Obstacle Collision Avoidance Path Modification

In section1.4.1a method for reference path segmentation and digital map implemen- tation in real-time application is described with the hypothesis that a high-level path planning strategy (not designed in the present work) provides a global way-points trajec- tory, defined by global coordinates Xexp and Yexp, without considering the presence of

external obstacles. The present section aims to show how the elastic band theory [59,60] can be applied for autonomous steering control and its effect in terms of reference path modification when an obstacle is detected (object detection and/or recognition is not the final purpose of the present thesis). The method consists in comparing the initial reference path to a series of springs that can be deformed by internal and external

forces applied to the node Ni (see Fig.1.27). The elastic band is not deformed when no

obstacles are detected and, consequently, the static balance of internal forces acting on each node Ni can be calculated as:

FI N T_{i ,i −1}+ FI N T_{i ,i +1}= ks(ri −1− ri) + ks(ri +1− ri) = 0 (1.148)

where ks is the spring stiffness, ri is the position vector of i-th node and F_{i ,i −1}I N T is the

internal force between Ni and Ni −1.

When an obstacle is detected, an external force FE X T_i is applied thus modifying the static relation expressed by Eq.1.148:

FI N T_{i ,i −1}+ FI N T_{i ,i +1}+ FE X T_i = ks(r∗_{i −1}− r∗_i) + ks(r∗_{i +1}− r∗_i) + FE X T_i =

= ks(ri −1+ ui −1− ri− ui) + ks(ri +1+ ui +1− ri− ui) + FE X Ti = 0

Ni

Ni-1

Ni+1

Figure 1.27 Reference path modification according to the elastic band theory: external (FE X T_i ) and internal (FI N T_{i ,i −1},FI N T_{i ,i +1}) forces applied to the i-th node Niprovokes a deformation ui

where r∗_i is the vector position of node Niafter the application of the external force FE X T_i .

By substituting Eq.1.148in Eq.1.149the following relation is obtained:

FE X T_i = −[ks(ui −1− ui) + ks(ui +1− ui)] = −ks(ui −1− 2ui+ ui +1) (1.150)

The elastic band method is not applied to all n nodes that constitutes the initial path, but it is limited to q < n nodes that are located within a desired circle with radius Rpr

defined as preview radius. The external force FE X T_i is selected according to the relative position between the obstacle and the initial path:

FE X T_i =   

−ke(∥ri∥ − rmax)_∥ri ,mi n_{ri ,mi n∥}, ∥ri∥ ≤ rmax

0, ∥ri∥ > rmax

(1.151)

where ri ,mi nis the minimum vector position ri between the obstacle and initial path. If

the obstacle is detected beyond a safety margin expressed by rmax, no external forces

are applied thus locally limiting the path deformation. An example of the external force distribution is shown in Fig.1.28as function of the initial path X and Y global coordinates. Eq.1.150can be reformulated into a matrix equation:

Figure 1.28 Force distribution applied to initial path according to the elastic band theory

FE X T = ksKu (1.152)

where FE X T, u and K have respectively q x2, q x2 and q xq dimensions:

K =        2 −1 0 0 0 · · · 0 −1 2 −1 0 0 · · · 0 .. . ... ... ... ... · · · ... 0 0 0 · · · 0 −1 2        , u =           u1,x u1,y .. . ... ui ,x ui ,y .. . ... uq,x uq,y           , FE X T =           F_1,xE X T F_1,yE X T .. . ... F_{i ,x}E X T F_{i ,y}E X T .. . ... F_q,xE X T F_q,yE X T          

F_{i ,x}E X T and F_{i ,y}E X T are X and Y axis components of FE X T_i meanwhile ui ,x and ui ,y are the

same components of ui. The deformation vector u can be obtained by inverting the

matrix K:

u = 1

ks

i nv(K)FE X T (1.153) thus generating the modified reference path defined by X_exp∗ and Y_exp∗ coordinates:

[X_exp∗ Y_exp∗ ] = [XexpYexp] + [0 0;u;0 0] (1.154)

An example of reference path deformation as a consequence of different obstacle posi- tions is reported in Fig.1.29by adopting the following parameters:

• Rpr = 25 m

• rmax= 7.5 m

• ks= 520000 N /m

• ke= 3 N /m

Figure 1.29 Initial path deformation for four obstacle positions according to the elastic band theory

When the obstacle is closer to the initial path a larger deformation is requested in order to overtake the safety region. Moreover, the deformation of the elastic band can change its sign when the obstacle changes its relative position with respect the reference path. In the present activity, the safety region bounded by rmaxis selected by assuming only

static obstacles. The value of maximum external force amplitude ke and the preview

radius Rpr are properly designed in order to have a smooth curvature change when the

initial path deformation occurs. In particular, Fig.1.30shows the effect of different ratios

Tp = Rpr

ke on the path deformation: a high Tp value provokes a smoother deformation

thus requiring a less aggressive steering action when the vehicle begins and completes the obstacle avoidance maneuver.