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CAD-based approach for identification of elasto-static parameters of robotic manipulators

Alexandr Klimchik, Anatol Pashkevich, Damien Chablat

arXiv:1311.6685v1cs.RO

TL;DR

Accurate stiffness identification matters because manipulator stiffness is tied to design accuracy, while prior FEA-based identification still leaves experiment-setting and numerical-processing questions. The paper develops CAD-based identification of link stiffness matrices and reports about 0.1% accuracy in evaluating their elements.

  • Problem

    FEA-based stiffness identification still has open questions about virtual-experiment settings and numerical algorithms, despite stiffness being tied to manipulator design accuracy.

    Source: §1 · pp. 1, 3

  • Method

    The paper develops a CAD-based approach that identifies link stiffness matrices from FEA virtual experiments and numerical processing.

    Source: §3 · p. 5, §4.1 · p. 5, §4.3 · p. 8

  • Results

    About 0.1% accuracy in evaluating stiffness matrix elements is reported for the developed approach.

    Source: §8 · p. 17

  • Takeaways & Limitations

    The identified matrices account for real link shape, coupled rotational and translational deflections, and joint particularities within a VJM model used for computational efficiency.

    Source: §1 · p. 2, §8 · p. 17

    FEA modeling errors and nontrivial modeling-option choices affect accuracy, although the paper reports that a practically acceptable 0.1% level can be achieved with standard computing facilities.

    Source: §6.3 · p. 15

Abstract

from arXiv · show

The paper presents an approach for the identification of elasto-static parameters of a robotic manipulator using the virtual experiments in a CAD environment. It is based on the numerical processing of the data extracted from the finite element analysis results, which are obtained for isolated manipulator links. This approach allows to obtain the desired stiffness matrices taking into account the complex shape of the links, couplings between rotational/translational deflections and particularities of the joints connecting adjacent links. These matrices are integral parts of the manipulator lumped stiffness model that are widely used in robotics due to its high computational efficiency. To improve the identification accuracy, recommendations for optimal settings of the virtual experiments are given, as well as relevant statistical processing techniques are proposed. Efficiency of the developed approach is confirmed by a simulation study that shows that the accuracy in evaluating the stiffness matrix elements is about 0.1%.

1 Introduction

Reducing manipulator mass can improve dynamic performance, but stiffness constrains that reduction because it is tied to design accuracy. The paper addresses unresolved choices in FEA-based link-stiffness identification while combining FEA accuracy with the computational efficiency of VJM models.

1 Introduction

  • Stiffness constrains manipulator mass reduction because it is directly related to robot accuracy defined by design specifications.Reducing moving masses is pursued to achieve high dynamic performance with relatively small actuators and low energy consumption.Source: §1 · p. 1
  • FEA-based link stiffness identification can supply VJM and MSA models while avoiding intensive computations across different manipulator configurations.The cited hybrid approach applies FEA to each link rather than repeatedly recomputing a whole-manipulator FEA for each posture.Source: §1 · pp. 2–3
  • Prior link models often used regular-beam approximations and decoupled one-dimensional springs, while complex-shaped links require accurate non-diagonal stiffness matrices.The non-diagonal matrix represents translational and rotational compliance and coupling between deflections.Source: §1 · p. 2
  • Open issues include choosing virtual-experiment settings and improving numerical algorithms' robustness to FEA errors and their handling of zero versus small matrix elements.The paper identifies these issues as underexplored in prior work.Source: §1 · p. 3

2 Problem statement

Simple beam approximations can be inaccurate for links with complex geometry, motivating stiffness identification from CAD-based virtual experiments. The paper develops a workflow and numerical processing approach, with planned assessment of experiment settings, error reduction, and manipulator examples.

2 Problem statement

  • For the Orthoglide foot, even a four-beam approximation gives only 30–50% accuracy, illustrating the difficulty of modeling complex link shapes with beams.The paper notes that industrial links may be nonhomogeneous, complex-shaped, and have non-constant cross-sections.Source: §2 · p. 3
  • The workflow models an isolated CAD link, fixes one end, applies force or torque at the other, and extracts reference-point-neighborhood deflections from repeated FEA experiments.The extracted deflection field is processed to obtain the link stiffness matrix.Source: §2 · p. 3
  • Modeling joint elasticity and contact-area compliance can reduce some stiffness-matrix elements by a factor of 10–12 in the paper's case study.The described approach incorporates joint particularities that analytical expressions do not account for.Source: §2 · p. 3
  • The stated aims include developing FEA-based identification that accounts for link shape, coupled deflections, and joint particularities, plus a numerical matrix estimator from extracted points.The method's accuracy is to be evaluated across virtual-experiment settings and virtual-sensor size and shape.Source: §2 · p. 4
  • The paper also targets statistical reduction of identification errors and validation on typical parallel-manipulator examples against regular-shape approximation models.The planned statistical processing includes outlier elimination, confidence intervals, and detection of zero stiffness-matrix elements.Source: §2 · p. 4

3 FEA-based approach for identification of link stiffness matrix

The approach identifies link stiffness from virtual CAD/FEA loading experiments, combining measured deflections across load directions to estimate a matrix that can represent translation–rotation coupling. It targets links whose complex shapes make primitive-based stiffness estimates inaccurate.

3 FEA-based approach for identification of link stiffness matrix

  • FEA experiments apply different force and torque directions to an isolated elastic link, and their deflections are combined to estimate its stiffness or compliance matrix.At least six experiments are used; with more than six, the matrix inverse is replaced by a pseudoinverse.Source: §3 · pp. 4–5
  • The 6×6 stiffness matrix relates small translational and rotational displacements to static forces and torques, with off-diagonal terms representing their coupling.Its inverse is the compliance matrix.Source: §3 · p. 4
  • For sophisticated link shapes, primitive-based approximations have relatively low accuracy, motivating FEA-based stiffness estimation for individual links.The approach builds on computing manipulator stiffness from the stiffness matrices of its separate components.Source: §3 · p. 5
  • Increasing the number of virtual experiments can reduce estimation error, but classical optimal experiment design cannot be applied directly because virtual-experiment errors cannot be modeled as additive terms.The paper notes that some aspects of this design problem are studied in Section 5.4.Source: §3 · p. 5

4 Numerical technique for evaluating the stiffness matrix elements

The section estimates link translation and rotation from FEA displacement fields by fitting a rigid transformation. It presents exact SVD and linearized alternatives, with the linearized method offering computational savings and simplifications for regular point patterns.

4.1 Related optimization problem

  • Deflections are estimated by fitting a rigid transformation to nearby FEA nodes, treating their positions and displacements as a displacement field.The fit minimizes squared residuals subject to an orthogonal rotation matrix.Source: §4.1 · pp. 5–6

4.2 SVD-based solution

  • The exact solution uses an orthogonal Procrustes procedure to estimate rotation and translation, then extracts rotation angles from the rotation matrix.The rotation-angle extraction formulas are formally equivalent but may differ in robustness to round-off error.Source: §4.2 · pp. 6–7

4.3 LIN-based solution

  • Linearizing the rotation matrix avoids SVD, and the resulting expression is more computationally efficient than the SVD-based expressions.For some regular FEA meshing patterns, the expression can be simplified further.Source: §4.3 · pp. 7–8

4.4 Analytical expressions for typical case studies

  • Regularly distributed points allow analytical inversion and simple deflection expressions for symmetrical, uniformly meshed cubic, and planar-square fields.For the cubic field, the matrix is a scalar multiple of the identity under the stated symmetry and uniform-meshing conditions.Source: §4.4 · p. 8

4 Numerical technique for evaluating the stiffness matrix elements

  • The proposed technique estimates translational and rotational deflections simultaneously from an FEA displacement field using a computationally attractive numerical procedure.The section evaluates its precision and robustness in the following section.Source: §4 · p. 5, §4.4 · p. 8

Accuracy of the stiffness matrix identification

Identification accuracy depends on FEA discretization and numerical errors, while deflection amplitude and sampled-point geometry affect the error estimates. Simulations support rotational deflections of 0.01–0.2° for about 0.2% identification accuracy, with translation errors governed by FEA noise.

5.1 Error sources and their impact

  • FEA discretization, mesh choices, numerical algorithms, word length, and round-off are principal sources of estimation error, with technical constraints limiting remedies.Reducing mesh size and increasing numerical precision could improve accuracy, but cannot remove the practical constraints.Source: §5.1 · p. 9

5.2 Influence of linearization and round-offs

  • For rotations, linearization and round-off errors are practically acceptable below 1.0°, with errors up to 0.01°, while translation errors are below 10^-14 mm.The comparison supports the LIN-based technique as more robust and less computationally complex than the SVD-based method.Source: §5.2 · p. 9

5.3 Influence of FEA-modeling errors

  • Rotational estimation uncertainty depends on the spatial locations of sampled points, whereas translational uncertainty depends only on their number.For a cubic field of size a × a × a, rotational standard deviation is approximately aσ/√(6n).Source: §5.3 · p. 10
  • The residual-based noise estimate can aggregate squared residuals across FEA experiments, and residual filtering removes points with the largest residuals.Filtering addresses potential outliers from anomalous meshing or joint boundary-condition violations.Source: §5.3 · p. 10

5.4 Optimal settings for virtual experiments

  • Translational identification error has a standard deviation of about 1.36×10^-6 mm, depends on FEA-induced noise, and is independent of translation amplitude.The FEA-induced component is about 1.37×10^-6 mm; the linearization and round-off component is less than 10^-14 mm.Source: §5.4 · p. 10, Figure 7 · p. 11
  • 0.01–0.2° rotational deflections yield about 0.2% identification accuracy, and rotational error depends strongly on deflection amplitude.At 0.1°, the identification-error standard deviation is about 8.4×10^-5 deg; Figure 8 compares errors across amplitudes.Source: §5.4 · p. 11, Figure 8 · p. 11

6 Minimization of the identification errors

The section benchmarks the identification procedure and examines how FEA settings and statistical processing affect compliance-matrix accuracy. It reports practical choices for meshing and virtual-sensor size, plus procedures for outlier removal and zero-element detection.

6.1 Benchmark example

  • The benchmark compares the cantilever beam’s numerically identified compliance matrix with an analytical solution using six sequential load cases.Each load case produces one matrix column; the beam’s opposite end is clamped.Source: §6 · p. 11, §6.1 · p. 12, Figure 9 · p. 12

6.2 Optimal selection of FEA-modeling options

  • 0.1% maximum compliance-matrix error is achieved with a 2 mm parabolic mesh, versus 12% for the best tested linear mesh at 1 mm.The position-accuracy comparison also supports the 2 mm parabolic mesh for this case study; further mesh-step reduction brings no essential error reduction.Source: §6.2 · pp. 12–13, Table 5 · p. 12, Figure 10 · p. 13
  • 0.1% identification error is achieved with a one-layer virtual sensor at the beam’s rare edge; a 10×10×10 mm3 sensor gives about 0.08%.The authors recommend the square-type virtual sensor in practice.Source: §6.2 · pp. 13–14, Table 6 · p. 13

6.3 Statistical processing of FEA-based data

  • Filtering the 10% of nodes with the highest residuals reduced the maximum compliance-matrix element error from 0.1% to 0.05%.Outliers appeared in 2 of 6 experiments and were concentrated at beam edges; a node was removed if any of its three residuals was an outlier.Source: §6.3 · p. 14, Figure 12 · p. 14, Figure 13 · p. 15
  • Confidence intervals containing zero identified all 26 zero elements in the 36-element compliance matrix, while all nonzero elements remained significant.The reported safety factor for nonzero elements was 10^2 or higher.Source: §6.3 · p. 14, Table 7 · p. 15
  • The cantilever example reports some disagreement between applied loads and values extracted from the modeling protocol, alongside nontrivial modeling-option choices.The authors state that 0.1% accuracy is achievable with standard computing facilities and that matrix symmetrization may further improve results.Source: §6.3 · p. 15

7 Application example: stiffness matrices for Orthoglide links

The Orthoglide application identifies stiffness matrices for flexible links using 2 mm parabolic-mesh virtual experiments and statistical processing. Comparisons show substantial differences from other methods, with joint-specific force distribution identified as a reason.

7 Application example: stiffness matrices for Orthoglide links

  • The identified link matrices use 2 mm parabolic meshing and 1 N and 1 N·m test loads, settings reported to yield about 0.1% accuracy.The matrices include zero elements identified by the proposed technique and are symmetrical.Source: §7 · p. 16, Table 8 · p. 16
  • For the parallelogram bar, torsional compliance k44 differs by a factor of 13 across the compared methods.The authors attribute the difference mainly to accounting for joint particularities that determine force distribution; prior results localized loading at the reference point.Source: §7 · p. 16, Table 9 · p. 17
  • The method accounts for complex link shapes and joint-related force/torque distribution, producing symmetrical matrices with physically consistent zero elements.The matrices are intended for subsequent VJM-based stiffness modeling of parallel manipulators.Source: §7 · pp. 15–16

8 Conclusions

The paper presents a CAD-based approach for identifying manipulator elasto-static parameters through virtual experiments and statistical data processing. Case studies report identification errors of about 0.1%; future work is directed toward automation and CAD integration.

8 Conclusions

  • The approach combines virtual experiments in CAD with numerical processing to identify stiffness matrices, accounting for link shape, coupled deflections, and joint particularities.The paper reports about 0.1% accuracy for stiffness-matrix elements and Orthoglide link case studies.Source: §8 · p. 17
  • The reported Orthoglide case studies reduced identification errors to 0.1%.The conclusion describes this as confirmation of the approach’s advantages.Source: §8 · p. 17
  • Further development is planned to automate operations currently performed interactively and integrate the technique fully into an existing CAD environment.The paper specifically proposes routines for CAD-to-computation data exchange and virtual-experiment parameter definition.Source: §8 · p. 17
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