## Abstract

Incremental sheet metal forming is a highly versatile die-less forming process for manufacturing complex sheet metal components. Robot-assisted incremental sheet forming, or roboforming, allows a wider range of tool motion, providing the capability to shape intricate components. This makes roboforming the most flexible variant of the incremental forming method. However, the serial arrangement of links and joints in a robotic manipulator results in low positional accuracy under forming loads due to insufficient structural stiffness. The stiffness of the machine frame and tool directly impacts the accuracy of the final formed profile. The impact of machine compliance on component shape in incremental sheet forming is substantial in roboforming. This work presents a methodology for systematic analysis of the factors contributing to the errors in the geometric shape of robot-based forming. Experimental and numerical methods are used to estimate the material springback, tool/tool holder deflections, and errors due to machine compliance. The CNC machine frame is relatively stiffer than the industrial robots, such that material springback is estimated based on the experimental trials on CNC for cone and variable wall angle cone profiles. Tool and tool holder deflections are estimated using finite element simulations. The analytical method using the Virtual Joint Model is used to model the joint stiffness, and consequently, the robot Cartesian stiffness is estimated to predict path deviation contributing to geometric shape errors. The proportional contribution of each factor in the overall deviation in the Roboforming is also quantified.

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## Introduction

Incremental sheet metal forming (ISF) is a versatile die-less forming process. Reduced forming loads, improved formability, and implementation of quick design changes with simple tooling make ISF an attractive sheet metal forming process. The approach can be used as a solution for rapid prototyping tasks with reduced costs associated with tooling and dies. This is especially suitable for low-volume production with smaller batch sizes and higher levels of customization, which is relevant in today’s manufacturing trend.

In ISF, the part is formed continuously with progressive local deformations across a sheet metal using a general-purpose tool. The development of local necks is suppressed in ISF due to the dynamic tool contact zone, which allows higher overall deformation. However, the gradual thinning^{1} of the material may eventually lead to fracture. Fracture-forming limit lines (FFL) are used to represent formability limits in ISF^{2}. Along with material thinning and fracture, the geometric deviation in the final shape of the formed component is one of the critical limitations of ISF^{3,4,5,6}. As pointed out by Ren et al.^{7,8}, improving geometric accuracy is one of the primary challenges for successful industrial adaptation of the ISF process.

Attempts have been made to improve the geometric accuracy of the formed component by including error compensation in the toolpath or by altering the forming sequence^{9}. Most compensation approaches rely on empirical observations^{10,11}, theoretical predictions^{12}, or regression models based on finite element simulations^{13}. Various assisted techniques, such as local heating of the deformation zone^{14,15}, and use of additional tools (double-sided incremental forming) or external support^{16,17} have also been tried to improve the geometric accuracy. Möllensiep et al.^{18} used a regression-based compensation approach through a prediction-based modification of the toolpath to reduce the geometric inaccuracy for ISF-formed parts at elevated temperatures. Asghar et al.^{9} proposed a mechanics-based approach for estimating and correcting sheet and tool deflections in ISF. The approach was validated for conical profiles and free-form shapes. The tool and sheet deflection compensation methodology by Asghar et al.^{9} was adopted by Rakesh et al.^{19} for double-sided incremental forming (DSIF).

ISF process has evolved over time, and several variants have been developed ever since it was first patented, as discussed by Jeswiet et al.^{20,21}. CNC-based ISF is the most common adaptation as it is easier to develop and implement. However, it is challenging to produce parts with complex designs due to restricted degrees of freedom and limited workspace. Robot-assisted incremental forming, also referred to as “Roboforming”, is the most flexible variant of ISF. The robot offers higher degrees of freedom (usually 6 DOF) for tool positioning and maneuverability. It makes it easier to accommodate complex part designs and offers a larger workspace. Although flexible, the effective stiffness associated with the serial arrangement of the industrial robot links and joints is low and should be accounted^{22,23,24} for practical robot-based applications. Lower stiffness makes robots more compliant compared to CNC machines^{25}, which reduces the geometric accuracy^{26} of the final formed component under high forming loads. In addition to machine compliance, the geometric deviation can be attributed to the material springback. The net geometric inaccuracy is the accumulative effect of both machine compliance and material springback. Geometric deviation due to material springback is commonly studied. Ambrogio et al.^{10} studied the effect of process parameters (depth, tool diameter) and geometrical parameters (wall angle, depth of component, thickness of sheet material) on the profile deviation of a truncated pyramid component. The study indicated that the material springback^{27}and bending of the unsupported sheet are the significant factors contributing to the observed geometric inaccuracy. Overall, machine compliance in ISF can influence part geometry by affecting springback behavior and forming accuracy. Therefore, the geometric error for the formed component should be associated with the accumulated error due to machine compliance and possible deflections in tool assembly along with material springback^{4,20,28}. Fig. 1 is a schematic depicting possible error-causing factors in Roboforming. Many research efforts focus on isolating the impact of robot compliance on robot-forming inaccuracies. The robot-compliant model, derived from supplementary calibration experiments^{29}, is subsequently employed for path compensation. The force data essential for assessing deviations can be obtained through either Finite Element analysis^{30} or real-time force measurements^{31}. Although various studies examine the specific error contributions of factors such as material springback^{32}, elastic deflection of the robot, and tool assembly^{9} in forming profile deviations, they utilize diverse experimental conditions and methodologies. There is no standardized procedure in the literature to determine these individual error contributions.

This work systematically investigates sources of inaccuracies affecting the geometrical shape in ISF. The current work envisages proposing a hybrid methodology combining experimental, numerical, and analytical techniques to quantify the deviations occurring during roboforming. The focus of this work is more towards developing a comprehensive methodology rather than measuring the actual deviations as such. As discussed, both the material and machine tool affect the geometric deviation. The CNC machine tool is relatively rigid, and the deviations in the component formed using CNC can be attributed primarily to the material springback. The deviation due to the tool holder is estimated numerically using finite element method. The compliance error in the machine tool, especially in the robot, is modeled analytically using a reduced-order virtual joint model (VJM). The compliance error is sensitive to the robot size and more pronounced when smaller robots with relatively less payload capacity are used for incremental forming. The presented methodology can be seamlessly applied to a wide range of machine tools employed for incremental forming.

## Methods

The geometric form error in incremental forming is the deviation between the formed and the intended part profile. This geometric inaccuracy is primarily observed at the (a) unsupported regions adjacent to the active deforming area in the sheet due to the bending and (b) deformed section of the part (wall region). The geometric error in the deformed region (\(\delta R\)) is caused by (i) machine compliance (\(\delta _c\)), (ii) springback (\(\delta _s\)), and (iii) tool compliance (tool (\(\delta _t\)) and tool holder(\(\delta _{th}\)) ). Since all these deviations are small enough in practice, we can use their superposition to estimate the profile geometric error:

This work proposes a methodology to analyze individual contributions to forming profile deviations due to the above-mentioned factors in roboformed technological operations. Two different workpiece profiles were used both in CNC and robot-based incremental forming (Fig. 2). Both parts, i.e. variable wall angle conical frustum (VWACF) and constant wall angle cone (CWA) induce a unique load condition for geometric deviation study.

The experimental setup is shown in Fig. 3. Consistent forming conditions, i.e., toolpath, tool diameter, lubrication, blank size, and material, were maintained in both cases (CNC and Roboforming). Incremental forming (ISF) experiments were carried out over a ‘Jyoti Huron KMill’ CNC machining centre and an IRB7600 ABB robot with 500 kg payload capacity; using AA1050 blanks of dimension 150 mm \(\times\) 150 mm \(\times\) 1.2 mm. A simple sheet metal fixture with a work area (100 mm \(\times\) 100 mm) was used, as shown in Fig. 3. Hydraulic oil was used for lubrication. The tool was maintained normal to the fixture plane for the entire period of ISF. The fixture ensures a rigid constraint to prevent material ‘draw-in’ during incremental forming. Hydraulic oil provided sufficient lubrication to minimize the interface friction during forming. A simple ball end tool with a 10 mm diameter was used with a constant feed rate of 750 mm/min. A step toolpath with a constant 0.5 mm step depth was used. A six-axis FT Omega191 Force/Torque sensor (mounted at the robot flange) with a maximum load capacity of 9000 N in the z-direction ) and a Kystler multi-component dynamometer with max \(F_z\) of 1000 N was used to measure forces and torques during roboforming and CNC-based incremental forming respectively (Fig. 3). Both the sensors were synced to the same frequency of data logging for consistency in force comparison and post-processing.

The formed components were scanned using a laser scanner (HEXAGON|8325-7 with RS5) with 25 µm accuracy. The scanned components were compared with the nominal CAD profiles to estimate overall profile errors in each case.

The proposed methodology estimates the overall deviations in the roboformed component by comparing it with the nominal CAD profile; it also isolates the error due to springback, tool deflection, tool holder deflection, and robot compliance. The deviation due to tool holder and tool deflections are estimated using static FE analysis. The springback is estimated from the experimental observations of deviations in CNC-formed components. Due to the considerable stiffness gap between CNC and robots^{25,33,34}, the deviations in CNC-formed components are assumed to be mainly due to springback after reducing the numerically estimated tool deflections. The error due to robot deflections (robot compliance) is deduced experimentally and validated using the virtual joint method. The steps indicating the individual treatment of each parameter for the estimation of deviations are presented in Fig. 4. The details of each step are thoroughly discussed in the following subsections.

### Roboforming setup

The actuation of the forming tool is achieved through the relative motion of the machine’s linkages and joints. The position of the tool in space is described relative to a defined reference coordinate frame. In the context of the roboforming setup, several such reference frames are located. A schematic of all the reference coordinate frames to be defined in ISF is shown in Fig. 5. The \(O_b\), \(O_e\),\(O_t\), \(O_h\),\(O_w\), and \(O_f\) are the origins for the robot base, end effector, tool, work object, and force sensor coordinate frames, respectively.

All the experimental measures (deviations and forces) in ISF are obtained with respect to the work coordinate frame with the origin assigned to the center \(O_w(0,0,0)\) of the sheet metal blank, as shown in Fig. 5. \({\,}_{j}^{i}{\textbf{T}}\) denotes the transformation of measured parameters (forces, moments and displacements) from reference frame *i* to *j* (the transformation matrices and steps followed are presented in supplementary information Appendix (**A**).

In the ISF process, the forming tool moves along a pre-defined toolpath generated for a target geometry. The tool axis is aligned perpendicular to the sheet plane (*xy*-plane), as shown in Fig. 6. For a spherical tool with a constant radius, the moment experienced by the tooltip varies due to the shifting contact location on the part profile. In the incremental forming process, different tool-tip profiles may be used based on the complexity of the part profile^{35}. Therefore, the tool and the part profile may influence the moments developed at the forming tool (see Appendix **A**). Here, the forming forces at each point (\(\textbf{p}(x,y,z)\)) along the toolpath are estimated by transforming the forces and moments (\(\textbf{F}_f,\textbf{M}_f\)) recorded using a dynamo-meter (make: Kystler) placed under the fixture (Fig. 3) to the tooltip (\(\textbf{F}_e, \textbf{M}_e\)) using the transformation steps discussed in the Appendix **A**.

### Stiffness model of robotic setup

The entire roboforming setup can be modeled as a serial arrangement of rigid components, including the tool and the tool holder. The effective compliance (\({\textbf {k}}_{ef}\)) of the entire assembly is defined in a common reference frame (here, tool tip reference frame (\(O_{t}\))) by :

where \({\textbf {k}}_{t}\) and \({\textbf {k}}_{th}\) are the tool and tool holder compliance obtained by FE analyses. \({\textbf {J}}_{r},{\textbf {J}}_{th}\) and \({\textbf {J}}_{t}\) define the transformations from the local coordinate systems to the \(O_{t}\). Corresponding transformations can be computed using the following matrix expression:

where **I** is \(3 \times 3\) identity matrix, **0** is \(3 \times 3\) zero matrix and \(({\textbf {v}}_i \times )\) transforms vectors \({\textbf {v}}_i\) into skew-symmetric matrix for \(i=\{t,th,r\}\). Corresponding vectors \({\textbf {v}}_i\) are given in Table 1.

The compliance matrix \({\textbf {k}}_{ef}\) is identified based on the experimental force and deflection data measured during roboforming experiments. Similarly, robot compliance (\({\textbf {k}}_r\)) can be derived using the following expression:

While the tool is designed to follow a predetermined path, the deflections or errors in its position change at different locations due to the changing loads on the tool. The recorded reaction force and moments at the fixture base are transformed to the tool contact point for estimating the deflections. The estimated global tool (\({{\textbf {k}}}_{t}\)) and tool holder (\({{\textbf {k}}}_{th}\)) compliance matrices (Table 2) are used to calculate the corresponding deflections in ISF during forming. The detailed methodologies for all stiffness parameter identification from simulation and experimental studies are presented in the subsequent sections.

### FEA-based approach for identification of tool and tool holder deflection

The roboforming tool assembly consists of two components: the forming tool and the tool holder (Fig. 7), both with varied cross-sections and material properties. It is necessary to determine the compliance of the tool assembly during the roboforming. In this regard, the individual stiffness matrix of the forming tool (\({\textbf {K}}_{t}\)) and tool holder (\({\textbf {K}}_{th}\)) are determined using a static finite element analysis (FEA). The details of the FE model setup and boundary conditions are given in Fig. 8. The tool and tool holder are modeled using C3D20R solid elements. The tooltip surface nodes are coupled to a reference node defined by the tooltip using Kinematic coupling constraints. The trial forces and moments are applied to the reference node. The displacements at the base are constrained.

The FEA methodology is based on identifying corresponding nodal displacements for a set of external forces and moments. Distributed surface force and moments are applied at the tool hemispherical ball end (Fig. 8) to approximate the contact conditions at the tool-end during forming. The nodal displacements of the deformed tool are extracted for a set of surface nodes at the tool end, as shown in the Fig. 8. To effectively reduces the inaccuracies associated with local deformation of the nodes, the methodology proposed by^{36} is employed. The proposed methodology involves considering a displacement field around the reference point. The displacement field was defined for the nodes surrounding the ball end of the tool Fig. 8, and node displacements were treated as rigid transformations with displacement vector \(\textbf{p}\) and rotational displacement \(\mathbf {\delta \phi }\) represented by a rotation matrix \(\textbf{R}\):

where the initial location of the node is \(\mathbf {p_i}\), and \(\Delta \mathbf { p_i}\) represents the node’s displacement due to the applied wrench for the *n* number of considered nodes.

To find a set of desired parameters, a least squares optimization is used for the error function *f*:

This leads to the orthogonal Procrustes problem^{37} and involves the minimization of the sum of squared residuals and singular value decomposition in order to identify the translational and rotational parameters within a distinct set of virtual experiments. Further, an indicator for the parameter-to-noise ratio^{38} was adopted to assess the impact of errors introduced by mesh distortion and round-off error. This indicator was derived from covariance analysis of the translation and rotation parameters, and the estimation accuracy was evaluated using a common confidence interval technique. The procedure is presented in detail in the work by Klimchik et al.^{38}.

Wrench vectors (force and moment) in virtual experiments differ in the direction of application to develop a complete compliance matrix (Table 2, supplementary information Appendix **C**). The deflections and rotations are assumed to be relatively small, and the stiffness \({\textbf {K}}\) represents a linear relation between the three-dimensional static forces/moments **F**=\((F_x,F_y,F_z)^T\), **M** = \((M_x,M_y,M_z)^T\) and the corresponding translations/rotation displacements \(\varvec{\delta }\)=\((\delta _x,\delta _y,\delta _z)^T\), \(\varvec{\theta }\)=\((\theta _x,\theta _y,\theta _z)^T\). \({\textbf {K}}\) is the \(6\times 6\) stiffness matrix and the inverse of \({\textbf {K}}\) is called the compliance matrix (\({\textbf {k}}\)), defined as ([\(\textbf {k}\)]=[\({\textbf {K}}]^{-1}\)).

A similar procedure is followed to identify the stiffness matrix of the tool \({{\textbf {K}}_{t}}\) and tool holder \({{\textbf {K}}_{th}}\).

### Robot deflection modelling

This section describes an approach for computing the Cartesian stiffness and robot deflection using an analytical model. In order to evaluate the stiffness of the robot, a lumped model called the Virtual Joint Model (VJM) is used^{39}. According to this model, certain virtual joints are introduced in the joint space to describe the control loop and joint/link flexibility^{38}. The following relation represents the transformation from the base to the tool of the robot.

Here, \(\textbf{T}_{Base}\) represents the transformation from the base of the manipulator to the first actuated joint, and \(\textbf{T}_{Tool}\) represents the transformation from the flange to the tool. Transformation of the flexible actuated joint is represented by \({\textbf {T}}_{joint}^i(q^i+\theta _{joint}^i)\) where \(q^i\) and \(\theta _{joint}^i\) are the joint and virtual joint variables. Link geometry is described by \({\textbf {T}}_{Link}^i\). In the VJM model, flexible links are replaced by a combination of rigid links and virtual joints^{39}.

Studies show that for heavy industrial manipulators, joint elasticity dominates link elasticity^{40}. The effective stiffness of the robot, which is generally defined by the Cartesian stiffness matrix \({\textbf {K}}_{c}\), depends on joint and link stiffness. The Cartesian stiffness matrix of the robot can be expressed as a lumped stiffness of joint space stiffness \({\textbf {K}}_{\theta }\) according to the following relation^{41}:

Here, \({\textbf {J}}_{\theta }\) is the configuration-dependent Jacobian of the robot, describing the mapping between Cartesian and joint spaces. The configurations ( joint angles \(\theta _1\) to \(\theta _6\) ) of the robot corresponding to each Cartesian coordinate location on the toolpath are retrieved from the robot controller. \({\textbf {K}}_{\theta }\) corresponds to the joint stiffness matrix. The impact of configurations on the accuracy of incremental forming was outlined in our earlier study^{42}. A schematic diagram of the VJM model of an industrial robot with a tool and tool holder is shown in Fig. 9. Here, each one-dimensional virtual spring represents the compliance of the actuated joint (\({\textbf {k}}_{i}\)) where \(i=(1-6)\), while the compliance of the tool and tool holder are described by the spring located prior to them.

A set of auxiliary elasto-static experiments was carried out to identify robot stiffness properties (the stiffness coefficients of the robot virtual joints \(\textbf{K}_i\) used in VJM) following the procedure discussed in^{38}. Identified joint stiffness is given in the Table 3. These coefficients compose a joint stiffness matrix

where operator \(diag ({\textbf {K}}_i)\) creates stiffness matrix from the vector \({\textbf {K}}_i\) with \(K_i\) components on the main diagonal, vector \({\textbf {K}}_i\) composed of virtual joint compliance \({\textbf {K}}_i\), \(i=1.. 6\).

Deviation due to robot compliance (\({\varvec{{k}}}_c\)) during the forming experiment was estimated using the following static relation

where the compliance matrix \({\varvec{{k}}}_{c}\) is obtained by taking the inverse of the Cartesian stiffness matrix (Eq. 8), \({\textbf {F}}\) was calculated based on the wrench data measurements from the force sensor (originally measured in its coordinate system) and related geometric transformations. Thereby, the force was transformed into the work object coordinate system of the robot according to the methodology presented in the supplementary information Appendix (**A**).

In the following section, we will use the above-described methodology to analyze the contributions of all inaccuracy factors in the roboforming process.

## Results and discussion

This study is performed using a 1.22 mm thick sheet of commercially pure aluminum alloy (AA1050). The mechanical properties of the material are presented in Table 1, (Supplementary information Appendix **B**). In a typical incremental sheet metal forming, sheet and tool deflection are the major factors for the geometric errors. However, as mentioned earlier, it is important to consider the effect of machine stiffness in roboforming. The estimated deflections for conical profiles with constant wall angle (cone) and variable wall angle (VWA) are discussed in the following subsections.

### A constant wall angle cone part

A cone with a constant wall angle (CWA) of 71°, an opening diameter of 87 mm, and a depth of 102.97 mm is developed using both CNC and the robot. In both cases, the step toolpath is employed with a constant increment (step size) of 0.5 mm.

The measured forming load components (\(F_x\),\(F_y\),\(F_z\)) demonstrate comparable closeness in CNC and roboforming experiments (Fig. 10). It can be observed that the forming loads gradually saturate after a sharp rise during the initial phase of deformation. The fluctuation in the forming force can be attributed to the observed local thinning of the sheet around the cone opening (*z* = 0 to 20 mm).

The CNC and Roboformed parts are scanned and compared with the nominal CAD profiles to analyze deviations in the part geometry. Figure 11 presents the total deviations along the x, y, and z axes. Significant deviations occur near the cone opening (from *z* = 0 to −10 mm) due to the bending of the unsupported sheet (Fig. 11b). The tool stretches the material in contact, creating a pulling force on the forming neighborhood and inducing minor curvature in the area lacking external support. Bending stresses balance the majority of the external load. The data analyzed in this study (from z = −20 mm depth) do not consider the effects of sheet bending on the total deviations.

When the part is unclamped from the fixture post-deformation, it experiences geometric distortion due to material shrinkage caused by recovery of the elastic strains. This leads to a geometric error known as a springback error. Along with material springback, tool deflections are considered significant to the overall geometric deviations in the CNC-formed component. All deviations (\(\delta _x\), \(\delta _y\), and \(\delta _z\)) are measured on an unclamped sheet, considering only global springback values. It is assumed that CNC machines are structurally more rigid than serial robots. Thereby, the springback error can be determined by subtracting the tool deflections from the total deviations in the CNC-formed part.

The deviations due to tool deflections are calculated using finite element (FE) simulations, as depicted in Fig. 12(a). It is assumed that tool deflections are comparable in both CNC and roboforming, considering that the measured forming forces are the same. Notably, the tool holder experiences the least deflection of 0.0003 mm (\(\delta _{th}\)) along the tool axis (*z*-axis). The average deflection of the tool is approximately 0.136 mm along the *x* and *y* axes. The main cause of tool and tool holder deflections (Fig. 12(a,b)) is the in-plane forces (\(F_x, F_y\)) (Fig. 10). This deviation is significant for a 1.22 mm thick sheet across the part cross-section. Accounting for this tool deflection when developing an ISF toolpath can enhance the geometric accuracy of the formed component.

Similarly, the average springback measured along the *xy* plane (0.72 mm) is larger in magnitude compared to the tool axis (0.49 mm). The springback tends to follow the force trend and the variation in the part profile. In the roboforming setup, the tool is mounted on a slender tool holder (Fig. 7). The errors due to robot compliance are evaluated using the approach illustrated in Fig. 4. The average robot compliance error in the part cross-section varies between 0.3 mm near the cone opening (z = −10 mm) to 1.3 mm around the base region (z = −100 mm). The robot is the least compliant along the tool axis, with a maximum deflection of 0.16 mm. A representative plot of the planar distribution of the radial deviation for cross-sections at three z-heights along the part depth is given in the supplementary information Appendix (**D**).

The thickness distribution of the formed cone is shown in Fig. 13. The wall thickness of the cone reduced sharply from 1.22 mm to about 0.25 mm in the initial phase of deformation. It is expected due to the abrupt change in the part wall angle from \(0^{\circ }\) to \(71^{\circ }\) at *z* = 0. This sharp change in the profile angles leads to the accumulation of strains in the thickness direction^{43} proportional to the degree (rate) of change in wall angles. The thickness (*t*) reached a stable value of \(t = 0.4\) mm after a depth of \(z = 20\) mm. The wall thickness variances between CNC and roboformed parts are negligible at the mid-depth, with a marginal disparity of 0.05 mm at the upper and lower sections of the formed part.

### A variable wall angle cone part

To evaluate the deflections for a different load profile, a truncated cone with varying wall angle (VWACF) ranging from \(0^{\circ }\) to \(90^{\circ }\) is developed. The VWACF has an opening diameter of 88.15 mm and is formed using both CNC and Robot. A toolpath with a 0.5 mm step size and a 10 mm diameter ball end tool is used. The dimensions of the VWACF are shown in Fig. 2b.

The evolution of forming force for VWACF for both CNC and robot is illustrated in Fig. 14. The forming loads rise gradually to the maximum value of 475 N and eventually drop, leading to fracture. The forces profile in roboforming is close to CNC, with a minor variation. The forces recorded in the robot ISF are slightly less than in CNC during the initial stage of deformation but progressively increase with depth. It should be mentioned that the robot’s effective stiffness is configuration-dependent. A more compliant robot configuration results in less force being transmitted to the workpiece. These variations can lead to deviations in the tool contact position from the intended toolpath, leading to geometrical inaccuracy.

The overall geometric deviations in VWACF from the nominal CAD profile are shown in Fig. 15. The variations around the part opening are rather insignificant in the case of VWACF compared to the cone profile.

Furthermore, as shown in Fig. 16c, the wall thickness decreases gradually with depth reaching a minimum at fracture. The formed depth for parts obtained by the robot and CNC are presented in Fig. 16. The reduced bending deviation and absence of local thinning in VWACF can be due to the gradual change in the wall angle of the part profile, unlike the cone with a constant wall angle.

Figure 17 presents deviations from the tool, tool holder, springback, and robot compliance. It can be observed that the average tool (0.002 mm) and tool holder (0.0004 mm) deviations are negligible along the tool’s axis (z-axis). However, in-plane tool deviations are significant, reaching up to 0.239 mm as depicted in Fig. 17a. It can be suggested that altering the force direction by adjusting the tool’s orientation relative to the sheet plane might reduce these deviations. Additionally, Fig. 17c shows geometric deviation due to springback. Similar to the conical profile, the deviation due to springback is more pronounced in the x–y plane, up to 1.1 mm. Springback in VWACF increases at a part depth of *z*= (30–35 mm) and diminishes near the fracture area. A significant range of springback (minimum 0.004 mm, maximum 0.364 mm) was observed in VWACF Although the absolute mean springback error is less compared to cone, the variation in springback along the part depth may indicate that the geometric profile significantly influences the springback error. The deviation due to robot compliance is shown in Fig. 17d. The robot appears to be more compliant towards the beginning of forming, which is also apparent in the load profile (Fig. 14) with an absolute mean deviation of 0.502 mm along the tool axis. The robot deviation tends to reduce up to 0.0005 mm towards the end of forming. The absolute magnitude of mean inplane deviations (\(\delta _x\), \(\delta _y\)) due to robot compliance reaches up to 0.75 mm and 0.522 mm, respectively. A representative plot of the planar distribution of the radial deviation for cross-sections at three z-heights along the part depth is given in the supplementary information Appendix (**E**).

## VJM prediction of deviation due to robot compliance

The deviations due to robot compliance are analytically predicted using the VJM model (Eq. 10) and compared with the experimental observation, as shown in Figs. 18 and 19 for cone and VWACF, respectively. The margin of error (\(\Delta = \delta _{Exp.} -\delta _{VJM}\)) in the average deviation predicted by VJM and measured experimentally is comparable for both cone (\(\Delta _x = 0.311, \Delta _y = 0.5, \Delta _z = -0.840\)) and VWACF (\(\Delta _x = 0.36, \Delta _y = 0.4, \Delta _z = -0.6\)). However, VJM over-predicts the average absolute deviation along the z-axis. The average resultant error (\(\Delta _{cone} = 1.02\) mm) is higher than the VWACF (\(\Delta _{VWACF} = 0.8\) mm), which may indicate that the VJM model predictions are influenced by the part geometry. It is observed that the deviations due to robot compliance are largely a function of part profile and the robot configuration.

The specific combination of load and robot joint configuration is unique to the shape of a particular part. The interaction between the robot configuration, part shape, and end effector load introduces a process non-linearity that is challenging to model. The VJM model effectively captures the variation in deviation, which is directly related to the end effector load. When working with certain part shapes (such as Cone and VWACF) in our case study, the range of achievable joint configurations may be limited due to the small size of the component, leading to a narrow range of active configurations. For clarity, a specific dataset compares predictions and experimental results for a single toolpath contour at z = −11 mm, as shown in Fig. 20.

A polar plot (Fig. 20) shows the experimental error, VJM predictions, and the mean error distribution along the toolpath profile at −11 mm depth for the chosen toolpath points. For this specific case, it can be observed that the VJM model over-predicts the errors along the x and z-axis and under-predicts along the y-axis. The discrepancies between the predicted results and the experimental deviations do not seem to be angle-dependent along the toolpath contour in general. Although the range of forces and moments along x &y axes are similar (see Table 4), one can observe (Fig. 20) a significant difference in VJM predictions for robot deflection in x and y directions (\(\delta x\) and \(\delta y\)) both for cone and VWACF profile. These variations are caused by the mutual dependency of the Cartesian deflections due to the high impact of off-diagonal components of the robot compliance matrix for the selected workpiece location. It should be stressed that variations in the compliance matrix components are not essential:

where the means and standard deviations of each component of the Cartesian compliance matrix are given for all robot configurations used in the cone experiment. The compliance matrix shows that the robot’s deflection in the y-direction is significantly influenced by the higher force component \(F_z\) due to the \(k_{23}\) component of the compliance matrix. This influence brings the y-direction deflection close to zero.

Hence, the variations in the robot end-effector deviation in the VJM modelling are related to the robot’s Cartesian compliance. To demonstrate the contribution of each force component (due to compliance), Table 4 contains an estimation (exact values can slightly vary due to variation of compliance matrix components) of a range of deflections induced by each force component; total deflections correspond to the absolute deflection range obtained from the VJM-based simulation presented in Fig. 20. It should be mentioned that moment contributions are negligible to any deflection and lower than \(\pm 0.0001\) mm. \(F_z\) has a major contribution to all Cartesian deflections while for \(\delta _y\) it contributes with a negative sign reducing the contribution of the main force component \(F_y\).

As mentioned earlier, robot compliance is sensitive to changes in joint configurations, and each part profile directs the robot’s motions and changes in configurations. One of the main factors that can also affect the results could be partial compensation of robot compliance due to the springback effect. Going deeper into these aspects might provide additional insights and is an area of further investigation. Based on the experimental observations, it is implied that there is a scope for improvement in error prediction by carefully incorporating the possible nonlinearities due to material and part profile. Such overlapping interactions between the process parameters make it difficult to model with simplified criteria. It is important to emphasize that an accurate analytical approach can eliminate the need to experimentally determine the geometric deviation caused by robot compliance. One of the simplifying assumptions of rigid link used in this study may be insufficient to justify the discrepancies completely, but it could not be ignored and should be investigated further.

In addition, Table 5 presents the summary of the absolute resultant deviations and the percentage contribution of each factor (tool, tool holder, springback, and robot compliance) in the geometric inaccuracy of roboformed components (cone and variable wall angle cone). The resultant deviation \(R_m\) is computed as follows:

Here, \(\bar{\delta _x}\),\(\bar{\delta _y}\) and \(\bar{\delta _z}\) are the mean of the absolute deviations along *x*, *y* and *z* axes. The \(R_m\) value is used to present the percentage contribution of each factor in the overall geometric deviations.

Based on the data in Table 5, it is clear that robot compliance plays a significant role, contributing to 40.28% of the geometric deviation in cone and 59.98% in variable wall angle cone (VWACF). Additionally, Table 5 also suggests that the part profile affects the percentage contribution of robot compliance to the overall geometric deviation. This can be attributed to the fact that robot compliance varies depending on the active joint configuration, resulting in different robot deflections for different part profiles. Springback is another important factor, which appears to be complementary to robot compliance and is influenced by the part geometry. A larger springback of 1.132 mm is observed for Cone compared to 0.530 mm for VWACF. The key insight here is the superposition of the springback and robot compliance. One of the main factors contributing to this could be the partial compensation of the robot compliance due to springback. In the present work, the springback error is estimated based on the difference between CNC and roboforming. Similarly, other errors are also obtained experimentally by additive superposition. To perform a parametric study, a finite element simulation of the entire roboforming process is necessary. This analysis is complex not only from the numerical aspect but also from the choice of constitutive model, for the estimation of springback for a dynamic process like incremental forming is not well established. The variability in part profiles or the extent of change in the robot configuration from its previous position may lead to a non-linear accumulation of compliance-related parameters. The overall springback in the VWACF is nearly half of that in the cone, and the robot deflection is higher in VWACF compared to the cone (Table 5). This raises an interesting discussion on the impact of part stiffness due to its geometric profile.

Additionally, factors such as material thickness are expected to have a magnifying effect on forming forces^{44}. For instance, during incremental forming of AA2024-O sheets, an increase of sheet thickness from 0.8 mm to 1.4 mm led to an increase in the peak axial forming forces by up to 36.38% for a tool diameter of 15.66 mm^{45}. It should be noted that the trend of forming force evolution depends more on the part profile (as presented in this work Figs. 10 and 14) and less on the material thickness. As shown by Kumar et al.^{45}, a \(2^{\circ }\) increase in the wall angle of the profile can lead to a 15% increase in the forming force. Such increased forming forces due to material and part profile parameters can lead to an overall increase in compliance-related deviations. On the other hand, material properties such as Young’s modulus (E), strain hardening exponent (n), and sheet metal thickness can have a direct effect on geometric deviation due to springback^{46}. For materials with a higher yield stress, the area of the elastically deformed regions is greater, which can contribute to a larger springback^{47,48}. The thickness of the material has an inverse effect on the springback due to increased region of plastic deformation^{49}.

The data provided in the Table 5 is valuable for roboforming applications as it helps in making decisions on quantifying the approximate compensations in the toolpath related to robot deflections. This data presents a general idea of the possible compensations needed for each factor to improve the geometric accuracy in the roboformed component. It should be noted that the springback and deflections in the robot are part profile and robot configuration dependent and which vary throughout the roboforming operation. However, mean values are presented here for quantitative comparison.

## Conclusion

This study proposes a systematic analysis of factors contributing to the geometric shape error in roboforming. Experimental and numerical approaches were used to estimate material springback, tool/tool holder, and robot link deflections. Since CNC machines are relatively stiffer than industrial robots, material springback is estimated from parts (cone and variable wall angle conical frustum) developed in CNC. Finite element simulations with an ABAQUS static solver estimate tool and tool holder deflections. Using a Virtual Joint Model (VJM), robot elasto-static behavior is modeled, and virtual joint deflections that cause geometric shape errors are estimated. A comprehensive framework is presented to account for the geometric deviations that arise during roboforming. The difference in a geometric deviation between the CNC and robot-formed components indicates that the robot stiffness has a significant role, contributing to 40.28% in the cone and 59.98% in the variable wall angle cone (VWACF) of the overall geometric error. It is observed that the material springback varies with formed depth and is sensitive to the part design. Robot deflections are not constant during the roboforming of a part; the effective stiffness of the robot is sensitive to the robot link configurations. Also, part design affects the instantaneous evolution of tool contact force in roboforming. In the present work, the difference in recorded deviations is rather small, as the payload capacity of the robot, as well as its workspace compared to the CNC machine that was used, causing minimum deviation. Nevertheless, the procedure presented is more important than the absolute deviation, as practical roboforming experiments often employ a mid-range robot, including one with lower stiffness. The simplified VJM model used in the present work can be extended to include link stiffness in the future.

## Data availability

Data analysed for this work is available within the supplementary information with this article.

## References

Malhotra, R., Xue, L., Belytschko, T. & Cao, J. Mechanics of fracture in single point incremental forming.

*J. Mater. Process. Technol.***212**, 1573–1590. https://doi.org/10.1016/j.jmatprotec.2012.02.021 (2012).Bharti, S., Gupta, A., Krishnaswamy, H., Panigrahi, S. & Lee, M.-G. Evaluation of uncoupled ductile damage models for fracture prediction in incremental sheet metal forming.

*CIRP J. Manuf. Sci. Technol.***37**, 499–517. https://doi.org/10.1016/j.cirpj.2022.02.023 (2022).Bârsan, A., Racz, S.-G., Breaz, R. & Crenganis, M. Evaluation of the dimensional accuracy through 3d optical scanning in incremental sheet forming.

*Mater. Today Proc.***93**, 594–598. https://doi.org/10.1016/j.matpr.2023.03.175 (2023).Reddy, N., Lingam, R. & Cao, J. Incremental metal forming processes in manufacturing.

*Handbook Manufact. Eng. Technol.*411–452 (2014).Behera, A. K., de Sousa, R. A., Ingarao, G. & Oleksik, V. Single point incremental forming: An assessment of the progress and technology trends from 2005 to 2015.

*J. Manuf. Process.***27**, 37–62. https://doi.org/10.1016/j.jmapro.2017.03.014 (2017).Gatea, S., Ou, H. & McCartney, G. Review on the influence of process parameters in incremental sheet forming.

*Int. J. Adv. Manuf. Technol.***87**, 479–499. https://doi.org/10.1007/s00170-016-8426-6 (2016).Ren, H.

*et al.*General contact force control algorithm in double-sided incremental forming.*CIRP Ann.***67**, 381–384. https://doi.org/10.1016/j.cirp.2018.04.057 (2018).Ren, H.

*et al.*In-situ springback compensation in incremental sheet forming.*CIRP Ann.***68**, 317–320. https://doi.org/10.1016/j.cirp.2019.04.042 (2019).Asghar, J., Lingam, R., Shibin, E. & Reddy, N. V. Tool path design for enhancement of accuracy in single-point incremental forming.

*Proc. Inst. Mech. Eng. Part B: J. Eng. Manuf.***228**, 1027–1035 (2014).Ambrogio, G.

*et al.*Influence of some relevant process parameters on the dimensional accuracy in incremental forming: A numerical and experimental investigation.*J. Mater. Process. Technol.***153–154**, 501–507. https://doi.org/10.1016/j.jmatprotec.2004.04.139 (2004) (**Proceedings of the International Conference in Advances in Materials and Processing Technologies**).Hirt, G., Junk, S., Bambach, M. & Chouvalova, I. Process limits and material behaviour in incremental sheet forming with cnc-tools.

*Mater. Sci. Forum***426**, 3825–3830. https://doi.org/10.4028/www.scientific.net/MSF.426-432.3825 (2003).Lingam, R., Bansal, A. & Reddy, N. V. Analytical prediction of formed geometry in multi-stage single point incremental forming.

*Int. J. Mater. Form.***9**, 1960–6214. https://doi.org/10.1007/s12289-015-1226-y (2016).Ren, H.

*et al.*In-situ springback compensation in incremental sheet forming.*CIRP Ann.***68**, 317–320. https://doi.org/10.1016/j.cirp.2019.04.042 (2019).Duflou, J., Callebaut, B., Verbert, J. & De Baerdemaeker, H. Laser assisted incremental forming: Formability and accuracy improvement.

*CIRP Ann.***56**, 273–276. https://doi.org/10.1016/j.cirp.2007.05.063 (2007).Duflou, J., Callebaut, B., Verbert, J. & De Baerdemaeker, H. Improved spif performance through dynamic local heating.

*Int. J. Mach. Tools Manuf***48**, 543–549. https://doi.org/10.1016/j.ijmachtools.2007.08.010 (2008).Malhotra, R.

*et al.*Improvement of geometric accuracy in incremental forming by using a squeezing toolpath strategy with two forming tools.*J. Manuf. Sci. Eng.***133**, 5179. https://doi.org/10.1115/1.4005179 (2011).Malhotra, R.

*et al.*Accumulative-dsif strategy for enhancing process capabilities in incremental forming.*CIRP Ann.***61**, 251–254. https://doi.org/10.1016/j.cirp.2012.03.093 (2012).Möllensiep, D., Kulessa, P., Thyssen, L. & Kuhlenkötter, B. Regression-based compensation of part inaccuracies in incremental sheet forming at elevated temperatures.

*Int. J. Adv. Manuf. Technol.***109**, 1917–1928. https://doi.org/10.1007/s00170-020-05625-y (2020) (**Cited by: 6; All Open Access, Green Open Access, Hybrid Gold Open Access**).Rakesh, L., Amit, S. & Reddy, N. V. Deflection compensations for tool path to enhance accuracy during double-sided incremental forming.

*J. Manuf. Sci. Eng.***138**, 091008. https://doi.org/10.1115/1.4033956 (2016).Jeswiet, J. Asymmetric incremental sheet forming.

*Adv. Mater. Res.***6**, 35–58. https://doi.org/10.4028/www.scientific.net/AMR.6-8.35 (2005).Emmens, W., Sebastiani, G. & van den Boogaard, A. The technology of incremental sheet forming—A brief review of the history.

*J. Mater. Process. Technol.***210**, 981–997. https://doi.org/10.1016/j.jmatprotec.2010.02.014 (2010).Boby, R. A. Identification of elasto-static parameters of an industrial robot using monocular camera.

*Robot. Comput. Integr. Manuf.***74**, 102276. https://doi.org/10.1016/j.rcim.2021.102276 (2022).Petric, T. & Žlajpah, L. Kinematic model calibration of a collaborative redundant robot using a closed kinematic chain.

*Sci. Rep.***13**, 17804. https://doi.org/10.1038/s41598-023-45156-6 (2023).Li, Z., Li, S. & Luo, X. An overview of calibration technology of industrial robots.

*IEEE/CAA J. Autom. Sin.***8**, 23–36. https://doi.org/10.1109/JAS.2020.1003381 (2021).Belchior, J.

*et al.*Off-line compensation of the tool path deviations on robotic machining: Application to incremental sheet forming.*Robot. Comput. Integr. Manuf.***29**, 58–69. https://doi.org/10.1016/j.rcim.2012.10.008 (2013).Meier, H., Buff, B., Laurischkat, R. & Smukala, V. Increasing the part accuracy in dieless robot-based incremental sheet metal forming.

*CIRP Ann.***58**, 233–238. https://doi.org/10.1016/j.cirp.2009.03.056 (2009).Shaker, W. K. & Klimchik, A. Towards single point incremental forming accuracy: An approach for the springback effect compensation. In

*2023 IEEE 19th International Conference on Automation Science and Engineering (CASE)*, 1–6. https://doi.org/10.1109/CASE56687.2023.10260568 (2023).Lu, H., Liu, H. & Wang, C. Review on strategies for geometric accuracy improvement in incremental sheet forming.

*Int. J. Adv. Manuf. Technol.***102**, 3381–3417. https://doi.org/10.1007/s00170-019-03348-3 (2019).Liu, Y.

*et al.*An iterative path compensation method for double-sided robotic roller forming of compact thin-walled profiles.*Robot. Comput. Integr. Manuf.***86**, 102689. https://doi.org/10.1016/j.rcim.2023.102689 (2024).Meier, H., Laurischkat, R. & Zhu, J. A model based approach to increase the part accuracy in robot based incremental sheet metal forming.

*AIP Conf. Proc.***1315**, 1407–1412. https://doi.org/10.1063/1.3552383 (2011).Möllensiep, D.

*et al.*Real-time stiffness compensation and force control of cooperating robots in robot-based double sided incremental sheet forming.*Prod. Eng. Res. Dev.***15**, 683–699. https://doi.org/10.1007/s11740-021-01052-4 (2021).Shaker, W. K. & Klimchik, A. Comparative analysis of springback compensation for various profiles in incremental forming. In

*2023 International Russian Automation Conference (RusAutoCon)*, 1040–1045. https://doi.org/10.1109/RusAutoCon58002.2023.10272754 (2023).Bo Tao, X. Z. H. D. Mobile-robotic machining for large complex components: A review study.

*SCIENCE CHINA Technol. Sci.***62**, 1869–1900. https://doi.org/10.1007/s11431-019-9510-1 (2019).Yuan, L., Pan, Z., Ding, D., Sun, S. & Li, W. A review on chatter in robotic machining process regarding both regenerative and mode coupling mechanism.

*IEEE/ASME Trans. Mechatron.***23**, 2240–2251. https://doi.org/10.1109/TMECH.2018.2864652 (2018).John Timothy Roth, G.C, Tyler John Grimm. New incremental forming tools and method. (2017). https://patents.google.com/patent/US20190099799A1/en

Klimchik, A., Pashkevich, A. & Chablat, D. Cad-based approach for identification of elasto-static parameters of robotic manipulators.

*Finite Elem. Anal. Des.***75**, 19–30. https://doi.org/10.1016/j.finel.2013.06.008 (2013).Schönemann, P. H. A generalized solution of the orthogonal procrustes problem.

*Psychometrika***31**, 1–10. https://doi.org/10.1007/BF02289451 (1966).Klimchik, A., Furet, B., Caro, S. & Pashkevich, A. Identification of the manipulator stiffness model parameters in industrial environment.

*Mech. Mach. Theory***90**, 1–22. https://doi.org/10.1016/j.mechmachtheory.2015.03.002 (2015).Pashkevich, A., Klimchik, A. & Chablat, D. Enhanced stiffness modeling of manipulators with passive joints.

*Mech. Mach. Theory***46**, 662–679. https://doi.org/10.1016/j.mechmachtheory.2010.12.008 (2011).Dumas, C., Caro, S., Garnier, S. & Furet, B. Joint stiffness identification of six-revolute industrial serial robots.

*Robot. Comput. Integr. Manuf.***27**, 881–888. https://doi.org/10.1016/j.rcim.2011.02.003 (2011) (**Conference papers of Flexible Automation and Intelligent Manufacturing**).Salisbury, J. K. Active stiffness control of a manipulator in cartesian coordinates.

*1980 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes*, 95–100, https://doi.org/10.1109/CDC.1980.272026 (1980).Paul, E.

*et al.*Evaluation of deviations due to robot configuration for robot-based incremental sheet metal forming. In*Proceedings of the 2023 6th International Conference on Advances in Robotics*, AIR’23 (Association for Computing Machinery, New York, NY, USA, 2023), https://doi.org/10.1145/3610419.3610471.Bharti, S., Karvaje, K. S., Krishnaswamy, H., Agrawal, A. & Panigrahi, S. K. Investigation of feature-based and space-filling tool path strategies for formability in incremental sheet metal forming.

*Int. J. Mater. Form.***16**, 60. https://doi.org/10.1007/s12289-023-01781-0 (2023).Shaker, W. K. & Klimchik, A. Comparative analysis of springback compensation for various profiles in incremental forming. In

*2023 International Russian Automation Conference (RusAutoCon)*, 1040–1045 (IEEE, 2023).Kumar, A., Gulati, V. & Kumar, P. Investigation of process variables on forming forces in incremental sheet forming.

*Int. J. Eng. Technol.***10**, 680–684 (2018).Buang, M. S., Abdullah, S. A. & Saedon, J. An overview of the impacts of material parameters on springback.

*Appl. Mech. Mater.***564**, 323–328 (2014).Gan, W., Babu, S., Kapustka, N. & Wagoner, R. H. Microstructural effects on the springback of advanced high-strength steel.

*Metall. Mater. Trans. A.***37**, 3221–3231 (2006).Slota, J., Jurčišin, M. & Spišák, E. Numerical and experimental springback determination of sheet metals in an air bending process.

*Acta Metall. Slovaca***18**, 200–209 (2012).Mohameed, B. R., Mubarak, E. M. M. & Alsalihi, N. H. Effect of die’s shape, sheet thickness and type of alloy on the springback phenomenon.

*Eng. Technol. J. textbf***30**, 2012 (2012).

## Acknowledgements

The authors would like to express their gratitude to the Department of Science and Technology for providing financial assistance via Project No. DST/INT/RUS/RSF/P-50/2021 titled “Enhancing the accuracy of roboforming through prediction and compensation of elastic behavior using Artificial Intelligence techniques”. Part of this work was supported by the RSF (Project Number 22-41-02006).

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All authors contributed to the study’s conception and design. S.B., E.P., and A.U. performed material preparation, data collection, and analysis. S.B. and E.P. wrote the first draft of the manuscript; K.H., A.K., and R.A. supervised the experiments and corrected the manuscript. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Bharti, S., Paul, E., Uthaman, A. *et al.* Systematic analysis of geometric inaccuracy and its contributing factors in roboforming.
*Sci Rep* **14**, 20291 (2024). https://doi.org/10.1038/s41598-024-70746-3

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DOI: https://doi.org/10.1038/s41598-024-70746-3