[Paper Review] Sensitivity Analysis of the Orthoglide, a 3-DOF Translational Parallel Kinematic Machine

Stéphane Caro, Philippe Wenger+2 more|arXiv (Cornell University)|Aug 31, 2007

Robotic Mechanisms and Dynamics4 citations

TL;DR

This paper presents two complementary sensitivity analysis methods for the Orthoglide, a 3-DOF translational parallel kinematic machine (PKM): a linkage kinematic analysis and a differential vector method. The key finding is that pose sensitivity to geometric variations is minimized at the kinematic isotropic configuration and maximized near singular configurations, with orientation being more sensitive to parallelogram side parallelism errors than position.

ABSTRACT

This paper presents a sensitivity analysis of the Orthoglide, a 3-DOF translational Parallel Kinematic Machine. Two complementary methods are developed to analyze its sensitivity to its dimensional and angular variations. First, a linkage kinematic analysis method is used to have a rough idea of the influence of the dimensional variations on the location of the end-effector. Besides, this method shows that variations in the design parameters of the same type from one leg to the other have the same influence on the end-effector. However, this method does not take into account the variations in the parallelograms. Thus, a differential vector method is used to study the influence of the dimensional and angular variations in the parts of the manipulator on the position and orientation of the end-effector, and particularly the influence of the variations in the parallelograms. It turns out that the kinematic isotropic configuration of the manipulator is the least sensitive one to its dimensional and angular variations. On the contrary, the closest configurations to its kinematic singular configurations are the most sensitive ones to geometrical variations.

Motivation & Objective

• To analyze the sensitivity of the Orthoglide PKM to geometric and kinematic variations in its design parameters.
• To identify configurations where pose errors are minimized or maximized due to manufacturing and assembly tolerances.
• To develop generalizable methods applicable to other 3-DOF Delta-Linear PKMs with orthogonal or parallel linear joints.
• To support tolerance synthesis by quantifying the impact of length, angular, and parallelism errors on end-effector accuracy.
• To investigate the relationship between kinematic isotropy, singularities, and sensitivity in PKM design.

Proposed method

• A linkage kinematic analysis method is used to estimate the influence of length variations in prismatic joints and parallelograms on end-effector position.
• A differential vector method models the propagation of position and orientation errors from joint and link variations, including angular and parallelism deviations.
• The differential vector method explicitly accounts for variations in parallelogram links, such as length changes and misalignments in the small and long sides.
• Error propagation is analyzed using vector-based sensitivity models derived from the manipulator’s kinematic Jacobian and closure equations.
• Statistical sampling with normally distributed geometric variations is applied to compute probability density functions of position and orientation errors.
• The isotropic and singular configurations are identified via kinematic analysis to compare sensitivity across different manipulator poses.

Experimental results

Research questions

• RQ1 How do variations in the lengths of prismatic joints and parallelograms affect the position and orientation of the Orthoglide’s end-effector?
• RQ2 What is the relative influence of parallelism errors in the short and long sides of the parallelograms on end-effector pose accuracy?
• RQ3 How does the sensitivity of the end-effector pose vary between the kinematic isotropic configuration and singular configurations?
• RQ4 To what extent do identical geometric variations in corresponding legs of the manipulator produce symmetric effects on end-effector error?
• RQ5 Can the proposed sensitivity analysis methods be generalized to other 3-DOF Delta-Linear PKMs with different joint arrangements?

Key findings

• The sensitivity of the end-effector’s pose to geometric variations is minimized at the kinematic isotropic configuration of the Orthoglide.
• Sensitivity to geometric variations increases significantly near kinematic singular configurations, where error amplification is most pronounced.
• The probability of achieving a position error below 0.3 mm is highest (96.83%) in the isotropic configuration, compared to 84.68% in $Q_1$ and 72.76% in $Q_2$.
• The probability of achieving an orientation error below 0.25° is highest in the isotropic configuration (96.90%) and slightly lower in $Q_2$ (94.53%), indicating greater orientation sensitivity in non-isotropic poses.
• The orientation of the end-effector is more sensitive to parallelism errors in the short sides of the parallelograms than in the long sides.
• Variations in the same type of design parameter (e.g., length or angular deviation) across different legs have identical effects on the end-effector position due to symmetric kinematic architecture.