# Object-Independent Grasping in Heavy Clutter
## 2 Dbp
### Grasp Pose Evaluator
#### 2.2.2 Evaluation Model by GMM  
To evaluate the grasp candidates, a model based on the GMM was designed, using three Gaussian models and DDI values corresponding to the grasp candidates, as shown in Figure 6. Additionally, a cost function was developed using the three parameters defined in the following. In Figure 6, the three Gaussian models are ordered according to their \(x\) values and denoted 1, 2, and 3. G\({}_{2}\) can be interpreted as the area where the object exists, and G\({}_{1}\) and G\({}_{3}\) can be considered as spaces to the left and right of the object, respectively. According to the Gaussian models obtained from the GMM,  
Figure 4: Example of DDI (Depth difference image). the proportion difference, height difference, and width are defined, and the cost function is constructed by multiplying or dividing them. Figure 6a,b shows the DDI and its profile along the grasp candidate indicated by the red line in Figure 6a, respectively. In (b), point (\(x_{i}\), \(D_{i}\)) and \(h_{i}\) are the average point and the maximum value of each Gaussian model estimated by the GMM, respectively. The proportion difference index \(d_{p}\) is defined as the ratio of G\({}_{1}\) to G\({}_{3}\). The proportion \(p\) is defined: as  
\[p_{i}=\frac{\text{number of data belonging to Gaussian model }i}{\text{number of total data}}; \tag{1}\]  
Figure 5: DDI and result of the Sobel operator in two and three dimensions. Figure 6: Evaluation model composed of three Gaussian models: examples of (**a**) the DDI, (**b**) the DDI profile along the grasp candidate, (**c**) the height difference \(D_{d}\), and (**d**) the width \(x_{d}\). thus, \(p_{1}+p_{2}+p_{3}=1\). For example, in Figure 6, \(p_{1}=0.25\), \(p_{2}=0.5\), and \(p_{3}=0.25\). If \(p_{1}\) and \(p_{3}\) have different values, there is space only on one side (left or right) of the object. Therefore, to select the cases in which both the left and right sides of the object are wide, the proportion difference \(d_{p}\) is defined as:  
\[d_{p}=\frac{\left|p_{1}-p_{3}\right|}{\max(p_{1},p_{2},p_{3})}, \tag{2}\]  
where the difference between \(p_{1}\) and \(p_{3}\) is divided by the largest value among \(p_{1}\), \(p_{2}\), and \(p_{3}\) for normalization. Thus, \(d_{p}\) indicates whether \(p_{1}\) and \(p_{3}\) are similar. However, even if they are, the spaces may not be large enough. Then, the height difference, which is the second evaluation index, is used. The height difference index \(d_{h}\) is defined as:  
\[d_{h}=\frac{D_{d}}{\max(h_{1},h_{2},h_{3})}=\frac{\left|h_{2}-\min(h_{1},h_{3} )\right|}{\max(h_{1},h_{2},h_{3})}, \tag{3}\]  
where the height difference \(D_{d}\) of Figure 6c is divided by the largest value among \(h_{1}\), \(h_{2}\), and \(h_{3}\) for normalization. Thus, \(d_{h}\) indicates the depth of the space around the object. A larger \(d_{h}\) indicates a deeper space around the object. In summary, \(d_{p}\) and \(d_{h}\) indicate the width and depth of the space around the object, respectively. The width index \(w_{c}\) is defined by the difference between the \(x\) values of G\({}_{1}\) and G\({}_{3}\) of Figure 6d, i.e.,  
\[w_{c}=\left\{ \begin{array}{l}\infty\text{ for }x_{d}=0\\ -\log(x/c)+1\text{ for }0<x_{d}<c\\ 1\text{ for }x_{d}=c\\ 2^{x/c}-1\text{ for }c<x_{d}<x_{l}\\ \infty\text{ for }x_{d}>x_{l}\end{array} \right.\text{ where }x_{d}=x_{3}-x_{1}. \tag{4}\]  
Here, \(x_{d}\) represents the distance between \(G_{1}\) and \(G_{3}\), and \(x_{l}\) represents the maximum width of the gripper. \(c\) is proportional to the width of the gripper and the camera resolution and inversely proportional to the distance. Because the opening of the gripper is limited, \(w_{c}\) is infinite when \(x_{d}>x_{l}\). Additionally, it is assumed that the gripper has an optimal width to grasp an object; thus, \(w_{c}\) is minimized at a specific \(c\) value and increases rapidly as \(c\) deviates from this value.