# Elasto-geometrical modeling and calibration of redundantly actuated PKMs

Abstract: Redundantly actuated parallel kinematic machines (PKMs) offer a number of advantages compared to classical non-redundant PKMs. Particularly, they show a better stiffness thanks to singularity avoidance and they have an improved repeatability due to a better behavior against backlashes. The main problem with the calibration of these machines is that the redundancy leads to some mechanical strains in their structure. This makes it difficult to identify the geometrical errors of their structure without taking into account the effects of the elastic deformations. The main originality of this work is to propose an efficient elasto-geometrical and calibration method that allows the identification of both the geometrical and stiffness parameters of redundantly actuated parallel mechanisms with slender links. The first part of the paper explains the proposed method through its application on a simple redundant planar mechanism. The second part deals with its experimental application to the redundant Scissors Kinematics machine.

## Summary (6 min read)

### 1 Introduction

- Redundantly actuated Parallel Kinematic Machines (PKMs) have recently attracted interest of researchers because they allow the reduction of some ∗ Corresponding author.
- Whereas for classical PKMs am insufficient knowledge of the mechanism geometrical properties, such as link length or joint position/orientation, leads exclusively to Cartesian position inaccuracies at the Tool Center Point (TCP) [9], in the case of redundant PKMs, such errors also lead to internal constraints.
- First, the proposed methodology that is used to derive the elasto-geometrical models of parallel mechanisms with one or more actuated redundant chains is described for planar mechanisms.
- Then, the calibration strategy that has been used to perform the geometrical and stiffness parameter identification of the obtained models is explained.

### 2.1 Method description

- The high dynamics of PKMs suppose low moving masses, i.e., slender elements and light joints [11] [12], which are then subject to elastic deforma- tions.
- For calibration purposes, these elastic deformations that depend on the PKM configuration [13] have to be calculated in order to be compensated.
- For this purpose, an analytical finite-element modeling using beam elements is proposed to describe redundant PKMs.
- The analytical finite-element method allows a more accurate calculation of the platform situation because all possible deformation effects are taken into account [18],[19].
- Calculation of all stiffness matrices into the global reference frame of the structure.

### 2.2 Elasto-geometrical modeling of redundant planar mechanisms

- In this section the description of the method is presented for planar mechanisms.
- For this purpose, the values of the geometrical parameters are considered as nominal, that is to say without any errors.
- As a result, the position and orientation of the platform associated frame is identical and can be calculated with any non-redundant subsystems of the structure.

### 2.2.1 Modeling of the structure links

- The slender links of the planar structures are considered as rods that can be described by planar 2-node beams.
- The stiffness of each two-node beam is first expressed in the beam local reference frame ℜij as [20]: Kij = . (3) L denotes the element’s length, S its cross-section area, Iz its quadratic moment along the z-axis and E the Young’s modulus of its material.
- The wrench of external actions applied on node i is Fi = (fx,i fy,i mz,i) T, where fx,i and fy,i are the forces along x and y and mz,i is the moment around z, in the local beam axes.

### 2.2.2 Modeling of the structure joints

- The solution that is used here to describe a structure with joints is to consider them as beam elements with coincident nodes.
- The parameters kx and ky stand respectively for the radial stiffness coefficients along x and y-axes.
- A passive revolute joint of axis z would be described with a beam element with a very small stiffness value for the rotation around the z-axis (krz ≈ 0) and a high stiffness value along the other directions.
- The small and high stiffness coefficients must be chosen so that they are as far as possible from the other stiffness coefficients and that the numerical accuracy is still valid.
- Its advantages reside in the ease of implementation and in particular the fact that its associated matrix is positive definite.

### 2.2.3 Mapping and assembly of all stiffness matrices

- During this step, the stiffness contributions of all links and joints are integrated into a global stiffness matrix that describes the whole structure stiffness.
- For this purpose, the local stiffness matrices of all the elements have to be expressed within the global reference frame gRij 03 03 gRij where gRij is the rotation matrix from the local frame.
- It is to be noted that the elastic deformations that are induced by the structure’s own weight are considered by reporting the weight of each link to its two associated nodes and then by merging the resulting equivalent efforts to the vector of external actions F g.

### 2.2.4 Modeling of links that are connected to the base or the ground

- The proposed method requires differenciating between the modeling of the passive and actuated links that connect the PKM structure to its base or to the ground.
- To illustrate this, one can consider the structure of Fig. 2 for which the length of the actuated redundant leg varies thanks to the actuator q3.
- To relate those conditions, the equations associated to the translational displacements of node 8 along x and y are removed from the equation system (7). .
- In a context of calibration, the modeling of the structure’s actuated links that are connected to the base or to the ground is done by considering that the elements related to them are fixed to the base (as for example the carriers q1 and q2 on the structure of Fig. 2).
- However those elements can still be considered as flexible and thus they will be described by using a 1-node beam element.

### 2.3.2 Modeling of Triglide links and joints

- The model of the Redundant Triglide structure and the numbering of the corresponding nodes are described in Fig.
- The link (for example between nodes 1 and 2) and the carriers (for example the element linked to node 3) are represented by one beam element each as explained in Section 2.2.4.
- These elements are linked together by 5 revolute joints to which stiffness coefficients are assigned.
- The axial rotational rigidity coefficients of all passive revolute joints is considered to be small enough to have no influence on the deformations of the Triglide structure.
- As a result, the two joints that correspond to the TCP (linking nodes 1 and 4 on one hand and nodes 1 and 7 on the other hand) are modeled as ideal revolute joints, whereas the revolute joints between the carriers and the rods have a given radial stiffness.

### 2.3.4 Modeling of the links connected to the base

- All links connecting the mechanism structure and its base are actuated.
- The three carriers are described by 1-node beam elements which means that no boundary conditions need to be applied for the definition of Kr.

### 2.3.5 Calculation of all nodal displacements

- The TCP displacement due to elastic deformations is ∆Xe = dx1 dy1.
- The node rotational displacement around the z-axis is not considered.
- The final TCP position is calculated through the resulting forward elastogeometrical model X = fgm(qnr, ξ) + ∆Xe.

### 2.4 Generalization of the proposed method to three-dimensional mechanisms

- The extension of the elasto-geometrical modeling method is presented for three-dimensional mechanisms.
- For this purpose, the nodal force and displacement vectors are modified to consider the six degree-of-freedom as follows: Fi = (fx,i fy,i fz,i mx,i my,i mz,i) T Ui = (dx,i dy,i dz,i rx,i ry,i rz,i) T (16) where fz,i is the force along z and mx,i and my,i are the moments around x and y, in the local beam axis.
- The displacements are dx,i, dy,i, dz,i, for the local displacements along x, y and z respectively, and rx,i, ry,i and rz,i, for the local rotations around these same axes.
- For the modeling of the structure bodies, the 12× 12 stiffness matrix that is used to describe a 2-node link between the nodes i and j is: Kij =.
- The difference with the modeling of planar mechanisms is that the submatrix Kdkl must be extended as Kdkl = diag(kx, ky, kz, krx, kry, krz).

### 3 Elasto-geometrical calibration of redundant PKMs

- In the first part of this section the elasto-geometrical modeling method that has been proposed previously is modified to be used for calibration.
- This leads to the error model that will be involved for the identification of both geometrical and stiffness parameters of the PKM structure.
- In the second part of the section, the whole calibration methodology will be illustrated by using again the Redundant Triglide.
- Qset are the actuator set values, ∆X is the vector of measurement errors obtained by the difference between Xmod and Xmeas, J ∗ is the pseudo-inverse of the Jacobian matrix of the calibration and ξcal is the vector of calibrated parameters.

### 3.1 Error model for the calibration of redundant PKMs

- In order to derive the calibration error model, the situation of the PKM platform has to be calculated with both nominal and real parameters, respectively ξnom and ξ = ξnom+∆ξ.
- The consideration of some geometrical errors ∆ξ leads to the fact that the nodes of some joints involved in the elastic modeling are not coincident anymore and the proposed elasto-geometrical method has to be modified.
- To illustrate this problem with the Redundant Triglide, the authors first consider that the position of all actuators is calculated for a given TCP position with the nominal geometrical parameters.
- If some geometrical errors exist, the distance between the two nodes of some joints is non-null before application of the external forces and they will behave as beam elements with an initial length δ0.
- Their node displacements calculated through the resolution of (7) will not fit with reality.

### 3.1.1 Method of joint internal forces

- The proposed solution to solve the problem of non-coincident joints consists in the following steps: (1) Calculating the platform position/orientation with the forward geometrical model Xnr = fgm(qnr, ξ) of any non-redundant substructure of the machine where qnr is the actuator position vector of the non-redundant subsystem.
- (2) Calculating the position of the nodes of the redundant link(s).
- The nominal geometrical parameters ξnom have to be used for this calculation and the distance between two nodes of a joint has to be minimal.
- For step 3, the joint stiffness is given by (5) and, since the distance δ0 between the two nodes 8 and 9 is non null (Fig. 9), an internal force proportional to the distance has to be applied and added to the right-hand side of the equation system as follows: F ′89 = F89 + K89.
- For step 4, the system (7) is solved to derive all nodal displacements and then the position of the Triglide’s TCP (Fig. 10).

### 3.2 Global Jacobian matrix for the calibration

- In order to perform the calibration, the local Jacobian matrix Ji that gives for a configuration i the relationship between the variations of the geometrical/stiffness parameters and the variations of the PKM platform situation is calculated as follows: Ji =.
- Since the finite-element method that is used to derive this forward elasto-geometrical model (fegm) requires a numerical solution of the equation system (7), those partial derivatives are calculated as a finite-difference of (11).
- Then, the 6m× p global Jacobian matrix J that will be involved further for the sensitivity and observability analyses is obtained as the concatenation of the local Jacobian matrices Ji calculated for the m configurations of measurement.

### 3.3 Sensitivity and observability analyses

- The range in which each of those parameters can vary can be set based on considerations related to the machine part manufacturing and assembly.
- Js from which only the columns of the selected parameters are kept.
- In order to calculate the global Jacobian matrix J and to perform the sensitivity and observability studies, 231 measurement points are taken over the entire workspace.
- The angle γ is the most influent parameter for both redundant and non-redundant mechanisms.
- The calibration of the non-redundant PKM will then tend to be more stable and accurate.

### 3.4 Identification

- Simulations were then carried out for the Redundant Triglide.
- The simulated measured TCP positions along the x and y-directions were obtained through the forward elasto-geometrical model with the real geometrical/stiffness parameters ξ. Figure 12(a) shows the mean final parameter error for the calibration of both mechanisms with respect to the number of measurement points (a Gaussian noise with a standard deviation of 10µm is added to each measurement point).
- The mean TCP positioning accuracy was calculated for a set of 10 points taken within the workspace.

### 4.1 Description and elasto-geometrical modeling of the Scissors-Kinematics machine

- The method that has been proposed for the elasto-geometrical modeling of redundant PKMs has been tested on the Scissors-Kinematics machine developed at the Fraunhofer Institute for Machine Tools and Forming Technology IWU in Chemnitz for tool and die machining [27].
- The elasto-geometrical modeling method that has been developed so far involves mechanisms with one redundant branch.
- Its application is also suitable for mechanisms with a higher order of redundancy and, therefore, it has been applied to the Scissors-Kinematics architecture.
- The Scissors-Kinematics redundant parallel structure includes a moving plat- form, four linear actuators along the y-axis and five fixed-length rods (Fig. 14(a)).

### 4.2 Elasto-Geometrical Modeling Validation

- For several TCP positions along the x-axis a variation was applied on one of the parameters, the new actuator positions were calculated with this modified parameter set and all actuators were driven to these positions.
- The scanning head is mounted directly in the spindle and the plate is fixed on the machine table.
- The TCP displacements are too small to be measurable.
- The average deviation between the measured TCP displacements and those obtained by the elasto-geometrical model for all parameters and over the whole x-movement is 14 %.
- For the redundant part, it is 100 % because these parameters are not taken into account in the model.

### 4.3 Elasto-geometrical calibration validation

- The calibration of the Scissors-Kinematics was carried out using the gridencoder to measure 80 TCP positions over the whole workspace.
- They do not appear in the list of parameters to be identified because of the column normalization of the Jacobian matrix used during the optimization process.
- The mean position difference for the redundant actuators between the results of the forward geometrical model with the calibrated parameters and the measurements are ∆q2 = 0.873 mm and ∆q3 = 0.305 mm for actuators 2 and 3, respectively.
- The measurement data are then excluded as being the cause of the discrepancy between the results of the elasto-geometrical calibration and the real mechanism.

### 5 Conclusion

- The external forces and the internal constraints linked to the actuation redundancy.
- The method uses a partly analytical finiteelement analysis based on beam elements, so that it is quick enough for an on-line implementation.
- This method can also be applied for the study of the influence of machine parameter errors on the TCP, thus it is adapted for the calibration of such mechanisms.
- The calibration simulations revealed the complementary facts that redundant PKMs are more robust to parameter errors and that for this reason they are more difficult to calibrate than the classical non-redundant PKMs.
- A self-calibration strategy could be used where the redundant actuators would act as extra measuring systems or extra sensors could be used on passive joints.

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