# Abstract

This contribution is concerned with the design of observers for a single mast stacker crane, which is used, e. g., for storage and removal of loads in automated warehouses. As the mast of such stacker cranes is typically a lightweight construction, the system under consideration is described by ordinary as well as partial differential equations, i. e., the system exhibits a mixed finite-/infinite-dimensional character. We will present two different observer designs, an Extended Kalman Filter based on a finite-dimensional system approximation, using the Rayleigh-Ritz method and an approach exploiting the port-Hamiltonian system representation for the mixed finite-/infinite-dimensional scenario where in particular the observer-error system should be formulated in the port-Hamiltonian framework. The mixed-dimensional observer and the Kalman Filter are employed to estimate the deflection of the beam based on signals acquired by an inertial measurement unit at the beam tip. Such an approach considerably simplifies mechatronic integration as it renders strain-gauges at the base of the mast obsolete. Finally, measurement results demonstrate the capability of these approaches for monitoring and vibration-rejection purposes.

# Zusammenfassung

Dieser Beitrag beschäftigt sich mit dem Entwurf von Beobachtern für ein Hochregalbediengerät in automatisierten Lagerhallen. Da es sich bei dem Mast typischerschweise um eine Leichtbaukonstruktion handelt, wird das betrachtete Modell von gewöhnlichen und partiellen Differentialgleichungen beschrieben, d. h. das Modell weist einen gemischt finite-/infinite-dimensionalen Charakter auf. Im Rahmen dieser Arbeit werden zwei verschiedene Beobachterentwürfe vorgestellt, ein Extended Kalman Filter, das auf einer finite-dimensionalen Systemapproximation basiert, unter Verwendung der Rayleigh-Ritz-Methode, und einen Ansatz, der die port-Hamiltonsche Systemdarstellung für das finite-/infinite-dimensionale Szenario ausnutzt, wobei insbesondere das Beobachterfehlersystem als port-Hamiltonsches System formuliert wird. Der finite-/infinite-dimensionale Beobachter sowie das Kalman Filter werden zur Schätzung der Balkenverformung herangezogen, wobei lokale Vibrationen durch einen an der Mastspritze angebrachten Beschleunigungssensor erfasst werden. Schlussendlich soll anhand von Messungen an einem realen Labormodell die Leistungsfähigkeit dieser Ansätze in Kombination mit einem Beschleunigungssensor für die Überwachung kritischer Größen und die Möglichkeit der Schwingungsunterdrückung demonstriert werden.

## 1 Introduction

Single mast stacker cranes (SMCs), also called rack feeders, are deployed in automated warehouses or logistic centers in order to move payloads to desired positions. From an economic point of view, to improve the productivity the intention is to decrease the access time of such stacker cranes. To this end, stacker cranes are typically build as lightweight constructions, consisting of driving units in fixed ducts and flexible mast structures. Therefore, a mathematical model of the SMC is characteristically governed by a combination of ordinary differential equations (ODEs) and partial differential equations (PDEs). However, as a consequence of the lightweight construction undesirable beam vibrations occur. This problem is of increasing relevance in practical applications as stacker cranes in automated warehouses are ever growing in height due to floor space constraints. Therefore, the idea is to address the problem from a control engineering point of view. With regard to appropriate control methods, for the governing mathematical model it is necessary to distinguish between scenarios allowing for an active lifting unit or not. Note that an active lifting unit implies a more complex model, however, it allows to decrease the access time since the driving and lifting unit move simultaneously. For example, in [5] a flatness-based feedforward control for the scenario of an SMC with an inactive lifting unit – i. e., the lifting unit remains at a constant position – has been presented, where the flatness of the mixed-dimensional problem has been shown. In contrast, in [1] and [3] a flatness-based feedforward control for a finite-dimensional system approximation, which was obtained by the Rayleigh-Ritz method, has been derived, where an active lifting unit has been taken into account. Furthermore, in [3] a passivity-based controller has been proposed for the stabilization of the infinite-dimensional error system, whereas in [1], [2] a dynamic controller based on the so-called energy-Casimir method has been derived for that purpose.

In this contribution, instead of discussing a further control methodology for SMCs, we focus on the derivation of proper observation strategies. A well-known approach for linear systems governed by ODEs is the Kalman Filter. Moreover, extensions of the Kalman Filter, like the Extended Kalman Filter (EKF), have become widely used for the state estimation of nonlinear systems, see, e. g., [4]. Therefore, the EKF can also be applied for the SMC by considering its finite-dimensional approximation.

A further relevant research topic is the observer design for infinite-dimensional systems, see, e. g., [6], where an approach based on the so-called backstepping methodology has been presented. Furthermore, in [7] a dissipativity-based observer-design strategy has been discussed. However, note that compared to systems governed by ODEs, in the distributed-parameter scenario there is an enormous rise of complexity, as, besides others, the stability analysis of the infinite-dimensional observer-error systems requires special attention. In fact, functional analytic methods are used to show the well-posedness of the governing PDE system, see [8] exemplarily for a detailed closed-loop stability investigation of a gantry crane with a heavy chain. To ensure that the observer state indeed converges to the system state, additionally the asymptotic stability has to be verified, see, e. g., [9], where LaSalle’s invariance principle for infinite-dimensional systems was used.

In this contribution, we exploit a certain port-Hamiltonian (pH) system representation for the observer design. It should be noted that for the infinite-dimensional case, in particular the so-called Stokes-Dirac scenario, see, e. g., [11], as well as an approach based on jet-bundle structures [13], [14] have turned out to be especially suitable. The main difference of these approaches is the choice of the states – energy variables versus derivative coordinates – see [10] for a detailed comparison by means of the Mindlin plate. Furthermore, in [12] the pH system representation based on Stokes-Dirac structures, where strain variables are exploited for mechanical systems, has been used for the observer design. The SMC can be modelled in a variational framework, and furthermore, since the beam deflection is of particular interest in our approach, we base our considerations on the jet-bundle scenario.

For the purpose of structure health monitoring and vibration rejection at least partial information about the current distortion of the beam is required. The mentioned controllers for the SMC presented in [1], [2] and [3] rely on the ability to measure the bending moment at the beam base by the use of a strain gauge. Integrating strain-gauges for directly measuring these quantities of interest can be challenging. Therefore, the proposed observer estimates the corresponding signals based on acceleration measurements acquired at the tip of the beam.

Thus, the main contributions of this paper are as follows. In Section 3, we present two observers for the model of an SMC with active lifting unit, which is briefly introduced in Section 2. For the finite-dimensional system approximation of the SMC, which is obtained by the Rayleigh-Ritz method, we propose an Extended Kalman Filter. Furthermore, based on the pH system representation we derive a mixed finite-/infinite-dimensional observer. In Section 4 we provide an experimental validation of both observers on a lab demonstrator, where each observer is tested in two selected scenarios. In Scenario A we confine ourselves to an inactive lifting unit, whereas in Scenario B an active lifting unit is considered and furthermore, also a closed-loop experiment based on a damping injection controller is presented. Section 5 is dedicated to a sketch of the stability proof regarding the asymptotic behaviour of the observer-error system of the finite-/infinite-dimensional observer where for simplicity we restrict ourselves to Scenario A with an inactive lifting unit at the top.

## 2 Mathematical modelling

In this section we briefly recapitulate the derivation of the mixed finite-/infinite-dimensional model based on the variational principle together with its port-Hamiltonian representation, as well as a finite-dimensional system approximation obtained by means of the Rayleigh-Ritz method, see [1], [3] and [15] for more details. The considered physical model, see Fig. 1, consists of three main mechanical parts. The position of the rigid driving unit with mass *L*, the mass density *w*, and its spatial derivatives are represented with the subscript *Y*, where

### 2.1 Finite-/infinite-dimensional model

Under the assumptions mentioned above the overall kinetic energy of the single mast stacker crane reads as

while the overall potential energy is given by

Therefore, the Lagrangian is considered to be of the form

the ordinary differential equations of the driving unit, the lifting unit and the tip mass

*M*and the shear force

*Q*evaluated at specific spatial positions

### Figure 1

### 2.2 Port-Hamiltonian representation

In this section the finite-/infinite-dimensional problem, given in Section 2.1, is represented as a port-Hamiltonian system. For this purpose, we introduce by means of regular Legendre mappings the momenta

The partial derivative of the finite-dimensional part with the Hamiltonian

and

with

Here, the canonical skew-symmetric matrix

describe the internal power flow of the finite-dimensional part and allows to incorporate the constraint forces

Consequently, the output of the port-Hamiltonian system (7) collocated to the external input

whereas those of the constraint forces are given by

### Remark 1.

*Note that due to the expression* *in the matrix* *it becomes obvious that the system possesses a nonlinear characteristic regarding the coupling expression in the principle of the conversation of momentum* (3c)*.*

### 2.3 Finite-dimensional approximation

A finite-dimensional model can be obtained by approximating the deformation of the beam

with a spatial ansatz function

where the mass matrix, with the ansatz abbreviation

with the constant mass abbreviations

Therefore, with

At this point, only the determination of the ansatz function

with *ω*. This can be achieved by substituting the ansatz (14) in the adapted boundary constraints (3e) and (4) to obtain a system of equations *ω*, we obtain a non trivial solution for *K* and therefore a proper ansatz function

## 3 Observer design

The focus in this section is on the design of two observers for the SMC. To begin with, we briefly present an Extended Kalman Filter for the finite-dimensional approximated model. Afterwards, in Section 3.2 a pH observer based on the mixed finite-/infinite-dimensional characteristics of the SMC is presented.

### 3.1 Extended Kalman Filter

This section is dedicated to the observer design based on the finite-dimensional system approximation (11). As mentioned, the aim is to design an algorithm in order to estimate the shear force and the bending moment at the bottom of the flexible beam, respectively, which are of particular importance with regard to vibration suppression. If the ansatz function is used for the deflections in (5), the problem corresponds to estimate the Ritz variable

With the objective to do state estimation without measuring the bending moment by the use of a strain gauge, our effort is to develop an observation strategy with the acceleration *P*,

*k*, the system state

*a posteriori*and

*a priori estimation*with

### 3.2 Infinite-dimensional observer

The underlying idea is to exploit the port-Hamiltonian system representation and its collocated outputs for the observer design, by using a copy of the plant extended by an observer-correction term

### Remark 2.

*It should be stressed, that due to the measurement* *we omit the corresponding balance of linear momentum* (3c)*. However, as* (3d) *remains, the coupling of the active lifting unit and the beam is still valid.*

### Table 1

Symbol | Value | Symbol | Value | Symbol | Value | Symbol | Value |

13.1 kg | L |
0.53 m | 0.32 kg | α |
3000 | ||

0.86 kg | 14.97 N m^{2} |
2.1 kg/m | β |
20 |

The observer-correction terms

with

the constraint shear forces

the error system with state

where the outputs (21) are collocated to the observer-correction terms

Subsequently, we show that by a proper choice of

### Remark 3.

*Note that the necessity of the additional term in* (22) *turns out within detailed well-posedness investigations. If we omit this term, we are not able to detect steady state errors since we have no information about the error deflection. Furthermore, the equivalence of the energy norm* *, where* *can be used as a Lyapunov functional with regard to stability investigations, and the natural norm* *of an appropriate function space* *, where* *, cannot be shown. This fact implies that the observer-error system would not be well-posed. To overcome this, we incorporate the term* *.*

If we investigate the formal change of the Lyapunov functional (22), which corresponds to the collocated energy ports and the derivation of the incorporated term of Remark 3,

## 4 Experimental results

In the following, the application of both observers are validated by measurement results obtained from a laboratory model of a single mast stacker crane, where the parameters are stated in Table 1. It should be mentioned that several control strategies for equilibrium point stabilization as well as for trajectory tracking purposes have been successfully implemented for the finite and mixed-dimensional problem, see, e. g., [1] and [3]. Exemplarily, an IDA-PBC law from [3] is given by

*d*represents reference trajectories, and

*Q*, see (15). Moreover, the Ritz coordinate is also associated with the relative deflection

*Y*and the shear force. The laboratory model possesses a strain gauge in order to measure the bending moment and the shear force, respectively, although, it is not a favoured and common tool. Instead, IMUs for measuring the acceleration at the tip of the beam may be a viable alternative for high bay racks with unguided masts. Therefore, an obvious intention is to apply acceleration measurements for the abstraction of the shear force and the bending moment. It is well known that double integration of noisy acceleration signals is not advisable. It should be mentioned that there exists methods, such as complementary filters, see, e. g., [19], to determine the position from acceleration measurements. However, for the sensor configuration of the laboratory model under consideration it turned out that this approach is not suitable.

In fact, our intention is to exploit filtered measurements of the beam tip acceleration

with the cutoff frequencies

**Scenario A (inactive lifting unit):** In the first experiment, we propose a scenario of an SMC with inactive lifting unit moving horizontally from

### Figure 2

**Scenario B (active lifting unit):** In the second experiment, we demonstrate the capability of both observers for an active lifting unit, where the driving unit as well as the lifting unit were driven simultaneously from

## Remark 4.

*In general, spillover phenomena in early-lumping approaches can lead to drastic problems, e. g., the excitation of high-frequency natural oscillations that can provoke closed-loop instability, see* [15]*. However, for the EKF, which is based on the finite-dimensional model, no such effects turned out to be crucial.*

### Figure 3

## 5 Stability investigations

The aim in this section is to present a sketch of the well-posedness and stability investigations of the mixed-dimensional observer, where we confine ourselves to Scenario A, where the lifting unit is located at the top, i. e.,

The error principle unit for the driving unit follows to

whereas those of the lifting unit can be omitted.

### 5.1 Well-posedness investigations

First, the observer-error dynamics are reformulated as an abstract Cauchy problem, which allows to apply the variant of the Lumer-Phillips theorem according to [16, Theorem 1.2.4]. Therefore, we consider the state space *l* are square integrable. Thus, the state space

with

where we have the equivalence

where

Regarding the Lumer-Phillips theorem [16, Theorem 1.2.4] it has to be shown that the domain

### 5.2 Asymptotic stability

Next, regarding the investigation of the asymptotic stability of the observer error we use LaSalle’s invariance principle for infinite-dimensional systems, see [17, Theorem 3.64, 3.65], which can be applied since the solution trajectories are precompact. This follows from the fact that the boundedness of

## 6 Conclusion

In this contribution, two different observer strategies for a single mast stacker crane have been presented. First, based on the finite-dimensional system approximation an EKF has been proposed. Further, considering the mixed-dimensional character of the system, an observer has been derived by means of a port-Hamiltonian system representation. The collocated observer inputs were chosen such that the energy functional of the observer-error is non-increasing. A sketch of well-posedness and asymptotic stability investigations is given for the special scenario with inactive lifting mass at the beam tip. Both observers were verified in experiments on a laboratory model, where an acceleration measurement of the beam tip has been exploited. Finally, we want to stress that the estimated quantities can be successfully used for vibration rejection.

**Funding source: **Austrian Science Fund

**Award Identifier / Grant number: **P 29964-N32

**Funding statement: **This work has been supported by the Austrian Science Fund (FWF) under grant number P 29964-N32.

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**Received:**2021-01-27

**Accepted:**2021-08-09

**Published Online:**2021-09-09

**Published in Print:**2021-09-27

© 2021 Ecker et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.