research paper on model order reduction

Investigation and Implementation of Model Order Reduction Technique for Large Scale Dynamical Systems

• Review article
• Published: 14 January 2022
• Volume 29 , pages 3087–3108, ( 2022 )

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• Santosh Kumar Suman   ORCID: orcid.org/0000-0003-1178-3234 1 &
• Awadhesh Kumar 1  

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Model Order Reduction (MOR) has demonstrated its robustness and wide use in engineering and science for the simulation of large-scale mathematical systems over the last few decades. MOR is currently being intensively optimized for dynamic systems that are becoming increasingly complex. MOR broad applications have been identified not only in the modeling but also for optimization and control engineering applications. In the present article, various methods related to MOR for large-scale linear and nonlinear dynamic systems have been analyzed, mainly pertaining to electrical power systems, control engineering, computational system theory, and design. The paper focuses on a detailed theoretic Perspective for MOR of the large-scale dynamical system to address the key challenges to the approximation along with their application. Firstly, a complete description of the literature search for various approximation techniques has been presented and the various inferences have been mentioned as the outcome. One of the drawbacks that we have found out of the investigation, has been taken as a sample problem. The key demerit of the balanced truncation approach is that the ROM steady-state values do not correspond with the higher-order system (HOS). This drawback has been eliminated in the proposed approach, which leads to the hybridization of balanced truncation and singular perturbation approximation (SPA) into a novel reduction method without the loss of retaining its dynamic behavior. The reduced system has been so designed to preserve the complete parameters of the original system with reasonable accuracy. The approach is based on the retention of the dominant states or modes of the system and is comparatively less important once. The reduced system evolves from the preservation of the dominant states or modes of the original system and thus retains stability intact. The methodology presented has been tested on two typical numerical examples taken from the literature review, to examine the performance, precision, and comparison with other available order reduction methods.

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Suman, S.K., Kumar, A. Investigation and Implementation of Model Order Reduction Technique for Large Scale Dynamical Systems. Arch Computat Methods Eng 29 , 3087–3108 (2022). https://doi.org/10.1007/s11831-021-09690-8

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Reduced order modeling and model order reduction for continuum manipulators: an overview

S.m.h. sadati.

1 Robotics and Vision in Medicine (RViM) Lab, School of Biomedical Engineering and Imaging Sciences, King’s College London, London, United kingdom

S. Elnaz Naghibi

2 Department of Aeronautics, Faculty of Engineering, Imperial College London, London, England, United kingdom

Lyndon da Cruz

3 Wellcome/EPSRC Centre for Interventional and Surgical Sciences, University College London, London, England, United kingdom

4 Moorfields Eye Hospital, London, United kingdom

Christos Bergeles

Sarmad Mehrdad , New York University, United States

Phuc Bui , University of Tulsa, United States

Soft robot’s natural dynamics calls for the development of tailored modeling techniques for control. However, the high-dimensional configuration space of the geometrically exact modeling approaches for soft robots, i.e., Cosserat rod and Finite Element Methods (FEM), has been identified as a key obstacle in controller design. To address this challenge, Reduced Order Modeling (ROM), i.e., the approximation of the full-order models, and Model Order Reduction (MOR), i.e., reducing the state space dimension of a high fidelity FEM-based model, are enjoying extensive research. Although both techniques serve a similar purpose and their terms have been used interchangeably in the literature, they are different in their assumptions and implementation. This review paper provides the first in-depth survey of ROM and MOR techniques in the continuum and soft robotics landscape to aid Soft Robotics researchers in selecting computationally efficient models for their specific tasks.

1 Introduction

Robots with continuum structures can mimic highly dexterous and deformable biological bodies, follow curved paths through body lumens (e.g., vessel interiors) in minimally invasive medical interventions, enable safe interactions with the environment ( Rus and Tolley, 2015 ). Considering compliant structures, however, brings forth issues such as uncertain deformations, limited control feedback, reduced control bandwidth and slow response, stability problems, underdamped modes causing persistent vibrations, and lack of precision under external loads ( Cianchetti et al., 2013 ; Blanc et al., 2017 ).

Soft Roboticists have approached such challenges by proposing new theoretical frameworks and tailored software packages ( Sadati et al., 2020a ). On one hand, geometrically exact Cosserat rod Variable Curvature kinematics and Finite Element methods (FEM) have proven to be of high fidelity ( Trivedi et al., 2008 ; Coevoet et al., 2017 ; Grazioso et al., 2019 ), but bring forth computational obstacles when used in real-time applications, such as control, are considered. On the other hand, Reduced Order Modeling (ROM) and Model Order Reduction (MOR) techniques, where low-dimensional continuous or discreterepresentations are used to approximate the full-order system (i.e., variational) kinematics representation or FEM solution for a continuum structure, can both address modeling challenges and improve computational performance ( Book, 1990 ; Della Santina et al., 2110 ; Goury, Duriez ). Such methods have pioneered methods for flexible-link structures ( Book, 1990 ) and highly articulated ( Chirikjian, 1993 ) robots and, more recently, have been permeating the soft robotics community (Goury, Duriez; Sadati et al., 2018 ; Singh et al., 2018 ).

Although Reduced Order Modeling and Model Order Reduction terms are utilized interchangeably in the literature, this paper makes a distinction and highlights their different assumptions, implementation steps, and capabilities, in order to help with deciding the most appropriate method for a given application. To this end, as instances of Reduced Order Modeling, we review discretization-based, modal, and strain techniques, where the continuum kinematics is constructed based on simplifying assumptions. Alternatively, schemes where the high dimensionality of a continuum robot is reduced, e.g., neglecting the less dominant deformation modes in a FEM solution, are considered as instances of Model Order Reduction. In this paper, we don’t consider reduced order integration schemes, e.g., ( Boyer et al., 2020 ; Godage et al., 2016 ; Caasenbrood et al., 1089 ), which may simplify the derivation of the dynamics without necessarily reducing the modeling state space.

Detailed review papers have recently discussed modeling of continuum robots and slender rods, and readers, novice or expert, can find details on modeling assumptions therein ( Della Santina et al., 2110 ; Rao et al., 2020 ; Gilbert, 2020 ; Sadati et al., 2017a ; Armanini et al., 2023 ). Our paper instead provides a classification based on classic continuum mechanics that highlights how to combine assumptions for a soft robot kinematics, governing equations, material mechanics, and solution strategies. More specifically, we reflect on the Reduced Order modeling techniques, as an emerging trend in the Soft Robotics community, and highlight the need for further studies to explore their full potential. Our goal is to provide a “recipe” that supports researchers, with moderate knowledge of the field and an interest in Reduced Order Modeling techniques, in identifying the most suitable modeling assumptions and solution strategy for their application.

The rest of this article is organized as follows. A brief review of the necessary elements of a soft robot model is discussed in Section 2 . A detailed review of Reduced Order Modeling and Model Order Reduction methods and complementary states space coordination and solution strategies is provided in Section 3 . The main features, advantages, and shortcomings of their different implementations are listed and contrasted in Tables 2 – 5 . The manuscript concludes in Section 4 . Table 1 lists the acronyms in literature, although we mostly use their expanded from in the rest of this manuscript.

Acronyms used in the literature for the simplified kinematic representation of a continuum robot.

Reduced Order Models and Model Order Reduction for continuum and soft robots in literature. SF is Shape (Basis) Function.

Summary of the findings from comparative studies of different modeling methods as presented in ( Sadati et al., 2017a ; Sadati et al., 2018 ; Sadati et al., 2019a ; Sadati et al., 2020a ; Rao et al., 2020 ).

2 Soft robot modeling: assumptions and solution strategies

This section categorizes the key elements of a continuum manipulator model from continuum mechanics theory point of view. This is to better understand how Reduced Order Modeling techniques fit in the soft manipulator modeling framework. The two key elements in theoretical modeling of a soft robot (except for deep learning approaches) are:

• A. system kinematics ,
• B. system mechanics ( governing equations or conservational law ),
• C. material mechanics ( constitutional law ); and
• II. the solution strategy (direct or indirect) for the resultant system of differential equations.

In the rest of this paper, we utilized the term “state space” to refer to the time-variant states of a continuum manipulator model. This is to distinguish between the time-variant states of a Reduced Order model (e.g., modal contribution or fitting function coefficients) with the robot configuration (e.g., strains, position, and orientation) and joint (e.g., joint angles and displacements in a Pseudo Rigid Body model) spaces.

2.1 Continuum robot kinematics

The kinematics of a continuum robot can be represented based on:

• a. Variable Curvature (VC), also called Continuum, Differential, Cosserat Rod, Geometrically Exact Kinematics ( Trivedi et al., 2008 ; Rucker and Webster, 2011 ; Till et al., 2019 ),
• 2. Shape (Basis) function-based, also called Geometrical Kinematics ( Armanini et al., 2023 ), solutions that approximate the bending angle ( Gravagne and Walker, 2000 ; Gravagne and Walker, 2002 ; Gravagne et al., 2003 ; Gravagne and Walker, 2012 ), strain field ( Orekhov and Simaan, 2008 ; Della Santina et al., 2020a ; Della Santina and Rus, 2020 ) or shape ( Godage et al., 2011a ; Godage et al., 2011b ; Godage et al., 2011b ; Godage et al., 2015a ; Sadati et al., 2018 ; Singh et al., 2018 ; Sadati et al., 2020a ; Rao et al., 2022 ) of the entire robot backbone via modal or fitting approaches,
• a. Pseudo Rigid Body (PRB) approach that approximates the robot kinematics as a highly-articulated serial rigid-link mechanism with states defined in relative coordinates, i.e., strains as rotation and translation of rigid-robot joints ( Howell and Midha, 1995 ; Howell et al., 1996 ; Yu et al., 2005 ; Sadati et al., 2017a ; Shiva et al., 2019 ; Sadati et al., 2020a ; Rao et al., 2020 ; Schultz et al., 2022 ),
• b. Piecewise Constant Curvature (PCC) that discretizes the backbone as a series of Constant Curvature infinitesimal segments with states defined in relative coordinates ( Sadati et al., 2020a ; Caasenbrood et al., 1089 ; Rao et al., 2020 ; Shiva et al., 2019 ; Renda et al., 2018 ; Renda, Giorgio-Serchi, Boyer, Laschi, Dias, Seneviratne),
• c. Piecewise Variable Curvature (PVC), with basis function-based (e.g., a polynomial) estimation of curvature value along each segment ( Boyer et al., 2020 ), or
• d. Absolute Nodal Coordinate Formulation (ANCF), based on position and orientation of finite number of nodes along the backbone as the system states defined in absolute coordinates ( Shabana, 2018 ; Sadati et al., 2020a ; Huang et al., 2021 ; Shabana and Eldeeb, 2021 ).

Constant Curvature (CC) kinematics can considered as a special case of basis function-based fitting for a continuum segment ( Webster and Jones, 2010 ; Mustaza et al., 2019 ; Della Santina et al., 2020b ). In the case of Piecewise Constant Curvature kinematics, the transformations between the segments can be derived based on rotation matrices ( Sadati et al., 2017a ; Shiva et al., 2019 ), quaternions ( Sadati et al., 2020a ), or screw theory ( Renda et al., 2018 ; Renda, Giorgio-Serchi, Boyer, Laschi, Dias, Seneviratne). The underlying transformation can be based on Constant Curvature assumptions ( Sadati et al., 2017a ; Sadati et al., 2017b ; Della Santina et al., 2020b ; Rao et al., 2020 ), discretization of Cosserat Rod (i.e., Variable Curvature) kinematics ( Renda et al., 2018 ; Shiva et al., 2019 ; Sadati et al., 2020a ), or deformation map of an Euler-Bernoulli beam ( Sadati et al., 2020a ). In the case of Absolute Nodal Coordinate Formulation, such transformations can be based on a basis function such as a polynomial ( Huang et al., 2021 ) or deformation map of an Euler-Bernoulli beam ( Sadati et al., 2020a ). Furthermore, fitting solutions are proposed for approximating the kinematics of a soft robot cross section geometry deformation, showing 1%–3% increase in the accuracy of a pneumatic soft appendage known as STIFF-FLOP ( Sadati et al., 2017b ).

Variable Curvature (also referred to as Continuum representation ( Armanini et al., 2023 ) provides a geometrically exact kinematic representation at the cost of a system with infinite number of states not suitable for controller design. Basis function-based methods present a low-dimensional continuous formulation suited for controller design with high modeling accuracy, however, are prone to simulation errors. The Pseudo Rigid Body method enables the usage of rigid-body robotic modeling and control techniques for flexible and continuum robots but is usually applicable for a small system (with small number, less than five, number of discretized segments) and usually suffers from singularity issues. This is due to the exponential growth of the derivations for the distal segments in a series rigid link mechanism. The Piecewise Constant Curvature techniques improve the modeling accuracy of a Pseudo Rigid Body representation by considering the structure deformation as a circular arc with limited applicability to large systems (with large number of discretized segments) and with external or body loadings. The Piecewise Variable Curvature technique provides a better estimate of the beam deformation with a polynomial curvature representation but is still not applicable to large systems due to the complex derivations for distal segments, similar to the case of the Pseudo Rigid Body method. Absolute Nodal Coordinate Formulation is the most accurate method for modeling a large system, via optimal sparse matrix formulation of the system dynamic equations, and for incorporating different material mechanics models and actuation forces ( Shabana, 2018 ), at the cost of a large state space dimension and complex models for internal forces and actuation inputs. Figure 1 presents the different kinematic representations for a continuum manipulator in the Soft Robotics literature.

Different kinematics representations for a continuum manipulator in Soft Robotics literature. Here R and p denote the local frame rotation matrix and position vector, m and I are the mass and inertia, v and u are the local translational (shear and elongation) and rotational (bending and torsion) strains, ϵ and α are the translation and Euler angular rotation of an element in the local frame, s is the unit length along the backbone, C and S are the coefficient and basis function for the basis function-based kinematics, i is a free index, δ and ′ are operators denoting infinitesimal change and partial differentiation along and w.r.t. s .

2.2 System mechanics (governing equations) derivation

The system mechanics can be derived for:

• 1. Euler-Bernoulli Beam Theory (static), i.e., based on Euler-Bernoulli beam formulation ( Wegiriya et al., 2019 ; Yu et al., 2021 ),
• 2. Cosserat rod method (static and dynamic) ( Trivedi et al., 2008 ; Rucker and Webster, 2011 ; Burgner-Kahrs et al., 2015 ; Sadati et al., 2017a ; Sadati et al., 2017b ; Gazzola et al., 2018 ; Till et al., 2019 ; Rao et al., 2020 ),
• 3. Principle of Virtual Work (PVW, static and dynamic) ( Sadati et al., 2017a ; Sadati et al., 2017b ),
• 4. Lagrangian Formulation (dynamic) based on shape function-based kinematics ( Godage et al., 2016 ; Mustaza et al., 2019 ; Della Santina et al., 2020a ; Della Santina and Rus, 2020 ), or discretized kinematics ( Godage et al., 2015b ; Sadati et al., 2020a ; Della Santina et al., 2020b ; Habibi et al., 2020 ; Schultz et al., 2022 ),
• 5. TMT Dynamics (dynamic) ( Sadati et al., 2018 ; Sadati et al., 2019a ; Sadati et al., 2020a ; Sadati et al., 2022 ), which is the vector format for the Principle of Virtual Work that derives the system dynamics using a vector formalism with fewer derivation steps compared to the Lagrangian approach, and
• 6. Recursive Computation Scheme (dynamic) ( Godage et al., 2016 ) (also called Reduced Order Integration scheme ( Caasenbrood et al., 1089 ) or Inverse Dynamic Model ( Boyer et al., 2020 ), for forming the equation of motion in a vector formalism by recursively evaluating the nonlinear equations of motion based on Lagrangian or Cosserat rod dynamics.

Euler-Bernoulli Beam theory provides the simplest mechanics formulation for a continuum robot but is only applicable to small deformation cases. Cosserat rod mechanics provides an exact solution in the form of a boundary value problem, known to be computationally expensive to solve via numerical methods. The Principle of Virtual Work is a powerful method of deriving governing equations to incorporate a robot structural complexity and inhomogeneities but challenging to derive for dynamic cases. Lagrangian dynamics (i.e., derivation of the equations of motion using Lagrangian methods) is the commonly utilized method for dynamic cases but results in complex nonlinear relations that are hard to rearrange in a vector formalism, i.e., linear w.r.t. the acceleration terms, when using a standard symbolic mathematical software, e.g., Matlab. Identifying repetitive structures in the Lagrangian formulation can simplify this problem (35).

TMT and Recursive Computation Scheme are proposed to derive such a form. TMT method, which is named after the formulation of the generalized mass matrix in a dynamic system ( Sadati et al., 2020a ), is an easy-to-interpret analytical derivation for the system dynamic equation of motion via direct evaluation of the terms in a vector format. However, it can be analytically challenging to derive the vector format of the virtual work associated with some of the terms in a continuum system dynamics. It is worth noting that the Principle of Virtual Work, Lagrangian, and TMT Dynamic formulations result in the same set of final derivations, but with different derivation steps. The Recursive Computation Scheme simplifies the vector-format formulation of a complex dynamic model based on the numerical evaluation of the equation of motion terms by selectively setting the inertial, damping, gravitational, and elastic terms to zero. As a result, it is more computationally expensive due to multiple recursive numerical evaluations of the governing equations.

2.3 Material mechanics (constitutional law)

The material mechanics for a continuum robot is usually derived based on:

• 1. Hooke’s law ( Trivedi et al., 2008 ; Rucker and Webster, 2011 ; Godage et al., 2016 ; Sadati et al., 2017a ; Sadati et al., 2017b ; Till et al., 2019 ) (i.e., linear elasticity, infinitesimal strain theory) for elastic materials (e.g., Nitinol),
• a. Neo-Hookean ( Trivedi et al., 2008 ; Sadati et al., 2017a ; Sadati et al., 2017b ; Shiva et al., 2019 ),
• b. Mooney-Rivlin ( Gopesh and Friend, 2020 ),
• c. Gent ( Shiva et al., 2019 ), or
• a. Non-Newtonian fluid via viscosity power law ( Sadati et al., 2018 ; Mustaza et al., 2019 ; Sadati et al., 2020a ) and
• b. Kelvin-Voigt method ( Mustaza et al., 2019 ).

Hooke’s law is the commonly used material model with acceptable accuracy for slender systems with small strains. Finite strain theory is used for cases experiencing large strains but cannot capture the material hysteresis and creep effects. The neo-Hookean method is the standard technique for hyper-elastic materials but with limited accuracy in practical cases. Mooney-Rivlin method offers a more accurate model with a standard identification procedure but is still limited to strains less than 100%. The Gent method is the standard technique for larger strain cases and is commonly used for modeling rubber structures. Modeling the material hysteresis is possible by considering a hyper-viscoelastic model. The viscosity power law is a simple method but is more suitable for fluid mediums. The Kelvin-Voigt method can provide a good estimate of the material behavior with a standard process to identify the material properties. The nonlinear formulation of hyper-elastic and hyper-viscoelastic models makes their implementation challenging in a continuum robot model. The Principle of Virtual Work is the most powerful method of deriving the system constitutional law to incorporate different material mechanics representations in a continuum robot model.

2.4 Model formation and solution

A variety of combinations of the above choices for the system kinematics, mechanics, and material mechanics are possible to model a continuum robot. As a result, a system of Ordinary (ODE, mostly for dynamic models with discretized or reduced order kinematics) or Partial Differential Equations (PDE, mostly for quasi-static models, most forms of FEM, and Cosserat rod dynamics) is formed to be solved numerically considering the system’s initial and/or boundary conditions. An ODE formulation is possible for a model with variable curvature kinematics and Cosserat rod mechanics with general loading if load Boundary Conditions (e.g., moment, shear, and tension) are known ( Xu et al., 2013 ; Sadati et al., 2020b ) or for quasi-static models based on discretized kinematics in the absence of distributed, body, or external loads ( Shiva et al., 2019 ).

The common solution strategies, as for any dynamic system, are:

• a. Temporal (time-domain) discretization & integration of a dynamic model (most common) ( Godage et al., 2015b ; Godage et al., 2016 ; Sadati et al., 2020a ), or
• b. Spatial (spatial-domain, e.g., along the backbone curve) discretization and integration of a dynamic model ( Till, 2019 ; Till et al., 2019 ; Till et al., 2019 ; Till et al., 2020 ) or a static model (special cases as explained above) ( Xu et al., 2013 ; Shiva et al., 2019 ; Sadati et al., 2020b );
• a. Shooting (optimization-based) methods for static or Cosserat rod dynamic models which relies on guessing the solution vector and Jacobian-based numerical optimization techniques ( Godage et al., 2011a ; Rucker and Webster, 2011 ; Sadati et al., 2017a ; Sadati et al., 2017b ; Sadati et al., 2018 ; Aloi and Rucker, 2019 ; Till et al., 2019 ; Sadati et al., 2020a ),
• b. Ritz-Galerkin ( Tunay, 2013 ; Sadati et al., 2018 ) method for basis function-based solutions where the system of equations is weighted (i.e., multiplied) by the basis functions vector to achieve higher estimation accuracy.

Direct methods are numerically more efficient because they do not rely on an initial guess, but they are limited to system models in the form of ODEs. Temporal domain integration is the standard technique used in the community but may suffer from time-stepping issues for systems with highly oscillatory states. Spatial domain integration techniques can better handle a system with highly oscillatory states but are not as developed as the temporal domain integration schemes. Indirect schemes are suitable for geometrically exact models that are usually in the form of a BVP. Shooting methods are the standard methods to solve such systems at the cost of computational performance, resulting in slow simulations. The Ritz-Galerkin method can turn a BVP into an ODE to lower the simulation computational cost at the cost of a more complex derivation, challenges with selecting suitable basis functions, and convergence guarantee studies.

2.5 Data-driven and learning-based models

Although data-driven techniques do not necessarily fit in our presented categories above, a short survey of their advantages and limitations can help better justify the usage of Reduced Order Modeling techniques for soft manipulators. Analytical models for continuum robots present a robust solution in the presence of unknown conditions. However, it is not easy to capture commonly observed phenomena in soft robot real behavior such as fabrication inconsistencies, visco-hyper-elastic material properties, material creep, and hysteresis. Learnt solutions (when trained) are on the other hand real-time, accurate (within the training dataset domain), capable of handling complex structural geometries and nonlinearities, and easy to implement by not requiring significant knowledge of continuum robot theory ( Thuruthel et al., 2017 ; Thuruthel et al., 2018a ; Jolaei et al., 2020 ; Truby et al., 2020 ; Kim et al., 2021 ; Wang et al., 2021 ; Wang et al., 2022 ). However, the validity of a deep learning-based solution without a conservational law is limited to the richness and generality of the dataset used for learning, hence lacking robustness in presence of new conditions, contacts, and external loads which are not tested before (i.e., available in the dataset). Lack of generality and robustness guarantee are the main drawbacks of data-driven methods.

The learning-based solution drawbacks limit the usability of such methods to the case of improving a soft robot modeling accuracy for repetitive tasks but challenging to apply in highly varying and dynamic environments such as medical surgery. Koopman operator techniques ( Bruder et al., 2019 ; Bruder et al., 2021a ; Bruder et al., 2021b ), learning the residual (unmodelled) system dynamics, and a combination of learning-based forward models with analytical controller designs can address some of these issues ( Della Santina et al., 2110 ). Alternatively, training such models based on high-fidelity analytical or FEM-based models may solve the robustness and lack of generality issues of data-driven approaches by providing them with rich enough datasets that are hard to gather with a real robot ( Iyengar et al., 2020 ; Iyengar and Stoyanov, 2021 ). A more detailed review of the topic is presented in ( Della Santina et al., 2110 ; Thuruthel et al., 2018b ). The rest of this paper provides a review of Reduced Order Modeling and Model Order Reduction techniques for modeling soft manipulators.

3 Reduced order modeling and model order reduction

Any effort to approximate the differential evolution of states, i.e., differential kinematics of a Cosserat rod, in terms of a representation with finite number of states, e.g., Pseudo Rigid Body kinematics, is often called Model Order Reduction approach in classic continuum mechanics. Although commonly employed for FEM ( Tunay, 2013 ), Computational Fluid Dynamics ( Naghibi et al., 2019 ), and in control theory ( Moore, 1981 ; Antoulas, 2005 ; Van Dooren et al., 2008 ), the application of lightweight slender rigid manipulators for space explorations, undergoing infinitesimal deformations, has motivated the usage of Model Order Reduction techniques in robotics research since 1980s ( Book, 1990 ). More recently, Model Order Reduction (also referred to as Reduction or, interchangeably, as Reduced Order Modeling) techniques are utilized for Soft Robotics research to capture the complex dynamics of active highly deformable structures. In this section, we follow the naming conventions used by ( Thieffry et al., 2018a ; Sadati et al., 2018 ) in which Reduced Order Modeling refers to the methods that construct a reduced order kinematics for a soft robot, while Model Order Reduction is used when compression techniques are utilized the reduce the state space dimension of a FEM model.

The main features, advantages, and shortcomings of the different implementations of Reduced Order Modeling and Model Order Reduction are presented and compared with each other in Table 2 . A brief review of these methods in Soft Robotics research is presented in this section. Relevant coordinate formulations and solution strategy are discussed at the end of this section and summarized in Tables 3 , ​ ,4, 4 , respectively.

State Space Coordination in continuum and Soft Robot modeling.

Solution strategies for reduced order modeling of soft robots.

3.1 Reduced Order Modeling (ROM) for kinematic formulation

Cosserat rod formulation of continuum robots results in a set of Partial Differential Equations (PDE) that form a Boundary Value Problem (BVP) in continuous domain. Traditionally, the Cosserat rod BVP formulation is turned into a discretized BVP with finite number of states, e.g., via Pseudo Rigid Body or Reduced Order kinematics, to be solved by an optimization-based (e.g., shooting) numerical method. Alternatively, weak-form solutions can be sought for a BVP, a common practice in applied mathematics and FEM analysis. To this end, the separation of variables method is used to decouple the temporal (time-dependent, i.e., states evolution over time) and spatial (space-dependent, i.e., states evolution along the robot backbone arc length) characteristics. As a result, the system of PDEs is replaced by two decoupled Ordinary Differential Equations (ODE), each with respect to either the spatial (arc length) or temporal (time) variables. Hence, a solution can be provided using an ODE solver (either analytical or numerical). The former method results in a kinematic formulation similar to rigid body robots, while the former approach requires fewer modeling states, hence is more computationally efficient. Seeking a solution via forward integration and minimal state space dimension of a Reduced Order Model weak solution are desirable for Soft Robotics controller design ( Della Santina et al., 2110 ). We have categorized the different weak-form solutions to approximate a soft robot kinematics in Table 2 .

A basis function-based (modal and fitting) approach can simply the complexity and high-dimensionality of the state space of a soft robot model via:

• 1. Modal approaches where the system kinematics is approximated by a weighted sum of the dominant deformation modes, or
• 2. fitting where the system kinematics is interpolated based on a basis function with control parameters.

As a result, the time-variant weights (modal contributions) in the modal approach or the control parameters of the fitting approach are the new system modeling states, and the mode shapes or fitting basis function are the spatial (i.e., w.r.t. curve length) basis functions. A basis function-based approach separates the variables of the original PDE system and enables seeking an ODE solution. Similar to the application of this method in flexible-arm robotics ( Book, 1990 ), mode shapes (shape or basis functions) can be found based on:

• a. spectral (e.g., Fourier) ( Chirikjian, 1992 ; Chirikjian and Burdick, 1994 ) or power (e.g., Taylor, i.e., simple polynomials or splines) ( Sadati et al., 2018 ; Della Santina et al., 2020a ; Sadati et al., 2022 ) series expansions for a modal approach, and
• b. configuration fitting via Lagrange, Hermite Bezier, Pythagorean Hodograph (PH), B-Spline, NURBS polynomial) ( Singh et al., 2018 ) for a fitting approach;
• 2. eigenfunctions based on an eigenvalue problem (via Singular Value Decomposition) after linearizing the system;
• 3. experimental-based mode shape (via a modal test) or basis shape function selection based on the actual system tests ( Rao et al., 2021 ; Rao et al., 2022 ).

Although investigated for flexible-arm robots in the 1980s ( Book, 1990 ), cases (2) and (3) are relatively unexplored for soft robots. The case of the Eigenvector solution to a finite element problem is discussed as a part of Model Order Reduction techniques in the following sections ( Book, 1990 ).

Modal approach results in a truncated series solution (with a finite number of terms that are adequate to achieve the desired simulation accuracy) with a finite number of terms (time-variant coefficient and space-variant basis functions). The fitting approach does not necessarily result in a series solution format as it employs a predefined basis function. In both cases, the solution for the system kinematics should satisfy the system initial (i.e., initial stationary state) and boundary (e.g., right angle at the robot base) conditions. As a result, the approximate solution complies with the overall system kinematics (e.g., configuration ( Sadati et al., 2020a ) or curvature ( Della Santina and Rus, 2020 ) along a continuum arm backbone), and overall system velocity field. The basis function-based kinematics approximation improves the simulation numerical performance and results in a system of acceptable dimensions for controller design. This way, rather than attacking the general modeling problem directly, a low-dimensional approximate model is introduced and studied in detail ( Della Santina, 2020 ).

Reduced Order Modeling in the form of assumed continuous modes is a common approach in computer graphics for reducing the rendering computational cost of large scenes where modeling accuracy is not a priority ( Barbic et al., 2009 ; Badías et al., 2018 ). In the robotic community, a similar approach has been taken since the 1980s for modeling and control of flexible-arm ( Book and Majette, 1983 ; Tsujisawa and Book, 1989 ; Book, 1990 ; Moberg, 2010 ; De Luca and Book, 2016 ) and hyper-redundant under-actuated ( Chirikjian, 1992 ; Chirikjian, 1993 ; Chirikjian and Burdick, 1994 ; Chirikjian, 1995 ; Chirikjian and Burdick, 1995 ; Chirikjian, 1997 ) robots. All the reported instances of Reduced Order Modeling in this survey have utilized an elastic material mechanics based on Hooke’s law. In the rest of this section, we summarized the main different Reduced Order formulations presented for a soft robot kinematics.

3.1.1 Modal approach

Modal approach is equivalent to interpolating a precomputed basis (the dominant deformations) to model the run-time pose of an object ( Xu and Barbič, 2016 ). In a FEM-based framework, the complex deformation of a continuum object can be reduced to a weighted summation over a finite-dimensional subset of all possible deformations of an object, which is referred to as the dominant deformation subset ( Wang et al., 2015 ) or assumed modes. Such a method achieves good computational performance but at the cost of accuracy, which can be improved by carefully selecting the dominant deformation subset and increasing the number of terms in the summation series. Unlike mode shapes that are stand-alone analytical solutions from the modal analysis of a system, assumed modes (basis or shape functions), despite satisfying the system BCs, may not represent a modal solution for the system and only provide a basis for approximating the system solution, to be verified numerically. For example, although assuming a polynomial solution for the deformation of a continuum manipulator can provide adequate simulation accuracy and computational efficiency, it is not a solution for the modal analysis of the same system. A modal representation of a soft robot kinematics simplifies the infinite-state-space-dimension of the system variable curvature kinematics and transform it to a finite dimensional representation, i.e., a fitting problem with a few weighting coefficients as the system states, and hence results in less theoretical complexity.

The modal approach is a common method for dynamic modeling of flexible-link robots and structures ( Book, 1990 ; Sayahkarajy, 2018 ). For example, Wu et al. have investigated nonlinear model order reduction based on the system modal derivatives ( Wu and Tiso, 2016 ) limited to simple geometries. Alternatively, Weeger et al. utilized pre-defined shapes in the form of 4D NURBS curves to express the configuration and quaternion orientation of a Cosserat rod centerline in large deformation ( Weeger et al., 2017 ). Liu et al. utilized a polynomial collocation spectral method (in which the series index is the basis function wave numbers, i.e., frequency in the spatial domain) for a Kirchhoff’s rod discretized on a set of Chebyshev or Legendre Gauss-Lobatto control points ( Liu et al., 2018 ). They showed the spectral method precision exponentially increases as the number of the nodes increases.

Modal approaches have been preliminarily investigated for Soft Robots in the form of run-time fitting techniques based on a precomputed basis (i.e., assumed deformation modes) as presented by Wang et al. ( Wang et al., 2015 ) and Xu et al. ( Xu and Barbič, 2016 ). Della Santina et al. (1806) utilized a robot with compliant joints stabilized via sub-manifolds on the system state space (i.e., system nonlinear continuation of linear eigenspaces, called nonlinear normal modes) on which the system would naturally evolve for motion generation. Alternatively, one could bypass the constitutional and conservational laws (system and material mechanics) by matching the basis functions from the experimental observations as the ground truth as presented by Chirikjian et al. for modeling volume preserving 3D deformations of soft simple geometries ( Chirikjian, 1995 ). A similar method has been presented for characterization of worm locomotion kinematics ( Digumarti et al., 2019 ). SOFA (Software Open Framework Architecture), as a computationally efficient simulation tool for evaluating random deformations of soft objects, can serve as an ideal platform in such method to identify an object’s dominant deformation basis.

3.1.2 Polynomial shape (PS) fitting for configuration space characterization

Amongst the early works on Reduced Order Modeling in Soft Robotics, the shape fitting concept was utilized by Godage et al. since 2011 as a singularity free representation of a continuum manipulator kinematics in which the backbone Cartesian configuration is expressed in terms of a power series with basis functions based on a manipulator actuation chambers length ( Godage et al., 2011a ; Godage et al., 2016 ). However, their representation could not capture the backbone twist and the choice of a hard-to-interpret basis function resulted in a series solution with many coefficients to identify (e.g., 165 ( Godage et al., 2011a ) and a complicated system mechanics derivations ( Godage et al., 2011a ; Godage et al., 2011b ; Godage et al., 2016 ). Sadati et al. showed the advantageous numerical performance and accuracy of using a simpler alternative based on parametrization of the configuration space, i.e., the robot backbone shape ( Sadati et al., 2018 ). A truncated Lagrange polynomial passing through a number of equally spaced control points was used to describe a continuum manipulator shape (configuration or kinematics) with the Cartesian position of the control points as the system modeling states. This method was referred to as Polynomial Shape fitting for soft robot kinematics.

Although it could successfully handle hyper-elastic deformation of the robot cross section, considerations for braided pneumatic actuators, and general loading cases for planar motions, the use of the Frenet-Serret tangent frame formulation invalidates the 3D solutions in the case of a twisted backbone due to external loading. The issue lies with a non-physical twist predicted based on the mathematical definition of a Frenet–Serret curvilinear frame. The same issue presents itself in the work by Singh et al. ( Singh, 2018 ; Singh et al., 2018 ) and utilized by Wiese et al. ( Wiese et al., 2019 ). Higher modeling and control accuracy could perhaps be achieved by considering this effect, noting that neglecting to do so may result in severe errors in the case of out-of-plane bending. Furthermore, the convergence of a Lagrange polynomial series solution (i.e., solution accuracy increase by the addition of control points) is not guaranteed with a set of equally-spaced points ( Sadati et al., 2018 ). This is due to Runge’s phenomenon (oscillation problem of the fitted curve at the ends of an interval) that can be addressed by using Chebyshev or Legendre Guass-Lobatto techniques ( Liu et al., 2018 ).

To address these issues, a general 3D method is presented in ( Sadati et al., 2019a ; Sadati et al., 2020a ) in which a 7D series solution (i.e., with seven independent series, three for the cartesian coordinates vector and four for the orientation quaternion vector) represents a continuum rod backbone’s 3D Cartesian position and quaternion orientation. This approach is similar to that of (98) but employs simple polynomials instead of NURBS curves. Instead of using Cartesian locations as system states as in (12), the new method uses the polynomial coefficients themselves, thereby improving convergence otherwise affected by Runge’s phenomenon. This also simplifies the derivation complexity caused by the nonlinear form of Lagrange polynomials.

However, in the 7D solution, the conformity of the local frame to the robot backbone tangent vector is governed by the system conservational law (i.e., compliance potential field). These constraints may not perfectly hold for a coarse spatial integration step. The exact formulation of the deformed frame twist (or local curvilinear frame in general) is needed for modeling instability-induced rapid snapping motions that are observed in some continuum robots such as Concentric Tube Robots ( Mitros et al., 2021 ). This is addressed based on a Bishop’s frame formulation for the tangent frame in (55) by introducing a correction twist on an arbitrary tangent frame to the robot backbone. As a result, the system kinematics can be described by a 4D series solution (three for the cartesian coordinates and one for the correction twist) at the cost of slightly more complex derivation of the governing equations. A spline may be used instead of a polynomial when certain degree of continuity is required along the robot overall backbone, e.g., due to the change in the robot structure as in a Concentric Tube Robot. The transition between the spline segments can be continuously implemented based on logistic functions. This representation is used to present the first dynamic model of an everting growing (vine) robot in 3D motion ( Berthet-Rayne et al., 2021 ). The dynamic modeling and controller design of hybrid rigid-soft robots based on the Polynomial Shape kinematic are implemented in an open-source Matlab software package, called “TMTDyn” ( Sadati et al., 2020a ).

Although being the most common choice, polynomials aren’t the only options to describe the shape of a soft robot. Singh et al. compared Hermite, Bezier, Pythagorean Hodograph (PH), BSplines, NURBS curves for this purpose and showcased, in simulation and experiments, the advantages of a PH curve representation for the controller design of a continuum arm ( Singh, 2018 ; Singh et al., 2018 ). This representation is later adopted for potential field-based obstacle avoidance of continuum arm on a mobile platform while navigating an unconstructed environment ( Mbakop et al., 2021 ).

3.1.3 Polynomial curvature (PC) fitting for curvilinear space characterization

Amongst the early works, shape approximation techniques were more common to model flexible arm robots ( Book, 1990 ). On the other hand, polynomial approximation of the strain (curvature and torsion) along a continuum backbone (Polynomial Curvature kinematics) was investigated to model planar motion of a hyper-redundant robot and later a DNA strand by Chirikjian et al. ( Chirikjian, 1992 ; Chirikjian, 1993 ; Chirikjian and Burdick, 1994 ; Chirikjian and Burdick, 1995 ; Chirikjian, 1997 ) who referred to it as “Modal Approach.” It is introduced to constrain a hyper redundant arm curvature to a representation with a few DOFs based on trigonometric (sine and cosine) functions as the assumed mode shapes for the curvature values, and time-variant coefficients as the modal participation factors.

In soft robotics research, Polynomial Curvature kinematics and Lagrange dynamics of a planar Kirchhoff rod (i.e., infinite shear) were studied by Della Santina et al. (who, for the first time, referred to it as Polynomial Curvature kinematics) highlighting their advantages and challenges in controller design ( Della Santina et al., 2110 ; Della Santina and Rus, 2020 ; Della Santina et al., 2020a ; Della Santina, 2020 ). A 1D series solution was used for the curvature of a planar beam backbone to overcome the robot vibration and reduced performance associated with a Constant Curvature-based controller. A similar method with Cosserat rod mechanics was studied for quasi-static 3D modeling of continuum rod in (27) focusing on the compatibility of the curvilinear constraints such as those seen in a Concentric Tube Robots ( Sears and Dupont, 2006 ; Webster et al., 2006 ; Mitros et al., 2021 ). More recently, 3D implementation of the Polynomial Curvature kinematics is showcased and verified in standard simulation scenarios by Boyer et al. ( Boyer et al., 2020 ), who referred to their presented kinematics as “Piecewise Variable Curvature”. There, the problem of deriving the complex governing equations for the system dynamics is addressed based on a recursive computational scheme which they referred to as “Inverse Dynamic Model.” Later, this method is showcased for modeling closed-chain (parallel) soft robots ( Armanini et al., 2021 ) and made available in an open-source Matlab software package, called “SoRoSim” ( Teejo Mathew et al., 2021 ).

Alternatively, the right basis function candidate can be identified based on the fitting performance of different shape functions to capture the shape of a real soft robot. As an instant of Polynomial Curvature method, Rao et al. have demonstrated that an Euler curve (a curve whose curvature varies linearly along the arc length) is a viable candidate to capture the planar deformation of a continuum appendage under external tip load ( Rao et al., 2021 ; Rao et al., 2022 ). Similarly, modal test results on the actual soft robots can be used to construct the basis functions for the robot’s reduced order kinematics, as practiced for flexible arm robotics in 1980s ( Book, 1990 ).

The curvilinear space, i.e., strains (shear, elongation, curvature, and torsion) requires one fewer state compared to the rectilinear space (i.e., absolute Cartesian position and orientation) to describe continuum backbone kinematics, since the curvilinear space is derived through spatial differentiation from the rectilinear parameters. However, deriving the rectilinear parameters is still required for calculating the local effect of external loading and the system inertial dynamics. This requires spatial integration of a set of non-integrable trigonometric functions. These integrations are handled via analytical approximation (e.g., Taylor series expansion ( Della Santina, 2020 ; Della Santina and Rus, 2020 )) of the integrands ( Della Santina et al., 2020a ; Della Santina, 2020 ; Della Santina and Rus, 2020 ) or numerical integration methods ( Orekhov and Simaan, 2008 ). The former method is complex and inaccurate for large deformations (outside the Taylor series convergence radius) but necessary for integration in Lagrangian dynamics. The latter method is only compatible with optimization-based solutions (e.g., shooting method). More accurate solutions for these integrals are proposed based on the formulation of the closed-form integrals with 0th-order Bessel functions for the 2D case ( Chirikjian, 1992 ) or, more generally, the use of integrable basis functions such as helical functions ( Grazioso et al., 2019 ). The Polynomial Shap approach overcomes these limitations (i.e., integration complexity, inaccuracy, and lack of generality) by relying on a differentiation step to obtain the curvilinear space parameters (required to calculate the strain actions) instead of the integration step required by the polynomial Curvature method to obtain the rectilinear space parameters.

However, unlike the Polynomial Shape method, the definition of curvilinear frame is not problematic in the case of Polynomial Curvature method due to the direct parametrization of curvature and torsion. The only exception is the method presented by Grazioso et al. ( Grazioso et al., 2019 ) in which the deformation map derivation relies on a curve in the Frenet-Serret frame that is constructed by analytical integration of helical basis functions for the curvilinear (curvature) space. There. similar to the works by Singh et al. ( Singh, 2018 ; Singh et al., 2018 ), the problem with an deformed local frame twist isn’t addressed.

Most of the recent Polynomial Curvature related studies are limited to theoretical investigations ( Orekhov and Simaan, 2008 ; Grazioso et al., 2019 ; Della Santina et al., 2020a ; Della Santina, 2020 ; Della Santina and Rus, 2020 ). Future experimental studies on soft robot hardware and a comparison with Polynomial Shape methods can help to achieve a fair comparison and conclusion on the most appropriate method for a given Soft Robotics application.

3.2 Model Order Reduction (MOR) for finite element models

In this paper, we used the Reduced Order Modeling term for the techniques that provide an approximate solution (usually in a basis function-based from) for the system kinematics. The system kinematics and material mechanics are then fed into the system mechanics to derive the system’s equation of motions. Alternatively, we employed the term Model Order Reduction (MOR) to refer to the techniques that reduce the state space dimension of an already developed model (usually based on FEM) for a soft system. Although Reduced Order Modeling and Model Order Reduction are employed interchangeably in some literature, the reason for the use of different terms in this paper is that Model Order Reduction, unlike Reduced Order Models for a robot kinematics, is not a fundamental block (i.e., the system kinematics, system mechanics, and material mechanics) of a continuum robot model. Model Order Reduction is also referred to as System Compression.

The Model Order Reduction utilizes Principal Orthogonal Decomposition or Principal Component Analysis techniques which are widely used in computational mechanics. There, Singular Value Decomposition is performed on the state variables of several sample deformations of a soft robot (minimum 2 within each actuation range). The results are truncated up to a tolerance to define the system reduced basis and to neglect insignificant system states. The insignificant states are hard to reach (require large excitation energy) or hard to observe (produce small observable energy) ( Antoulas, 2005 ). Such basis need not necessarily have any physical meaning. Measures are considered to achieve desired tolerance while preserving the system properties such as stability, passivity, and contractility. The compressed (reduced) system can be used for nonlinear controller and observer design with stability guarantee ( Chenevier et al., 2018 ).

Principle Orthogonal Decomposition, although being the only technique suitable for a nonlinear system ( Thieffry et al., 2018a ), is computationally expensive and therefore not real-time. To achieve real-time performance, Thriftey et al. ( Thieffry et al., 2018a ; Thieffry et al., 2018b ; Thieffry et al., 2019 ; Thieffry et al., 2020 ) performed the same analysis on a linearized FEM model around an equilibrium point, and utilized their reduced system to formulate a stable observer and controller. As a result, their reduced model was 3–100 times faster in simulations than the original FEM model. Alternatively, Goury et al. (Goury, Duriez) computed a reduced integration domain (and weights associated with it) by computing the reduced basis on the sample deformation space and simultaneously storing the elements contributions based on an assumed tolerance. The Galerkin method was then used to solve the resulted system. Elastic material mechanics based on co-rotational linear model ( Chenevier et al., 2018 ) and hype-elastic material mechanics based on neo-Hookean (Goury, Duriez; Chenevier et al., 2018 ) and Mooney-Rivlin (Goury, Duriez) models have been utilized in different Model Order Reduction implementations in the literature.

Model Order Reduction method is implemented in the “Soft Robotics Toolkit,” an open-source simulation package based on SOFA-framework. This implementation benefits from the real-time performance, open-source (hence expandable), accurate, and configurable numerical methods of SOFA and is experimentally verified for nonlinear controller design of soft robots of general shape (Goury, Duriez; Thieffry et al., 2018a ; Bieze et al., 2018 ; Thieffry et al., 2018b ; Thieffry et al., 2019 ; Thieffry et al., 2020 ; Katzschmann et al., 2019 ). However, the underlying FEM model results in high computational cost, lack of theoretical clarity (i.e., non-interpretability of the modes and complex nonlinear formulation), and complexity in implementing new assumptions (e.g., constraint) and elements (e.g., braided pneumatic actuators). Furthermore, the current implementation of SOFA-framework cannot handle Coriolis terms in dynamics of rigid bodies and hence poses difficulty in modeling hybrid rigid-continuum body structures. The plugin enables implementing a variety of material laws that are already available in SOFA.

3.3 State space coordination and dimensionality

A continuum system can be modeled based on states defined in relative (i.e., the continuum manipulator local tangent frame) and absolute (i.e., ground fixed reference/inertial frame) coordinates, which are called “relative states” and “absolute states,” respectively, in the rest of this paper. Relative states are usually referred to local strain or free parameters of a fitting method that describes a single segment w.r.t. the previous segment (local frame) in a series chain (e.g., in the case of Constant Curvature case). An absolute state refers to the robot shape itself (e.g., node position and quaternion orientation), participation coefficients of the robot shape modal representation, or free parameters of an fitting method that describes a robot segment w.r.t. the global frame (e.g., Hermite, Bezier, PH, or NURBS fitting of a robot backbone shape). Recently, a hybrid-state model is presented that combines the absolute (e.g., position) and relative (e.g., local correction twist angle to represent a Bishop’s frame) states (e.g., in the case of Concentric Tube Robots) ( Sadati et al., 2022 ).

Except for FEM research, relative state representations are dominant for modeling continuum robots due to simplicity in dealing with system elastic energy, internal forces, and actuation inputs. Modal representations with relative states require lower order basis functions too, e.g., a polynomial curvatures representation relies on 2 nd -order polynomials while polynomial shape representations require a 3 rd -order polynomial to capture the backbone basic motion and satisfy the boundary conditions (e.g., backbone right angle at the robot base). However, implementation of external force fields, contacts, and constraints are challenging. Furthermore, the system dynamic derivation involves an integration stage to account for the inertial effects that is usually dealt with based on approximate numerical solutions that may reduce the overall accuracy and increase the numerical cost of the system simulation. The complexity of the equations of motion increases as the number of elements, states, and segments in a system increases, making them unsuitable for large-scale systems. The similarity of such representations with the models for a rigid-link serial mechanism makes it easier to utilize well-developed techniques from rigid-body robotics. As a result, relative states are favorable for control system design in continuum robotics research.

On the other hand, absolute states result in an easier derivation of inertial terms, external forces, contact, and constraint (e.g., for parallel mechanisms). The system dynamic derivation involves a differentiation stage that can be accurately handled analytically, and the equation of motion derivation complexity does not significantly increase as the number of elements in the system increases, making it suitable for large-scale systems. As a result, absolute states are favored for FEM modeling in the Soft Robotics community. Furthermore, considering the robot shape in task space (configuration in global frame), as the system absolute states, makes such methods more suitable for trajectory tracking, shape and/or tip motion control.

Absolute representation of position and orientation does not kinematically constrain the local frame to the backbone curve tangent and this should be enforced by the governing equations, i.e., due to balance of shear and bending virtual energies. A hybrid state space method is recently proposed that benefits from absolute states for the robot position while defines a relative correction twist angle to define a local Bishop’s frame based on an arbitrary tangent frame to the resultant curve. As a result, for the first time, a seamless framework is derived that can handle instability instances in the dynamics of a continuum system, in this case a Concentric Tube Robot ( Sadati et al., 2022 ). Such a hybrid presentation results in a more complex derivation of the Coriolis terms due to the need for derivation of the backbone curve tangent frame. However, it benefits from a smaller state space since the orientation is defined based on a single correction twist angle instead of three states for Euler angles or for states for a quaternion representation.

In the case of geometrically exact modeling, variable Curvature kinematics result in an infinite-dimensional states space based on strains (relative states) along the robot backbone. Alternatively, FEM and Absolute Nodal Coordinate Formulations, which result in a large-dimensional state space based on only position (FEA ( Coevoet et al., 2017 ) or both position and orientation (Absolute Nodal Coordinate Formulations ( Shabana, 2018 ; Huang et al., 2021 ; Shabana and Eldeeb, 2021 ) of a finite number of nodes along the robot backbone, are instances of cases with absolute states. On the reduced order modeling front, Constant Curvature assumptions, modal, and fitting-based representation of a continuum backbone strain domain deal with relative states. On the other hand, modal, and fitting-base approximation of the robot shape present a reduced order representation of a system with absolute states. Recent formulation of a Concentric Tube Robot kinematics based on polynomial shape approximation in reference coordinates for the backbone position and Bishop’s frame representation for orientation in relative coordinates is an example of hybrid state formulation. The main features, advantages, and shortcomings of different state space coordination employed in continuum and Soft Robotics research are summarized in Table 3 .

3.4 Solution strategies

The resulted approximate solution for the system kinematic or mechanics based on reduced order modeling or model order reduction is called a weak solution in continuum mechanics and applied mathematics. Such solution is then substituted in the system governing equations. Afterwards, the Ritz or Ritz-Galerkin method can be used to solve the resulting system ( Hesthaven et al., 2007 ).

Substituting an approximate (e.g., basis function-based) solution for the system deformation map. i.e., kinematics, in the system governing equations is called the Ritz approach to solve a system of complex PDEs. Although it is not referenced in most Soft Robotics research works, most of the proposed models for soft robots based on a basis function based approximation utilize the Ritz method. Later, the spatial domain of the resulted decoupled system can be numerically integrated while the time-varying coefficients are optimized to minimize the governing equation residual error.

Alternatively, the coefficients can be considered as the system’s EOM states for which the time series can be found from the numerical integration in time ( Sadati et al., 2018 ). The Galerkin method of weighted residuals, known as Ritz-Galerkin method, may provide a better approximation where the weighted residual of the system is minimized instead of the residual function itself ( Tunay, 2013 ). In simple words, both sides of the governing equations resulted from the Ritz method is multiplied by a vector, called weighting vector, which usually consists of the shape (basis) functions. Tunay et al. ( Tunay, 2013 ), Sadati et al. ( Sadati et al., 2018 ), and Goury et al. (Goury, Duriez) have investigated Ritz and Ritz-Galerkin solutions for continuum robots in the soft robotics community. Table 4 summarizes the main features of these solution strategies.

3.5 Controller design implications

Standard control architectures (e.g., Jacobian-based inverse dynamics ( Coevoet et al., 2017 ; Goury, Duriez; Thieffry et al., 2018a ; Thieffry et al., 2018b ; Katzschmann et al., 2019 ; Thieffry et al., 2019 ; Thieffry et al., 2020 ), feedback linearization ( Della Santina et al., 2020a ; Sadati et al., 2020a ; Della Santina and Rus, 2020 ), potential field planning ( Mbakop et al., 2021 ) for finite-state space systems, such as rigid body robots, have been integrated with Reduced Order models of soft manipulators (see Table 2 ). From the formulation perspective, the use of absolute states with simpler mapping between the control system states and the robot workspace (e.g., shape position and orientation) may simplify the controller design for tracking and planning scenarios in the robot task space. On the other hand, the use of relative states, with a simpler mapping between the system states and the actuation inputs, may simplify the configuration space controller design. This is useful for tasks involving vibration attenuation or under-actuated system control design. A detailed discussion on model-based control of soft robots is presented in (9). A detailed comparative study of different control strategies for Reduced Order Model of soft manipulators is needed to better highlight the performance of different combinations of the Reduced Order modeling and control technique.

3.6 Comparative studies and software packages

To reach a fair comparison between the different methods in the literature, it is important to minimize the effect of experimental hardware and procedures on the results between different studies. To this end, comparative studies of different methods based on experimental results with a unique setup can be helpful. We have conducted a series of such comparative studies based on experiments with a soft appendage with pneumatic braided actuators, known as STIFF-FLOP (STIFFness controllable Flexible and Learn-able Manipulator for surgical OPerations) in ( Sadati et al., 2017a ; Sadati et al., 2018 ; Sadati et al., 2019a ; Sadati et al., 2020a ). A more recent study by Rao et al. (2020) compared the accuracy of different modeling techniques (Constant Curvature, Pseudo Rigid Body, Piecewise Constant Curvature, and Variable Curvature) for a tendon-driven continuum robot. To help with choosing the right elements for modeling a soft manipulator, Table 5 provides a summary of findings based on these comparative studies.

There has been an increasing interest in the application and contribution of reduced order basis function-based methods handling singularity issues ( Godage et al., 2011a ; Godage et al., 2011b ; Godage et al., 2015a ; Godage et al., 2016 ), general loading cases ( Sadati et al., 2018 ; Sadati et al., 2019a ; Sadati et al., 2020a ; Rao et al., 2021 ; Rao et al., 2022 ; Sadati et al., 2022 ), kinematic constraints ( Orekhov and Simaan, 2008 ), force observation ( Aloi and Rucker, 2019 ; Aloi et al., 2022 ), controller design ( Singh et al., 2018 ; Della Santina et al., 2020a ; Della Santina, 2020 ; Della Santina and Rus, 2020 ), and path planning ( Mbakop et al., 2021 ) emphasizing the ever-increasing importance of Reduced Order Modeling methods in the Soft Robotics community.

Tailored software packages are developed mostly for FEM-based ( Naughton et al., 2009 ; Coevoet et al., 2017 ; Gazzola et al., 2018 ; Sadati et al., 2019b ; Grazioso et al., 2019 ; Hu et al., 2019 ; Mathew, Hmida, Renda; Huang et al., 2020 ; Li et al., 2020 ; Munawar et al., 2020 ; Bern and Rus, 2021 ; Graule et al., 2021 ) implementations targeting fast simulation of complex scenarios for realistic graphical representations and learning-based research. The exceptions are TMTDyn (based on the TMT method) ( Sadati et al., 2019b ; Sadati et al., 2020a ) and SoRoSim (based on a Recursive Computational Scheme) (Mathew, Hmida, Renda), which are Matlab packages for the theoretical derivation of a continuum robot Lagrangian dynamics suited for theoretical dynamical system analysis and nonlinear controller design. Although, similar functionalities have been or can be achieved by further developments around FEM-based packages such as SOFA ( Coevoet et al., 2017 ), PyElastica ( Naughton et al., 2009 ), and DiffPD ( Ma et al., 2021 ). Ease of use, integration with existing tools, and suitability to research objectives alongside excellent accuracy, robustness, and real-time computational performance are key to ensuring wide acceptability.

4 Conclusion and future directions

Reduced Order Modeling and Model Order Reduction techniques have been extensively investigated to obtain models of acceptable dimension for Soft Robotics control. In this paper, we reviewed and compared the techniques identified in the literature, summarizing our findings in Tables 2 , Table 3 , Table 4 , Table 5 . Although, Reduced Order Modeling and Model Order Reduction techniques are promising paths towards Soft Robotics control, they require further theoretical development, experimental validation, and comparative study.

The quest for finding a better shape fitting technique for soft manipulators is still ongoing with recent studies on Euler curves and Magnus expansion technique. Modeling complex soft manipulators with parallel structure, braided, growing, and concentric elements are the most recent trends in relevant modeling research. On the control research front, application of shape fitting techniques for shape control, path planning, and obstacle avoidance have been among the most recent studies.

Amongst the Reduced Order Modeling techniques, the Polynomial Shape method has been primarily investigated for modeling complex continuum systems, such as Concentric Tube and Eversion Growing Robots. On the other hand, the Polynomial Curvature method has been mostly studied for the control problem of simple continuum manipulators. Both the Polynomial Shape and Polynomial Curvature methods for soft manipulators are now supported by open-source software, dynamical implementation of general deformation and loading cases, experimental studies, and controller designs, but require further theoretical developments. Such developments need to focus on integrating nonlinear material mechanics laws, simulation stability analysis for modeling hyper-elastic material models, and further experimental investigations. On the other hand, Model Order Reduction implementations for soft robots, which relies on a preliminary FEM model, will benefit from more collaborative work, dissemination, and support for implementation by experimentalists.

Immediate next research steps are simulation numerical stability analysis for high bandwidth and unstable rapid motions of soft manipulators, high-bandwidth motion and vibration attenuation control, force and stiffness regulation, external force and impact observation, multi-arm coordination, robot planning for self-collision and obstacle avoidance, and path following based on techniques such as follow-the-leader. The Polynomial Shape and Curvature methods are yet to be explored for parallel structure and path planning problems respectively. Finally, the potential of run-time fitting techniques based on modal test results on the actual system, or a precomputed basis (normal nonlinear modes) are untapped.

In the long run, it is interesting to investigate the Reduced Order Modeling techniques for the emerging research directions in the Soft Robotics such as multi-physics simulations (e.g., soft swimming or flying robots), machine learning and data-driven techniques, and task automation. Finally, developing software toolkits and experimental comparative studies are essential to broaden the reach and impact of the Reduced order Modeling techniques and to help identify each method’s potential from an application-oriented and control perspective.

Acknowledgments

The authors would like to thank Ian D. Walker from Clemson University, United States, Cosimo Della Santina from Delft University, Netherland, and Gregory Chirikjian from National University of Singapore for helpful discussions and feedback on the manuscript.

Funding Statement

This research was supported by an ERC Starting Grant [714562], by core funding from the Wellcome/EPSRC Centre for Medical Engineering [WT203148/Z/16/Z]. The authors acknowledge support from the Department of Health and Social Care (DHSC) through the National Institute for Health and Care Research (NIHR) MedTech Co-operative award for Cardiovascular Diseases to Guy’s and St Thomas NHS Foundation Trust in partnership with King’s College London. For the purpose of Open Access, the Author has applied a CC BY public copyright license to any Author Accepted Manuscript version arising from this submission.

Author contributions

SS have authored the manuscript. SN has contributed in the discussions lead to effective categorization of the theoretical techniques presented in this review article. LC and CB supervised the efforts lead to the preparation of this draft and contributed to the discussions in improving the quality of the review article. All authors contributed to the article and approved the submitted version.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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• Published: 26 November 2023

Impact of industrial robots on environmental pollution: evidence from China

• Yanfang Liu 1  

Scientific Reports volume  13 , Article number:  20769 ( 2023 ) Cite this article

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• Environmental sciences
• Environmental social sciences

The application of industrial robots is considered a significant factor affecting environmental pollution. Selecting industrial wastewater discharge, industrial SO 2 emissions and industrial soot emissions as the evaluation indicators of environmental pollution, this paper uses the panel data model and mediation effect model to empirically examine the impact of industrial robots on environmental pollution and its mechanisms. The conclusions are as follows: (1) Industrial robots can significantly reduce environmental pollution. (2) Industrial robots can reduce environmental pollution by improving the level of green technology innovation and optimizing the structure of employment skills. (3) With the increase in emissions of industrial wastewater, industrial SO 2 , and industrial dust, the impacts generated by industrial robots are exhibiting trends of a “W” shape, gradual intensification, and progressive weakening. (4) Regarding regional heterogeneity, industrial robots in the eastern region have the greatest negative impact on environmental pollution, followed by the central region, and the western region has the least negative impact on environmental pollution. Regarding time heterogeneity, the emission reduction effect of industrial robots after 2013 is greater than that before 2013. Based on the above conclusions, this paper suggests that the Chinese government and enterprises should increase investment in the robot industry. Using industrial robots to drive innovation in green technology and optimize employment skill structures, reducing environmental pollution.

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Introduction

Since the reform and opening up, China’s rapid economic growth has created a world-renowned “economic growth miracle” 1 . With the rapid economic growth, China’s environmental pollution problem is becoming more and more serious 2 . According to the “ Global Environmental Performance Index Report ” released by Yale University in the United States in 2022, China’s environmental performance index scores 28.4 points, ranking 160th out of 180 participating countries. The aggravation of environmental pollution not only affects residents’ health 3 , but also affects the efficiency of economic operation 4 . According to calculation of the General Administration of Environmental Protection, the World Bank and the Chinese Academy of Sciences, China’s annual losses caused by environmental pollution account for about 10% of GDP. Exploring the factors that affect environmental pollution and seeking ways to reduce environmental pollution are conducive to the development of economy within the scope of environment.

Industrial robots are machines that can be automatically controlled, repeatedly programmed, and multi-purpose 5 . They replace the low-skilled labor force engaged in procedural work 6 , reducing the raw materials required for manual operation. Industrial robots improve the clean technology level and energy efficiency of coal combustion, reducing pollutant emissions in front-end production. Industrial robots also monitor the energy consumption and sewage discharge in the production process in real time. The excessive discharge behavior of enterprises in the production process is regulated, reducing the emission of pollutants in the end treatment. Based on the selection and coding of literature (Appendix A ), this paper uses the meta-analysis method to compare the impacts of multiple factors such as economics, population, technology, and policy on environmental pollution. As shown in Table 1 , compared to other factors, industrial robots demonstrate greater advantages in reducing environmental pollution. There is a lack of research on the relationship between industrial robots and environmental pollution in China. With the advent of artificial intelligence era, China’s industrial robot industry has developed rapidly. According to data released by the International Federation of Robotics (IFR), from 1999 to 2019, China’s industrial robot ownership and installation shows an increasing trend year by year (Fig.  1 ). In 2013 and 2016, China’s industrial robot installation (36,560) and ownership (349,470) exceeds Japan for the first time, becoming the world’s largest country in terms of installation and ownership of industrial robots. Whether the application of industrial robots in China contributes to the reduction of environmental pollution? What is the mechanism of the impact of China’s industrial robots on environmental pollution? Researching this issue is crucial for filling the gaps in existing research and providing a reference for other countries to achieve emission reduction driven by robots.

Industrial robot installations in the world’s top five industrial robot markets from 1999 to 2019.

Based on the above analysis, this paper innovatively incorporates industrial robots and environmental pollution into a unified framework. Based on the panel data of 30 provinces in China from 2006 to 2019, this paper uses the ordinary panel model and mediating effect model to empirically test the impact of industrial robots on China’s environmental pollution and its transmission channels. The panel quantile model is used to empirically analyze the heterogeneous impact of industrial robots on environmental pollution under different environmental pollution levels.

Literature review

A large number of scholars have begun to study the problem of environmental pollution. Its research content mainly includes two aspects: The measurement of environmental pollution and its influencing factors. Regarding the measurement, some scholars have used SO 2 emissions 7 , industrial soot emissions 8 and PM2.5 concentration 9 and other single indicators to measure the degree of environmental pollution. The single indicator cannot fully and scientifically reflect the degree of environmental pollution. To make up for this defect, some scholars have included industrial SO 2 emissions, industrial wastewater discharge and industrial soot emissions into the environmental pollution evaluation system, and used the entropy method to measure environmental pollution level 10 . This method ignores the different characteristics and temporal and spatial trends of different pollutants, which makes the analysis one-sided. Regarding the influencing factors, economic factors such as economic development level 11 , foreign direct investment 12 and income 13 , population factors such as population size 14 and urbanization level 15 , energy consumption 16 all have an impact on environmental pollution. Specifically, economic development and technological innovation can effectively reduce environmental pollution 17 . The expansion of population size can aggravate environmental pollution. Income inequality can reduce environmental pollution, but higher income inequality may aggravate environmental pollution 18 . There are “pollution heaven hypothesis” and “pollution halo hypothesis” between foreign direct investment and environmental pollution 19 . Technological factors also have a non-negligible impact on environmental pollution 20 .

With continuous deepening of research, scholars have begun to focus on the impact of automation technology, especially industrial robot technology, on the environment. Ghobakhloo et al. 21 theoretically analyzed the impact of industrial robots on energy sustainability, contending that the application of industrial robots could foster sustainable development of energy. Using data from multiple countries, a few scholars have empirically analyzed the effect of industrial robots on environmental pollution (Table 2 ). Luan et al. 22 used panel data from 73 countries between 1993 and 2019 to empirically analyze the impact of industrial robots on air pollution, finding that the use of industrial robots intensifies environmental pollution. Using panel data from 66 countries from 1993 to 2018, Wang et al. 23 analyzed the impact of industrial robots on carbon intensity and found that industrial robots can reduce carbon intensity. On the basis of analyzing the overall impact of industrial robots on environmental pollution, some scholars conducted in-depth exploration of its mechanism. Based on data from 72 countries between 1993 and 2019, Chen et al. 5 explored the impact of industrial robots on the ecological footprint, discovering that industrial robots can reduce the ecological footprint through time saving effect, green employment effect and energy upgrading effect. Using panel data from 35 countries between 1993 and 2017, Li et al. 24 empirically examined the carbon emission reduction effect of industrial robots, finding that industrial robots can effectively reduce carbon emissions by increasing green total factor productivity and reducing energy intensity. Although the above studies have successfully estimated the overall impact of industrial robots on environmental pollution and its mechanisms, they have not fully considered the role of technological progress, labor structure and other factors in the relationship between the two. These studies all chose data from multiple countries as research samples and lack research on the relationship between industrial robots and environmental pollution in China, an emerging country.

The above literature provides inspiration for this study, but there are still shortcomings in the following aspects: Firstly, there is a lack of research on the relationship between industrial robots and environmental pollution in emerging countries. There are significant differences between emerging and developed countries in terms of institutional background and the degree of environmental pollution. As a representative emerging country, research on the relationship between industrial robots and environmental pollution in China can provide reliable references for other emerging countries. Secondly, theoretically, the study of the impact of industrial robots on environmental pollution is still in its initial stage. There are few studies that deeply explore its impact mechanism, and there is a lack of analysis of the role of technological progress and labor structure in the relationship between the two.

The innovations of this paper are as follows: (1) In terms of sample selection, this paper selects panel data from 30 provinces in China from 2006 to 2019 as research samples to explore the relationship between industrial robots and environmental pollution in China, providing references for other emerging countries to improve environmental quality using industrial robots. (2) In terms of theory, this paper is not limited to revealing the superficial relationship between industrial robots and environmental pollution. it starts from a new perspective and provides an in-depth analysis of how industrial robots affect environmental pollution through employment skill structure and green technology innovation. This not only enriches research in the fields of industrial robots and the environment, but is also of great significance in guiding the direction of industrial policy and technology research and development.

Theoretical analysis and hypothesis

Industrial robots and environmental pollution.

As shown in Fig.  2 , the impact of industrial robots on environmental pollution is mainly reflected in two aspects: Front-end production and end treatment. In front-end production, industrial robots enable artificial substitution effects 25 . Manual operation is replaced by machine operation, reducing the raw materials needed for manual operation. Through the specific program setting of industrial robots, clean energy is applied to industrial production 26 . The use of traditional fuels such as coal and oil is reduced. In terms of end treatment, the traditional pollutant concentration tester only measures a single type of pollutant. Its data cannot be obtained in time. It is easy to cause pollution incidents. Industrial robots can measure a variety of pollutants, and have the function of remote unmanned operation and warning. It reflects the pollution situation in time, reducing the probability of pollution incidents. The use of robots can upgrade sewage treatment equipment and improve the accuracy of pollution treatment, reducing pollutant emissions. Based on the above analysis, this paper proposes hypothesis 1.

The impact of industrial robots on environmental pollution.

Hypothesis 1

The use of industrial robots can reduce environmental pollution.

Mediating effect of green technology innovation

Industrial robots can affect environmental pollution by promoting green technology innovation. The transmission path of “industrial robots-green technology innovation-environmental pollution” is formed. Industrial robots are the materialization of technological progress in the field of enterprise R&D. Its impact on green technology innovation is mainly manifested in the following two aspects: Firstly, industrial robots classify known knowledge, which helps enterprises to integrate internal and external knowledge 27 . The development of green technology innovation activities of enterprises is promoted. Secondly, enterprises can simulate existing green technologies through industrial robots. The shortcomings of green technology in each link are found. Based on this, enterprises can improve and perfect green technology in a targeted manner. Industrial robots can collect and organize data, which enables enterprises to predict production costs and raw material consumption. Excessive procurement by enterprises can occupy working capital. Inventory backlog leads to warehousing, logistics and other expenses, increasing storage costs 28 . Forecasting the consumption of raw materials allows enterprises to purchase precisely, preventing over-procurement and inventory backlog, thereby reducing the use of working capital and storage costs 29 . The production cost of enterprises is reduced. Enterprises have more funds for green technology research and development.

The continuous innovation of green technology is helpful to solve the problem of environmental pollution. Firstly, green technology innovation helps use resources better 30 , lowers dependence on old energy, and reduces environmental damage. Secondly, green technology innovation promotes the greening of enterprises in manufacturing, sales and after-sales 31 . The emission of pollutants in production process is reduced. Finally, green technology innovation improves the advantages of enterprises in market competition 32 . The production possibility curve expands outward, which encourages enterprises to carry out intensive production. Based on the above analysis, this paper proposes hypothesis 2.

Hypothesis 2

Industrial robots can reduce environmental pollution through green technology innovation.

Mediating effect of employment skill structure

Industrial robots can affect environmental pollution through employment skill structure. The transmission path of “industrial robots-employment skill structure-environmental pollution” is formed. Industrial robots have substitution effect and creation effect on the labor force, improving the employment skill structure. Regarding the substitution effect, enterprises use industrial robots to complete simple and repetitive tasks to improve production efficiency, which crowds out low-skilled labor 6 . Regarding the creation effect, industrial robots create a demand for new job roles that matches automation, such as robot engineers, data analysts, machine repairers, which increases the number of highly skilled labor 33 . The reduction of low-skilled labor and increase of high-skilled labor improve employment skill structure.

High-skilled labor is reflected in the level of education 34 . Its essence is to have a higher level of skills and environmental awareness, which is the key to reducing environmental pollution. Compared with low-skilled labor, high-skilled labor has stronger ability to acquire knowledge and understand skills, which improves the efficiency of cleaning equipment and promotes emission reduction. The interaction and communication between highly skilled labor is also crucial for emission reduction. The excessive wage gap between employees brings high communication costs, which hinders the exchange of knowledge and technology between different employees. The increase in the proportion of high-skilled labor can solve this problem and improve the production efficiency of enterprises 35 . The improvement of production efficiency enables more investment in emission reduction research, decreasing pollutant emissions. Based on the above analysis, this paper proposes hypothesis 3.

Hypothesis 3

Industrial robots can reduce environmental pollution by optimizing employment skills structure.

Model construction and variable selection

Model construction, panel data model.

The panel data model is a significant statistical method, first introduced by Mundlak 36 . Subsequently, numerous scholars have used this model to examine the baseline relationships between core explanatory variables and explained variables 37 . To test the impact of industrial robots on environmental pollution, this paper sets the following panel data model:

In formula ( 1 ), Y it is the explained variable, indicating the degree of environmental pollution in region i in year t . IR it is the core explanatory variable, indicating the installation density of industrial robots in region i in year t . X it is a series of control variables, including economic development level (GDP), urbanization level (URB), industrial structure (EC), government intervention (GOV) and environmental regulation (ER). \(\lambda i\) is the regional factor. \(\varphi t\) is the time factor. \(\varepsilon it\) is the disturbance term.

Mediating effect model

To test the transmission mechanism of industrial robots affecting environmental pollution, this paper sets the following mediating effect model:

In formula ( 2 ), M is the mediating variable, which mainly includes green technology innovation and employment skill structure. Formula ( 2 ) measures the impact of industrial robots on mediating variables. Formula ( 3 ) measures the impact of intermediary variables on environmental pollution. According to the principle of mediating effect 38 , the direct effect \(\theta 1\) , mediating effect \(\beta 1 \times \theta 2\) and total effect \(\alpha 1\) satisfy \(\alpha 1 = \theta 1 + \beta 1 \times \theta 2\) .

Panel quantile model

The panel quantile model was first proposed by Koenke and Bassett 39 . It is mainly used to analyze the impact of core explanatory variables on the explained variables under different quantiles 40 . To empirically test the heterogeneous impact of industrial robots on environmental pollution under different levels of environmental pollution, this paper sets the following panel quantile model:

In formula ( 4 ), \(\tau\) represents the quantile value. \(\gamma 1\) reflects the difference in the impact of industrial robots on environmental pollution at different quantiles. \(\gamma 2\) indicates the different effects of control variables at different quantiles.

Variable selection

Explained variable.

The explained variable is environmental pollution. Considering the timeliness and availability of data, this paper selects industrial wastewater discharge, industrial SO 2 emissions and industrial soot emissions as indicators of environmental pollution.

Explanatory variable

According to production theory, industrial robots can enhance production efficiency 41 . Efficient production implies reduced energy wastage, which in turn decreases the emission of pollutants. Industrial robots can upgrade pollution control equipment, heightening the precision in pollution treatment and reducing pollutant discharge. Referring to Acemoglu and Restrepo 25 , this paper selects the installation density of industrial robots as a measure. The specific formula is as follows:

In formula ( 5 ), Labor ji is the number of labor force in industry j in region i . IR jt is the stock of industrial robot use in industry j in the year t .

Mediating variable

Green technology innovation. Industrial robots can increase the demand for highly-skilled labor 42 , subsequently influencing green technology innovation. Compared to ordinary labor, highly-skilled labor possesses a richer knowledge base and technological learning capability, improving the level of green technology innovation. Green technology innovation can improve energy efficiency 43 , reducing pollution generated by energy consumption. The measurement methods of green technology innovation mainly include three kinds: The first method is to use simple technology invention patents as measurement indicators. Some of technical invention patents are not applied to the production process of enterprise, they cannot fully reflect the level of technological innovation. The second method is to use green product innovation and green process innovation as measurement indicators. The third method is to use the number of green patent applications or authorizations as a measure 44 . This paper selects the number of green patent applications as a measure of green technology innovation.

Employment skill structure. The use of industrial robots reduces the demand for labor performing simple repetitive tasks and increases the need for engineers, technicians, and other specialized skilled personnel, improving the employment skill structure 45 . Compared to ordinary workers, highly-skilled laborers typically have a stronger environmental awareness 46 . Such environmental consciousness may influence corporate decisions, prompting companies to adopt eco-friendly production methods, thus reducing environmental pollution. There are two main methods to measure the structure of employment skills: One is to use the proportion of employees with college degree or above in the total number of employees as a measure. The other is to use the proportion of researchers as a measure. The educational level can better reflect the skill differences of workers. This paper uses the first method to measure the employment skill structure.

Control variable

Economic development level. According to the EKC hypothesis 47 , in the initial stage of economic development, economic development mainly depends on input of production factors, which aggravates environmental pollution. With the continuous development of economy, people begin to put forward higher requirements for environmental quality. The government also begins to adopt more stringent policies to control environmental pollution, which can reduce the level of environmental pollution. According to Liu and Lin 48 , This paper uses per capita GDP to measure economic development level.

Urbanization level. The improvement of urbanization level has both positive and negative effects on pollution. Urbanization can improve the agglomeration effect of cities. The improvement of agglomeration effect can not only promote the sharing of public resources such as infrastructure, health care, but also facilitate the centralized treatment of pollution. The efficiency of environmental governance is improved 49 . The acceleration of urbanization can increase the demand for housing, home appliances and private cars, which increases pollutant emissions 50 . This paper uses the proportion of urban population to total population to measure the level of urbanization.

Industrial structure. Industrial structure is one of the key factors that determine the quality of a country’s environmental conditions 51 . The increase in the proportion of capital and technology-intensive industries can effectively improve resource utilization efficiency and improve resource waste 52 . This paper selects the ratio of the added value of the tertiary industry to the secondary industry to measure industrial structure.

Government intervention. Government intervention mainly affects environmental pollution from the following two aspects: Firstly, the government can give high-tech, energy-saving and consumption-reducing enterprises relevant preferential policies, which promotes the development of emission reduction technologies for these enterprises 53 . Secondly, the government strengthens environmental regulation by increasing investment in environmental law enforcement funds, thus forcing enterprises to save energy and reduce emissions 54 . This paper selects the proportion of government expenditure in GDP to measure government intervention.

Environmental regulation. The investment in environmental pollution control is conducive to the development of clean and environmental protection technology, optimizing the process flow and improving the green production efficiency of enterprises 55 . Pollutant emissions are reduced. This paper selects the proportion of investment in pollution control to GDP to measure environmental regulation.

Data sources and descriptive statistics

This paper selects the panel data of 30 provinces in China from 2006 to 2019 as the research sample. Among them, the installation data of industrial robots are derived from International Federation of Robotics (IFR). The data of labor force and employees with college degree or above are from China Labor Statistics Yearbook . Other data are from the China Statistical Yearbook . The descriptive statistics of variables are shown in Table 3 . Considering the breadth of application and the reliability of analysis capabilities, this paper uses Stata 16 for regression analysis.

Results analysis

Spatial and temporal characteristics of environmental pollution and industrial robots in china, environmental pollution.

Figure  3 a shows the overall trend of average industrial wastewater discharge in China from 2006 to 2019. From 2006 to 2019, the discharge of industrial wastewater shows a fluctuating downward trend, mainly due to the improvement of wastewater treatment facilities and the improvement of treatment capacity. Figure  3 b shows the changing trend of average industrial wastewater discharge in 30 provinces of China from 2006 to 2019. Industrial wastewater discharge in most provinces has declined. There are also some provinces such as Fujian, Guizhou and Qinghai, which have increased industrial wastewater discharge. Their emission reduction task is very arduous.

Industrial wastewater discharge from 2006 to 2019.

Figure  4 a shows the overall trend of average industrial SO 2 emissions in China from 2006 to 2019. From 2006 to 2019, industrial SO 2 emissions shows a fluctuating downward trend, indicating that air pollution control and supervision are effective. Figure  4 b shows the trend of average industrial SO 2 emissions in 30 provinces of China from 2006 to 2019. Similar to industrial wastewater, industrial SO 2 emissions decrease in most provinces.

Industrial SO 2 emissions from 2006 to 2019.

Figure  5 a shows the overall trend of average industrial soot emissions in China from 2006 to 2019. Different from industrial wastewater and industrial SO 2 , the emission of industrial soot is increasing year by year. From the perspective of governance investment structure, compared with industrial wastewater and industrial SO 2 , the investment proportion of industrial soot is low. From the perspective of source, industrial soot mainly comes from urban operation, industrial manufacturing and so on. The acceleration of urbanization and the expansion of manufacturing scale have led to an increase in industrial soot emissions. Figure  5 b shows the trend of industrial soot emissions in 30 provinces in China from 2006 to 2019. The industrial soot emissions in most provinces have increased.

Industrial soot emissions from 2006 to 2019.

Figure  6 shows the spatial distribution characteristics of industrial wastewater, industrial SO 2 and industrial soot emissions. The three types of pollutant emissions in the central region are the largest, followed by the eastern region, and the three types of pollutant emissions in the western region are the smallest. Due to resource conditions and geographical location, the central region is mainly dominated by heavy industry. The extensive development model of high input and consumption makes its pollutant emissions higher than the eastern and western regions. The eastern region is mainly capital-intensive and technology-intensive industries, which makes its pollutant emissions lower than the central region. Although the leading industry in the western region is heavy industry, its factory production and transportation scale are not large, which produces less pollutants.

Spatial distribution characteristics of industrial wastewater, industrial SO 2 and industrial soot.

Industrial robots

Figure  7 a shows the overall trend of installation density of industrial robots in China from 2006 to 2019. From 2006 to 2019, the installation density of industrial robots in China shows an increasing trend year by year. The increase of labor cost and the decrease of industrial robot cost make enterprises use more industrial robots, which has a substitution effect on labor force. The installation density of industrial robots is increased. Figure  7 b shows the trend of installation density of industrial robots in 30 provinces of China from 2006 to 2019. The installation density of industrial robots in most provinces has increased. Among them, the installation density of industrial robots in Guangdong Province has the largest growth rate. The installation density of industrial robots in Heilongjiang Province has the smallest growth rate.

Installation density of industrial robots from 2006 to 2019.

Figure  8 shows the spatial distribution characteristics of installation density of industrial robots. The installation density of industrial robots in the eastern region is the largest, followed by the central region, and the installation density of industrial robots in the western region is the smallest. The eastern region is economically developed and attracts lots of talents to gather here, which provides talent support for the development of industrial robots. Advanced technology also leads to the rapid development of industrial robots in the eastern region. The economy of western region is backward, which inhibits the development of industrial robots.

Spatial distribution characteristics of industrial robots.

Benchmark regression results

Table 4 reports the estimation results of the ordinary panel model. Among them, the F test and LM test show that the mixed OLS model should not be used. The Hausman test shows that the fixed effect model should be selected in the fixed effect model and random effect model. This paper selects the estimation results of the fixed effect model to explain.

Regarding the core explanatory variable, industrial robots have a significant negative impact on the emissions of industrial wastewater, industrial SO 2 and industrial soot. Specifically, industrial robots have the greatest negative impact on industrial soot emissions, with a coefficient of -0.277 and passing the 1% significance level. The negative impact of industrial robots on industrial wastewater discharge is second, with an estimated coefficient of -0.242, which also passes the 1% significance level. The negative impact of industrial robots on industrial SO 2 emissions is the smallest, with an estimated coefficient of -0.0875 and passing the 10% significant level. Compared with industrial wastewater and SO 2 , industrial robots have some unique advantages in reducing industrial soot emissions. Firstly, in terms of emission sources, industrial soot emissions mainly come from physical processes such as cutting. These processes can be significantly improved through precise control of industrial robots. Industrial SO 2 comes from the combustion process. Industrial wastewater originates from various industrial processes. It is difficult for industrial robots to directly control these processes. Secondly, in terms of source control and terminal treatment, industrial robots can reduce excessive processing and waste of raw materials, thereby controlling industrial soot emissions at the source. For industrial SO 2 and industrial wastewater, industrial robots mainly play a role in terminal treatment. Since the terminal treatment of industrial SO 2 and industrial wastewater often involves complex chemical treatment processes, it is difficult for industrial robot technology to fully participate in these processes. This makes the impact of industrial robots in the field of industrial SO 2 and industrial wastewater more limited than that in the field of industrial soot.

Regarding the control variables, the level of economic development has a significant inhibitory effect on industrial SO 2 emissions. The higher the level of economic development, the stronger the residents’ awareness of environmental protection, which constrains the pollution behavior of enterprises. The government also adopts strict policies to control pollutant emissions. The impact of urbanization level on the discharge of industrial wastewater, industrial SO 2 and industrial soot is significantly negative. The improvement of urbanization level can improve the efficiency of resource sharing and the centralized treatment of pollutants, reducing environmental pollution. The industrial structure significantly reduces industrial SO 2 and industrial soot emissions. The upgrading of industrial structure not only reduces the demand for energy, but also improves the efficiency of resource utilization. The degree of government intervention only significantly reduces the discharge of industrial wastewater. The possible reason is that to promote economic development, the government invests more money in high-yield areas, which crowds out investment in the environmental field. Similar to the degree of government intervention, environmental regulation has a negative impact on industrial wastewater discharge. The government’s environmental governance investment has not given some support to the enterprise’s clean technology research, which makes the pollution control investment not produce good emission reduction effect.

Mediation effect regression results

Green technology innovation.

Table 5 reports the results of intermediary effect model when green technology innovation is used as an intermediary variable. Industrial robots can have a positive impact on green technology innovation. For every 1% increase in the installation density of industrial robots, the level of green technology innovation increases by 0.722%. After adding the green technology innovation, the estimated coefficient of industrial robots has decreased, which shows that the intermediary variable is effective.

In the impact of industrial robots on industrial wastewater discharge, the mediating effect of green technology innovation accounts for 8.17% of the total effect. In the impact of industrial robots on industrial SO 2 emissions, the mediating effect of green technology innovation accounts for 11.8% of the total effect. In the impact of industrial robots on industrial soot emissions, the mediating effect of green technology innovation accounts for 3.72% of the total effect.

Employment skill structure

Table 6 reports the results of intermediary effect model when the employment skill structure is used as an intermediary variable. Industrial robots have a positive impact on the employment skill structure. For every 1% increase in the installation density of industrial robots, the employment skill structure is improved by 0.0837%. Similar to green technology innovation, the intermediary variable of employment skill structure is also effective.

In the impact of industrial robots on industrial wastewater discharge, the mediating effect of employment skill structure accounts for 6.67% of the total effect. In the impact of industrial robots on industrial SO 2 emissions, the mediating effect of employment skill structure accounts for 20.66% of the total effect. In the impact of industrial robots on industrial soot emissions, the mediating effect of employment skill structure accounts for 15.53% of the total effect.

Robustness test and endogeneity problem

Robustness test.

To ensure the robustness of the regression results, this paper tests the robustness by replacing core explanatory variables, shrinking tail and replacing sample. Regarding the replacement of core explanatory variables, in the benchmark regression, the installation density of industrial robots is measured by the stock of industrial robots. Replacing the industrial robot stock with the industrial robot installation quantity, this paper re-measures the industrial robot installation density. Regarding the tail reduction processing, this paper reduces the extreme outliers of all variables in the upper and lower 1% to eliminate the influence of extreme outliers. Regarding the replacement of samples, this paper removes the four municipalities from the sample. The estimation results are shown in Table 7 . Industrial robots still have a significant negative impact on environmental pollution, which confirms the robustness of benchmark regression results.

Endogeneity problem

Logically speaking, although the use of industrial robots can reduce environmental pollution, there may be reverse causality. Enterprises may increase the use of industrial robots to meet emission reduction standards, which increases the use of industrial robots in a region. Due to the existence of reverse causality, there is an endogenous problem that cannot be ignored between industrial robots and environmental pollution.

To solve the impact of endogenous problems on the estimation results, this paper uses the instrumental variable method to estimate. According to the selection criteria of instrumental variables, this paper selects the installation density of industrial robots in the United States as the instrumental variable. The trend of the installation density of industrial robots in the United States during the sample period is similar to that of China, which is consistent with the correlation characteristics of instrumental variables. The application of industrial robots in the United States is rarely affected by China’s economic and social factors, and cannot affect China’s environmental pollution, which is in line with the exogenous characteristics of instrumental variables.

Table 8 reports the estimation results of instrumental variable method. Among them, the column (1) is listed as the first stage regression result. The estimated coefficient of instrumental variable is significantly positive, which is consistent with the correlation. Column (2), column (3) and column (4) of Table 8 are the second stage regression results of industrial wastewater, industrial SO 2 and industrial soot emissions as explanatory variables. The estimated coefficients of industrial robots are significantly negative, which again verifies the hypothesis that industrial robots can reduce environmental pollution. Compared with Table 4 , the absolute value of estimated coefficient of industrial robots is reduced, which indicates that the endogenous problems caused by industrial robots overestimate the emission reduction effect of industrial robots. The test results prove the validity of the instrumental variables.

Panel quantile regression results

Traditional panel data models might obscure the differential impacts of industrial robots at specific pollution levels. To address this issue, this paper uses a panel quantile regression model to empirically analyze the effects of industrial robots across different environmental pollution levels.

Table 9 shows that industrial robots have a negative impact on industrial wastewater discharge. With the increase of the quantile of industrial wastewater discharge, the regression coefficient of industrial robots shows a W-shaped change. Specifically, when the industrial wastewater discharge is in the 0.1 quantile, the regression coefficient of industrial robot is − 0.229, and it passes the 1% significant level. When the industrial wastewater discharge is in the 0.25 quantile, the impact of industrial robots on industrial wastewater discharge is gradually enhanced. Its regression coefficient decreases from − 0.229 to − 0.256. When the industrial wastewater discharge is in the 0.5 quantile, the regression coefficient of industrial robot increases from − 0.256 to − 0.152. When the industrial wastewater discharge is at the 0.75 quantile, the regression coefficient of industrial robot decreases from − 0.152 to − 0.211. When the industrial wastewater discharge is in the 0.9 quantile, the regression coefficient of industrial robot increases from − 0.211 to − 0.188. For every 1% increase in the installation density of industrial robots, the discharge of industrial wastewater is reduced by 0.188%.

When industrial wastewater discharge is at a low percentile, the use of industrial robots can replace traditional production methods, reducing energy waste and wastewater discharge. As industrial wastewater discharge increases, the production process becomes more complex. Industrial robots may be involved in high-pollution, high-emission productions, diminishing the robots’ emission-reducing effects. When industrial wastewater discharge reaches high levels, pressured enterprises seek environmentally friendly production methods and use eco-friendly industrial robots to reduce wastewater discharge. As wastewater discharge continues to rise, enterprises tend to prioritize production efficiency over emission control, weakening the negative impact of industrial robots on wastewater discharge. When wastewater discharge is at a high percentile, enterprises should balance production efficiency and environmental protection needs, by introducing eco-friendly industrial robots to reduce wastewater discharge.

Table 10 shows that with the increase of industrial SO 2 emission quantile level, the negative impact of industrial robots on industrial SO 2 emissions gradually increases. Specifically, when industrial SO 2 emissions are below 0.5 quantile, the impact of industrial robots on industrial SO 2 emissions is not significant. When the industrial SO 2 emissions are above 0.5 quantile, the negative impact of industrial robots on industrial SO 2 emissions gradually appears.

When industrial SO 2 emissions are at a low percentile, the application of industrial robots primarily aims to enhance production efficiency, not to reduce SO 2 emissions. Enterprises should invest in the development of eco-friendly industrial robots, ensuring they are readily available for deployment when a reduction in industrial SO 2 emissions is necessary. As industrial SO 2 emissions continue to rise, both the government and the public pay increasing attention to the issue of SO 2 emissions. To meet stringent environmental standards, enterprises begin to use industrial robots to optimize the production process, reduce reliance on sulfur fuels, and consequently decrease SO 2 emissions. Enterprises should regularly evaluate the emission reduction effectiveness of industrial robots, using the assessment data to upgrade and modify the robots’ emission reduction technologies.

Table 11 shows that with the increase of industrial soot emissions quantile level, the negative impact of industrial robots on industrial soot emissions gradually weakens. Specifically, when industrial soot emissions are below 0.75 quantile, industrial robots have a significant negative impact on industrial soot emissions. This negative effect decreases with the increase of industrial soot emissions. When the industrial soot emissions are above 0.75 quantile, the negative impact of industrial robots on industrial soot emissions gradually disappears.

When industrial soot emissions are at a low percentile, they come from a few sources easily managed by industrial robots. As industrial soot emissions increase, the sources become more diverse and complex, making it harder for industrial robots to control. Even with growing environmental awareness, it may take time to effectively use robots in high-emission production processes and control industrial soot emissions. Enterprises should focus on researching how to better integrate industrial robot technology with production processes that have high soot emission levels. The government should provide financial and technical support to enterprises, assisting them in using industrial robots more effectively for emission reduction.

Figure  9 intuitively reflects the trend of the regression coefficient of industrial robots with the changes of industrial wastewater, industrial SO 2 and industrial soot emissions. Figure  9 a shows that with the increase of industrial wastewater discharge, the regression coefficient of industrial robots shows a W-shaped trend. Figure  9 b shows that with the increase of industrial SO 2 emissions, the regression coefficient of industrial robots gradually decreases. The negative impact of industrial robots on industrial SO 2 emissions is gradually increasing. Figure  9 c shows that with the increase of industrial soot emissions, the regression coefficient of industrial robots shows a gradual increasing trend. The negative impact of industrial robots on industrial soot emissions has gradually weakened. Figure  9 a, b and c confirm the estimation results of Tables 9 , 10 and 11 .

Change of quantile regression coefficient.

Heterogeneity analysis

Regional heterogeneity.

This paper divides China into three regions: Eastern, central and western regions according to geographical location. The estimated results are shown in Table 12 . The industrial robots in eastern region have the greatest negative impact on three pollutants, followed by central region, and the industrial robots in western region have the least negative impact on three pollutants. The use of industrial robots in eastern region far exceeds that in central and western regions. The eastern region is far more than central and western regions in terms of human capital, technological innovation and financial support. Compared with central and western regions, the artificial substitution effect, upgrading of sewage treatment equipment and improvement of energy utilization efficiency brought by industrial robots in eastern region are more obvious.

Time heterogeneity

The development of industrial robots is closely related to policy support 56 . In 2013, the Ministry of Industry and Information Technology issued the “ Guiding Opinions on Promoting the Development of Industrial Robot Industry ”. This document proposes: By 2020, 3 to 5 internationally competitive leading enterprises and 8 to 10 supporting industrial clusters are cultivated. In terms of high-end robots, domestic robots account for about 45% of the market share, which provides policy support for the development of industrial robots. Based on this, this paper divides the total sample into two periods: 2006–2012 and 2013–2019, and analyzes the heterogeneous impact of industrial robots on environmental pollution in different periods. The estimation results are shown in Table 13 . Compared with 2006–2012, the emission reduction effect of industrial robots during 2013–2019 is greater.

The use of industrial robots can effectively reduce environmental pollution, which is consistent with hypothesis 1. This is contrary to the findings of Luan et al. 22 , who believed that the use of industrial robots would exacerbate air pollution. The inconsistency in research conclusions may be due to differences in research focus, sample size, and maturity of industrial robot technology. In terms of research focus, this paper mainly focuses on the role of industrial robots in reducing pollutant emissions during industrial production processes. Their research focuses more on the energy consumption caused by the production and use of industrial robots, which could aggravate environmental pollution. In terms of sample size, the sample size of this paper is 30 provinces in China from 2006 to 2019. These regions share consistency in economic development, industrial policies and environmental regulations. Their sample size is 74 countries from 1993 to 2019. These countries cover different geographical, economic and industrial development stages, affecting the combined effect of robots on environmental pollution. In terms of the maturity of industrial robots, the maturity of industrial robot technology has undergone tremendous changes from 1993 to 2019. In the early stages, industrial robot technology was immature, which might cause environmental pollution. In recent years, industrial robot technology has gradually matured, and its operating characteristics have become environmentally friendly. Their impact on environmental pollution has gradually improved. This paper mainly conducts research on the mature stage of industrial robot technology. Their research covers the transition period from immature to mature industrial robot technology. The primary reason that the use of industrial robots can reduce environmental pollution is: The use of industrial robots has a substitution effect on labor force, which reduces the raw materials needed for manual operation. For example, in the industrial spraying of manufacturing industry, the spraying robot can improve the spraying quality and material utilization rate, thereby reducing the waste of raw materials by manual operation. Zhang et al. 57 argued that energy consumption has been the primary source of environmental pollution. Coal is the main energy in China, and the proportion of clean energy is low 58 . In 2022, clean energy such as natural gas, hydropower, wind power and solar power in China accounts for only 25.9% of the total energy consumption, which can cause serious environmental pollution problems. Industrial robots can promote the use of clean energy in industrial production and the upgrading of energy structure 24 . The reduction of raw materials and the upgrading of energy structure can control pollutant emissions in front-end production. On September 1, 2021, the World Economic Forum (WEF) released the report “ Using Artificial Intelligence to Accelerate Energy Transformation ”. The report points out that industrial robots can upgrade pollution monitoring equipment and sewage equipment, which reduces pollutant emissions in end-of-pipe treatment. Ye et al. 59 also share the same viewpoint.

The use of industrial robots can reduce environmental pollution through green technology innovation, which is consistent with hypothesis 2. Industrial robots promote the integration of knowledge, which helps enterprises to carry out green technology innovation activities. Meanwhile, Jung et al. 60 suggested that industrial robots can lower production costs for companies, allowing them to invest in green technology research. The level of green technology innovation is improved. Green technology innovation reduces environmental pollution through the following three aspects: Firstly, the improvement of energy utilization efficiency. China’s utilization efficiency of traditional energy sources such as coal is not high. The report of “ 2013-Global Energy Industry Efficiency Research ” points out that China’s energy utilization rate is only ranked 74th in the world in 2013. Low energy efficiency brings serious environmental pollution problems 61 . Du et al. 62 found that the innovation of green technologies, such as clean coal, can enhance energy efficiency and decrease environmental pollution. Secondly, the production of green products. Green technology innovation accelerates the green and recyclable process of production, thereby reducing the pollutants generated in production process. Thirdly, the improvement of enterprise competitive advantage. Green technology innovation can enable enterprises to gain greater competitive advantage in green development 63 . The supply of environmentally friendly products increases, which not only meets the green consumption needs of consumers, but also reduces the emission of pollutants.

Industrial robots can reduce environmental pollution by optimizing the structure of employment skills, which is consistent with hypothesis 3. Autor et al. 64 contended that industrial robots would replace conventional manual labor positions, reducing the demand for low-skilled labor. Industrial robots represent the development of numerical intelligence. With the continuous development of digital intelligence, the demand for high-skilled labor in enterprises has increased. Koch et al. 65 demonstrated that the use of industrial robots in Spanish manufacturing firms leads to an increase in the number of skilled workers. In February 2020, the Ministry of Human Resources and Social Security, the State Administration of Market Supervision and the National Bureau of Statistics jointly issues 16 new professions such as intelligent manufacturing engineering and technical personnel, industrial Internet engineering and technical personnel, and virtual reality engineering and technical personnel to the society. These new occupations increase the demand for highly skilled labor. The reduction of low-skilled labor and increase of high-skilled labor optimize the structure of employment skills. The optimization of employment skill structure narrows the wage gap between employees, reducing the communication cost of employees. Employees learn and exchange technology with each other, which not only improves the absorption capacity of clean technology. It also improves the production efficiency of enterprises and increases corporate profits, so that enterprises can use more funds for clean technology research and development, thereby reducing environmental pollution.

Conclusions and policy recommendations

Based on the panel data of 30 provinces in China from 2006 to 2019, this paper uses the panel data model and mediating effect model to empirically test the impact of industrial robots on environmental pollution and its transmission mechanism. This paper uses panel quantile model, regional samples and time samples to further analyze the heterogeneous impact of industrial robots on environmental pollution. The conclusions are as follows: (1) Industrial robots can significantly reduce environmental pollution. For every 1% increase in industrial robots, the emissions of industrial wastewater, industrial SO 2 , and industrial dust and smoke decrease by − 0.242%, − 0.0875%, and − 0.277%. This finding is contrary to that of Luan et al. 22 , who argued that the use of industrial robots exacerbates air pollution. The results of this paper provide a contrasting perspective, highlighting the potential value of industrial robots in mitigating environmental pollution. (2) Industrial robots can reduce environmental pollution by improving green technology innovation level and optimizing employment skills structure. In the impact of industrial robots on industrial wastewater discharge, the mediating effect of green technology innovation accounts for 8.17% of total effect. The mediating effect of employment skill structure accounts for 6.67% of total effect. In the impact of industrial robots on industrial SO 2 emissions, the mediating effect of green technology innovation accounts for 11.8% of total effect. The mediating effect of employment skill structure accounts for 20.66% of total effect. In the impact of industrial robots on industrial soot emissions, the mediating effect of green technology innovation accounts for 3.72% of total effect. The mediating effect of employment skill structure accounts for 15.53% of total effect. While Obobisa et al. 66 and Zhang et al. 67 highlighted the role of green technological innovation in addressing environmental pollution. Chiacchio et al. 68 and Dekle 69 focused on the effects of industrial robots on employment. The mediating impact of technology and employment in the context of robots affecting pollution hasn’t been addressed. Our research provides the first in-depth exploration of this crucial intersection. (3) Under different environmental pollution levels, the impact of industrial robots on environmental pollution is different. Among them, with the increase of industrial wastewater discharge, the impact of industrial robots on industrial wastewater discharge shows a “W-shaped” change. With the increase of industrial SO 2 emissions, the negative impact of industrial robots on industrial SO 2 emissions is gradually increasing. On the contrary, with the increase of industrial soot emissions, the negative impact of industrial robots on industrial soot emissions gradually weakens. (4) Industrial robots in different regions and different periods have heterogeneous effects on environmental pollution. Regarding regional heterogeneity, industrial robots in eastern region have the greatest negative impact on environmental pollution, followed by central region, and western region has the least negative impact on environmental pollution. Regarding time heterogeneity, the negative impact of industrial robots on environmental pollution in 2013–2019 is greater than that in 2006–2012. Chen et al. 5 and Li et al. 24 both examined the overarching impact of industrial robots on environmental pollution. They did not consider the varying effects of robots on pollution across different regions and time periods. Breaking away from the limitations of previous holistic approaches, our study offers scholars a deeper understanding of the diverse environmental effects of industrial robots.

According to the above research conclusions, this paper believes that the government and enterprises can promote emission reduction through industrial robots from the following aspects.

Increase the scale of investment in robot industry and promote the development of robot industry. China’s industrial robot ownership ranks first in the world. Its industrial robot installation density is lower than that of developed countries such as the United States, Japan and South Korea. The Chinese government should give some financial support to robot industry and promote the development of robot industry, so as to effectively reduce environmental pollution. The R&D investment of industrial robots should be increased so that they can play a full role in reducing raw material consumption, improving energy efficiency and sewage treatment capacity.

Give full play to the role of industrial robots in promoting green technology innovation. Industrial robots can reduce environmental pollution through green technology innovation. The role of industrial robots in innovation should be highly valued. The advantages of knowledge integration and data processing of industrial robots should be fully utilized. Meanwhile, the government should support high-polluting enterprises that do not have industrial robots from the aspects of capital, talents and technology, so as to open up the channels for these enterprises to develop and improve clean technology by using industrial robots.

Give full play to the role of industrial robots in optimizing employment skills structure. The use of industrial robots can create jobs with higher skill requirements and increase the demand for highly skilled talents. China is relatively short of talents in the field of emerging technologies. The education department should actively build disciplines related to industrial robots to provide talent support for high-skilled positions. Enterprises can also improve the skill level of the existing labor force through on-the-job training and job competition.

Data availability

The datasets used or analyzed during the current study are available from Yanfang Liu on reasonable request.

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Liu, Y. Impact of industrial robots on environmental pollution: evidence from China. Sci Rep 13 , 20769 (2023). https://doi.org/10.1038/s41598-023-47380-6

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ORIGINAL RESEARCH article

This article is part of the research topic.

Low-Carbon Economy and Sustainable Development: Driving Force, Synergistic Mechanism, and Implementation Path

resources on carbon emissions reduction in China Provisionally Accepted

• 1 University of South China, China

The final, formatted version of the article will be published soon.

The Accountability Audit of Natural Resources (AANR) is a major institutional arrangement for advancing the construction of an ecological civilization in China. Based on the panel data of 271 cities in China from 2005 to 2017, this paper investigates the relationship between the AANR and carbon dioxide (CO2) emissions using a multiperiod difference-in-differences (DID) model. The results show that AANR significantly increases the CO2 emissions reduction rate by 0.009 units at the 5% significance level. The results still hold after a series of robustness tests. Given all else being equal, this significant effect is 0.001. Further analyses show that AANR improves pilot cities' CO2 emissions reduction rate mainly by enhancing their green innovation capability. The mediating effect of cities' green technology innovation capability plays a role of 96.00%, while the AANR's direct effect only accounts for 4.00%. The AANR has significantly positive effects of 0.017 and 0.029 for western cities and cities with high fiscal pressure at the 5% and 1% significance levels, respectively. Therefore, strengthening AANR implementation through enhancing the mediating efficiency of cities' green technology innovations, and implementing dynamically differentiated AANR policies in Chinese meso-cities will contribute to the achievement of China's carbon peaking and carbon neutrality targets.

Keywords: AANR, CO2 emissions reduction rate, government concern for the environment, Green technology innovation capability, multiperiod DID model

Received: 21 Nov 2023; Accepted: 18 Apr 2024.

Copyright: © 2024 Tang, Shu and Li. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Mx. Xuefeng Li, University of South China, Hengyang, 421001, Hunan Province, China

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