Correction: Montgomery-Smith, S.; Shy, C. Using Lie Derivatives with Dual Quaternions for Parallel Robots. Machines 2023, 11, 1056

1. Error in Equation

In the published paper [1], there is an error in Equation (33) that should be corrected as follows:

Original version:

$$\widehat{\eta }={\left|\eta \right|}^{-1}\eta ={\eta \left|\eta \right|}^{-1}={Q/\left|Q\right|+ϵ(B-(B·Q)Q/|Q|}^2).$$

(33)

New version:

$$\widehat{\eta }={\left|\eta \right|}^{-1}\eta ={\eta \left|\eta \right|}^{-1}={Q/\left|Q\right|+ϵ(B/|Q|-(B·Q)Q/|Q|}^3).$$

(33)

2. Correction in Text

The correction to Equation (33) leads to a small difference in the results in Section 10.3. Therefore, some corrections have been made to Section 10.3 “Results of Simulations for Forward Kinematics” as follows:

Original version:

All runs were successful. The average number of required iterations was about 4.8. The run time for each forwards kinematics calculation was a little under 100 microseconds, using a fairly modern but low-end laptop. Increasing the allowed angle to 45° gave a failure rate of 2 in 10,000.

New version:

Only one run out of 10,000 failed. The average number of required iterations was about 5.1. The run time for each forward kinematics calculation was a little under 100 microseconds, using a fairly modern but low-end laptop. Increasing the allowed angle to 45° also gave a failure rate of one in 10,000.

Original version:

Increasing the allowable difference between the initial guess and the original pose to 5% resulted in only about 88% of the runs being successful.

New version:

Increasing the allowable difference between the initial guess and the original pose to 5% resulted in only about 89% of the runs being successful.

The authors state that their scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.