Adjusting the leading tip position in Step 2

After the leading tip reaches p, as shown in Fig. 5.3(b), one of the univariant procedures to minimize the repulsive potential, which allows lnkn to adjust its location but not its

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Figure 5.5: Translating lnk1 to slide p on GP₁⁰ to reduce the repulsive potential.

orientation, is performed in Step 2 of Articulated Robot to GP. In order to speed up the computation, a sequential planning strategy [60] which plan the motion of links sequentially is adopted in Step 2. Accordingly, the procedures of Step 2 first minimize the potential of lnk₁ whose translation is constrained in two dimensions by GP. As for the motion of lnk_i (i > 1), as in [60], it is assumed that all links up to lnk_i−1 has been planned, i.e., one end of lnk_i is fixed in position. Therefore, only the orientation of lnk_i, which has three degrees of freedom, need to be adjusted for minimum potential.

Consider the forces exerted on lnk₁, as shown in more detail in Fig. 5.5. Since the minimization is constrained by p ∈ GP , only the projection of the resultant force experienced by lnk₁ on GP is taken into account. Let f₁ be the repulsive force exerted on lnk₁ due to the repulsion between lnk₁ and the obstacles. For a univariant minimization approach, only one variable is adjusted at a time. To determine the direction in which lnk₁ should translate to slide p on GP to reduce the repulsive potential, the projection of the resultant force exerted on lnk₁ along an arbitrary −→u k GP , f_1,u is calculated.A gradient-based binary search for the minimum potential location of p along −→u can then be performed. The initial step of sliding is arbitrarily chosen as 10% of the workspace size. If the movement of lnk1 along −→u or f1,u

is negligible, e.g., the movement is smaller than 1% of workspace size, another minimization of potential along −→v k GP , which is orthogonal to −→u , is performed to minimize f1,v. Step 2 ends when two consecutive binary searches along the two orthogonal directions result in negligible movement of lnk1. Once the optimal configuration of lnk1 is determined, the procedure for adjusting the rest links is similar to that in Step 3, as discussed next.

5.3.3 Adjusting joint angle in Step 3

Once the minimum potential position of lnk_n is determined with Step 2 of Articu-lated Robot to GP, another univariant procedure, which allows lnk₁ to adjust its orientation with leading tip p fixed in position is performed to reduce the potential further, as shown in Fig. 5.3(c). Under such a constraint, lnk₁ can rotate with respect to p to reduce the repulsive potential. The direction in which lnk₁ should rotate is determined by the repulsive torque experienced by lnk₁ with respect to p.

Let τ₁ be the repulsive torque experienced by lnk₁ with respect to p due to the repulsion between lnk₁ and obstacles. To find the minimum potential orientation of lnk₁ for p fixed in position, gradient-based binary searches are performed repeatedly using the projection of τ₁ along three orthogonal axes of rotation, e.g., τ_1,u, τ_1,v and τ_1,w, respectively. For each binary search, the initial rotating angle is arbitrarily chosen as 5^o, while the accuracy in identifying the 1-D potential minimum is chosen as 0.5^o. Step 3 ends if a binary search results in a negligible change in the orientation of lnk₁, i.e., less than 0.5^o.

5.4 Simulation Results

In this section, simulation results are presented for path planning performed on Pentium III (500MHz) personal computer for articulated robots in 3-D environments. Fig. 5.6(a) shows the initial configuration of the path obtained for a 3-link articulated robot in a 3-GP workspace wherein the GPs are shown as black polygons. Fig. 5.6(b) shows the complete trajectory of the articulated robot which reaches the final GP safely. Since collisions occur frequently near the 90^o turn of the passage for the δ₀ chosen, more configurations of the articulated robot are planned near the turn than those near start and goal. Due to the repulsive potential model, the trajectory of the articulated robot is smooth and lies near the middle of the workspace. The simulation takes a total of 18.266 seconds to plan the 12-configuration collision-free path.

Figs. 5.7(a)(b) show the initial configuration and the trajectory, respectively, of the path obtained for another 3-link articulated robot in a 2-GP workspace. The simulation takes 26.638 seconds to planning the 8-configuration path within a workspace with 48 triangles of obstacles. While the length of the passage shown in Fig. 5.6 is shorter than that in Fig. 5.7, less robot configurations is generated for the path of the letter because the passage

is spatially more smooth.

Figs. 5.8(a)(b) show the initial configuration and final trajectory for the path planning of a 4-link articulated robot moving into a winding passage with 4 GPs. Fig. 5.8(a) shows the initial configuration of the articulated robot which lies at the entrance of the passage.

The trajectory shown in Fig. 5.8(b) indicates that the articulated robot reaches the final GP safely. While the robot motions in Figs. 5.6 and 5.7 are essentially two-dimensional, the robot trajectory planned in Fig. 5.8 requires three-dimensional maneuvering. The simulation takes a total of 68.999 seconds to plan the 11-configuration collision-free path.

Figs. 5.9(a)(b) show the initial configuration and final trajectory of the path obtained for another 4-link articulated robot moving in a turnaround passage. It can be seen clearly from Fig. 5.9(b) that the articulated robot traverses the five GPs safely and smoothly. The simulation takes a total of 151.518 seconds to plan the 24-configuration collision-free path.

Since the passage of this example is more crooked, more intermediate GPs are added into the path, which in turn increases the computation time.

(a) (b)

Figure 5.6: A path planning example for a 3-link articulated robot in a 3-GP workspace. (a) The initial configuration. (b) The trajectory.

(a) (b)

Figure 5.7: A path planning example for a 3-link articulated robot in a 2-GP workspace. (a) The initial configuration. (b) The final trajectory.(c) The 248-triangle tunnel.