Calculating optimal trajectories from contact transitions

The learning-from-demonstration method is focused on for a novel robot-programming style. It consists of two parts: to recognize human performance from observation as sequential motion primitives and to execute the same performance. We already proposed a method to recognize assembly tasks. However, execution requires the ability to convert motion primitives to collision free paths. In this paper, we describe a method to calculate collision free paths. Many researchers have proposed to calculate collision free paths using analytical methods, potential fields or probabilistic methods. Potential and probabilistic methods are very powerful tools on a computer, but their solutions are not optimal. We propose a method to calculate optimal collision free paths analytically. 1 I n t r o d u c t i o n Recently, the method called "Learning from demonstration," an easy robot-programming style, has been focused on[l, 2, 3, 4, 5, 6, 7]. Its main characteristic is that it is enough to perform a desired action in front of the robot's vision system to make it repeat the action. Therefore, even people with little robotics knowledge can use the method. It recognizes an action as a sequence of symbolic motion primitives. Symbolization enables a robot to execute the same action more flexibly. We proposed a method to symbolize and sequence assembly tasks[8]. The "Learning from Demonstration" method requires control parameters for a robot to perform the desired actions, because the actions have already been converted to symbolic motion primitives, that do not have such parameters. In this paper, we especially consider trajectories. First, trajectories must be collision-free. LozanoPerez et al. proposed a method to generally calculate collision-free trajectories using the concept of Configuration Space(C-space)[9]. However, it may be impossible to analytically calculate such trajectories in a high-dimensional C-space, even for six dimensions, which is equivalent to our living world. Kavraki et al. proposed the "Probabilistic Roadmap Method(PRM)[10]" to calculate such trajectories using a randomized algorithm. (~) (b) Figure 1: Optimal trajectory Next, increasing the success rate of assembly tasks usually requires calculating a trajectory that maintains a contact relation. Hirukawa proposed a method to analytically formulate and calculate a trajectory in a spatial motion case[11]. However, it is not always possible to solve such formulations, the complexity of the problem depending on the user's skill at choosing adequate ones. Ji and Xiao proposed a method to solve the formulations using PRM[12]. We propose a method to analytically and easily calculate an optimal trajectory by first calculating an axis direction of rotation. The method is based on the fact that we prefer the rotation in which the direction of the axis is constant. The reader may wonder why we solve a trajectory by such a method, instead of simply using PRM. Consider the case as shown in Figure 1. The two trajectories, (a) and (b), satisfy the condition to move to the next contact transition maintaining the contact relation. However, the trajectory (b) includes wasteful motion. Although the P RM method may not be able to find the waste, our method can, because the trajectory obtained by our method is equal to the trajectory (a). The contents of this paper are as follows: Section 2 easily illustrates our method to recognize assembly tasks proposed in [8]. Section 3 presents the method to calculate optimal trajectories. Section 4 verifies the validity of the method. Section 5 concludes this paper. 2 R e c o g n i t i o n of a s s e m b l y tasks In [81, we proposed a method to describe assembly tasks as a sequence of "sub-skills". These are essential motion primitives in assembly tasks, representing transitions between contact relations. Performing assembly tasks means to achieve the de-