[Paper Review] Kinematic analysis of the 3-RPR parallel manipulator

Damien Chablat, Philippe Wenger+1 more|ArXiv.org|Aug 29, 2007

Robotic Mechanisms and Dynamics7 references3 citations

TL;DR

This paper presents a comprehensive kinematic analysis of a planar 3-RPR parallel manipulator with actuated revolute joints on the base. It derives the direct and inverse kinematics using vector and matrix formulations, characterizes both parallel and serial singularities, and identifies conditions under which the manipulator's motion mimics a Reuleaux straight-line mechanism, achieving a precisely known straight-line displacement of length 2 under specific joint angle configurations.

ABSTRACT

The aim of this paper is the kinematic study of a 3-RPR planar parallel manipulator where the three fixed revolute joints are actuated. The direct and inverse kinematic problem as well as the singular configuration is characterized. On parallel singular configurations, the motion produce by the mobile platform can be compared to the Reuleaux straight-line mechanism.

Motivation & Objective

• To analyze the kinematics of a 3-RPR planar parallel manipulator with actuated revolute joints on the base.
• To characterize the direct and inverse kinematic problems using vector and matrix formulations.
• To identify and classify both parallel and serial singularities in the manipulator’s workspace.
• To investigate conditions under which the mobile platform’s motion replicates a Reuleaux straight-line mechanism.
• To determine the geometric and kinematic conditions leading to degenerate Cardanic curves and infinite solution sets.

Proposed method

• Derives the velocity of the mobile platform’s operation point P using three leg-based kinematic chains, each involving actuated joint rates and platform orientation.
• Eliminates idle joint rates by dot-multiplying velocity equations with rotated unit vectors, leading to a system of linear equations.
• Formulates the direct-kinematics matrix A and inverse-kinematics matrix B, enabling the computation of the manipulator’s Jacobian J = A⁻¹B.
• Identifies parallel singularities as configurations where det(A) = 0, indicating loss of stiffness and control.
• Identifies serial singularities as configurations where det(B) = 0, particularly when any ρᵢ = 0, marking the boundary of the reachable workspace.
• Uses geometric and trigonometric methods to solve the direct kinematic problem by finding the intersection of Cardanic curves with prismatic joint constraints.

Experimental results

Research questions

• RQ1 Under what conditions does the 3-RPR manipulator exhibit parallel singularities, and how do they affect controllability and stiffness?
• RQ2 How can the direct and inverse kinematics of the 3-RPR manipulator be analytically formulated using vector and matrix methods?
• RQ3 When does the Cardanic curve degenerate, and what kinematic behavior emerges in such cases?
• RQ4 What specific joint angle configurations cause the mobile platform’s motion to emulate a Reuleaux straight-line mechanism?
• RQ5 What is the precise displacement amplitude of the operation point P when the manipulator behaves as a Reuleaux mechanism?

Key findings

• Parallel singularities occur when the three axes normal to the prismatic joints intersect, and det(A) = 0, leading to loss of control and inability to resist applied wrenches.
• The determinant of matrix A is given by a trigonometric expression involving sin(θ₃−θ₁−θ₂), sin(θ₃−θ₁+θ₂), sin(θ₃+θ₁−θ₂), cos(θ₃+θ₁−θ₂), and cos(θ₃−θ₁+θ₂), with singularities occurring when this expression vanishes.
• When θ₁ = θ₂ = θ₃, the singularity occurs at infinity, allowing pure translational motion along the prismatic joint axes.
• Serial singularities occur when any ρᵢ = 0, restricting the manipulator’s ability to produce motion in certain Cartesian directions.
• For θ₂−θ₁ = π/3 or θ₁−θ₂ = π/3, and θ₃−θ₁ = π/3 or θ₁−θ₃ = π/3 with θ₁ ≠ θ₂ ≠ θ₃, the Cardanic curve degenerates, leading to infinite solutions and Reuleaux mechanism behavior.
• Under Reuleaux mechanism conditions, the mobile platform’s operation point P moves along a straight line of length 2, with the displacement magnitude at A₁ and A₂ both equal to 4√3/3.