The application of the potential model reviewed in the previous section for path planning of manipulators will be discussed in this section. Unlike some c-space based approaches, which often require expensive preprocessing to construct the c-space, the proposed approach uses the workspace information directly. The approach computes repulsive force and torque experienced by each rigid component, e.g., a link, of a manipulator. A collision-free path of the manipulator can then be obtained by locally adjusting its configuration along the path
GP1
GP2
(a)
GP1
GP2
(b)
Figure 4.1: A manipulator is moved toward the goal (not shown) by sequentially traversing a sequence of GPs.
(a) (b)
Figure 4.2: The generalized cylinder presentation of a passage.
for minimum potential using these force and torque. In this thesis, the spherical joint is adopted to connect links of a manipulator since its high DOFs can take full advantage of the proposed potential minimization algorithm.
For a rough description of manipulator path, the proposed approach uses one or more guide planes (GPs) as final or intermediate goals in the 3-D workspace. The GPs are polygons among obstacles in the free space, providing the manipulator a general direction to move forward (see Fig. 4.1). A collision-free traversal of a given sequence of GPs by the end-effector is regarded as a global solution of the path planning problem of a manipulator.
4.2.1 Generation and Selection of initial GPs
In the proposed algorithm, the GPs provide the articulated robot a general direction to move forward. The selection of the initial GPs may base on (i) their density along the passage, (ii) the visibility between two adjacent GPs, or (iii) the angular variation of two adjacent GPs. For the examples considered in this chapter, the initial GPs are selected arbitrarily and the algorithm seems to work reasonably well in terms of the sensitivity of the planned path to the selection of initial GPs. Often, these initial GPs can also be obtained from the Generalized Cylinder (GC) [56] representation as cross-sections perpendicular to the GC axis. Figs. 4.2(a) shows a passage which has approximately rectangular cross-sections, and an axis of its GC representation. In general, there is no limit on the number of cross-sections and their shapes are not explicitly specified in advance. More details about generalized cylinder representation can be found in Appendix A.
In the proposed approach, the shape of the GPs is usually not critical as long as the GPs are confined in the free space. In face, only the normal direction of a GP, e.g., the direction of the GC axis, is important. Such an axial representation can also be derived from (i) the global navigation function as in [57][5][58], as well as (ii) the tree structure representation of free space obtained from the wavefront expansion presented in [59]. While the gradient of (i) leads a point object to the goal, global connectively of free space is available in (ii) and is used in [59] to solve a planning problem in low dimensions.
4.2.2 Basic Procedure of Path Planning
In this chapter, the proposed path planning approach derives a series of minimum potential configurations along the path of a manipulator by locally adjusting its configuration for minimum potential using the results given in Sec. 2.2. Assuming that a guide plane GP1 is given as an intermediate goal, the basic path planning procedure for moving the end-effector p0 of a manipulator onto GP1 include (see Fig. 4.3):
(i) Translate the distal links1 of the manipulator to move its end-effector p0 toward the GP1. If p0 can not reach GP1 directly, e.g., due to collision, a virtual intermediate plane GP10
1In step (i), an intermediate simple solution of the inverse kinematics problem is obtained by translating of all manipulator links except for the two base links, the base link and the link connected to it. For each translation, the two base links, together, can have at most three DOFs. In step (ii), the problem is solved by finding a sequence of sub-optimal solutions with monotonically decreasing potential. Finally, the minimal potential solution is found in (iii).
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Figure 4.3: Basic path planning procedure for a given GP (see text).
is inserted. (Fig. 4.3(a))
(ii) Search for the minimum potential configuration of the manipulator for p ∈ GP10 by repeatedly executing :
(a) Search for the minimum potential configuration of the manipulator with the distal link fixed in orientation. The minimization is performed by sliding p along two orthogonal directions on GP10, e.g., −→u and −→v . (Fig. 4.3(b))
(b) Search for the minimum potential configuration of the manipulator with the end-effector p fixed in position. The minimization is performed by changing the ori-entation of the distal link with p as its rotation center. (Fig. 4.3(c))
(iii) Repeat (i) and (ii) until p reaches GP1.
In general, there are different ways to change the manipulator configuration to move p toward GP1. A simple translation of distal links is adopted in (i) as a preliminary implemen-tation of our algorithm. As shown in Fig. 4.3(a), the translation of the distal links is carried out to move the end-effector from p0 to c, where c is the centroid of GP1. If collision occurs or if the connectivity of manipulator can not be maintained, the distance of the translation is reduced until the translation is collision-free while the manipulator remain connected.
Accordingly, a new GP is inserted, e.g., GP10. No configuration improvement to reduce the repulsive potential is considered at this stage.
As for the search for the minimum potential configuration of the manipulator in (ii), links of the manipulator are adjusted from the distal link to the base link using the repulsion experienced by the manipulator. The distal link has five DOFs, i.e., two for its location for p ∈ GP10 and three for its orientation. While each of other distal links has three DOFs for
its orientation, the two base links, together, have at most three DOFs with their connecting joint being constrained to lie on a circle.
In (ii), the associated constrained optimization problem is divided into two iterative univariant optimization procedures, as in (ii-a) and (ii-b). In (ii-a), the distal link is fixed in its orientation (see Fig. 4.3(b)) as p slides on GP10 to search for the minimal potential configuration and other distal links are sequentially adjusted in orientation, staring from the link connected to the distal link. In (ii-b), the distal link is adjusted in orientation while fixed in position (see Fig. 4.3(c)) and the procedure for adjusting the rest links is similar to that in (ii-a). For a particular GP, say GP10 in Fig. 4.3, (ii-a) and (ii-b) are repeatedly performed until negligible changes in the manipulator configuration is obtained. Then, another intermediate GP which is closer to GP1 is obtained with (i) and the process repeats. The path planning algorithm, as summarized below, ends as the end-effector reaches the given GP1 or exist abnormally for an infeasible problem. Detailed implementation of the algorithm is presented in the next section.
Algorithm End Effector to GP
Step 0 Initialize δ = δ0 = p0c and GP ← GP1.
Step 1 Translate the distal links of the manipulator with distance δ to move p0 to p along the direction of −→
p0c. Find the smallest n ≥ 0 such that δ ←− δ/2n corresponds to a feasible and collision-free translation.
Step 2 Translate the distal link by sliding p on GP to minimize the potential.
Step 3 Adjust joint angle of the manipulator for the minimum potential configuration with p fixed in position.
Step 4 Go to Step 2 if the translation in Step 2 or the joint angle adjustment in Step 3 is not negligible.
Step 5 If p reaches GP1, the planning is completed. Otherwise, p0 ← p and go to Step 1 with δ = δ0.
For path planning involving multiple GPs, the above algorithm will be executed for each of them sequentially. It is assumed that the planning for a GP starts as the planning of the previous GP is accomplished. The path planning ends as the end-effector reach the goal, which is usually a (goal) GP in the path planning problems considered in this chapter.