3,434 research outputs found
Analysis of minimum required sliding friction coefficient in the brachistochronic motion of a mechanical system with nonlinear nonholonomic constraint
This paper analyzes the problem of brachistochronic planar motion of a mechanical system with nonlinear nonholonomic constraint. The nonholonomic system is represented by two Chaplygin blades [3,4,5,6] of negligible dimensions, which impose nonlinear constraint in the form of perpendicularity of velocities. The brachistrochronic planar motion is considered, with specified initial and terminal positions, at unchanged value of mechanical energy during motion. Differential equations of motion, where the reactions of nonholonomic constraints and control forces figure, are obtained on the basis of general theorems of mechanics [8]. Here, this is more convenient to use than some other methods of analytical mechanics applied to nonholonomic mechanical systems [9], where a subsequent physical interpretation of the multipliers of constraints is required. The formulated brachistochrone problem, with adequately chosen quantities of state, is solved as simple a task of optimal control as possible in this case [6,7] by applying the Pontryagin maximum principle [1]. The corresponding two-point boundary value problem of the system of ordinary nonlinear differential equations is obtained, which has to be numerically solved [2]. Numerical procedure for solving the two-point boundary value problem is performed by the method of shooting. On the basis of thus obtained brachistochronic motion, the active control forces, along with the reactions of nonholonomic constraints, are defined. Using the Coulomb friction laws [8,9], a minimum required value of the coefficient of sliding friction is defined [10], so that the considered system is moving in accordance with nonholonomic bilateral constraints
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
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
Nilpotent Bases for a Class of Non-Integrable Distributions with Applications to Trajectory Generation for Nonholonomic Systems
This paper develops a constructive method for finding a nilpotent basis for a special class of smooth nonholonomic distributions. The main tool is the use of the Goursat normal form theorem which arises in the study of exterior differential systems. The results are applied to the problem of finding a set of nilpotent input vector fields for a nonholonomic control system, which can then used to construct explicit trajectories to drive the system between any two points. A kinematic model of a rolling penny is used to illustrate this approach. The methods presented here extend previous work using "chained form" and cast that work into a coordinate-free setting
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
A new approach to the vakonomic mechanics
The aim of this paper is to show that the Lagrange-d'Alembert and its
equivalent the Gauss and Appel principle are not the only way to deduce the
equations of motion of the nonholonomic systems. Instead of them, here we
consider the generalization of the Hamiltonian principle for nonholonomic
systems with nonzero transpositional relations.
By applying this variational principle which takes into the account
transpositional relations different from the classical ones we deduce the
equations of motion for the nonholonomic systems with constraints that in
general are nonlinear in the velocity. These equations of motion coincide,
except perhaps in a zero Lebesgue measure set, with the classical differential
equations deduced with d'Alembert-Lagrange principle.
We provide a new point of view on the transpositional relations for the
constrained mechanical systems: the virtual variations can produce zero or
non-zero transpositional relations. In particular the independent virtual
variations can produce non-zero transpositional relations. For the
unconstrained mechanical systems the virtual variations always produce zero
transpositional relations.
We conjecture that the existence of the nonlinear constraints in the velocity
must be sought outside of the Newtonian model.
All our results are illustrated with precise examples
Generalized Hamilton-Jacobi equations for nonholonomic dynamics
Employing a suitable nonlinear Lagrange functional, we derive generalized
Hamilton-Jacobi equations for dynamical systems subject to linear velocity
constraints. As long as a solution of the generalized Hamilton-Jacobi equation
exists, the action is actually minimized (not just extremized)
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