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integration through computationacadia 2011 _proceedings
This paper proposes and describes a new methodology for the design, fabrication,
and construction of unreinforced thin-shell stone vaulting through the use of
algorithmic form-finding techniques and multi-axis robotic water jet cutting. The
techniques build upon traditional thin-shell masonry vaulting tectonics to produce
a masonry system capable of self-support during construction. The proposed
methodology expands the application of thin-shell vaulting to irregular forms,
has the potential to reduce the labor cost of vault construction, and opens
the possibility of response to external factors such as siting constraints and
environmental criteria. The intent of the research is to reignite and reanimate
unreinforced compressive masonry vaulting as a contemporary building practice.
Keywords: masonry vaulting; robotic fabrication; water-jet cutting; multi-axis
fabrication; dynamic relaxation; file-to-factory, form-finding; self-supporting;
parametric modeling, computational design.
Maciej P. Kaczynski
University of Michigan
Studio Big Obvious
Wes McGee
University of Michigan
Matter Design
David Pigram
University of Technology Sydney
Supermanoeuvre
ABSTRACT
Robotically Fabricated Thin-shell Vaulting
a method for the integration of multi-axis fabrication processes with algorithmic
form-finding techniques
115
1 Introduction
Traditional vaulting techniques (both thin-shell and voussoir vaults) often achieved
efficiency by employing regular geometries, regular constituent parts, or both. Spanish
architect, Rafael Guastavino transplanted traditional Catalan vaulting techniques to the
United States and built hundreds of expansive yet efficient thin-shell tile vaults during
the late nineteenth and early twentieth centuries (Ochsendorf 2010). As a tectonic, thin-
shell Catalan vaulting can indeed achieve irregular geometries, but remained relatively
regularized in Guastavino’s body of work. Although historically present, asymmetrical
and irregular vaults have largely disappeared from contemporary construction due to
the relatively high cost of skilled labor and the complexity of necessary formwork/
guidework.
The research is founded on the ambition to reignite and reanimate unreinforced
compressive masonry vaulting and to expand its application through the addition
of multiple performance criteria. The authors intend to achieve these goals by [1]
increasing the practicality of thin-shell vault construction and [2] imbuing thin-shell
masonry vault tectonic with new formal possibilities beyond the regularity required by
traditional models. The research seeks to ask: can algorithmic form-finding techniques
when deployed in concert with multi-axis robotic fabrication reestablish thin-shell
masonry vaulting as a viable and desirable contemporary building technique?
2 Approach
This section describes the stages of the digital workflow and the panel design
process responsible for the creation of unique, non-standard panels. The stages
include [1] form-finding (dynamic relaxation method), [2] surface description (planar
rationalization), [3] robotic fabrication, and [4] self-supporting assembly.
The methodology incorporates custom-written form-finding software, which utilizes
the dynamic relaxation method to achieve viable funicular geometries. The workflow
involves the generation and simulation of cutting paths in 3D design software, as
well as the direct output of machine instruction codes. Paired with an open-source
fabrication script library, this workflow eliminates the manual translation of part
drawings into CAM software, increasing efficiency and providing direct feedback of
fabrication constraints.
The fabrication process is based upon the use of an abrasive water jet to cut unique
panels nested into standard stone slabs. The tectonic design strategy of the vault
utilizes a hyperbolic-paraboloid (ruled) surface joint that accurately positions each
component of the self-supporting assembly, minimizing the need for centering or
formwork. The twisting geometry of the self-aligning panel edges necessitates a 5
+ axis machine, which can tilt the orientation of the high-pressure water jet cutting
stream. The increase in fabrication cost is offset by the minimized formwork costs, a
decreased assembly time, as well as an increased formal adaptability that contributes
to potential gains in structural, thermal, and acoustical performance.
Thus far, the potentials of the research have been demonstrated through a full-scale
six-panel prototype constructed of two-inch thick Berea sandstone (Figure 1). A much
larger and non-uniform vault is the subject of current continuing research.
2.1 TECTONIC DESIGN
Before each of the steps of the digital workflow is described in full, it is worthwhile to
explain the structural behavior behind the design of the self-supporting panels. The
wedge-shaped blocks of a voussoir vault are arranged radially and achieve stability only
once the keystone is in place. The matching faces of the stone blocks within a voussoir
vault are typically planar and, given their radial distribution, would quickly collapse
without the help of robust centering. This research instead proposes a tectonic which
utilizes stone panels where many of the matching edges are constructed from twisting
(hyperbolic paraboloid) surfaces. The ruled edge-faces provide a counter force to
resist each panel’s tendency to rotate and collapse inwards.
Fig. 1
Figure 1. Assembled Self-supporting Stone
Prototype
fabrication and prouction techniques
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integration through computationacadia 2011 _proceedings
Figure 2. Interlocking Ruled Edge-faces of Stone Panels
Fig. 2
The edge-face geometry allows each successive course of vault panels to cantilever
from the previous without formwork. Each stone panel sits upon two panels from the
lower course. The cantilever of the upper course creates a moment force that tries
to rotate the panel out of position. This force is resisted by the shear reaction that
occurs due to the ruled edge geometry. Effectively, the panel can only be removed by
lifting it in the direction of its own major axis, and the self-weight serves to lock the
panel in place (Figure 2). As each course becomes more complete and ‘closes,’ more
panels act in strict compression, as each panel in the same course presses against its
neighbor, attempting to fall inward. This is always true in areas of synclastic curvature,
and the thickness of the stone means that it can also hold true in areas of moderate
anticlastic curvature.
The proposed tectonic shares characteristics with both thin-shell tile vaulting and
stone voussoir vaulting, acting as a hybrid of the two tectonic methods. It behaves like
a thin-shell Catalan vault, in that it can largely support itself when partially complete
during construction, therefore minimizing (in some cases eliminating) the need for
temporary formwork. But, like the wedge-shaped voussoirs of a stone vault, it supports
itself using gravity-induced friction rather than adhesion resistance or aggressive mortar
(as is necessary with tile vaulting) (Collins 1968). The major differentiating feature of
the proposed method is that it uses unique geometric components as compared to the
generic tile or voussoir stones of traditional vaulting techniques. Certainly, this hybrid
vault tectonic could be composed of regular geometry, but the research proposes
its use as a means to construct irregular vault geometries without extensive, digitally
fabricated formwork. In order to achieve this, the final vault form is embedded into the
uniquely prefabricated components, which self-align to generate the final, structurally
optimized vault geometry.
The ruled surfaces of the edge-faces of the stone panels, like any ruled geometry,
are comprised from a series of straight line segments gradually twisting relative to
one another (Pottmann et al. 2007). This geometric composition makes the edge-
faces ideally suited to the constraints of robotically controlled water jet fabrication, as
explained in section 2.4.2.
2.2 FORM-FINDING
As a formalized process, form-finding techniques have nineteenth century origins
in the work of Catalan architect Antoni Gaudi. Gaudi, and his twentieth century
intellectual compatriot, Frei Otto, carefully crafted intricate inverted models of catenary
nets and soap-film surfaces, respectively, to study and measure the effect of force
upon form. Once restrained to the computational capacity made possible through the
material properties of such models, form-finding techniques now enjoy much greater
flexibility, adjustability, and accessibility to engineers and designers with the aid of
computer calculation. Yet the art and practice of finding-form remains largely confined
to structural problem solving. Form in nature is dramatically more complex, resulting
from the interactions of myriad environmental constraints in continual feedback. Formal
solutions are found through adaptive responses to a rich network of influences rather
than from a singular input.
This research seeks to diversify traditional form-finding techniques beyond the
monoculture of structure by emulating nature’s iterative model through the addition
of design criteria, i.e., constraints. While acknowledging that the forms produced
by Gaudi and Otto were indeed instructed by criteria beyond structural performance
(programmatic, material, financial) these additional criteria were not applied in an
equivalently explicit manner.
In order to develop a structural system that maximizes material efficiency, custom-
written dynamic relaxation software was used to “form-find” the resulting double
curved surface geometry. While relaxation itself is far from novel, this implementation
(written in Java) incorporated a series of valuable additions tailored to its specific use
as a vault design tool. These additions were primarily interface related, designed to
allow for, and to take advantage of, realtime adjustments to parameters during the
form-finding process (Figure 3).
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Figure 3. Custom Form-finding Software Used for
Dynamic Surface Relaxation
Figure 4. Hexagonal Panelization Strategy
Fig. 3
Initial member geometry can be separated into a series of layers/groups, so that during
the relaxation each group of members can have its own slack-length varied via a series
of sliders. This functionality allows for the development of a sophisticated intuition as
to the complex emergent behavior of the system. It also allows for a nuanced control of
the overall form (while maintaining structural ‘purity’); an opportunity not usually enjoyed
in dynamic relaxation processes, which can tend to be overly initial-state dependent.
Along with realtime sub-system tuning via sliders, a series of different data-driven display
modes allow the designer to appreciate various aspects of the form-finding system
including: surface-normals, offset surfaces or volumes, and color codification based on
member extension, node connectivity, panel area and crease angles/curvature.
2.3 GEOMETRIC RATIONALIZATION
The resultant form-found mesh becomes the input geometry for the generation of
individual stone panel geometries by rationalizing a continuous surface into discrete
elements. At this stage of the workflow, the surface is broken into individual panel
geometries to prepare for robotic fabrication through the use of custom-written
software scripts. This process requires a sensitive and strict control over the surface-
description (rationalization) of the funicular surface geometry to reliably produce
appropriately oriented hexagonal panels. Although the described prototype utilized
hexagons, the tectonic strategy does not require that the panels be six-sided. Rather,
the primary constraint is to provide an arrangement of planar polygonal panels in
which the lower-courses provide restraining edge-faces to be matched by the edge-
faces from panels of the upper-course (Figure 4). In this case, two edge-faces of
a single upper-course panel match the edge-faces from two lower-course panels.
Hexagonal tilings provide exactly this arrangement, but it is not difficult to imagine
other tesselations that satisfy the general requirements.
Quarried stone is typically processed and distributed as sheet stock of standardized
thicknesses. The research accepts this constraint and prioritizes the preservation
of planarity during the panelization process while retaining the potential to construct
irregular surface geometries using standardized sheet stock as a material source. Cutting
planar panels from standardized stock achieves a greater material efficiency, with less
waste material, than panels with non-uniform thickness. Furthermore, the fabrication of
variable thickness panels would require multiple production operations, including sawing
and milling (described in the next section). Rather than machine volumetric stone with
multiple operations, the panels here achieve uniqueness (and thus the ability to describe
irregular curvature) from the varying geometry of the edge-faces.
Non-planar hexagonal surface-description strategies have already been employed
as a successful panelization technique to approximate irregular surfaces (Lachauer
et al. 2010). Unlike previous research, the proposed tectonic necessitates that the
hexagons describing the form-found surface retain strict planarity.
Fig. 4
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integration through computationacadia 2011 _proceedings
According to lower-bound analysis, the stability of an unreinforced masonry vault
requires that the path of compressive network forces lie within the section of
the panel geometry (Block and Ochsendorf 2008). Because the proposed planar
rationalization is an approximation of the form-found surface geometry, it implies
deviation from the ideal funicular surface and will include among other deviations, a
degree of stepping at the joints. The amount of deviation (tolerance) is an important
consideration during the panelization process in order to insure that the panel
arrangement retain the force network within its section (preferably the middle third
of the section) or otherwise lose equilibrium.
Ultimately, total deviations due to planarization can be reduced though a reduction
in panel size particularly in areas of greatest curvature. This, however, corresponds
to increases in fabrication and installation times and will likely complicate hexagonal
panelization. Alternately, increasing the thickness of the stone would in turn
increase the dimension of acceptable deviations including stepping; allowing for
larger panels. In any vault constructed following this proposed method, a balance
will need to be found which resolves these opposing demands.
2.4.1 ABRASIVE WATER JET CUTTING
The contemporary masonry industry already relies heavily upon CNC tools to
process stone. There are several methods by which natural stone slabs can be
processed, including CNC sawing, milling, and abrasive water jet cutting. Sawing
is generally used for thicker sections with straight cuts, where the diamond saw
blade produces the most efficient cut per unit cost of tooling. Milling holds the
advantage where partial depth machining is needed, and requires machines with
very high rigidity, which comes at a high cost of equipment. Abrasive water jet
cutting utilizes a high-pressure stream of water (60K psi) mixed with garnet abrasive
to erode material, producing a thin kerf. It is one of the most flexible of all machining
processes, as it can be used to cut a wide variety of materials and thicknesses. It
is especially useful for cutting brittle materials, as the absence of vibration prevents
crack generation. Additionally, the process generates very low reaction forces in
the part, and requires minimal fixturing to secure the stock to the machine bed.
The success of the research relies heavily upon the use of multi-axis robotic
water jet cutting to advance the design of stone fabrication beyond conventional
3-axis limitations. While research is being conducted on abrasive water jet milling
processes (those which can produce pocket or partial-depth cuts) in general, the
process is limited to full-depth cuts that pass completely through the material.
Due to the dynamic nature of abrasive water jet cutting, the process is considerably
more speed sensitive than milling or saw cutting. The kerf of the cutting path is
very thin, typically around 0.03-0.04”. Variations in speed can cause the kerf to
taper through the thickness of the material. The latest commercial control systems
have the ability to compensate for this taper using proprietary algorithms to control
speed and tilt of the nozzle based on geometry, even with 5-axis cutting paths.
2.4.2 ROBOTIC FABRICATION
In order to create the ruled geometry on the edge-faces of the sandstone panels,
a manipulator with five or more axes is necessary to control the abrasive water
jet stream. While five-axis CNC water jet cutting systems exist, their high cost
and programming complexity makes them relatively rare, primarily confined to
aerospace manufacturing where they are used for trimming composite parts.
Six+ axis robotic manipulators have also been used extensively in aerospace and
automotive manufacturing with abrasive water jet technologies. Serial kinematic
robots are uniquely suited to water jet cutting, as the process generates relatively
low reaction forces; robots cannot match the rigidity of gantry CNC systems for
machining. For an equivalent working envelope, robotic systems generally cost
around 50-70% less than gantry based CNC machines. Lower cost gantry CNC
water jet options are becoming increasingly available, and in applications requiring
tight tolerances they hold a distinct advantage.
Figure 5. Six Axes of the Kuka KR 100 and
External Axis (Rail)
Figure 6. Seven Axis Robot with Custom-built
Abrasive Water Jet Nozzle
Fig. 5
Fig. 6
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For this project a seven-axis Kuka KR 100 HA (high accuracy) robot was used to
manipulate a custom-built abrasive water jet nozzle for the stone cutting (Figures 5, 6).
The system utilizes an auto-loading end-of-arm toolchanger to rapidly switch between
various fabrication methods. Laser calibration has been performed on the system to yield
approximately 0.8mm error throughout the 10m x 5m half-cylindrical work envelope.
While industrial robots have been an active component in advanced manufacturing
processes for decades, only recently have advances in offline programming techniques
and software made them a viable option for custom component manufacturing. This
research forms part of a larger body of work related to the development of integrated
robotic fabrication methodologies.
For multi-axis water jet cutting, the preferred geometric input uses two driving curves,
extracted from the ruled edge-face of the panel. Custom scripts are used to generate the
toolpath as a series of positions along the upper curve, with the axis of the tool following
the ruling lines of the edge surface, while compensating for kerf width of the cutting
process (Figure 7). Suitable feedrates for the thickness and hardness of the stone are
chosen, as currently the scripts do not implement a geometry based feedrate control.
Toolpaths are then translated directly into KRL code (Kuka Robot Language). Because of
the additional degrees of freedom that exist in the robotic workcell, user input is required
at this stage to correctly pose the robot. It is important to note that the scripts are written
to be modular. The toolpath could just as directly be translated into 5-axis CNC code by
writing the correct postprocessor, which takes into account specific machine geometry
and programming requirements.
Robotic water jet cutting of ruled surface geometry in stone is not without precedent.
This fabrication strategy was employed to fabricate panel geometry for the curvilinear
marble wall that now resides in the backyard of Harvard University’s Graduate School of
Design. The tectonic strategy of the wall cleverly hides the post-tensioned reinforcement
cables behind the ruled edge-faces of marble panels (Bechthold 2009). The research
outlined here proposes a different use of robotically water jet cut ruled surface geometry
for structural performance during construction rather than act as a visual barrier.
2.5 ASSEMBLY
Due to the wedge geometry of the matching edge-faces, panel arrangement is self-solving
or self-aligning during construction. Simply placing the stone panel into its proper position
provides the necessary information for assembly. Unlike a generic brick, the unique stone
panel of the proposed tectonic contains its own set of implied instructions for placement.
Speaking about a similar self-solving tectonic strategy, Fabian Scheurer writes that “a pile
of components contains most of the [assembly] information, since it is embedded in the
individual shape of every single component. The effort to assemble the final structure is
much lower, provided that the [parts] are correctly numbered. The neighbors just snap
into place” (2008). Because every stone panel has only one predetermined position, it is
necessary for every panel to have its own unique label for proper identification. Here, the
process of cutting the unique ID into the panels is combined with the process of cutting
the panels from larger sheet stock, thus minimizing the possibility of mislabeling.
fabrication and prouction techniques
Fig. 7
Figure 7. Fabrication Sequence of a Water Jet Cut
Stone Panel
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integration through computationacadia 2011 _proceedings
Figure 8. Guidework for Thin-shell Catalan Vaulting
(Ramage et al. 2010)
So, while the six-panel prototype was assembled in a matter of minutes without the use of formwork or
grouting, it is very likely that larger prototypes will require some form of light-duty template ‘guidework’ to
ensure the overall form adheres to the funicular shape. Similar to thin-shell Catalan vaulting, the proposed
tectonic would likely benet from guidework despite its unique ability to self-align. A successful example
of the Catalan technique can be seen in the construction of the recently completed Mapungubwe
National Park Interpretive Centre in South Africa (Figure 8) (Ramage et al. 2010). Alternatively, advances
in the use of large-volume 3D measurement devices, like laser metrology systems, could provide a
means of rapidly checking the positions of panels during placement, prompting minor adjustments to
maintain the structurally dened form.
3 Findings
The previously described digital workow and fabrication processes were tested and evaluated through
the production of a full-scale portion of a vault in sandstone. Rather than using scaled models, the
prototype was designed to test structural and performance assumptions of the tectonic with specied
materials. Given the success of the six-part prototype, the proposed tectonic holds promise to be
propagated over a much larger surface for future iterations. Important ndings and lessons learned during
the construction of the prototype are included below.
3.1 STRUCTURAL PERFORMANCE
The self-supporting holding strength and frictional resistance of the edge-face geometry is inversely
related to the corner angle of the panel, that is, as the corner angle increases (as seen in the higher
courses), the matching edge-faces decrease in surface area, consequently applying less frictional
resistance. This is important because current surface rationalization schemes can produce hexagonal
panels with large corner angles in areas of greater curvature, possibly compromising the self-supporting
ability of the proposed tectonic in specic areas of some test vaults. Thus, a signicant and on-going
component of the research extending this method is related to surface panelization strategies that can
best mediate between the dynamic relaxation form-nding process and the ruled surface joint tectonic.
3.2 FABRICATION TOLERANCES
While the six-part stone prototype was assembled without complication, it is reasonable to assume the
importance of strict tolerances. The tectonic strategy proposed in this research requires very accurate
component tolerances. By denition, the nal structural form is intrinsic to the cutting of the hyperbolic
edge-faces of panels. Additionally, the accuracy of the edge-face cuts directly determines the ability
of the panels to provide self-support as courses are assembled. Angular deviations along the edge-
faces of panels will cause a number of problems. Poor matching between panels can lead to rises in
local stress and failure of the stone in compression. Compounding errors in the overall assembly could
produce a row of panels unable to ‘close’ a course. Such errors could also result in a structure that
deviates from the ideal form-found shape, which is critical for structural stability of thin-shell vaulting.
The robotic fabrication process used in this research, while accurate by serial kinematic standards,
cannot match the accuracy of gantry type CNC machines. The robotic system used for the fabrication
of the prototype has been calibrated to approximately 0.8 mm maximum spherical error. Combined with
kerf and stream lag errors, edge geometry can have local deviation on the order of 2 mm and +/- 2
degrees of taper. Advanced 5-axis CNC water jet machines hold tolerances on the order of 0.1 mm
volumetrically and 0.1 degree angular error, which is a signicantly smaller margin.
3.3 CONSTRUCTION CONSIDERATIONS
The self-supporting ability of this methodology is also expected to provide a useful advantage for the
subsequent construction of larger vault prototypes with irregular overall form. Regular surface geometries
comprising regular component parts can often be assembled with courses of panels lying neatly in
horizontally bands. This is not necessarily true of the rationalization applied to irregular (form-found)
surface geometries and the resultant panels. Therefore, the panels of irregular courses may not have as
much support from neighboring panels as the regularized equivalent. The self-supporting ability of the
panels is expected to compensate for any such irregularities.
Perfectly matching edge-face geometry would preclude the use of grout, providing an idealized load
transfer between stone panels. Instead, even fractional degree errors of the ruled edge-faces may
produce points of stress concentration. While small scale prototypes have already proven the viability
Fig. 8
121
of friction tting panels without grout, future prototypes will likely require the use of an adhesive bond or
grouting to provide full connection and load transfer between matching panels (Figures 9, 10).
3.4 MATERIAL LIMITATIONS
Theoretically, the orientation of the stone panels could be cantilevered as far as horizontal (potentially
past horizontal with frictional resistance). The primary limitation of the cantilever would then be the
compressive and shear capacity of the subject stone. It is conceivable that the complete vault could
be constructed from differing kinds of stone, with varying compressive strengths, built into appropriate
locations where necessary.
4 Conclusions
The proposed method of hybrid vaulting possesses the potential to reignite and reanimate unreinforced
compressive masonry vaulting as a contemporary building process. It does so because it attains a
greater practicality by minimizing the need for formwork or centering while simultaneously minimizing
the necessity for skilled labor, each contributing to potential cost reductions. The proposed tectonic
dramatically simplies the craft of assembly, which is essentially reduced to the task of identifying a
panel and placing it appropriately. Additionally, the timing of skilled labor becomes front-loaded with
programming and geometric coordination compared to the traditional model; back- loaded with highly-
skilled construction labor. Finally, the research imbues thin-shell masonry vault tectonics with new formal
possibilities beyond the regularity required by traditional models. By allowing for irregular thin-shell vault
forms, the method provides the possibility of formal responses to a greater set of performance criteria
and therefore potential gains in structural, thermal, and acoustical performance.
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fabrication and prouction techniques
Fig. 9
Fig. 10
Figure 9. Detail of Grout-less Prototype Panel
Joinery
Figure 10. Detail of Edge-faces Cut with the
Robotic Abrasive Water Jet
