# Inversion and Control of Non-minimumphase Multibody Systems

### Description

Lightweight robots allow faster motion compared to conventional robots due to lower inertial forces. Moreover, less energy is required for operation of the lightweight robots. On the other hand, the reduction in mass leads to less stiffness. While this is sometimes even desired and part of the robot design, the reduced stiffness often leads to undesirable oscillation modes. Such oscillations cannot be actuated directly with the available actuators, since the actuators are usually placed on the robots joints. Therefore, this kind of robots is underactuated with possibly many unactuated degrees of freedom. Such robots are often called flexible robots and their control is still a challenging task.

A two-degree-of freedom structure with a feedforward and feedback part is a typical control strategy for such systems. Thereby, the feedforward control is an inverse model of the system. It is responsible for large motions and trajectory tracking. It is usually combined with a feedback controller, which compensates modelling errors and disturbances. In this framework, many feedback controllers are applicable.

Our focus is on the design of inverse models for trajectory tracking of the end effector of flexible robots. Flexible robots are often non-minimum phase systems. In that case, the inverse model would yield unbounded system inputs when integrated forward in time. This property can be analyzed by transforming the equations of motion into input-ouput-normal form. This extracts the so-called internal dynamics, which need to be analyzed with respect to its stability. Systems with unstable internal dynamics are called non-minimum phase systems. This can be the case even for a simple two-arm robot with one passive joint. In order to avoid unbounded solutions, stable inversion is applied. Thereby, a boundary value problem is stated and solved for a stable solution of the internal dynamics. This comes at the cost of pre- and postactuation phases before and after the actual desired trajectory. Thus, the solution is non-causal.

In the video, a flexible manipulator is controlled with a feedforward controller. It is supposed to follow a smooth trajectory from 0° to 30°. The system input is an actuator on the left end of the manipulator. The system output is the angle between the right end of the manipulator and the joint at the left end. The manipulator in red shows the forward simulation with a system input obtained from stable inversion. The manipulator follows the desired trajectory exactly and there are no oscillations around the final position. In contrast, the manipulator in blue shows a solution for a system input which is obtained from inverting the equivalent rigid system (rigid arm). It induces oscillations in the system. Therefore, inversion of the rigid system is not sufficient and flexibilities should be considered.

In this project, we compare available methods for stable inversion and analyze them with respect to numerical efficiency. Then, we intend to extend the methods and apply them to robots with many unactuated degrees of freedom. Moreover, experiments are planned to validate the theoretical work.

Moreover, the feedforward control based on the inverse models should be combined with an adaptive feedback controller to account for disturbances and modelling errors. Thereby, the modelfree funnel controller is a favored choice. The combination of both approaches is analyzed in the framework of the joint research project with Thomas Berger funded by the DFG (GEPRIS).

### Selected Publications

*Berger, T., Otto, S., Reis, T. and Seifried, R.:*Combined open-loop and funnel control for underactuated multibody systems, Nonlinear Dynamics, 2018, online first [DOI:10.1007/s11071-018-4672-5]