Control Of Nonholonomic Mechanical Systems Research Articles

Constraint Control of Nonholonomic Mechanical Systems

We derive an optimal control formulation for a nonholonomic mechanical system using the nonholonomic constraint itself as the control. We focus on Suslov's problem, which is defined as the motion of a rigid body with a vanishing projection of the body frame angular velocity on a given direction $\boldsymbol{\xi}$. We derive the optimal control formulation, first for an arbitrary group, and then in the classical realization of Suslov's problem for the rotation group $SO(3)$. We show that it is possible to control the system using the constraint $\boldsymbol{\xi}(t)$ and demonstrate numerical examples in which the system tracks quite complex trajectories such as a spiral.

Passive Configuration Decomposition and Passivity-Based Control of Nonholonomic Mechanical Systems

We propose a novel passivity-based stabilization control framework for a certain class of nonholonomic mechanical systems. First, we derive the notion of passive configuration decomposition, which enables us to configuration-level decompose the system's Lagrange–D’Alembert dynamics into two systems, each evolving on their respective configuration spaces and also individually inheriting Lagrangian structure and passivity from the original dynamics. We then propose passivity-based time-varying and passivity-based switching control schemes that, by utilizing some control actions defined on each of the configuration spaces of the two systems and also their interplay with the nonholonomic constraint, can achieve stabilization while exploiting the nonlinear dynamics of the decomposed dynamics. We also provide conditions under which the passive configuration decomposition is possible, establish an equivalence between certain conditions for the proposed controls and kinematic controllability, and elucidate robustness property of the proposed controls based on passivity. Finally, examples are provided along with their simulation and experimental results to illustrate the theory.

Adaptive motion/force control of nonholonomic mechanical systems with affine constraints

Journal provides a multidisciplinary forum for scientists, researchers and engineers involved in research and design of nonlinear processes and phenomena, including the nonlinear modelling of phenomena of the nature.

Motion/force tracking control of nonholonomic mechanical systems via combining cascaded design and backstepping

This paper considers motion/force tracking control of a class of Lagrange mechanical systems with classical nonholonomic constraints. A tracking control method is proposed by combining cascaded methods and backstepping techniques. The main results of this paper include three parts: (1) error dynamics between the kinematic system and the desired paths are transformed into a cascaded system consisting of two subsystems and an interconnection function; (2) under the framework of cascaded methods, virtual controllers for the subsystems are designed to stabilize the error dynamics; (3) the tracking controller is designed for the overall mechanical systems using backstepping techniques.

Passive Decomposition and Control of Nonholonomic Mechanical Systems

We propose nonholonomic passive decomposition, which enables us to decompose the Lagrange–D’Alembert dynamics of multiple (or a single) nonholonomic mechanical systems with a formation-specifying (holonomic) map <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$h$</tex></formula> into 1) shape system, describing the dynamics of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$h(q)$</tex></formula> (i.e., formation aspect), where <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$q \in \Re^n$</tex></formula> is the systems’ configuration; 2) locked system, describing the systems’ motion on the level set of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$h$</tex></formula> with the formation aspect <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$h(q)$</tex></formula> being fixed (i.e., maneuver aspect); 3) quotient system, whose nonzero motion perturbs both the formation and maneuver aspects simultaneously; and 4) energetically conservative inertia-induced coupling among them. All the locked, shape, and quotient systems individually inherit Lagrangian dynamics-like structure and passivity, which facilitate their control design/analysis. Canceling out the coupling, regulating the quotient system, and controlling the locked and shape systems individually, we can drive the formation and maneuver aspects simultaneously and separately. Notions of formation/maneuver decoupled controllability are introduced to address limitations imposed by the nonholonomic constraint, along with passivity-based formation/maneuver control design examples. Numerical simulations are performed to illustrate the theory. Extension to kinematic nonholonomic systems is also presented.

Passive Decomposition and Control of Nonholonomic Mechanical Systems

We propose nonholonomic passive decomposition, which enables us to decompose the Lagrange-D'Alembert dynamics of multiple (or a single) nonholonomic mechanical systems with a formation-specifying (holonomic) map h into 1) shape system, describing the dynamics of h(q) (i.e., formation aspect), where q ∈ ℜn is the systems' configuration; 2) locked system, describing the systems' motion on the level set of h with the formation aspect h(q) being fixed (i.e., maneuver aspect); 3) quotient system, whose nonzero motion perturbs both the formation and maneuver aspects simultaneously; and 4) energetically conservative inertia-induced coupling among them. All the locked, shape, and quotient systems individually inherit Lagrangian dynamics-like structure and passivity, which facilitate their control design/analysis. Canceling out the coupling, regulating the quotient system, and controlling the locked and shape systems individually, we can drive the formation and maneuver aspects simultaneously and separately. Notions of formation/maneuver decoupled controllability are introduced to address limitations imposed by the nonholonomic constraint, along with passivity-based formation/maneuver control design examples. Numerical simulations are performed to illustrate the theory. Extension to kinematic nonholonomic systems is also presented.

Advanced Programmed Motion Tracking Control of Nonholonomic Mechanical Systems

Trajectory tracking control of nonholonomic systems has been extended to tracking a desired motion. The desired motion is specified by equations of constraints, referred to as programmed, which may be differential equations of high order and may be nonholonomic. The strategy enables motion tracking control under the assumption that the system dynamics are accurately known. It is referred to as a model reference tracking control strategy for programmed motion. In this paper, adaptive and repetitive extensions of the strategy are proposed. Two selected advanced tracking control algorithms, i.e., the desired compensation adaptation law and the repetitive control law, which were originally dedicated to holonomic systems, are adapted to motion tracking control of nonholonomic systems. Simulation studies that illustrate programmed motion tracking control of systems with unknown parameters and the performance of repetitive motions are provided. A new performance measure to evaluate a programmed motion tracking performance is introduced.

Control of Nonholonomic Mechanical Systems

Control of Nonholonomic Mechanical Systems

Control of nonholonomic mechanical systems using reduction and adaptation

This paper considers the problem of controlling the motion of nonholonomic mechanical systems in the presence of uncertainty regarding the system model and state. It is proposed that a simple and effective solution to this problem can be obtained by first using a reduction procedure to obtain a lower dimensional system which retains the mechanical system structure of the original system, and then adaptively controlling the reduced system in such a way that the complete system is driven to the goal configuration. This approach is shown to be easy to implement and to ensure accurate motion control despite measurement and model uncertainty. The efficacy of the proposed control strategy is illustrated through computer simulations and preliminary hardware experiments with nonholonomic mechanical systems arising from both explicit kinematic constraints and symmetries of the system dynamics.

Decentralized adaptive control of nonholonomic mechanical systems

This paper considers the problem of controlling the motion of nonholonomic mechanical systems in the presence of incomplete information concerning the system model and state, and presents a class of decentralized adaptive controllers as a solution to this problem. The proposed control strategies provide simple and robust solutions to a number of important nonholonomic system control problems, including stabilization to an equilibrium manifold, motion control to an equilibrium point, and trajectory tracking control. All of the schemes are computationally efficient, are implementable without system dynamic model or rate information, and ensure uniform boundedness of all signals and accurate motion control. Computer simulation results are provided to complement the theoretical developments.

Sliding Mode Control of Nonholonomic Mechanical Systems: Underactuated Manipulators Case

The control of mechanical systems subject to a set of second-order nonholonomic constraints represents an important class of control problem. In this paper underactuated robots as an example of such nonholonomic systems are studied. Based on the results that smooth feedback stabilization to a single equilibrium point is not possible, a feedback scheme providing stability to a desired manifold is proposed.