The influence of dimensional parameters on the kinetic range of four systems of deployable structures with hinged nodes and rigid rods

Abstract

In several previous studies regarding foldable structures comprised of rods and knots, it has been usual to simplify their geometry to lines and points respectively. However, this study assumes that the dimensions that these elements necessarily acquire (due to their material condition) constitute an important issue in the folding and unfolding process. To test this hypothesis, the research first defines the concept of kinetic range and its various variants and studies how variations in the radius of the rods, the eccentricity of the knots and the opening angle influence this parameter. Furthermore, this study will demonstrate that this parameter is extensible to bar bundle systems and scissors systems and finally will verify that the results obtained by using algorithm-driven models are also verifiable by mathematical methods and experimentation using physical models. As a result, the methodology followed in this article allows the inclusion of a comparative table that classifies the performance of the structural systems studied in order to establish which is the optimum in relation to their kinetic range.

Keywords

Scissor structure bundle system folding angle opening angle radius eccentricity folding structure

Introduction

Deployable structures trace their roots back to ancient times, where they were ingeniously employed by nomadic communities to fashion portable shelters tailored to their exigencies. Examples abound, from the iconic Bedouin tents dotting the Middle Eastern desert landscapes to the time-honored yurts adorning the vast expanses of Central Asia.^1,2 However, it was not until the 1950s with the beginning of the USA versus USSR “Space Race,” that these types of structures were studied more systematically by engineers and architects and who generated a large number of classification systems and typologies that continue to be used in the present day.^2
–5

In this text we will focus on the study of the so-called rigid bar and articulated nodes/knots structures⁴ (the term used to refer to knots often varies from author to author^6
–8 to refer to specific features of these elements, in this text we will use the terms knot, joint or pivot interchangeably). Closely linked to this category of structures, are the Spanish pioneers Emilio Pérez Piñero and Félix Escrig, whose patents, on bar bundle structures and scissor structures (respectively), are reference documents in this field of study.^9,10

Within these patent documents, both architects represented their structures with different levels of detail (Figure 1). As an example, some drawings simplify and reduce the structure to an abstract geometric pattern, composed only of lines and dots (which symbolize the bars and the nodes respectively). However, in other illustrations, the authors go much further by defining how these nodes (or per this paper, defined as “knots”) could be materialized or how the bars should look and even suggesting the materials with which they could be built and the dimensions they could acquire.^9,10

Figure 1.

Pages of Félix Escrig’s patent. Left: Drawing of the deployable structure as an abstract geometric pattern of dots and lines. Right: drawings of the details where the material qualities of the proposed solutions can be appreciated.

It is precisely this material condition of deployable structures that this text attempts to evaluate more deeply and thus also build upon the work of previous architects who often could only rely upon building physical models and prototypes to test, verify, and understand further the behavior of such structures. This is in fact the main contribution of this research. While authors of deployable structures of the 20th century had to build and test them (physical or at most mathematical models) today we have algorithm programming that allows to foresee the dynamic behavior of these structures. Accordingly, and since this study’s broader questions remain elusive to resolution by traditional simplified drawing techniques, one of the key challenges to overcome, was to understand why certain elements of the various foldable structures would unexpectedly clash with other adjacent elements and thus prevent or obstruct the folding or unfolding process from completing fully or as originally designed and envisaged.

Until only very recently, most of the programs used to study the behavior of such structures with rigid bars and articulated nodes have required the development of simplified models, in which again, the rods are represented as lines and the nodes as points.^11
–13

Notwithstanding, other recent works^14
–17 have attempted to approach the problem of foldability and deployability by focusing on the length of the adjacent bars and the positions of the nodes that articulate them. However, whilst these investigations did stem from an analytical point of view, they hold limitations by only working with the longitudinal variable and do not address other factors derived from considering the material reality of these structures such as the thickness of these bars or the size of their nodes.^14
–17

Others works do study the influence of variables related to the material reality of these structures. Even Gantes in his book “Deployable Structures”⁵ has a section dedicated to this idea. Other researchers consider member cross-sectional sizes and joint sizes related to the linkage of the structures. Zhao et al.⁶ clarifies the self-locking mechanism of foldable grid structures and develops an approximate expression for evaluating the self-locking capability of their structural units. Georgiu⁸ provides a comparative analysis of deployable and reconfigurable rigid-bar linkage systems, while Pérez-Valcárcel et al.⁷ focus on the use of reciprocal linkages in deployable cylindrical vaults for emergency buildings.

The present research is in line with these previous studies that collectively highlight the potential of deployable rigid bar structures in various architectural contexts as well as the usefulness of locking these structures by means of collision between their bars (reciprocal linkages).

These structures will be able to move as long as their bars do not collide with each other, and reciprocal linkage does not occur. Therefore, the study of the conditions in which these bars collide with each other allows us to know their influence on the range of movement (kinetic range). Nowadays, newer, and more powerful parametric design or algorithm-assisted design tools allow us to study the trajectories followed by the elements (nodes and bars) of these structures and to also check at much larger “real-world” scales, what happens when the thickness of their constituent elements is varied. Hence, such variations can have an important influence in determining the limits at which an assembly changes from acting as a mechanism to acting as a structure and thus establish the limits of the kinetic behavior of the structure.¹⁸ The use of these parametric design tools has made it possible in this study to obtain and precisely define the kinetic limits of the structures, even with a three-dimensional representation of the results.

Objectives

The prime objective of this paper is to understand to a higher degree, how variations in the dimensions of the nodes (eccentricity) and the radius (or thickness) of the structural bars, limit the behavior of the mechanisms of a sample of four different systems of deployable structures: two configurations comprised of bundle beam arrangement, and two configurations formed of scissors type arrangements.

The purpose of using this mix of four different structures is to identify similarities and differences across their various behaviors when in motion and that might help us to better compare the potential merits of one design to another and therefore reinforce possible conclusions, we might be able to make about their overall structural limits and efficiencies.

In parallel to the above, we will also seek to further elaborate more detail into existing concepts and variables as well as define other new ones which when linked to the issues related to the dimensional geometry of the elements used (such as radius, eccentricity, and folding angle), will further contribute to the characterization of their behavior in the folding-unfolding process in both qualitative and quantitative ways (2. Definitions and variables). Specifically, we will define the concepts of:

Kinetic range.

Kinetic range for constant radius.

Kinetic range for constant eccentricity.

Kinetic range for constant angle.

Lastly, from undertaking a range of research studies based on the four different selected systems for the new concepts defined, we will seek to gather and validate a set of various results that can be compared with each other. These studies will be derived from three different but complementary methodological approaches (Methodology). From this point of view, this paper aims to combine the work carried out using the following three types of models: parametric models digitally simulated by algorithms; Analytical models governed by mathematical expressions and physical models elaborated with digital fabrication tools.

Definitions and variables

To develop this research, several key concepts, and variables must be clearly defined beforehand. To do so, we will distinguish between the (a) Initial data and independent variables versus the (b) Dependent variables and/or concepts related to them.

Initial data and independent variables

Below, we describe some basic parameters and concepts that will serve as a baseline for this research and will allow us to go into more detail later.

Systems under study

In this research we will work with four systems of deployable structures. Depending on how the bars are joined together at the central joints of a module, we will have two types of systems:

bar bundle systems,^4,9,19
–28

scissors systems.^4,20,29

In a bar bundle system, all the bars of the same module articulate at a singular central point, generating a basic module whose geometry is inscribed within a prism or, in some cases, within an antiprism. In scissors systems, the bars are articulated in pairs across every face of the prism/polygonal extrusion that defines the module.¹⁰

First, we will consider two beam systems, whose basic modules consist of three and four bars, which we will call EP1 and EP2, respectively (as described in Emilio Perez Piñero’s patent, Figure 2).⁹

Figure 2.

Basic modules of the systems studied and graphical representation of some of their variables.

In parallel, we will also study two deployable scissors systems, with basic modules composed of three and four pairs of scissors, which we will call FE1 and FE2 (as described in Félix Escrig’s patent, Figure 2) respectively.¹⁰

Figure 2 shows, conceptually, the geometry reduced to lines and points of each of these basic modules. And Figure 3, represents plan diagrams showing how these basic modules are concatenated to form a larger structure, in each case.

Figure 3.

Plan of the modules of the four systems considered. Top left: EP1; top right: EP2. Bottom left: FE1; bottom right: FE2.

Knots

In this paper, we elected to use the word “knot” as it was considered to be a more appropriate piece of terminology to describe the name of the junctions where the structural bars (regardless of whether in bundle or scissor formations), intersect, interface, and join on to other bars. In addition, it is important to note two other things regarding this terminology adopted as follows. The use of the word “knot” has no connection or relevance to the concept of knots as studied in the field of mathematics.

We felt it was more appropriate to use the word knot, instead of node due to the fact that a knot by its own virtue, is something which is tied and thus has an element of closure and which as a device, can limit/lock or tie something down—for example, as a concept similar to the limitations imposed on the movements of the bars of the structures being studied under this paper.

Rod length (L)

For the purpose of this research, we will consider that the rods behave as rigid non-deformable solids and therefore assume that the length of the bars of every one of the systems studied have the same length, which we will abbreviate as L (Figure 2). The parameter L, thus defined, will be considered as a unit of length and be the basis for the definition of the other linear variables that we will define and detail in the next sections, namely, total eccentricity of the system (E), partial eccentricity (e), and the radius of the bars (r).

For clarity, in the case studies used in this investigation, the articulation of the bars together to form a module (in the case of bundle systems) or two by two (in the case of scissors systems) occurs at the midpoint of their length, that is, at L/2 from each end (Figure 2).

Folding angle of the structure (θ)

Numerous precedent studies have defined the degree of deployment (or folding) of scissor structures by the angle formed by the rods, as shown in Figure 4.^30
–33

Figure 4.

The deployment of a set of blades controlled by the angle they form between the rods.

However, this angle is not appropriate for defining the degree of folding of the bundle structures, because this angle does not bear any relation to the horizontal base plane that the structure is resting upon. Alternatively, the deployment degree can also be controlled by the angle that the bars form with the horizontal base plane.^34,35 This angle, as shown in Figure 2, is denoted by the Greek letter θ. In addition, defining this parameter with respect to the base plane of the module is more appropriate for comparisons between the beam and scissors systems we have considered. Accordingly, when the structure is fully deployed, θ takes a value of 0°. In contrast, the value of θ would be 90°, for example, when the system is fully folded.^36,37

In our research, the angle θ will be one of the main independent variables to be considered and its variation will influence the values reached by the rest of the variables and concepts that we will define.

Rod radius (r)

Like the folding angle, for this research the bar radius (r) is also considered a relevant independent variable, therefore it will also be analyzed against the other chosen parameters alongside its general effects on the behavior of the structure during the folding and unfolding processes (Figure 5). This variable will be dimensionally referred as a function of the length of the bars (L). In addition, for this research, we will also assume that the bars are cylindrical. Whilst it is true that this is not the only shape that the bars of the systems we are studying can adopt (it is also common to use square or rectangular section bars). It is always possible to draw cylinders that are inscribed or circumscribed to these other shapes and then transfer the results obtained.

Figure 5.

Definition of the radius (r) for the circular section of the rods.

System eccentricity (E) and partial eccentricity (e)

In the present research, we will define the total eccentricity of the system or simply the eccentricity of the system (E) as the distance between the longitudinal axis of the bars (all of which will be considered equal, straight and of constant section along their entire length) and the center point of the node, regardless of the geometry the node adopts, as shown in Figure 6.

Figure 6.

Relationship between total eccentricity (E), partial eccentricity (e), and radius (r).

It is also useful to consider the eccentricity of the system from the partial eccentricity (e), defined as the smallest distance between the surface of the bars and the center point of the node (Figure 6).

Thus, we can define the system eccentricity (E), as the sum of the partial eccentricity (e), and the radius (r) of the bars given in the following equation.

E = e + r

(1)

Where,

e; is the partial eccentricity, defined as the smaller of the distances between the points on the surface of the bars and the center point of the node.

r; is the radius of the bars.

In our research, we will consider the eccentricity of the system (E) and the associated partial eccentricity (e), as independent variables that can influence the values generated by the other variables and concepts defined in this text’s overall Section “Initial data and independent variables.”

Minimum representative sample

Due to the modular nature of the deployable systems considered, it is necessary to define a minimum sample size for each system to standardize the size of models used and tested. In our research, the minimum size chosen will be 3 × 3 modules per system which we consider sufficient to allow the observation of the phenomena of internal and external collisions (as defined in Section “Internal and external collisions”).

For further clarification in this matter, we have considered that samples with a number of modules smaller than the representative sample may not guarantee that the behavior of their mechanisms (when at their opening and folding limits) will correspond to that of a complete hinged bar system. Furthermore, the broader level at which this study currently stands, would entail a much higher computational cost if an algorithm of a system composed of a larger number of modules were to be used, a factor which at this stage we need to be limited by.

Internal and external collisions

In this study, we will say that collisions have occurred between bars, or that two or more bars have collided, when during the folding-unfolding process, bars come close enough to each other to make their respective surfaces come into contact.

In addition, these phenomena can be between bars of the same module, or between bars of adjacent modules. A collision between bars of the same module is referred to as an internal collision (Figure 7) and a collision between bars of adjacent modules is referred to as an external collision (Figure 8).¹⁸

Figure 7.

Internal collisions between members of the basic module of the EP1 system with an intermediate eccentricity e = 0.025 L for a very low folding angle θ = 10°. In red the intersection curves between the bars.

Figure 8.

External collisions between adjacent module members for three EP1 modules with an eccentricity e = 0.20 L for an angle θ = 60°. The radius of the spheres represents the eccentricity e. In the foreground the image shows the bars of a module and the sphere representing the eccentricity at the central node of the module, all in semi-transparent mode. In the background, the bars of two adjacent modules are represented in line mode, in blue and purple colors respectively. In red the intersection curves between bars.

Dependent variables

In this section we define in more detail the dependent variables of our research and several concepts that are either related to the results obtained by the studies and tests undertaken or that will allow comparisons to be made between the systems under study.

Minimum opening angle of the structure (θ_min)

The minimum opening angle for each system, or in short θ_min, is the minor value of the folding angle that can be obtained without collision (Figure 9).

Figure 9.

System minimum opening angle of the structure (θ_min), due to collision between members of the same module at opening for EP2. In this illustration the radius takes the value r = 0.018 L and the eccentricity adopt e = 0.05 L.

The value of this variable is a good indicator of the efficiency of a system when deployed, since the lower the value of the minimum opening angle, the larger the area that the system is able to cover with the same number of modules.^18,38

Maximum folding angle of the structure (θ_max)

The maximum folding angle or closure angle of each system or (θ_max) is the highest value of the folding angle that can be obtained without the systems elements (knots and rods) colliding with each other (Figure 10).

Figure 10.

Configuration of the structure with the folding angle adopts its maximum value, maximum folding angle (θ_max), due to collisions among members of adjacent modules at folding state for EP1. In this illustration the radius takes the value r = 0.025 L and the eccentricity adopt e = 0.20 L.

The value of this variable relates to the efficiency of the system when folded. As an example when considered from a logistically intensive point of view such as an emergency situation (e.g. that might require the rapid supply of temporary roofing structures over large and small spaces) a system that is more efficient (e.g. a higher maximum folding angle) can be assumed to be more agile, sustainable and economically viable to manufacture and transport.^18,38

Folding width

The folding width Δθ is the difference between the maximum folding angle, θ_max and the minimum opening angle, θ_min.

Δ θ = θ \max - θ \min

(2)

The higher the value of this variable, the more versatile the folding and unfolding process, but neither this parameter, nor the maximum folding angle, nor the minimum opening angle are as complete when evaluating a folding structure as the kinetic range, which we will define below.¹⁸

Overall kinetic range (KR)

We define the overall kinetic range (simply the kinetic range—KR) of a deployable structural system as the set of values of opening angle (θ), bar radius (r), and node eccentricity (E), for which neither internal nor external collisions occur in that system. That is, for that set of values of θ, E, and r, the system acts as a mechanism “without fault.” This parameter, although with certain restrictions, has already been defined by Pérez-Egea et al.¹⁸

In this research we intend to generalize this concept whilst also considering the influence of the variation of the radius of the bars. Later, in the section dedicated to discussing the results obtained by our research, we will refine this definition later on by adding the concepts of kinetic range for a constant value of the opening angle (KR_θ), kinetic range for a constant value of the eccentricity (KR_e) and kinetic range for a constant value of the radius (KR_r).

Methodology

The methodology we have used in this research is divided into three main phases, which are summarized below:

-First, the systems are modeled using parametric design tools and visual programming, specifically Grasshopper³⁹ on Rhinoceros Version 5.⁴⁰ In addition, the algorithms applied allow us to generate digital 3D models of the systems that we want to consider and which have the capability of performing their folding and unfolding process in multiple iterations of modified dimensional variables (e.g. of the elements that compose the 3D models/their constituent bars and nodes) (Figure 11). These 3D models make it possible to detect immediately whether or not collisions (both internal and external) occur and to plot these results graphically on a system of coordinate axes (Oθ, Oe, and Or). As a side note to this, whilst it is possible to consider the matter of structural deformation when using digital models, the scope of this research will remain focused on a geometrical level at this stage.

Figure 11.

Screenshot of the algorithms defined in Grasshopper to digitally model the drop-down structures studied.

As we will see in the following section discussing the results, this graphic representation will prove to be a very eloquent way of observing the behavior of the variables defined above, and consequently, when comparing the systems with each other.

-Second, by using mathematical formulae, we can analyze further the results obtained in the first method from another perspective. These expressions will enable us to explain with much more fidelity, the most notable aspects detected in the preceding phase from the perspective of differential calculus and analytical geometry.

-Third, and finally, we will fabricate physical models of the deployable structural systems that will allow us to consider in an almost “real world/real time” manner, the results obtained in the previous phases and anticipate their practical applications.

Method 1. Using parametric design tools based on algorithms

The results obtained with method 1, with simulations produced from the use of parametric design tools based on algorithms, can be divided into two steps.

The variables defined previously can be modified in real time within the parametric tool (Grashopper 3D) by implementing components such as sliders that affect the movement and features of the structures. More specifically, these components control the radius of the rod (r), the opening angle (θ), and the eccentricity of the systems (e).

Firstly, from a purely instrumental point of view, we find the set of algorithms that allow us to operate a virtual model with which to observe the movements of each system in the folding and unfolding process (Figure 12).

Figure 12.

Algorithmically governed three-dimensional model of the EP1 system for a value of eccentricity E = 0.1 L and three values of θ = 10°, 25°, and 40°. Left: Plan view of the system construction Right: Perspective of the system construction.

Next, another set of algorithms allows us to graphically represent, on three coordinate axes (θ, e, and r), the values of these variables for the cases in which there are no collisions between the elements of the systems, neither internal nor external and thus obtaining a point cloud.

Due to the obvious difficulties that a point cloud presents when visualizing these types of results, we finish the algorithm by adding a “graphical surface” that will help us to better see how the numerical values of the results inter-relate (Figure 13). In the following section, the mathematical expressions will allow us to discuss the nature of this surface and the relevance of the approximation in more depth (5. Results obtained with method 2, using mathematical formulae).

Figure 13.

Left: point cloud of the EP1 system, for the pairs of values (θ, e) of free kinetics colored according to the radius r. Right: Superposition of the point cloud with the enveloping surfaces of the point cloud.

Similarities between the graphs of the four studied systems

The graphs of results and values generated by each system (Figure 14), although different from each other, show certain common features that are described below:

The volume enclosed by the envelope surface generated and the coordinate planes, represents the set of points (θ, e, r) for which no collision occurs and therefore constitute a graphical representation of the extents to which the sequence of folding- unfolding mechanisms are limited by. Accordingly, this volume represents the kinetic range (KR) of the system and can be attributed a measurable value (Figures 14 and 15).

In all cases, the envelope assigned to each system is subdivided into two facets or slopes that are clearly separated by a curved edge. This curved edge represents an intersection, and we will call this line the collision edge since it is formed by points where both internal and external collisions occur (Figure 15).

The facet that is located from this collision edge toward increasing values of the eccentricity represents the external Collision Limit Surface of the system (eCLS) (Figure 15). That is, configurations of the structural system with combinations of θ, e, and r located on or above the points that integrate this surface would imply collisions between the elements of a module and those of adjacent modules.

Similarly, the surface located from the collision edge toward decreasing values of the eccentricity represents the internal Collision Limit Surface of the system (iCLS) of the system (Figure 15). Combinations of θ, e, and r located on or above the points that integrate this surface would imply collisions between the elements of the same module.

In all the cases studied, the volume that defines the kinetic range has a shape like that of a conoid, being defined by ruled surfaces. This conoid is such that its sharpest part points toward increasing values of r. This issue reveals that, indeed, as the radius of the rods increases both internal and external collisions occur in intervals of values of e and θ which has gradually less amplitude. That is, the folding width is progressively reduced as the value of r increases (Figure 15).

It is observed that the collision edge has a maximum value, which we will call critical radius ( r_c ) and this has a characteristic value for each system. When this value is reached, the system, although it can be conceived and built as a deployable structure, loses its capacity to act as a mechanism, since internal and external collisions occur, regardless of the value acquired by the other variables e and θ.

In the four cases studied, it is observed that, for values of the folding angle and radius relatively low or close to zero, maximum eccentricity values are obtained.

Figure 14.

Graphical representation of the volume bounded by the internal (iCLS) and external (eCLS) collision boundary surfaces for the four systems studied.

Figure 15.

Notable elements common to all four results graphs.

Comparison between the graphs of the four systems studied

If we compare the graphs that describe the kinetic range of each of the four systems, we also observe some notable issues that allow us to differentiate some systems from others according to the values acquired by the dependent variables described above.

Kinetic range (KR)

The graphical representation of the kinetic range as a function of θ, e, and r, allows us to establish that there is a magnitude specific to each system which can be calculated from its associated volume (e.g. the space enclosed between the envelope and the coordinate axes). This is easily done by adding a counter to the algorithm represented. For example, for EP1 in Figure 16.

Figure 16.

Algorithm for the calculation of KR, applied to the EP1 system.

Accordingly, the system with the highest kinetic range is FE2 (1.207), compared to EP2 (0.352) whose value is the lowest of the four, with EP1 (1.099) and FE1 (0.355) in intermediate positions. These values are represented in the following graph (Figure 17).

Figure 17.

Radial graph with the values of the Kinetic Range (KR) for the four systems under study, in L2°.

To better understand the variations of the KR in each system as a function of the variables involved, it is necessary to consider other concepts that help to study the behavior of the kinetic range of each system in comparison with the rest. As mentioned in Section “Overall kinetic range (KR),” these such other concepts are the kinetic range for a constant radius (KR_r), the kinetic range for a constant eccentricity (KR_e) and the kinetic range for a constant opening angle (KR_θ).

Kinetic range for a constant radius (KR_r)

We use this term to refer to the kinetic range of the system for a given value of the radius (r_o). In the geometrical representation we can define it intuitively as the intersection of the envelope that defines the overall kinetic range of the system with a plane parallel to the θ-e plane (horizontal in our graph), at the height of r_o. This intersection gives rise to a region, which acquires a tapered shape, bounded by two continuous curves. We will call them iCC_r curve of internal collisions for constant radius and eCC_r curve of external collisions for constant radius, respectively. Likewise, the intersection of this plane with the collision edge would account for the discontinuity vertices shown by this tapered shape (Figure 18). These observations would generalize the results published in the work by XXX et al.,¹⁸ showing agreement with the rest of the observations regarding these lines and other notable parameters, collected in that research.¹⁸

Figure 18.

Graphical representation of the kinetic range for constant radius (KR_r) in the four systems studied. Left: Intersection curves between the constant radius planes and collision surfaces of the four systems for the series of radius values, r = r₀. Right: Definition of the radial limits and kinetic range for r = r₀.

Figure 19 shows the values of the kinetic ranges for constant radius (KR_r), for a set of values of r = {L/40, 2L/40, 3L/40, 4L/40, 5L/40}. It shows that the FE2 system has a higher kinetic range for constant radius KR_r for radius values between r = 0 and r = 2·L/40, while EP1 has the highest KR_r in the range between r = 3·L/40 and r = 4·L/40. The EP2 and FE1 systems are those with the lowest KR_r, EP2 being the one with the lowest KR_r in the range between r = 0 and r = L/40, while the FE1 system is the one with the lowest KR_r in the range between r = 2·L/40.

Figure 19.

Values of kinetic ranges for constant radius (KR_r).

Kinetic range for constant eccentricity (KR_e)

We use this term to refer to the kinetic range of the system for a given value of the eccentricity (e_o). In the graphical representation we can define it intuitively as the intersection of the envelope defining the kinetic range of each system with a plane parallel to the r-θ plane (vertical in our graph), at the distance of e_o, which gives rise to a region that acquires a variable shape, delimited by a variable number of curves depending on the case, These curves are the result of the intersection of the plane located at the height of the corresponding value of e_o with the internal (iCC_e) and external (eCC_e) collision limit surfaces of the system and by one more straight line which is the intersection with the θ-e plane (Figure 20).

Figure 20.

Graphical representation of the kinetic range for constant eccentricity (KR_e) in the four systems studied. Left: Intersection curves between the constant eccentricity planes and collision surfaces of the four systems for the series of partial eccentricity values, e = e₀. Right: Definition of the eccentric limits and kinetic range for e = e₀.

In the graph of Figure 21 below we can see how the FE2 system is, in general terms, the system with the highest kinetic range for a constant eccentricity (KR_e), only slightly bettered by EP2 in the partial eccentricity range between e = 0.10 L and e = 0.15 L.

Figure 21.

Values of the kinetic ranges for constant eccentricity (KR_e).

Kinetic range for constant angle (KR_θ)

We use this term to refer to the kinetic range of the system for a given value of the opening angle (θ_o). In the graphical representation we can define it intuitively as the intersection of the envelope defining the kinetic range of each system with a plane parallel to the r-E plane (vertical in our graph), at a distance of θ_o, which gives rise to a region that acquires a sensibly triangular shape delimited by two straight lines, which we will call iCC_θ (the line that delimits the internal collisions) and eCC_θ (the line that delimits the external collisions), and by a third line that is the intersection with the θ-e plane. Likewise, we obtain an upper vertex. This point indicates the largest radius that can be available for that value of the opening angle (θ_o). But it should be noted that, for these radius values, internal and external collisions occur simultaneously (Figure 22).

Figure 22.

Graphical representation of the kinetic range for constant angle KR_θ in the four systems studied. Left: Intersection curves between the constant angle planes and collision surfaces of the four systems for the series of angle values, θ = θ₀. Right: Definition of the angular limits and kinetic range for θ = θ₀.

Figure 23 shows how the FE2 system has a higher kinetic range for constant angle (KR_θ) for angle values between θ = 10° and θ = 40°, while EP1 presents a higher KR_θ for angle values between θ = 50° and θ = 70°, with the KR_θ of the FE2 system again being higher at angle θ = 80⁰.

Figure 23.

Values of the kinetic ranges for constant angle (KR_θ).

Method 2. Using mathematical formulae

In this section we make an analytical approach to the most notable elements of the graphical representation. First, we will obtain the equations of the internal Collision Limit Surface (iCLS) and the external Collision Limit Surface (eCLS). Then we will indicate the formulae of the collision edge, the global kinetic range as well as the rest of the concepts associated to it pointed out in the previous section.

Collision boundary surfaces

We will obtain the expressions of the internal and external collision limit surfaces as a function of θ and E, iCLS (θ, E) and eCLS (θ, E) respectively.

We will start with the bar bundle systems, but we must study separately the moduli of square-plan systems and triangular-plan systems. In both cases we start from considering the expressions that provide the distance between two intersecting straight lines (s and t) in Euclidean space. These straight lines are the axes of the bars. We will also consider the points, O, P, and Q. The first one corresponds to the center of the knot (O), P is the point of s that is closest to t and Q is the point of t that is closest to s.

First, in the case of square plan systems, the axis of the bars is at a distance from the center point of the node (O) equivalent to the eccentricity of the system E = e + r, as shown in Figure 24. The straight lines s and t form an angle θ with the reference XY plane and are contained in two vertical planes perpendicular to each other.

Figure 24.

Geometric conditions at the deployment boundary for four-rod systems.

Let P and Q be the crossing points of s and t through the XY plane:

P (E, 0, 0); Q (0, E, 0)

(3)

Let $\vec{P Q}$ be the vector of the direction of the segment joining points P and Q, and let $\vec{u_{1}}$ and $\vec{u_{2}}$ be the unit vectors of s and t, respectively:

\vec{u_{1}} (0, \cos θ, \sin θ); \vec{u_{2}} (- \cos θ, 0, \sin θ)

(4)

Under these conditions the distance between two lines intersecting in space will be the mixed product of the three vectors $\vec{P Q}$ , $\vec{u_{1}}$ , and $\vec{u_{2}}$ , divided by the modulus of the vector product of $\vec{u_{1}}$ by $\vec{u_{2}}$ . The distance between the bars when contact occurs between them will be twice the radius (2r). Then:

2 r = \frac{| \vec{P Q} \cdot \vec{u_{1}} \vec{\cdot u_{2}} |}{| \vec{u_{1}} \times \vec{u_{2}} |} = \frac{E \cdot \sin θ}{\sqrt{\sin^{2} θ + 1}}

(5)

In the case of three-rods triangular modules, the formulation is analogous, it is only necessary to consider that the angle formed by the vertical planes containing s and t will be 60⁰ and not 90⁰ as in the previous case. (Figure 25)

Figure 25.

Geometric conditions at the deployment boundary for three-rod systems.

Let P and Q be the crossing points of s and t, respectively:

P (E, 0, 0); Q (- \frac{E}{2}, \frac{\sqrt{3}}{2} E, 0)

(6)

And let $\vec{u_{1}}$ and $\vec{u_{2}}$ be the unit vectors of s and t, respectively:

\vec{u_{1}} (0, \cos θ, \sin θ); \vec{u_{2}} (- \frac{\sqrt{3}}{2} \cos θ, \frac{1}{2} \cos θ, \sin θ)

(7)

In these conditions the distance is defined as in the previous case and will be twice the radius (2r).

2 r = \frac{| \vec{P Q} \cdot \vec{u_{1}} \vec{\cdot u_{2}} |}{| \vec{u_{1}} \times \vec{u_{2}} |} = \frac{E \cdot \sin θ}{\sqrt{\sin^{2} θ + \frac{3}{4}}}

(8)

The equations of the scissor systems FE1 and FE2 are formulated in a similar way, but considering that the rod s’, which forms the scissor together with the rod s, can be obtained by rotating around the Y and Z axis the straight-line s an angle of value π, respectively.

Meanwhile, point Q is obtained by shifting it in the positive direction of the X-axis with a translation vector:

\vec{w} = (L \cdot c o s θ, 0, 0)

(9)

Once we have defined the straight lines of the bars that compose one of the scissors, s and s’, we will construct the straight lines of the adjacent scissors t and t’ by rotating around the Z axis of the respective bars s and s’. Under these conditions we are interested in studying the collision between the straight-line t’ and the straight-line s.

The director angle of the system, being $φ = \frac{π}{3}$ for FE1, and $φ = \frac{π}{2}$ for FE2.

And the director vector of t′ will be: $\vec{v_{t^{'}}} = (\frac{1}{2} \cdot c o s θ, \frac{- \sqrt{3}}{2} \cos θ, - \sin θ)$ for FE1.

And : $\vec{v_{t^{'}}} = (\frac{\sqrt{2}}{2} \cdot c o s θ, \frac{- \sqrt{2}}{2} \cos θ, - \sin θ)$ for FE2.

Under these conditions, for FE1.

2 \cdot r = \frac{| s i n θ \cdot (- \sqrt{3} / 2 \cdot L \cdot \cos θ - 3 \cdot E) |}{\sqrt{2 \cdot \sin^{2} θ + 1}}

(10)

And for FE2

2 \cdot r = \frac{| s i n θ \cdot (- 2 \cdot E) |}{\sqrt{\sin^{2} θ + 1}}

(11)

Then the expressions of eCLS and iCLS, as a function of the opening angle θ and eccentricity E, are given in Table 1.

Table 1.

Equations of the internal and external collision limit surfaces (eCLS and iCLS) for each system under study.

System	Ecuation for eCLS and ICLS	Ecuation Number
EP1
External	$e C L S (E, θ) = \frac{(\frac{3}{4} \cdot L \cdot \cos θ - \sqrt{3} \cdot E)}{2 \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{\cos^{2} θ}{\sin^{2} θ}}}$	(12)
Internal	$i C L S (E, θ) = \frac{\sqrt{3} \cdot E}{2 \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{\cos^{2} θ}{\sin^{2} θ}}}$	(13)
EP2
External	$e C L S (E, θ) = L \cdot \frac{\cos θ}{4} - E$	(14)
Internal	$i C L S (E, θ) = E \cdot \frac{\sin θ}{\sqrt{\sin^{2} θ + 1}}$	(15)
FE1
External	$e C L S (E, θ) = (E - L \cdot \frac{\sqrt{3}}{3} \cdot \frac{\cos θ}{2}) \cdot \frac{3}{2} \cdot \frac{\sin θ}{\sqrt{2 \cdot \sin^{2} θ + 1}}$	(16)
Internal	$i C L S (E, θ) = \frac{E}{\frac{2}{\sqrt{3 \cdot \sin^{2} θ + 1}} + \frac{\sin^{2} θ + 1}{\sqrt{4 \cdot \sin^{4} θ + \cos^{2} θ}}}$	(17)
FE2
External	$e C L S (E, θ) = (E - L \cdot \frac{\cos θ}{2}) \cdot \frac{\sin θ}{\sqrt{\sin^{2} θ + 1}}$	(18)
Internal	$i C L S (E, θ) = \frac{E}{\frac{1}{\| \sqrt{\sin^{2} θ + 1} \|} + \frac{\sin^{2} θ}{\| \sqrt{(4 \cdot \sin^{4} θ - 3 \sin^{2} θ + 1)} \|}}$	(19)

As we can see, in all cases, if we assume that the value of θ is constant, the expression is reduced to the expression of a straight line whose independent variable is E, which allows us to verify that the surfaces obtained are indeed ruled surfaces.

Collision edge as a function of θ and E

To obtain the expression corresponding to the collision edge, it would be enough to equal the equations of iCLS and eCLS in each of the previous cases, since the collision edge is the intersection of both surfaces. The results are shown in Table 2:

Table 2.

Equations of the collision edge of the four studied systems.

System	Ecuation for the collision edge	Ecuation Number
EP1	$\frac{\sqrt{3} \cdot E}{2 \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{c o s^{2} θ}{s i n^{2} θ}}} = \frac{(\frac{3}{4} \cdot L \cdot c o s θ - \sqrt{3} \cdot E)}{2 \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{c o s^{2} θ}{s i n^{2} θ}}}$	(20)
EP2	$E \cdot \frac{s i n θ}{\sqrt{s i n^{2} θ + 1}} = L \cdot \frac{c o s θ}{4} - E$	(21)
FE1	$\frac{E}{\frac{2}{\sqrt{3 \cdot \sin^{2} θ + 1}} + \frac{\sin^{2} θ + 1}{\sqrt{4 \cdot \sin^{4} θ + \cos^{2} θ}}} = (E - L \cdot \frac{\sqrt{3}}{3} \cdot \frac{\cos θ}{2}) \cdot \frac{3}{2} \cdot \frac{\sin θ}{\sqrt{2 \cdot \sin^{2} θ + 1}}$	(22)
FE2	$\frac{E}{\frac{1}{\| \sqrt{\sin^{2} θ + 1} \|} + \frac{\sin^{2} θ}{\| \sqrt{(4 \cdot \sin^{4} θ - 3 \sin^{2} θ + 1)} \|}} = (E - L \cdot \frac{\cos θ}{2}) \cdot \frac{\sin θ}{\sqrt{\sin^{2} θ + 1}}$	(23)

Global Kinetic Range (KR)

According to what has been explained so far, to obtain the kinetic range of each system we have to calculate the volume enclosed by the corresponding iCLS and eCLS and the θ-E plane. Consequently, its expression can be obtained as the sum of the double integral of the functions of the internal and external collision surfaces, respectively:

K G = \int_{e_{i n t}}^{e_{m a x}} \int_{θ_{m i n}}^{θ_{m a x}} i C L S \cdot d (θ) \cdot d (e) + \int_{0}^{e_{i n t}} \int_{θ_{m i n}}^{θ_{m a x}} e C L S \cdot d (θ) \cdot d (e)

(24)

Kinetic Range for Constant Radius (KR_r)

To obtain the expression that determines the kinetic range for constant radius, KR_r, we must start from the expressions that define iCLS and eCLS and assume that the radius is constant (r = r_o). By doing this these surfaces would have been reduced to curves iCC_r(θ) and eCC_r(θ) respectively, and the KR_r would correspond to the surface of the region enclosed between them, so the corresponding expression would be as follows:

K R_{r} = \int_{θ_{m i n, r}}^{θ_{m a x, r}} e C C_{r} (θ) d (θ) - \int_{θ_{m i n, r}}^{θ_{m a x, r}} i C C_{r} (θ) d (θ)

(25)

The integral must be performed considering that the angle θ is the variable with respect to which we integrate. This integral will be defined in the interval between the minimum opening angle (θ_min) and the maximum folding angle (θ_max) obtained for this assumed constant value of r (Figure 26).

Figure 26.

Example of the region representing the kinetic range for constant radius (KR_r) in the four systems, for r = 0.025 L.

Kinetic Range for Constant Eccentricity (KR_e)

The value of the kinetic range for constant eccentricity (KR_e) can be calculated from the expressions defining iCLS and eCLS. By particularizing each of them, for a constant value of eccentricity (e = e_o), these expressions are assimilated to the equations of two curves iCC_e(θ) and eCC_e(θ), whose independent variable is the opening angle (Figure 27).

Figure 27.

Definition of the kinetic range for constant eccentricity (KR_e) in the four systems. For e = 0.05 L.

We can see here from Figure 27 that to calculate the Kinetic Range for constant eccentricity (KR_e), we can now reduce the equation to that shown below, in order to calculate the area of the region enclosed between these curves and the O θ axis:

K R_{e} = \int_{0}^{θ_{c o l, e}} i C C_{e} (θ) d (θ) + \int_{θ_{c o l, e}}^{θ_{m a x, e}} e C C_{e} (θ) d (θ)

(26)

Being,

θ_col,e the angle for eccentricity e at the point of intersection of iCC_e(θ) and eCC_e(θ), obtained by equating both expressions.

θ_max,e the limiting angle of folding of the system for eccentricity e.

Kinetic Range for Constant Angle (KR_θ)

The expression that determines the kinetic range for a constant value of the opening angle (KR_θ) can be obtained from the expressions defining iCLS and eCLS.

By particularizing each of them, for a constant value of the folding angle (θ = θ_o), these expressions are assimilated to those of linear equations of a straight line, so that the problem is reduced to calculating the area of a triangle (Figure 28). If we denote as iCC_θ(e) the line bounding the internal collisions and eCC_θ(e) the line bounding the external collisions for the set value of θ, we obtain the following expression:

K R_{θ} = \int_{0}^{e_{c o l, θ}} i C C_{θ} (e) d (e) + \int_{e_{c o l, θ}}^{e_{m a x}} e C C_{θ} (e) d (e)

(27)

Figure 28.

Definition of the kinetic range for constant angle (KR_θ) in the four systems, for θ = 30⁰.

Being e_col, the eccentricity of the system at the point of intersection of iCC_θ(e) and eCC_θ(e) (obtained by equating both expressions).

Considering the triangular shape of the area to be calculated, simplistically this equation is equivalent to the following:

K R_{θ} = \frac{e_{m a x} \cdot r_{m a x}}{2}

(28)

Developing this expression for all systems we obtain the equations of Table 3.

Table 3.

Equations of the KR_θ, for each system.

System	Ecuation for KRθ	Ecuation Number
EP1	$K R_{θ} = \frac{3}{16} \cdot \frac{\sqrt{3}}{8} \cdot L^{2} \cdot \frac{\cos^{2} θ}{\sqrt{1 + \frac{1}{4} \cdot \frac{\cos^{2} θ}{\sin^{2} θ}}}$	(29)
EP2	$K R_{θ} = \frac{L^{2}}{32} \cdot \cos^{2} θ \cdot (1 - \frac{1}{\frac{\sin θ}{\sqrt{\sin^{2} θ + 1}} + 1})$	(31)
FE1	$K R_{θ} = \frac{L^{2}}{16} \cdot \cos^{2} θ \cdot \frac{\sin θ}{\sqrt{2 \cdot \sin^{2} θ + 1}} \cdot (\frac{\frac{3}{2} \cdot \frac{\sin θ}{\sqrt{2 \cdot \sin^{2} θ + 1}}}{(\frac{2}{\sqrt{3 \cdot \sin^{2} θ + 1}} + \frac{\sin^{2} θ + 1}{\sqrt{4 \cdot \sin^{4} θ + \cos^{2} θ}} - 1)} - 1)$	(31)
FE2	$K R_{θ} = \frac{L^{2}}{8} \cdot \cos^{2} θ \cdot \frac{\sin θ}{\sqrt{\sin^{2} θ + 1}} \cdot (\frac{\frac{\sin θ}{\sqrt{\sin^{2} θ + 1}}}{(\frac{1}{\| \sqrt{\sin^{2} θ + 1} \|} + \frac{\sin^{2} θ}{\| \sqrt{(4 \cdot \sin^{4} θ - 3 \sin^{2} θ + 1)} \|} - 1)} - 1)$	(32)

Method 3. Using physical models

For this study we have produced a number of three-dimensional models from PLA filament using an Artillery SideWinder X2 3D printer with automatic ABL calibration and a Titan Direct Drive extruder working at 180-240 degrees. An additional feature of the models is that they incorporate a series of interchangeable sleeves that allow us to modify the radius of the bars quickly and easily in order to observe the effects of this variable (Figure 29).

Figure 29.

Left: constituent elements of the physical models, nodes (EP1 e = 5, EP2 e = 3, FE1 e = 7, FE2 = 6, top), bars (length L = 120, r = 1,5, centre) and sleeves (radii r = 6, 5, 4, and 3, bottom). Centre: elements used to build the models shown in the left column. Right: physical models of the EP1 (r = 4, E = 6,5), EP2 (r = 4, E = 4,5), FE1 (r = 4, E = 8,5) and FE2 (r = 5, E = 7,5) systems (from top to bottom). All units in mm.

By performing measurements on these physical models, we have been able to obtain a series of data that confirm the assessments made by the other two methods described above, with relatively acceptable tolerance margins (Table 4).

Table 4.

Comparison between the empirical results obtained by the three methods.

Table 4 shows in the first two columns of the first group (Method 1 = 2), boundary angles for methods 1 and 2. The third column shows the value of the height of the basic modulus of the system L.senθ, corresponding to θ_min, and the fourth column shows the value of the aperture of the basic modulus of the L.cosθ system, corresponding to θ_max.

In Method 3 group, range of measurements obtained on the physical models for the height (H) and width (W) of the modules, respectively. The values of L.senθ and L.cosθ are within the ranges for height (H) and width (W).

The results show a certain deviation between method 3 and methods 1 and 2. Whilst the first two methods are more consistent to each other, the third method is known to be subject to various errors such as manufacturing tolerances, accuracy in data acquisition, clearances in the knots and deformability of the physical parts, all of which are non-existent in the first two methods and which become quite clear to see when compared in the results table.

Notwithstanding, it is not entirely impossible to consider that with better conditions of model production (e.g. where there is a higher manufacturing quality), the results of method 3 could much more closely align with that of method 1 (algorithm) and method 2 (mathematical) if the impact of defects are minimized sufficiently enough. A possible solution to this problem could be the use of metal rods that reduces their deformation compared to plastic ones and more precise manufacturing techniques such as laser cutting. Indeed, manufacturing errors in the small-scale knots (due, among other things, to the stresses introduced by the relatively high temperatures needed for PLA fusion) would be eliminated when considering full-scale models or prototypes, where these imperfections are practically irrelevant compared to the overall size of the structure. Nevertheless, the errors obtained in the data collection of the physical models corroborate the results obtained in the other methods with a margin of error that we consider acceptable.

Another interesting feature evidenced more closely as a result of working with the physical models, is that the phenomena of restrictions in opening angles due to collisions, could for certain applications, serve as a useful feature when we understand that it is possible (with further research), to calculate the opening angle to a required or pre-defined “limiting force,” for use as a “device.” For example, rather than having to introduce a separate component such as a “mechanical strut” to limit the opening angle in a scissor structure. Nevertheless, such solutions would of course introduce stresses in these types of nodes and would probably only be valid for simple or lightweight structures.

Conversely, it is worth noting that in addition to the detailed considerations we have established regarding the collisions between bars, the phenomenon of collisions between the bars and nodes is also a possibility, however such issues are limitations under the scope and level of this study but could be a rich source of any extra angles of consideration to develop further lines of research. Such considerations could include the size, type, geometry, formation, and connection methods of the bars to their nodes and the overall various interactions that these additional parameters might reveal.

One of the most key limitations we can now confirm, is the need to maintain these studies to a purely geometric and dimensional scope, and leave aside secondary (at this stage) considerations of structural behavior (such as forces exerted, tension and compression in the bars etc.) of the assemblies used that otherwise would constitute a series of questions that relate to numerous other preliminary conditions which overall, may detract from the original focus of this study.

Conclusions

The results obtained allow us to establish a series of conclusions linked to the objectives initially set out. In relation to the first one, our study allows us to order the four systems studied, according to the kinetic range of each one. Thus, it is found that the FE2 system has the highest overall kinetic range closely followed by EP1. Thereafter and to a lesser degree follow EP2 and finally FE1. In general terms, this ranking also applies for the rest of the partial kinetic range parameters, but with certain point-worthy notes.

It should be noted in relation to Table 5 that both the bar and beam systems and their plan arrangements (triangular or square) are interspersed in the ranking.

Table 5.

Heatmap with classification of systems according to total and partial kinetic range where the darker colors equals the more favorable results.

In parallel, in relation to the second of our objectives, we have been able to verify that the magnitudes KR_e, KR_r, and KR_θ, defined in this research, allow us to complement in a valuable way the information provided by the kinetic range. Especially when it comes to establishing design criteria for the subsequent construction of these systems or adapting them to previously defined dimensional requirements.

Thus, considering the kinetic range for a constant value of the eccentricity (KR_e), we can observe the minimum opening angle and the maximum folding angle for a given value of the radius of the bars to be used. This is very interesting when considering the transport conditions and the possible size of the system once deployed.

Similarly, when considering the kinetic range for a constant radius (KR_r) value, we can select the knot size (depending on the eccentricity) and obtain the range of values for which the structure would work as a mechanism.

By considering the kinetic range for a constant value of the opening angle (KR_θ), we can establish a knot size (minimum value of the eccentricity) together with a value of the radius of the bars, for which the structure ceases to be deployable, limiting its own movement without the need for additional parts (stoppers), having a behavior closer to that of the so-called reciprocal structures, that is, the nodes move away from their ideal behavior as articulated joints that allow the free rotation of the bars.

Finally, we have also been able to verify that the methods used offer convergent and complementary results. The parametric method provides versatility and economy of material means, since it can be applied to various systems obtaining data in a massive way. The mathematical method allows a better understanding of the relationships between the data obtained by the previous method, although the equations obtained sometimes require the use of numerical tools for their resolution. Finally, the physical modeling method is the one that verifies and contrasts the results obtained by the previous methods and even allows us to perceive details of execution that escape them. allowing us to establish new research questions.

Thereafter, in terms of potential future lines of research, this study has found that the level of detail or resolution of the nodes could comprise further useful parameters—for example considering the geometry of the knot, as it might hold numerous correlations between itself, the kinetic range, the type of system, its geometry in plan and the number of bars making up the basic modules of the system.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This paper is embedded within the work of a broader study, which is part of a research project entitled “Application of Parametric and Generative Design for the Analysis and Optimization of Deployable Spatial Structures”, financed with public funds obtained as part of a competitive tender within the 2017–2019 period at UPCT (Polytechnical University of Cartagena). Funding was provided by UPCT (Universidad Politécnica de Cartagena. Grant no. PRIPRO_2017_2453).

ORCID iDs

Adolfo Pérez-Egea

Pedro García Martínez

Manuel Alejandro Ródenas-López

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The influence of dimensional parameters on the kinetic range of four systems of deployable structures with hinged nodes and rigid rods

Abstract

Keywords

Introduction

Objectives

Definitions and variables

Initial data and independent variables

Systems under study

Knots

Rod length (L)

Folding angle of the structure (θ)

Rod radius (r)

System eccentricity (E) and partial eccentricity (e)

Minimum representative sample

Internal and external collisions

Dependent variables

Minimum opening angle of the structure (θmin)

Maximum folding angle of the structure (θmax)

Folding width

Overall kinetic range (KR)

Methodology

Method 1. Using parametric design tools based on algorithms

Similarities between the graphs of the four studied systems

Comparison between the graphs of the four systems studied

Kinetic range (KR)

Kinetic range for a constant radius (KR r )

Kinetic range for constant eccentricity (KR e )

Kinetic range for constant angle (KRθ)

Method 2. Using mathematical formulae

Collision boundary surfaces

Collision edge as a function of θ and E

Global Kinetic Range (KR)

Kinetic Range for Constant Radius (KR r )

Kinetic Range for Constant Eccentricity (KR e )

Kinetic Range for Constant Angle (KRθ)

Method 3. Using physical models

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References

Minimum opening angle of the structure (θ_min)

Maximum folding angle of the structure (θ_max)

Kinetic range for a constant radius (KR_r)

Kinetic range for constant eccentricity (KR_e)

Kinetic range for constant angle (KR_θ)

Kinetic Range for Constant Radius (KR_r)

Kinetic Range for Constant Eccentricity (KR_e)

Kinetic Range for Constant Angle (KR_θ)