Eduardo Bayo

A finite-element approach to control the end-point motion of a single-link flexible robot

A structural finite-element technique based on Bernoulli-Euler beam theory is presented which will permit the finding of the torques (or forces) that are necessary to apply at one end of a flexible link to produce a desired motion at the other end. This technique is suitable for the open loop control of the tip motion. It may also provide a good control law for feedback control. The finite-element method is used to discretize the equations of motion. This method has a major advantage in the fact that different material properties and boundary conditions like hubs, tip loads, changes in cross sections, etc., can be handled in a very simple and straightforward manner. The resulting differential equations are integrated via the frequency domain. This allows for the expansion of the desired end motion into its harmonic components and helps to visualize the complex wave propagation nature of the problem. The performance of the proposed technique is illustrated in the solution of a practical example. Results point out the potential that this technique has in the study of the dynamics and control not only of flexible robots, but also of any other flexible mechanisms like those used in biomechanics, where high precision at the tip of very light flexible arms is required.

Inverse Dynamics and Kinematics of Multi- Link Elastic Robots: An Iterative Frequency Domain Approach

A technique is presented and experimentally validated for solving the inverse dynamics and kinematics of multi-link flexible robots. The proposed method finds the joint torques necessary to produce a specified end-effector motion. Since the inverse dynamic problem in elastic manipulators is closely coupled to the inverse kinematic problem, the solution of the first also renders the displacements and rotations at any point of the manipulator, including the joints. Further more the formulation is complete in the sense that it includes all the nonlinear terms due to the large rotation of the links. The Timoshenko beam theory is used to model the elastic characteristics, and the resulting equations of motion are discretized using the finite element method. An iterative solu tion scheme is proposed that relies on local linearization of the problem. The solution of each linearization is carried out in the frequency domain. The performance and capabilities of this technique are tested, first through simulation analysis, and second through experimental validation using feed-forward control. Results show the potential use of this method not only for open-loop control, but also for incorporation in feedback control strate gies. 1. This section contains a revised and corrected formulation of open-chain case. A previous method reported by the first author in: Computed Torque for the Fosition Control of Open-Chain Flexible Robots. Proc. 1988 IEEE International Conference in Robotics and Automation contained an error.

A modified Lagrangian formulation for the dynamic analysis of constrained mechanical systems

Abstract A modified Lagrangian formulation is presented for the dynamic analysis of constraint mechanisms. The proposed method is based on a Hamiltonian description of the dynamics which leads to the Lagranges equations. However, the constraint conditions are not appended to the Lagranges equations in the form of algebraic or differential constraints, but instead inserted in them by means of a penalty formulation, and therefore the number of equations of the system does not increase. In addition, this approach directly leads to a system of ordinary differential equations, as opposed to the classical Lagranges formulation which results in differential algebraic equations. The resulting set of equations is of the form dot y =g(y,t) , which can be integrated by standard numerical algorithms. Finally, the proposed method is very systematic and general, and can model any type of constraint conditions, either holonomic or nonholonomic. A series of illustrative examples are analyzed. The results demonstrate the capabilities of the proposed method for simulation analysis.

Augmented lagrangian and mass-orthogonal projection methods for constrained multibody dynamics

This paper presents a new method for the integration of the equations of motion of constrained multibody systems in descriptor form. The method is based on the penalty-Augmented Lagrangian formulation and uses massorthogonal projections for the solution to satisfy the kinematic constraint conditions. The number of equations being solved is equal to the number of states, and does not depend on the number of constraint conditions. Therefore, the method is particularly suitable for systems with redundant constraints, singular configurations or topology changes. The major advantage of the new method relies on the fact that for a low computational cost, the constraints in positions, velocities and accelerations are satisfied to machine precision during the numerical integration. This process is efficiently done by means of a mass-orthogonal projection without the need for coordinate partitioning or reduction to a minimum set of coordinates. The projection scheme allows for a more accurate and robust integration of the equations of motion since constraint violations constitute one of the primary sources of numerical errors and instabilities during the integration process. The proposed projection is also applied to the classical Lagrangian approach, thus eliminating the need for further stabilization as well as the selection of parameters in Baumgartes method.

Computed torque for the position control of open-chain flexible robots

A technique is presented for the solution of the inverse dynamics of open-chain flexible robots. The proposed method finds the joint torques necessary to produce a specified end-effector motion. The formulation includes all the nonlinear terms due to the large rotation of the links, together with Timoshenko beam theory to model their elastic characteristics. The finite-element method is used to discretize the equations of motion. The performance and capabilities of this technique are tested through a simulation analysis. Results show the potential of the method not only for feedforward control, but also for incorporation in feedback control strategies.<<ETX>>

Exponentially Stable Tracking Control for Multijoint Flexible-Link Manipulators

In this paper we describe a new tracking control law for multijoint flexible-link manipulators. The scheme is a synthesis of the inverse dynamics solution for flexible manipulators developed by Bayo at UCSB and tracking control theory for rigid-link manipulators put forth by Bayard, Wen and others. We show that passive joint controllers, together with the feedforward of nominal joint torques corresponding to a desired end-effector trajectory, results in exponentially stable tracking control

Modeling and Solution Methods for Efficient Real-Time Simulation of Multibody Dynamics

Current simulation tools for multibody dynamics arenot problem dependent, they use the same modelingprocess to all cases regardless of theircharacteristics. In addition, real-time simulation ofsmall multibody systems is achievable by existingsimulation tools, however, real-time simulation oflarge and complex systems is not possible withexisting methods. This is a challenge that needs to beaddressed before further advances in mechanicalsimulation with hardware-in-the-loop andman-in-the-loop, as well as virtual prototyping aremade possible.This paper addresses the issue of how the modelingprocess – dependent versus independent co-ordinates, anddescriptor form versus state-space form of theequations of motion – affects the dynamic simulation ofmultibody systems and how it could be taken intoaccount to define the concept of intelligentsimulation. With this new concept all the factorsinvolved in the simulation process – modeling,equations, solution, etc. – are chosen and combineddepending upon the characteristics of the system to besimulated. It is envisioned that this concept willlead to faster and more robust real-time simulators.

Intelligent Simulation of Multibody Dynamics : Space-State and Descriptor Methods in Sequential and Parallel Computing Environments

Real-time dynamic simulation of large, realistic and complexmultibody systems is essential in developing modern technologies such asvirtual prototyping, man-in-the-loop simulators and intelligent vehiclecontrol systems. In order to achieve real-time performance, currentcommercial codes require the use of large costly computers, thuslimiting the number of potential users.This paper shows thatreal-time can be achieved on medium-size workstations if, on the onehand, an adequate combination of modeling, dynamic formulation, andnumerical integration scheme is selected and, on the other hand,advantage is taken of sparse matrix technology and parallel computing. Astudy of space-state and descriptor methods involving the dynamics of awhole vehicle model is carried out and in conclusion, two methods areproposed as the best candidates for real-time simulation.

Singularity-Free Augmented Lagrangian Algorithms for Constrained Multibody Dynamics

After a general review of the methods currently available for the dynamics of constrained multibody systems in the context of numerical efficiency and ability to solve the differential equations of motion in singular positions, we examine the acceleration based augmented Lagrangian formulations, and propose a new one for holonomic and non-holonomic systems that is based on the canonical equations of Hamilton. This new one proves to be more stable and accurate that the acceleration based counterpart under repetitive singular positions. The proposed algorithms are numerically efficient, can use standard conditionally stable numerical integrators and do not fail in singular positions, as the classical formulations do. The reason for the numerical efficiency and better behavior under singularities relies on the fact that the leading matrix of the resultant system of ODEs is sparse, symmetric, positive definite, and its rank is independent of that of the Jacobian of the constraint equations. The latter fact makes the proposed method particularly suitable for singular configurations.

On trajectory generation for flexible robots

A trajectory based on a Gaussian velocity profile is presented as an alternative to the double square pulse acceleration profile. The new trajectory leads to the fast positioning of the tip of a flexible robot with a minimal excitation of high-frequency modes. The torques necessary to move the robot according to this trajectory show a very smooth behavior. The absence of high-frequency content, present when double square pulse accelerations are considered, eliminates the Occurrence of undesired residual vibrations produced by modeling uncemnties at high frequencies. The excellent results obtained suggest the use of this new trajectory for fast and precise positioning of flexible robots.