Discrete & Continuous Dynamical Systems - S
March 2010 , Volume 3 , Issue 1
Special Issue on Nonholonomic Constraints in Mechanics and Optimal Control Theory
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Nonholonomic mechanics describes the motion of systems with constraints on velocities that are not derivable from position or geometric constraints. The best known examples of such systems are rolling balls or disks and the sliding skate. Nonholonomic constraints have been the subject of deep analysis since the dawn of Analytical Mechanics. Recently, many authors have shown a new interest in that theory and also in its relation to the new developments in control theory, subriemannian geometry, robotics, etc. The main characteristic of this period is that geometry has been used in a systematic way.
Optimal control theory and geometry have also strongly influenced each other. This relationship begins with the formulation of the Maximum principle in the 1950's, initiating a geometrization program of optimal control theory which continues nowadays. It is now usual to work in optimal control theory using a differential geometric language (Lie algebras and Lie groups, integral manifolds, symplectic structures, riemannian and subriemannian geometries, homogeneous spaces...).
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In optimal control problems, there exist different kinds of extremals; that is, curves candidates to be solution: abnormal, normal and strictly abnormal. The key point for this classification is how those extremals depend on the cost function. We focus on control systems such as nonholonomic control mechanical systems and the associated kinematic control systems as long as they are equivalent.
With all this in mind, first we study conditions to relate an optimal control problem for the mechanical system with another one for the associated kinematic system. Then, Pontryagin's Maximum Principle will be used to connect the abnormal extremals of both optimal control problems.
An example is given to glimpse what the abnormal solutions for kinematic systems become when they are considered as extremals to the optimal control problem for the corresponding nonholonomic mechanical systems.
Controlled Lagrangian and matching techniques are developed for the stabilization of relative equilibria of discrete mechanical systems with symmetry and equilibria of discrete mechanical systems with broken symmetry. Unexpected phenomena arise in the controlled Lagrangian approach in the discrete context that are not present in the continuous theory. In particular, to make the discrete theory effective, one can make an appropriate selection of momentum levels or, alternatively, introduce a new parameter into the controlled Lagrangian to complete the kinetic shaping procedure. New terms in the controlled shape equation that are necessary for potential shaping in the discrete setting are introduced. The theory is illustrated with the problem of stabilization of the cart-pendulum system on an incline, and the application of the theory to the construction of digital feedback controllers is also discussed.
We construct different almost Poisson brackets for nonholonomic systems than those existing in the literature and study their reduction. Such brackets are built by considering non-canonical two-forms on the cotangent bundle of configuration space and then carrying out a projection onto the constraint space that encodes the Lagrange-D'Alembert principle. We justify the need for this type of brackets by working out the reduction of the celebrated Chaplygin sphere rolling problem. Our construction provides a geometric explanation of the Hamiltonization of the problem given by A. V. Borisov and I. S. Mamaev.
The paper develops discretization schemes for mechanical systems for integration and optimization purposes through a discrete geometric approach. We focus on systems with symmetries, controllable shape (internal variables), and nonholonomic constraints. Motivated by the abundance of important models from science and engineering with such properties, we propose numerical methods specifically designed to account for their special geometric structure. At the core of the formulation lies a discrete variational principle that respects the structure of the state space and provides a framework for constructing accurate and numerically stable integrators. The dynamics of the systems we study is derived by vertical and horizontal splitting of the variational principle with respect to a nonholonomic connection that encodes the kinematic constraints and symmetries. We formulate a discrete analog of this principle by evaluating the Lagrangian and the connection at selected points along a discretized trajectory and derive discrete momentum equation and discrete reduced Euler-Lagrange equations resulting from the splitting of this principle. A family of nonholonomic integrators that are general, yet simple and easy to implement, are then obtained and applied to two examples-the steered robotic car and the snakeboard. Their numerical advantages are confirmed through comparisons with standard methods.
In this paper the further investigation and development for the simplified mathematical model of a skateboard with a rider are obtained. This model was first proposed by Mont Hubbard [12, 13]. It is supposed that there is no rider’s control of the skateboard motion. To derive equations of motion of the skateboard the Gibbs-Appell method is used. The problem of integrability of the obtained equations is studied and their stability analysis is fulfilled. The effect of varying vehicle parameters on dynamics and stability of its motion is examined.
In this paper we discuss variational constrained mechanics (vakonomic mechanics) on Lie affgebroids. We obtain the dynamical equations and the aff-Poisson bracket associated with a vakonomic system on a Lie affgebroid $\mathcal A$. We devote special attention to the particular case when the nonholonomic constraints are given by an affine subbundle of $\mathcal A$ and we discuss the variational character of the theory. Finally, we apply the results obtained to several examples.
The mechanical system of two disks, moving freely in the plane, while in contact and rolling against each other without slipping, may be written as a Lagrangian system with three degrees of freedom and one holonomic rolling constraint. We derive simple geometric criteria for the rotational relative equilibria and their stability. Extending to three dimensions, we derive the kinematics of the analogous system where two spheres replace two disks, and we verify that the rolling disk system occurs as a holonomic subsystem of the rolling sphere system.
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