[Paper Review] A Topological Kinematic Workspace Analysis of the Canfield Joint

Robert Short, Alan Hylton|arXiv (Cornell University)|Sep 6, 2018

Robotic Mechanisms and Dynamics2 references3 citations

TL;DR

This paper presents a topological kinematic analysis of the Canfield Joint, a 3-DOF spatial linkage used in aerospace applications, to determine its workspace when one actuator fails (i.e., one base angle is fixed). Using a novel hybrid control scheme combining forward and inverse kinematics, the study reveals that the workspace reduces to a spherical cap whose size and position depend on the locked angle, with motion range dropping from nearly a quarter-sphere at 100° to less than a sixth at 160°.

ABSTRACT

We use topological techniques to do a workspace analysis of the Canfield Joint, a mechanical linkage constructed with two plates connected by three legs. The Canfield Joint has three degrees of freedom and can be controlled using three actuators attached to the base in strategic positions. In the process of performing the workspace analysis, we describe a new method of controlling the Joint which includes elements of both forward and inverse kinematics. This control process is then used to answer the question of how the workspace of the joint changes in the possibility of a failure mode where one degree of freedom is lost.

Motivation & Objective

• To analyze the kinematic workspace of the Canfield Joint when one of its three actuators fails, i.e., one base angle is fixed.
• To develop a new hybrid control scheme that integrates forward and inverse kinematics for accurate simulation of the joint's motion under failure conditions.
• To determine how the shape, size, and position of the workspace change depending on which arm is locked and at what angle.
• To identify geometric and kinematic constraints that prevent valid configurations in the control space, especially near maximum plunge distance.
• To provide a foundation for robust control and failure-tolerant design in applications like deep-space optical communications using the iROC project's antenna pointing systems.

Proposed method

• Model the Canfield Joint as a symmetric spatial linkage with two triangular plates connected by three equal-length arms, each composed of two bars joined by a spherical joint.
• Define key geometric parameters: arm length ℓ, base triangle side length b, plunge distance p, and distance d from joint center to hinge (d = √(p² + b²/3)).
• Introduce a novel control scheme using spherical coordinates (θ, p, φ) to parameterize the joint’s configuration, enabling simulation of fixed-angle failure modes.
• Use topological techniques from Kevin Walker’s work to analyze the structure of the workspace under constraints, particularly when one θi is fixed.
• Implement numerical simulations in GeoGebra to visualize the center of the distal plate’s reachable positions when one base angle is locked at various values (e.g., 80°, 100°, 160°).
• Analyze geometric consistency by checking whether the midpoints of the arms can form valid spherical joints under given ℓ and b values, identifying physical bounds on valid configurations.

Experimental results

Research questions

• RQ1 What is the kinematic workspace of the Canfield Joint when one of its three base angles is fixed due to actuator failure?
• RQ2 How does the size and location of the reduced workspace vary with the value of the locked base angle?
• RQ3 What geometric constraints prevent certain configurations from being physically realizable, especially near the maximum plunge distance?
• RQ4 How does the proposed (θ, p, φ) control scheme compare to the standard (θ₁, θ₂, θ₃) scheme in terms of robustness and usability under failure conditions?
• RQ5 Can topological methods be effectively used to analyze and predict the behavior of redundant or partially failed mechanical linkages like the Canfield Joint?

Key findings

• When one arm is locked at 100°, the workspace spans nearly a quarter of a sphere, indicating substantial residual mobility.
• When locked at 160°, the workspace is reduced to less than one-sixth of a sphere, indicating significant loss of motion capability.
• The shape of the reduced workspace resembles an orange peel, with the angular extent and orientation dependent on the locked angle.
• The position of the workspace on the sphere shifts depending on which base angle is fixed, indicating asymmetric failure behavior.
• Geometric constraints were observed where certain (θ, p, φ) configurations were physically unrealizable due to non-intersecting mid-arm joint circles, especially when ℓ was small relative to b.
• The control scheme (θ, p, φ) enables more intuitive simulation of failure modes and could simplify future programming and physical bound determination for the joint.