Department of Mathematics and Statistics: Dr. Andrew D. LewisSelected preprints of Dr. Andrew D. Lewishttps://hdl.handle.net/1974/8https://qspace.library.queensu.ca/retrieve/8470071b-fad9-47ea-b436-02aff7ff75fd/Math_and_Stats.jpg2024-04-22T23:48:39Z2024-04-22T23:48:39Z331Affine connection control systemsLewis, Andrew D.https://hdl.handle.net/1974/452016-11-29T14:38:26Z1999-01-01T00:00:00Zdc.title: Affine connection control systems
dc.contributor.author: Lewis, Andrew D.
dc.description.abstract: The affine connection formalism provides a useful framework for the
investigation of a large class of mechanical systems. Mechanical systems
with kinetic energy Lagrangians and possibly with nonholonomic constraints
are fit naturally into the formalism, and some results are stated in the
areas of controllability and optimal control for affine connection control
systems.
1999-01-01T00:00:00ZAffine connections and distributionsLewis, Andrew D.https://hdl.handle.net/1974/402016-11-29T14:38:28Z1996-01-01T00:00:00Zdc.title: Affine connections and distributions
dc.contributor.author: Lewis, Andrew D.
dc.description.abstract: We investigate various aspects of the interplay of an affine connection with
a distribution. When the affine connection restricts to the distribution, we
discuss torsion, curvature, and holonomy of the affine connection. We also
investigate transformations which respect both the affine connection and the
distribution. A stronger notion than that of restricting to a distribution
is that of geodesic invariance. This is a natural generalisation to a
distribution of the idea of a totally geodesic submanifold. We provide a
product for vector fields which allows one to test for geodesic invariance in
the same way one uses the Lie bracket to test for integrability. If the
affine connection does not restrict to the distribution, we are able to
define its restriction and in the process generalise the notion of the second
fundamental form for submanifolds. We characterise some transformations of
these restricted connections and derive conservation laws in the case when
the original connection is the Levi-Civita connection associated with a
Riemannian metric.
dc.description: Appears as ``Affine connections and distributions with applications
to nonholonomic mechanics'' in Reports on Mathematical Physics
42(1/2), pages 135-164, 1998 (Proceedings for the workshop on
Non-Holonomic Constraints in Dynamics held in Calgary in August
1997)
1996-01-01T00:00:00ZAn example with interesting controllability and stabilisation propertiesHirschorn, Ron M.Lewis, Andrew D.https://hdl.handle.net/1974/2122019-08-08T18:25:28Z2005-01-01T00:00:00Zdc.title: An example with interesting controllability and stabilisation properties
dc.contributor.author: Hirschorn, Ron M.; Lewis, Andrew D.
dc.description.abstract: A simple three-state system with two inputs is considered. The system's controllability is determined using properties of vector-valued quadratic forms. The quadratic structure is then used as the basis for the design of a homogeneous, discontinuous, stabilising feedback controller. The paper should be seen as an attempt to relate controllability of a system from a point to stabilisability of the system to the same point.
2005-01-01T00:00:00ZAspects of Geometric Mechanics and Control of Mechanical
SystemsLewis, Andrew D.https://hdl.handle.net/1974/392016-11-29T14:38:29Z1995-01-01T00:00:00Zdc.title: Aspects of Geometric Mechanics and Control of Mechanical
Systems
dc.contributor.author: Lewis, Andrew D.
dc.description.abstract: Many interesting control systems are mechanical control systems. In spite of
this, there has not been much effort to develop methods which use the special
structure of mechanical systems to obtain analysis tools which are
suitable for these systems. In this thesis we take the first steps towards a
methodical treatment of mechanical control systems.
First we begin to develop a framework for analysis of certain classes of
mechanical control systems. In the Lagrangian formulation we study "simple
mechanical control systems" whose Lagrangian is "kinetic energy minus
potential energy." We propose a new and useful definition of
controllability for these systems and obtain a computable set of conditions
for this new version of controllability. We also obtain decompositions of
simple mechanical systems in the case when they are not controllable. In the
Hamiltonian formulation we study systems whose control vector fields are
Hamiltonian. We obtain decompositions which describe the controllable and
uncontrollable dynamics. In each case, the dynamics are shown to be
Hamiltonian in a suitably general sense.
Next we develop intrinsic descriptions of Lagrangian and Hamiltonian
mechanics in the presence of external inputs. This development is a first
step towards a control theory for general Lagrangian and Hamiltonian
control systems. We also study systems with constraints. We first give a
thorough overview of variational methods including a comparison of the
"nonholonomic" and "vakonomic" methods. We also give a generalised
definition for a constraint and, with this more general definition, we are
able to give some preliminary controllability results for constrained systems.
dc.description: Ph.D. Thesis, defended 21 April, 1995,
1995-01-01T00:00:00ZConfiguration controllability of simple mechanical control
systemsLewis, Andrew D.Murray, Richard M.https://hdl.handle.net/1974/372016-11-29T17:04:35Z1995-01-01T00:00:00Zdc.title: Configuration controllability of simple mechanical control
systems
dc.contributor.author: Lewis, Andrew D.; Murray, Richard M.
dc.description.abstract: In this paper we present a definition of "configuration controllability" for
mechanical systems whose Lagrangian is kinetic energy with respect to a
Riemannian metric minus potential energy. A computable test for this new
version of controllability is also derived. This condition involves a new
object which we call the symmetric product. Of particular interest
is a definition of "equilibrium controllability" for which we are able
to derive computable sufficient conditions. Examples illustrate the
theory.
dc.description:
1995-01-01T00:00:00ZControllability and motion algorithms for underactuated Lagrangian systems on Lie groupsBullo, FrancescoLeonard, Naomi E.Lewis, Andrew D.https://hdl.handle.net/1974/2062019-08-08T18:25:28Z2000-01-01T00:00:00Zdc.title: Controllability and motion algorithms for underactuated Lagrangian systems on Lie groups
dc.contributor.author: Bullo, Francesco; Leonard, Naomi E.; Lewis, Andrew D.
dc.description.abstract: In this paper, we provide controllability tests and motion control algorithms for underactuated mechanical control systems on Lie groups with Lagrangian equal to kinetic energy. Examples include satellite and underwater vehicle control systems with the number of control inputs less than the dimension of the configuration space. Local controllability properties of these systems are characterised, and two algebraic tests are derived in terms of the symmetric product and the Lie bracket of the input vector fields. Perturbation theory is applied to compute approximate solutions for the system under small-amplitude forcing; in-phase signals play a crucial role in achieving motion along symmetric product directions. Motion control algorithms are then designed to solve problems of point-to-point reconfiguration, static interpolation and exponential stabilisation. We illustrate the theoretical results and the algorithms with applications to models of planar rigid bodies, satellites and underwater vehicles.
dc.description: The IEEE has asked that authors display copyright information for the online versions of IEEE publications. Here's my announcement.
This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.
2000-01-01T00:00:00ZControllability of a hovercraft model (and two general results)Lewis, Andrew D.Tyner, David R.https://hdl.handle.net/1974/602016-11-29T14:38:33Z2003-01-01T00:00:00Zdc.title: Controllability of a hovercraft model (and two general results)
dc.contributor.author: Lewis, Andrew D.; Tyner, David R.
dc.description.abstract: Modelling and controllability studies of a hovercraft system are undertaken.
The system studied is a little more complicated than some in the literature
in that the inertial dynamics of the thrust fan are taken into account. The
system is shown to be representative of a large class of systems that are
controllable only a set described by the zeros of a nontrivial analytic
function. Recent results for controllability using vector-valued quadratic
forms are useful in arriving at the stated conclusions. As part of the
development, two new controllability results of independent interest are
proved.
This paper was rejected for CDC'03 because of an erroneous review. We did,
and still do, think this is an interesting paper. Therefore, to maintain the
integrity of this online version, I am putting the review in question, and
our response, online. No malice towards the reviewer is intended; we merely
wish to reaffirm these results. Our response is in italics.
This was an interesting paper that I spent a lot of time on. It has many
drawbacks in terms of explanations, typos, etc. but was interesting and at
first blush appeared to be a nice use of the modern results of Lewis, Bullo,
Leonard, Murray, and others. The claimed result, however, is
counter-intuitive and caused me to dive into the proofs.
We are happy the reviewer found the results counter-intuitive. Indeed,
one's intuition can be difficult to trust in these sorts of problems.
The primary problem, and the reason I had to reject the paper, is that the
proof of the main result- Thm. 4.3- is incorrect. There are problems
throughout the proof but the main issue is as follows. It is established that
B_Y_q0 is NEVER definite- this is a fairly easy thing to show (although I
have problems with how it is shown in the proof but the basic idea is right).
In the final part of the proof is the line:
``It remains to show that when B_Y_q is indefinite only on a proper analytic
subset S, then from q0 in Q/S the system is not STLCC. This, however,
follows from Thm. 4.2.''
Unfortunately Thm 4.2 states that the system is not STLCC if B_Y_q is
definite. Since we've aready shown B_Y_q is not definite everywhere that thm
does not apply. Thiis invalidates the thm and prop. 4.7.
This completely misses the point of Theorem 4.2 which says that
$B_{Y_q}$ need only be semidefinite in order to conclude lack of
controllability, provided that the semidefiniteness holds, with a fixed rank,
in a neighbourhood of $q$.
Also, in the cases for the two linear maps: what do you mean by
dim(ker(L11(1))) = n-1? L11 maps R^(n-2) to R and thus the domain is only
n-2 dim'l. How is the dimension of the kernel larger than that of the
domain?
This is a typo. We were replacing in our mind $n-2$ with $n$.
Other problems throughout the paper to take a look at:
sec. 2.2, right at the start "...we note that Q=SE(2)xSO(2) is a principal
fibre bundle..." No, it isn't- not until you give me the base space. I know
it's obvious but the technical details should be correct. It is written
correctly later on (pi:Q -> SO(2))
This is correct, in some sense. We might have said
``$Q=SE(2)\times SO(2)$ is the total space of a principal fibre bundle.''
However, the reviewer left off the remainder of the sentence which says what
the group action is. This diminishes the technical offence to close to
nothing, we think.
sec. 2.2, proof of Lemma 2.1 You state that G-invariance is given by
DPhi(a,phi)'[g(a,phi)]DPhi(a,phi) = [g(abar.a,phi)]
This is incorrect- the inner product matrix on the LHS should be evaluated at
the translated point and on the RHS at the original point. Perhaps a better
expression would be g(Phi(a,phi))(TPhi.X,TPhi.Y) = g(a,phi)(X,Y) for all X,Y
in TqQ The same error is made in the proof of Lemma 2.2. Note that the
Lemmas are correct.
This is quite right. Another typo.
dc.description: This was submitted to CDC'03 and rejected due to an erroneous review.
See abstract for details.
2003-01-01T00:00:00ZControllable kinematic reductions for mechanical systems: concepts,
computational tools, and examplesBullo, FrancescoLewis, Andrew D.Lynch, Kevin M.https://hdl.handle.net/1974/492016-11-29T14:38:35Z2001-01-01T00:00:00Zdc.title: Controllable kinematic reductions for mechanical systems: concepts,
computational tools, and examples
dc.contributor.author: Bullo, Francesco; Lewis, Andrew D.; Lynch, Kevin M.
dc.description.abstract: This paper introduces the novel notion of kinematic reductions for mechanical
systems and studies their controllability properties. We focus on the class
of simple mechanical control systems with constraints and model them as
affine connection control systems. For these systems, a kinematic reduction
is a driftless control system whose controlled trajectories are also
solutions to the full dynamic model under appropriate controls. We present a
comprehensive treatment of local controllability properties of mechanical
systems and their kinematic reductions. Remarkably, a number of interesting
reduction and controllability conditions can be characterized in terms of a
certain vector-valued quadratic form. We conclude with a catalog of example
systems and their kinematic reductions.
dc.description:
2001-01-01T00:00:00ZDecompositions of control systems on manifolds with an affine connectionLewis, Andrew D.Murray, Richard M.https://hdl.handle.net/1974/2042019-08-08T18:25:28Z1997-01-01T00:00:00Zdc.title: Decompositions of control systems on manifolds with an affine connection
dc.contributor.author: Lewis, Andrew D.; Murray, Richard M.
dc.description.abstract: In this letter we present a decomposition for control systems whose drift vector field is the geodesic spray associated with an affine connection. With the geometric insight gained with this decomposition, we are able to easily prove some special results for this class of control systems. Examples illustrate the theory.
1997-01-01T00:00:00ZDiscussion on: ``Dynamic Sliding Mode Control for a Class of Systems with Mismatched Uncertainty''Lewis, Andrew D.https://hdl.handle.net/1974/2102019-08-08T18:25:28Z2005-01-01T00:00:00Zdc.title: Discussion on: ``Dynamic Sliding Mode Control for a Class of Systems with Mismatched Uncertainty''
dc.contributor.author: Lewis, Andrew D.
dc.description.abstract: This is an invited discussion paper on the paper ``Dynamic Sliding Mode Control for a Class of Systems with Mismatched Uncertainty'' by Xing-Gang Yan, Sarah K. Spurgeon, and Christopher Edwards, that will appear in the European Journal of Control.
2005-01-01T00:00:00Z