4,161 research outputs found
Dynamics and control of a class of underactuated mechanical systems
This paper presents a theoretical framework for the dynamics and control of underactuated mechanical systems, defined as systems with fewer inputs than degrees of freedom. Control system formulation of underactuated mechanical systems is addressed and a class of underactuated systems characterized by nonintegrable dynamics relations is identified. Controllability and stabilizability results are derived for this class of underactuated systems. Examples are included to illustrate the results; these examples are of underactuated mechanical systems that are not linearly controllable or smoothly stabilizable
On stabilization of nonlinear systems with drift by time-varying feedback laws
This paper deals with the stabilization problem for nonlinear control-affine
systems with the use of oscillating feedback controls. We assume that the local
controllability around the origin is guaranteed by the rank condition with Lie
brackets of length up to 3. This class of systems includes, in particular,
mathematical models of rotating rigid bodies. We propose an explicit control
design scheme with time-varying trigonometric polynomials whose coefficients
depend on the state of the system. The above coefficients are computed in terms
of the inversion of the matrix appearing in the controllability condition. It
is shown that the proposed controllers can be used to solve the stabilization
problem by exploiting the Chen-Fliess expansion of solutions of the closed-loop
system. We also present results of numerical simulations for controlled Euler's
equations and a mathematical model of underwater vehicle to illustrate the
efficiency of the obtained controllers.Comment: This is the author's version of the manuscript accepted for
publication in the Proceedings of the 12th International Workshop on Robot
Motion Control (RoMoCo'19
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
Feedback Synthesis for Controllable Underactuated Systems using Sequential Second Order Actions
This paper derives nonlinear feedback control synthesis for general control
affine systems using second-order actions---the needle variations of optimal
control---as the basis for choosing each control response to the current state.
A second result of the paper is that the method provably exploits the nonlinear
controllability of a system by virtue of an explicit dependence of the
second-order needle variation on the Lie bracket between vector fields. As a
result, each control decision necessarily decreases the objective when the
system is nonlinearly controllable using first-order Lie brackets. Simulation
results using a differential drive cart, an underactuated kinematic vehicle in
three dimensions, and an underactuated dynamic model of an underwater vehicle
demonstrate that the method finds control solutions when the first-order
analysis is singular. Moreover, the simulated examples demonstrate superior
convergence when compared to synthesis based on first-order needle variations.
Lastly, the underactuated dynamic underwater vehicle model demonstrates the
convergence even in the presence of a velocity field.Comment: 9 page
Singular trajectories of control-affine systems
When applying methods of optimal control to motion planning or stabilization
problems, some theoretical or numerical difficulties may arise, due to the
presence of specific trajectories, namely, singular minimizing trajectories of
the underlying optimal control problem. In this article, we provide
characterizations for singular trajectories of control-affine systems. We prove
that, under generic assumptions, such trajectories share nice properties,
related to computational aspects; more precisely, we show that, for a generic
system -- with respect to the Whitney topology --, all nontrivial singular
trajectories are of minimal order and of corank one. These results, established
both for driftless and for control-affine systems, extend previous results. As
a consequence, for generic systems having more than two vector fields, and for
a fixed cost, there do not exist minimizing singular trajectories. We also
prove that, given a control system satisfying the LARC, singular trajectories
are strictly abnormal, generically with respect to the cost. We then show how
these results can be used to derive regularity results for the value function
and in the theory of Hamilton-Jacobi equations, which in turn have applications
for stabilization and motion planning, both from the theoretical and
implementation issues
Stabilization of non-admissible curves for a class of nonholonomic systems
The problem of tracking an arbitrary curve in the state space is considered
for underactuated driftless control-affine systems. This problem is formulated
as the stabilization of a time-varying family of sets associated with a
neighborhood of the reference curve. An explicit control design scheme is
proposed for the class of controllable systems whose degree of nonholonomy is
equal to 1. It is shown that the trajectories of the closed-loop system
converge exponentially to any given neighborhood of the reference curve
provided that the solutions are defined in the sense of sampling. This
convergence property is also illustrated numerically by several examples of
nonholonomic systems of degrees 1 and 2.Comment: This is the author's version of the manuscript accepted for
publication in the Proceedings of the 2019 European Control Conference
(ECC'19
On Exponential Stabilization of Nonholonomic Systems with Time-Varying Drift
A class of nonlinear control-affine systems with bounded time-varying drift
is considered. It is assumed that the control vector fields together with their
iterated Lie brackets satisfy Hormander's condition in a neighborhood of the
origin. Then the problem of exponential stabilization is treated by exploiting
periodic time-varying feedback controls. An explicit parametrization of such
controllers is proposed under a suitable non-resonance assumption. It is shown
that these controllers ensure the exponential stability of the closed-loop
system provided that the period is small enough. The proposed control design
methodology is applied for the stabilization of an underwater vehicle model and
a front-wheel drive car.Comment: This is the author's version of the manuscript accepted for
publication in the Proceedings of the Joint 8th IFAC Symposium on Mechatronic
Systems and 11th IFAC Symposium on Nonlinear Control Systems (MECHATRONICS &
NOLCOS 2019
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