The remainder of the thesis is structured as follows:
Chapter 2 provides a detailed kinematic analysis of the coaxial SPM under study. It begins with a description of the general terminology and notation,
followed by the presentation of computation approaches for obtaining unique solutions to the forward and inverse kinematic problems of the SPM. The chap- ter also includes a discussion of the infinite rotation motion of the coaxial SPM.
Chapter 3 describes the 3D-printed mechanical prototype developed for con- ducting experimental tests. It also demonstrates the simulation model designed in the CopelliaSim robotic simulator to verify link collision detection and control strategies used later in the thesis.
Chapter 4 presents the numerical methodology used to generate the Carte- sian workspace and configuration space for the coaxial SPM, including singularity analysis and link collision detection approaches. Three case studies with the simu- lation model and the real prototype are presented to demonstrate the application of the developed configuration space in motion trajectory generation.
Chapter 5 demonstrates the orientation control developed for the coaxial SPM prototype with external reference tracking and object stabilization capabilities.
Two cases are presented to demonstrate the application of the developed orien- tation control.
Chapter 6 concludes the thesis by highlighting its main achievements and contributions, as well as informing on the limitations of the conducted research and providing recommendations for future studies and experiments.
Chapter 2
Kinematic Analysis
2.1 Overview
The design process of any robotic mechanism should start with its kinematic modeling and analysis, which is followed by the detailed characterization of its Cartesian workspace and its joint space. The operation and control of robot ma- nipulators are based on the results of their kinematic analysis. Due to the complex architecture of PMs, their kinematic analysis is usually a lot more complicated if compared to SMs. For example, forward and inverse kinematic problems of general 3-DOF 3-RRR SPM, and coaxial SPM in particular, result in at most 8 solutions each [65]. As is the case for most PMs, inverse kinematics of 3-RRR SPMs is easier to solve and can be solved analytically as derived in [61, 64], though the problem of obtaining several solutions still exists. Unfortunately, forward kinematics of 3-RRR SPMs has no closed-form solutions and is usually solved using numerical methods, that are often computationally demanding.
An approach to solve SPM’s forward kinematics based on the derivation of the polynomial characteristic equation was described in [66]. It was demonstrated that this polynomial is of 8-th order and is minimal; solving it results in at most 8 real solutions to the SPM forward kinematic problem. In this approach, the orientation of the mobile platform is described by means of Euler angles.
Another approach for describing the mobile platform’s orientation is based on using the input-output equations of spherical four-bar linkages as reported in [232]. Solving obtained trigonometric equations using a semi-graphical approach
was outlined in [233]. Other methods based on solving polynomial expressions of forward kinematics of 3-RRR SPMs include [68,276]. A method that involves training neural networks was attempted in [277]. It is also a common practice, to design special kinematic architectures that result in simplified forward kinematics as in [65, 78]. In such designs, geometric parameters are selected in a way that allows for simplifying trigonometric equations. For example, the special geometry of Agile Eye as in [78] allows obtaining a closed-form solution for the forward kinematics and succeeded by its application in the real-time control. Redundant additional sensors such as external video cameras and rotary position sensors can also be used to simplify the forward kinematic analysis and estimate the actual orientation of the mobile platform [149, 278–284]. This allows for determining a single solution corresponding to the manipulator’s assembly mode uniquely, which is desirable for fast and correct control algorithms. However, it requires the addition of extra sensors, which in some occasions is not desired as it can increase the cost of the system. In this regard, a numerical approach for obtaining unique solutions, corresponding to a working or assembly mode of an SPM’s prototype, with no extra sensors use, was proposed in [89].
The kinematic analysis of the coaxial SPM studied in this thesis follows a similar numerical approach as in [89]. Being a specific case of the general SPM, coaxial SPM’s kinematics is based on the same fundamentals adapted from [66, 77, 232, 233]. This Chapter briefly introduces these fundamentals (Section 2.2), as well as an approach for obtaining unique forward and inverse kinematic solutions of the coaxial SPM (Sections 2.3 and 2.4, respectively). Methodology and algorithms presented in these Sections were published in a conference paper titled "Computation of Unique Kinematic Solutions of a Spherical Parallel Manipulator with Coaxial Input Shafts" [285], and presented at the 2019 IEEE 15th International Conference on Automation Science and Engineering. This pa- per was later extended with an approach and algorithm for generating infinite rotational motion of the coaxial SPM (Section 2.5). This work was published in a conference paper titled "Infinite Torsional Motion Generation of a Spherical Parallel Manipulator with Coaxial Input Axes" [286], and presented at the 2020 IEEE/ASME International Conference on Advanced Intelligent Mechatronics.