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Machine knitting is a well-established fabrication technique for complex soft objects, and both companies and researchers have developed tools for generating machine knitting patterns. However, existing representations for machine knitted objects are incomplete (do not cover the complete domain of machine knittable objects) or overly specific (do not account for symmetries and equivalences among knitting instruction sequences). This makes it difficult to define correctness in machine knitting, let alone verify the correctness of a given program or program transformation. The major contribution of this work is a formal semantics for knitout, a low-level Domain Specific Language for knitting machines. We accomplish this by using what we call the "fenced tangle," which extends concepts from knot theory to allow for a mathematical definition of knitting program equivalence that matches the intuition behind knit objects. Finally, using this formal representation, we prove the correctness of a sequence of rewrite rules; and demonstrate how these rewrite rules can form the foundation for higher-level tasks such as compiling a program for a specific machine and optimizing for time/reliability, all while provably generating the same knit object under our proposed semantics. By establishing formal definitions of correctness, this work provides a strong foundation for compiling and optimizing knit programs.
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UFO Instruction Graphs Are Machine Knittable
Programming low-level controls for knitting machines is a meticulous, time-consuming task that demands specialized expertise. Recently, there has been a shift towards automatically generating low-level knitting machine programs from high-level knit representations that describe knit objects in a more intuitive, user-friendly way. Current high-level systems trade off expressivity for ease-of-use, requiring ad-hoc trapdoors to access the full space of machine capabilities, or eschewing completeness in the name of utility. Thus, advanced techniques either require ad-hoc extensions from domain experts, or are entirely unsupported. Furthermore, errors may emerge during the compilation from knit object representations to machine instructions. While the generated program may describe a valid machine control sequence, the fabricated object is topologically different from the specified input, with little recourse for understanding and fixing the issue. To address these limitations, we introduce instruction graphs, an intermediate representation capable of capturing the full range of machine knitting programs. We define a semantic mapping from instruction graphs to fenced tangles, which make them compatible with the established formal semantics for machine knitting instructions. We establish a semantics-preserving bijection between machine knittable instruction graphs and knit programs that proves three properties - upward, forward, and ordered (UFO) - are both necessary and sufficient to ensure the existence of a machine knitting program that can fabricate the fenced tangle denoted by the graph. As a proof-of-concept, we implement an instruction graph editor and compiler that allows a user to transform an instruction graph into UFO presentation and then compile it to a machine program, all while maintaining semantic equivalence. In addition, we use the UFO properties to more precisely characterize the limitations of existing compilers. This work lays the groundwork for more expressive and reliable automated knitting machine programming systems by providing a formal characterization of machine knittability.
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- PAR ID:
- 10635922
- Publisher / Repository:
- ACM
- Date Published:
- Journal Name:
- ACM Transactions on Graphics
- Volume:
- 43
- Issue:
- 6
- ISSN:
- 0730-0301
- Page Range / eLocation ID:
- 1 to 22
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.1145/3687948
