Nonholonomic Mechanical Systems Research Articles

Lagrangian reduction of nonholonomic discrete mechanical systems by stages

In this work we introduce a category $LDP_d$ of discrete-time dynamical systems, that we call discrete Lagrange--D'Alembert--Poincar\'e systems, and study some of its elementary properties. Examples of objects of $LDP_d$ are nonholonomic discrete mechanical systems as well as their lagrangian reductions and, also, discrete Lagrange-Poincar\'e systems. We also introduce a notion of symmetry group for objects of $LDP_d$ and a process of reduction when symmetries are present. This reduction process extends the reduction process of discrete Lagrange--Poincar\'e systems as well as the one defined for nonholonomic discrete mechanical systems. In addition, we prove that, under some conditions, the two-stage reduction process (first by a closed and normal subgroup of the symmetry group and, then, by the residual symmetry group) produces a system that is isomorphic in $LDP_d$ to the system obtained by a one-stage reduction by the full symmetry group.

Geometric Optimal Trajectory Tracking of Nonholonomic Mechanical Systems

We study the tracking of a trajectory for a nonholonomic system by recasting the problem as a constrained optimal control problem. The cost function is chosen to minimize the error in positions and...

Integrability and Chaos in Figure Skating

We derive and analyze a three dimensional model of a figure skater. We model the skater as a three-dimensional body moving in space subject to a non-holonomic constraint enforcing movement along the skate's direction and holonomic constraints of continuous contact with ice and pitch constancy of the skate. For a static (non-articulated) skater, we show that the system is integrable if and only if the projection of the center of mass on skate's direction coincides with the contact point with ice and some mild (and realistic) assumptions on the directions of inertia's axes. The integrability is proved by showing the existence of two new constants of motion linear in momenta, providing a new and highly nontrivial example of an integrable non-holonomic mechanical system. We also consider the case when the projection of the center of mass on skate's direction does not coincide with the contact point and show that this non-integrable case exhibits apparent chaotic behavior, by studying the divergence of nearby trajectories We also demonstrate the intricate behavior during the transition from the integrable to chaotic case. Our model shows many features of real-life skating, especially figure skating, and we conjecture that real-life skaters may intuitively use the discovered mechanical properties of the system for the control of the performance on ice.

Reduction of a Hamilton — Jacobi Equation for Nonholonomic Systems

Nonholonomic mechanical systems have been attracting more interest in recent years because of their rich geometric properties and their applications in Engineering. In all generality, we discuss the reduction of a Hamilton-Jacobi theory for systems subject to nonholonomic constraints and that are invariant under the action of a group of symmetries. We consider nonholonomic systems subject to linear or nonlinear constraints, with different positioning with respect to the symmetries. We describe the reduction procedure first, to later reconstruct solutions in the unreduced picture, by starting from a reduced Hamilton-Jacobi equation. Examples can be depicted in a wide range of scenarios: from free particles with linear constraints, to vehicle motion.

Stable control strategy for a second-order nonholonomic planar underactuated mechanical system

This paper presents a stable control strategy for a planar four-link active–passive–active–active (APAA) underactuated mechanical system. The control objective is to move its end-point from any initial position to any target position. First, the controllers are designed to move the first link to the target angle which ensures the target position is within the reachable area of the planar three-link passive–active–active (PAA) system. Meantime, the system is reduced to a planar virtual Pendubot. Then, the periodic controllers are designed based on the nilpotent approximation model to make the first link return to the target angle and all angular velocities converge to zeroes, where the fuzzy modulating is added to adjust the convergence of angular velocities. Thus, the system is reduced to a planar virtual PAA system. Next, the control is divided into two stages, and based on the angle constraint relationships, the target angles of the planar virtual PAA system for the target position are obtained by using the online particle swarm optimisation algorithm. Each stage controllers are designed according to the target angles, so the end-point of the planar APAA system is controlled to the target position. Finally, the validity of the control strategy is demonstrated via simulations.

Robust Adaptive Sliding Mode Controller for a Nonholonomic Mobile Platform

In this paper, a robust adaptive sliding mode controller is designed for a mobile platform trajectory tracking. The mobile platform is an example of a nonholonomic mechanical system. The presence of holonomic constraints reduces the number of degree of freedom that represents the system model, while the nonholonomic constraints reduce the differentiable degree of freedom. The mathematical model was derived here for the mobile platform, considering the existence of one holonomic and two nonholonomic constraints imposed on system dynamics. The partial feedback linearization method was used to get the input-output relation, where the output is the error functions between the position of a certain point on the platform and the desired path. The dynamic error model was considered uncertain and subjected to friction torques on the wheels. The adaptive sliding mode control was utilized to design a robust controller, that will force the platform to follow the desired trajectory. The simulation of the proposed controller was done via MATLAB to reveal the ability of the robust adaptive sliding mode controller applied as a trajectory tracker for various path shapes.

Moving frames for Lie symmetries reduction of nonholonomic systems

Given a Lie group of finite-dimensional transformations acting on a manifold, there is always an action known as a long-acting group action. This action describes the fundamental basis of the Lie theory connecting groups of symmetry in differential equations. Differential invariants emerge as constants of the action of the prolongation of a group. Elie Cartan extended this in the twentieth century involving the geometry of the action of this group, grounding the so-called moving frame theory. With this theory, various applications are possible and detailed in the literature, such as symmetries of variational problems, conservation laws, invariant differential forms, and group invariant solutions. In order to demonstrate the approach, two nonholonomic constrained mechanical systems are exemplified to obtain either the general closed-solution in explicit form, when possible, or an order reduction provided by the Lie symmetries via moving frames. The first example is a coin with mass m rolling without slipping and takes on an inclined plane (x, y) with angle $$\alpha $$ and nonlinear constraint. The second example is a Chetaev type described by a dog pursuing a man in a plane surface with a nonholonomic restriction. A full detailed analysis is addressed to define the Lie symmetries and the corresponding moving frames obtained in both examples.

A Jet Bundle Approach to the Variational Structure of Nonholonomic Mechanical Systems

The aim of this paper is to study the geometry of nonholonomic mechanical systems in the realms of geometric mechanics and geometric analysis. This work was intended as an attempt at bringing together these two areas, in which geometric methods play the major role, in the study of nonholonomic systems. In this paper, we explore the geometry of Lagrangian mechanical systems subject to nonholonomic constraints using various bundle and variational structures intrinsically present in the nonholonomic setting. We consider the constrained Hamel equations of motion in a way that aids the analysis and helps to highlight the variational structure of such equations. To illustrate results of this work and as an application of the constrained Hamel formalism discussed in this paper, we conclude with taking the balanced Tennessee racer into consideration.

Energy-Preserving Integrators Applied to Nonholonomic Systems

We introduce energy-preserving integrators for nonholonomic mechanical systems. We will see that the nonholonomic dynamics is completely determined by a triple $$({{\mathcal {D}}}^*, \varPi , \mathcal {H})$$ , where $${{\mathcal {D}}}^*$$ is the dual of the vector bundle determined by the nonholonomic constraints, $$\varPi $$ is an almost-Poisson bracket (the nonholonomic bracket) and $$ \mathcal {H}: {{\mathcal {D}}}^*\rightarrow \mathbb {R}$$ is a Hamiltonian function. For this triple, we can apply energy-preserving integrators, in particular, we show that discrete gradients can be used in the numerical integration of nonholonomic dynamics. By construction, we achieve preservation of the constraints and of the energy of the nonholonomic system. Moreover, to facilitate their applicability to complex systems which cannot be easily transformed into the aforementioned almost-Poisson form, we rewrite our integrators using just the initial information of the nonholonomic system. The derived procedures are tested on several examples: a chaotic quartic nonholonomic mechanical system, the Chaplygin sleigh system, the Suslov problem and a continuous gearbox driven by an asymmetric pendulum. Their performance is compared with other standard methods in nonholonomic dynamics, and their merits verified in practice.

Forward and Inverse Dynamics of a Unicycle-Like Mobile Robot

In this research work, a new method for solving forward and inverse dynamic problems of mechanical systems having an underactuated structure and subjected to holonomic and/or nonholonomic constraints is developed. The method devised in this paper is based on the combination of the Udwadia-Kalaba Equations with the Underactuation Equivalence Principle. First, an analytical method based on the Udwadia-Kalaba Equations is employed in the paper for handling dynamic and control problems of nonlinear nonholonomic mechanical systems in the same computational framework. Subsequently, the Underactuation Equivalence Principle is used for extending the capabilities of the Udwadia-Kalaba Equations from fully actuated mechanical systems to underactuated mechanical systems. The Underactuation Equivalence Principle represents an efficient method recently developed in the field of classical mechanics. The Underactuation Equivalence Principle is used in this paper for mathematically formalizing the underactuation property of a mechanical system considering a particular set of nonholonomic algebraic constraints defined at the acceleration level. On the other hand, in this study, the Udwadia-Kalaba Equations are analytically reformulated in a mathematical form suitable for treating inverse dynamic problems. By doing so, the Udwadia-Kalaba Equations are employed in conjunction with the Underactuation Equivalence Principle for developing a nonlinear control method based on an inverse dynamic approach. As shown in detail in this investigation, the proposed method can be used for analytically solving in an explicit manner the forward and inverse dynamic problems of several nonholonomic mechanical systems. In particular, the tracking control of the unicycle-like mobile robot is considered in this investigation as a benchmark example. Numerical experiments on the dynamic model of the unicycle-like mobile robot confirm the effectiveness of the nonlinear dynamic and control approaches developed in this work.

Improving E. Cartan considerations on the invariance of nonholonomic mechanics

This paper concerns an intrinsic formulation of nonholonomic mechanics. Our point of departure is the paper [6], by Koiller et al., revisiting E. Cartan's address at the International Congress of Mathematics held in 1928 at Bologna, Italy ([3]). Two notions of equivalence for nonholonomic mechanical systems $ ( {\mathsf{{M}}}, {{\mathsf{{g}}}}, {\mathscr{D}}) $ are introduced and studied. According to [6], the notions of equivalence considered in this paper coincide. A counterexample is presented here showing that this coincidence is not always true.

Realizing brachistochronic planar motion of a variable mass nonholonomic mechanical system by an ideal holonomic constraint with restricted reaction

The paper considers realization of the brachistochronic motion of a nonholonomic mechanical system, composed of variable mass particles, by means of an ideal holonomic constraint with restricted reaction. It is assumed that the system performs planar motion in an arbitrary field of forces and that it has two degrees of freedom. In addition, the laws of the time-rate of mass variation of the particles, as well as relative velocities of the expelled and gained particles, respectively, are known. Restricted reaction of the holonomic constraint is taken for the scalar control. Applying Pontryagin?s maximum principle and singular optimal control theory, the problem of brachistochronic motion is solved as a two-point boundary value problem (TPBVP). Since the reaction of the constraint is restricted, different types of control structures are examined, from singular to totally nonsingular. The considerations are illustrated via an example.

A Riemannian Geometric Approach to Output Tracking for Nonholonomic Systems

The problem of designing coordinate-invariant output tracking control laws for nonholonomic mechanical systems is addressed. The velocity constrained Euler-Lagrange equations of motion are expressed through a constrained affine connection which is compatible with the kinetic energy Riemannian metric. This formalism is used in designing an output tracking control law via backstepping, which is shown to guarantee exponential stability when the initial distance between the output and reference trajectory is within injectivity radius of the output manifold. In particular this enables almost-global tracking when the output manifold is a rank-1 symmetric space. The control law is intrinsic to the Riemannian structure, and is explicitly constructed. The control law is applied to the problem of tracking the reduced attitude of a rigid body with a nonholonomic velocity constraint. Numerical simulations illustrating the tracking performance are presented.

Feedback Integrators for Nonholonomic Mechanical Systems

The theory of feedback integrators is extended to handle mechanical systems with nonholonomic constraints with or without symmetry, so as to produce numerical integrators that preserve the nonholonomic constraints as well as other conserved quantities. To extend the feedback integrators, we develop a suitable extension theory for nonholonomic systems, and also a corresponding reduction theory for systems with symmetry. It is then applied to various nonholonomic systems such as the Suslov problem on SO(3), the knife edge, the Chaplygin sleigh, the vertical rolling disk, the roller racer, the Heisenberg system, and the nonholonomic oscillator.

Analysis of multi-degree-freedom nonholonomic constraint mechanical systems using canonical coordinate’s method

The multi-degree-freedom nonholonomic constraint systems were analyzed using method of canonical coordinates. The theorem and characteristic equations of the canonical coordinate’s equations for multi-degree-freedom nonholonomic systems are obtained. The conserved quantities and general solutions of the systems are deduced directly by choosing appropriate canonical coordinates. An example is given to illustrate application of the results.

On the dynamics and control of underactuated nonholonomic mechanical systems and applications to mobile robots

This paper deals with the dynamics and control of underactuated nonholonomic mechanical systems. It is shown in this investigation that the same analytical methods can be used for effectively solving both the forward and the inverse dynamic problems relative to underactuated mechanical systems subjected to a general set of holonomic and/or nonholonomic algebraic constraint equations. The approach developed in this work is based on the combination of two fundamental methods of analytical dynamics, namely the Udwadia–Kalaba equations and the Underactuation Equivalence Principle. While the Udwadia–Kalaba equations represent a fundamental mathematical tool of classical mechanics, the Underactuation Equivalence Principle is a new method recently discovered in the field of analytical dynamics and is associated with nonholonomic mechanical systems. In the paper, these two important analytical methods are discussed in detail. Furthermore, numerical experiments are performed in this investigation in order to demonstrate the effectiveness of the proposed approach considering as an illustrative example of a dynamic model a mobile robot.

Dynamic modeling and trajectory tracking control of unmanned tracked vehicles

Tracked vehicles have inherent advantages over wheeled vehicles, as the former provide stable locomotion on loose and uneven terrain. However, compared with the latter, the slippage generated due to the complex, nonlinear track-terrain interactions during skid-steering to follow a curve, brings about difficulties preventing the accurate prediction of their motions. The key to improving the accuracy of trajectory-following is the “proper” motion control methodology that can accurately factor-in the slippage behavior. In this paper, the authors propose a novel approach to the dynamic modeling and motion control of tracked vehicles undergoing skid-steering on horizontal, hard terrain, under nonholonomic constraints. Due to the skew-symmetry property of nonholonomic mechanical systems, the control methodology is established using the backstepping method based on a modified Proportional–Integral–Derivative (PID) computed-torque control. A key element in the control strategy proposed here is the reliable estimation of the pose – position and orientation – of the vehicle platform and its twist—point velocity and angular velocity. It is assumed that the vehicle is suitably instrumented to allow for accurate-enough pose and twist estimates. Validated via a numerical example, the proposed controller is proven to be effective in controlling an unmanned tracked vehicle.

New methods to find solutions and analyze stability of equilibrium of nonholonomic mechanical systems

A large proportion of constrained mechanical systems result in nonlinear ordinary differential equations, for which it is quite difficult to find analytical solutions. The initial motions method proposed by Whittaker is effective to deal with such problems for various constrained mechanical systems, including the nonholonomic systems discussed in the first part of this paper, where in addition to differential equations of motion, nonholonomic constraints apply. The final equations of motion for these systems are obtained in the form of corresponding power series. Also, an alternative, direct method to determine the initial values of higher-order derivatives $${\ddot{q}}_0 ,{{\dddot{q}{} }}_{\!0} ,\ldots $$ is proposed, being different from that of Whittaker. The second part of this work analyzes the stability of equilibrium of less complex, nonholonomic mechanical systems represented by gradient systems. We discuss the stability of equilibrium of such systems based on the properties of the gradient system. The advantage of this novel method is its avoidance of the difficulty of directly establishing Lyapunov functions aimed at such unsteady nonlinear systems. Finally, these theoretical considerations are illustrated through four examples.

A geometric approach to the transpositional relations in dynamics of nonholonomic mechanical systems

Exploring the geometry of mechanical systems subject to nonholonomic constraints and using various bundle and variational structures intrinsically present in the nonholonomic setting, we study the structure of the equations of motion in a way that aids the analysis and helps to isolate the important geometric objects that govern the motion of such systems. Furthermore, we show that considering different sets of transpositional relations corresponding to different transitivity choices provides different variational structures associated with nonholonomic dynamics, but the derived equations (being referred to as the generalized Hamel–Voronets equations) are equivalent to the Lagrange–d’Alembert equations. To illustrate results of this work and as some applications of the generalized Hamel–Voronets formalisms discussed in this paper, we conclude with considering the balanced Tennessee racer, as well as its modification being referred to as the generalized nonholonomic cart, and an [Formula: see text]-snake with three wheeled planar platforms whose snake-like motion is induced by shape variations of the system.

Distributed optimal control for nonholonomic systems with input constraints and uncertain interconnections

This paper studies a distributed optimal tracking control method for nonholonomic mobile mechanical multi-agent systems under complex conditions such as input constraints, the presence of both kinematic and dynamic disturbances, and uncertain interconnections. Initially, novel feed-forward control inputs are proposed to transform the inherently separate systems of kinematics and dynamics into an equivalent integrated system. Successively, an online distributed $$\mathcal {L}_2$$ -bounded optimal control algorithm is designed by utilizing adaptive dynamic programming and the theory of cooperative differential graphical games. In the algorithm, a single neural network instead of three for each agent is chosen, and the online weight-tuning laws for which are designed without identifying uncertain parameters directly or indirectly. Additionally, the optimal control and worst disturbance policies are synchronously updated in only one iterative loop. It is shown that during the convergence of the value functions to the approximate optimal values when the agents perform the algorithm, overall tracking and function approximation errors are uniformly ultimately bounded. Finally, as a successful application of the study, control of the wheeled mobile multi-robot system is discussed through simulations.