Robotic Fabrication of Parametric Chairs

I just finished the design and fabrication of a series of chairs. The chairs were made by hot wire cutting Expanded Polystyrene (EPS) foam using the 7-axis robot at Taubman College of Architecture at the University of Michigan.

To do the cutting the robot holds a rigid frame with a wire stretched between the ends. A voltage is applied to the wire which heats it up allowing it to move through the foam. A spring is used to allow the wire to stretch a bit as it cuts.

Here's a a video of the cutting process. The first section shows the cross sections of the chair being cut. Those cross sections are glued together and then additional operations are done. The video is sped up by a factor of 3 or 4  depending on the operation shown.


For me, the question was – how can a straight line be used to achieve pleasing, comfortable curvature in the design of the chair? To understand the answer requires a bit of study of ruled surface geometry.

Ruled Surface Geometry - Definitions
• Ruled Surface: A ruled surface is a surface swept out by a straight line as it moves through space. For example, a cylinder is formed by moving a straight line around a curve in a plane, keeping it perpendicular to the plane at all times.

A cone is formed by moving a line so that it stays fixed at one point.

A helicoid is formed by moving a straight line along another straight line, keeping it perpendicular but rotating it as it moves.

• Doubly Ruled Surface: A surface is doubly ruled if through every one of its points there are two distinct lines that lie on the surface. The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces.

• Hyperboloid:  A doubly-ruled surface generated by a set of straight wires, whose ends span two parallel circles rotated relative to one another.

Parametric Chair Development
I began by creating a basic chair form I liked by manually modeling the chair section curve in Rhino. This is the main generating factor for the chair. The section can be edited to alter the shape of the chair. Thus the section is another parameter.

Next I decided what properties of the chair were going to be parameters. The principal ones are:

Seat Height, Seat Angle, Seat Center Depression, Back Height, Back Angle,
Back Pattern Scale, Cutter Angle, Cutter Bulge, Cutter Height, Cutter Width, Cutter Z Start

Variations
A variety of chairs – these all have the same initial cross section. Only the parameters were altered to create these variations:

Two chairs with surface variations applied to the back – this creates a stark contrast from front to back and makes for some dramatic ruled surfaces:

Parametric Model - Grasshopper
 I developed the parametric chair using Grasshopper. Grasshopper is a graphical algorithm editor integrated with Rhino’s 3D modeling tools. All the modifications are done using simple geometric transformations: translation, rotation, scaling.

A variety of user-interface controls are available. I used all sliders - here are a portion of them used to affect the chair:

Shaded model – typical of what’s seen while adjusting the model in Grasshopper:

Shown below are the original generator curve (green) and the final section curves:

Grasshopper also does an initial layout of the parts on the foam block. The block is correctly position in world space for cutting on the robot:

The parts are manually rotated to optimally position them on the foam for cutting:

Additional geometry is used to make the cuts possible by removing material that would hit the frame. Also additional cuts are used to free the parts from the foam.

Mastercam/Robotmaster Setup
The next step is to import the geometry into Mastercam/Robotmaster and to establish the sequence of cuts and position of the robot during cutting. It takes experience to know how to rearrange the position of the arm during cuts to make it work. The motion of the robot can be simulated to test if interference will happen. The robot has no knowledge about the shape of the tool it is holding. Therefore it is critical that I carefully verified the paths in the simulation prior to cutting. I needed to make sure the frame never hit the robot nor touched the foam block in areas which hadn't been removed yet.

Fabrication
First the four section pieces are cut.

These are then glued up using polyurethane glue.

After that cures the chairs are placed on a vacuum table and extra cuts are made to trim the sides and holes. It's necessary to raise the chair on blocks so the robot can reach all the way to the bottom of the chair.

Completing the second side cut:

Here the robot is preparing to make the hole cut:

Here are the chairs after all cuts have been made. These are just prototypes - to make a fully functional chair they'd need to be covered in a more durable material. They are strong enough to sit on and test however.

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UMMA Installation - Photoformance

I recently completed working on an installation at the University of Michigan Museum of Art. Here’s a link to the web site for it: Photoformance: An Empathic Environment, and here's a review: UMMA's 'Photoformance' installation finds several top talents exploring notions of 'skin'. 

Shown below are the base surfaces designed by Monica Ponce de Leon and Maciej Kaczynski. My role was to help write the computer code to triangulate the surface, generate "tabs" so the triangles could be fastened together, and lay them out flat so they could be fabricated.

Here is the result of our computer code – a triangulated surface. This is a crude computer rendering – the actual material is lighter in color.

Next, all the triangle were "unrolled" flat so they could be cut on the CNC router. The final step of the code, written by Maciej, was adding the details to each tab so they can be folded, and attached to one another. There are 2072 parts – every one unique – and all carefully numbered!

This is the material they are produced from – 0.02” thick plastic. The blue is a peel off cover over each sheet.

Here are the parts being cut on the CNC router:

They come off the router quite dusty and need to be cleaned: 

Here parts are folded up and assembled into groups of six triangles:

These get transported to the site…

… and assembled…

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Surfacing With Planar Polygons

The last project in the Digital Fabrication course at U of M was to panel a surface. Out team choose planar polygons as the method. The team was myself, Bob Bradfield, Bruce Findling, and Clay Montgomery.


Here's the surface that we paneled. To get an idea of scale the gray base is 60"x24".

Here are a few photos of the resulting surface paneled in 24 gauge steel:

Each piece is a tabbed, water jet cut polygon. The under side was left to rust. The upper surface has been sanded to a bright finish:

The tabs are perforated where they meet the polygon to allow easy folding. This makes interesting patterns along the edges:

 Each polygon is spot-welded to the tabs of its neighbors.

Algorithm
This section discusses the technique used to generate the polygons and prepare them for fabrication. The entire process was generated using code written in RhinoScript (essentially Visual Basic for Rhino).


The first part of the algorithm is to generate a single polygon. This is accomplished one vertex at a time. The user chooses a surface to panel and enters the number of sides for the polygons. They also enter a maximum allowable deviation from the surface. That is, what is the maximum amount that the planar polygon is allowed to rise above or fall below the curved surface.
To get a vertex the code steps along a straight line on the surface and measures the distance along the way. As soon as the distance from the surface is exceeded that point is fixed.

Doing this through 360 degrees generates a planar polygon with the specified number of sides. In this case the polygon is fairly symmetric because it was chosen in an area of relatively uniform curvature. 

Next, we need to generate more polygons. This is done by projecting out from the center of the original polygon through the midpoint of each edge. The user can specify the amount as a ration of the polygon radius. From the projected point a new polygon is generated.

This is done through each edge of the source polygon. And here's the result. You can't build that!

So the code checks for intersections with other polygons. It won't generate points inside another polygon and also snaps to the edges of existing ones.

The propagation continues in this manner. Note that the polygons are also clamped to keep them relatively uniform - you can see this as some of the points lie outside of the actual polygon that was generated. 

With the points not shown the paneling looks like this:

Sample Surfaces
Here are a few sample surfaces:

Merge Panels

The next step is to merge the panels so they are pulled tightly together. The step iterates over every vertex of every polygon and compares it to the other vertices. It moves the vertex a portion of the way closer to the nearest vertex. It does this a user specified number of times. You can see that number listed in the diagram below. By the 25th iteration the polygons are merged closer together to prevent overlap and give more area for welding adjacent polygons. 

Remove Coincident and Collinear Points

The next operation is to remove extra vertices which are coincident or nearly so. Also removed are verts which are colinear or nearly so. This is done so wider tabs are generated. 

Generating Tabs

The next step is to generate the tabs on the sides of the polygon which allows them to be connected to one another. They are generated using an offset surface. 

 The surface is offset more towards the center of the span and less at the edges:

The technique starts by finding the closest point on the surface to the midpoint of each edge. The vector between the midpoint and the closest point is copied over to the corners of the edge and lengthened. These vectors are then intersected with the surface. The intersection points are shown as yellow dots below:

This is done for each edge of the polygon. The verts are connected by lines which form the bottoms of the planar tabs. However we want to make adjacent tabs match up at their corners.

To do this we calculate the intersection of each edge with the adjacent edge. Of course two lines in space do not necessarily intersect at a point. So we find the closest line segment which connects the lines and then take the midpoint of that line. Those points are shown as green dots below.

Those new corner points are then used as the ends of the tabs. Doing this introduces a very small break from planarity however it is insignificant in practice.

The resulting polygon is shown in yellow. The blue lines indicate the tabs after being rotated into a plane. This closed polygon is then used for fabrication on the water jet. 

Layout for Fabrication
The next step is to generate tabs for every polygon and project them onto the ground plane so they can be cut from the water jet. Here the polygons are shown on each tab:

 These are oriented onto the ground plane and arrayed. Here are the 158 polygons in a 10"x10" grid.

Here's a close-up look at each polygon. The tabs are a single closed polyline. The original planar polygon has been redraw as a perforated line. The water jet cuts these perforations for easier folding. 

Each polygon is group together for easier dragging inside Rhino. This makes it easy to generate the sheets for cutting on the water jet.

Finally, each polygon is numbered on the 3D model. These correspond to the numbers on the sheet to guide fabrication.

Fabrication

Fabrication begins by cutting sheets of 24 gauge steel on the abrasive watejet cutter. The parts are then cut from the sheets with a dremel tool and organized into piles of 10 polygons, 0-9, 10-19, etc. 

This is the first polygon generated by the code. It is nearly symmetrical because it was unaffected by intersection with other polygons and was positioned on a portion of the surface with nearly uniform curvature. 

Here's the spot welder used to attach the polygons together: 

Wooden forms were used to aide in the shaping the curvature of the span as well as where the form meets the ground. 

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