1,677 research outputs found
Obstacle Avoidance Problem for Second Degree Nonholonomic Systems
In this paper, we propose a new control design scheme for solving the
obstacle avoidance problem for nonlinear driftless control-affine systems. The
class of systems under consideration satisfies controllability conditions with
iterated Lie brackets up to the second order. The time-varying control strategy
is defined explicitly in terms of the gradient of a potential function. It is
shown that the limit behavior of the closed-loop system is characterized by the
set of critical points of the potential function. The proposed control design
method can be used under rather general assumptions on potential functions, and
particular applications with navigation functions are illustrated by numerical
examples.Comment: This is the author's accepted version of the paper to appear in: 2018
IEEE Conference on Decision and Control (CDC), (c) IEE
Nonholonomic Dynamics
Nonholonomic systems are, roughly speaking, mechanical
systems with constraints on their velocity
that are not derivable from position constraints.
They arise, for instance, in mechanical systems
that have rolling contact (for example, the rolling
of wheels without slipping) or certain kinds of sliding
contact (such as the sliding of skates). They are
a remarkable generalization of classical Lagrangian
and Hamiltonian systems in which one allows position
constraints only.
There are some fascinating differences between
nonholonomic systems and classical Hamiltonian
or Lagrangian systems. Among other things: nonholonomic
systems are nonvariational—they arise
from the Lagrange-d’Alembert principle and not
from Hamilton’s principle; while energy is preserved
for nonholonomic systems, momentum is
not always preserved for systems with symmetry
(i.e., there is nontrivial dynamics associated with
the nonholonomic generalization of Noether’s
theorem); nonholonomic systems are almost Poisson
but not Poisson (i.e., there is a bracket that together
with the energy on the phase space defines
the motion, but the bracket generally does not satisfy
the Jacobi identity); and finally, unlike the
Hamiltonian setting, volume may not be preserved
in the phase space, leading to interesting asymptotic
stability in some cases, despite energy conservation.
The purpose of this article is to engage
the reader’s interest by highlighting some of these
differences along with some current research in the
area. There has been some confusion in the literature
for quite some time over issues such as the
variational character of nonholonomic systems, so
it is appropriate that we begin with a brief review
of the history of the subject
Exponential stabilization of driftless nonlinear control systems using homogeneous feedback
This paper focuses on the problem of exponential stabilization of controllable, driftless systems using time-varying, homogeneous feedback. The analysis is performed with respect to a homogeneous norm in a nonstandard dilation that is compatible with the algebraic structure of the control Lie algebra. It can be shown that any continuous, time-varying controller that achieves exponential stability relative to the Euclidean norm is necessarily non-Lipschitz. Despite these restrictions, we provide a set of constructive, sufficient conditions for extending smooth, asymptotic stabilizers to homogeneous, exponential stabilizers. The modified feedbacks are everywhere continuous, smooth away from the origin, and can be extended to a large class of systems with torque inputs. The feedback laws are applied to an experimental mobile robot and show significant improvement in convergence rate over smooth stabilizers
The energy–momentum method for the stability of non-holonomic systems
In this paper we analyze the stability of relative equilibria of nonholonomic systems (that is, mechanical systems with nonintegrable constraints such as rolling constraints). In the absence of external dissipation, such systems conserve energy, but nonetheless can exhibit
both neutrally stable and asymptotically stable, as well as linearly unstable relative equilibria. To carry out the stability analysis, we use a generalization of the energy-momentum method combined with the Lyapunov-Malkin theorem and the center manifold theorem. While this approach is consistent with the energy-momentum method for
holonomic systems, it extends it in substantial ways. The theory is illustrated with several examples, including the the rolling disk, the roller racer, and the rattleback top
The power dissipation method and kinematic reducibility of multiple-model robotic systems
This paper develops a formal connection between the power dissipation method (PDM) and Lagrangian mechanics, with specific application to robotic systems. Such a connection is necessary for understanding how some of the successes in motion planning and stabilization for smooth kinematic robotic systems can be extended to systems with frictional interactions and overconstrained systems. We establish this connection using the idea of a multiple-model system, and then show that multiple-model systems arise naturally in a number of instances, including those arising in cases traditionally addressed using the PDM. We then give necessary and sufficient conditions for a dynamic multiple-model system to be reducible to a kinematic multiple-model system. We use this result to show that solutions to the PDM are actually kinematic reductions of solutions to the Euler-Lagrange equations. We are particularly motivated by mechanical systems undergoing multiple intermittent frictional contacts, such as distributed manipulators, overconstrained wheeled vehicles, and objects that are manipulated by grasping or pushing. Examples illustrate how these results can provide insight into the analysis and control of physical systems
Nonholonomic motion planning: steering using sinusoids
Methods for steering systems with nonholonomic constraints between arbitrary configurations are investigated. Suboptimal trajectories are derived for systems that are not in canonical form. Systems in which it takes more than one level of bracketing to achieve controllability are considered. The trajectories use sinusoids at integrally related frequencies to achieve motion at a given bracketing level. A class of systems that can be steered using sinusoids (claimed systems) is defined. Conditions under which a class of two-input systems can be converted into this form are given
Formation of Multiple Groups of Mobile Robots Using Sliding Mode Control
Formation control of multiple groups of agents finds application in large
area navigation by generating different geometric patterns and shapes, and also
in carrying large objects. In this paper, Centroid Based Transformation (CBT)
\cite{c39}, has been applied to decompose the combined dynamics of wheeled
mobile robots (WMRs) into three subsystems: intra and inter group shape
dynamics, and the dynamics of the centroid. Separate controllers have been
designed for each subsystem. The gains of the controllers are such chosen that
the overall system becomes singularly perturbed system. Then sliding mode
controllers are designed on the singularly perturbed system to drive the
subsystems on sliding surfaces in finite time. Negative gradient of a potential
based function has been added to the sliding surface to ensure collision
avoidance among the robots in finite time. The efficacy of the proposed
controller is established through simulation results.Comment: 8 pages, 5 figure
Configuration Controllability of Simple Mechanical Control Systems
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 derived. This condition involves an object that 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
- …