Udwadia Kalaba Equations Research Articles

Dynamic Modeling and Control of a Flexible Planar Three-Link Mechanism Using Joint and Piezoelectric Actuators

Abstract This paper presents a new method of controlling the end effector position and orientation of a flexible planar three-link mechanism. The coupled dynamic model was formulated using a method based on the Udwadia–Kalaba equations of motion for constrained systems. Lyapunov's theory was used to develop a nonlinear control law using piezoelectric actuators and the unconstrained link dynamic models. Numerical simulation was used to demonstrate the system tracking performance of the end effector using a reduced order controller applied to a higher order truth dynamic model.

Planning and distributed control for cooperative transportation of a non-uniform slung-load by multiple quadrotors

This paper presents a novel control and planning algorithm for cooperative slung-load transportation using multiple quadrotors. Each aerial robot modeled as a quadrotor is connected to a non-uniform payload by the cables. Udwadia-Kalaba equations are used for calculating the forces on the cables acting on each robot. A novel leaderless distributed adaptive control algorithm is presented which not only ensures the safe transport of payload in the generated path in the presence of external disturbances but also manages to solve the problems related to the interaction of payload with the aerial robots. Moreover, the payload dynamics is not considered in the design of the controller which facilitates the load transportation in any size, shape, or weight by the robotic team without compromising performance. Then, an improved rapidly exploring random tree star (RRT*) algorithm is developed to generate desired path for each aerial robot in order to avoid known obstacles for cooperative payload transportation problem. The proposed path planning algorithm is capable of finding a free collision path in any situation, specially where high maneuverability is crucial to avoid collision of the payload with obstacles. Three numerical simulations are conducted to verify the proposed planning and control scheme. Additionally, in the distributed control algorithm simulation, two indexes including ISE and IADU demonstrate the improved performance of the distributed control over independent control of robots. The superior performance of the proposed controller in terms of reduced tracking errors in presence of external disturbances is also shown in comparison with a distributed nonlinear H∞ design.

Multibody modeling and nonlinear control of the pantograph/catenary system

In this paper, a closed-chain multibody model of a pantograph/catenary system is developed and used for the optimal design of a nonlinear controller based on an open-loop control architecture. The goal of the nonlinear controller is the reduction of the contact force arising from the pantograph/catenary interaction and, at the same time, the suppression of the mechanical vibrations of the pantograph mechanism. The analytical formulation employed in this paper for describing the nonlinear dynamics of the pantograph/catenary multibody system considers a Lagrangian approach and is based on a redundant set of generalized coordinates. The contact forces generated by the pantograph/catenary interaction are modeled in this work employing an elastic force element collocated between the pantograph pan-head and a moving support. The external support follows a prescribed motion law that simulates the periodic deployment of the catenary system. On the other hand, in this investigation, the algebraic constraints arising from the closed-loop topology of the pantograph multibody system are enforced employing a method based on the Udwadia–Kalaba equations recently developed in the field of analytical dynamics. Furthermore, the problem of the determination of an effective feedforward controller for reducing the pantograph/catenary contact force is formulated in this work as a nonlinear optimal control problem. For this purpose, the solution of the control optimization problem is carried out by using an adjoint-based computational procedure. Numerical simulations demonstrate the effectiveness of the nonlinear controller obtained in this investigation for the pantograph/catenary multibody system.

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.

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.

A comparative study of the principal methods for the analytical formulation and the numerical solution of the equations of motion of rigid multibody systems

The goal of this investigation is to perform a comparative analysis of the principal methodologies employed for the analytical formulation and the numerical solution of the equations of motion of rigid multibody mechanical systems. In particular, three formulation approaches are considered in this work for the analytical formulation of the equations of motion. The multibody formulation strategies discussed in this paper are the Reference Point Coordinate Formulation with Euler Angles (RPCF-EA), the Reference Point Coordinate Formulation with Euler Parameters (RPCF-EP), and the Natural Absolute Coordinate Formulation (NACF). Moreover, five computational algorithms are considered in this investigation for the development of effective and efficient solution procedures suitable for the numerical solution of the equations of motion. The multibody computational algorithms discussed in this paper are the Augmented Formulation (AF), the Embedding Technique (ET), the Amalgamated Formulation (AMF), the Projection Method (PM), and the Udwadia-Kalaba Equations (UKE). The multibody formulation approaches and solution procedures analyzed in this work are compared in terms of generality, versatility, ease of implementation, accuracy, effectiveness, and efficiency. In order to perform a general comparative study, four benchmark multibody systems are considered as numerical examples. The comparative study carried out in this paper demonstrates that all the methodologies considered can handle general multibody problems, are computationally effective and efficient, and lead to consistent numerical solutions.

On the Computational Methods for Solving the Differential-Algebraic Equations of Motion of Multibody Systems

In this investigation, different computational methods for the analytical development and the computer implementation of the differential-algebraic dynamic equations of rigid multibody systems are examined. The analytical formulations considered in this paper are the Reference Point Coordinate Formulation based on Euler Parameters (RPCF-EP) and the Natural Absolute Coordinate Formulation (NACF). Moreover, the solution approaches of interest for this study are the Augmented Formulation (AF) and the Udwadia–Kalaba Equations (UKE). As shown in this paper, the combination of all the methodologies analyzed in this work leads to general, effective, and efficient multibody algorithms that can be readily implemented in a general-purpose computer code for analyzing the time evolution of mechanical systems constrained by kinematic joints. This study demonstrates that multibody algorithm based on the combination of the NACF with the UKE turned out to be the most effective and efficient computational method. The conclusions drawn in this paper are based on the numerical results obtained for a benchmark multibody system analyzed by means of dynamical simulations.

On the Lagrange multipliers of the intrinsic constraint equations of rigid multibody mechanical systems

This paper deals with the Lagrange multipliers corresponding to the intrinsic constraint equations of rigid multibody mechanical systems. The intrinsic constraint equations are algebraic equations that are associated with nonminimal sets of orientation parameters employed for the kinematic representation of large finite rotations. Two coordinate formulations are analyzed in this investigation, namely the reference point coordinate formulation (RPCF) with Euler parameters and the natural absolute coordinate formulation (NACF). While the RPCF with Euler parameters employs the four components of a unit quaternion as rotational coordinates, the NACF directly uses the orthonormal set of nine direction cosines for describing the orientation of a rigid body in the three-dimensional space. In the multibody approaches based on the RPCF with Euler parameters and on the NACF, the use of a nonminimal set of rotational coordinates facilitates a general and systematic formulation of the differential–algebraic equations of motion. Considering the basic equations of classical mechanics, the fundamental problem of constrained motion is formalized and solved in this paper by using a special form of the Udwadia–Kalaba method. By doing so, the Udwadia–Kalaba equations are employed for obtaining closed-form analytical solutions for the Lagrange multipliers associated with the intrinsic constraint equations that appear in the differential–algebraic dynamic equations developed by using the RPCF with Euler parameters and the NACF multibody approaches. Two simple numerical examples support the analytical results found in this paper.

Relationship between the Udwadia–Kalaba equations and the generalized Lagrange and Maggi equations

In their paper “A New Perspective on Constrained Motion,” F. E. Udwadia and R. E. Kalaba propose a new form of matrix equations of motion for nonholonomic systems subject to linear nonholonomic second-order constraints. These equations contain all of the generalized coordinates of the mechanical system in question and, at the same time, they do not involve the forces of constraint. The equations under study have been shown to follow naturally from the generalized Lagrange and Maggi equations; they can be also obtained using the contravariant form of the motion equations of a mechanical system subjected to nonholonomic linear constraints of second order. It has been noted that a similar method of eliminating the forces of constraint from differential equations is usually useful for practical purposes in the study of motion of mechanical systems subjected to holonomic or classical nonholonomic constraints of first order. As a result, one obtains motion equations that involve only generalized coordinates of a mechanical system, which corresponds to the equations in the Udwadia–Kalaba form.

A constrained motion perspective of railway vehicles collision

Impacts, friction, and normal contact forces occur in the railway vehicles couplers. This paper presents a novel nonsmooth model of the train collision; until now, only penalized models have been approached. The train dynamics is described by an equality of measures formulated at the velocity level. The equations of motion are integrated using the Moreau time-stepping algorithm. Impulsive and normal contact forces are described by a set-valued law of Signorini type, while friction forces are described by a set-valued law of Coulomb type. The constrained forces are computed deducing a particular, simplified formulation of the Udwadia–Kalaba equations. The resulting algorithm is simple and straightforward. Both impulsive and nonimpulsive dynamics are casted in the same framework. Any feature or situation regarding train collisions may be modeled. A demonstrative application is presented. Simulations reveal nonsmooth phenomena like simultaneous multiple collisions, stick-slip, captures, and offset in the final equilibrium position.

New Approach to Mission Design Based on the Fundamental Equations of Motion

The complexity of space missions has been steadily increasing as new missions are expected to be designed to fly under more and more stringent constraints. This paper shows the use of a new, simple methodology that is useful for complex mission design. It deals with determining the “exact” control force needed to cause a spacecraft to move at a given inclination in a circular orbit around an oblate rotating body whose gravitational field is nonuniform. The methodology uses the explicit Udwadia–Kalaba equations of motions showing the simplicity, accuracy, and power of the approach, and the ease with which the control forces can be explicitly obtained.