The robot has to define the number, locations and sizes of obstacles to achieve successful dynamic obstacle avoidance. Hence, it is crucial to segment the sensor data using clustering methods. This section focuses on the clustering of 2D laser range finder (LRF) data, which consists of N points and each point can be described by pairs of polar coordinates (r,a) or in Cartesian coordinates (x,y), as shown in Figure 2.22.
>Z&y
z
KďũĞĐƚ
Figure 2.22: Schematic representation of scan laser data
The clustering process involves dividing the sensor data into separate groups which repre- sent the existing objects in the robot’s workspace. Clustering methods can be divided into two types; point-distance-based methods (PDBS) and Kalman-filter-based methods (KFBS) [Nunes and U., 2005; Rebai et al., 2009]. For the PDBS methods, the distance between two
CHAPTER 2. LITERATURE REVIEW
consequent points is given as follows:
d(ri,ri+1) =
r2
i +r2i+1− 2riri+1cosa (2.7)
where r andΔa define the calculated distance and the angular resolution.
The distance d(ri,ri+1) is compared with a threshold value Dth, and if the condition d(ri,ri+1)<
Dthis met, the two points belong to the same object; otherwise they belong to two different
objects. However, different methods can be used to determine the threshold distance, which can be held constant, but then the clustering process does not segment the sensor data prop- erly because the distance between the laser points changes depending on distance from the laser sensor. In [Lee, 2001], the threshold is calculated as:
Dth=|r|ri− ri+1| i+ri+1|
(2.8) It should be noted that the threshold distance is calculated using the distance between the detected object and the laser sensor, which can make separating the objects more difficult as they are further away. In [Dietmayer et al., 2001], the threshold distance is given according to the following equation:
Dth=C0+C1min(ri,ri+1) (2.9)
where C0is a constant value used to reduce noise, and C1is given as follows:
C1=
d(ri,ri+1)
ri
(2.10) In this method the threshold distance depends on the distance to the sensor. In [Santos et al., 2003], a new parameter β is introduced to reduce the effect of object distances, where the threshold distance is defined as follows:
Dth=C0+
C1min(ri,ri+1)
CHAPTER 2. LITERATURE REVIEW
It is very important to tuneβ, since a large value can segment an object into two, while two objects can be interpreted as one object for a small value.
The K-means clustering algorithm is another PDBS method, but it does not depend on the dis- tance between the object and sensor [Napoleon and Lakshmi, 2010]. The K-means algorithm consists of five steps:
1. Define the number of clusters K. 2. Initialize the cluster centres randomly.
3. Define the class membership of points by assigning them to the nearest cluster centre. 4. Recalculate the cluster centre using the mean value of memberships.
5. Repeat steps 3 and 4 until no point changes its class membership.
The K-means clustering is easy, simple, and efficient, however, it is not suitable for clustering non-convex shapes, and the number of clusters K needs to be defined in advance. The K- means algorithm was improved to tune the clusters number automatically using the following steps [Marron et al., 2004]:
1. Select a random number of clusters K. 2. Initialize the cluster centres randomly.
3. Define the class memberships of the N points by assigning them to the nearest cluster centre.
4. If the distance between a point to its cluster centre is bigger than a threshold, a new cluster is added.
5. Recalculate the centres of clusters using the mean value of memberships. 6. Remove cluster centres which have very few points.
7. Repeat steps 3, 4, 5 and 6 until there are no changes in class membership.
KFBS methods are used to segment laser data by applying the following algorithm for all laser points pi[Kmiotek, 2009]:
CHAPTER 2. LITERATURE REVIEW
1. Filter initialization: in this stage, the system state vector x is assigned to the first laser point p0, and the covariance matrix is initialized P = P0.
2. Implement the prediction stage of the KF algorithm.
3. Test the gating situation for point pi: if it is located inside the gate, the update stage is implemented; otherwise piis selected as a break point.
The laser point piis considered to be inside the gate if the condition pTiS−1pi< Dth is met,
where S−1 is the measurement covariance matrix. For 2D laser data, the linear dynamic equations can be written as follows:
rt+1=rt+a.drt da (2.12) drt+1 da = drt da (2.13) xt= [rt drt da] T (2.14)
where r and a represent the length and the angle of a laser line respectively, while xt is the state vector. For the Extended Kalman filter the dynamic equations are given as follows:
φt+1=φt+a (2.15) rt+1= sin(φt) sin(φt+a)rt (2.16) xt= [rt φt] (2.17)
whereφ represents the angle between the laser line and the object surface, as shown in Figure 2.23. The extended Kalman filter gives more precise estimations than the standard Kalman filter. However, these methods are more complex than distance-based methods and require large computation.
CHAPTER 2. LITERATURE REVIEW y z
Figure 2.23: KF-based methods (KFBS)