Milan Anderle

On Feedback Architectures With Zero-Vibration Signal Shapers

The classical and recent feedback solutions for signal shapers are reviewed and critically analyzed in this paper. Based on this assessment, a new effective feedback control architecture is proposed, suitable for manipulation of weakly damped flexible structures. More specifically, we propose to include the inverse of a signal shapers dynamics in the feedback path, and we justify this architecture by analysis of important feedback loop channels, namely the feedback responses from input disturbance to output and from reference to output. Both the classical ZV shaper with one lumped delay, and more recent shapers with distributed delays are considered and analyzed. The most appreciated features of suitable distributed-delay shapers are retarded characteristics-the closed-loop spectrum does not exhibit high-frequency roots close to the imaginary axis (in contrast to ZV shapers)-and at the same time reduced sensitivity of the closed loop interconnection to output disturbances. Feedback performance is demonstrated by two laboratory experiments.

Feedback design for the Acrobot walking-like trajectory tracking and computational test of its exponential stability

This paper aims to the further improve of the previously developed design for the Acrobot walking based on the partial exact feedback linearization of order 3. Namely, such an exact system transformation leads to an almost linear system where error dynamics along trajectory to be tracked is a 4 dimensional linear time varying system having 3 time varying entries only. Unlike previous approaches treating time varying entries as uncertainties with various extent of conservatism, the present paper takes into the account an information about these time varying functions including their derivatives up to order 4. Using that, the time varying state and the feedback transformation enable to design a fundamental matrix of the error dynamics in an explicit form and pre-designed stability properties. In particular, product of that fundamental matrix at the end of the single support walking phase by the impact map Jacobian enables directly prove stability of the hybrid cyclic walking like trajectory by computing certain 4×4 matrix and determining numerically whether its eigenvalues lie within the unit circle. This combination of analytical and numerical computations provides the justification of the exponential stability of the walking-like trajectory tracking. Moreover, it is supported by numerical simulations showing practically unlimited number of steps of the Acrobot “walking”.

Double oscillatory mode compensation by inverse signal shaper with distributed delays

Signal shaping technique can be distinguished as a very effective tool for suppressing both single and multiple oscillatory modes of a system linked with the shaper. Utilizing the concept of inverse signal shapers placed in the feedback loop, the double oscillatory mode suppression is addressed, considering that the oscillations induced by both the reference signal and the disturbance changes are to be compensated. In order to avoid the closed loop neutrality of the dynamics, the lumped delay that is usually considered in the shaper structure is substituted by a distributed delay. Next to the theoretical analysis, both the numerical and experimental examples are included.

Approximate feedback linearization of the Acrobot tracking error dynamics with application to its walking-like trajectory tracking

The purpose of this paper is to provide theoretical framework enabling to design tracking feedback for a general Acrobot trajectory which allows rigorous convergence proof. It is based on the partial exact feedback linearization of the Acrobot model followed by further approximate feedback linearization of the tracking error dynamics for arbitrary target trajectory. The approach presented here enables to prove the convergence in a rigorous way at least for small initial tracking errors. The slight novelty here is that neglecting is made with respect to tracking error along any general trajectory to be tracked, not just in some neighborhood of fixed working point. To demonstrate viability of this approach, simulations of a tracking of a walking-like cyclic trajectory are presented. The walking includes several steps including impacts between them. As a matter of fact, the exponentially stable tracking during the swing phase only is capable to stabilize overall walking, including the effect of the impacts.

Time-Delay Algorithms for Damping Oscillations of Suspended Payload by Adjusting the Cable Length

This paper deals with damping oscillations of a payload suspended to a fixed position base (e.g., a standing crane trolley). The damping is done by utilizing the Coriolis force generated by synchronizing translational and rotational motion of the payload. After revising existing approaches mainly based on approximating the payload motion by the motion of a mathematical pendulum, several nonlinear time-delay algorithms are proposed to handle the given task efficiently. The key benefit of the proposed methods is the feedback character of the algorithms guaranteeing the required synchronization of the cable length adjustment with the payload swing motion under the varying oscillation period and even under the effect of external disturbances. Note that the preceding algorithms were of open-loop character where such a synchronization was not guaranteed. The theoretical design is validated by simulations. The implementation aspects are discussed by relaxing the simplifying assumptions. Finally, the proposed time-delay feedback method is experimentally verified on a laboratory setup. The achieved results demonstrate viability of the proposed method for payload oscillation damping, e.g., in crane applications.

On the collocated virtual holonomic constraints in Lagrangian systems

In this paper, the collocated virtual holonomic constraints are introduced in the coordinate free manner and their flat coordinates representation is derived. Flat coordinates are those where the virtual holonomic constraints are given by a subset of coordinates components put to be zero. The flat coordinates representation enables a clear description of both the constraint system and the feedback imposing it. Results are illustrated by swinging up the mechanical four link chain. Possible outlooks for walking design are discussed as well.

Zero vibration derivative shaper with distributed delay for both feed-forward and feedback interconnections

Using a distributed delay as the key element, a novel zero vibration derivative (DαZVD) shaper is introduced with the objective to suppress effectively and robustly an undesirable oscillatory mode of a controlled system. Compared to the classical ZVD shaper with a lumped delay, the DαZVD shaper is less sensitive to system-model mismatch in higher frequency range. Various aspects of the novel shaper are studied, particularly the spectral patterns, time responses, and sensitivity features. Next, the closed loop architecture with an inverse implementation of the shaper is considered with the aim to suppress robustly the vibrations induced by both the set-point and disturbance changes.

Acrobot Stable Walking in Hybrid Systems Notation

The Acrobot walking is interesting and still challenging control problem. By virtue of an actuator location, the Acrobot is partially feedback linearizable up to order 3. This property is widely used both in a feedback tracking of a reference trajectory and in the reference trajectory design. Moreover, the walking of the Acrobot consists of continuous-time and discrete-time dynamics, therefore, the Acrobot belongs in a class of hybrid systems. The aim of this paper is to integrate continuous-time and discrete-time dynamics into a general model of hybrid systems and prove hybrid stability of the Acrobot walking by using both the already developed feedback control and the reference multi-step trajectory.