A dissertation submitted to the Faculty of Engineering and the Built Environment,
University of the Witwatersrand, Johannesburg, in fulfilment of the requirements
for the degree of Master of Science in Engineering in the Control Research Group
School of Electrical and Information Engineering, Johannesburg, 2017 / In this research, we observed a torque-related limitation in the swing-up control
of underactuated mechanical systems which had been integrated with viscous
damping in the unactuated joint. The objective of this research project was thus to
develop a practical work-around solution to this limitation.
The nth order underactuated robotic system is represented in this research as a
collection of compounded pendulums with n-1 actuators placed at each joint with
the exception of the first joint. This system is referred to as the PAn-1 robot (Passive
first joint, followed by n-1 Active joints), with the Acrobot (PA1 robot) and the PAA
robot (or PA2 robot) being among the most well-known examples. A number of friction
models exist in literature, which include, and are not exclusive to, the Coulomb
and the Stribeck effect models, but the viscous damping model was selected for
this research since it is more extensively covered in existing literature. The effectiveness
of swing-up control using Lyapunov’s direct method when applied on the
undamped PAn-1 robot has been vigorously demonstrated in existing literature, but
there is no literature that discusses the swing-up control of viscously damped systems.
We show, however, that the application of satisfactory swing-up control using
Lyapunov’s direct method is constrained to underactuated systems that are either
undamped or actively damped (viscous damping integrated into the actuated joints
only). The violation of this constraint results in the derivation of a torque expression
that cannot be solved for (invertibility problem, for systems described by n > 2) or a
torque expression which contains a conditional singularity (singularity problem, for
systems with n = 2). This constraint is formally summarised as the matched damping
condition, and highlights a clear limitation in the Lyapunov-related swing-up control
of underactuated mechanical systems. This condition has significant implications
on the practical realisation of the swing-up control of underactuated mechanical
systems, which justifies the investigation into the possibility of a work-around. We
thus show that the limitation highlighted by the matched damping condition can be
overcome through the implementation of the partial feedback linearisation (PFL)
technique. Two key contributions are generated from this research as a result, which
iii
include the gain selection criterion (for Traditional Collocated PFL), and the convergence
algorithm (for noncollocated PFL).
The gain selection criterion is an analytical solution that is composed of a set of
inequalities that map out a geometric region of appropriate gains in the swing-up
gain space. Selecting a gain combination within this region will ensure that the
fully-pendent equilibrium point (FPEP) is unstable, which is a necessary condition
for swing-up control when the system is initialised near the FPEP. The convergence
algorithm is an experimental solution that, once executed, will provide information
about the distal pendulum’s angular initial condition that is required to swing-up a
robot with a particular angular initial condition for the proximal pendulum, along
with the minimum gain that is required to execute the swing-up control in this
particular configuration. Significant future contributions on this topic may result
from the inclusion of more complex friction models. Additionally, the degree of
actuation of the system may be reduced through the implementation of energy
storing components, such as torsional springs, at the joint.
In summary, we present two contributions in the form of the gain selection criterion
and the convergence algorithm which accommodate the circumnavigation of the
limitation formalised as the matched damping condition. This condition pertains to the
Lyapunov-related swing-up control of underactuated mechanical systems that have
been integrated with viscous damping in the unactuated joint. / CK2018
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:wits/oai:wiredspace.wits.ac.za:10539/24790 |
| Date | January 2018 |
| Creators | De Almeida, Ricardo Galhardo |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | Online resource (xxi, 358 leaves), application/pdf, application/pdf |
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