Shock and Vibration

Volume 2015, Article ID 294157, 13 pages

http://dx.doi.org/10.1155/2015/294157

## Payload Mass Identification of a Single-Link Flexible Arm Moving under Gravity: An Algebraic Identification Approach

Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain

Received 8 December 2014; Accepted 27 May 2015

Academic Editor: Reza Jazar

Copyright © 2015 Juan Carlos Cambera et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We deal with the online identification of the payload mass carried by a single-link flexible arm that moves on a vertical plane and therefore is affected by the gravity force. Specifically, we follow a frequency domain design methodology to develop an algebraic identifier. This identifier is capable of achieving robust and efficient mass estimates even in the presence of sensor noise. In order to highlight its performance, the proposed estimator is experimentally tested and compared with other classical methods in several situations that resemble the most typical operation of a manipulator.

#### 1. Introduction

Flexible link robotics is a research field focused on building robots with better performance than the conventional robots. The flexibility of these robots is a consequence of using links with lower sectional area and lighter materials than its rigid counterpart. Higher operational speed, lower energy consumption, better transportability, and lower cost are only a few advantages of the flexible link robots over the traditional rigid manipulators. These advantages can only be obtained by facing very challenging problems on modelling and control that after four decades have not been completely solved.

The flexible link robots are systems characterised by nonlinear ordinary, coupled, and partial differential equations, whose exact solution is not viable practically. This had led to look for models with manageable complexity but still reliable and useful for the design of controllers. From the control perspective, the same characteristics that improve the performance of these robots have led to vibration problems that undermine the positioning of the end effector. The solution to these problems can be very difficult considering the complexity of the model and, in the most general case, its nonminimum phase nature. Surveys dealing with dynamic modelling and control of flexible link robots can be found in [1–3].

In flexible link robotics, to guarantee an accurate positioning in pick and place tasks is a very important problem to solve. One of the main obstacles to overcome is to design a control algorithm capable of cancelling the vibrations when the dynamics is affected by changes in the payload mass. When these changes are not considered in the control design, the algorithm may lose accuracy and effectiveness in the vibration suppression and, in some cases, may become unstable.

Several works have addressed the problem from the adaptive control point of view. Most of them rely on the indirect methods category, which consists of two clearly differentiated stages (there are some early research works that use a direct approach; Siciliano et al. [4] and Yuh [5] applied the Model Reference Adaptive Control (MRAC)). In the first stage, an online identification of the system parameters is needed. In the second one, the parameters identified in the first stage are used to adjust the adaptive control law, in such a way that the overall performance of the system is improved. This paper is devoted to the real time characterization of the parameter that is most likely to change in a robot: the payload. In particular, we want to identify the tip mass, as we assume that the payload polar moment of inertia is negligible. Once this parameter has been identified, to update the dynamic model of the arm is immediate, and to recompute the controller parameters is straightforward.

A payload change affects in two ways a flexible link robot, it changes the vibration frequencies of the links, and it changes the motor torques demanded for a specific maneuver. Hence the identification algorithms can be classed, on a similar way, into frequency based approaches and model based approaches.

The frequency based approaches, normally, do not depend explicitly on the robot’s model but on the output where the vibrations appear and in some cases on the input. The more classical approaches are normally based on the FFT [6]. The adaptive notch filter [7] is one of the preferred methods because of its fast convergence rates and its low computational burden. Other approaches, like [8], have considered adaptive observers to perform simultaneously frequency and states estimation. A more recent work uses algebraic identification to estimate amplitude, frequency, and phase of a sinusoidal signal in the presence of noise and DC-offsets [9], which are two common problems not explicitly considered in the methods aforementioned.

In the second category, which we referred to as model based approaches, the payload mass is identified by using the dynamic model of the robot and the input and output signals. Least square based techniques, like [10–12], cover most of the work carried out under this category, but there are some other alternatives based on algebraic manipulations of the model transfer functions [13], Kalman filtering [14], or the already mentioned algebraic identification technique [15] that are also worth mentioning.

From the vibration control point of view, the frequency based approaches are at a disadvantage. The frequency based approaches require the system to vibrate at least a cycle fraction before the identification can be carried out. This condition goes against the main goal of the control algorithm, the vibration suppression, where a very fast identification is required. In order to update the controller as soon as possible during the trajectory execution, on the other hand, the model based approaches have proved to be highly reliable in problems concerning dynamic linear systems, but its applicability to nonlinear systems is not straightforward and may imply in some cases numerical differentiation of noisy signals.

In this paper, we will focus on the problem of real time identification of the tip mass of a single-link flexible arm that moves in a vertical plane under the effects of the gravity. The algorithm follows a model based approach and it is based on the algebraic identification framework, proposed in [16], and it generalizes a previous research work presented in [15]. Unlike the previous research work, where only movements in the horizontal plane were considered and then a linear model was used, in this paper we deal with a nonlinear dynamic model as a consequence of taking the gravity into account. This leads to a more complex problem, where most of the real time identification techniques developed up to date cannot be applied. Preliminary results of this identification algorithm were presented in [17]. This paper details our new identification algorithm, proposes several improvements, and presents a comparative analysis with other identification methods.

The algebraic identification provides a very fast and simple solution for online parameter estimation in systems where the parameters are piecewise constant (change from one constant value to another unpredictably). This methodology is fundamentally different from other approaches in some basics aspects: (a) it does not require any statistical knowledge of the noise corrupting the signals; therefore, classical Gaussian noise assumptions are not necessary; (b) it does not need to compute iterative time derivatives of noise corrupted signals; (c) it is not an asymptotic approach; and (d) it does not require persistently exciting inputs in order to make the system identifiable [18].

This paper is organised as follows. Section 2 presents the dynamic model of the flexible-link robot. In Section 3, the design of the algebraic identification algorithm is presented. Section 4 is devoted to the analysis and comparison of the experimental results. Conclusions and future work are presented in Section 5.

#### 2. Dynamic Model

Figure 1 shows a schematic representation of the flexible link robot. It consists of a motor and a flexible beam that bends on the vertical plane and therefore is affected by the gravitational force. One end of the beam is clamped to the shaft of the motor, while the other end moves freely and carries a payload. The model we introduce in this section was prepared according to the lumped mass method presented in [19] and relies on the following assumptions.(i)The link mass is negligible in comparison to the tip mass.(ii)The payload is considered as a point mass; therefore, its polar moment of inertia can be neglected.(iii)The deflections are elastic and small in relation to the link’s length, so that geometrical linearity can be assumed.(iv)Torsion and compression effects of the link are small in relation to the deflections.