Articulated Robots with 2-DOF Joints

In [61], a potential-based path planning algorithm was proposed for articulated robots with 3-DOF joints, spherical joints, as shown in Fig. 5.14(a). Because the joint has highest DOFs, the robot can take full advantage of the proposed potential minimization algorithm in reducing the potential. In this thesis, an improved algorithm of [61] is proposed for a robot with 2-DOF joints. While a robot with active 3-DOF joints is not trivial in practice, a robot with 2-DOF joints is usually adopted. The Hooke joint, which has two orthogonal rotation axes as shown in Fig. 5.14(b), is adopted in this thesis to establish the connection between two links.

Consider joint J_i which connects two links, lnk_i and lnk_i−1, as shown in Fig. 5.14(b).

(a) (b)

Figure 5.11: The computation time will be proportional to the number of triangles of obsta-cles, and the number of robot configurations calculated. (a) Computation time vs. Number of triangles of obstacles. (b)Computation time vs. Number of links.

Assume R_i−1 represents the rotation axis of lnk_i−1 after a potential minimization procedure while R^A_i is the rotation axis of lnk_i (which is to be connected to lnk_i−1such that a potential minimization for lnk_i can take place), as shown in Fig.5.15. Because of the constraint of the 2-DOF joint (R_i ⊥ R_i−1), lnk_i can not be connected directly to lnk_i−1 through pure translation if the orientation of lnk_i−1 has been changed due to potential minimization.

Let S_i be the skeleton axis of lnk_i. It is obviously that two rotation axes are not always orthogonal if two links are connected arbitrarily. Such a configuration of these two links is illegal for a robot with 2-DOF joints. Therefore, R_i should be moved from R^A_i to R^B_i and then orthogonal to R_i−1. R^B_i can be determined as following. Assume all possible positions of R_i are lying on Plant P if lnk_i rotates about the axis S_i. R^⊥_i−1 is the projection of R_i−1 on P . If a vector R^B_i of P is orthogonal to R^⊥_i−1, it will be orthogonal to R_i−1 too. Thus, lnk_i should rotate about S_i by θ to let R_i moves to R_i^B. After the previous adjusting lnk_i orientation, the two links are connected legally with a 2 DOF joint. And, then, the further potential minimization can take place.

A Comparison Between 2-DOF Joints and 3-DOF Joints

In this section, the simulation results of the 2-DOF-joint algorithm is compared with the ones of 3-DOF-joints algorithm. In order to show the difference of algorithms clearly, the third configurations of robots with 2-DOF joints and 3-DOF joints of a simple example are shown in Figs. 5.16(a)(b), respectively. In Fig. 5.16(b), the second link is twisted to min-imize the configuration potential due to the additional DOF, rotating with respect to the

(a) (b) (c)

Figure 5.12: Path planning examples for a 4-link articulated robot in an U-shaped workspace with (a) 3 (b) 4 and (c) 5 initial GPs.

skeleton. In general, the algorithm with 2-DOF joints takes more time than a 3-DOF one to plan a collision-free path due to less DOFs to minimize the potential. Figs. 5.17(a)(b) show trajectories of a complex example of algorithms with 2-DOF joints and 3-DOF joints, respectively. While Fig. 5.17(a) takes 91.34 seconds to planning the 24-configuration path, Fig. 5.17(b) taking 88.16 seconds to planning the 18-configuration path. Due to the less free-dom, the 2-DOF-joint algorithm needs more time to plan more intermediate configurations.

Since the 3-DOF-joint robot has additional DOFs to minimize potential than the former, the planned trajectory is more smoothly. Though there are different between two trajectories of Figs. 5.17, the difference is not so significant. Thus, the proposed algorithm with 2-DOF joints is work well as the algorithm with 3-DOF joints.

5.6 Summary

In this thesis, a potential-based path planning of articulated robots with moving bases is proposed. According to the adopted potential model, surfaces of workspace obstacles are assumed to be uniformly charged and the links of the articulated robot are represented as a set of charged sampling points. The repulsive force and torque between links and obstacles thus modeled are analytically tractable. It is assumed that a sequence of GPs to be traversed by an articulated robot are given in advance in the workspace providing the general direction of the path. According to the motion constraints established by the

(a) (b)

Figure 5.13: Successful paths of Figs. 5.12(b)(c), respectively.

GPs, the proposed approach derives the path for the articulated robot by adjusting its configurations at different locations along the path to minimize the potential using the above force and torque. Simulation results show that a path thus derived always stays away from obstacles and is spatially smooth.

(a) (b)

Figure 5.14: A path planning example for a 3-link articulated robot in a 3-GP workspace.

(a) The initial configuration. (b) The trajectory.

Figure 5.15: Since T_i−1 is perpendicular with T i, the lnk_i should rotate with respect to S_i to let T_i⁰ moves to T_i.

(a) (b)

Figure 5.16: Two planned paths for a 3-link articulated robots with 2-DOF and 3-DOF Joints. (a) The third configuration of the robot with 2-DOF joints. (b) The third configu-ration of the robot with 3-DOF joints.

(a) (b)

Figure 5.17: Two planned paths for a 3-link articulated robots with 2-DOF and 3-DOF Joints in the same environment.(a) A 24-configuration path of the robot with 2-DOF joints.

(b) A 24-configuration path of the robot with 3-DOF joints.

Chapter 6

Potential-based Shape Matching and Recognition of 3-D objects

6.1 Overview

In this chapter, the potential-based shape matching approach presented in [50] is extended to three dimensions using a generalized potential model presented in [1]. The goal is to develop a potential-based 3-D shape matching algorithm which, as its 2-D counterpart, can correctly and efficiently perform the matching without knowing the exact information about the location, orientation and size of the input object. Furthermore, the proposed shape matching scheme is not based on any hypothesis of feature correspondence. Therefore, feature extraction of an input object which is required for a structured object representation is not needed. According to such a model, the repulsion between two 3-D objects, one with polyhedral description and the other represented by point samples on its surface, can be evaluated analytically.

In 6.2, a potential-based shape matching algorithm using these analytic results is devel-oped. Some computer implementation details of the matching algorithm are also presented.

Simulation results for the shape matching of some 3-D objects are presented in 6.3. 6.4 presents some concluding remarks.