Review of Industrial Robot Stiffness Identification and Modelling

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(1) Static stiffness modelling methods are classified and summarized. Each method is studied with regards to algorithms, implementation, and limitations; (2) A single degree of freedom and multiple degrees of freedom are proposed to describe the modelling methods in the joint space. (3) Regarding the relationship between external force and deformation, linear and nonlinear modelling methods are discussed.

Abstract

Due to their high flexibility, large workspace, and high repeatability, industrial robots are widely used in roughing and semifinishing fields. However, their low machining accuracy and low stability limit further development of industrial robots in the machining field, with low stiffness being the most significant factor. The stiffness of industrial robots is affected by the joint deformation, transmission mechanism, friction, environment, and coupling of these factors. Moreover, the stiffness of a robot has a nonlinear distribution throughout the workspace, and external forces during processing cause irregular deviations of the robot, thereby affecting the machining accuracy and surface quality of the workpiece. Many scholars have researched identifying the stiffness of industrial robots and have proposed methods for improving the performance of industrial robots, mainly by optimizing the body structure of the robot and compensating for deformation errors with stiffness models. This paper reviews recent research on the stiffness modelling of industrial robots, which can be broadly classified as finite element analysis (FEA), matrix structure analysis (MSA), and virtual joint modelling (VJM) methods. Each method is studied from three aspects: algorithms, implementation, and limitations. In addition, common measurement techniques have been introduced for measuring deformation. Further research directions are also discussed.

1. Introduction

Since precision has improved and manufacturing costs have decreased, robots have been widely adopted in different industrial fields. Compared to computer numerical control (CNC) machine tools, industrial robots have a low cost, high flexibility, and large workspace, and they have gradually become popular in the manufacturing industry. An industrial robot can replace traditional CNC machine tools to complete part of a machining task [1,2]. At present, industrial robots can perform primary machining processes, including milling, drilling, grinding, cutting, deburring, polishing, and friction stir welding [3,4]. Industrial robots are used in industries such as aerospace, automobile manufacturing, casting, medical treatment, plastics, and wood processing [5,6,7,8]. Especially for machining large-scale components, e.g., drilling holes on the inboard flaps [9] and milling operations for windmills [7], industrial robots are more efficient than machine tools. Machining is an important process from raw materials to finished products [10] that is mainly realized using machine tools with high precision and stability. Although industrial robots have advantages, their low stiffness characteristics can cause unexpected deformations and vibration/chatter during the machining process [11,12]. Two types of industrial robots, parallel robots and serial robots, are being investigated for machining. A parallel robot is a type of closed-loop structure (Figure 1a), which is composed of a mobile platform connected to a fixed base by a set of identical or different parallel kinematic chains (legs) [13]. A serial robot is a type of open-loop structure, as shown in Figure 1b. It is constituted by a mechanical system consisting of robust links (arms) coupled with either rotational or translating joints. Different tools can be fixed on the flange for machining tasks. Due to the closed-loop structure, the parallel robot has the characteristics of high stiffness and position accuracy compared to the serial robot. However, this structure limits the robot’s operational workspace and flexibility. As a result, more research has been conducted to improve the stiffness of the serial robot. In the machining process, a cutting force of 500 N introduces a position error of 1 mm to a serial robot, while the error introduced to a CNC machine tool is less than 0.01 mm [4,14,15]. The experimental results indicate that the static stiffness of industrial serial robots is in the range of ${10}^5{\text{~}10}^6\mathrm{N}/\mathrm{m}$ [16,17,18]. In addition, the stiffness changes as the robot’s posture configuration changes, resulting in a nonlinear distribution throughout the workspace, which leads to inaccurate machining [19]. Therefore, most industrial robots operate in fields with low precision requirements and no external contact forces. The number of industrial robots in high-value-added, high-precision industries is far lower than the number of CNC machine tools.

The stiffness of industrial robots depends on a variety of factors. The primary sources are elastic deformation caused by the geometric and material properties of various parts, including the joints, connecting links, bases, actuators, and other transmission components of robots. Furthermore, the active stiffness provided by the position control system of robots requires consideration [20,21,22,23]. Joint deformation results in most errors. Under the weight or wrench of a robotic arm, an additional angle increment is generated by the joint deformation. Then, the angle increment $\Delta \mathit{\theta }$ multiplied by the long-length mechanical arm would result in serious errors in the end effector (EE) and reduce the precision and machining quality. Therefore, traditional research usually ignores other deformations and focuses solely on joint deformation.

Nevertheless, for many engineering applications, other deformation factors cannot be ignored, such as heavy load conditions, hard material removal, and high-temperature operations [24,25,26,27,28].

• To improve the stiffness of a robot, manufacturers typically increase the cross-sectional area of the arm, which increases the weight of the robot. In addition, large-part production requires the use of large robots. Due to the gravity of the connecting link and the high load on the EE, each joint of the robot has a large deflection, and the running trajectory of the EE has a large deviation [27,29,30].
• Significant variation in robot stiffness affects the stability of the machining process. In this case, machining chattering problems are particularly prominent [6,31,32,33].
• High temperatures usually cause thermal deformation and expansion in robot parts. Model parameter errors can occur during the stiffness identification process [28].

This paper summarizes the development and application status of industrial robots, as well as the challenges that industrial robots face during machining processing. It provides a comprehensive review of robot stiffness research. The remainder of this paper is structured as follows. Section 2 introduces methods for improving the stiffness of industrial robots. Section 3, Section 4 and Section 5 introduce in detail the FEA, MSA, and VJM research results and deformation measurement, respectively. In Section 6, the common measuring techniques are introduced for measuring the deformation. Section 7 summarizes the research results and proposes future outlooks.

2. Methods for Improving Robot Processing Accuracy

To improve the accuracy of robotic machining, two main approaches are being studied to overcome the above problems. The first approach is optimizing the structure of industrial robots to improve their inherent stiffness properties. Fang et al. developed a new type of industrial robot and improved its stiffness using a hybrid structural design with open-loop and closed-loop motion chains [34]. Specialized end-effectors, such as antennas and pressure feet, were designed to improve the machining stability and stiffness of the robot during drilling and boring operations [35,36].

Manufacturers of industrial robots have also optimized the mechanism design so industrial robots meet the stiffness requirements of production processing. For example, ABB produced IRB 6660 for press tending and machining, which has an additional parallel arm to increase the stiffness of the robot and a special dual-bearing design to overcome the fluctuating process forces during machining [37]. KUKA offered KR 500 FORTEC MT (machining tooling), which was designed for high-payload milling applications [38]. In addition, some researchers have proposed improving the structural stiffness of robots by optimizing the materials used for robot joints and manipulator arms [39,40,41].

The second approach is to compensate for deformation during processing with a static stiffness model. The principle of compensation control is shown in Figure 2. First, an external force is acquired and transformed with sensors installed on industrial robots. When combined with the transformed data and joint angles directly acquired from the robot system, the stiffness model outputs the deformation after calculation to calibrate the robot [23]. In general, robot stiffness parameters are not provided by the manufacturers of industrial robots. For this reason, a stiffness model must be developed using dedicated experimental research [42].

The stages of compensation execution can be divided into online compensation and offline compensation, with the main difference being whether the error model input needs to be obtained online.

Model-based online error compensation relies on the acquisition of the actual position of a robot, which includes the 3D/6D Cartesian coordinates of the robot tool central point (TCP) or each joint angle. The TCP position is normally measured using optical metrology equipment such as a laser tracker, coordinate measurement machine (CMM), and 3D camera systems [43,44,45]. The joint angle is measured using a secondary encoder mounted at the output of each joint [9]. Then, real-time compensation is carried out based on the errors between the actual and theoretical coordinates [25,27,46,47,48,49,50]. Since the measurement accuracy of the metrology equipment is much higher than the positioning accuracy of robots, this method has the advantage of high precision. At the same time, some scholars combine the algorithm model with the stiffness evaluation to improve the accuracy of the stiffness evaluation [51,52], including the controller strategy, recursive algorithm, etc. For example, Flacco et al. combined a residual-based estimator of the torque at flexible transmission and a recursive least squares stiffness estimator based on a parametric model to conduct an online stiffness evaluation. Without a joint torque sensor and speed measurement, this method exhibits good robustness [53]. However, it also presents the following problems [54,55,56]:

• Online error compensation is difficult to implement, has high hardware requirements, high sensor costs, and a long operating time.
• Sensors have some inherent limitations in the application process due to sensor data noise, communication delays between the sensor and the robot controller, and the interference suppression bandwidth of the robot EE.
• The measurement system is susceptible to external factors, which greatly reduce the measurement accuracy. For example, a chip generated by cutting hinders the real-time measurement of the optical measurement system.

Compared with online error compensation, model-based offline compensation requires less auxiliary hardware. Therefore, it reduces the costs and maintains the flexibility of the robot [57,58]. At present, the main methods for robot stiffness modelling are FEA, MSA, and VJM [21,59,60,61,62,63,64,65,66].

Due to its high precision and high computational cost, FEA is used in the structural design stage. MSA simplifies the robot’s structure but reduces the modelling accuracy [67]. Moreover, FEA and MSA focus on structural design and stiffness performance analysis. VJM adds virtual axes to traditional rigid-body modelling to describe the deformation of key structures, manipulators, drive motors, and transmission structures, resulting in a reasonable balance between model accuracy and computational complexity that has been widely applied in industrial fields [68,69,70]. The static Cartesian stiffness of a robot in a workspace and at different postures can be determined by studying stiffness models. Different types of performance indicators have been introduced to evaluate the stiffness of robots [71,72,73,74,75,76], including the Jacobian eigenvalue, norm index, operability index, isotropic compliance property, and stiffness ellipse. Some scholars have compared and evaluated some stiffness performance indicators, which is convenient for selecting the corresponding indicators [77]. For milling, grinding, drilling, and spraying, redundant axes expand the workspace and provide greater flexibility for trajectory optimization [78,79]. In the following sections, the FEA, MSA, and VJM stiffness modelling methods and relevant research results are introduced.

3. FEA

FEA is considered to be an accurate method for modelling robot stiffness [62,63]. Due to multidimensional deformations, additional degrees of freedom (DOFs) are introduced into the robot system. Therefore, the FEA method is used to divide a continuous body with infinite DOFs into a specific number of flexible elements with mass and stiffness characteristics, reducing the problem to a structural problem suitable for numerical solutions. Moreover, based on the invariable mechanical properties, a certain distribution relation can be established for the displacement of the elements, the dynamics equations of the elements can be derived, and the dynamics equation of the robot system can finally be obtained through the element group [80]. A schematic diagram of a robot processed by the FEA method is shown in Figure 3.

In FEA modelling, the accuracy of the analysis depends on the design of the CAD model, the boundary conditions of the parts, and the parameter settings of the simulation. A specific discussion of these three factors is as follows [31]:

• A well-designed robot CAD model can improve the accuracy and calculation efficiency of the stiffness model. First, a 3D geometric model of each manipulator is established. Then, the transmission components, such as the bearings, motors, and gears, are removed. Mass units are used to replace the matching in each robot joint. This process simplifies the robot joint fit and reduces the calculation. Additionally, parts and features that have little influence on the stiffness of the robot, such as the pinhole and fillet of the non-bearing parts and rounded corners, are simplified.
• The material properties, including the material of the mechanical arm and the type of heat-treatment process, are specific to the CAD model parts of the robot.
• The parameter settings of the simulation model affect the complexity, calculation speed, and accuracy of the system modelling. Generally, reasonable mesh cell type, mesh density, and mesh quality settings not only save calculation costs but also maintain calculation accuracy.

Starting with a simple beam model and ignoring complex shapes, Corradini et al. established a stiffness model of the H4 parallel robot using the FEA method. In good agreement with the experimental measurement, FEA was used as an effective and accurate tool for obtaining the mechanical parameters of the robot [81]. Bouzgarrou et al. used CAD models of each mechanical leg, added link parts and defined each rotary joint, and then established an FEA model of the parallel platform mechanism T3R1. Based on this model, the static stiffness of the robot was calculated and represented in the whole workspace [82]. Pashkevich et al. introduced a local 6-DOF virtual spring to replace the flexibility of a connecting link and used FEA modelling to determine the high-precision stiffness parameters of the spring. In their experiment, they added auxiliary objects to the CAD model of each link as a reference to evaluate the compliance of the link. In addition, unlike FEA modelling of the entire manipulator, this method only evaluated a single link at a time, which greatly reduced the FEA calculation cost [83]. Furthermore, Klimchik et al. proposed a method for obtaining the stiffness parameters of the manipulator. Each robot connecting link was modelled using the FEA method. During the modelling process, the CAD model considered six parameters: the shape, cross-section, material distribution, physical properties of the connecting link, specificity of the connecting link joints, and coupling of translational/rotational deflection. In addition, the method reduced the error of FEA modelling. The reason for this is that additional random error was assumed in the “stiffness transformation”, and the abnormal values of the experimental data were eliminated. Based on experiments using the Orthoglide series-parallel manipulator, the accuracy of this model reached 0.1% when using a 2 mm parabolic mesh, and the test force and test moment were 1 $\mathrm{N}$ and 1 $\mathrm{N}·\mathrm{m}$, respectively [59].

In addition to directly applying the FEA method in robot stiffness modelling, it can be used to prove the validity of other models. In [21,62,64,84,85,86,87], the FEA method was used to verify the accuracy of the MSA method, while in [29], it was used to verify a VJM model. More complex models can also be verified using FEA [88,89].

The FEA method has unique limitations. As the number of finite elements increases, the continuous calculation of mesh nodes has higher requirements for computer hardware, the calculation time increases, and calculating the inversion of high-dimension matrices becomes increasingly critical [59,62].

4. MSA

The MSA method is a common technical method in the machinery industry. When analysed from the perspective of structural mechanics, a link of a robot can be regarded as a single beam structural unit (Figure 4). It assumes that the materials and structures of the links have perfectly linear behaviour, and dynamical effects are negligible. The deformation $\Delta \mathit{t}$ at the free end under an external load (force/torque) $\mathit{W}$ can be defined as (1).

$$\mathit{W}=\mathit{K}·\Delta \mathit{t}$$

(1)

where K is the $6\times 6$ stiffness matrix. $\Delta \mathit{t}$ is a 6-dimensional deformation vector including three translational displacements $\Delta \mathit{p}={\left[\Delta {\mathit{p}}_{\mathit{x}}, \Delta {\mathit{p}}_{\mathit{y}},\Delta {\mathit{p}}_{\mathit{z}}\right]}^{\mathit{T}}$ and three rotational angles $\Delta \mathit{\phi }={\left[\Delta {\mathit{\phi }}_{\mathit{x}}, \Delta {\mathit{\phi }}_{\mathit{y}},\Delta {\mathit{\phi }}_{\mathit{z}}\right]}^{\mathit{T}}$. Correspondingly, the external load W is also a six-dimensional vector containing the force $\mathit{F}={\left[{\mathit{F}}_{\mathit{x}}, {\mathit{F}}_{\mathit{y}},{\mathit{F}}_{\mathit{z}}\right]}^{\mathit{T}}$ and torque $\mathbf{M}={\left[{\mathit{M}}_{\mathit{x}}, {\mathit{M}}_{\mathit{y}},{\mathit{M}}_{\mathit{z}}\right]}^{\mathit{T}}$. For typical beams with regular cross-sections, the stiffness matrix K can be calculated analytically with information on the beam properties (beam cross-section area and Young’s modulus and shear modulus of the beam material). When beam units are combined by joints to build a robot structure, the stiffness matrix K can be obtained based on the static equilibrium of the structure. Furthermore, joint link inertia and transverse and axial deformations must be considered [62]. Subsequently, considering the rotation matrix and boundary condition constraints between each beam unit, the local stiffness matrices can be assembled into a global stiffness matrix for the robot [90,91,92]. Based on the existing research results and the characteristics of the MSA method, it has been widely used in the stiffness modelling of parallel robots.

Clinton et al. used the MSA method to model the stiffness of the Stewart parallel robot for the first time. Two assumptions were made in the experiments. First, the connecting links of the robot were only subjected to tensile deformation without bending. The second was that the stiffness of the robot frame was much stronger than that of the connecting link. The experimental results demonstrated the effectiveness of the MSA method for stiffness modelling [93]. For further study, Goncalves et al. first obtained the stiffness matrix of a 6-RSS parallel robot by considering the stiffness characteristics of elements such as the connecting links, actuators, and joints. Additionally, the stiffness of the joints and brakes were ignored by this simplified model [21]. To investigate the influence of joint stiffness, Deblaise et al. established two PKM stiffness models based on MSA. The first model assumed that the robot’s joints were rigid and only considered the deformation of the connecting links, while the second model considered the deformation of both the robot’s connecting links and joints. Load experiments showed that the second model was closer to the true elastic deformation of the robot [63]. Azulay et al. also considered the flexibility of the robot’s connecting links and joints. The MSA method was applied to derive a high-precision static stiffness model of a 9-DOF redundant and reconfigurable 3 × PPPRS parallel mechanism [91]. Based on the assumption of 12 DOFs for each link (Figure 5), Shi et al. proposed an MSA model of a complex EAST Articulated Maintenance Arm (EAMA). The model matrix was simplified based on boundary constraints. Compared with the FEA method, the prediction error of this method was less than 5%, which is within the acceptable prediction error range, and the computational efficiency was improved [87].

For different types of connections, the stiffness matrix can be different. In Figure 6, rigid and elastic connections are used in the two-link serial system. A $12\times 12$ Cartesian stiffness matrix $K_C^r$ for rigid connection can be deduced. This matrix $K_C^r$ is non-singular and invertible. When the connection is elastic, the Cartesian stiffness matrix $K_C^e$ is rank-deficient and singular. To efficiently apply the MSA method to serial and parallel robots, Klimchik et al. [63] developed an approach to model robot stiffness with a set of two groups of matrix equations describing the elasticity of separate links and connections between links. The general components and structures in a robot were modelled with this approach. A three-degrees-of-freedom planar parallel manipulator was selected for demonstration. The proposed MSA-based approach was approved to be in good agreement with FEA modelling. A level of 10 ms of computational efficiency can be achieved for stiffness modelling per manipulator configuration.

Detert et al. viewed the original MSA elements as having a limitation that could be entirely represented by beam theory. Therefore, an extended program was developed to automatically calculate the global stiffness matrix and deformation in a given structure composed of different elements under external loads. In this method, the nonlinear stiffness of the complex joints, links, and rolling contact bearings were all considered. The nonlinear stiffness characteristics were explained through several iterations of the linear model. After conducting experiments on the reconfigurable modular multiarm robot system PARAGRIP, their model demonstrated effective deformation prediction in the X, Y, and Z directions. The Z-direction had the best prediction effect, which can be used to compensate for robot errors due to gravity [65]. Lu et al. introduced a dynamic external load and inertial external wrench into the overall stiffness model of a parallel robot [94].

Moreover, some scholars have applied the MSA method to model the stiffness of serial robots. Kefer et al. proposed nonlinear flexible stiffness modelling of 6-DOF serial robots based on the MSA method, which achieved comparable accuracy to the FEA method. The model considered the flexible connecting link and the nonlinear characteristics of the joints as the posture changed. Finally, an energy index $\mathit{\delta }\mathit{D}$ (2) was introduced to evaluate the precision of the model, with $\mathit{f}$ and $\mathit{u}$ representing the induced force vector or wrench and the corresponding displacement vector, respectively. Compared with the FEA method, the deformation energy index was less than 0.03% in the whole workspace. In addition, the calculation time was greatly reduced. However, actual measurement experiments were not performed, and the model was only applicable for robot error predictions in stationary states [62].

$$\mathit{\delta }\mathit{D}={\mathit{f}}^{\mathit{T}}\mathit{u}$$

(2)

Compared with the FEA method, the MSA method has clear advantages when used with large flexible components. Many component elements are simplified to some degree, which effectively reduces the calculation cost and achieves a reasonable balance between calculation time and accuracy [83,84]. Unlike the VJM method, the MSA method does not use differential equations. Therefore, when describing the complex structure of a robot with a large number of closed loops and crosslinks, the MSA method reduces the complexity of the calculation [95]. However, more details need to be considered in MSA models, such as the nontrivial actions of the passive and elastic joints [63].

5. VJM

The traditional VJM method first defines each joint of a robot as a virtual spring and assumes that the connecting link has no deformation (Figure 7). It establishes a rotational stiffness matrix ${\mathit{K}}_{\mathit{\theta }}$ for the robot joint space. The Cartesian stiffness ${\mathit{K}}_{\mathit{c}}$ is calculated based on the transformation of the kinematic Jacobian matrix ${\mathit{J}}_{\mathit{\theta }}$. The mapping relation of the stiffness matrix in Cartesian space to joint space (3) was first proposed by Salisbury [68] and has been applied in the industry [42,72,90].

$${\mathit{J}}_{\mathit{\theta }}^{\mathit{T}}·{\mathit{K}}_{\mathit{c}}·{\mathit{J}}_{\mathit{\theta }}={\mathit{K}}_{\mathit{\theta }}$$

(3)

To improve the accuracy of stiffness prediction based on the traditional VJM method, researchers have considered the working conditions of various industrial robots and further developed the VJM method. The stiffness of an industrial robot is affected by various factors, including joint flexibility, link flexibility, and transmission structure flexibility. To some extent, each factor can be defined as a DOF in robot joint stiffness research. This paper divides the research into single-DOF and multiple-DOF stiffness models and describes them in detail.

5.1. Single DOF

The most commonly used and simplest stiffness modelling method is single-DOF VJM modelling. As mentioned above, joint deformation is the most significant of the various deformations. Many studies on the joint stiffness of a robot with a single DOF focus on joint deformation. The torque of the corresponding joint is required in the model. However, different measurement methods have revealed different accuracies for torque measurements under joint deformation.

The majority of researchers use force analysis to approximate joint torque. According to the static equilibrium equation, wrenches on an EE can be decomposed to calculate the torque of each joint [29]. Some scholars calculate the torque of a joint by obtaining the motor current corresponding to each joint. The reason for this is that each joint of an industrial robot is driven by an independent drive motor due to the reducer and transmission structure [96,97,98]. However, deriving the torque from the motor current usually gives inaccurate results.

The single-DOF joint stiffness model can be divided into linear stiffness and nonlinear stiffness based on the characteristics of the hypothetical virtual springs.

5.1.1. Linear Stiffness

Linear stiffness simplifies the elastic deformation of each joint to a constant joint rotational stiffness coefficient. Therefore, ${\mathit{K}}_{\theta }$ in Equation (2) is a diagonal matrix and each joint stiffness coefficient in the ${\mathit{K}}_{\theta }$ matrix is a constant.

Pan et al. adopted this method to establish a stiffness model for the ABB IRB6400 robot. The measurement position was defined in the working space. After loading the motor spindle at the end of the robot, the 3D displacement and 6D wrench data were measured. In addition to these data, the stiffness of each axis joint was calculated using the least squares method, and an end-milling experiment on the cylinder head was carried out. The real-time deformation compensation based on the stiffness model was effectively reduced from the original surface dimensional accuracy of 0.9 mm to 0.3 mm [99,100]. Eberhard Abele et al. [101] used two methods to obtain the Cartesian stiffness distribution of a five-joint industrial robot. The first method directly measured the stiffness of each joint. When measuring one joint, the other joints were fixed to prevent a coupling effect between all the joints. With Equation (3), the Cartesian stiffness was calculated using a rotational stiffness model. The second method was to directly measure the stiffness of the robot end. Four 800 mm × 800 mm planes were selected in the Z-direction of the robot coordinate system (Figure 8), and 16 positions on each plane were selected to measure the X, Y, and Z stiffnesses of the robot EE. The stiffness of the entire plane was fitted through cubic interpolation. The experiment indicated that the second method estimated the stiffness better. The main reason was that the first method approximated the joint as a linear torsion spring. This stiffness estimation introduced errors due to other deformations. The second method, which directly measured the stiffness of the end, considered the deformation of the whole joint and the robot links [102]. Subsequently, this method was applied to model the stiffness of five-axis robots and PA-10 seven-axis robots, achieving good results in error compensation. However, this method requires a large quantity of experimental data, and the nonlinear difference estimation of the unmeasured position introduces certain errors [103,104].

A further study on the traditional VJM method was conducted by Shih-Feng Chen and Imin Kao. They found that the traditional method included a nonconservative transformation, and (3) was used on a robot without an external load. Therefore, based on the traditional method, a supplementary stiffness matrix:

$${\mathit{K}}_{\mathit{g}}=\left[(\frac{\mathit{\partial }{\mathit{J}}_{\mathit{\theta }}^{\mathit{T}}}{\mathit{\partial }{\mathit{\theta }}_{\mathbf{1}}}\mathit{f}) (\frac{\mathit{\partial }{\mathit{J}}_{\mathit{\theta }}^{\mathit{T}}}{\mathit{\partial }{\mathit{\theta }}_{\mathbf{2}}}\mathit{f}) \cdots (\frac{\mathit{\partial }{\mathit{J}}_{\mathit{\theta }}^{\mathit{T}}}{\mathit{\partial }{\mathit{\theta }}_{\mathit{n}}}\mathit{f})\right]$$

(4)

Equation (4) was introduced to describe the structural change caused by the external load $\mathit{f}$ on the robot EE, where ${\mathit{\theta }}_{\mathit{i}}$ is the ith joint angle and $\frac{\mathit{\partial }{\mathit{J}}_{\mathit{\theta }}^{\mathit{T}}}{\mathit{\partial }{\mathit{\theta }}_{\mathit{i}}}\mathit{f}$ is an $n\times 1$ column vector, with $i=1, \dots , n$and i is the number of the joint. Therefore, ${\mathit{K}}_{\mathit{g}}$ is an $n\times n$ matrix. Besides, as shown in (5):

$${\mathit{J}}_{\mathit{\theta }}^{\mathit{T}}·{\mathit{K}}_{\mathit{c}}·{\mathit{J}}_{\mathit{\theta }}={\mathit{K}}_{\mathit{\theta }}-{\mathit{K}}_{\mathit{g}}$$

(5)

A conservative congruence transformation (CCT) was proposed between the rotational stiffness and Cartesian stiffness of the robot. Based on a digital simulation of the double-link planar manipulator, the work of the robot in the Cartesian space and the joint space was calculated. The results indicated that the CCT method was closer to the actual situation than the traditional method [70,105,106]. Gürsel Alici and Bijan Shirinzadeh used experimental methods to calculate the axis joint stiffness and Cartesian stiffness of a six-axis continuous articulated robot, further verifying the effectiveness of the CCT method [107].

Based on the CCT method, Dumas et al. identified the rotational stiffness ${\mathit{K}}_{\mathit{\theta }}$ of the KUKA KR240-2 robot. The effect of the supplementary stiffness matrix ${\mathit{K}}_{\mathit{g}}$ on ${\mathit{K}}_{\mathit{c}}$ was analysed. It was found that the larger the torque applied to the robot EE, the greater the effect of ${\mathit{K}}_{\mathit{g}}$ on ${\mathit{K}}_{\mathit{c}}$. Therefore, when EE is subjected to a large force and torque, the stiffness of the robot will have a large deviation from the actual situation. At a specific location, however, ${\mathit{K}}_{\mathit{g}}$ could be ignored relative to ${\mathit{K}}_{\mathit{\theta }}$. For this reason, the conditional number $K_F\left(M\right)$ based on the Frobenius norm (6):

$${\mathit{K}}_{\mathit{F}}\left(\mathit{M}\right)=\frac{\mathbf{1}}{\mathit{m}}\sqrt{\mathit{t}\mathit{r}\left({\mathit{M}}^{\mathit{T}}\mathit{M}\right)\mathit{t}\mathit{r}\left[{\left({\mathit{M}}^{\mathit{T}}\mathit{M}\right)}^{-\mathbf{1}}\right]}$$

(6)

was defined to identify and determine the measuring position where the effect of ${\mathit{K}}_{\mathit{g}}$ on ${\mathit{K}}_{\mathit{c}}$ could be ignored [72,108]. Therefore, Equation (7) let ${\mathit{J}}_{\mathit{N}}$ become a standard Jacobian matrix:

$${\mathit{J}}_{\mathit{N}}=\left[\begin{array}{cc}\frac{\mathbf{1}}{\mathit{L}}{\mathit{I}}_{\mathbf{3}\times \mathbf{3}}& {\mathit{O}}_{\mathbf{3}\times \mathbf{3}}\\ {\mathit{O}}_{\mathbf{3}\times \mathbf{3}}& {\mathit{I}}_{\mathbf{3}\times \mathbf{3}}\end{array}\right]{\mathit{J}}_{\mathit{\theta }}$$

(7)

${\mathit{I}}_{\mathbf{3}\times \mathbf{3}}$ is a $3\times 3$ identity matrix and ${\mathit{O}}_{\mathbf{3}\times \mathbf{3}}$ is a $3\times 3$ zero matrix. Moreover, $\mathit{L}$ is the characteristic length, obtained by means of the methodology proposed in [16], and it normalizes the units in the Jacobian matrix. Then, the inverse condition number of ${\mathit{J}}_{\mathit{N}}$ is illustrated in Figure 9 [72]. The lower ${\mathit{K}}_{\mathit{F}}{\left({\mathit{J}}_{\mathit{N}}\right)}^{-\mathbf{1}}$ is, the closer the robot is to singularities. As the robot pose is close to the singularity, the determinant of ${\mathrm{J}}_{\mathit{\theta }}$ approaches zero, which leads to abnormally high Cartesian stiffness ${\mathit{K}}_{\mathit{c}}$. Therefore, the VJM method is not suitable for the estimation of the Cartesian stiffness near the singular position [109].

By applying the CCT method and using Equation (4), Yang et al. identified the rotational stiffness of a robot in the entire workspace. The results of the study indicated that the effect of ${\mathit{K}}_{\mathit{g}}$ on ${\mathit{K}}_{\mathit{c}}$ was greater than 10% in a small part of the workspace, whereas its effect in the rest of the workspace was less than 5%. This result was consistent with Dumas’s conclusion [98]. Huseyin Celikag et al. also used the CCT method to establish a stiffness model of the ABB IRB 6640 robot and applied the model to milling processing. Given the characteristics of milling, only five DOFs were necessary to satisfy the positioning requirement. Hence, the DOF of rotation about the tool axis became functionally redundant. Due to the redundant DOF, the robot’s posture can be changed without changing its position. In the milling experiment, the 2D plane of the working space was evenly divided into positions 1 cm apart. While keeping the direction of the tool axis unchanged, the tool axis was planned in 0.01 rad increments in the interval (0, 2π). Based on the stiffness model, the stiffness of each robot posture was identified. According to a comparative analysis, the change rate of the robot Cartesian stiffness at each position in the working plane was between 58% and 161% [19]. For a six-DOF industrial robot, in general, the first three joints determine the robot’s position in joint space, while the last three joints determine the robot’s posture. Ambiehl et al. defined the first three joints as arm joints and the last three joints as wrist joints. In their research, two bottlenecks were observed. First, the stiffness value of each arm joint was 10 times greater than the stiffness value of each wrist joint. This ratio was too high to cause joint stiffness identification errors. In addition, with and without loading, the translational/rotational deflection was not homogeneous. To resolve this issue, a decoupled partial pose method (DPP) was proposed based on a metrological measurement. After “flange transformation” proposed by Ambiehl, the load variability shifted to the centre of the wrist. Using experiments on a dual-encoding industrial robot, the DDP method achieved 84% deflection error compensation under local posture decoupling [110]. Due to the effectiveness and convenience of the CCT method, it is widely used in single-DOF VJMs [17,20,75,111,112,113,114,115,116].

5.1.2. Nonlinear Stiffness

In contrast with linear stiffness models, the joint rotational stiffness coefficients in nonlinear stiffness models are not constant. In a nonlinear stiffness model, each joint stiffness coefficient in the ${\mathit{K}}_{\mathit{\theta }}$ matrix changes with the joint angle. The joint stiffness exhibits nonlinearity, which is more realistic.

Schneider et al. identified the stiffness of the six DOFs of industrial robot joints and displayed the nonlinear characteristics of the joint. First, force and torque sensors were installed on the end flange of the robot. Wrenches on the robot EE were measured from the maximum load in the negative direction to the maximum load in the positive direction. An optical measurement system was used to detect the 6-DOF deformation of each joint in the loaded and unloaded states. The distribution diagram of the rotational stiffness of joint four based on the measurements of four different positions and postures is shown in Figure 10a. The stiffness identification results demonstrated that (i) the amplitude of the displacement deformation in the X, Y, and Z directions was within the measured noise range, and there was no clear trend change; and (ii) the joint deformation was mainly rotation angle deformation. The torques in the X- and Y-directions had linear relationships with the torsional deformation in the X- and Y-directions, while the torque in the Z-direction had an approximate cubic relationship with the torsional deformation in the Z-direction. The rotational stiffness of the six joints is shown in Figure 10b. Therefore, the nonlinear least squares method was used to fit the rotational stiffness data and establish a stiffness model. Through static wrench loading, the position of the robot EE in the Cartesian coordinate system was measured. The error between the predicted value and the measured value based on the stiffness model was less than 36%. Moreover, there was no model compensation delay. To better verify the performance of the model, a machining experiment was carried out to mill a circle with a diameter of 70 mm on a steel workpiece. Comparing the milling performance with and without stiffness compensation, the absolute average position deviation was reduced from 0.39 mm to 0.2 mm [117,118].

5.2. Multiple DOFs

Due to the flexibility of the joints, other factors can be introduced to develop multiple-DOF VJM stiffness models.

5.2.1. Three DOFs

A three-DOF VJM not only considers the joint deformation but also introduces two additional deformations to the stiffness model. Abele et al. proposed a modified stiffness modelling method. In addition to the joint deformation, both the bearing tilt stiffness and the deformation of the connecting link were considered, as shown in Figure 11. With this method, each joint was increased from a single DOF to three DOFs. With the D-H method, the abovementioned model of the robot was increased to 15 DOFs, and the Cartesian stiffness value was calculated using Equation (2). Compared with the basic stiffness model, the predicted value of the modified stiffness model was closer to the actual measured value. The maximum error was less than 30% [101].

5.2.2. Six DOFs

Klimchik et al. considered the deformation of all components for stiffness identification. Each driving joint and connecting link were described by a 6 × 6 stiffness matrix. Hence, 258 parameters were used for a traditional 6-axis robot. To solve this hypothetical model, a simplified model was proposed to reduce the number of identification parameters while maintaining the accuracy of the stiffness model identification [66]. In further research, Klimchik et al. introduced the gravity G of a robot arm and an external load F into stiffness modelling (Figure 12). The relationship between the wrench and the displacement of the robot EE was linearized. Based on the static balance equation, the Cartesian stiffness ${\mathit{K}}_{\mathit{c}}$ was calculated. When this model was applied for deformation compensation control during the robot milling process, the static deviation was effectively improved [119].

In the general case, the stiffness matrix ${\mathit{K}}_{\mathit{c}}$ can be expressed by (8), with Hessian ${\mathit{H}}_{\mathit{\theta }\mathit{\theta }}$ expressed using (9). A small change in the Jacobian matrix of the robot is expressed by its second derivative $\delta {\mathit{J}}_{\mathit{\theta }}^{\left(\mathit{F}\right)}={\mathit{H}}_{\mathit{\theta }\mathit{\theta }}^{\left(\mathit{F}\right)}·\mathit{\delta }\mathit{\theta },\mathit{\delta }{\mathit{J}}_{\mathit{\theta }}^{\left(\mathit{G}\right)}={\mathit{H}}_{\mathit{\theta }\mathit{\theta }}^{\left(\mathit{G}\right)}·\mathit{\delta }\mathit{\theta }$. Similarly, a slight change in the gravity load G is represented by its first derivative $\mathit{\delta }\mathit{G}=\frac{\mathit{\partial }\mathit{G}}{\mathit{\partial }\mathit{\theta }}\mathit{\delta }\mathit{\theta }$. Among them, ${\mathit{H}}_{\mathit{\theta }\mathit{\theta }}^{\left(\mathit{G}\right)}$ and ${\mathit{H}}_{\mathit{\theta }\mathit{\theta }}^{\left(\mathit{F}\right)}$ are expressed in (9) and (11) [119].

$${\mathit{K}}_{\mathit{c}}={\left({\mathit{J}}_{\mathit{\theta }}^{\left(\mathit{F}\right)}·{\left({\mathit{K}}_{\mathit{\theta }}\mathit{-}{\mathit{H}}_{\mathit{\theta }\mathit{\theta }}\right)}^{-\mathbf{1}}{\mathit{J}}_{\mathit{\theta }}^{\left(\mathit{F}\right)\mathit{T}}\right)}^{-\mathbf{1}}$$

(8)

$${\mathit{H}}_{\mathit{\theta }\mathit{\theta }}={\mathit{H}}_{\mathit{\theta }\mathit{\theta }}^{\left(\mathit{F}\right)}+{\mathit{H}}_{\mathit{\theta }\mathit{\theta }}^{\left(\mathit{G}\right)}+{\mathit{J}}_{\mathit{\theta }}^{\left(\mathit{G}\right)\mathit{T}}·\frac{\mathit{\partial }\mathit{G}}{\mathit{\partial }\mathit{\theta }}$$

(9)

$${\mathit{H}}_{\mathit{\theta }\mathit{\theta }}^{\left(\mathit{G}\right)}={\sum }_{\mathit{j}=\mathit{1}}^{\mathit{n}}\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{g}}_{\mathit{j}}^{\mathit{T}}{\mathit{G}}_{\mathit{j}}}{\mathit{\partial }{\mathit{\theta }}^{\mathbf{2}}}$$

(10)

$${\mathit{H}}_{\mathit{\theta }\mathit{\theta }}^{\left(\mathit{F}\right)}=\frac{{\mathit{\partial }}^{\mathbf{2}}{\mathit{g}}^{\mathit{T}}\mathit{F}}{\mathit{\partial }{\mathit{\theta }}^{\mathbf{2}}}$$

(11)

Based on this stiffness model, Klimchik et al. further analysed the structure of a gravity compensator in heavy industrial robots. A gravity compensator was attached to joint 1 and joint 2 and formed a closed loop, which was defined as a linear spring structure and generated torque at joint 2. Combined with the stiffness of joint 2, the stiffness of the compensator was transferred to joint 2 to determine the equivalent stiffness of this joint. Thereafter, the geometric parameters of the compensator and the rotational stiffness were identified through experiments [30,42]. Before and after loading, the deformation data of each joint were directly read by a secondary angle encoder [121], and a rotational stiffness model was established based on these data. Finally, three methods for position compensation feedback control were established, which used information from a single-angle encoder, a double-angle encoder, and a modified stiffness model. The analysis found that compared with traditional robot feedback control (angle feedback from the main angle encoder), the modified stiffness model effectively improved the position accuracy of the robot. The deformation error was reduced by more than 80%. However, the model was complex and computationally intensive. Therefore, the numerical iteration process of the static equilibrium structure took a long time to converge. For heavy industrial robots with gravity compensators, Peng Xu et al. proposed an efficient stiffness model. This model processed each joint/link as a virtual spring with six DOFs. In the model, all kinds of deformation were considered, including joint/link flexibility, link weights, and gravity compensators. First, the expressions for the reaction forces of each joint were derived from joint 6 to joint 1 for static equilibrium. Then, a stiffness model was established in conjunction with the CAD model and FEA method. An experiment on the deformation of the KUKA KR 500 industrial robot (KUKA, Augsburg, Germany) was used to verify the correctness of the model. The theoretical value of the model was close to the measured value when an external load was added to links 4 and 5. Furthermore, when the first and second derivatives decreased, the computation time of the model was greatly reduced. In addition, EE deformation, which is caused by different factors, including the external wrench, gravity compensator, and link weights, can be separated under certain conditions [29].

6. Deflection Measurement

Normally, external loads (such as mass in Figure 13) would be exerted on the robot EE to bring about the deformation. The measurement technique has a significant influence on the parameter identification and selection of the stiffness modelling method. Several techniques can be used to measure deformation, including laser trackers, optical CMMs, laser interferometers, and digital image correlation (DIC) techniques [122]. The common equipment for deformation measurement in robot stiffness modelling contains a laser tracker and optical CMM [123], which are described in detail as follows.

6.1. Laser Tracker

The laser tracker is most commonly used for tracking the robot’s EE position. It projects a laser beam onto an optical target that is in contact with an object to be measured and determines the three-dimensional position of the target. The optical targets, known as spherically mounted retroreflectors (SMRs), reflect the laser beam to its emitted origin point. As shown in Figure 13, an EE was built to hold three SMRs, which were tracked by a FARO laser tracker. Each SMR’s three translational position coordinates referring to the laser tracker coordinate system were recorded. If a six-dimensional (three translational positions plus three rotational angles) pose needed to be measured, at least three SMRs were required to form the six-dimensional pose [109]. The layout of the SMRs on the EE should be considered when the robot moves to the planned poses, which could hinder the laser beam reflecting from SMRs. To ensure that the SMRs were in the range of the laser tracker, offline simulation was needed to plan the location of the laser tracker and the robot poses. In addition, more than three SMRs could be mounted on the EE to ensure that at least three SMRs could be tracked at arbitrary poses. As shown in Figure 14, four SMRs were symmetrically located on a square plate for position tracking. The laser tracker manufacturer also supplied a six-dimensional position tracking device, such as a Leica T-Mac (Leica, Wetzlar, Germany) [124]. Depending on the distance between the SMRs and laser tracker, the absolute measuring accuracy reached 10 µm.

6.2. Optical CMM

Compared to the laser tracker, optical CMM uses cameras to locate the 3D position of the target referring to the camera coordinate system. There are two types of optical CMMs regarding the target mechanism.

• A black and white ring is used as the target for position recognition, which is shown as markers in Figure 15. These markers are normally adopted by 3D camera systems. In addition to the stereo cameras, illuminations are provided to offer appropriate light for marker recognition. According to the measurement volume, the measurement accuracy is up to 6 µm [45].
• An infrared light-emitting diode (LED) is used as the target. As shown in Figure 16, three infrared line scan cameras are used to estimate the 3D position of the LEDs. The product specification of the Nikon Metrology K610 (Nikon, Tokyo, Japan) states a volumetric precision of 60 µm, depending on the measurement situation (distance and inclination towards the camera, ambient light conditions) [44].

Figure 15. 3D camera measurement system [45].

Figure 15. 3D camera measurement system [45].

Figure 16. All system components of the Nikon Metrology K610 [44].

Figure 16. All system components of the Nikon Metrology K610 [44].

Both types of targets can be very convenient to stick to the measuring object. In particular, black and white rings are very suitable for measuring a large number of components. When the 3D/6D deformation of each arm needs to be measured, at least three targets are attached to the arms (Figure 17).

Both measuring techniques have their advantages and disadvantages. The laser tracker has a larger measuring range and is not affected by ambient light. Due to the size and price of the SMR, a laser tracker is normally used to measure the displacement of the robot EE. Optical CMM is adopted to measure not only the displacement of the robot EE but also the deformation between the arms (joint deformation). When the stiffness modelling is built with the VJM method, the joint stiffness needs to be identified. Direct joint deformation measurement provides more precise information for joint stiffness identification. When only robot EE deformation is provided, the joint deformation is estimated by decoupling the robot EE deformation to each joint based on the inverse kinematics of the robot. However, the ambient light and the measuring volume should be considered if the optical CMM is appropriate for the size of the robot. In addition, the targets should be confirmed to be recognized when the robot moves to different positions.

7. Conclusions and Outlook

In various machining processes, the low stiffness of industrial robots inevitably leads to processing deformation. This problem is not prominent in ordinary handling, welding, laser cutting, etc. However, continuously high processing forces occur during machining, such as milling, drilling, and incremental sheet forming. The low stiffness of a robot causes the robot joints, mechanical arms, transmission structure, and other structures to deform, which greatly reduces the positioning accuracy of the robot; thus, it is difficult to use robots in high-precision machining [103,108,115]. To overcome this problem, deformation compensation based on a stiffness model is used to improve the machining accuracy of industrial robots. Different stiffness modelling methods were discussed and summarized in this paper.

Robot deformation compensation based on a stiffness model encounters the following issues:

• Simplifying the stiffness identification process while maintaining accuracy.
• In the process of robot processing, there are possible delays from measured wrenches due to the calculated error deformation and the feedback position compensation of the system.
• Position deviations are caused by external wrench impacts when a robot enters and exits the machining process. The dynamic characteristics of the stiffness compensation should be considered.

There are multiple challenging problems in modelling the stiffness of industrial robots. To improve stiffness models, further research can be conducted in the following areas:

• Standardization of stiffness modelling. At present, there is no standard procedure for establishing a stiffness model. It is difficult to select and apply one type of modelling method conveniently and quickly. With in-depth research on stiffness modelling methods, a modelling process with standard modelling principles, evaluation indicators, and measuring techniques could be developed.
• Automating measuring and modelling. In terms of actual measurements and stiffness modelling, the workload of manual participation can be reduced, and a set of automated systems for measuring and modelling various scenarios can be developed.
• Application of machine learning (ML) methods for stiffness modelling [43,126,127]. A high-precision stiffness model can be obtained after processing the experimental data with various artificial neural networks (ANNs). A self-learning process for stiffness modelling can also be developed and automated based on ML.