## 2 Materials and Methods
### Computational Design of Lattices and Compliant Cross-Axis joints  
The CCAJ was first proposed in [38]. Here, computational models for CCAJ and LFHS were built using Solidworks(v2019, Dassault Systemes, Waltham, MA, USA). The macro-dimensions of the flexure hinges, shown in Figure 2a, were kept constant as follows: length (\(l\)) 42.23 mm, width (\(w\)) 20 mm, and depth 2 mm (\(e\)). The lattice honeycomb patterns studied here, i.e., re-entrant, hexagonal, and square, are shown in Figure 2b-d. To compare the mechanical responses, the representative unit cell dimensions (\(a\)) were adjusted to obtain close values for the volume fraction (relative density) among the different topologies. The values for \(a\) used were 5.5 mm for the re-entrant LFH and 3.5 mm for the hexagonal and square. Finally, for the internal unit cell dimensions, a constant thickness (\(t\)) of 0.5 mm was used. An internal angle (\(o\)) equal to 50\({}^{\circ}\) was used for the re-entrant lattice (Figure 2d). The resulting volume fractions used were 0.27, 0.30, and 0.32 for the square, hexagonal, and re-entrant, respectively. To study the effect of the lattice orientation with respect to the principal axis of the flexures, lattice arrangements were rotated to orientations from 0\({}^{\circ}\) to 80\({}^{\circ}\) with intervals of 20\({}^{\circ}\). Examples of these for the re-entrant topology are depicted in Figure 3.  
The complete CCAJ is shown in Figure 4, and it consisted of four main removable parts: a fixed add-in, a fixed part, two removable FHs or LFHs, and the acting part. The fixed part was designed to fit into a modified fixed add-in that was screwed into the base of the experimental setup (more details are given in the upcoming subsections). A \"T\" joining system was proposed to assemble the flexure to the fixed and acting parts. The acting part included a prismatic extension for the load to be applied. The part labeled as  
Figure 1: Flowchart with the 6 representative stages of the process used in this work for the compliant cross-axis joint.  
_flexure hinges_ was replaced with the _lattice_ hinges, resulting in a lattice-compliant cross-axis joint (LCCA)). Samples here are labeled with the following scheme: lattice topology (RH) for re-entrant, HH for hexagonal, and SH for square)-unit cell dimension _a_-orientation (\({}^{\circ}\)). Hence, a sample labeled as RH-5.5-80\({}^{\circ}\) is that corresponding to the re-entrant lattice with a unit cell dimension of 5.5 mm oriented at 80\({}^{\circ}\).  
Figure 3: Example of re-entrant lattice-flexure hinge schematics with a unit cell dimension of 5.5 mm at orientations of 0\({}^{\circ}\), 20\({}^{\circ}\), 40\({}^{\circ}\), 60\({}^{\circ}\), and 80\({}^{\circ}\) with respect to their principal axes.  
Figure 2: Schematics of the lattice-flexure hinges, showing topology designs and variables. (**a**) Constant dimensions of the flexure hinge. (**b**) Hexagonal lattice-flexure hinge with a unit cell dimension of 3.5 mm and an orientation of 0\({}^{\circ}\). (**c**) Square lattice-flexure hinge with a unit cell dimension of 3.5 mm and an orientation of 0\({}^{\circ}\). (**d**) Re-entrant lattice-flexure hinge with a unit cell dimension of 5.5 mm and an orientation of 0\({}^{\circ}\).