G. Le Vey

Stable inversion of SISO nonminimum phase linear systems through output planning: an experimental application to the one-link flexible manipulator

In this paper, we introduce a new technique that allows for causal stable inversion of linear nonminimum phase systems. This approach is presented through its application to a flexible arm robot. In contrast with the available techniques, which amounts to finding a particular initial value for the inverse system to have bounded state trajectories, the presented scheme starts from any free initial conditions of the inverse dynamics, and searches for proper output trajectories with a polynomial form. Those output trajectories are computed such that the effect of the unstable zeros are completely cancelled. Furthermore, the scheme deals equally with hyperbolic as well as nonhyperbolic systems. The stable inverse is then incorporated into an output tracking controller, using a classical static state feedback. This global controller permits an exact tracking of the planned output. Experiments for a one-link flexible arm show good end-effector tracking results.

Flexible Links Manipulators: from Modelling to Control

This paper relates recent results obtained in the field of modelling and control of flexible link manipulators and proposes an investigation of the problem raised by this type of systems (at least in the planar case). First, adopting the modal floating frame approach and the Newton–Euler formalism, we propose an extension of the models for control to the case of fast dynamics and finite deformations. This dynamic model is based on a nonlinear generalisation of the standard Euler–Bernoulli kinematics. Then, based on the models recalled we treat the end-effector tracking problem for the one-link case as well as for the planar multi-link case. For the one-link system, we propose two methods, the first one is based on causal stable inversion of linear non-minimum phase model via output trajectory planning. The other one is an algebraic scheme, based on the parametrization of linear differential operators. For the planar multi-link case the control law proposed is based on causal stable inversion over a bounded time domain of nonlinear non-minimum phase systems. Numerical tests are presented together with experimental results, displaying the well behaved of these approaches.

Some remarks on solvability and various indices for implicit differential equations

It has been shown [17,18,21] that the notion of index for DAEs (Differential Algebraic Equations), or more generally implicit differential equations, could be interpreted in the framework of the formal theory of PDEs. Such an approach has at least two decisive advantages: on the one hand, its definition is not restricted to a “state-space” formulation (order one systems), so that it may be computed on “natural” model equations coming from physics (which can be, for example, second or fourth order in mechanics, second order in electricity, etc.) and there is no need to destroy this natural way through a first order rewriting. On the other hand, this formal framework allows for a straightforward generalization of the index to the case of PDEs (either “ordinary” or “algebraic”). In the present work, we analyze several notions of index that appeared in the literature and give a simple interpretation of each of them in the same general framework and exhibit the links they have with each other, from the formal point of view. Namely, we shall revisit the notions of differential, perturbation, local, global indices and try to give some clarification on the solvability of DAEs, with examples on time-varying implicit linear DAEs. No algorithmic results will be given here (see [34,35] for computational issues) but it has to be said that the complexity of computing the index, whatever approach is taken, is that of differential elimination, which makes it a difficult problem. We show that in fact one essential concept for our approach is that of formal integrability for usual DAEs and that of involution for PDEs. We concentrate here on the first, for the sake of simplicity. Last, because of the huge amount of work on DAEs in the past two decades, we shall mainly mention the most recent results.It has been shown [17,18,21] that the notion of index for DAEs (Differential Algebraic Equations), or more generally implicit differential equations, could be interpreted in the framework of the formal theory of PDEs. Such an approach has at least two decisive advantages: on the one hand, its definition is not restricted to a “state-space” formulation (order one systems), so that it may be computed on “natural” model equations coming from physics (which can be, for example, second or fourth order in mechanics, second order in electricity, etc.) and there is no need to destroy this natural way through a first order rewriting. On the other hand, this formal framework allows for a straightforward generalization of the index to the case of PDEs (either “ordinary” or “algebraic”). In the present work, we analyze several notions of index that appeared in the literature and give a simple interpretation of each of them in the same general framework and exhibit the links they have with each other, from the formal point of view. Namely, we shall revisit the notions of differential, perturbation, local, global indices and try to give some clarification on the solvability of DAEs, with examples on time-varying implicit linear DAEs. No algorithmic results will be given here (see [34,35] for computational issues) but it has to be said that the complexity of computing the index, whatever approach is taken, is that of differential elimination, which makes it a difficult problem. We show that in fact one essential concept for our approach is that of formal integrability for usual DAEs and that of involution for PDEs. We concentrate here on the first, for the sake of simplicity. Last, because of the huge amount of work on DAEs in the past two decades, we shall mainly mention the most recent results.

Accurate Trajectory Tracking of Flexible Arm End-Point

Abstract This paper discusses the design of an open-loop control law for a flexible one-link robot. The obtained torque permits to track with low errors any smooth end-effector trajectory. This feedforward control torque is obtained using a new control approach, based on the parameterization of differential operators, which gave good results on the articular tracking (Benosman and Le Vey, 20006). We present a feedback extension of the obtained nominal torque and test the robustness of the closed-loop law with respect to unstructured uncertainties, parametric errors and measurement noises.

Stable Inversion of a Nonminimum Phase System Through Nonlinear Boundary Value Problem Formulation

Abstract In this work, we study the model inversion problem of a particular nonlinear nonminimum-phase system, namely the planar multilink flexible manipulator, with the end-effector position as the tracked output. The inverse method is based on nonlinear boundary value problem formulation. We give the general formulation for a discretized model obtained from the Euler-Bernouilli PDE with an assumed modes approach and present the efficiency of this approach for tracking problems, through simulation results for a two-links flexible manipulator.

End-effector Motion Planning for One-Link Flexible Robot

Abstract The problem studied here amounts to designing a feasible trajectory, defined for the end-effector of a flexible link robot. A trajectory planning scheme is presented, that allows for computing a path leading to a bounded required feed forward torque, implying an exact reproduction of this trajectory. The main contribution of the present work is that the presented planning technique, yields trajectories permitting the direct inversion of a non minimum phase system, leading to open loop smooth control torque. A theoretical stability study is presented for the closed-loop extension of the obtained open-loop control law.

Rest-to-rest motion for planar multi-link flexible manipulator

Abstract In this work we consider the problem of rest-to-rest motion for planar flexible multi-link manipulators. We introduce a simple idea permitting the cancellation of end-effector residual vibration when reaching a desired angular equilibrium position, in a fixed travelling time. No internal elastic damping effect is considered and the structure of the control is simply the sum of feedforward and joint feedback terms. The new approach concerns the computation of the feedforward control, which is based on backward integration of the elastic dynamics, starting from a rest position of the flexible arms. For fast rest-to-rest motion, the feedback compensator fails to drive the system states along the desired trajectories, due to the relatively large initial elastic error. To overcome this limitation, proper joint motion is planned between the desired initial and final positions through optimization techniques. This scheme is validated via numerical tests on a two-link planar manipulator

Kinematics of Free-Floating Systems through Optimal Control Theory

This work presents a new method for solving the inverse kinematical problem for freefloating space manipulators. It is based on a novel formulation of the problem within the framework of optimal control theory for discrete linear systems, thanks to a reconsideration of the energy conservation property of this class of systems. This way, inverse kinematics is obtained in a purely deductive manner, quite analogous to previous works by the author about dynamical forward and backward models, for either multibody or continuous hyperredundant actuated mechanical systems. One important by-product of the approach is an effective way for detecting dynamical singularities for the considered class of systems.

Experiments on the end-effector control for a one-link flexible manipulator

Abstract The problem studied here concerns the end-effector rest-to-rest motion in a desired fixed time for the one-link flexible arm. The experimental results obtained by the application of an algebraic scheme introduced in (Benosman and Le Vey, 2000c), are presented. Application of this method to a discretized model of the flexible arm, gives a closed-forme open-loop control torque leading, when added to a simple joint feedback , to rest-to-rest motion with elimination of the tip oscillation at the final time.

Experimental results for the end-effector trajectory tracking of a two-link flexible manipulator

Abstract In the present work, experimental results for end-effector tracking of a planar two-link flexible manipulators, are presented. The basic idea of the method, introduced in (Benosman and Le Vey, 2001 b ) in the general case of any finite number of links, is to use a nonlinear boundary value problem formulation, taking into account the full nonlinear model. One salient feature of the method is that it deals equally with hyperbolic and non-hyperbolic (e.g. with vanishing internal damping of the flexible links) zero dynamics equilibrium point (Benosman and Le Vey, 2001 a ; Benosman, 2002), a significant improvement over other approaches to the dealt with problem. Also, as we do not use noncausal computations, the method introduces no preactuation. The efficiency of this formulation is demonstrated, through experimental results for a two-link flexible manipulator.