Inverse dynamic control via “simulation of feedback control” by artificial neural networks for a crane system

Control Engineering Practice 94 (2020) 104203

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Inverse dynamic control via ‘‘simulation of feedback control’’ by artificial neural networks for a crane system Liz Rincon ∗, Yuta Kubota, Gentiane Venture, Yasutaka Tagawa Tokyo University of Agriculture and Technology, 2-24-16, Nakacho, Koganei-shi 184-8588, Tokyo, Japan

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Keywords: Machine learning Nonlinear crane system Dynamic artificial neural network Inverse dynamic control simulation (IDCS)

ABSTRACT Cranes are systems highly used in industrial applications to transport heavy loads. The nonlinear behavior, and the crane own dynamic produce vibrations during the motion. In order to improve the system performance and specifically to ensure stability and a control with minimum vibrations, this work proposes a new strategy based on IDCS (Inverse Dynamic Control Simulation) using machine learning with Artificial Neural Networks ANN that gives the possibility to learn the inverse dynamic model of the crane and apply the feedback information based on the inverse dynamic control simulation. IDCS allows a suitable signal control avoiding noise and need for extra sensors in the feedback loop. The ANN creates the inverse dynamic model of the crane. The training architecture is developed to learn the inverse dynamic model of the crane in different operational points, and is used as feedforward control with the feedback dynamic in the IDCS scheme. Simulations and experiments are conducted with an industrial crane, and results show that the proposed method decreases vibration and position error. The proposed ANN-IDCS showed suitable performance compared to other controllers such as analytical inverse model control (AIC), Dual Matching Control (DMM) and shaped reference in feedforward (FRC).

1. Introduction In construction sites, warehouses, factories or harbors, cranes are systems often encountered and widely used to lift and transport heavy loads (Ramli, Mohamed, Abdullahi, Jaafar, & Lazim, 2017; Ramli, Mohamed, & Jaafar, 2018). The crane is constituted mainly by hoisting and support mechanisms that generate the dynamic behavior of its structure (Abdel-Rahman, Nayfeh, & Masoud, 2003; Tagawa, Mori, Wada, Kawajiri, & Nouzuka, 2016). The support mechanism produces the suspension motion to move the load around the space where the crane reaches, and the hoisting mechanism produces the lifting or lowering motion of the load. The characteristics of the components (the trolley car, the hoisting elements, the transport rolls, the guides for moving the load) and their interrelations are responsible of the dynamic behavior of the total system, also involving the external forces and perturbations that can be generated during the transportation of the load. One of the critical issues of these systems are the vibrations that occur in the positioning of the load during the transportation task. The vibrations depend on the dynamic responses of the overall system and can result in unstable motions (Kacalak, Budniak, & Majewski, 2017). The strong relationship between cable (wire characteristics), payload assembly and suspension mechanisms affects the dynamic response. External perturbations, such as wind, can affect the crane by producing continuous oscillations of the load. In addition, during the motion of

the crane other inertial forces can affect the load pendulation (AbdelRahman et al., 2003). Vibrations of the load are therefore the result of the mechanical structure vibrations, load characteristics, transportation and hoisting velocities and accelerations of the load. Usually, cranes are operated by humans. The operator controls the velocity and position to drive the load to the final target avoiding extreme vibrations. However, the crane system can be altered by the natural frequency of the vibration that changes with the variation of the wire length during the motion and require the operator to be an expert (Tagawa et al., 2016). The unstructured environment, the diverse behaviors of different kinds of loads and the dependency on the operator actions are issues to take into account when the crane is operated. The proposal of effective controllers that minimize the vibrations and avoid the instabilities is important to ensure a safe and efficient operation of the crane, and relief the human operator. Control architectures for cranes could generally be divided into three categories: feedforward, feedback and hybrid controllers. Feedforward control architecture are subdivided into input shaping controllers, which use filters or commands by smoothing control. Some examples are presented in Benhellal, Hamerlain, Ouiguini, and Rahmani (2014) and Ramli et al. (2017), where authors proposed to remove vibrations using input shaping to control the crane velocity. In Singhose, Porter, Kenison, and Kriikku (2000), Singhose et al. propose the command

∗ Corresponding author. E-mail address: [email protected] (L. Rincon).

https://doi.org/10.1016/j.conengprac.2019.104203 Received 28 December 2018; Received in revised form 17 October 2019; Accepted 17 October 2019 Available online xxxx 0967-0661/© 2019 Elsevier Ltd. All rights reserved.

L. Rincon, Y. Kubota, G. Venture et al.

Control Engineering Practice 94 (2020) 104203

generation by input shaping to reduce the residual vibration taking into account the effect produced by the dynamic of the hoisting wire length. The feedback controllers can be regrouped in linear, intelligent, adaptive, optimal and hybrid controllers (Ramli et al., 2017). Some examples are using genetic algorithm (Önen & Çakan, 2017), fuzzy control system (Rong, Rui, Tao, & Wang, 2018), neural networks based on PID and multi-objective optimization to tune the control parameters (Diep & Khoa, 2014). In optimal control for cranes, controllers with Model Predictive Control (MPC) (Wu, Xia, & Zhu, 2015), Linear Quadratic Gaussian (LQG) and Linear Quadratic Regulation (LQR) methods are implemented running the optimization algorithms to find the controllers for load positioning with reduced vibrations (Jafari, Ghazal, & Nazemizadeh, 2014; Santhi & Beebi, 2014). Other techniques as Sliding Mode Controller (SMC) (Liu, Yi, Zhao, & Wang, 2005) are implemented to increase the versatility and accuracy of these controllers (Ramli et al., 2017). Adaptive controllers for crane based on integral barrier Lyapunov function (IBLF) are proposed in He, Zhang, and Ge (2014) to suppress the vibrations of a flexible crane using boundary output constraints. The adaptation is developed to handle the uncertainties in the system’s parameters. This control guarantees stability of the closed loop. The problem with the output constraints is solved by applying a Lyapunov Function for a SISO nonlinear system in a closed feedback loop. Other controllers using machine learning techniques like neural network as a controller with genetic algorithms are demonstrated with simulations in a feedback configuration (Nakazono, Ohnishi, Kinjo, & Yamamoto, 2008). In Lee, Huang, Shih, Chiang, and Chang (2014), artificial neural networks (ANN) and variable structure systems to estimate the control signal are presented. This structure is defined using Sliding Mode Control to tune the feedback parameters. All these techniques apply feedback control scheme, i.e. the control is calculated with the closed loop signals measured from the real crane during the task. The crane feedback control is sensitive to noise during the load transportation. If the crane signals show sensitivity to external perturbations, it can generate an unstable signal, which is critical for the load vibration control. Furthermore, the error and instability can increase when the load is positioned horizontally, while the length of hoisting wire is changed (Tagawa et al., 2016). As a result, the cost in technical extra issues to refit the crane with the equipment is increased, and the suitable sensors for the measurement are not always possible. For that, it is crucial to define other control techniques that allow reducing the vibration with stable motions in real-time, and increase the performance for efficient load transportation, while insuring safety, robustness, stability, and minimum vibration. This work focuses on the development of the inverse dynamic control by feedback simulation (IDCS) where the inverse control is achieved by machine learning, and applying the controller to a real crane. IDCS (Tagawa, Tu, & Stoten, 2011) (Inverse Dynamic Control Simulation) is a technique that proposes the inverse dynamic model calculation using the feedback compensation via simulation, with the advantages of the implementation of the feedforward control in a feedback control simulation scheme. It allows to avoid noise in the control input (Tagawa et al., 2016; Wada, Mori, Tagawa, & Honma, 2017) reducing unstable responses. The feedforward controller is implemented in the scheme of sensor-less vibration control system and only a stabilizing feedback control is applied. IDCS, however, relies heavily on the accuracy of the model used to generate the feedforward control input. As mentioned earlier, cranes have a complex dynamics that depends on many parameters. To avoid issues with classic linear modeling and strengthen the model on dynamic effects difficult to model, this work proposes to use machine learning (ML) to calculate the inverse dynamics model. Techniques based on ML to identify complex systems have been studied using Artificial Neural Networks (ANN). For the identification of dynamic systems, Hossain, Ong, Ismail, Noroozi, and Khoo (2017) explained the dynamic of vibration in systems based on the inverse

parametric identification. The most conventional ANN are the MLPBN (Multi-Layer Perceptron with Batch Normalization), RBFN (Radial Basis Function Network) and GRNN (General regression Neural Network), showing good trade-off between quantity of data needed for training, accuracy of the final model and simplicity of implementation. Other alternatives to control cranes are the generation of the desired trajectory (input shaping) using ANN. This new trajectory is applied as the motion control signal (Abe, 2011). ML can be used to generate dynamic models taking information of the system, and also can be extended to learn effective control laws for complex systems (Duriez, Brunton, & Noack, 2017; Suykens & Bersini, 1996). In Lewis and Ge (2005), Lewis et al. propose different structures with ANN applied in feedback or feedforward control configurations for general systems. The benefits of these structures to learn nonlinearities, and the potential to apply these controllers as an extension of adaptive techniques for nonlinearly parameterized learning systems were demonstrated. Other works using ANN for tuning the controllers or replacing the control in mechanical structures are presented in Omidvar and Elliott (1997) and Yousefian and Kamalasadan (2017). Vibration neurocontrol for small mechanical structures (Abdeljaber, Avci, & Inman, 2016) or neurocontrollers for robotic systems (Yan & Li, 1997) also show effective performances. Some of the difficulties to design the inverse controller for the cranes using analytical models are the improper characteristics of the system, the nonlinearities, and the not proximity models to represent it and its inverse dynamic, showing inconsistent solutions (Blajer & Kołodziejczyk, 2006, 2011). For these reasons, ML is suggested as a versatile and robust proposal. 2. Contributions This work proposes a new control strategy using inverse dynamic control (IDCS) with Artificial Neural Networks acting in real-time for industrial cranes. The neurocontroller works in real-time with the crane. The control signal for the crane (trolley velocity) is calculated by the inverse dynamic controller, and generated by the feedback simulation model in the IDCS scheme. An overview of this work is showed in Fig. 1. This control is able to learn the dynamics of the crane, and also to minimize the vibration of the load. It has the advantage to give the system a signal noise-free with precise control information. The proposed method reduces vibrations and increases the performances with robustness during the motion. This proposal also improves the system stability to guarantee a suitable operation on the real system. The paper is structured as follows. In Section 3, the concept of the IDCS and structure of the system applying machine learning is explained. In Section 4, the dynamic modeling for crane systems is developed. In Section 5, the inverse modeling is explained using ANN in IDCS scheme. In Sections 6 and 7, simulation and experimental results are showed for a real crane. Finally, discussions and conclusions are given according to the work developed. 3. Inverse dynamic control simulation (IDCS) by machine learning The inverse dynamic compensation via ‘‘simulation of feedback control systems’’ (IDCS) is a numerical method proposed by Tagawa and Stoten in Tagawa et al. (2011). This method proposes the calculation of the inverse dynamic model of the system in a feedback control by simulation. Usually, the implementation of this scheme is based on the mathematical model of the system to calculate in simulation the inverse dynamics and used it as the control law. The proposal in this work aims to develop intelligent controllers where the models for IDCS are learned with an Artificial Neural Network (ANN) trained on simulated or on real crane’s data. The IDCS strategy is used to guarantee a safe and stable control due to the perfect simulation in an ‘‘ideal control’’. The performance and stability of the controller can be complemented with a feedback 2

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Fig. 1. Overview of the proposed inverse model control by machine learning via IDCS. 𝑦∗ (𝑡) is the load velocity output, 𝐲∗ (𝑡) is the output vector of the load velocity after calculated by the regressors, 𝑢(𝑡) is the control signal output from the ANN controller to the real crane, and 𝐮(𝐭) is the output vector of the control signal after calculated by the regressors.

The IDCS control framework by machine learning uses the dataset from the crane to learn and create the forward and inverse models with ANN architectures. 4. Crane system: Dynamic modeling The crane system is constituted mainly by the trolley car, the girder, the wire for hanging a load, and a load. According to Tagawa et al. (2016) three motions characterize the crane system: traveling motion that is a motion in a girder along the rails, traversing motion that is the motion of the other girders installed in the system, and hoisting/lowering motion, which is related with the load height. The load can be transported with the combinations of horizontal and vertical motions. The load that the crane transports can be modified during the motion, producing vibrations. In addition, the variation in the wire length can produce different vibrations (Tagawa et al., 2016). The analytical model is shown in Fig. 3. The load–displacement is x, the trolley displacement is u, the length wire is l, the angular position is ̇ the angular acceleration is 𝜃, ̈ and the load 𝜃, the angular velocity is 𝜃, mass is m. The equation of motion is presented in Eq. (1).

Fig. 2. General scheme for feedback and Inverse Dynamic Control Simulation IDCS: (a) Feedback control, (b) Main concept of IDCS control (Tagawa et al., 2011).

when the models cannot be perfectly obtained generating inaccurate controllers (Norgaard, Ravn, Poulsen, & Hansen, 2000). The implementation of these controllers based on IDCS avoids unstable signals, and guarantee stable controllers using the perfect control signal obtained in the simulated environment. The concept of the IDCS is shown in Fig. 2b, where 𝑟(𝑡) is the reference for the system 𝑃 . 𝑢(𝑡) is the control input, 𝑦(𝑡) is the output of the system, and 𝐾 is any controller. For this work, the controller 𝐾 (modeled as the inverse model) is determined by the inverse dynamic 𝑃 −1 of the crane. Fig. 2b shows the feedback control with the highperformance controller 𝐾, the output 𝑦∗ (𝑡) is the output of the system that will be ideally close to the reference r(t) in a simulated environment with minimum error. The output y(t) will be closed to 𝑦∗ (𝑡) if 𝑃𝑀 is an approximated enough model for the system.

𝑚𝑙2 𝜃̈ = −2𝑚𝑙𝑙̇ 𝜃̇ − 𝑚𝑔𝑙 sin 𝜃 − 𝑚𝑢𝑙 ̈ cos 𝜃

(1)

For the control validation, the ANN controllers work in a feedforward scheme connected with the analytic model of the crane (P) using Eq. (1) and the geometric configuration of Fig. 3. The feedback is measured directly from the output of the ANN-IDCS controller. Another factor that influences the crane response is the mechanical structure. It is constituted by a mechanical drum, a sheave with radius

Fig. 3. (a) model of the crane (Tagawa et al., 2016), (b) components of the crane system.

3

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optimizing the learning law in a corresponding inverse model structure. The model for the crane can be represented by the load velocity y(t) as the output, and the trolley velocity as the input u(t) with 𝑛 as the quantity of past inputs, and 𝑚 the quantity of the past outputs in the model configuration by 𝑦(𝑡 + 1) = 𝑔[𝑦(𝑡), … , 𝑦(𝑡 − 𝑛 + 1), 𝑢(𝑡), … , 𝑢(𝑡 − 𝑚)]

(3)

The inverse dynamics model configuration for training is showed by: ̂ 𝑢(𝑡) = 𝑔̂−1 [𝑦(𝑡 + 1), … , 𝑦(𝑡 − 𝑛 + 1), 𝑢(𝑡 − 1), … , 𝑢(𝑡 − 𝑚)]

(4)

The output y(t + 1) of the inverse dynamic model is replaced with the input reference r(t + 1) when it is applied as the controller.

Fig. 4. Scheme for the mechanical structure of the crane: (a) crane with the basic mechanical components, (b) relation of motion between the encoder and the drum, (c) position of the encoder, the guides and wire for the load lifted by the crane.

5.1. Architecture of the neural network controller The ANN can represent complex dynamic models and deal with the nonlinear control problem. The architecture of the neural network can be configured with many different structures in static and dynamic representations (Yousefian & Kamalasadan, 2017). The static networks are characterized with memoryless nodes, the Multi-Layer Perceptron Neural Network (MLPNN) has an efficient function approximation, feedforward architecture and simple distribution of the layers and neurons, and the Radial Basis Functions Neural Network (RBFNN) are used frequently for classification and pattern recognition (Yousefian & Kamalasadan, 2017). The architecture for the ANN can be represented by inputs, hidden layers, outputs, neurons, and the interconnections with the weights and bias information. The configuration for the ANN in an inverse dynamic control is presented in Fig. 6. The general representation of the MLP network is by Norgaard et al. (2000). 𝑛

𝑛ℎ 𝜁 ∑ ∑ 𝑦̂𝑖 (𝑡) = 𝑔𝑖 [𝜁 , 𝜃] = 𝐹𝑖 [ 𝑊𝑖,𝑗 𝑓𝑗 ( 𝑤𝑖,𝑗 𝜁𝑙 + 𝑤𝑖,0 ) + 𝑊𝑖,0 ]

Fig. 5. Crane system components: (a) Angles and lengths involved in the load position, (b) Angle position error in the crane.

where 𝑦̂𝑖 (𝑡) is the output of the network related to the parameter 𝜃, which has the information of the tuning values of the network, and the vector of network’s inputs 𝜁, which is constituted by the quantity of inputs 𝜁𝑙 . 𝑤𝑖,𝑗 and 𝑊𝑖,𝑗 are the weights and bias of the network. f and F are the activation functions respectively for the neurons in each layer. The 𝑛ℎ and 𝑛𝜁 parameters are the quantity of neurons by each layer 𝑗 = 1...j. 𝑦̂𝑖 (𝑡) is related to the output of the crane: load velocity, and 𝜁𝑙 with the input: trolley velocity.

𝑟1 , and a hook mechanism with radius 𝑟2 where the load is lifted as shown in Figs. 4 and 5. The error occurring during load transportation is related to the mechanical structure when the length changes and the load position is moved. This error can be expressed by the angles 𝜃1 and 𝜃2 : 𝑒 = 𝜃2 − 𝜃1 𝑒=

(𝑟 −𝑟 ) 𝑡𝑎𝑛−1 1𝑙 2 2

− 𝑡𝑎𝑛−1

(𝑟1 −𝑟2 ) 𝑙1

(5)

𝑙=1

𝑗=1

(2) 5.2. Training of the inverse dynamic model

Structurally, as the wire length changes during the load transportation, the convergence point also changes from the point that was initially set to 0 producing small variations in the steady-state of the load angle.

The ANN was trained using the Levenberg–Marquardt algorithm that minimizes the mean square error of the signal (Norgaard et al., 2000). The search direction of the algorithm is determined by approximation of 𝐿(𝑖) (𝜃), which is validated in the neighborhood values. A spherical surface for the optimization is selected with radius 𝛿 (𝑖) and the optimization is formulated by

5. Inverse dynamic control by machine learning in IDCS scheme The control of the crane as an under-actuated system is a complex task (Blajer & Kołodziejczyk, 2011). The inverse dynamics model can be developed by the calculation of an analytical model. It needs to be precisely formulated according to the dynamics of the system. The analytical model can be approximated by numerical calculations. However, this model does not always guarantee the control performance. The proposed controller solves the crane’s inverse dynamic using ANN. The ANN algorithms have the potential to learn the dynamics of the system involving the nonlinearities and other externals or internal factors that are not easy to model. As proposed by Norgaard et al. (2000) the inverse dynamic model can be obtained directly configuring the parameters for an ANN, training and validating it using the dynamics information of the system, and

𝜃 (𝑖+1) = 𝑎𝑟𝑔 min

𝐿(𝑖) (𝜃)

s.t.

|𝜃 − 𝜃 (𝑖) |

𝜃

≤

𝛿 (𝑖)

(6)

The rules to update the values in the training and solving this constrained optimization are determined by [

𝜃 (𝑖+1) = 𝜃 (𝑖) + 𝑓 (𝑖) ] (𝑖) 𝑓 = −𝐺(𝜃 (𝑖) )

𝑅(𝜃 (𝑖) ) + 𝜆(𝑖) 𝐼

(7)

where 𝑓 (𝑖) is the search direction, 𝑅(𝜃 (𝑖) ) is the Gauss–Newton Hessian, 𝜆 is the optimization balance parameter and 𝐺(𝜃 (𝑖) ) is the gradient. The training can be formulated in terms of the dataset 𝑍 𝑁 , and candidate models 𝑦(𝑡), such as in Eqs. (8) and (9) (Norgaard et al., 2000), to do ̂ T is the vector of the quantity of inputs and the mapping 𝑍 𝑁 ⟶ 𝜃. 4

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Fig. 6. Inverse dynamic control by ANN controller with the IDCS feedback regressor signal for the crane system.

by Ljung (1999) can be shifted between ±(𝑝) with the following mean and covariance 𝑝 1 ∑𝑀 |𝑀 𝑡=1 𝑝(𝑡)| = 𝑀 (13) 1 ∑𝑀 𝑅𝑢 (𝑘) = 𝑀 𝑡=1 𝑝(𝑡)𝑝(𝑡 + 𝑘) 2

) With 𝑅𝑢 (𝑘)=(𝑝2 ) for 𝑘 = 0, ±𝑀, ±2𝑀..., or 𝑅𝑢 (𝑘)= −(𝑝 in elsewhere. 𝑀 The period of the PRBS signal is 𝑀 = 2𝑛 − 1. The power spectrum of the PRBS signal can be calculated by ∑ (14) 𝛷𝑝 (𝜔) = 2𝜋𝑝 𝑀−1 𝑘=1 𝛿(𝜔 − 2𝜋𝑘∕𝑀), 0 ≤ 𝜔 ≤ 2𝜋 𝑀

The 𝑀 − 1 frequency peaks appear in the interval of −𝜋 ≤ 𝜔 ≤ 𝜋. The PRBS has whole periods. This PRBS input was designed to obtain in simulation the data of the crane and make the training of the ANN as the inverse model. Fig. 7 shows a sample of the input and output crane’s data used for training of the dynamic inverse model. 5.3. Data collection for training the ANN controllers

Fig. 7. Training signal: Extracted signal of the excited input PRBS signal, and the crane output load velocity.

For the ANN training, crane’s load velocity is computed in simulation when the crane is excited by a PRBS signal. This signal is prepared according to the procedure explained in the previous section. Precise models must be trained with sufficient dynamics data (data maximization). However, a real crane has mechanical restrictions and cannot be excited safely in different frequencies with different displacements and velocities. For these reasons, the data collection is based on simulation responses to create different options in frequencies and changes the physics of the crane to produce outputs rich in dynamics information. The couples input/output for training and validation of the ANN were defined as data vectors with the trolley velocity as the input and load velocity as the output. The dataset is constituted by 20001 input points and the corresponding 20001 output points.

outputs used for the training, and N is the limit of the dataset according to the quantity of the inputs/outputs in the training dataset. 𝑍 𝑁 = [𝑢(𝑡), 𝑦(𝑡)], 𝑇 = 1, … , 𝑁

(8)

𝑦(𝑡) = 𝑦̂(𝑡|𝜃) + 𝑒(𝑡) = 𝑔[𝑡, 𝜃] + 𝑒(𝑡)

(9)

e(t) is the noise signal taken into account during the training of the network, it is calculated as a Gaussian noise distribution. The error used to formulate the optimization is 𝑉𝑁 (𝜃, 𝑍 𝑁 ) =

𝑁 1 ∑ [𝑦(𝑡) − 𝑦̂(𝑡|𝜃)]2 2𝑁 𝑡=1

(10)

5.4. ANN neurocontrol in IDCS configuration

The network as inverse dynamics model uses the error (Eq. (10)) in an inverse training configuration, minimizing the cost function defined by 𝐽 (𝜃, 𝑍 𝑁 ) =

𝑁 1 ∑ [𝑢(𝑡) − ̂ 𝑢(𝑡|𝜃)]2 2𝑁 𝑡=1

The inverse dynamic model of the crane created by ANN (Section 5) is tested as a feedforward controller inside of the IDCS scheme. Fig. 6 shows the configuration of this controller. After training and validating the inverse dynamics, the model is configured as the control of the crane. The main input reference for this neurocontroller is the signal for the trolley velocity 𝑣(𝑡), and the output is the control signal for the trolley 𝑢(𝑡). The inverse controller has as inputs: the trolley velocity reference 𝑟(𝑡), the structure of the regressors [𝑞 −1 , 𝑞 −2 , … , 𝑞 −𝑛 ] with 𝑛 the quantity of past inputs, and [𝑞 −2 , … , 𝑞 −𝑚 ] with 𝑚 quantity of the past outputs, which define the regression vector structure in the model configuration, and the load velocity 𝑣𝑙𝑜𝑎𝑑 (t). The output 𝑢(𝑡) is the response of the neuroncontroller. The control law is calculated by the responses of the functions of each neuron and layer configuration in the neurocontroller following the velocity’s reference (Fig. 8). Fig. 1 shows the proposed scheme of the controller where 𝑢(𝑡) ̂ expressed in Eq. (4) is applied for the real crane as the trolley command velocity input.

(11)

Excitation signal for the ANN training: To perform the training and validation process, the crane system is excited with a predefined signal using a PRBS (Pseudo Random Binary Signal) configured by the classic method proposed by Ljung (1999). The PRBS signal is a periodic and deterministic signal with noise properties. It is generated by 𝑝(𝑡) = 𝛽(𝐴(𝑞)𝑝(𝑡), 2) 𝑝(𝑡) = 𝛽(𝑎1 ∗ 𝑝(𝑡 − 1) + ⋯ + 𝑎𝑛 ∗ 𝑝(𝑡 − 𝑛), 2)

(12)

𝑝(𝑡) can take 0 or 1 values as the binary states. 𝑝(𝑡) must be a periodic signal with period 2𝑛 . The maximum length in the PRBS as discussed 5

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Fig. 8. Predicted output signal and predicted error during the validation of the ANN models for the controllers. Table 1 Results of the crane inverse models for different ANN architectures used for the neurocontrollers during the training-validation process.

6. ANN control validation: Simulation setup and results The ANN neurocontrol is validated in simulation and in real experimentation (Section 7). In simulation, the following items are presented: (i) training process for ANN models learning as the inverse dynamics of the crane, (ii) control validation with ANN controllers, (iii) robustness validation of the controller, (iv) stability analysis with ANN controllers and (v) comparative results with other controllers. 6.1. Training of the ANN models as the inverse dynamic of the crane The training setup was configured with the following parameters: the optimal criteria is calculated with the error measured in the trolley velocity. The architecture of the ANN is developed with MLPBN scheme using 3 layers, one input and one output. In the hidden layer, the neurons and the quantity of regressors are changed according to Table 1. The parameters for the training (explained in Section 5) were configured with maximum number of iterations 3000, stop parameters with minimum criteria 0, maximum criteria 1 × 10−7 , for values in the gradient larger than 1 × 10−4 , the weight decay 𝛼 = 0, 𝜆 = 1, and the step size of the back-propagation algorithm 𝜂 = 1 × 10−4 . The ANN controllers were configured with different types of regressors [𝑚 𝑛 1] and experiments regressor A = [2 2 1], regressor B = [3 3 1], regressor C = [4 4 1] and regressor D = [5 5 1]. During the training, the sample time is 𝑇 𝑠 = 0, 01 s; the initial control signal is 𝑢0 = 0, the initial output is 𝑦0 = 0; the limits for the minimum and maximum control signal are 𝑢𝑙𝑖𝑚𝑚𝑖𝑛 = −1 and 𝑢𝑙𝑖𝑚𝑚𝑎𝑥 = 1. 20001 samples data were collected for training. The prediction error and its correlation are calculated (Norgaard et al., 2000). Fig. 8 shows one example of the output of the prediction error for the model calculated, and its correlation between prediction error and input. During the training process, 20 different ANN configurations for the inverse crane controllers were tested. Results are presented in Table 1. The training and validation parameters were set up with the same values for all experiments. Table 1 presents the results of the different models with the prediction errors, and the % of the data correlation. During the validation of the models, the predictive errors were analyzed, for errors lower than 10−9 or 10−8 the crane control responses present more oscillations and a risk of instability. With models calculated with error higher than 10−6 and more quantity of regressors, the system does not have a suitable correlation, and the control is unstable. The controller cannot follow and stabilize the load position of the crane. Due to these crane’s behaviors, the models fitting with error around 10−6 to 10−9 were calculated and summarized in Table 1.

Model

Neuron

Regressor

Pred. error

Correlation

ANN1 ANN2 ANN3 ANN4 ANN5 ANN6 ANN7 ANN8 ANN9 ANN10 ANN11 ANN12 ANN13 ANN14 ANN15 ANN16 ANN17 ANN18 ANN19 ANN20

10 10 10 10 20 20 20 20 50 50 50 50 100 100 100 100 200 200 200 200

[2 [3 [4 [5 [2 [3 [4 [5 [2 [3 [4 [5 [2 [3 [4 [5 [2 [3 [4 [5

6, 8 × 10−8 1, 3 × 10−8 1, 3 × 10−8 1, 4 × 10−8 4, 4 × 10−8 1, 3 × 10−8 1, 4 × 10−8 1, 3 × 10−8 3, 0 × 10−7 1, 8 × 10−8 3, 7 × 10−9 1, 2 × 10−8 9, 6 × 10−8 1, 2 × 10−7 1, 4 × 10−8 9, 8 × 10−9 2, 9 × 10−8 1, 5 × 10−6 8, 3 × 10−8 8, 6 × 10−9

0,839 0,022 0,025 0,024 0,142 0,038 0,028 0,025 0,839 0,235 0,193 0,639 0,589 0,840 0,066 0,086 0,960 0,942 0,807 0,934

2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5

1] 1] 1] 1] 1] 1] 1] 1] 1] 1] 1] 1] 1] 1] 1] 1] 1] 1] 1] 1]

6.2. Control validation with ANN controllers The controllers during the training process were tested using the models from Table 1 and structures of Figs. 1 and 6. The inverse model was configured as feedback controller with the reference input, past control signals, and current and past output signals. The reference 𝑟(𝑡) was programmed with a trapezoid profile input shaping for the trolley velocity with maximum acceleration of 0,33 m/s2 and acceleration time of 3 s (profile input often used to control cranes). A low-pass filter was implemented in the reference to avoid the high acceleration and velocity when is changing the reference states. The crane model parameters were configured using the same values of the real crane in normal operation (hoisting velocity in 0,166 m/s, wire length in 2 m and maximum acceleration in 0,33 m/s2 ). The responses of the crane were analyzed quantitatively using RMSE of the load position, and the % of vibration suppression index 𝜖 during the transportation of the load. The qualitative responses are analyzed directly with the crane time series responses of the load position, load velocity, trolley velocity, and control signal. The vibration index is calculated using (Tagawa et al., 2016). 𝐵𝑚𝑒𝑎𝑛 is the mean signal using the ANN control, and 𝐴𝑚𝑒𝑎𝑛 is the mean signal without controller. 𝜖= 6

𝐵𝑚𝑒𝑎𝑛 𝐴𝑚𝑒𝑎𝑛

× 100%

(15)

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Fig. 9. Simulations results: Angle response for regressors A and C with 10 and 200-ANN controllers.

Fig. 10. Simulations results: Load velocity response for regressors A and C with 10 and 200-ANN controllers.

Fig. 11. Simulations results: Load position response for regressors A and C with 10 and 200-ANN controllers.

The representative dynamic responses of the ANN controllers during the load transportation are presented in Fig. 9 showing the load angle for regressors [2 2 1] and [4 4 1]. Fig. 10 presents the load velocity’s responses, Fig. 11 shows the load position, Fig. 12 represents the trolley velocity’s responses, and Fig. 13 summarizes the RMSE error in load velocity for all controllers.

obtained using highest quantity of regressors type D, and also when the quantity of neurons is increased. Table 2 summarizes the results of the % of vibration index. Almost, all controllers keep a vibration suppression around 15%, only with the highest quantity of regressors type D and more neurons this factor is increased (see Fig. 14).

The responses obtained with controllers ANN1-ANN20 (Table 1) for different regressors (A, B, C and D) are presented in Figs. 18 and 19. These results showed similar characteristics in time response.

6.3. Stability analysis with ANN controllers The stability of the ANN controllers is analyzed using the qualitative dynamic responses of the angle and velocity during the transportation of the load (Figs. 9–12, and 18 and 19). The stability can be analyzed in the range of the bound reference signal used to excite the crane

The highest error is found with the ANN configured with 20 neurons and regressor type B, the minimum error is found with 10 neurons and regressor type C. Highest quantity of vibration and instability are 7

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Fig. 12. Simulations results: Trolley velocity response for regressors A and C with 10 and 200-ANN controllers. Table 2 Vibration control performance: Simulation results for the ANN controllers validating the % of vibration index 𝜖 for the different neurocontrollers and regressors [A, B, C, D]. Control

𝜖𝐴

𝜖𝐵

𝜖𝐶

𝜖𝐷

ANN1-4 (10N) ANN5-8 (20N) ANN9-12 (50N) ANN13-16 (100N) ANN17-20 (200N)

15% 15% 15% 15% 15%

15% 15% 15% 15% 15%

15% 15% 28% 15% 15%

15% 15% – – –

The stability in closed loop is verified in the IDCS using the simulation feedback, and guarantying a stable control signal for the real crane. The stability is tested in simulation for different physical characteristics of the system, introducing model error and variation in the input profile. In all cases the stability of the controllers was verified directly by the load position and velocity of the crane keeping within the operational limits.

Fig. 13. Control performance: RMSE error in the load velocity using different ANN controllers (neurons and regressors), with the operational conditions in the crane (wire length = 2 m, hoisting velocity = 0,166 m/s and trolley acceleration = 0,33 m/s).

6.4. Robustness validation with ANN controllers The robustness of the controllers is analyzed changing the parameters of the crane system, in order to create variations when the real crane is operated. The parameters were changed in the system with the following configurations: the wire length in 1, 1,5, 2 and 10 m, the maximum trolley velocity in 0,33 and 0,42 m/s, and the hoisting velocity in 0,1, 0,5 and 1 m/s. The robustness was validated with the controllers using 10 neurons, 50 neurons and 100 neurons with the regressors A, B, C and D. The error calculation (RMSE) for variations of the maximum trolley velocity and hoisting velocity are presented in Table 3. The maximum errors are shown for the 50NN controller using more quantity of regressors (C and D). The controllers presented more sensibility to

for the transportation of the load, and the output is supervised from the vibration suppression as a bounded signal, but not saturated. The unstable responses can be described by not constant and incremental oscillations, that means non-bounded or saturated signals produced by the neurocontrol responses (Norgaard et al., 2000). Though the inverse models were trained guarantying minimum error and high correlation with the learning crane data, it is not enough to guarantee the control performance and stability. Controllers with more neurons and regressors, trained with the same parameters of the other cases, presented larger oscillations and in some cases instability. The controllers ANN11, ANN12, and ANN15 (Tables 1 and 2) with 50 and 100 neurons, and regressors type C and D, showed the most unstable responses.

Fig. 14. Robustness control validation: Mean of the RMSE errors of the load velocity with variation of the wire length (1 m, 1,5 m, 2 m, and 10 m) using the hoisting velocity = 0,167 m/s and trolley acceleration = 0,33 m/s2 for different ANN-IDCS controllers.

8

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Fig. 15. Robustness validation of the controllers: RMSE error in the load velocity with variation in the wire length for 10 NN and 50 NN.

Fig. 16. Robustness control validation: RMSE errors of the load velocity with variation of the wire length (1 m, 1,5 m, 2 m, and 10 m) using the hoisting velocity = 0,167 m/s and trolley acceleration = 0, 33 m/s2 for different ANN-IDCS controllers. Table 3 Robustness validation of the controllers: Results of the RMSE in the load velocity with variation in the maximum trolley velocity (𝑇𝑣 a = 0,33 m/s, b = 0,42 m/s) and hoisting velocity (𝐻𝑣 a = 0,1 m/s, b = 0,5 m/s and c = 1 m/s) for regressors type A , B, C and D. Parameter

RMSE A Reg. A

RMSE B Reg. B

RMSE C Reg. C

RMSE D Reg. D

𝑇 𝑣10𝑁 a 𝑇 𝑣10𝑁 b

1, 9 × 10−7 1, 8 × 10−7

1, 4 × 10−6 1, 4 × 10−6

4, 9 × 10−8 4, 9 × 10−8

2, 2 × 10−6 2, 2 × 10−6

𝑇 𝑣50𝑁 a 𝑇 𝑣50𝑁 b

1, 9 × 10−8 2, 0 × 10−8

3, 9 × 10−7 3, 9 × 10−7

4, 0 × 10−2 4, 0 × 10−2

2, 0 × 10−2 2, 0 × 10−2

𝐻𝑣10𝑁 a 𝐻𝑣10𝑁 b 𝐻𝑣10𝑁 c

1, 8 × 10−7 1, 8 × 10−7 1, 8 × 10−7

1, 4 × 10−6 1, 4 × 10−6 1, 0 × 10−3

5, 2 × 10−8 5, 1 × 10−8 4, 0 × 10−3

2, 2 × 10−6 2, 2 × 10−6 1, 0 × 10−3

𝐻𝑣50𝑁 a 𝐻𝑣50𝑁 b 𝐻𝑣50𝑁 c

2, 0 × 10−8 2, 0 × 10−8 3, 6 × 10−5

3, 9 × 10−7 3, 9 × 10−7 1, 0 × 10−3

4, 0 × 10−2 1, 0 × 10−2 4, 0 × 10−2

3, 0 × 10−2 2, 0 × 10−2 2, 0 × 10−2

is changed. With variations of the maximum trolley velocity and more regressors, the responses showed more vibration in all controllers. The results show that the minimum error is found with 10NN controller with a mean error of 9, 5 × 10−7 and standard deviation of 2, 6 × 10−9 . Fig. 15 shows the variation of the RMSE using 10NN and 50NN controllers, and the robustness response for the % of vibration index using the 10NN controller. The index of vibration is around 15% when changing the lengths and the regressors in the controller. Fig. 16 shows the results of robustness validation (RMSE error) using different lengths (1 m, 1,5 m, 2 m and 10 m) with the ANNIDCS controllers (10NN, 50NN and 100NN). The error with the 10NN controller kept less variation when it is applied to the crane. With 50NN and 100NN controllers, the error in the load velocity increased, and it is more affected by the variation of the wire length with regressors C and D.

Table 4 Robustness validation: Results of % Vibration Index 𝜖 with variation in the maximum trolley velocity (𝑇𝑣 a = 0,33 m/s, b = 0,42 m/s) and hoisting velocity (𝐻𝑣 a = 0,1 m/s, b = 0,5 m/s and c = 1 m/s) for regressors type A, B, C and D. Parameter

% 𝜖𝐴 Reg. A

% 𝜖𝐵 Reg. B

% 𝜖𝐶 Reg. C

% 𝜖𝐷 Reg. D

𝑇 𝑣10𝑁 a 𝑇 𝑣10𝑁 b

15, 15 18, 94

15, 15 15, 15

15, 15 15, 15

15, 15 15, 15

𝑇 𝑣50𝑁 a 𝑇 𝑣50𝑁 b

15, 15 18, 94

15, 15 18, 94

− −

− −

𝐻𝑣10𝑁 a 𝐻𝑣10𝑁 b 𝐻𝑣10𝑁 c

18, 94 18, 85 18, 66

18, 94 18, 86 −

18, 94 18, 86 −

18, 94 18, 86 −

𝐻𝑣50𝑁 a 𝐻𝑣50𝑁 b 𝐻𝑣50𝑁 c

18, 95 18, 86 46, 70

18, 95 18, 86 −

− − −

− − −

6.5. Comparative simulation results with other controllers In order to verify the performance and potential of the ANN-IDCS control, it is compared with other controllers. Simulations and experiments were developed applying DMM controllers (Tagawa et al., 2011), analytical inverse model control (AIC), and feedforward control signal (FRC) using the shaped reference for the trolley velocity applied directly in the crane. Figs. 20–23 show the results in simulation for all controllers. Figs. 24–27 show the experimental results with the real crane, and the robustness validation with the ANN-IDCS. All proposed controllers were tuned and tested in simulations first before being used in the real crane. The simulations were set up as following: (i) Condition 1 (default): wire length changing from 2,0 m to 1,5 m, hoist type is up motion, and trolley velocity in 0,35 m/s, (ii) Condition 2 (changing wire length): wire length 1 m, 1,5 m, 2 m, 10 m and 20 m, hoist type is up motion, and trolley velocity in 0,35 m/s, (iii) Condition 3 (changing max trolley velocity): wire length from 2,0 m to 1,5 m, hoist type motion is up, and trolley velocity is 0,35 m/s, 0,38 m/s and 1 m/s. The reference for the trolley velocity is generated using a second order filter. It reshapes the signal obtaining an operative input suitable to command the crane. This filter adjusts the input avoiding high acceleration and sudden changes in the crane. The filter was tuned for the best condition and performance considering that it has influence

the wire length and hoisting velocity with the 50NN controller. With 10NN controller, the variation of the parameters did not alter the final response in the crane, and suggests the best possibility to be implemented in the real crane. Table 4 shows the % of vibration suppression with different controllers and regressors. Almost, all controllers keep the % of vibration around 15% and 18%, with more variability when the hoisting velocity 9

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Fig. 17. Overcrane control system components and real crane used in the experimentation.

Table 5 Simulation results with the calculation of the RMSE errors with ANN inverse dynamic control, DMM1 control (with poles in high frequency 𝜔 = 100 and 𝜁 = 0, 8), FRC feedforward reference, and AIC analytical inverse control, the condition 2 is divided in different options according to the variation of the wire length L = 1,5, 2 , 10, 20 m, and the condition 3 for variation of the trolley velocity (T). The filter for the controllers is tuned in 𝜔 = 2 and 𝜁 = 0, 9. Parameters

Condition Condition Condition Condition Condition Condition Condition Condition

1 2a 2b 2c 2d 3a 3b 3c

RMSE

(L = 1,5 m) (L = 2 m) (L = 10 m) (L = 20 m) (T = 0,35 m/s) (T = 0,38 m/s) (T = 1 m/s)

DMM1

ANN

AIC

FRC

DMM2

FRC𝑓

1, 5 × 10−3 2, 2 × 10−3 1, 5 × 10−3 2, 5 × 10−4 1, 2 × 10−4 1, 5 × 10−3 1, 6 × 10−3 1, 9 × 10−3

2, 5 × 10−7 2, 6 × 10−7 2, 6 × 10−7 2, 7 × 10−7 2, 9 × 10−7 2, 6 × 10−7 2, 5 × 10−7 2, 3 × 10−7

4, 6 × 10−5 9, 8 × 10−5 4, 6 × 10−5 7, 7 × 10−4 3, 2 × 10−3 4, 6 × 10−5 6, 0 × 10−6 1, 6 × 10−4

1, 5 × 10−4 4, 3 × 10−5 1, 5 × 10−4 2, 9 × 10−3 1, 0 × 10−2 1, 5 × 10−4 9, 5 × 10−5 1, 7 × 10−4

2, 4 × 10−6 4, 9 × 10−5 2, 4 × 10−6 4, 9 × 10−6 2, 4 × 10−6 2, 4 × 10−6 3, 5 × 10−5 2, 4 × 10−6

6, 1 × 10−2 6, 1 × 10−2 6, 1 × 10−2 6, 3 × 10−2 5, 7 × 10−2 6, 1 × 10−2 5, 4 × 10−2 1, 4 × 10−4

Table 6 Technical specifications for the overhead crane system.

on the controllers related with the damping and frequency responses. For low frequency 𝜔, the input reference is slow, and with high 𝜔 the time response increases. Increasing the damping factor 𝜁, the input presents less over-damping in the output. For these factors, the filter in the input to reshape the reference is calculated and projected with the controllers for the ANN, Dual Matching Control (DMM), Analytic Inverse control (AIC), and Feedforward Reference Control (FRC). The filter for the reference in the crane trolley velocity is tuned for 𝜔 = 2 and 𝜁 = 0,9 Eq. (16). 𝛷𝑝 (𝜔) =

𝜔2 𝑠2 +2𝜁𝜔𝑠+𝜔2

Parameter

Hoisting

Traverse

Rated capacity Hoisting height Power supply Wire rope

2,8 t 12 m 3-phase, 3-wire 200 V 60 Hz JIS G3525 6 × 𝐹 𝑖 (29) Z twist 𝛷9 × 4 break power 48,0 kN 0,167 m/s –

2,8 t 12 m

Maximum speed Acceleration time

0,417 m/s 4 s

(16) the variation of the wire length. DMM1, AIC and FRC showed highest error during the load transportation in all conditions. The highest load vibrations is obtained with AIC and FRC. With ANN a short delay can be seen in the response. It is due to the capacity of the current equipment for the calculation of the neural network controller. The DMM controller presented in all conditions more overshoot in the trolley and load velocity responses which is not convenient during the operation of the crane. The major error in all conditions is evidenced with the FRC and maximum in the variation of the length with 1, 01 × 10−2 . The ANN and DMM showed less % vibration index in the angle compared to other controllers. The ANN presented robust responses with variation of the conditions with an error between 2, 3 × 10−7 − 2, 9 × 10−7 .

The reshaped input reference is applied for the controllers with the IDCS. DMM, and AIC controllers were tuned for the best operative conditions. The simulation results are summarized in Table 5 (comparative analysis) and Figs. 20–23. The DMM1 control is tuned with poles in high frequency 𝜔 = 100 and 𝜁 = 0, 8, DMM2 control is tuned with poles in low frequency 𝜔 = 3 and 𝜁 = 0, 8. The feedforward reference control FRC used directly the trapezoidal reference for the trolley velocity, FRC𝑓 is a reference signal shaped with a filter, and the analytical inverse control AIC is calculated for the model with the wire length assumed in the operational range around 5 m in the model. 6.6. Discussion about the simulation results

7. Experimental results using the ANN-IDCS controllers Different conditions were tested in the crane in order to analyze the performance of the controllers and implement the most suitable controllers for the real experimentation. Simulations results (Table 5) showed that the ANN presented the lowest error performance compared with the other controllers with values around 2, 6 × 10−7 . The DMM2 also presented low error values, however the error increases during

In experiments, only the controllers with the best conditions were tested. The ANN controllers in the IDCS structure (Fig. 1) were tested on a real crane using the controllers calculated in the previous section. The technical specifications of the crane are presented in Tables 6–8. For the experimental setup, the components are presented in Fig. 17. 10

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Fig. 18. Simulations results with different ANN-IDCS controllers (10 N–200 N) with the number of regressors type A: (a) angle (b) zoom of angle, (c) load velocity, (d) zoom of load velocity, (e)trolley velocity, (f) zoom of trolley velocity.

Table 7 Technical specifications for inverter controller 𝑆𝐽 700 − 007𝐿𝐹 𝐹 2 in the crane system. Parameter

Value

Rating capacity 200 V Rated output current Rated output voltage Frequency range Rated input AC voltage

1,7 kVA 5A 3-phase 3-wire 200–240 V 0,1–400 Hz 200–240 V, 50/60 Hz

stability and performance during the load transportation in a normal operative state (Table 6). For these experiments, the controller with 10 neurons was chosen for its suitable performance results calculation cost. The experimental results for default parameters are showed in Fig. 24, and Tables 9–11 present RMSE and %vibration index for condition 1 (default parameters in a normal operation) and the other conditions. The second part of the experiments was the validation of robustness of the controller when different parameters are changed. For this part, four experiments were conducted based on the robustness validation. The experiments were setup as following: (i) Condition 1 (default): wire length changing from 2,0 m to 1,5 m, hoist type is up motion, and trolley velocity in 0,35 m/s, (ii) Condition 2 (change wire length shorter): wire length from 1,5 m to 1,0 m, hoist type is up motion, and trolley velocity in 0,35 m/s, (iii) Condition 3 (change hoist type

The ANN controllers are programmed directly in a DSP to command the trolley velocity. Two main experiments were conducted to validate the control performance using the operational specifications with a load of 500 kg. The first part of the experiment was conducted to validate the 11

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Fig. 19. Simulations results with different ANN-IDCS controllers (10 N–200 N) with the number of regressors type A: (a) Load position (b) zoom of load position, (c) load angle, (d) zoom of load angle. Table 8 Technical specifications for digital signal processor (DSP). Parameter

Variable

Value

Controller

CPU OS Memory CPU Operating freq. Resolution Input range Convert time Resolution Convert time Max. current Min. current Max frequency Output current Max frequency

Intel ATON 1,6 GHz Linux 2,6x SDRAM 1 GB SH-4H 600 MHz 16 bit ±10 V 2 μs 16 bit 15 μs 15 mA 5 mA 5 MHz 30 mA max 1 kHz

Real-time controller Analog input

Analog output Digital input

Digital output

To see more details about the DMM controller refer to Tagawa et al. (2016). Table 9 presents the margin of amplitude error, phase error and combined error for the experimental results. Table 10 presents the RMSE error load velocity calculated for the different conditions. Figs. 20–27 show the results for all controllers and conditions. These figures represent the controllers with minimum error in magnitude 𝑀𝑆𝐺 , phase error 𝑃 and combined error 𝐶𝑆𝐺 (ANN-10NN[3 3 1], DMM = DMM2 (poles in low frequency 𝜔 = 3 and 𝜁 = 0, 8), FRC and 𝐹 𝑅𝐶𝑓 (filter in 𝜔 = 2 and 𝜁 = 0, 9) and AIC (wire length of 5 m). The %vibration index is presented in Table 11. Table 9 shows the results for the comparison between ANN, DMM, FRC and AIC controllers using the metrics of magnitude, phase and the combination of these measurements. Based on (Schwer, 2007) the Sprague and Geers metrics to calculate the measurement errors is used to determine the performance between controllers. 𝑚(𝑡) is the measured values and 𝑐(𝑡) is the reference in this application, then the following integrals are used to determine the values

motion): wire length from 2,0 m to 1,5 m and 2,0 m, hoist type motion

𝑡

𝑉𝑚𝑚 = (𝑡2 − 𝑡1 )−1 ∫𝑡 2 𝑚2 (𝑡)𝑑𝑡 1

up and down, and trolley velocity in 0,35 m/s, (iv) Condition 4 (change

𝑡

𝑉𝑐𝑐 = (𝑡2 − 𝑡1 )−1 ∫𝑡 2 𝑐 2 (𝑡)𝑑𝑡

max trolley velocity): wire length from 2,0 m to 1,5 m, hoist type

1

(17)

𝑡2

𝑉𝑚𝑐 = (𝑡2 − 𝑡1 )−1 ∫𝑡 𝑚(𝑡)𝑐(𝑡)𝑑𝑡

motion is up, and trolley velocity is 0,38 m/s.

1

The error in magnitude which is insensitive to phase variation is given by √ (18) 𝑀𝑆𝐺 = 𝑉 𝑐𝑐∕𝑉 𝑚𝑚 − 1

All conditions were tested with ANN controllers, DMM controller, AIC and FRC controllers in order to compare performance and vibration suppression. The experimental results are presented in Figs. 24–27. 12

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Fig. 20. Simulation results by condition 1 in default parameters: wire length from 2,0 m to 1,5 m, hoist type is up motion, and trolley velocity in 0,35 m/s.

Fig. 21. Simulation results in condition 2 with variation of the wire length: wire length from 1,5 m to 1,0 m, hoist type is up motion, and trolley velocity in 0,35 m/s,.

The phase error which is insensitive to the magnitude variation is

Finally, the Comprehensive Error Factor (𝐶𝑆𝐺 ) is calculated which represents the combined magnitude and phase differences. √ 2 + 𝑃2 (20) 𝐶𝑆𝐺 = 𝑀𝑆𝐺

expressed by 𝑃 =

√ 𝑉 𝑚𝑚 ∗ 𝑉 𝑐𝑐)

1 𝑐𝑜𝑠−1 (𝑉 𝑚𝑐∕ 𝜋

(19) 13

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Fig. 22. Simulation results in condition 3 with variation of hoist type motion: wire length from 2,0 m to 1,5 m and 2,0 m, hoist type motion up and down, and trolley velocity in 0,35 m/s.

Fig. 23. Simulation results in condition 4 with variation of trolley velocity: wire length from 2,0 m to 1,5 m, hoist type motion is up, and trolley velocity is 0,38 m/s.

8. Discussion

different quantity of neurons and regressors were validated using the error and correlation of the data.

For the data collection the PRBS signal was applied in the crane as the trolley velocity. The results from the inverse models generated with

However, the predictive error and correlation of the error increase means that the consideration for the training parameters setup must be 14

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Fig. 24. Experimental results in condition 1 in default parameters.

Fig. 25. Experimental results in condition 2 with variation of the wire length.

changed according to this architecture for the algorithm to converge

or 10−8 guarantying a margin of the correlation between 0,02–0,03 as a

inside the same range of the correlation error. The models with high

good performance for the crane controllers. During the training process,

correlation error showed a control signal performance increasing over

the increase in the quantity of regressors and the number of neurons

peaks and points with accelerations that can affect the motion load in

implied more calculation time, and these augmented architectures did

the crane. The prediction error of the models must be kept around 10−9

not always result in the best control performance. The training needed 15

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Control Engineering Practice 94 (2020) 104203

Fig. 26. Experimental results in condition 3 with variation of hoist type motion.

Fig. 27. Experimental results in condition 4 with variation of trolley velocity.

to be optimized and pre-configured for each model setup to guarantee the quality in the control signal. In order to create and validate models that guarantee stability and good control signal that avoid vibrations, the predictive error and correlation must be set in the optimization of the ANN with the constraints of the dynamics of the crane. According to the data, the

best condition was obtained when the error is about 0,02–0,03, other high values present spikes in the control signal implying more vibration in the load motion. To reduce the vibration index the dynamics has to be calculated using the relation between the minimum vibration and setup condition for the model correlation. 16

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Table 9 Summary of the performance of the experimental results with ANN inverse dynamic control, DMM1 control (with poles in high frequency 𝜔 = 100 and 𝜁 = 0, 8), DMM2 control (with poles in low frequency 𝜔 = 3 and 𝜁 = 0, 8), FRC feedforward reference control, AIC analytical inverse control. Results based on analysis of 𝑀𝑆𝐺 error of magnitude, 𝑃 error of phase, and combined error 𝐶𝑆𝐺 - Comprehensive Error Factor, using the filter in the reference 𝜔 = 2 and 𝜁 = 0, 9. Parameters

Condition Condition Condition Condition

1 2 3 4

ANN

(Default parameters) (Wire length 1,5 m to 1 m) (Hoisting type motion up/down) (Max trolley veloc 0,38 m/s)

Parameters

Condition Condition Condition Condition

1 2 3 4

1 2 3 4

𝑃

𝐶𝑆𝐺

𝑀𝑆𝐺

𝑃

𝐶𝑆𝐺

0,0438 0,0433 0,0434 0,0441

0,0425 0,0446 0,0478 0,0432

0,0610 0,0621 0,0646 0,0617

0,0616 0,0722 0,0513 0,0611

0,0230 0,0239 0,0278 0,0230

0,0658 0,0761 0,0584 0,0653

𝑀𝑆𝐺

𝑃

𝐶𝑆𝐺

𝑀𝑆𝐺

𝑃

𝐶𝑆𝐺

0,0273 0,0284 0,0238 0,0285

0,0335 0,0284 0,0388 0,0325

0,0432 0,0402 0,0455 0,0432

−0,0103 0,0183 −0,0199 −0,0188

0,1046 0,0855 0,1137 0,1130

0,1051 0,0874 0,1155 0,1145

𝑀𝑆𝐺

𝑃

𝐶𝑆𝐺

𝑀𝑆𝐺

𝑃

𝐶𝑆𝐺

0,0677 0,0743 0,0687 0,0623

0,0522 0,0504 0,0540 0,0611

0,0855 0,0898 0,0874 0,0873

0,0272 0,0344 0,0212 0,0252

0,0467 0,0322 0,0571 0,0501

0,0540 0,0471 0,0610 0,0560

DMM2

(Default parameters) (Wire length 1,5 m to 1 m) (Hoisting type motion up/down) (Max trolley veloc 0,38 m/s)

Parameters

Condition Condition Condition Condition

DMM1

𝑀𝑆𝐺

FRC

AIC

(Default parameters) (Wire length 1,5 m to 1 m) (Hoisting type motion up/down) (Max trolley veloc 0,38 m/s)

FRC𝑓

Table 10 Experimental results: RMSE errors with ANN inverse dynamic control, DMM control, feedforward reference and AIC analytical inverse control. Parameters

RMSE Cond. 1

Cond. 2

Cond. 3

Cond. 4

ANN DMM1 DMM2 FRC FRCf AIC

0,0400 0,0471 0,0496 0,0354 0,0139 0,0580

0,0276 0,0475 0,0511 0,0341 0,0151 0,0595

0,0461 0,0460 0,0501 0,0363 0,0150 0,0586

0,0499 0,0503 0,0543 0,0374 0,0149 0,0615

during the load transportation compared to the other controllers (DMM, AIC, FRC). Table 9 presents the results of the error with the calculation of the Magnitude error 𝑀𝑆𝐺 , phase error 𝑃 and combined error 𝐶𝑆𝐺 . ANN presented less error in the combined error compared to DMM1, the case in which the controller is tuned with poles in high frequency. DMM2 presented less error in the combined error, however, RMSE and vibration index were higher compared ANN controllers. ANN and DMM1 presented less % vibration which is suitable for the crane operation. ANN presented lower vibration in all conditions with minimum vibration in condition 4 (changing the maximum trolley velocity). For all controllers, the highest % vibration index is obtained when the wire length is changing, an important condition to take into account during the operation of the crane. AIC showed more error in the 𝐶𝑆𝐺 , and the FRC has small error during the load transportation. The controllers AIC, FRC and 𝐹 𝑅𝐶𝑓 presented the highest % vibration index compared with the other controllers. DMM presented higher overshoot in the trolley velocity in the variation of all conditions. ANN control presents robustness in all conditions with low variation when the trolley velocity and hoist type motion are changed. In addition, the ANN was easier to tune when the parameters were calibrated, while other controllers needed to be tuned each time when the conditions were changed to guarantee suitable results and operative conditions in the real crane.

Table 11 Experimental results: % Vibration Index (VI) with ANN inverse dynamic control, DMM control, feedforward reference and AIC analytical inverse control. Parameters

% Vibration Index (VI) Cond. 1

Cond. 2

Cond. 3

Cond. 4

ANN DMM1 DMM2 FRC FRCf AIC

0,187 0,051 0,280 1,000 0,444 0,603

0,420 0,116 0,398 1,000 0,510 1,076

0,268 0,099 0,339 1,000 0,543 0,499

0,157 0,085 0,203 1,000 0,423 0,569

When the inverse models are applied as feedforward control, RMSE and % vibration index keep a mean 15% with different configurations (Table 2). The most critical point with the controllers was the time needed to calculate the control signal. That means with a small configuration in neurons and regressors it is possible to get suitable performance compared with the high quantity of neurons or regressors. To implement in a hardware and real operative industrial cranes, controllers designed for the minimal time with minimum computational cost guarantying robustness are more suitable for the load transportation operations. ANN controllers presented smooth motion transportation avoiding oscillations in the position angle. The load velocity also kept the trajectory profile according to the reference specifications. The trolley velocity presented some segments with more accelerations, that could result in low performance of the crane. In order to avoid high acceleration, the controllers were adjusted directly with the model. The robustness of the crane was analyzed using the variation of the wire length, type of hoisting, and maximum trolley velocity. The ANN controllers showed a vibration control reduced with maintained stability

9. Conclusion In this work, a new scheme for the crane control using machine learning for nonlinear inverse model in a IDCS control (Inverse Dynamic Simulation Control) is proposed. The models were trained by artificial neural networks ANN in different configurations and the inverse control was developed from this modeling. The variation in the number of neurons has not high impact in the predictive error, effective model and signal control. However, the computational cost to train and determine the final model implies the final consuming time in the control processing. Simulations for the control validation determined a satisfactory development with the different neural networks in the IDCS structure. The error performance and vibration index keep in very small values for every controller. The most critical part from the control validation was the time consuming to determinate the final control signal using the different machine learning models. This time is directly related with the dimensions of the neural network and the parameters with the learning model processes. The ANN controllers presented the minimum error approximately of 5 × 10−8 and 15% of vibration 17

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performance, keeping this value in the robustness test. Experiments in real-time on a real crane system were conducted with three ANN controllers applying the control with the best minimum error and minimum calculation time. The results with the proposed controller showed that the reduction of the vibration is possible, avoiding noise of the feedback sensor signals. The crane controllers responded well to load positioning according to the velocity profile, keeping a good motion tracking avoiding high vibrations. In the real crane it is to be noted that a high number of neurons in the architecture is not suitable for this application due to calculation time of the control signal. The ANN showed good robustness when the wire length, maximum trolley velocity and hoisting motion were changed, which was not evidenced with classical controllers or feedforward reference control. The possibility to implement other ANN controllers is open according to the hardware and the needs to process models with more precision. Using ANN control with feedback simulation allowed to introduce a new strategy of machine intelligence control for industrial equipment and operative risky tasks.

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