The framework for dynamic load prediction proposed in this paper draws inspiration from various fields, including the resistive force model, classical mechanics, and advanced machine learning methods, such as PINNs.

### 2.1 Problem Definition

As a complex multidisciplinary system, an ECS consists of three major assemblies, namely, the upper assembly, lower assembly, and front-end attachments, as illustrated in Figure 1. Among them, the attachments, consisting of the boom, hoist ropes, crowd machinery, dipper handle, and dipper, etc., are the main operating mechanism that directly contacts the media to complete the digging task.

While digging, two types of motions are simultaneously performed by the dipper handle: extension/retraction motion in the direction parallel to the major axes of dipper handle and circular motion around the axis of crowd gear. Therefore, the digging mechanism of ECS can be considered as a two degree of freedom (DOF) mechanism. Based on the motion characteristics, a polar coordinate system is established to describe the dynamic system, as illustrated in Figure 2, where the axis of the crowd gear, *O*, is set to be the origin; the stretching length of dipper handle is set to be the polar diameter (*r*), and the angle between vertical direction and axes of dipper handle is set to be the independent variable (*ψ*).

Generally, three main factors affect the dynamic load of ECS in practice: trajectory parameters (e.g., velocity and acceleration), ore pile parameters (e.g., material mechanical properties and terrain of the ore pile), and structural parameters (e.g., dipper width and boom length). Since the structural parameters are static and remain relatively unchanged during the digging process, the mapping relationship among the digging trajectory, ore pile parameters, and corresponding dynamic load is considered in this paper.

Based on sensor measurements, the dipper motion information (consisting of the angle between the vertical direction and the axis of dipper handle \(\psi\), stretching length of dipper handle \(r\), the angular velocity of dipper handle \(\dot{\psi }\), stretching velocity of dipper handle \(\dot{r}\), the angular acceleration of dipper handle \(\ddot{\psi }\), the stretching acceleration of dipper handle \(\ddot{r}\)), and the corresponding digging forces (consisting of the hoist force \(F_{r}\) and crowd force \(F_{h}\)) can be synchronously acquired at every moment. A 3D scanner is used to measure the profile of the ore pile being excavated, and the obtained laser scanning data can be applied to establish an accurate geometric model, which is capable of describing the dynamic shape of the ore pile, as shown in Figure 3. The digging is a dynamic process, and based on the obtained 3D model, the digging depth *d* corresponding to the digging trajectory can be obtained. In the digging process, the medium loaded into the dipper continues to accumulate. Therefore, the mass, moment of inertia, and centroid of the dipper-material system are constantly changing, and the required digging force depends not only on the current working conditions but also on the historical digging trajectory. Essentially, dynamic load prediction can be summarized as a time series modeling task. For simplicity, let \({\varvec{x}} = [\psi ,r,\dot{\psi },\dot{r},\ddot{\psi },\ddot{r},d]\) and \({\varvec{y}}{\mathbf{ = }}[F_{r} ,F_{h} ,F_{t} ,F_{n} ]\). The goal of load prediction is to find a suitable function \(f\) that can efficiently and accurately map from input \({\varvec{x}}\) to digging forces \({\varvec{y}}\), e.g.,

$$ ({\varvec{x}}_{0} ,{\varvec{x}}_{{1}} , \cdots ,{\varvec{x}}_{t - 1} ,{\varvec{x}}_{t} )\to ^{{\varvec{f}}} {\varvec{y}}_{t} . $$

(1)

In addition, the non-observable variable (e.g., the resistive force due to media-dipper interactions) which is not measured in data, and cannot be directly modeled and solved. We can introduce available prior physics knowledge (e.g., Lagrangian mechanics, the conservation of energy) as extra constraints into the function \(f\) for indirectly solving the latent variable.

### 2.2 Digging Resistance Model

During the digging process, the dipper interacts with the excavated material, and the excavated material produces a great resistance to the dipper originating from the presence of surcharge, cohesion, etc. As mentioned, empirical equations of modeling the resistance of complex digging process may not be reliable, but give insights on digging resistant force prediction. Figure 4 shows the various forces components due to the interaction of media tools that the dipper needs to overcome during the digging process. \(F_{t}\) is the tangential resistance parallel to the direction of tip motion; \(F_{n}\) is the normal resistance normal to the direction of tip motion, and \(G\) is the gravity of the medium loaded in the dipper.

Herein, the dynamic prediction model of the digging resistant force based on the method of trial wedges proposed by McKyes et al. [6] is selected to predict the tangential resistance. With this method, the tangential resistance can be divided into three parts, including the cutting resistance tangential components \(F_{c}\), the velocity effect resistance \(F_{v}\), and the resistance caused by the extrusion from the two sides of the dipper \(F_{s}\).

$$ F_{t} = F_{c} + F_{v} + F_{s} , $$

(2)

where \(F_{c}\) can be obtained as Eq. [6].

$$ \left\{ \begin{gathered} F_{c} = \omega \left( {\gamma gd^{2} N_{\gamma } + cdN_{c} + \gamma v^{2} dN_{a} } \right), \hfill \\ N_{\gamma } = 0.5\left( {\cot \beta + \cot \rho } \right)/E_{N} , \hfill \\ N_{c} = \left[ {1 + \cot \rho \cot \left( {\rho + \varphi } \right)} \right]/E_{N} , \hfill \\ N_{a} = \left[ {\tan \rho + \cot \left( {\rho + \varphi } \right)} \right]/\left[ {1 + \tan \rho \cot \beta /E_{N} } \right], \hfill \\ E_{N} = \cos \left( {\beta + \delta } \right) + \sin \left( {\beta + \delta } \right)\cot \left( {\rho + \varphi } \right), \hfill \\ \rho = \left( {\uppi - \varphi } \right), \hfill \\ \end{gathered} \right. $$

(3)

where *ω* is the dipper width, *γ* is medium specific mass, *d* is the digging depth, *c* is the medium cohesion, *v* is the speed of dipper teeth, *β* is the digging angle, *ρ* is the failure plane angle, *ψ* denotes the internal friction angle of the medium, and *δ* denotes the external friction angle.\(F_{v}\) and \(F_{s}\) can be obtained through Eqs. (4) and (5).

$$ F_{v} = \frac{{\omega d\nu^{2} \gamma [\tan \rho \sin (\rho + \varphi ) + \cos (\rho + \varphi )]}}{\sin (\beta + \delta + \rho + \varphi )(1 + \tan \rho \cot \beta )}, $$

(4)

$$ F_{s} = \frac{{2d^{3} \gamma (\cot \beta + \cot \rho )\sin (\beta + \delta )\sqrt {\cot^{2} \rho + \cot \beta \cot \rho } }}{3\omega \sin (\beta + \rho + \varphi + \delta )}. $$

(5)

When the dipper cuts through the medium, the bottom of the tip compresses the medium, thus, the normal resistance due to the extrusion reaction arises whose orientation is perpendicular to the speed of dipper teeth. And it’s difficult to obtain an analytical expression of the normal resistance \(F_{n}\) because it depends on both digging operations and the medium’s hardness. Usually, the value is obtained by multiplying the tangential cutting resistance by a factor obtained from experience [7]. However, in this article, we use a neural network to represent the normal resistance:

$$ F_{n} = f_{\theta } \left( {{\varvec{x}}_{{\varvec{0}}} ,{\varvec{x}}_{{\varvec{1}}} , \cdots ,{\varvec{x}}_{{\user2{t - 1}}} ,{\varvec{x}}_{{\varvec{t}}} } \right). $$

(6)

### 2.3 Deep Lagrangian Networks (DeLaN)

The purpose of establishing dynamic models of the digging process is to predict how the state of the system evolves over time by a vector of generalized variables \({\varvec{q}} \in {\mathbb{R}}^{N}\) and velocities \(\dot{\user2{q}} \in {\mathbb{R}}^{N}\), where *N* is the number of coordinates. DeLaN uses knowledge originating from the Euler-Lagrange equation and encodes this prior within a flexible deep learning architecture [21]. Based on this architecture, all learned models adhere to Lagrangian mechanics. Specifically, the Lagrangian of a rigid body is generally defined as

where \(T = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\dot{\user2{q}}\user2{H}({\varvec{q}})\dot{\user2{q}}\) is the kinetic energy, *V* is the potential energy, which can be defined as a scalar function *V*(*q*), and *H* is the positive definite mass matrix. Substituting *L* into the Euler-Lagrange differential equation yields the second order ordinary differential equation (ODE) described by:

$$ {\varvec{H}}{(}{\varvec{q}}{)}\user2{\ddot{q}} + \underbrace {{\dot{\user2{H}}{(}{\varvec{q}}{)}\dot{\user2{q}} - \frac{1}{2}\left( {\frac{\partial }{{\partial {\varvec{q}}}}\left( {\dot{\user2{q}}^{{\text{T}}} {\varvec{H}}{(}{\varvec{q}}{)}\dot{\user2{q}}} \right)} \right)^{{\text{T}}} }}_{{: = {\varvec{c}}({\varvec{q}}{,}\dot{\user2{q}})}} - \frac{\partial V}{{\partial {\varvec{q}}}} = {\varvec{\tau}}, $$

(8)

where *τ* represents the nonconservative generalized forces, for ECS system, mainly refers to motors force effects and the resistive force due to media-dipper interaction, *c* describes Coriolis and centripetal effects, and \({{ - \partial V} \mathord{\left/ {\vphantom {{ - \partial V} {\partial {\varvec{q}}}}} \right. \kern-\nulldelimiterspace} {\partial {\varvec{q}}}}\) is gravity [25]. In DeLaN, the unknown functions \({\varvec{H}}({\varvec{q}})\) and \(V({\varvec{q}})\) are represented as a feed-forward network, i.e.,

$$\left\{ \begin{gathered} \user2{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{H} }{ = }\user2{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{L} }{(}{\varvec{q}}{;}\theta_{1} {)}\user2{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{L} }^{{\text{T}}} {(}{\varvec{q}}{;}\theta_{1} {), } \hfill \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{V} = {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{V} (}}{\varvec{q}}{;}\theta_{2} {),} \hfill \\ \end{gathered} \right.$$

(9)

where \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{.}\) refers to an approximation, \(\user2{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{L} }\) is a lower triangular matrix with a nonnegative diagonal, and *θ*_{1} and *θ*_{2} are the network parameters, and one can encode the ODE by exploiting the full differentiability of the neural networks. The parameters *θ*_{1} and *θ*_{2} can be learned online and end-to-end by minimizing the violation of the physical law described by the ODE. The basic architecture of the DeLaN [26] can be found in Figure 5.

### 2.4 Long-Short-Term Memory Networks (LSTM)

LSTMs have achieved state-of-the-art performance in a range of different domains comprising sequential data, such as natural language processing (NLP) [27], load prediction [28], and remaining useful life (RUL) estimation [29]. In Figure 6, we show a typical structure of an LSTM's hidden nodes incorporating four interacting units, including an internal cell, an input gate, a forget gate, and an output gate [30]. The internal cell memorizes the cell state at the previous time step via a self-recurrent connection. The input gate controls the flow of input activation into the internal cell state. The output gate regulates the flow of output activation into the LSTM cell output. The forget gate scales the internal cell state, enabling the LSTM cell to adaptively forget or reset the cell’s memory [24]. Specifically, given the previous hidden output \(h_{{t{ - 1}}}\), cell state memory \(C_{t - 1}\), and current input \(x_{t}\), the current hidden output \(h_{t}\) can be computed in the following way:

$$ \left\{ \begin{gathered} {\varvec{i}}_{t} = \sigma \left( {{\varvec{U}}^{[i]} {\varvec{x}}_{t} + {\varvec{W}}^{[i]} {\varvec{h}}_{t - 1} } \right), \hfill \\ {\varvec{f}}_{t} = \sigma \left( {{\varvec{U}}^{[f]} {\varvec{x}}_{t} + {\varvec{W}}^{[f]} {\varvec{h}}_{t - 1} } \right), \hfill \\ {\varvec{o}}_{t} = \sigma \left( {{\varvec{U}}^{[o]} {\varvec{x}}_{t} + {\varvec{W}}^{[o]} {\varvec{h}}_{t - 1} } \right), \hfill \\ \tilde{\user2{c}}_{t} = \tanh \left( {{\varvec{U}}^{[g]} {\varvec{x}}_{t} + {\varvec{W}}^{[g]} {\varvec{h}}_{t - 1} } \right), \hfill \\ {\varvec{c}}_{t} = {\varvec{c}}_{t - 1} \odot {\varvec{f}}_{t} + \tilde{\user2{c}}_{t} \odot {\varvec{i}}_{t} , \hfill \\ {\varvec{h}}_{t} = \tanh ({\varvec{c}}_{t} ) \odot {\varvec{o}}_{t} , \hfill \\ \end{gathered} \right. $$

(10)

where \(\sigma\) is the logistic sigmoid function, \(\odot\) represents for the Hadamard product, and \({\varvec{U}}^{[\zeta ]}\) (\({\varvec{W}}^{[\zeta ]}\)) denotes the weight matrix between the current input \({\varvec{x}}_{t}\)(\({\varvec{h}}_{t - 1}\)) and the operations \(\zeta (\zeta \in i,f,o,g)\).