Locally supervised neural networks for approximating terramechanics models

doi:10.1016/j.ymssp.2015.12.028

Mechanical Systems and Signal Processing

Volume 75, 15 June 2016, Pages 57-74

https://doi.org/10.1016/j.ymssp.2015.12.028 Get rights and content

Highlights

•
For the highly complex systems, it is difficult to build a robust global model by neural networks, and efficiently managing the large amounts of experimental data is often required in real-time applications.
•
In this paper, an effective method for building local models is proposed to enhance robustness and learning speed in globally supervised neural networks.
•
Furthermore, each local neural network is learned in the same manner as a Gaussian process (GP), because GP produces prediction that captures the uncertainty inherent in actual systems, and typically provides superior results.
•
A mixture of local model is created and then augmented using weighted regression.
•
This method, referred to as LGPN, is utilized for approximating the complex terramechanics models under fixed soil parameters.
•
The prediction results show that the proposed method yields significant robustness, modeling accuracy, and rapid learning speed.

Abstract

Neural networks (NNs) have been widely implemented for identifying nonlinear models, and predicting the distribution of targets, due to their ability to store and learn training samples. However, for highly complex systems, it is difficult to build a robust global network model, and efficiently managing the large amounts of experimental data is often required in real-time applications. In this paper, an effective method for building local models is proposed to enhance robustness and learning speed in globally supervised NNs. Unlike NNs, Gaussian processes (GP) produce predictions that capture the uncertainty inherent in actual systems, and typically provides superior results. Therefore, in this study, each local NN is learned in the same manner as a Gaussian process. A mixture of local model NNs is created and then augmented using weighted regression. This proposed method, referred to as locally supervised NN for weighted regression like GP, is abbreviated as “LGPN”, is utilized for approximating a wheel–terrain interaction model under fixed soil parameters. The prediction results show that the proposed method yields significant robustness, modeling accuracy, and rapid learning speed.

Introduction

Recently, research involving the development of motion control systems for planetary rovers has generated increased interest in wheel–terrain interactional systems [1]. Research in this field typically involves the identification of soil parameters and the building of physical models [2]. These proposed terramechanics models are both highly nonlinear and include multi-coupling equations that use many soil parameters with multivariable integration. Thus, they are difficult to apply under unknown soil parameters. Therefore, developing an effective and intelligent modeling algorithm is desirable. It is generally accepted that neural network (NN), as a popular machine learning method, exhibits potent learning abilities for approximating nonlinear functions and predicting outputs [3], [4]. Although there may be no prior knowledge of a system׳s dependences, a special neural network model can still effectively approximate the response of the system from input to output by updating the weights [5], [6]. In particular, when a sufficient number of training samples are obtained, an NN using supervised learning generally produces better prediction results, compared to NNs that use unsupervised learning [7]. To accurately approximate a highly complex nonlinear system, large amounts of experimental data are necessary. In that case, building an appropriate NN structure for a global model is quite difficult. This is due to the fact that a substantial number of neurons and weights must be learned, and iteratively tuning of all NN model parameters produces delays due to slow learning speeds [8]. This limitation prevents the use of NNs in real-time applications for learning complex nonlinear systems, which require large amounts of training data to be rapidly processed. For example, an online approximation of a nonlinear dynamics model for the purpose of robot control requires highly efficient online regression technology.

In recent years, various local learning methods have been proposed to resolve the aforementioned problems [9], [10]. The objective of local learning methods is to express complex global data as simple subsets, based on clustering techniques. Local learning that is considered to be effective locally, would not necessarily be effective globally [11]. The main advantage of local learning is the ability to process training data locally on the multiple individual components, which significantly reduces computation time and provides local robustness [12]. Fast local learning is more effective than a globally optimized model in real-time applications. To partition a full set of training data into smaller local subsets, many clustering algorithms have been proposed. The k-means clustering algorithm has been widely implemented in many fields to partition complex data sets into simpler data subsets [13], [14]. This method requires the number of local models k to be given, and randomly locates the initial k centers. Mixture of Experts (ME) method has been implemented to divide global training models into much simpler local models [15], [16]. However, to facilitate training data clustering, this method depends on the selection of an optimal number of local models k for the particular data set. Therefore, an effective method which can rapidly cluster a large amount of data, and conveniently adjusting the number of local models based on the special requirement.

Although training speed and robustness can be improved by local learning methods, this mixture of local models has remaining drawbacks. In particular, this supervised NN does not utilize an uncertainty model during the learning phase. For each local NN model configuration, an input variable is given, and the corresponding target of the network is an output variable; no other data are considered. However, a real dynamic system is often dominated by uncertainty, and real model predictions that do not include uncertainty may be of limited value and undesirable [17]. A Gaussian process (GP) with the nonparametric Bayesian model, which provides an explicit uncertainty measure, has developed into a useful machine learning tool [18], [19], [20]. It has shown a powerful learning capability, which enables it to accurately approximate complex nonlinear models in high-dimensional space [21], and to identify complex dynamic systems [22]. GP provides a fully generative model without significant formal requirements for training data distribution. As its main advantage, it considers both the noise in the system and uncertainty in the model. When sufficient training data are provided, the GP model can approximate the parametric models accurately.

The contribution of this paper lies in that a novel method is proposed by merging the intersected ε-region, which can conveniently adjusting the parameter ε based on the requirement; a locally supervised NN model is considered to train with uncertainty like GP, and then uses the locally weighted regression [11] to obtain the final predictive distribution.

This article is organized as follows. In Section 2, a novel cluster algorithm is proposed to partition the global NN model into simpler local models. Then, a locally supervised NN model is presented in Section 3. The advantage of a Gaussian process model is introduced, and each local NN is performed with a Gaussian process in Section 4. Predictions are performed by locally weighting each local NN model in Section 5. Finally, the performance of this algorithm is demonstrated by learning the model of a wheel–terrain interaction system in Section 6; the paper is concluded in Section 7.

Section snippets

The task of local learning

An observed set of n data $X = {x_{i}}_{i = 1}^{n}$ from a specially nonlinear system is used as input; the corresponding target output set is $y = {y_{i}}_{i = 1}^{n}$ , with p-dimension input $x_{i} \in R^{p}$ and scalar output $y_{i} \in R$ . In more realistic modeling situations, the outputs are the noisy observations of a latent function $f (x_{i})$ . The objective of using a neural network model is to build a mapping function from input data to output data, in order to predict the distribution of the new test data.

Once the training samples $D = {x_{i}, y_{i}$

Forward NN model

Here, the observed input data are considered as the centers of hidden neurons. For the kth local model, the requested nonlinear mapping relationship from the training subset $X_{k}$ to $f_{k}$ can be expressed by a simple forward NN model: $f_{k} (x_{k i}) = φ_{k} (x_{k i}) α_{k}, k = 1, \dots, M; i = 1, \dots, m_{k},$ $α_{k} = {[α_{k 1}, .. ., α_{k m}]}^{T},$ $φ_{k} (x_{k i}) = [φ_{k 1} (x_{k i}), .. ., φ_{k m} (x_{k i})],$ where $x_{k i} \in X_{k} = {[x_{k 1}, .. ., x_{k m_{k}}]}^{T}$ is the input point i of the kth model; m_k is the number of nodes in the hidden layer, and the corresponding output $f_{k} (x_{k i}) \in f_{k} = {[f_{k} (x_{k 1}), .. ., f_{k} (x_{k m_{k}})]}^{T}$ ; $α_{k} \in R^{m}$

Training NN model with uncertainty like GP

According to the training phase of supervised NNs, it is evident that the expression of predictive models is similar to the model-predicting mean of the Gaussian process. However, no uncertainty model was included in previously described NN implemented during the training phase. Since an actual dynamic system is dominated by uncertainty, a prediction without uncertainty is of limited value and undesirable. A Gaussian process for a regression (GPR) model provides an explicit uncertainty measure

Locally weighted NN models for prediction

Local learning typically depends on the notion of a “neighborhood,” which is always based on the prior measure of locality. Here, we introduce the weight measure $ω_{k}$ to ensure smooth transitions between the local models and the test point. It is a function of the Mahalanobis distance $d^{2} = {(x_{*} - x_{i})}^{T} Λ_{k}^{- 1} (x_{*} - x_{i})$ from the test point to a local model [27]. This distance is often selected from the test point to the center of a local model. However, this study utilizes the nearest Mahalanobis distance

Wheel–terrain interaction test platform

Research involving wheel–terrain interaction systems has an important role in the design of motion control systems for planetary rovers. However, most physical modeling performed during this research is highly nonlinear and includes multi-coupling equations. In this case, the locally supervised NN is applied to identify the optimal model for wheel–terrain interaction systems. The platform used for testing is the wheel–terrain interaction test system developed in the Chinese State Key Laboratory

Conclusion

Neural network based on supervised learning is capable of identifying nonlinear models and predicting the distribution of systems more accurately than models based on unsupervised learning. However, for highly complex nonlinear systems, building a global robust network model is quite difficult, since a large amount of experimental data must be rapidly processed, especially in real-time applications. To improve learning ability, this paper has proposed an effective clustering method to build a

Acknowledgments

This study was supported in part by the National Natural Science Foundation of China (Grant no. 61370033/51275106); National Basic Research Program of China (Grant no. 2013CB035502); Harbin Talent Programme for Distinguished Young Scholars (No. 2014RFYXJ001); Fundamental Research Funds for the Central Universities (Grant no. HIT.BRETIII.201411); Research Project of State Key Laboratory of Mechanical System and Vibration (No. MSV201610); and the “111 Project” (Grant no. B07018).

Xingguo Song is a PhD candidate in Mechanical Engineering at Harbin Institute of Technology, State Key Laboratory of Robotics and System, Harbin, China. His current research interests include stability of dynamical systems, neural networks, robotics and automatic control.

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Terramechanics models augmented by machine learning representations
2023, Journal of Terramechanics
The field of terramechanics focuses largely on two types of simulation approaches. First, the classical semi-empirical methods that rely on empirically determined soil parameters and equations to calculate the soil reaction forces acting on a wheel, track or tool. One major drawback to these methods is that they are only valid under steady-state conditions. The more flexible modelling approaches are discrete or finite element methods (DEM, FEM) that discretize the soil into elements. These computationally demanding approaches do away with the steady state assumption at the cost of including more model parameters that can be difficult to accurately tune. Model-free approaches in which machine learning algorithms are used to predict soil reaction forces have been explored in the past, but the use of these models comes at the cost of the valuable insight that the semi-empirical models provide. In this work, we presume that in a dynamic simulation, the soil reaction forces can be divided into a steady state component that can be captured using semi-empirical models and a dynamic component that cannot. We propose an augmented modelling approach in which a neural network is trained to predict the dynamic component of the reaction forces. We explore how this theory can be applied to the simulation of a soil-cutting blade using the Fundamental Earthmoving Equation and of a wheel driving over soft soil using the Bekker wheel-soil model.
Machine learning in planetary rovers: A survey of learning versus classical estimation methods in terramechanics for in situ exploration
2021, Journal of Terramechanics
Citation Excerpt :
However, due to the limited computational resources of planetary rovers, measurements of VO can only occur e.g. once every 90 s (Toupet et al., 2019), even proprioceptive sensor data attains a frequency of 8 Hz (Sullivan et al., 2011) typically. In traditional literature, classical (Bekker-derived) DT methods are both highly nonlinear and include multi-coupling equations, that use many soil parameters with multivariable integration (Song et al., 2016). Solving this problematic requires adopting simplification models to obtain closed-form analytical solutions for the forces and torques, that can be computable online (real-time) at the cost of increasing the modeling error.
For the design of space missions in the Moon and planets, analysis of mobility in robots is crucial and poor planning has led to abortion of missions in the past. To mitigate the risk of mission failure, improved algorithms relying intrinsically on fusing visual odometry with other sensory inputs are developed for slip detection and navigation. However, these approaches are significantly expensive computationally and difficult to meet for future space exploration robots. Hence, today the central question in the field is how to develop a novel framework for in situ estimation of rover mobility with available space hardware and low-computational demanding terramechanics predictors. Ranging from pure simulations up to experimentally validated studies, this paper surveys dozens of existing methodologies for detection of vehicle motion performance (wheel forces and torques), surface hazards (slip-sinkage) and other parameters (soil strenght constants) using classical terramechanics maps, and compare them with novel approaches introduced by machine learning, allowing to establish future directions of research towards distributed exteroceptive and proprioceptive sensing for visionless exploration in dynamic environments. To avoid making it challenging to collect all relevant studies expeditiously, we propose a global classification of terramechanics according most common practices in the field, allowing to form an structured framework that condense most works in the domain within three estimator categories (direct/forward or inverse terramechanics, and slip estimators). Likewise, from the experiences collected in previous MER (Mars Exploration Rover) missions, five overlooked problems are documented that will need to be addressed in next generation of planetary vehicles, along three research questions and few hypothesis that will pave the road towards future applications of machine learning-based terramechanics.
Towards in-situ characterization of regolith strength by inverse terramechanics and machine learning: A survey and applications to planetary rovers
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In the following, we try to clarify most of these aspects defining different learning paradigms. Neural networks (NNs) are one of the most popular machine learning methods (Chen, 1995; Song et al., 2016), and typically characterized according their architecture, node or learning rule characteristics. Although not all terramechanics applications of machine learning use them (e.g. (Gonzalez et al., 2018a; Bouguelia et al., 2017)), their generality will be useful to introduce certain concepts easier.
Characterization of planetary soils in past space missions has been a task confined mostly to specific instruments developed to assess the terrain. However, the non-standard techniques introduced by Sojourner Pathfinder and Mars Exploration Rovers (MER), have shown that analytical modeling of the wheel-soil interactions, combined with suspension-related telemetry acquired from the rovers, was sufficient to reveal natural restrictions of Mars terrains and strength properties, without the need of in-situ samplers. The progress on this field established since then is briefly summarized in this paper. Although dozens of approaches for in-situ characterization of planetary grounds with terramechanics have evolved over the years (defined here as inverse terramechanics estimators, or IT), most techniques are achievable in controlled environments but unsuitable for application in planetary rovers. This is because they generally require intensive onboard processing to obtain teal-time estimations like slip, which is not straightforward to perform in real conditions (MER must stop all functions to perform visual odometry). Consequently, beyond previous MER approaches, detection of martian or lunar terrain constants will not be realizable with most proposed empirical methods and available rover hardware. However, novel developments based in machine learning (ML) can quickly change this paradigm. ML-regression models can provide the available measurements (e.g. slip) for IT-approaches to work, even in challenging conditions such as dim light or eclipses. Nevertheless, ML-based methods typically do not include uncertainty and may require to incorporate gaussian process regression models to perform better. Hence we recommend novel research directions for IT to become achievable in practical applications. We further discuss how methods for planetary soil strength detection by IT have grouped over the years into basically three classifications (pure-empirical, learning-based, or combined frameworks), and relate the advantages of each according to the environment.
Environmental contact modeling for the earthworm-like robot via the novel elementary mechanical network
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Markov Chain Monte Carlo Parameter Estimation for Nonzero Slip Models of Wheeled Mobile Robots: A Skid-Steer Case Study
2021, Journal of Mechanisms and Robotics
Learning-Based Terrain Identification with Proprioceptive Sensors for Mobile Robots
2021, IEEE Transactions on Industrial Electronics

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Haibo Gao received his Ph.D. degree in Mechanical Engineering from Harbin Institute of Technology, Harbin, China, in 2003. His current research interests include aerospace mechanism and control. He is a Professor and Ph.D. advisor at the State Key Laboratory of Robotics and System, Harbin Institute of Technology. He won the second prize of National Award for Technological Invention.

Liang Ding received his Ph.D. degree in Mechanical Engineering from Harbin Institute of Technology, Harbin, China in 2010. His current research interests include control, simulation, and mechanics for mobile robots. He is an associate Professor and Ph.D. advisor at the State Key Laboratory of Robotics and System, Harbin Institute of Technology. He also is an IEEE member. He won the second prize of National Award for Technological Invention, and 2011 Excellence Award of Hiwin Doctoral Dissertation Award.

Pol D. Spanos is L.B. Ryon Chair in Engineering, Professor of Mechanical Engineering and Materials Science, and of Civil and Environmental Engineering, Rice University. His current research interests include digital signal processing, parameters identification, response determination, dynamical systems, performance and safety assessment for statics and dynamics problems.

Zongquan Deng is a vice-president, Professor and Ph.D. advisor at the State Key Laboratory of Robotics and System, Harbin Institute of Technology. His current research interests include robotics, planetary exploration rovers, aerospace mechanism and control.

View full text

Locally supervised neural networks for approximating terramechanics models

Highlights

Abstract

Introduction

Section snippets

The task of local learning

Forward NN model

Training NN model with uncertainty like GP

Locally weighted NN models for prediction

Wheel–terrain interaction test platform

Conclusion

Acknowledgments

Expert Syst. Appl.

Neurocomputing

Neural Netw.

Neural Netw.

J. Terramech.

Theory of Ground Vehicles

Interaction mechanics model for rigid driving wheels of planetary rovers moving on sandy terrain with consideration of multiple physical effects

J. Field Robot.

Real-time learning capability of neural networks

IEEE Trans. Neural Netw.

Real-time model predictive control using a self-organizing neural network

IEEE Trans. Neural Netw.

Global convergence of online BP training with dynamic learning rate

IEEE Trans. Neural Netw. Learn. Syst.

Adaptive optimal control of unknown constrained-input systems using policy iteration and neural networks

IEEE Trans. Neural Netw. Learn. Syst.

Local learning algorithms

Neural Comput.