference surface. GCCC is unable to change configurations within the same circuit, To violate Condition 3, the convex side of the coupler would which means that the convex side of the coupler cannot penetrate need to cross the reference surface. Because each link is shaped the surface before Condition 1 or 2 occur. to the reference surface (see Assumption B in Sec. 3), the only GRCR can reach both open and crossed configurations (and way that any point on the coupler link (link 3 in traditional invert the coupler). In this case, the mechanism can deploy off four-bar mechanisms) can cross the reference surface is if one or the surface, toggle to its open configuration, and then link 2 can more of the endpoints has already crossed, as shown in Fig. 9. re-conform to the surface, as shown in Fig. 7. We will show that Therefore, the intramobile motion for a regular cylindrical DM is when link 2 comes back to the conformed position, the convex bounded by the motion of links 2 and 4 (Conditions 1 and 2). side of the coupler has not penetrated the reference surface. In Class 2, α < π/2 (the angle between links 1 and 4) because link 4 must not cross over the center of the circle to maintain extramo- 4 Change-Point Mechanisms bility. The angle α subtends the same arc as the angle between links Because change-point mechanisms often have unique consider- 1 and 3 (β). Therefore, α = β due to the inscribed angle theorem, ations in their motion, and because many of them lie at the interface which states that any two angles that subtend the same arc on a between extramobile and intramobile classes of mechanisms, they circle will have the same value. Hence, β < π/2. are treated separately here. Change-point mechanisms exist when When link 2 re-conforms to the surface a symmetric polygon is formed by links 3 and 4 in their open and crossed positions. Follow- s+l=p+q (3) ing the same logic as Class 1, γ < π. It is then concluded for Class 2 that the defining limits to extramobility, given the assumptions where s is the shortest link, l is the longest link, and p and q are the above, are set by Conditions 1 and 2. remaining two links. Because of the unique geometry that exists Class 3. mechanisms are conformed in their crossed configura- within a change-point mechanism, there is often crossover tion (Fig. 5(c)). The only possible mechanism under Class 3 is between where they exist in terms of the three classes of extramo- GCRR (excluding change-point mechanisms). This mechanism bile and intramobile cylindrical DMs discussed above. This Fig. 7 Class 2 mechanism in its crossed (solid, conformed) and Fig. 9 The convex side of the coupler cannot cross the refer- open (dashed) configuration ence surface prior to an endpoint 011024-4 / Vol. 13, FEBRUARY 2021 Transactions of the ASME Table 1 Possible change point mechanisms in each class that Terms a, b, c, and d represent the four link lengths of a crossed exhibit only extramobile and intramobile behaviors. Names four-bar, in no particular order. Because each link must have a pos- follow Barker’s classifications [21] itive, non-zero length, P4 > 0. McCarthy and Soh showed that for a change-point mechanism, CPCRR/ the product P1 P2 P3 always equals 0 [23]. A combination of Eq. CPRRC CPCCC CPRCR CP2X CP3X 4 and McCarthy’s result suggests that a crossed change-point Open Class 1 ✓ ✓ ✓ ✗ ✗ may exist on a circle only if the links are all co-linear in the con- change-points formed position (the circle has infinite radius). However, Hyatt et al. showed a case where a circle may have a non-infinite radius Crossed Class 2 ✗ ✗ ✗ ✓ ✓ and still contain a crossed change-point mechanism at the con- change-points Class 3 ✗ ✗ ✗ ✓ ✓ formed position [24]. This is only possible if at least two of the links are the same length. Therefore, the only way to obtain a change-point mechanism in Class 2 or 3 is through a CP2X or CP3X mechanism, as shown in Table 1. section discusses where each type of change-point mechanism is Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/13/1/011024/6611384/jmr_13_1_011024.pdf by guest on 20 November 2021 found in the three classes of extramobile and intramobile mecha- nisms and provides logic on why these mechanisms are still 4.3 Condition 3 for Extramobile Motion unable to violate Condition 3 prior to Conditions 1 and 2. Table 1 Class 1. CPCCC, CPCRR/CPRRC, and CPRCR change-point summarizes these results and provides a reference tool to quickly mechanisms can be created depending on the location of the short- identify the possible change points in each class. est link relative to ground. CPCCC and CPRCR mechanisms are The considerations described in previous sections apply for unable to invert their coupler and therefore cannot violate Condition change-point mechanisms pertaining to Conditions 1 and 2 and 3 before Condition 1 or 2 is violated, as is discussed in Sec. 3.2.2. for intramobility. CPCRR/CPRRC mechanisms can invert their coupler to place its convex side adjacent to the reference surface. In each case, γ < π, which shows that Condition 3 has not been violated before Condi- 4.1 Open Change-Points. Change-point mechanisms that tion 1 or 2. Therefore, for Class 1 mechanisms, Condition 1 or 2 will exist in Class 1 must, by definition, exist in an open configuration be reached prior to Condition 3. in the conformed position. As shown in Fig. 6, the longest link l in a Class 1 mechanism lies closest to the center of the circle. Hence, no other link may be of equal length to l, making it impos- Class 2. Possible iterations of Class 2 change-point mechanisms sible to create a CP2X (two pairs of equal length links) or CP3X are shown in Fig. 10. The mechanism in Fig. 10(a) follows the same mechanism (all links have equal length) in a Class 1 mechanism. logic as other crossed mechanisms in Class 2. If the two crossed All other change-point mechanisms (CPCCC, CPCRR/CPRRC, links cross through the center of the reference surface, the toggled and CPRCR) [21] can be created depending on the location of the coupler remains tangent to the reference surface through all shortest link relative to ground, as shown in Table 1. motion in the open configuration, meaning the coupler at no time crosses the reference surface. To invert the coupler, the mechanisms in Figs. 10(b) and 10(c) 4.2 Crossed Change-Points. Mechanisms in Class 2 and 3 must rotate links 2 and 3 to be co-linear with links 3 and 4 before must be in a crossed configuration when conformed. Not all change- it may reach its open configuration. Once all links are co-linear, points are capable of existing in a crossed configuration while the mechanism may only move away from the reference surface mapped to a circle. According to Josefsson [22], the area of a without violating Condition 1 or 2. Condition 1 is also violated crossed cyclic quadrilateral (a crossed four-bar mapped to a prior to the convex side of the coupler contacting the reference circle) is found by surface. These results lead to the conclusion that all Class 2 change-point 1 

















mechanisms violate Conditions 1 or 2 prior to Condition 3 given the K= (P1 )(P2 )(P3 )(P4 ) (4) 4 above assumptions. where Class 3. Possible iterations of Class 3 change-point mechanisms P1 = −a + b + c − d are shown in Fig. 11. For a Class 3 mechanism, one of the grounded P2 = a − b + c − d links (link 2 in Fig. 5(c)) must be ≤ all other links. (5) Two iterations of the CP2X mechanism are possible; the first P3 = a + b − c − d occurring when equal link lengths exist between links 1 and 4 P4 = a + b + c + d and links 2 and 3 (Fig. 11(a)), and the second occurring when (a) (b) (c) Fig. 10 Class 2 change-point mechanisms in their conformed positions. Hatch marks indicate equal lengths: (a) CP2X where no links lie in the same position, (b) CP2X where links of the same length lie in the same position, and (c) CP3X where pairs of links lie in the same position and all links are the same length. Journal of Mechanisms and Robotics FEBRUARY 2021, Vol. 13 / 011024-5  (a) (b) (c) Fig. 11 Class 3 change-point mechanisms in their conformed positions. Hatch marks indicate equal lengths: (a) CP2X where r2 = r3 and r1 = r4, (b) CP2X where r1 = r2 and r3 = r4, and (c) CP3X where all links lie in the same position. Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/13/1/011024/6611384/jmr_13_1_011024.pdf by guest on 20 November 2021 equal link lengths exist between links 1 and 2 and links 3 and 4 cannot reach the reference surface as the coupler remains parallel (Fig. 11(b)). Additionally, each pair of links that are of equivalent with the grounded link throughout its motion. length must lie at the same location in the conformed position. These results lead to the observation that, for extramobile motion, For the CP2X in Fig. 11(a), only links 2 and 3 may move from all Class 3 change-point mechanisms violate Condition 1 or 2 prior the conformed position. Since they rotate together, they may to Condition 3 given the asserted assumptions. In summary, all pos- either rotate far enough away from the reference surface that they sible change-point mechanisms within all classes of extramobile cross the reference surface simultaneously, or they may stop and intramobile mechanisms will violate Conditions 1 and 2 prior when collinear with the other two links. When this occurs, the to Condition 3 given the same assumptions. mechanism moves into an open configuration, shown in Fig. 12(a). Because r2 ≤ r1, γ will never exceed π. To invert its coupler, links 2 and 3 of CP2X shown in Fig. 11(b), must first rotate to the change-point position. Figure 12(b) demon- 5 Toggle Positions strates how links 2 and 3 must make a rotation greater than π radians Conditions 1 and 2 determine the limits of motion for any cylin- to reach this position, violating Condition 1 prior to the mechanism drical developable mechanism given the stated assumptions. being able to move back toward the reference surface. However, the analysis of many mechanisms may be further simpli- The CP3X mechanism is obtained when all links are of the same fied if they have a toggle position (a position where two moving length, as shown in Fig. 11(c). The convex side of the coupler links become collinear) in their motion, as this can limit the mech- anism’s range of motion. Figure 13 demonstrates a mechanism in a toggle position, where links 2 and 3 are collinear. At this position, link 4 is at an extreme of its motion and has reached θ4min. Many mechanisms, including crank-rockers, double-rockers, and triple-rockers, can reach toggle positions. Condition 1 states that no link may rotate far enough to intersect the reference surface. Figure 14 shows these limits for a single link and that links two extreme positions. If a toggle position of a mech- anism is reached prior to Condition 1, such as is shown in Fig. 14, analysis of extramobile and intramobile behavior is further simpli- fied because Condition 1 cannot be violated for that link. When the relationship |θmin,max − θo | < δmax (6) is true, then the associated link is incapable of violating Condition 1 and is only able to violate Condition 2. It should be noted that a toggle position may be considered separately for extramobile and intramobile motions. For example, a toggle position exterior to (a) the reference surface may be reached prior to δextramobile,max while the correlating toggle position interior to the reference surface may exceed δintramobile,max. (b) Fig. 12 Class 3 CP2X mechanisms: (a) a class 3 CP2X (with r2 = r3 and r1 = r4) shown in its conformed and open configurations and (b) a class 3 CP2X (r1 = r2 and r3 = r4) shown in its conformed Fig. 13 A mechanism at a toggle position has reached an position and change-point position extreme value for one of its links 011024-6 / Vol. 13, FEBRUARY 2021 Transactions of the ASME Fig. 14 It is possible for the limits of a link’s motion to occur prior to δ Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/13/1/011024/6611384/jmr_13_1_011024.pdf by guest on 20 November 2021 These toggle positions may be found through the law of cosines. Because of symmetry and the constraint that θ1 = 0, the only possi- ble extramobile and intramobile mechanisms that may toggle are triple-rockers (RRR4), crank-rockers (GCRR), and double rockers (GRCR). In the case of a double crank, there are no toggle positions, making Condition 1 the only possible condition to violate, given the stated assumptions. Change-point mechanisms must align all links at a change-point position, making their analysis more complex than a comparison of θ and δ. The extreme angular positions for each grounded link (θmax and θmin for subclass A mechanisms due to symmetry) can be found through the law of cosines for each case. Similar results can be shown for subclass B mechanisms (symmetrical results for each of Class 1, 2, and 3). For triple rockers (RRR4), these limits are Fig. 16 A Class 3 intramobile mechanism moving from: (a) con- formed crossed position, (b) the toggle position, and (c) reaching the limit of intramobile motion, where the short grounded link has rotated δintramobile,max (Condition 1) given as follows:  2  r1 + r22 − (r4 − r3 )2 (a) θ2 min,RRR4 = arccos (7) 2r1 r2  2  r1 + r22 − (r4 − r3 )2 θ2 max,RRR4 = 2π − arccos (8) 2r1 r2  2  r1 + r42 − (r2 + r3 )2 θ4 min,RRR4 = π − arccos (9) 2r1 r4  2  r1 + r42 − (r2 + r3 )2 θ4 max,RRR4 = π + arccos (10) 2r1 r4 For double rockers (GRCR), the limits are θ2 min,GRCR = θ2 min,RRR4 (11)  2  r1 + r22 − (r3 + r4 )2 (b) θ2 max,GRCR = arccos (12) 2r1 r2 Fig. 15 A Class 2 extramobile mechanism demonstrating that the coupler does not cross the reference surface prior to a θ4 min,GRCR = θ4 min,RRR4 (13) grounded link moving back to the conformed position: (a) con- formed, crossed position and (b) limit of extramobile motion,  2  r1 + r42 − (r2 − r3 )2 where the coupler link has returned to the reference surface θ4 max,GRCR = π − arccos (14) (Condition 2) 2r1 r4 Journal of Mechanisms and Robotics FEBRUARY 2021, Vol. 13 / 011024-7  For crank-rockers (GCRR), link 2 is fully revolute. Since there is Conflict of Interest no extreme value associated with θ2, we will only consider θ4. There are no conflicts of interest. θ4 min,GCRR = θ4 min,RRR4 (15) θ4 max,GCRR = θ4 max,GRCR (16) Data Availability Statement The datasets generated and supporting the findings of this article 6 Physical Demonstration are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included Prototypes were created to demonstrate some of the concepts dis- in the paper. cussed in this work. Two different classes were selected to demon- strate both extramobile and intramobile motions. They also demonstrate that the coupler does not cross the surface while extra- mobile, and that a mechanism can be made to move intramobile and References reach the toggle position prior to touching the reference surface. [1] Cao, W.-a., Zhang, D., and Ding, H., 2020, “A Novel Two-Layer and Two-Loop Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/13/1/011024/6611384/jmr_13_1_011024.pdf by guest on 20 November 2021 Figure 15 shows a sample Class 2 mechanism conformed to the Deployable Linkage With Accurate Vertical Straight-Line Motion,” ASME J. Mech. Des., 142(10), p. 103301. exterior of a reference surface. The mechanism moves from a [2] Wei, G., and Dai, J. S., 2014, “A Spatial Eight-Bar Linkage and Its Association crossed, (a) conformed position into an (b) open position. The With the Deployable Platonic Mechanisms,” ASME J. Mech. Rob., 6(2), blue link in the image is the link that limits the extramobile p. 021010. motion as it moves away from the surface then back to its initial [3] Huang, H., Li, B., Zhang, T., Zhang, Z., Qi, X., and Hu, Y., 2019, “Design of Large Single-Mobility Surface-Deployable Mechanism Using Irregularly position (Condition 2). 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Des., 101(3), pp. 515–518. joints, mechanisms may be synthesized that reach desired or [21] Barker, C. R., 1985, “A Complete Classification of Planar Four-Bar Linkages,” stable configurations while remaining exterior or interior to the Mech. Mach. Theory., 20(6), pp. 535–554. reference surface. [22] Josefsson, M., 2017, “101.38 Metric Relations in Crossed Cyclic Quadrilaterals,” Math. Gazette, 101(552), pp. 499–502. [23] McCarthy, J. M., and Soh, G. S., 2010, Geometric Design of Linkages, Vol. 11, Springer Science & Business Media.Berlin. Acknowledgment [24] Hyatt, L. P., Greenwood, J. R., Butler, J. J., Magleby, S. P., and Howell, L. L., 2020, “Using Cyclic Quadrilaterals to Design Cylindrical Developable This work was supported by the U.S. National Science Founda- Mechanisms,” Proceedings of the 2020 USCToMM Symposium on Mechanical tion through NSF Grant No. 1663345 and the Utah NASA Space Systems and Robotics, P. Larochelle, , and J. M. McCarthy, eds., Springer Grant Consortium. International Publishing, Berlin, pp. 149–159. 011024-8 / Vol. 13, FEBRUARY 2021 Transactions of the ASME