# Retro-Reflective-Marker-Aided Target Pose Estimation in a Safety-Critical Environment
## 2 Methods
### Retro-Reflective Marker Design  
Retro-reflectors can be characterized by four main properties: retro-reflectance (ratio of retro-reflected to incident light), brilliancy (reflectance for a specific observation angle), divergence (maximum angular distribution of retro-reflected light), and angularity (brilliancy as a function of the incident angle) [30]. In many applications, the desired marker behavior resembles the ideal retro-reflector, which allows large entrance angles (high angularity) and returns as much light as possible in as small a cone of light as possible (high reflectance, high brilliancy, and low divergence). However, in applications where there is a displacement between the light source and the detector, higher divergence might be acceptable or even required. As an example, in traffic applications, where there is a displacement between the car's headlamps and the driver's eyes of about 60 cm, the useful retro-reflecting angle has been quantified to be from nearly 0 to approximately 3 degrees for distances of 12 to 122 m [31]. In this case, the aim is to optimize brilliancy at the observation angle rather than at the incident angle. This is often achieved by using a retro-reflector with higher divergence, though at the cost of some intensity loss. The same principle applies to photogrammetry and dictates the first functional requirement of an effective retro-reflector for our application. The useful retro-reflecting angle can be easily quantified for a specific setup as a function of the imaging distance (\(H\)) and the displacement between the camera and the light source (\(D\)) as follows:  
\[\delta=tan^{-1}\frac{H}{D}\,. \tag{1}\]  
For the setup used in this work, we calculate a maximum useful retro-reflecting angle of \(\pm 18^{\circ}\) for the shortest working distance and the maximum possible displacement between the camera and the source. The second functional requirement of a successful retro-reflector is the acceptance angle. We consider that an acceptance angle of \(\pm 10^{\circ}\) is sufficient for our application, since an operator can, with very little effort, make a preliminary target alignment within this range. Furthermore, there is no minimum specification for the retro-reflectance, but the aim is to maximize its value. Figure 1: Relevant coordinate frames and rigid body transformations. There is a relatively long list of environment requirements dictated by the ITER application when it comes to materials that stay in the divertor for a considerable amount of time. These include an operating temperature of up to 200 \({}^{\circ}\)C and adequate radiation tolerance. In practice, these restrictions deem unfeasible the use of most common materials, except for stainless steel (which makes up most of the other structures in the divertor) and fused silica glass (which is planned to be used in diagnostic windows and fiber optics). To the best of our knowledge, there is no adhesive material that can be used in this environment. Furthermore, the size of the developed marker should be rather small, such that the mounting of several specimens in a 200 \(\times\) 200 mm area is feasible and the embedment depth should be rather small so as not to compromise the structural integrity of the cassette. Retro-reflection is often achieved using one of two designs: a corner cube or a lens and a mirror. Corner-cube retro-reflectors (CCR) rely on the reflection of light on three mutually orthogonal surfaces. Generally, CCRs are manufactured either as a truncated cube corner of a transparent material (i.e., prism) or as flat reflective surfaces surrounding empty space. Cat's eye retro-reflectors consist of a primary lens with a secondary mirror located at its focus [30]. These can be easily implemented using a sphere of a transparent material. Spherical retro-reflectors generally have higher divergence due to spherical aberration and lower retro-reflectivity. On the other hand, corner-cube retro-reflectors usually have higher retro-reflectivity, while their fabrication is more difficult [30; 32].