Revisiting Stewart–Gough platform applications: A kinematic pavilion

Stewart


Stewart-Gough platform applications
In 1955, an automobile stability device was built in Dunlop Rubber Co. [1], which was described thoroughly in 1962 by V.E.Gough and S.G.Whitehall [2].The system consists of two platforms, a fixed and a moving one, joined by six loading jacks, acting as legs, with gauges connected through universal joints at both ends.The device allows for a wide range of motion of the wheel mounted on the moving platform in order to investigate tyre wear and tear.
When in 1965 D. Stewart, who was working for Elliott Automation Co., tried to publish his work on flight simulators [3], Gough was one of the reviewers [4].It turned out that Stewart was unaware of Gough's and Whitehall's work, which is similar regarding the kinematic behavior but mechanically different, [1].The moving platform is supported by three articulated legs comprised of hydraulic jacks.Each one of those legs is constrained by another jack, which is connected to the body of the legs creating a shape similar to a 90 • counter-clockwise rotated letter .
At the same time with Stewart in 1964, an engineer from the Franklin Institute, namely K. Cappel, filed a patent for a helicopter flight simulator, which was approved in 1967 [4] similar to Gough's and Whitehall's device.In this case, the arrangement of the six legs and two platforms creates an octahedron.
Gough, Whitehall, Stewart and Cappel established the basis of the 'Stewart-Gough platform' (SGP) research topic within the field of parallel coupled 6-DOF manipulators [4], even though their theoretical background dates back centuries [5].Since then, a wide range of SGP applications has been presented from various scientific fields mainly due to their accuracy, rigidity and adaptability [6].Typical configurations of the leg 1 arrangements of the SGP can be classified as 3-3 and 6-3, as shown Fig. 1.
More specifically, Morán et al. [7] modified the original SGP by restricting the linear motions along  and  axes to reduce economic costs.The modification resulted in a system with four degrees of https://doi.org/10.1016/j.engstruct.2021.113304Received 26 January 2021; Received in revised form 12 September 2021; Accepted 28 September 2021 freedom (DOF), instead of six.The purpose of the work was to develop a rotating platform able to recreate the disorientation effects of the pilots due to centripetal accelerations, which result in optical illusions.Along the same lines, Copeland et al. [8] built a full-scale prototype flight simulator for pilot training.The cockpit consists of a sphere based on Mecanum wheels without joints, which provide unbounded orientation workspace, while the translational actuation is provided by an SGP.In another attempt towards motion simulators, Advani et al. [9] suggested an SGP to replace the current gimbaled system in the Vertical Motion Simulator for NASA.Furthermore, Campos et al. [10] developed an active helideck system that allows helicopters to land on floating structures (e.g.ships) by limiting the movements of the landing platform based on a 3-6 SGP.Additionally, Ross et al. [11] used an SGP to simulate the motion of a ship stern in order to provide surge, pitch, heave and roll to test a zero visibility autonomous landing algorithm for unmanned aerial vehicles (UAV).
Liu and Wiersma [12] used a 3-3 SGP to support patient's head during radiosurgery in order to minimize the exposure of normal tissue to radiation [13].Furthermore, Akcali et al. [14] suggested an SGP as an external fixation for bone fractures.The proximal and distal fragments are attached to two ring platforms connected with six legs [15].Sevillano et al. [16] suggested a 3-3 SGP as a human gait simulator.The device is used to study the human gait to treat patients from motor injuries.In the same manner, Kalani et al. [17] used an SGP to simulate the human jaw system.Yang et al. [18] used an SGP miniature as active microsurgical manipulator.The device involves linear motors that provide a 4 mm by 4 mm workspace [19].Along the same lines, Wapler et al. [20] suggested a 3-3 SGP for precision surgery.In order to achieve positioning accuracy of 10 μm the authors suggest a teleoperated SGP operated by a man-machine interface system.Similar devices have been suggested for skull surgery by Kobler et al. [21], where the SGP is attached to the patient's skull to improve accuracy.Furthermore, Park et al. [22] suggested the use of SGP systems to correct the posture of the human spine.The authors used two 3-3 SGP systems in series with independent control of force and position.
Reinoso et al. [23] used parallel platforms to construct a climbing robot around tubular structures, such as pipes, bridge cables, towers and trunks.The six legs of the 3-3 SGP connect the two circular rings and allow climbing outside or inside tubular structures.Galvan-Pozos and Ocampo-Torres [24] suggested a new wave energy converter based on the SGP concept.The authors propose a 6-3 SGP, where three rigidly interconnected floats form an equilateral triangle, namely the upper platform, which is connected to the fixed base.Each leg consists of a linear electrical generator, which driven by the motion of the upper platform due to wave motion, produces energy.Another application in marine structures is the floating marine platform studied by Horoub et al. [25], which forms a 3-3 SGP connected to the seabed with six mooring cables allowing for three DOF motion.
Stewart-Gough platform has also been used in radio-telescopes to control the position of its secondary mirror [26,27].More precisely, Yingjie et al. [28] tested a feed positioning system for a 500 m aperture radio-telescope.The system comprises three components, namely a cable system, a trolley hanged on the cables and an SGP mounted on the trolley.The SGP acts as a vibration isolator mainly from wind disturbances, isolating the trolley-cable system from the lower platform, see also [29].Likewise, Premont et al. [30] suggested an active isolation system for space applications.The system consists of a 3-3 SGP with active isolated legs, to avoid low amplification at resonance and large attenuation at high frequency provided by passive systems [31,32].Iqbal et al. [33] studied the dynamics of SGP subjected to moving payload to investigate the behavior of satellite antennas with movable dishes.
Richter et al. [34] proposed the use of an inverted 3-3 SGP, which can be mounted to a crane to create a suspension and positioning tool on the construction site.The legs consist of six cables and the upper platform comprises three telescopic arms.Additionally, Salah et al. [35] used a similar SPG with cable-legs to build an automated storage and retrieval system for warehouses, see also [36].In another application for the construction field, Kim and Bernold [37] used a 3-3 SGP for handling and joining heavy pipe segments to increase safety and accuracy.Despite the hard control interface compared to the conventional electric winch and cable method, the SGP system provided sufficient accuracy during the pipe manipulation.Stewart-Gough platform has also been used for material handling and characterization [38,39].Additionally, Hansen et al. [40] proposed a mobile 3-6 SGP for machining components of the wind industry in a flexible process.The machining unit provides the possibility of different configurations to reduce its workspace.

Stewart-Gough platform as an architectural object
The historical review implemented in the previous section, puts some light on the diverse application possibilities of SGP within a wide variety of fields.In common use, an SGP is a parallel robot that consists of a mobile rigid platform, which is connected to a fixed base by six articulated legs as illustrated in Fig. 1, frequently investigated in literature with well-established design [42][43][44].Each of the legs, which typically have the same kinematic structure, is connected to the platform and the base by universal joints at points that are distributed in a geometrically symmetric way [22].Therefore, the legs are commonly arranged in symmetric constellations of (isosceles) triangles forming 3-3 or 6-3 configurations as shown in Fig. 1.In this paper such leg-arrangements are, not least from an architectural point of view, referred to as regular designs.In contrast, irregular designs are referring to leg-arrangements with all legs skew.Regular designs characterize the majority of application-related setups, which usually aim at a maximum workspace [45] and for ensuring the desired stability and functionality [46] of the mechanism such as achieving predefined paths, and to reach certain points or workspaces.
In the field of parallel robots the terminology of architecture, structure, modeling, and design is predominantly used to describe the components, joints and configuration of such systems and also to highlight the differences between different approaches such as parallel and/or serial robot settings with different DOF's [47].Inhere, architecture is referred to as both, the process and the product of conceptualizing, designing, planning and constructing structures with the aim to fulfill practical and expressive requirements, and thus it serves both utilitarian and aesthetic ends.Structure is a key aspect of architecture as a load-bearing, functional requirement and as an aspect of its safety.In this context architectural works, in the material form of built structures or as works of art, are often perceived as cultural symbols and expression of society.
Nowadays, we are on the move from a purely static thinking about architecture towards a more dynamic perception and idea of our built environment.In the light of present environmental and societal challenges, dynamically changing systems, building envelopes, and spaces beyond the scale of technical equipment can be of interest for architecture and engineering.
Kinetic architecture [48] and transformable design, adaptive and adaptable architectural spaces [49] can be seen as emerging topics in the field of innovative architectural and structural design [50,51].The concept of buildings that can reposition parts of their structure to change their appearance, to create different usable spaces or to respond to environmental conditions is of interest to architecture for long time.The ''velarium'' [52] of the Colosseum in Rome, Italy can be considered an early example of kinetic architecture.In 1924, architect Konstantin Melnikov proposed a five-story building, four floors of which revolve around a fixed core, in a competition for the construction of the Moscow branch of the Leningrad Pravda newspaper, [53,54].Other trends focus on spatial optimization [55], on touristic attractions [56], and on multi-functional design [57] by applying transformability to different scales from mobility concepts [58], to spatial configurations [59], to furniture.
During past decades, advancements in mechanical, electrical systems and robotics are opening up new frontiers with a focus on responsiveness and sustainability aspects.The integration of sensing and responsive technology systems pervades the architectural discourse.Selfregulating technological systems can regulate temperature, light, energy, safety, and other conditions.Even though the Villa Girasole [60,61] was probably the first building to follow the course of the sun, recent projects increasingly focus on the buildings' envelope.Pioneering examples include the Institut du Monde Arabe, 1987, [62], the Al Bahr Towers in Abu Dhabi [63][64][65] and the Penumbra system, 2014 [66], among others.Conversely, there exists a long genealogy of buildings that uses transformative and mobile elements to control the same set of comfort factors by means of analogue control compared to their abovementioned high-tech counterparts.These kinetic elements regulate the environmental conditioning of the interior as a fundamental design objective.The House With Balls, 2009, the Social Pavilion, 2006 [67], the Snowcone, 2015 [68] and the Shiver Pavilion, 2015 [69], among innumerous other projects can be seen in this context, where at a scale of a panel or a wall slide, fold, pivot and transform the building climatically and visually.
Even though the combination of sensors, processors and effectors of the movable, responsive or adaptable elements transform the perception and understanding of architecture from static into actively changing configurations [70], the mechanism itself and as it changes over time has only recently come into the focus of interest and consideration [71].The well-known, geometrical principles for stable trussed bearing structures which can also be folded by Santiago Calatrava [72] and the expanding structures of Chuck Hoberman [73] in the field of kinetic art have triggered new applications in engineering like tensegrity bridges [74] and scissor systems [75], in robotics and deployable mechanisms [76], in material sciences and nanotechnology [77] and in computational geometry and motion processing [71] among other fields.
Altogether, most of the mentioned projects and examples mainly or exclusively focus on the effects of the mechanisms.Conversely, mimicking the visual appearance of a 6-6 SGP with its separated, pinned legs and their irregular, skew arrangement can be found in architecture in several projects [78][79][80][81], among others.However, SGPs have not yet been explored much as large-scale structures and architectural design applications, and besides kinetic art, the creative discipline of architecture can open new avenues by the paired aspects of functionality and aesthetics of the transformation of structures.In this regard, the 2-Landscapes pavilion [82] that explores the transformability of spaces  and the associate movement of the aluminum roof panels by combining two layers of elastic grid-shells by an irregular 6-6 SGP, is the only architectural structure of its kind known to the authors.

Goal and overview
Our research aims to explore SGP as an architectural object and its coalesce structural performance, unlocking its functional and aesthetic potentials for these disciplines accordingly.More specifically, we suggest a kinematic pavilion as an architectural object of a 6-6 SGP.Our study and realized pavilion is inspired by [82][83][84][85] and therefore considers the mechanism as an architectural object from both perspectives: (i) as a constantly transforming, architectural space and (ii) as a structural system with changing geometry.
We have provided a historical review of SGPs in a wide variety of fields and we have explored SPG as architectural object.In Section 2 the project of the kinematic pavilion is presented, while in Sections 3 and 4 , the equilibrium equations along with the kinematics of the system are explained.In Section 5, the driving factors of the design process towards the decision for the telescopic leg selection of the kinematic pavilion are presented in more detail.The final conclusions of the design study are presented in Section 6.

The Zero Gravity pavilion project
The Zero Gravity pavilion (Fig. 2), emerged from a design-build [86] and a multidisciplinary co-creation process [87,88], and serves as a showcase considering the realized SGP as an architectural object that is constantly transforming the architectural space.The transformation is initiated by a human-structure interaction, which changes the role of the visitor to the role of an actively influencing user.Simultaneously, the pavilion is a structural system with changing geometry, which is among other requirements a key aspect for the design process.

Concept and design
Elegantly moving structures seem to visually overcome gravity.Along these lines, the design process of the Zero Gravity pavilion was initiated by firstly exploring scaled, physical models as shown in Fig. 3, and later by the parallel use of computational models for architectural purposes and structural evaluation.Two major, cyclical design phases -(i) finding a 6-6 starting configuration (pose) and (ii) finding the most appropriate leg for being actuated for the specific pose -were made for finding a multiobjective solution, which is technically and structurally feasible, architecturally suitable and aesthetically pleasing.The predominantly considered parameters for our design phases encompass (i) structural considerations including the stability of the structure, the distribution of the roof's selfweight on a nearly circular arrangement of the legs' top ends, the force distribution and the change of forces during motion, the speed of the motion (dynamic effect) in dependence of distance of legs' top ends, collision of the structure and self-collision of legs, (ii) architectural requirements characterized by functional aspects including the visitor flow and place-making, the clearance below the roof and between the columns, the size of the realized structure, the magnitude (workspace) and speed of motion of the roof, the transition and inclination of columns, (iii) aesthetic aspects such as the generated architectural open space at the base plane, the type, path and speed of motion, the human-structure interaction by the sensor-activated motion of the pavilion, and (iv) economic aspects characterized by the general low-budget resulting in decisions such as having two legs on the same support (I-beam), one actuated leg only, which is easy to produce and to maintain, and the overall size of the realized pavilion.
In a first design phase, the Stewart-Gough 3-3 configuration (Fig. 4(a)) was rearranged to an irregular 6-6 configuration (Fig. 4(b)).The initial 3-3 design was assembled from 3 pairs of legs with equal lengths and inclination, thus forming a stable structure from isosceles triangles and providing a platform (roof) parallel to its base.Along above-mentioned architectural, aesthetic and economic parameters, the 3-3 SGP structure was stepwise decomposed and modified to a 6-6 configuration by horizontal shifts and neither changing the legs' lengths nor the direction vector of the legs.All these horizontal shifts do not create any motion on the platform or the base.This way, the changes in the leg arrangement resulted in a stable overall structure, which was checked mathematically by computing the determinant of the rigidity matrix.(see Appendix A).Contrary, this increased the architectural freedom in the arrangement of leg elements and design.The 6-6 configuration (pose) as illustrated in Fig. 4(b) was slightly modified (Fig. 4(c)) to meet requirements such as nearly circular arrangement of the legs' top ends or having two legs on the same support (I-beam).Finally, the legs' lengths were marginally adjusted and only one leg was slightly rotated for architectural reasons, resulting in a satisfying 6-6 starting configuration (pose) as shown in Fig. 4(d).Consequently, the structure was checked against stability and collision in the starting position and during motion.
Early in the design process, it was decided to opt for only one of the six articulated legs as a linear telescopic actuator for several reasons including the following main aspects: (i) Due to clearance, a subject of safety, the roof was supposed to use a comparably limited workspace restricted to a linear actuation of +/− 0.50 m.
(ii) A 5+1-set-up of legs offers the possibility to separately investigate the motion of the platform depending on the actuated leg.Our selection   of a specific actuated leg is based on structural consequences and architectural requirements, visually appealing motion and transformation, and structural performance as described in Sections 5 and 6.
(iii) Linearly actuated by one leg, limited in length change, the platform follows a certain path that is part of a curve with degree 20.In the realized structure, the constant back and forth motion of the roof is not perceived by the viewer as repetitive.
(iv) One linearly actuated leg asks for one single needed motor that can be located in any of the six legs.Compared to linearly actuated columns, fixed length columns are cheap and easy to produce and to maintain.This has a positive impact on the cost, weight, and design of the leg elements.
The question of self-collision like leg interference is of interest and an important aspect in the design of such a kinematic pavilion.Usually, it is the aim to find self-collision-free realizations over the complete motion cycle, which is quite a challenging task [71].Even though there are many approaches for search strategies for finding a self-collision free setup, [76,89,90], the collision-avoidance of our structure is based on scaled physical models and on two separate computational models, see Section 5.
Even though this paper does not aim to provide a stepwise design procedure for other designers, altogether, the finally selected configuration resulted from an iterative exploration of the architectural and structural design space, as explained above.Most of the design-driving parameters can be rationalized, and their results can be compared and evaluated, supporting the decision-making process towards the final project.In addition, the design process also includes creative, intuitive and artistic aspects usually described in literature as the black box design process [91].However, also the creative process of architectural design is usually considered cyclical and iterative [92], as it narrows down the possibilities until all architectural drivers have been addressed in the creative design process to a feasible and meaningful solution.
At this crossroad, the Zero Gravity kinematic pavilion [41], see Fig. 2, was designed, developed, built and exhibited by the multidisciplinary team of Aalto University Structures and Architecture at the Väre building, Aalto University, Espoo, Finland in 2019.

Realization
The pavilion consists of three main parts, (i) a moving platform, namely the lightweight roof, (ii) six articulated columns, five of which with fixed lengths plus one linearly actuated column, and (iii) I-beams on the fixed base-plane (floor), serving as benches and as temporary heavy-weight foundations for the columns, Fig. 5.
For the realization of the full-scale structure we used thin 6.5 mm plywood for the central part of the roof with its cantilevering beams (Fig. 6) [93].The lateral stability and rigidity of the cantilevering part of the roof was achieved by a triangulated rope pattern (Figs. 2, 5), later replaced by a net-structure, see Fig. 8 right.These modifications were done for exploring spatial qualities during motion, visualized by the suspended ''hair'' and the ''falling leaves'' Fig. 9, but their shape generation and artistic meaning are not within the scope of this paper.This way, the Zero Gravity pavilion covered an area of about 40m 2 with a total weight of 150 kg for the roof.Thin unprocessed birchwood trees with a diameter of about 70 mm served as columns, see Fig. 7 left.Spherical joints were attached at both ends of the birch trunks, see Fig. 7 left, and the lightweight roof was connected to the columns by fork elements from steel, Fig. 7 right.One of the columns was executed as telescopic leg, by means of a servomotor-driven, metric screw.The screw provided the possibility to extend and to shorten the initial length of the column by 0.5 m, which represents a total change of length of 1 m.Metal connectors and spherical joints, see Fig. 7 right, were used for the manufacturing and the assembly process, which was supported by architecture and engineering students from Aalto University.Finally, the bottom ends of the columns were connected to steel I-beams, which functioned as both, the temporary heavy-weight foundation of the pavilion and as benches for the visitors.

Human-structure interaction
The motion of the structure allowed for an intended human-structure interaction of the Zero Gravity pavilion, by use of components from the Arduino open-source electronics platform [94], which controlled the sensors, the spotlights and the motor, which actuated the motion of the structure.The full cycle of motion was divided into six equal sequences, corresponding to the amount of cantilever beams of the roof.The sensors and spotlights were attached at each cantilever beam's free end (tip), see Fig. 8 right.When approaching the pavilion, the structure was standing still, but the visitor moreover the user found a randomly switched on spotlight.By stepping into the spotlight, the sensors recognized the user and actuated the first sequence of the structure's motion, but only if the user would stay within the range of the sensor.At the end of each sequence, the spotlight was switched off.The user could activate the next sequence of motion by stepping again into another randomly switched on spotlight and so forth.In this way, an either/or-process was initiated, which practically means that either the user or the structure would be in motion.Consequently, the user would explore the architectural space by moving through and around the structure, or the transformation of the space would be visible for the static user.The only way to by-pass the system and explore the transforming structure by simultaneously moving around it, is by working as a team of two or more users.However, the spotlights served as a guiding system, in the sense that after having activated all sensors and sequences by stepping into six consecutively switched on spotlights, the user has basically explored the space during a full cycle of motion from different viewpoints.The randomness in activated spotlights guaranteed that the user would experience different combinations of motions and standpoints, even after several visits.

Equilibrium equations
The system presented herein is a statically determinate system and therefore the six equations of equilibrium are enough to calculate the internal forces in each of the six columns by assuming that the roof behaves as a rigid body, Fig. 10, [95].It should also be noted that the motion of the structure is very slow and therefore the behavior of the system is assumed as static.
A column element  in space is defined by points  and  + 6, see Table 1 and Fig. 10, with unit vector ⃗  () , length vector ⃗  () , while its length is  () , see Fig. 11, Appendix A. The force vector ⃗  () of element  shown in Fig. 11 can be written as: where  () is the magnitude of internal axial force of element .The equilibrium equations of all normal forces in , ,  directions can be calculated as follows: where  = 150 kg is the resultant of the weight of the roof, which can be written in vector form as: The three additional equations are provided by the moments, which are calculated around point 7, namely top of column 1 see Fig. 10 by the following expression: where ⃗  1 is the position vector at the roof level and is defined by Eq. (A.4).
The equations of equilibrium regarding moments around axes , ,  are calculated as follows: The equilibrium equations are written in matrix form: where , ,  are defined in Appendix A. The system is solved with the linsolve function in MatLab: Fig. 11.An element  in space.
The system of Eq. ( 11) is linear and has unique solution when the determinant of the rigidity matrix  is different from zero, () ≠ 0.

Kinematics
In this section the kinematics of the SGP are solved.As stated by Husty [42], when only six distance measurements are available, the direct kinematic problem of parallel manipulators has forty solutions, some of which might be complex.Therefore, when solving the direct kinematics of SGP, a process that defines the unique solution of the platform configuration, should be followed.Along these lines in the current study, scaled models along with computational models have been used to define the starting configuration (pose) of the platform parallel to the base.As a first step, by using the inverse kinematics the leg lengths of the pavilion at the starting pose are defined.In the next step, only one length of the pavilion's legs changes very slightly, while the rest of the legs have constant length.In this way, the pose of the platform will not change drastically.Therefore, the so-called nearby pose of the platform is derived.Thus, we have a starting value when trying to solve numerically the new position of the platform.Consequently, we obtain the unique solution of the direct kinematics of the platform (as far as the determinant of matrix  is different from zero).All further changes of the leg length use the previous pose of the platform as a starting value of the numerical solution process.The computation of the determinant of matrix , the Jacobian matrix, has shown that none of the computed poses is close to a singularity (determinant of  is equal to zero), which means that no other solution of the direct kinematics is close to the obtained pose of the platform.
To solve this pose problem the approach presented in [42] in a slightly modified version is applied.In this approach, the displacement matrix is parametrized using Study coordinates.First of all two coordinate systems are attached to the roof (  ,   ,   ,   ) and the base (  ,   ,   ,   ) of the pavilion model.The coordinates of all points (base and roof points of the columns) in the base coordinate system are presented in Table 2. Additionally, the initial coordinates of the tips of the cantilevers are presented in Table 3 in the base coordinate system.The origin of the moving (roof) coordinate system lies at point (0,0,3) of the base coordinate system.The transformation matrix  between the moving and the fixed coordinate system in homogeneous Study coordinates is: The eight coordinates are homogeneous and linked by the so called Study equation: Following [42] the six distance equations can be written where   are the coordinates of the base attachment points of the legs (nodes 1, … , 6 in Table 2).The coordinates  +6 can be computed from Table 2 Initial coordinates of top and base points of column members (Fig. 10) in the base coordinate system (  ,   ,   ,   ).where  +6 are the coordinates of the moving nodes in the moving coordinate system (x,y coordinates of nodes 7, … , 12 in Table 2).Eqs. ( 15) are quartic equations in Study coordinates (  ,   ,  = 0, … , 3), but can be reduced without loss of generality [42] to quadratic equations by adding 4 2 .

Elements
where  () can be computed from   and  +6 from Table 2 and this yields In this case,  (4) is the leg which will be the telescopic one and therefore, a parameter  is added to its length.When other legs will be the telescopic ones, the parameter  will be added to their length.To solve the pose problem a set  of eight equations is produced from Eqs. (17) and  as follows: The parameter  0 can be set  0 = 1 without loss of generality because the Study coordinates are homogeneous and  0 = 0 would lead to an upside down assembly of the framework which is definitely not wanted for the architectural purpose.This system of equations completely solves the pose problem of the mechanical system.Especially it would allow to compute all the assembly modes.In the case of the pavilion this is not necessary.First of all it is easy to see that the numerical solution of  for the starting pose  = 0 is obtained by  which results in the transformation matrix Substituting e.g. = 0.05 into the system  and numerically solving with the starting value  0 = 1,  1 = 0,  2 = 0,  3 = 0,  0 = 0,  1 = 0,  2 = 0,  3 = −1.5 yields  0 = 1.000000000,  1 = −0.001669497310, 2 = 0.008559319407,  3 = 0.01475519880,  0 = 0.02211326466,  1 = −0.04031998574, 2 = 0.07689857451, with the transformation matrix and the position of the first node can be computed via where  17 ,  27 ,  37 represent the moving coordinates, ,  and  respectively, of point 7 in the moving coordinate system.Similar computation for the other nodes yields their coordinates in the base frame and it can be easily checked that the leg length constraint is fulfilled for all of them.Using these coordinates the determinant of the stiffness matrix  can be computed which is far from zero for all values of  and therefore all these poses are stable configurations of the framework.
The problem of kinematics was solved in MatLab, [96].Additionally, a geometric model was developed in Rhinoceros/Grasshopper environment, [97].The comparison of the kinematics between the model developed in MatLab and the one developed in Rhinoceros/Grasshopper, is shown in Fig. 12, which clearly demonstrates that the two models provide identical results.Finally, the validity of the current approach is proven by the comparison of the kinematics of the Rhinoceros/Grasshopper model with the Zero Gravity pavilion, shown in Fig. 13 at different time instances.

Kinematic pavilion application
In this chapter we investigate the behavior of the Zero Gravity system in order to establish the choice of the telescopic leg based upon four criteria: stability range, self-collision and leg interference, force distribution and path trajectory.

Stability range
In order to investigate the stability of the SGP, the behavior of matrix  of Eq. ( 11) is checked.If the determinant of matrix  is equal to zero, it follows that the system is not stable and behaves as a mechanism.On the other hand, if the determinant of matrix  is different than zero, then the system has only one solution and it is stable.According to that, Fig. 14 shows the stability range of the six telescopic columns individually, see also Table 4.More precisely, columns 2 and 6 have the shortest allowable length variation among the six columns, respectively −0.71 m and +0.72 m.Columns 1 and 5 have nearly symmetric span for elongation and shortening.More specifically, for column 1 the span is between +1.78 m and −2.24 m and for column 5 the span is between +1.77 m and −1.53 m.The ratio between the length elongation and the shortening is approximately equal to two thirds for columns 3 and 4. For column 4 the maximum length elongation is +0.94 m, while for column 3 it is equal to +1.86 m.Due to construction restrictions and in order to have all columns as a candidate option we limit our study to the length variation range between +0.50 m and −0.50 m.
Figure 15 shows the variation of the determinant of matrix  for each telescopic column.The largest variation of () occurs in columns 2 and 6.Additionally, as shown in Fig. 15 column 2 has a clear trend to instability through shortening, and column 6 through elongation.Columns 4 and 5 indicate a nearly symmetrical transition of (), which is a sign of balanced behavior.Finally, columns 1 and 3 show a trend to instability through elongation.

Self-collision and leg interference
Due to the above-described restricted length variation of the linear actuation and limited workspace consequently, self-collision and leg interference could easily be checked by moving our scaled physical models [71].Furthermore, A parametric and kinematic 3d-model was set up in the Rhinoceros/Grasshopper environment [98], which included all real dimensions and geometries (profiles) of the elements of the later realized structure, and which served for the architectural design.The later realized path of the platform and the motion of the columns were subdivided by splitting the 1 m range of the actuated column into 0.1m-steps.The closest points between the individual columns, as well as the distance between the roof and the columns were observed visually by the computational analysis.Additionally, a computational model set up in Matlab [96], provided a numerical analysis over the motion cycle of the structure.Like in the parametric model, the motion was split into small steps and the geometry was checked against collision accordingly.

Force distribution
During the length variation of the telescopic columns, the change of force distribution for each column is investigated.The results are presented in Fig. 16 and Table 5.In the case of telescopic column 1, the force variation of all columns is almost constant.The largest variation occurs in column 2, while column 5 is almost unloaded, Fig. 16(a).In the case of telescopic column 2, a wider range of force variation is observed, Fig. 16(b).More specifically, in column 3 a force variation of 0.68 occurs, while in column 2 the range is equal to 0.55 with load reversal from compression to tension.In addition to column 2, a load reversal occurs in columns 3, 5 and 6.In the case of telescopic column 3, the largest force variation occurs in column 3 itself and is equal to 0.82 , Fig. 16(c).In columns 2, 3 and 5 load reversals from compression to tension can be observed.In the case of telescopic column 4, the largest load variation occurs in column 4 itself and is equal to 0.31 , Fig. 16(d).In column 5, load reversal occurs from compression to tension.In case of telescopic column 5, the largest load variation occurs in column 5 itself and is equal to 0.56 , Fig. 16(e).Additionally, columns 2 and 5 exhibit load reversal.In the case of telescopic column 6, the largest load variation occurs in column 3 and it is equal to 0.37 , while load reversal occurs in columns 2 and 5, Fig. 16(e).Due to the fact that the roof of the structure is lightweight, the buckling of the columns was not considered.

Path of kinematic structure
The path of the motion of the SGP was validated by using different computational tools, namely MatLab [96] with the method described in previous sections.Additionally, Rhinoceros/Grasshopper [97] with Kangaroo plug-in [99] within the Rhinoceros environment [98] was used.The results of the path trajectory are presented in Figs.17,18,19,20,21,22, where the elongation step for all figures is 20 cm.Apart from the path trajectory, the distance between the lowest point of the roof and the virtual top-plane of all I-beams, namely the clearance, is investigated, see Table 6.The clearance shown in Table 6 accounts also for the height of the I-beams (30 cm), which act as foundation of the pavilion.This way, the project provided enough clearance for the visitors on the ground plane and even when standing on the I-beams.Telescopic columns 2 and 3 exhibit clearance of 1.71 m and 2.01 m, which is not acceptable.Columns 1 and 5 exhibit dominant rotational motion, while column 6 exhibits dominant translational motion along the vertical axis.In column 4, both rotational and translational motion along the vertical axis can be observed.This type of motion is more observable to the visitors of the pavilion, taking into consideration that the motion of the structure is very slow in order to reduce the dynamic effects and to increase the visitor's experience.

Selection of telescopic leg
The selection of the telescopic leg of the pavilion is based on the four aforementioned criteria.More specifically, the stability range criterion investigated the stability of the global, structural system of the SGP, depending on the length variations of the six telescopic legs, but separately for each of them.In addition to clearance issues (safety) and due to construction limitations (length of used screw) the length variation of each column was restricted to ±0.50 m.Leg interference and self-collision was thoroughly checked through the computational and physical models for each case separately.All telescopic columns were providing stable configurations within these limits, with columns 2 and 6 having the bounds closest to the limit.Columns 2 and 3 were excluded due to the insufficient clearance, see Table 6.Additionally, columns 2, 3 but also 5 were excluded due to undesirable load reversal.The load range distribution of column 5 is 0.56 P and it is larger compared to column 1 (0.15 P), column 4 (0.31 P) and 6 (0.37P).Therefore, the choice was to be made between columns 1, 4 and 6.The advantage of column 4 compared to columns 1 and 6 is that the motion is more expressive/visible and not limited as in the case of column 1.More specifically, the path trajectory of telescopic column 4 involves a balanced motion between rotational and translational movement around the vertical axis, compared to telescopic column 1, where the rotational component is dominant and to the telescopic column 6, where the translational movement is dominant.In the variation of force distribution in telescopic column 1, column 5 is almost unloaded, while in case of telescopic column 4, the force variation of column 5 is 3 times larger.Finally, due to the above-mentioned reasons, the telescopic column 4 was selected in the Zero Gravity pavilion realization.

Conclusions
In the current study, an SGP application of a kinematic pavilion is presented.After reviewing a wide range of SGP applications in various fields, SGP as an architectural object is explored (Section 1).The Zero Gravity structure was built in Espoo, Finland and its design   and fabrication processes are explained (Section 2) along their decision making criteria (Section 5), which are of architectural, functional and structural nature, after establishing the equilibrium equations (Section 3) and the kinematics (Section 4).More specifically, the presented case study is based on an irregular, 6-6 configuration derived from the Stewart-Gough 3-3 configuration.In this sense, the Zero Gravity pavilion represents one of many initially feasible options.Its irregularity is rooted in architectural decisions, whereas the increased freedom in arrangement of columns is used for the functionality including visitor flows and place-making, and it is accompanied by structural stability and strength.Together with the decision for having only one linearly actuated leg these aspects come at the expense of the maximum workspace of the structure.But contrary to an SGP that adjusts all legs simultaneously (parallel manipulator), the Zero Gravity pavilion demonstrates an increased freedom in architectural design paired with a reduction in self-weight and costs, for the fabrication, installation and maintenance of the leg elements.Technically, configurations like the Zero Gravity pavilion set-up result in a complex sequence of translation and rotation on their paths, but with reduced elaborate programming and system-control.The visually appealing motion of the roof and the   To present, the general formulation of SGP has been mostly used as a theoretical model for mathematical and geometrical description, but has not been instrumentalized much in an applied sense in architecture.By applying it in an architectural scale, where the precision of SGP is not the primary goal, but at the same time other parameters are also focused on and new application possibilities can be opened up.Furthermore, SGP principles can help develop the built environment  from static to mobile, flexible, adaptable and transformable structures and architectures including change of position, volume and adaptability of inhabitable spaces.In addition, sustainability aspects like heating, cooling, ventilation, light supply, space consumption will benefit from this approach, which are subject of our ongoing research.
Finally, it is worth highlighting that the Zero Gravity pavilion contributes to the field of experimental and innovative architecture from several perspectives.The project emerged from design-build process and a multidisciplinary co-creation setting in all stages from conceptual design to realization [100,101].Therefore, the architectural space and spatial experience was iteratively shaped by a creative team of structural engineers, mechanical engineers, and architects [86,88,102,103].The main contribution for architecture is seen in the shift of the SGP from a mechanism that facilitates motion, to a mechanism that The unit vector ⃗  () of element  is given by: where  () is the length of element  defined as follows: +6 −  ()  ) 2 + ( () +6 −  ()  ) 2 + ( () +6 −  ()  ) 2 (A.3) The position vector ⃗  1 at the roof level from the top of column 1 to the top of column , is obtained by the following expression: ⃗  1 = ( ()  +6 −  (1)  +6 ) ⃗  + ( () +6 −  (1)  +6 ) ⃗  + ( () +6 −  (1)  +6 ) ⃗  (A.4) The external force vector  is defined as: where the point  corresponds to the point of application of the load  .The vector  of the unknown internal forces in the columns is defined as: (1)   (2)   (3)   (4)   (5)   (6) The matrix  is defined as given in Box I:

Fig. 3 .
Fig. 3.A preliminary scaled model of the Zero Gravity project.

Fig. 4 .
Fig. 4. Design and concept development of the Zero Gravity project in top view.A 3-3 configuration (a) from 3 pairs of legs arranged in an isosceles triangulated form being decomposed and rearranged to an irregular 6-6 configuration (b) by horizontal shifts of the 6 legs only.Marginal modifications (leg lengths, leg rotation) due to architectural and structural requirements such as nearly circular arrangement of the legs' top ends (c), resulting in a satisfying 6-6 starting configuration (pose) (d).

Fig. 6 .
Fig. 6.Six cantilevers before assembly (left) and central grid structure of the roof (right).

Fig. 7 .
Fig. 7. Spherical joint detail of column (left) and fork detail that connects the roof with the column of the pavilion (right).

Fig. 8 .
Fig. 8. Installation of Arduino controlled sensor (left) and spotlights at the tip of the roof's cantilevers (right).

Fig. 10 .
Fig. 10.Structural system of the Zero Gravity pavilion, numbers in squares (blue) denote leg elements/columns, the rest (red) denote nodes such as pin joints from the legs or free ends (tips) of the cantilever beams and point  denotes the point of application of external load  .

Fig. 12 .Fig. 13 .
Fig. 12. Validation of the kinematics by the numerical model developed in MatLab through the one developed in Rhinoceros/Grasshopper.

Fig. 15 .
Fig. 15.Variation of determinant of matrix  for each telescopic leg.

Table 1
Number of nodes of column members , where  denotes the base point of the column and  + 6 the roof point, see Fig.10.

Table 3
Initial coordinates of cantilever members and point of application of the external load aP, see Fig.10.

Table 4
Stability range of telescopic column members.

Table 5
Extrema forces in  for each column for each telescopic column case.

Table 6
Clearance of the system for each telescopic column by adding the 30 cm of the I-beams where the columns are attached.