Reduced-Order Observer-Based Point-to-Point and Trajectory Controllers for Robot Manipulators

Abstract

This paper presents a design procedure for a reduced-order observer-based controller dedicated to n-joint robot manipulators. It is assumed that only the joint angular positions are measured. The joint angular velocities are estimated via an exponential reduced-order observer. Two types of control laws based on this observer are studied: point-to-point control with gravity compensation and trajectory control. Sufficient conditions to ensure the closed-loop stability are given. Performances of the reduced-order observer used with these two control laws are illustrated in a simulation study of a two-degrees-of-freedom robot manipulator.

Reduced-order observer design for a robotic manipulator

“…Theorem 3.1 Consider the manipulator system (5), let Assumptions 2.1 and 3.1 hold, and let η > 0 be fixed. Then the reduced-order observer ( 6) with linear gain function k(y) = k 0 y, where k 0 is assigned by (11), provides a locally exponentially stable estimation error ε = x 2 − x2 with a region of attraction that contains the set E defined in (12).…”

“…Considering a nonlinear gain function k(y), rather than a scalar and constant gain as in the case described earlier, might provide additional degrees of freedom in the observer design. For instance, it must be pointed out that the bound (11) is somewhat conservative and oversized as it is based on the least achievable eigenvalue for the inertia matrix. Furthermore, the spectrum of the inertia matrix can have large variations as the joint angles range over the admissible set, and the eigenvalues typically achieve their maximum and minimum for singular configurations of the joints.…”

“…Furthermore, the spectrum of the inertia matrix can have large variations as the joint angles range over the admissible set, and the eigenvalues typically achieve their maximum and minimum for singular configurations of the joints. As a consequence, in many scenarios, the observer is likely to provide an accurate estimation even if the gain is selected considerably lower than the bound given in (11). To overcome this issue and design an estimator with a tighter output injection gain, it is possible to adopt more sophisticated strategy.…”

“…Such design weakness is somewhat mitigated by the fact that the bound can be automatically enforced when the observer is used to implement feedback control laws, such as point-to-point regulation or trajectory tracking. Furthermore, available observers for robotic manipulators are typically full-order estimators, whereas the development of reduced-order schemes seems limited to a handful of papers, such as [11] which however still suffers from the requirement of a known velocity bound. Reduced-order observers are well recognized for having faster convergence rates and lower computational burden, as the only variables to be estimated are the ones which are actually not measured.…”

Reduced-order observer-based point-to-point and trajectory controllers for robot manipulators

“…The efficiency of the proposed reduced-order observer-based control schemes is shown by a comparison with the same state feedback controllers using the full-order state estimator proposed in (Deza and Gauthier, 1991) and (Deza, et al, 1993). A first version of this work has been previously published in an IFAC Conference (Khelfi, et al, 1995).…”