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Über dieses Buch

This book shows how, through certain geometric transformations, some of the standard joints used in parallel robots can be replaced with lockable or non-holonomic joints. These substitutions allow for reducing the number of legs, and hence the number of actuators needed to control the robot, without losing the robot's ability to bring its mobile platform to the desired configuration. The kinematics of the most representative examples of these new designs are analyzed and their theoretical features verified through simulations and practical implementations.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction: Lockable and Non-holonomic Joints

Abstract
A serial robot is a set of rigid bodies, or links, connected in series through actuated joints, which are typically either revolute (R) or prismatic (P) joints. One end of this serial chain of links is called the base and the other the end-effector. In a parallel robot, the end-effector (also known in this case as the moving platform) is connected to the fixed base through several serial chains. In this case, most of the joints are not actuated. These passive joints are typically either universal (U) or spherical (S) joints. Prismatic, revolute, universal, and spherical joints constitute the four major joints used in robotics (Fig.  1.1).
Patrick Grosch, Federico Thomas

Chapter 2. Parallel Robots with Lockable Revolute Joints

Abstract
This chapter describes a class of reconfigurable parallel robots consisting of a fixed base and a moving platform connected by n \(\text {R}_\text {b}\)RPS serial chains, with \(n<6\). Only the prismatic joints are actuated and the first revolute joint in each chain can be locked or released during operation. It will be shown how the introduction of these lockable joints allow the prismatic actuators to maneuver to approximate 6-degrees-of-freedom motions for the moving platform. An algorithm for generating these maneuvers is also described. Then, a motion planner, based on the generation of a probabilistic road map, whose nodes are connected using the described maneuvers, is presented. The generated trajectories are also designed to avoid both singularities and possible collisions between legs. Part of the work presented in this chapter appeared in [8].
Patrick Grosch, Federico Thomas

Chapter 3. Spherical Non-holonomic Joints

Abstract
In this chapter, a method to generate under-actuated robots by substituting ordinary spherical joints (S joints) with non-holonomic spherical joints is described, a topic that was already treated in [3, 4]. The practicalities of implementing non-holonomic spherical joints are also included.
Patrick Grosch, Federico Thomas

Chapter 4. Kinematics of the 3 PU Spatial Robot

Abstract
In this chapter, the under-actuated 3\(\text {S}_\text {n}\) PU parallel robot presented in [1] is derived from the 6-3 fully-parallel robot. A compact formulation for its kinetostatics is presented, an essential result for its design and control. Part of the work presented in this chapter was carried out in collaboration with professor Raffaele di Gregorio, from the University of Ferrara, Italy. Part of this work appeared in [5, 6].
Patrick Grosch, Federico Thomas

Chapter 5. Motion Planning for the 3 PU Robot

Abstract
The kinetostatics of the 3\(\text {S}_\text {n}\!\!\) PU robot allowed us, in the previous chapter, to prove that this robot is able to locally move its moving platform—within some regions—in a six-dimensional configuration space. In this chapter we go a step further by presenting a solution to the motion planning problem which can be adapted to other non-holonomic parallel robots. This chapter presents some results obtained in cooperation with professors Krzysztof Tchoń and Janusz Jakubiak from Wrocław University of Technology, Poland, which already appeared in [3].
Patrick Grosch, Federico Thomas

Chapter 6. Kinematics of the -2UPS Spherical Robot

Abstract
In this chapter, we analize the kinetostatics of a non-holonomic parallel spherical robot which can maneuver to reach any orientation for its moving platform. We show how by properly locating the actuators, and by representing the platform orientation using Euler parameters, the analysis admits a simple bilinear formulation after introducing a local feedback transformation. Interestingly enough, the singularities introduced by this transformation coincide with the singularities of the robot Jacobian. Thus, from the practical point of view, no extra singularities are added. A complete description of the robot’s workspace, which also takes into account the limits of all joints, is presented. Part of this work has previously been published in [4].
Patrick Grosch, Federico Thomas

Chapter 7. Motion Planning for the -2UPS Robot

Abstract
As explained in Chap. 6, any system with two inputs and up to four generalized coordinates can always be transformed into chained form. Then, since the \(\text {S}_\text {n}\)-2UPS non-holonomic robot has two inputs (its leg lengths) and three generalized coordinates (its orientation parameters), its kinematics can be formulated in chained form. Given a system in chain form, its motion planning problem can be solved using well-established procedures (see [7] and the references therein) which means the motion planning problem for the analyzed spherical robot can be solved using one of these procedures, as Jakubiak did in [6] for a particular configuration of the system. In this case, Jakubiak used control functions given by truncated trigonometric series, as we already did in Chap. 5 for the 3\(\text {S}_\text {n}\) PU robot. Nevertheless, the use of these procedures requires a good understanding of sophisticated methods in non-linear control whose technicalities have proven a challenge to many practitioners who are not familiar with them. As an alternative, geometric motion planners have been proposed, for example, in [1]. The main advantage of this kind of motion planners is that they are based on elementary kinematics arguments. However, they generate stepped maneuvers, that is, maneuvers with intermediate instants with zero velocity that guide the moving platform to the desired orientation. This chapter presents a geometric motion planner able to steer the robot to the desired orientation through a differentiable path in the space of actuation variables. Part of the work presented in this chapter appeared in [5].
Patrick Grosch, Federico Thomas

Chapter 8. Conclusions

Abstract
It has been shown how the substitution of ordinary joints with lockable revolute joints (\(\text {R}_\text {b}\) joints) or non-holonomic spherical joints (\(\text {S}_\text {n}\) joints), which we have referred here to as unconventional joints, provide lower-mobility parallel robots with interesting features. One important common characteristic of this kind of robots is that they can approximate, in general, trajectories for their moving platform in a configuration space of dimension higher than the number of their continuously actuated joints. In this monograph, the kinematics of the following three robots have been analyzed: the 4\(\text {R}_\text {b}\)RPS spatial reconfigurable robot (Chap. 2), the 3\(\text {S}_\text {n}\) PU spatial non-holonomic robot (Chaps. 4 and 5), and the S-2SPS spherical non-holonomic robot (Chaps. 6 and 7).
Patrick Grosch, Federico Thomas
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