Accuracy Optimization of Robotic Machining Using Grey-Box Modeling and Simulation Planning Assistance

Abstract

The aim of this paper is to develop an approach to increase the accuracy of industrial robots for machining processes. During machining tasks, process forces displace the end effector of the robot. A simulation of the various process influences is therefore necessary to ensure stable machining during production planning in optimizing the process parameters. Realistic simulations require precise dynamics and stiffness models of the robot. Regarding the dynamics, the frictional component is highly complex and difficult to model. Therefore, this paper follows a grey-box approach to combine the advantages of the state-of-the-art Lund–Grenoble model (white-box) with those of a data-driven one (black-box) in the first part. The resulting grey-box LuGre model proves to be superior to the white- and black-box models. In the second part, a model-based simulation planning assistance tool is developed, which makes use of the grey-box LuGre model. The simulation assistance provides the manufacturing planner with process knowledge using the identified robot and cutting force models. Furthermore, it provides optimization methods such as a switching point analysis. Finally, the assistance tool gives predictions about the machining result and a process evaluation. The third part of the paper shows the evaluation of the simulation assistance on a real machining process and workpiece, showing an increase in accuracy using the tool.

1. Introduction

1.1. Motivation

This paper is aimed at robot-based manufacturing processes with high path speeds and accelerations and simultaneously high demands on the positioning and path accuracy of the industrial robot (IR). Technologies such as high-speed cutting (HSC), laser cutting, or conventional additive manufacturing processes like fused deposition modeling (FDM) have these characteristics. Switching from machine tools, gantry robots, and special machines to conventional IR for the aforementioned tasks increases flexibility, requires lower investment costs, and leads to shorter delivery times, which is particularly advantageous for small and medium-sized businesses (SMEs) with high variance and complex components [1,2,3].

In addition to these advantages, savings in raw material and energy consumption can also be the motivation for implementing robot-assisted machining solutions [4]. As an example of the low energy consumption of IRs, according to Tallal et al., IRs carry out around 50% of the manufacturing processes in automotive production but consume only 1% of the energy in this sector [5]. The use of special-purpose machines also requires a considerable amount of floor space and therefore occupies large hall areas and usually requires significant maintenance and servicing costs [6].

1.2. Problem Description

Current research has so far focused on robot-assisted processes with low feed rates and the compensation of static displacement. Applications such as tape laying and machining processes are widely represented in the research environment. Reference processes have already been successfully investigated and further developed in various scenarios [7,8,9,10,11,12,13,14,15,16]. Due to the advantages mentioned, other tasks such as polishing or laser-induced hardening processes have also been implemented using IRs [17,18].

Solutions to the challenges of the past have also been developed in the industrial environment. For example, the programming of processing paths has been solved by some software and control providers, sometimes using different approaches [19,20]. In addition, IR hardware is also constantly evolving, which underlines the pressure to innovate that prevails in the industry. Consequently, as described in [21,22], robot-based machining processes are already being implemented in industry in isolated cases. The solutions developed to date cannot be transferred to highly dynamic processes (in this paper, dynamic machining processes are defined as tasks with complex movements and process forces due to mechanical workpiece–tool contact, which also require narrow tolerance fields for path accuracy) or can only be transferred in part. In addition, the effects of friction and gear backlash are generally neglected.

The path accuracy of an IR depends on its kinematic structure. Significant deviations of up to 0.5 mm between the target and the actual path have been observed when the robot is required to perform linear movements that are only caused by changes in the direction of the rotation of individual gears [23]. The critical backlash of the individual axes also has an influence on the path accuracy when the path direction of the tool changes. The frequency and position of the critical changes in the direction of rotation during machining can be influenced and are to be considered using planning methods such as computer-aided manufacturing (CAM).

Due to the kinematic structure of articulated robots, switching points can occur even with linear movements due to the rotary axes. The qualitative effect of switching points on the path deviation depends on the acting axis of rotation, i.e., the gear backlash of the installed gearbox. The position of the nominal/actual deviation due to the slack error in a linear movement and the frequency of switching points during the entire machining process depend on the clamping position and the machining direction and therefore the position of the path operations relative to the robot. Upstream consideration in the machining planning regarding a suitable clamping position and direction of the machining paths has the potential to increase process accuracy [24].

The influence of switching points not only affects the path accuracy of linear movements but also leads to path deviations in the case of rectangular outer contours; for example, if the tool must change direction in Cartesian space. At the switching point, individual rotary axes must reverse so that the tool tip deviates from the target position during the approach process. The automated specification of suitable approaches and switching point strategies for corresponding machining operations (e.g., external contours) can lead to quality improvements, as shown in [24].

1.3. Solution Approach

In previous works, a dynamics model has already been derived for the six DOF IR MAX100 by MABI Robotic AG using the recursive Newton–Euler method [25]. The results of the compensation for the first processing tests confirmed a considerable improvement in the final contour and validity of the approach. Furthermore, an approach to directly measuring and evaluating the deformation of robot bodies in the presence of process forces has been developed to identify the stiffness model of the MABI robot [26].

Since [25] did not consider frictional effects, this paper aims to close this gap. Due to the highly complex and nonlinear nature of the friction phenomenon, a grey-box modeling approach is used in the first part of the paper (Section 2). As a result, the benefits of analytical or white-box friction models such as the popular Lund–Grenoble (LuGre) model and of data-driven or black-box methods, such as neural networks (NNs), are combined. The former ensure the physical plausibility of the model while the latter can model highly nonlinear effects.

Next, a model-based assistance tool is developed in Section 3, which is able to integrate the resulting grey-box LuGre model along with the already identified dynamics and stiffness models of the IR. The simulation assistance supports the user in production planning through visualization and process knowledge as well as through automated optimization methods. The assistance tool consists of eight components, which are described in Section 3: toolpath interpolation, inverse kinematic transformation, material removal simulation, cutting force model, structural mechanical robot model, switching point analysis, and process evaluation. Here, a special focus lies on the analysis of switching points and the optimization thereof. Furthermore, the visualization of force curves and displacements enables the identification and elimination of critical path sections.

In Section 4, the final simulation planning assistance is tested on a machining process using the MABI robot(MABI Robotic AG, Veltheim, Switzerland) in an application- and industry-oriented manner. The processing results are examined and evaluated regarding their machining accuracy using standardized quality measurements. The three main parts of the paper are shown in Figure 1.

Since optimizations in work preparation cannot eliminate all the production deficits mentioned, a compensation strategy is also conceptualized in the discussion in Section 5. The compensation uses the developed dynamics, stiffness, and friction models to compensate for the predicted deviations from the target contour. Finally, the paper concludes in Section 6.

2. Grey-Box Friction Modeling for Industrial Robots

This chapter briefly describes the dynamics and stiffness model of the IR, which are used for the model-based simulation planning assistance tool before focusing on current friction modeling. Here, a grey-box model approach is suggested to improve the state-of-the-art friction models.

2.1. Dynamics and Stiffness Modeling

Dynamics is the mathematical description of body movements under the impact of forces and torques as a function of time. A dynamics model of a robot establishes a temporal relationship between the position $q$, velocity $\dot{q}$, and acceleration $\ddot{q}$ of the joint (joint variables) and the joint torques $\tau$ [27]. An inverse dynamics model of a serial robot calculates the joint torques from given joint variables.

The inverse dynamics model can be attained using two formulations: Newton–Euler or Lagrangian dynamics. The first method applies Newton’s and Euler’s dynamic equations, which lead to the computationally efficient recursive Newton–Euler algorithm (RNEA) to solve direct and inverse dynamics. The second method results from the kinetic and potential energy of the robot. Both approaches lead to the same result [28,29,30].

$$\tau =M\left(q\right)\ast \ddot{q}+C\left(q,\dot{q}\right)+G\left(q\right)+{J^T\left(q\right)\ast F}_{ext},$$

(1)

The product of the mass matrix and the joint acceleration $M\left(q\right)\ast \ddot{q}$ represents the part of the torque, which is caused by the acceleration of an axis subjected to mass inertia. $C\left(q,\dot{q}\right)$ is a vector containing torques caused by Coriolis and centripetal forces. $G\left(q\right)$ is the vector including gravitational torques [28]. In addition, forces on the end effector (e.g., machining forces) can be considered by adding a term, which entails the product of the Jacobi matrix of the robot $J^T\left(q\right)$ and the external forces $F_{ext}$.

The dynamics model (in combination with a stiffness model) calculates position deviations of the TCP in advance, considering the influences of gravity, dynamics, and external loads. In order to simulate a more precise position deviation, compliance should be integrated into the aforementioned model. The authors of [26] introduced an approach to directly measure and evaluate the deformation of robot bodies in the presence of process forces. The authors used a measurement setup containing multiple Integral Deformation Sensors (IDSs), which provide the change in length due to deformations of the respective body. The measurements are fed to a beam model, which can calculate the body’s 3D Cartesian deflections.

Based on this, Ref. [26] developed an axis-specific simulation model to predict the compliance behavior. It is suitable for analytically calculating the overall compliance on the TCP, taking into account the compliance of the gearbox, the bearings, and the swing arm for any axle configuration. In particular, the compliance of the swing arm is often neglected or only considered.

2.2. Friction Modeling

The dynamics model given in Equation (1) in the previous section did not include frictional effects. Friction is a complex nonlinear phenomenon and incorporating it into the inverse dynamics model is the subject of the current research. The effect of joint friction on the axis torques is highly significant and cannot be omitted. In addition to analytical or white-box friction models such as the LuGre model, data-driven or black-box models have been researched in the past years. Grey-box models have been introduced recently to combine the advantages of the former approaches.

2.2.1. Analytical and Data-Driven Models

Analytical friction models can be classified into two categories depending on how they consider static friction (minimal friction to set a resting body in motion). Static models only depend on the velocity and are not defined for a velocity of 0 rad/s. They assume that the system behavior is static while the objects are in a macroscopic resting state (gross-sliding regime or GSR). Dynamic models, however, depend on the velocity and the position and are also defined for a velocity of 0 rad/s. In contrast, they assume that even in a resting state, small pre-sliding (transition from static or non-sliding friction to kinetic or sliding) displacement friction is happening [31]. Therefore, dynamic models offer a more realistic representation of the friction behavior. Popular friction models are shortly introduced below: the Coulomb–Viscous, Stribeck, Dahl, and LuGre model.

The well-known Coulomb–Viscous model only depends on the angular velocity and describes the frictional torque ${\tau }_F$ using the following equation [32]:

$${\tau }_F=\mu \ast sgn\left(\dot{q}\right)+\nu \ast \dot{q},$$

(2)

with

$$\mu ,\nu >0.$$

(3)

The sign function $sgn$ returns 1, 0, or −1 depending on whether the input is positive, 0, or negative. The kinetic friction is modeled by the Coulomb parameter $\mu$ multiplied by $sgn$. The viscous friction is modeled by the viscous parameter $\nu$ multiplied by the axis velocity $\dot{q}$. This model is very simple and identifying the two parameters is relatively easy. A drawback is the inaccuracies at low speeds [33].

The Stribeck model combines a function for the Stribeck effect with the linear model for viscous friction. The Stribeck effect or negative viscous friction effect describes the nonlinear negative ratio between friction and velocity [33].

$${\tau }_F=S\left(\dot{q}\right)+\nu \ast \dot{q},$$

(4)

with

$$S=sign\left(\dot{q}\right)\ast \left[{(f}_c+\left(f_s-f_c\right)\ast \mathit{exp}\left(-{\left|{\displaystyle \frac{\dot{q}}{{\dot{q}}_S}}\right|}^{\delta }\right)\right],$$

(5)

and

$$f_c, f_s, {\dot{q}}_S>0 \mathrm{and} {f_s>f}_c.$$

(6)

The parameter $\delta$ determines the exponential course of the curve and can be positive or negative. How fast the Stribeck curve converges towards the viscous friction model is described by ${\dot{q}}_S$. The static friction is represented by $f_s$ and the Coulomb friction resistance by $f_c$. For many use cases, the Stribeck model is sufficiently accurate, but it has weaknesses when the relative velocity is changing fast. Also, the model is discontinuous around the velocity of 0, which can cause problems when an oscillatory system should be simulated [33].

Dahl also models the friction for the pre-sliding phase or regime (PSR) in comparison to the previous models. It describes static friction as a continuous function, but since there are two possible velocity direction changes, the friction cannot be described by a single curve. The result is a differential equation that depends on the position and the sign of the velocity [34]. The equation describes a hysteresis loop. The Lund–Grenoble model (LuGre) combines the findings of the Dahl model for the PSR with the Stribeck model for the GSR. A nonlinear differential equation was introduced to model the transition from PSR to GSR [35].

$${\tau }_F={\sigma }_0\ast z+{\sigma }_1\ast \dot{z}+\nu \ast \dot{q,}$$

(7)

with

$$\dot{z}=\dot{q}-{\sigma }_0\ast {\displaystyle \frac{\dot{q}}{S\left(\dot{q}\right)}}\ast z,$$

(8)

and

$${\sigma }_0,{\sigma }_1>0.$$

(9)

The model views the surfaces on a macroscopic level instead of on a microscopic level. This means that the individual connections between the surfaces are modeled. This is realized via a bristle model where each bristle represents a connection between the surfaces. Since the real contact points of the surfaces are unknown, the bristle model cannot model them as an exact replica, therefore the average deflection is described by the internal state $z$, the stiffness ${\sigma }_0$, and the damping ${\sigma }_1$.

Choosing a particular model depends on many factors such as the characteristics of the robot system and its actuators [36]. For robot arms in free space, motor friction dominates. Models for frictional torques caused by actuators only depend on the velocity of the nth joint and are therefore independent of other joints [37]. The authors of [38] identified and compared different analytical friction models for the first axis of the MABI robot. Here, the LuGre model performed best compared to the Coulomb or Stribeck model. However, its identification process is more complex, since two different trajectories are needed. In order to identify the Stribeck model, a trajectory with different constant velocities is needed. For identifying the dynamic parameters of the LuGre model, which are important for the PSR, a continuous trajectory at very low velocities and accelerations is needed. This is because pre-sliding displacements are only limited to a few degrees of movement. For the parameter identification process, genetic algorithms are used as proposed in the literature [39,40].

Due to the complexity of modeling friction and considering its many influencing factors such as load and temperature [41,42,43,44,45,46,47], many authors have used data-driven models such as NNs [48,49,50]. NNs try to find relationships between the in- and output data of a specific system. They can model complex nonlinearities (given sufficient data) while achieving high accuracies but without explicit knowledge about the physical behavior of the system. For the first axis of the MABI robot, [38] found that a long short-term memory (LSTM) network, a type of NN that uses recurrent connections and memory cells for remembering values or time, achieved similar results to the LuGre model. The LSTM model was trained using gradient descent, a standard optimization algorithm for NN [51]. However, data-driven models can possess millions of parameters compared to analytical ones. They are therefore referred to as black-box models as opposed to white-box models [52].

2.2.2. Grey-Box Models

Mixing white- and black-box models creates grey-box models, which are currently being explored. Other words are structured learning, physics-informed neural networks, or informed learning [52]. Grey-box models combine prior analytical knowledge with data-driven techniques, therefore harnessing the advantages of both approaches. MATLAB’s System Identification Toolbox™ offers a grey-box model estimation function if a system can be represented using ordinary differential or difference equations with unknown parameters [53]. Grey-box models have been developed for various use cases such as automotive engines, shock absorbers, gas bearings, the heat dynamics of buildings, wave loading, and ocean vessels [54,55,56,57,58]. The authors of [59] and [60] developed and evaluated grey-box friction models for a rotating arm and tribometer, respectively. The authors of [61] proposed the grey-box identification of parameters for the rigid body dynamics, friction, and flexibilities of an industrial robot. However, [61] only used a simple Coulomb–Viscous model, while the following authors used a LuGre model.

The authors of [62] combined a fast integral terminal sliding mode control (SMC) with a robust exact differentiator observer and a feedforward NN (FFNN). Their approach was implemented on an IR with five degrees of freedom. The FFNN uses as an input the joint velocity given by the observer, the information about the internal state, and the torque predicted by an optimized LuGre model. The output is a new torque prediction. The calculated torque is then suitable as input for the sliding mode controller. The SMC uses these values to create a control command for the robot [62]. The FFNN tries to model the inaccuracies of the LuGre model (residual learning). This is attempted by using one hidden layer with 10 neurons. As an activating function, the hidden layer has the tangents hyperbolics activation function, and the output layer has a linear activation function [62].

The authors of [63] developed a more complex LuGre model-based NN (MBNN) for a linear motor stage. An MBNN is based on the idea of integrating a NN into a linearized state space model. So, for the LuGre model, a second-order discrete time state space model is supported by three different FFNNs. One FFNN models everything that depends on the internal bristle states and the other two FFNNs model the differential equation of the internal bristle state. By discretizing Equation (5), one FFNN has, as input, the current and prior velocity and the other one has only the current velocity. The NNs are connected to form the LuGre MBNN. One hidden layer with ten neurons and a tangents hyperbolic activation function is used. As a training trajectory, the following velocity trajectory was used to model the motion start region. The LuGre MBNN was able to model the motion start region and the motion reverse region for the linear motor stage. It performs better than a system without friction compensation and can reduce the maximum tracking error by up to 65% [63].

The authors of [64] used a radial basis function NN (RBF NN) as an observer for the internal bristle state and gradient descent for parameter estimation. For the internal bristle state, one single hidden layer with the velocity and position as an input was used. A regressor simultaneously updates the other LuGre parameters. A permanent magnet synchronous AC servo motor with a gear and an inertial load was used on two different trajectories. First, a normal motion case with a sinusoidal trajectory was used, and later, a slow-motion trajectory based on a decreasing sinusoidal trajectory was tested. Besides performing better than the same approach without friction compensation in terms of tracking error, the authors express concerns about the high computation time and suggest simplifying the structure of the NN [64].

The authors of [65] used four RBF NNs to model the dynamics and friction of a two-link planar robot. One RBF NN was used to estimate the sum of the inertia matrix, the centrifugal and Coriolis force vector, and the vector of gravitational torques. The other three RBF NNs deal with the LuGre friction model. The input of the RBF NN is the internal bristle state and the velocity. In the initialization process, the internal state is set to zero, as a trajectory sinusoidal and a quadratic growing velocity curve were chosen. The whole RBF NN-adapted block is connected to a sliding mode controller and tested in a simulation environment only [65].

2.3. Grey-Box LuGre Model

This paper aims to identify a precise friction model for the MABI robot, which can be used in the simulation assistance tool for optimizing the accuracy of the machining process. In previous works, the LuGre model has been identified as the most suitable, followed by an LSTM model. To improve the LuGre model, this paper aims to use a grey-box approach similar to other authors. To reduce the complexity of the identification process, two suggestions are made. First, despite using two different trajectories, one trajectory is created to identify the grey-box LuGre model. Second, instead of using two optimizers, the grey-box LuGre model should be optimized by the gradient descent algorithm only.

To determine the LuGre parameters with gradient descent, the training procedure should be as simple as possible. Therefore, as much prior knowledge as available should be used. This should reduce the complexity of the model as fewer parameters must be optimized and therefore the stability of the system should improve. Prior knowledge refers to the understanding of friction mechanics and phenomena as utilized in the conventional approach. Some unnecessary parameters are, for example, the NN parameters of the structured learning approaches.

This concludes that the approaches of [63,65] are not promising as they use several NNs. This leads to increasing complexity and more parameters that must be determined. The approach of [64] is also inauspicious as it has one big disadvantage. In this approach, the calculated internal state $z$ only depends on the velocity and position and not on the velocity of the prior step. As a reminder, the internal state is described by a differential equation. So, for an exact prediction of $z$, the prior values of $z$ or the prior velocities must be known.

2.3.1. Methodology and Data Management

In order to simplify the LuGre model using a grey-box approach, the Stribeck function (5) is substituted by a simple FFNN with one hidden layer and ten nodes using a sigmoid activation function. Additionally, to the FFNN, all LuGre parameters are optimized simultaneously using gradient descent with an Adam optimizer while ensuring the constraints given in Equations (3), (6) and (9). The resulting model is shown in Figure 2 and compared to a conventional LuGre model [38] and the residual learning approach in [62]. The residual LuGre model possesses two hidden layers and ten neurons with a Leaky ReLU activation function.

As a training trajectory, the Universal Trajectory (UT) by [38] is suggested, which is designed to simulate the occurring friction behavior during the application-related use of the robot for its possible velocity range. The UT is an Up-Chirp signal multiplied by an exponential function. Therefore, it covers high to medium velocities as well as low ones prevalent in the pre-sliding regime. In addition, the UT of the robot must excite atypical and nonlinear friction behavior, as these friction phenomena are particularly difficult to model and thus provide a good metric for comparison. Since NNs tend to develop a bias for situations they learn more often than other sections, there are no repetitive intervals in the UT.

For a final comparison of the models, a testing trajectory with the same characteristics is chosen while differing from the UT by reasonable modifications such as changing amplitudes [38]. The trajectories are used to generate measurement data on the MABI robot. The data are prepared following [38] and results in the measured joint velocities and friction torques of axis one.

2.3.2. Results

A comparison of the identified dynamic LuGre parameters using the conventional method [38] and the grey-box approach is shown in

Table 1. Comparison of identified dynamic LuGre parameters for axis 1 by conventional and grey-box models.

Parameter/LuGre Model	Conventional	Grey-Box
${\sigma }_0 \left[Nm/°\right]$	294.14	404.792
${ \sigma }_1 \left[Nms/°\right]$	3.263	0.1594
$\nu \left[Nms\right]$	0.0473	0.0379

. For the latter, the constraints given in Equations (3) and (9) were considered in the gradient descent optimizer. Only parameter $\nu$ is similar for both models.

The grey-box LuGre model did not learn all dynamic parameters accurately. However, it shows the highest prediction accuracy compared to the conventional and residual model, as can be seen in Figure 2. The grey-box LuGre model shows the lowest root mean squared error (RMSE) and best R² value, which can be seen in the table to the right. The former is a metric that evaluates the difference between actual and predicted values, which should ideally be 0. The latter is a statistical metric used to evaluate the performance of regression models. The closer the fit is to 100%, the better the model matches the given data [66]. The grey-box model is followed by the residual model according to [62] and the conventional one according to [38].

Figure 3 shows a visual comparison of the predicted frictional torque by the grey-box and residual model. The closer the predicted torque is to the red line, the better the prediction. For high negative torques at $\tau \in [-3.5,-2]Nm$, the grey-box model outperforms the residual one. For high positive torques at $\tau \in [2, 3.5]Nm$, the behavior is inverted. However, both approaches have difficulty predicting the torques at $\tau \in [-1, 1]Nm$.

The results show the superiority of the grey-box LuGre model compared to the conventional and residual one. However, the experiments have only been conducted on the first axis of the MABI robot and only considered velocity-dependent friction. Further works will include load-dependent friction as well as analyze the remaining five axes to develop a complete grey-box LuGre model, which can be used in the model-based simulation approach described in the next chapter.

3. Model-Based Simulation Planning Assistance for Robot Machining

The measurements conducted, along with the models derived from them, form valuable building blocks for improving the machining accuracy of robot-based milling processes. However, to make the effects of the individual factors on the manufacturing outcome assessable, a holistic simulation approach as shown in Figure 4 is required in which the developed models are combined effectively. The primary audience for this approach is the production planning staff, who should be empowered to contribute their expertise to process optimization through a task-oriented presentation of the simulated results.

The material removal simulation operates on a given toolpath, cutting tool, and raw workpiece information. For time-discrete intermediate steps, positions of the currently used cutting tool are obtained by interpolating the toolpath description. The positions are defined in the workpiece coordinate system (WCS) and directly used to align cutting tool models relative to a previously generated volumetric workpiece model. Furthermore, the path positions are transformed in positions and velocities in the axis coordinate systems (ACS) of the robot using an inverse kinematic transformation.

Next, the overlapping volume between the cutting tool and the workpiece models is identified and removed from the latter model. Also, engagement values are derived from the removed volume and given to a force model. In turn, the force model predicts the instantaneous spatially cutting forces. Handed over to a structural mechanical model of the robot, the current deflection of the cutting tool is computed and used to alter the relative cutting tool position of the next discrete simulation step. Receptances of the cutting tool itself were neglected. To account for the position-dependent stiffness of the robot, the structural mechanical model is updated in each simulation step using the previously computed axis positions.

The axis positions are further processed by a friction model. In combination with the axis velocities, these are used to derive a path-dependent tracking error that is also added to the relative cutting tool position of the next simulation step. Once the simulation has finished, the resulting workpiece geometry contains the effects of robot-related tool deflections and friction-related tracking errors.

To also include effects that do not impact the geometry of the virtual workpiece in the current status of the simulation approach, the virtual workpiece surface is furthermore enhanced with additional process information. This is carried out as part of the material removal calculations, where affected areas are enriched with process variables. Within this scope, these include the special cutting forces and the spatial deflection as well as the path tracking error and axis stop or reversal points of each robot axis. The latter are identified through an additional switching point analysis. For quick access to responsible toolpath areas, the time stamp of the current simulation step is also added to the surface. The resulting virtual workpiece then reflects geometrical as well as non-geometrical effects on its surface and serves as the basis for evaluating the planned machining process and deriving suitable optimization measures.

Following the described information flow, all introduced components are described in more detail within the following subchapters. Finally, an assistance system suitable for process evaluations and optimizations is presented.

3.1. Toolpath Interpolation

The task of the tool path interpolation block is to provide discrete intermediate tool positions TCP_WCS and tool orientations TO_WCS in the WCS for the continuous simulation and display of the advancing machining progress. Traditional simulation approaches rely on an equidistant distribution of these positions to control or standardize the geometric effects of discrete material removal calculations [67,68] or to perform the calculation of tool engagement parameters for subsequent cutting force calculations uniformly [69]. However, a disadvantage of the equidistant distribution is that the dynamic behavior cannot be adequately considered in the simulation.

Since the dynamic behavior is to be included in the simulation process, an interpolation scheme based on velocity settings with continuous acceleration according to [70] is used. The left side of Figure 5 illustrates the resulting dynamic profiles for a linear path segment. The specific velocity control was chosen because it is implemented in modern NC controllers, such as the SIEMENS SINUMERIK 840 D sl of the robot, and therefore closely reflects the step value generation.

The calculation of intermediate points starts with the definition of a velocity profile with predictive speed control [70,71] simply considering global acceleration limits A and D as well as jerk limits J₁ and J₂, which are derived from the NC control. This extends the programmed feed rates F along paths with feed rates f_s or f_e at path transitions. Following the calculation schemes provided by [72], intermediate positions TCP_WCS and orientations TO_WCS can be calculated for any point in time using linear and spherical interpolations and end points of the tool path segments.

TCP_WCS and TO_WCS are then used within the material removal calculations. To overcome the accuracy issues mentioned in [68], Δ_t was set according to the interpolation cycle time of the NC controller (2 ms). Challenges due to varying engagements caused by the nonuniform interpolation along the toolpath are handled by the material removal simulation (see Section 3.3). On an Intel i7-11700 at 2.5 GHz, the computation time of a single intermediate position is less than 0.1 ms.

3.2. Inverse Kinematic Transformation

The task of an inverse kinematic block is to transform the given TCP_WCS and TO_WCS into positions θa,₁, …, θa,₆ and velocities va,₁, …, va,₆ in the ACS of a six-axis industrial robot. To do so, TCP_WCS and TO_WCS are first transformed to the base coordinate system (BCS) of the robot using a zero-point offset transformation T_ZP:

$$\left[\begin{array}{cc}{TO}_{BCS}& {TCP}_{BCS}\\ 0& 1\end{array}\right]=T_{ZP} · \left[\begin{array}{cc}T{O}_{WCS}& {TCP}_{WCS}\\ 0& 1\end{array}\right].$$

(10)

Subsequently, an algorithm is required that performs the inverse kinematic transformation from the BCS to the ACS. The starting point for such an algorithm builds around kinematic transformation $T_{BCS}^{ACS}$:

$$\left[\begin{array}{cc}{TO}_{BCS}& {TCP}_{BCS}\\ 0& 1\end{array}\right]=T_{BCS}^{ACS} · \left[\begin{array}{cc}0& 0\\ 0& 0\\ 1& 0\\ 0& 1\end{array}\right].$$

(11)

$T_{BCS}^{ACS}$ can be derived from joint-specific matrices $T_i^{i-1}$ that perform individual-basis changes from joint i − 1 to joint i:

$$T_{robot}=T_1^0·{ T}_2^1·{ T}_3^2 ·{ T}_4^3 ·{ T}_5^4 · T_6^5{ · T}_7^6=T_6^0· T_7^6.$$

(12)

The matrices $T_i^{i-1}$ are orthonormal matrices, which can be obtained from Denavit–Hartenberg (DH) parameters (θ_i, d_i, a_i, α_i) using the equivalent model shown in Figure 6 [73]. With given positions θa,_1, θa,_2, θa,_3, θa,_4, θa,_5, and θa,₆ the matrices $T_i^{i-1}$ can be parametrized to calculate the corresponding cutting tool positions TCP_BCS and orientations TO_BCS.

3.2.1. Calculating θ_a,1, θ_a,2 and θ_a,3

In the first step, the orientation TO_BCS of the cutting tool is determined. Since 2.5D milling processes are addressed, the orientation is fixed and only needs to be specified once. The matrix corresponds to the coordinate system of the end effector and is set up as follows:

$$T_7^0=\left[\begin{array}{cccc}0& -1& 0& {TCP}_{BCS,x}\\ -1& 0& 0& {TCP}_{BCS,y}\\ 0& 0& -1& {TCP}_{BCS,z}\\ 0& 0& 0& 1\end{array}\right].$$

(13)

The orientation TO_BCS can then be derived via

$${TO}_{BCS}=T_7^0· \left[\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right].$$

(14)

In the next step, the position P₅ of the spherical joint (Figure 6, joint 6) is calculated using a given TO_BCS and TCP_BCS. According to Figure 6, the distances d_7E and d₆₇ are measured and further used for the calculation:

$$P_5={TCP}_{BCS}-{TO}_{BCS} ·d_{7E}+T_7^0·\left[\begin{array}{c}0\\ 1\\ 0\\ 0\end{array}\right] ·d_{67}.$$

(15)

According to the top view shown in Figure 7 θa,₁ can finally be calculated from the distance proportions from joint 1 to joint 5:

$${\theta }_{a,1}=arctan2\left(P_{5,y},P_{5,x}\right).$$

(16)

Here, arctan2 is used to enable the calculation in all quadrants.

$$C_1=P_x-a_{12}, C_2=P_{5,z}-d_{12}, { C}_3=\sqrt{C_1^2-C_1^2} { and C}_4=\sqrt{a_{34}^2-d_{45}^2 ,}$$

(17)

with

$$P_x=\sqrt{P_{5,x}^2+P_{5,y}^2 },$$

(18)

the angles

$${\varphi }_A=arctan2 \left(C_1,C_2\right) and { \varphi }_B=arctan2 \left(D_1,\sqrt{1-D_1^2}\right),$$

(19)

as well as

$${\varphi }_C=arctan2 \left(D_2,\sqrt{1-D_2^2}\right) and { \varphi }_D=arctan2 \left(d_{45},a_{34}\right),$$

(20)

with

$$D_1={\displaystyle \frac{a_{23}^2+C_3^2-C_4^2}{2·a_{23}·C_3}} and D_2={\displaystyle \frac{a_{23}^2+C_4^2-C_3^2}{2·a_{23}·C_4}},$$

(21)

and measured values according to the equivalent model of Figure 6:

$${\theta }_{a,2}={\displaystyle \frac{\pi }{2}}-\left({\varphi }_A+{\varphi }_B\right),$$

(22)

and

$${\theta }_{a,3}=\pi -\left({\varphi }_C+{\varphi }_D\right).$$

(23)

3.2.2. Calculating θ_a,4, θ_a,5 and θ_a,6

The remaining angles θa,₄, θa,₅, and θa,₆ are calculated using the known matrices of the forward transformation. For this purpose, the entries of the transformation matrix $T_6^3$ are derived from the DH transformation matrices:

$${ T}_6^3=T_4^3·{ T}_5^4·T_6^5.$$

(24)

The transformation ${ T}_6^0$ is already given by the target pose. Using this and the angles θa,₁, θa,₂, and θa,_3, the transformation matrix ${ T}_3^0$ can be calculated using matrices from kinematic transformation and solving for $T_6^3$:

$$T_3^0·{ T}_6^3={ T}_6^0 \iff T_6^3={\left({ T}_3^0\right)}^{-1}· { T}_6^0.$$

(25)

Since these matrices are also orthogonal, the transposed matrix can be used instead of the inversed matrix. Thus, the equation can be reformulated as follows:

$$T_6^3={\left({ T}_3^0\right)}^T· { T}_6^0={ T}_0^3 ·{ T}_6^0.$$

(26)

The calculated matrix $T_6^3$ now contains only the angles θa,₄, θa,₅, and θa,₆, as it only describes the transformation from joint 3 to joint 6. When this matrix is with the findings of (24), the three missing angles can then be determined elementwise.

3.2.3. Calculating Axis Velocities v_a,1, v_a,2 and v_a,3

Given the axis angles v_a,1, …, v_a,6, and Δ_t, the angular velocities can be derived from consecutive positions for time steps t − 1 and t:

$$v_{a,i,t}= {\displaystyle \frac{{\theta }_{a,i,t}-{\theta }_{a,i,t-1}}{\mathsf{\Delta }t}} for i=1,\dots ,6.$$

(27)

3.3. Material Removal Simulation

The material removal simulation block has two tasks. The first task is to generate a virtual workpiece model, which encapsulates the necessary process information for downstream process evaluations and optimizations. The second task is to provide cutter engagement information to compute process forces using a cutting force model (see Section 3.4). On an Intel i7-11700 at 2.5 GHz, the computation time of an inverse kinematics transformation is less than 0.1 ms.

Generating a Virtual Workpiece Model

To fulfill the first task, the material removal simulation from [75] is used. The workpiece model is represented by a multi-dexel model according to [67] while cutting tools are represented by their macro geometry and modeled via triangular meshes. The macro geometries are obtained from cutting tool parameters of the CAM system and parameters according to ISO 13399 from cutting tool manufacturers [76].

Due to the discrete workpiece representation and the use of triangle meshes for cutting tools, the simulation can parallelize the calculations for material removal and shift them to modern GPU architectures. In total, this enables a calculation of 500–800 simulation steps per second, so that the simulation runs in real time in conjunction with the selected interpolation scheme (Section 3.1).

In every simulation step, the cutting tool model is positioned based on the given tool position TCP_WCS and cutting tool orientation TO_WCS as well as computed from the previous simulation step. Afterwards, material removal computations are performed. Here, dexels of the workpiece model are pruned, erased, or split into further dexels. An intermediate state of the simulation is shown on the left side of Figure 8. At the same time, newly created ends are assigned with current process information. Both are shown on the right side of Figure 8. After the simulation has finished, the status of the workpiece model is exported as a Virtual Workpiece. Using an RTX 4090 graphics card from the nVidia Corporation, the computation time of a single material removal step is less than 1 ms.

To fulfill the second task, the material removal simulation was extended with cutting-edge descriptions (micro geometry) for each cutting tool as well as cutter engagement maps with chip thickness information. The microgeometry, which means the shape of the cutting edges, was derived from the ISO 13399 parameters published by cutting tool vendors. Using the methodology of [69], mathematical representations for the cutting edges were derived from the given sets of parameters. These representations were then used to generate sets of linear segments with respect to a predefined chordal error. An exemplary result for an inserted cutter is shown on the left-hand side of Figure 9.

To derive local chip thicknesses for each cutting edge, the cutting edges are used to build a map of reachable chip thicknesses for each cutting edge. This is shown on the left-hand side of Figure 10. Between two consecutive simulation steps t − 1 and t, the current positions of the cutting edges are interpolated to create swept volumes, which are then mapped to the surface of the macro geometry of the cutting tool at timestep t, resulting in a Reachable Chip Thickness Map (RCTM) as shown on the right-hand side of Figure 10. The thickness values EM are then compared with the thickness values of the RCTM, resulting in a Chip Thickness Map (CTM) for each cutting edge. The CTM is then sampled.

Afterward, the RTM is subtracted from the EM of timestep t, resulting in a map of the residual material map that is added to the EM of the next timestep t + 1. The whole procedure makes it possible to obtain local chip thickness values for cutting edges in consecutive simulation steps t-1 and t, where the positional distances between TCP_WCS,t−1 and TCP_WCS,t−1 vary and are different from the programmed tooth feed. The thickness values are then handed to a force model for predicting the cutting forces. Using an RTX 4090 graphics card from the nVidia Corporation, the computation time of 20 engagement computations is less than 1 ms.

3.4. Cutting Force Model

The task of the cutting force model block is the computation of cutting forces, given the linear segments of the cutting edges as well as local chip thicknesses for each point in the edges. The computation used a linear cutting force model from [77], which was parameterized from orthogonal cutting data using [78] for each individual cutting edge geometry:

$$f_i=b ·K_i\left(h\right),$$

(28)

with

$$K_i\left(h\right)=k_{i,c }·h+{ k}_{i,e} and i\in \left\{r,a,t\right\},$$

(29)

and cutting force coefficients k_i,c and k_i,e.

For force computing, the set of linear segments is subdivided into evenly sized disks (Figure 11, left-hand side). For each disk, an average chip thickness is computed using the given chip thicknesses, and the force model is applied. The resulting axial, radial, and tangential force components f_a, f_r, and f_t are then combined with the local radial, axial, and tangential directional vectors F_r, F_a, and F_t (Figure 11, right-hand side). The resulting vectors are then integrated over all disks, resulting in a force vector in the WCS. Transformed into the robot’s coordinate system, the force vector is then used to compute force-induced cutting tool deflections caused by the robot with a structural mechanical model.

3.5. Structural Mechanical Robot Model

The structural mechanical model is based on the findings of [26] as introduced in Section 2.1 and computes a position-dependent, cartesian robot receptance for given axis positions θa,₁, …, θa,₆. Combined with additionally provided cutting forces, it further outputs force-induced cutting tool deflections that are added to the tool center point position of the subsequent simulation step. Furthermore, the force-induced tool deflection is sent to the material removal simulation where it is added to the virtually machined surface in the subsequent simulation step.

3.6. Friction Robot Model

The friction model computes a path tracking error based on given axis positions θa,₁, …, θa,₆ and axis velocities va,₁, …, va,₆. The tracking error is then added to the tool center point position of the subsequent simulation step. Furthermore, the tracking error is sent to the material removal simulation where it is added to the virtually machined surface in the subsequent simulation step. Friction is a highly complex phenomenon that is difficult to model and depends on many factors such as velocity and load, but its effects on the tracking error are crucial. Therefore, an accurate model is necessary. A suitable candidate is the grey-box LuGre model, as described in Section 2.2, which outperformed the other friction models. Since it has only been evaluated for the first axis of the robot and velocity-dependent friction, the model should be extended to include all axes as well as load-dependent friction and integrated into the simulation approach.

3.7. Switching Point Analysis

The task of the switching point analysis is to provide process indicators when an axis of the industrial robot stops or switches its direction. To do so, the time series of axis velocities are evaluated for time steps with zero velocity or sign changes. One or multiple occurrences are binary encoded by the joint number in an integer value that is handed over to the material removal simulation. Here, it is added to the virtually machined surface in the subsequent simulation step. Otherwise, a zero integer is handed over [79].

3.8. Assistance System for Process Evaluation

Based on a generated virtual workpiece, the assistance system in Figure 12 was developed. The virtual workpiece is visualized in a 3D environment and can be evaluated by an experienced CAM planner. To support the evaluation, the assigned process information can be visualized on the workpiece model’s surface using a variable color scale. In Figure 12, this is shown for force-induced deflections as well as cutting forces and axis switching points.

Furthermore, it is possible to select workpiece surface areas manually or use definable point clouds to gather a set of assigned timestamps. This provides the prerequisites for effective process optimizations.

4. Evaluation of the Simulation Planning Assistance

The effectiveness of the overall approach was validated by manufacturing the component shown in Figure 13.

The results are shown in Figure 14. As it turns out, the least force-induced positional deviation can be achieved with pose 1 (20 µm) and pose 2 (25 µm). Pose 3 leads to medium deviations (35 µm) while the highest deviation appears in pose 4 (25 µm).

In total, 23 test criteria were measured for each part according to the three manufacturing features circle, square, and slope defined by ISO 10791-7 [ISO]. ${}^z\uparrow M$ denotes the height of a feature, Ø is the diameter, and O is the roundness of the circle feature, respectively. $⟷$ is the length and − is the straightness of a linear feature, respectively. Finally, $\mathrm{\angle }$ is the angle between two straight lines. The bold values in

Table 2. Comparison of workpiece accuracy for four poses according to ISO 10791-7 [76]. Deviation values in bold adhere to the tolerance bounds. The colors dark green to red indicate the best to worst values when comparing the four poses regardless of tolerance bounds.

	Nominal (mm)	Tolerance	Deviation (mm)
	Circle		Pose 1	Pose 2	Pose 3	Pose 4
${}^z\uparrow M$	Z6	0.020	−0.1017	−0.0673	−0.0848	−0.0557
Ø	∅218	0.020	0.2342	0.2464	0.3099	1.1677
O	∅218	0.020	0.5492	0.5844	0.5596	0.8639
	Square
${}^z\uparrow M$	Z6	0.020	0.0142	−0.0041	0.0186	−0.0245
$⟷$	2 × 220	0.020	0.2520	0.3611	0.8089	1.6140
$-$	4 × 220	0.015	0.0095	0.0125	0.0398	0.0144
$\angle$	30°, 60°	0.020	0.1451	0.2661	−0.2688	0.1252
	Slope
${}^z\uparrow M$	Z6	0.020	0.0045	0.0161	0.0154	0.0633
$-$	320	0.015	0.0138	0.0195	0.0268	−0.3193
$\angle$	3°	0.020	0.1354	−0.3675	−0.2226	0.0430

are within the tolerance bounds. As can be seen, this is only true for a few values. When comparing the deviations of the four poses, it is observable that poses 1 and 2 perform better than poses 3 and 4, with minor exceptions. This confirms the predictions made by the simulation tool.

5. Discussion

Despite the optimal design of the robot-specific production parameters, it makes sense to use control technology and in-process compensation to increase production quality. Preliminary investigations at the Laboratory for Machine Tools and Production Engineering and other research institutes have shown that online compensation strategies can significantly improve the machining results [79]. Furthermore, the focus should lie on the compensation of disturbances in path accuracy due to process-independent variables such as friction, gear backlash, and stick-slip effects. Therefore, it is necessary to develop a control technology concept to reduce the aforementioned effects.

A model-based pre-control of the joint torques can be selected as the approach. Based on the developed model and the possibility of calculating acting joint torques because of the process and machine properties, compensatory torque curves can be calculated for the respective machine axes. In the current literature, there are various approaches and strategies for eliminating the effects that cannot be compensated for in conventional axis control with the aid of feedforward control. A control engineering approach should be developed with the developed model that can be integrated into a robot controller due to the required computing times and computing capacities. An approach to the control compensation of process-independent influencing variables on the path accuracy of an industrial robot should be aimed for.

Despite extensive planning assistance, not all machining errors can be compensated for with optimized path planning. The process forces may lead to a strong displacement of the TCP and effects such as switching points and gears with backlash cannot be eliminated without additional compensation in the process. An approach to a compensation strategy using offset correction values calculated offline is described in [80]. The simulated deviations from the previously defined CAM modules can be written directly into a table readable by the controller via an interface for each IPO cycle and queried by the controller during operation. In addition, the effects that are not compensated for by an offset correction (e.g., switching points and gears with backlash) can be reduced by torque pre-control.

For broad application, a standardized concept must be developed that can be applied to several control architectures. In addition, the desired objectives favor user-friendliness, as the compensation information is already formed from the CAM functionalities and can subsequently be further used by the user.

6. Conclusions

In this paper, an approach was developed to increase the accuracy of industrial robots in highly dynamic machining processes. The machining task and the associated manufacturing program or robot paths are of decisive importance. In addition, process forces can occur that displace the robot’s end effector. The simulation of and compensation for the various process influences are therefore necessary to ensure stable machining by optimizing the process parameters during production planning and processing. In previous works, the dynamics and stiffness models of the MABI MAX 100 robot were identified.

In the first part of the paper, a grey-box LuGre model was developed to combine the advantages of analytical and data-driven approaches and to model the missing frictional effects of the robot. The model outperformed white- and black-box models by achieving the lowest RMSE and best R² values, respectively. Since the model has only been developed for the first axis of the MABI robot and velocity-dependent friction, further works will focus on including the other axes as well as load-dependent friction in order to develop a complete grey-box friction model.

In the second part of the paper, a holistic model-based simulation planning assistance tool was developed and validated that uses the aforementioned models as input. This tool focused on identifying and optimizing critical robot paths and switching points. During the development of the solution approach, special attention lay on the transferability to the heterogeneous industrial control landscape. The simulation assistance was validated on a machining process and succeeded in increasing the process accuracy. In addition to the evaluation of a compensation strategy, the authors suggest the integration of a pose-dependent stiffness model of the robot. Further works will focus on integrating and evaluating the complete grey-box friction model in the simulation tool and experimental validation of the expected accuracy optimization for robotic machining processes.