Keywords

1 Introduction

The range of possibilities and expectations for robot applications in machining and pre-machining tasks is reflected in the large collection of reviews and papers in the specialized literature [1,2,3]. The major challenges facing machining with robots versus machining with machine tools remain today [4] the characterization of robot stiffness and robot configuration [5], path planning and dynamics, vibration during machining [6] and robot deformation and compensation [7].

One way to confirm whether a robot is capable of performing a machining operation, minimizing the use of expensive tests, is through an efficient model that simulates its dynamic behaviour. For this reason, a robot model has been developed using the multibody method with mixed natural coordinates (MBSmc). As a result of this application, a trajectory and its deviation as a function of the process forces can be simulated.

2 Methodology

For the modelling of the robot, the MBSmc method has been selected [8]. The MBSmc formulation allows to apply driving forces, torques and to evaluate positioning errors in each of the links. This results in relating the angular response of the link motion to its corresponding drive.

In this application of the MBSmc method, the joints are considered as flexible solids. The scheme of the proposed model is shown in Fig. 1.

Fig. 1.
figure 1

Model for the robot milling system.

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Since the main deformations are located at the joints (compared to the stiffness of the links), the direction of the forces and torques that will play out during the execution of the applied machining have a relevant influence, as well as the configuration of the robot.

3 Robot Model

The model presented is a simulation model of the robot’s behaviour that takes into account the forces of the milling process. For this purpose, it interacts with the process based on the estimation of how the evolution of the forces evolves with the robot’s deviations and, therefore, with the modification of the machining parameters.

In this way, the deviation of the robot path δrobot is defined as the difference between the nominal movement of the TCP and its real position (Eq. (1)).

This variation is caused by deviations in the positioning of the robot and by deviations caused by deformations of the robot as a consequence of the forces and torques generated during the process being performed. The calculation of the estimation of this deviation can be done by simulating the behaviour of the robot, determining the deviation as the difference between the nominal and simulated trajectories or positions.

$${\delta }_{robot}={q}^{TCP}-{q}_{real}^{TCP}$$
(1)

The position q for all points and vectors of the robot (see Fig. 2) is modelled using the MBSmc method. The robot motion is obtained with a trajectory generated by a discrete set for each programmed robot position through master positions. All robot positions must fulfil the function of the non-linear constraint equations for each instant t (Eq. (2)).

By first performing the function of Eq. (2) in the inverse dynamics and then using the results of the inverse dynamics in the direct dynamics, the coordinates of the robot’s TCP displacement are calculated for each trajectory configuration.

$$\Phi \left({q}_{sim},t\right)=0$$
(2)
Fig. 2.
figure 2

Multibody model of the robot with mixed natural coordinates.

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In the formulation of the simulated rotation angles Ψsim (Eq. (3)), the dependence of the theoretical rotation angles Ψ and the term φ [8] is proposed. The torques τ that each motor produces at its joint are related to its angular stiffness k (Eq. (4)). The negative sign explains the direction of the torques from machining, which are the opposite of the motor torques.

$${\psi }_{sim}=\psi +\Delta {\psi }_{Kpos}+\varphi $$
(3)
$$\varphi =-\frac{\tau }{k}$$
(4)

In order to know which are the torques τi that the motors produce in each joint as a consequence of the process forces that are playing, it is necessary to solve the inverse dynamic problem, using the virtual method power (Eq. (5)).

$$\tau ={R}^{T}InR\ddot{\psi }-{R}^{T}InSc-{R}^{T}Q-{\tau }_{mach}$$
(5)

Operating with the direct dynamic model, the corrected angular accelerations of the joints are obtained. By applying ODE-type integration methods, the angles and their corrections due to the gain at each position of the trajectory are calculated (Eq. (6)).

$${\left\{{\ddot{\psi }}^{T}+\Delta {\ddot{\psi }}_{Kpos}^{T},{\dot{\psi }}^{T}+\Delta {\dot{\psi }}_{Kpos}^{T}\right\}}_{t}\stackrel{ode113}{\to } {\left\{{\dot{\psi }}^{T}+\Delta {\dot{\psi }}_{Kpos}^{T},{\psi }^{T}+\Delta {\psi }_{Kpos}^{T}\right\}}_{t+\Delta t}$$
(6)

With the simulated angles Ψsim, the natural coordinates for all the points of the robot at each instant t of the trajectory are generated using the cubic spline method.

4 Interaction Between the Robot and the Process

Once the robot has been modelled, the process forces are introduced. In this way, the process is modelled taking into account the influence of the process forces and the influence of the robot’s deformations on them.

The deviation of the robot in the direction of motion affects the instantaneous cutting edge feed rate and thus the average chip thickness. Robot deflections in the transverse direction will affect the cutting width, the engagement angle and the chip thickness. The robot deviation in the perpendicular direction to the robot base modifies the axial depth of cut.

The fact that the cutting forces are modified means that the reaction of the robot will be different. Therefore, it must be known to what extent the values of the machining forces will be modified, for which a complete model of the process has been chosen as described in [9] for peripheral end milling operations.

The cutting force in the tangential direction is expressed according to Eq. (7). Analogously, the forces in the radial and axial directions can be obtained.

$${F}_{c}\left({\varphi }_{j}\right)={k}_{c0}\cdot {a}_{p}\cdot {\left[\frac{1}{{\varphi }_{j}-\left({\varphi }_{j}-{\varphi }_{pr}\right)} {f}_{z} \left(\mathrm{cos}\left({\varphi }_{j}-{\varphi }_{pr}\right)-\mathrm{cos}{\varphi }_{j}\right)\right]}^{1-{m}_{c}}$$
(7)

The corrections applied to each of the force components are a function of position.

The calculation of the feed rate fzi at each point on the path takes into account the nominal feed rate fz, the deformation of the robot in the feed direction at that point and what the previous edge at that point did not cut, according to Eq. (8).

$${fz}_{i}=fz+{\delta }_{robot i}+{\delta }_{prev\,cutting\, edge\, i}$$
(8)

With respect to the cutting width aei, also for each position of the path, an analogous expression is used (Eq. (9)) where the cutting width ae is affected by the effect of the transverse deformation of the robot at that point.

$${a}_{ei}={a}_{ei}+{\delta }_{robot\,i}$$
(9)

In both cases, the calculation of the robot deformation at each position takes into account the stiffness of the joints and is simulated by means of the direct dynamics along the entire path with the torques obtained from the inverse dynamics when the process forces are introduced.

Once the corrections have been made and fzi and aei have been obtained, the corrected forces for each point are obtained (Eq. (10)).

$${Q}_{sim\,forces}={Q}_{nom\,forces}\cdot \frac{{fz}_{i}}{fz}\cdot \frac{{ae}_{i}}{ae}$$
(10)

The designed algorithm calculates, individually for each point, the inverse and direct dynamics with the corrected forces. This process is repeated for each point until the result converges and the deviation of the robot can be obtained (Eq. (1)). The procedure is applied to each position in the trajectory.

5 Results

The modelled (see Fig. 3) and measured (see Fig. 4) milling forces applied at the TCP are shown, together with their correction by the model, in the transverse direction of the feed. The effect of the forces on the motion is shown in Fig. 5. As a consequence of the deformations of the robot and what the previous cutting edge has not cut, the process forces decrease. This deviation varies as the cutting edge rotates.

Fig. 3.
figure 3

Modelled (blue) and corrected (green) transverse forces at the TCP.

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Fig. 4.
figure 4

Measured (blue) and corrected (green) transverse forces at the TCP with runout.

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Fig. 5.
figure 5

Movement for the TCP due to measured forces (blue) and their corrections (green) with runout.

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6 Conclusions

From the results obtained it is concluded that:

  • The model allows knowing the deviations of the TCP when interacting with the machining forces.

  • The behaviour of the robot reflects the evolution of the variable forces in milling operations.

  • The corrections are calculated from the configuration of the robot and the machining at each position of the cutting edge, taking into account the situation of the previous cutting edge in the same position.

  • The established procedure makes it possible to predict the deviations of the robot, depending on its mechanical characteristics.