Flexures can be defined as elements that allow movement in specific degrees of freedom but maintain rigidity in all other directions of motion. Flexures are used in several day to day items, such as nail clippers and living hinges as shown in Figure 5.5. For optomechanical applications, flexures are commonly used to achieve precise motion for alignment stages by purposely deflecting flexible elements. Flexures can be difficult to design to isolate translations or rotations about any of the three principle axes, while also maintaining rigidity in all other directions. Maintaining rigidity in the other degrees of freedom is important in reducing the amount of parasitic motion. Parasitic motions or errors are deviations from a stage’s intended direction of travel [21]. Parasitic motion can be difficult to predict when designing flexures. With the appropriate actuator locations, the amount of parasitic motion can be minimized. However, knowing where to properly place actuators or creating a stiff flexure in the appropriate directions can be difficult for an inexperienced flexure designer. Luckily, a methodology has been developed to help engineers build a foundation for flexure design. By employing the FACT methodology, constraints and actuation directions can be defined to ensure isolated motion in the desired degrees of freedom [21].
Figure 5.5: Examples of Flexures in Common Items, Living Hinges (Left)
5.3.1 FACT Methodology
FACT stands for Freedom, Actuation and Constraint Topologies and is used for creating flexure designs [21]. FACT is a powerful tool, allowing the user to visualize the necessary constraints for desired ranges of motion when designing a flexure sys- tem. There are a few specific terms that need to be defined before using the FACT methodology: twists, wrenches, freedom and constraint spaces. Twists describe lines of action that are allowable degrees of freedom. These lines can describe pure ro- tations (shown in red) and translations (shown in black) or a combination of both as represented in Figure 5.6. Wrenches describe lines of constraint and can be rep- resentative of pure forces (shown in blue) or moments (shown in black) to prevent deflection in unwanted directions. A freedom space is the 2 or 3-dimensional repre- sentation of allowable motion for a flexure. Conversely, the constraint space is the culmination of wrenches that constrain a flexure in the remaining degrees of freedom. An example of a freedom space and its corresponding constraint space is shown in Figure 5.6.
Figure 5.6: Example of a 2-DOF, Freedom and Constraint Space [21]
In Figure 5.6 the parallel red lines represent the plane of allowable rotations along the Z-axis, in addition to allowable translation in the Y-direction. This system would be a 2 DOF system that would allow for translation in one direction and rotation about a single axis. To prevent movements in other directions, four independent
wrenches would need to be implemented. These wrenches could be applied parallel to the rotation axis as pure forces (indicated by the blue lines). The system can also be constrained by any of the pure torques that is perpendicular to the plane of the freedom space. Any combination of 4 independent wrenches out of these options would constrain the flexure entirely.
A designer must be careful to implement enough independent constraints to pre- vent unwanted motions in other degrees of freedom. Independent constraints can be difficult to visualize, but usually if all lines of constraint do not meet at the same point in space or are parallel to one another they can be considered independent. The number of independent wrenches necessary is found by the simple relationship:
6 =m+n (5.1)
Where n is the degrees of freedom and m is the number of independent wrenches. Redundant wrenches can be implemented in a system to increase the rigidity of the system but will affect the dynamics and load capacity of the system [21].
With the FACT methodology, Figure 5.7 can be used to quickly determine the constraint spaces required for the desired range of motion. The chart contains the freedom and constraint spaces to achieve any type of motion desired for a flexure system.
Figure 5.7: FACT Chart for Flexure Design Synthesis [21]
The FACT chart contains a bolded line boundary, which is referred to as the “parallel pyramid”. Within this pyramid, it is possible to design a parallel configu- ration flexure to achieve the desired range of motion. However, if the desired flexure design lies outside the pyramid, this would require a combination of parallel flexures stacked serially to achieve the range of motion. The concept of parallel and serial configurations for flexures are shown in Figure 5.8.
Figure 5.8: Examples of Parallel and Serial Configurations for Flexures
[21]
In flexure stages, parallel and serial configurations are commonly used to describe a stage. In a parallel configuration, multiple actuators can push against the same stage, and allow for the desired degrees of freedom. However, with a serial configuration
multiple parallel stages are nested together, and through the combined motion are able to achieve the desired motion. The nested stage is often referred to as an intermediate stage, which connects from the ground, or nonmoving component, to the desired motion stage.
As an example, utilizing the FACT chart shown previously in Figure 5.7, consider the flexure design at the top of the 2 degrees of freedom column. This flexure design allows for planar translation. Since the design lies outside of the parallel pyramid, two parallel designs must be selected from a lower degree of freedom. To achieve this range of motion, 2 parallel flexure designs in the 1 DOF column will be combined to allow for planar translation as shown in Figure 5.9. This would require the necessity of an intermediate stage with ends connecting to ground and the main stage.
Figure 5.9: Combination of 1-DOF Freedom Spaces to Produce a Planar 2-DOF Translation Freedom Space [21]
Flexure stages have been used for several applications at the NIF. This is pri- marily due to their high versatility and capabilities of achieving any desired degrees of freedom. After overcoming the initial learning curve, flexures can be a powerful design tool when designing high precision stages.