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Concept# Autonomous system (mathematics)

Summary

In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems.
Many laws in physics, where the independent variable is usually assumed to be time, are expressed as autonomous systems because it is assumed the laws of nature which hold now are identical to those for any point in the past or future.
Definition
An autonomous system is a system of ordinary differential equations of the form
\frac{d}{dt}x(t)=f(x(t))
where x takes values in n-dimensional Euclidean space; t is often interpreted as time.
It is distinguished from systems of differential equations of the form
\frac{d}{dt}x(t)=g(x(t),t)
in which the law governing the evolution of the system does not depend solely on the system's

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Nadia Barbara Figueroa Fernandez

Humans have a remarkable way of learning, adapting and mastering new manipulation
tasks. With the current advances in Machine Learning (ML), the promise of having
robots with such capabilities seems to be on the cusp of reality. Transferring human-level
skills to robots, however, is complicated as they involve a level of complexity that cannot
be tackled by classical ML methods in an unsupervised way. Such complexities involve:
(i) automatically decomposing tasks into control-oriented encodings, (ii) extracting
invariances and handling idiosyncrasies of data acquired from human demonstrations,
and (iii) learning models that guarantee stability and convergence. In this thesis, we
push the boundaries in the learning from demonstration (LfD) domain by addressing
these challenges with minimal human intervention and parameter tuning. We introduce
novel learning approaches based on Bayesian non-parametrics and kernel-methods while
encoding novel metrics and priors, leveraging them with dynamical systems (DS) theory.
In the first part of this thesis we focus on learning complex sequential manipulation tasks
from heterogeneous and unstructured demonstrations. Such tasks are those composed of a
sequence of discrete actions. The particular challenge is learning these tasks without any
prior knowledge on the number of actions (unstructuredness) or restrictions as to how
the human is demonstrating the task (heterogeneous), e.g. changes in reference frames.
We propose a Bayesian non-parametric learning framework that can jointly segment
and discover unique discrete actions (and their sequence) in continuous demonstrations
of the task. Hence, we learn an entire task from a continuous, unrestricted natural
demonstration. The learned structure of the complex tasks and the segmented data were
then used to parametrize a hybrid controller to execute two cooking tasks of increasing
complexity: (i) single-arm pizza dough rolling, and (ii) a dual-arm vegetable peeling task.
Throughout this thesis, we assume that both the human and robot motions are
driven by autonomous state-dependent DS. Hence, in the second part of this thesis we
offer two novel DS formulations to represent and execute a complex task. We begin
by proposing a DS-based motion generator formulation and learning scheme that is
capable of automatically encoding continuous and complex motions while ensuring global
asymptotic stability. The type of tasks that can be learned with this approach are
unparalleled to previous work in DS-based LfD and are validated on production line
and household activities. Further, we propose a novel DS formulation and learning
scheme that can encode both complex motions and varying impedance requirements
along the task; i.e., the robot must be compliant in some regions of the task, while stiff in others.
This approach is validated on trajectory tracking tasks where a robot arm
must precisely draw letters on a surface. Due to the generalization power and
straight-forward learning schemes of the proposed DS-based motion generators, we also
apply them to more complex application. Such applications include providing adaptive
(i) navigation strategies for mobile agents, and (ii) locomotion and co-manipulation tasks
of biped robots. These applications are particularly novel in the LfD domain, as most
work is solely focused on robotic arm control.
In the last part of the thesis, we explore learning complex behaviors in joint space for
single and multi-arm systems.