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# Freedom and constraint topologies

Freedom and constraint topologies (a.k.a., freedom, actuation, and constraint topologies).[1][2][3] is a design approach, most commonly referred to by the acronym FACT, that uses a complete library of specialized vector spaces, which are graphically depicted by intuitive geometric shapes, to guide designers in synthesizing the topology of flexible systems. Flexible systems are devices, mechanisms, or structures that deform to achieve a desired set of objectives. Examples include compliant mechanisms, precision flexures, soft robots, and mechanical metamaterials (a.k.a., architected materials) that achieve engineered properties according to the collective deformations of their constituent flexible elements.

## FACT's geometric shapes

The FACT design approach utilizes a comprehensive library of geometric shapes that embody the mathematics of screw theory, linear algebra, projective geometry, and exact-constraint design.

One set of geometric shapes within the FACT library are called freedom spaces[1][2][3]. Freedom spaces consist of sets of rotation lines, screw lines, and translation arrows, which represent all the ways a flexible system could deform with high compliance (i.e., they geometrically represent the combination of the system’s degrees of freedom). The motion lines that constitute freedom spaces are modeled using twist vectors.

Another set of complementary geometric shapes within the FACT library are called constraint spaces[1][2][3]. Constraint spaces consist of pure force lines (a.k.a., constraint lines), wrench lines, and moment lines that guide designers in knowing how best to arrange flexible elements (e.g., wire, blade, and notch flexures) within the topology of a flexible system so that the system is properly constrained to only be able to deform with the degrees of freedom represented by the constraint space’s complementary freedom space. The lines that constitute constraint spaces are modeled using wrench vectors.

Another set of geometric shapes, which are similar to the constraint spaces within the FACT library, are called actuation spaces[4][5]. Actuation spaces consist of pure force, wrench, and moment lines, which are also modeled using wrench vectors, but these spaces guide designers in knowing how to arrange the best number and kind of actuators within a flexible system to actuate or load it so that the system successfully deforms with the desired degrees of freedom represented by its freedom space.

## FACT design approach

By obeying simple rules, designers can rapidly navigate through the complete solution space of flexible system designs, embodied by the FACT library of geometric shapes, to consider the most promising topologies of any configuration that achieve a desired combination of degrees of freedom. The rules of FACT vary depending on the configuration of the flexible system desired.

All flexible systems can be organized according to three primary configurations – parallel, serial, and hybrid. Parallel systems [1][2][3] consist of two rigid bodies connected directly together by parallel flexible elements. Serial systems [5][6] consist of two or more parallel systems stacked or nested in a chain from one rigid body to the next. Hybrid systems[7] consist of any other configuration of parallel and serial system combinations. Interconnected hybrid systems[8] are a special kind of hybrid configuration where intermediate rigid bodies are also interconnected together by flexible elements, which create internal loops within the system that don’t include (i.e., pass through) the grounded body. Such configurations require graph theory in conjunction with the traditional principles of FACT to analyze and design[8].

The innumerable variety of flexible elements (i.e., the springs that deform to guide the motions of the rigid bodies that they join) are themselves organized within parallel, serial, and hybrid configurations. The constraint characteristics of these elements also require special rules to analyze and design that depend on whether they are parallel[9], serial[10], or hybrid[7].

The most basic FACT rules (i.e., systematic steps) for synthesizing parallel flexure systems that consist of wire flexure elements include the following:

Step (1): Determine what degrees of freedom you’d like the system to achieve. In the example shown, we wish to achieve the three orthogonal intersecting rotational degrees of freedom shown as red lines and the single translational degree of freedom that points in the same direction as one of the three rotations as shown by the black arrow.

Step (2): Identify from the FACT library which freedom space depicts every permissible motion that results from the combination of the degrees of freedom identified in the first step. The correct freedom space will result from the linear combination of the twist vectors that mathematically model the specified degrees of freedom. In the example shown, the freedom space is the space labeled 1 in the 4 DOF column of the FACT library. It consists of a sphere of all rotation lines that intersect the same point as the intersection point of the three desired rotational degrees of freedom specified in the first step. It also consists of a plane of red rotation lines that contains that intersection point and is perpendicular to the translational degree of freedom’s black arrow. There are also many green screw lines in the space, but these lines aren’t shown in the figure to help avoid visual clutter. Note also that the freedom space shown is oriented differently from the freedom space labeled 1 in the 4 DOF column of the FACT library.

Step (3): Use the FACT library to identify the constraint space that is complementary to the freedom space identified in the second step. The correct constraint space is the space provided on the right side of the freedom space shown in the FACT library. In the example shown, the constraint space is a single disk of blue pure force lines (i.e., constraint lines) that intersect the same point as the center of the freedom space’s sphere and lies on the same plane as the red plane of rotation lines in the same freedom space.

Step (4): Use the constraint space identified in the third step to select the desired number of flexible elements and arrange them within the final flexible system by aligning their axes (in the case of wire flexures) with the blue constraint lines in the constraint space. Each constraint space in the FACT library contains instructions (in the form of sub-constraint spaces[11]) about how many flexible elements are necessary to select from their geometry and how to select them so that the resulting system will be exactly-constrained (i.e., contains independent flexible elements that each perform the unique job of constraining a single unwanted degree of freedom). If it is desired that the system be over-constrained (i.e., possess redundant constraints that don’t change the desired degrees of freedom of the system but stiffen and increase its load capacity by using extra dependent flexible elements), designers can select any number of wire flexures beyond the number selected to exactly-constrain the system and they can arrange the redundant wires anywhere within the constraint space as long as the wires’ axes align with the blue constraint lines in the space. In the constraint space of this section’s example, at least two wires must be selected within the disk such that their axes each align with two blue constraint lines that are not colinear as shown to exactly-constrain the system. If more than two wire flexures were selected, the resulting system would still achieve the desired degrees of freedom but it would be over-constrained and would possess redundant wires that perform the same job of constraining the unwanted motions outside of the system’s freedom space.

Step (5): Once the number, kind, and arrangement of flexible elements has been determined, the final step is to design the parallel system’s two rigid bodies and determine which end of each element connects to which body. The two bodies can be made into any shape as long as they remain rigid throughout (i.e. they do not deform appreciably compared with the flexible elements when the system is loaded) and each body must connect to each flexible element but at only one end of each element. In the example of this section, two possible ground designs are shown. Note that regardless of which body is held fixed as the system’s ground, the other rigid body will be able achieve the four desired degrees of freedom specified in the first step.

The rules of FACT have been extended to also enable the design of flexible transmission mechanisms[12], flexure systems that achieve decoupled actuators[13], and mechanical metamaterials that achieve desired directions of compliance[14]

## History

The FACT design approach was created in 2005 by Jonathan Brigham Hopkins while a Master’s student in Professor Martin L. Culpepper’s Precision Compliant Systems Laboratory at Massachusetts Institute of Technology. FACT was first published in a short conference paper in the 2006 proceedings of the 21st Annual Meeting of the American Society for Precision Engineering[15], but was later published in depth for the first time in Hopkins’ 2007 Master's thesis[11].