gooch@portia.Stanford.EDU (Carl Gooch) (02/15/91)
In article <puchm.666615329@cutmcvax.cs.curtin.edu.au> puchm@cutmcvax.cs.curtin.edu.au (RichardPuchmayer) writes: > > Dear netters, > > The following question refers to : > > Bayliss C. McInnis and Cheng-Kang Frank Liu (1986) "Kinematics > and Dynamics in Robotics: A Tutorial Based Upon Classical > Concepts of Vectorial Mechanics",IEEE Journal of Robotics > and Automation. Vol:RA-2, No:4, December 1986. > > If you don't have this article please ignore this posting. > > The authors differentiate equation (2) on page 181 to get > equation (3). > > (2): > Vp = Vi + P(o)i/i + Wi X Pi > > to get > > (3): > Ap = Ai + P(oo)i/i + 2Wi X P(o)i + W(dot)i X Pi + > Wi X ( Wi X Pi) ^ > | > | > I only get one of these. ie. a one instead of a two. Well, I haven't seen the article, but I can tell you where that term comes from. When you take the time derivative of a vector which is in a rotating frame, what you get is: dA/dt = (dA/dt)rel + omega x A That is, the time rate of change of A is equal to the time rate of change of A with respect to the rotating coordinate system plus a term which accounts for the fact that the vector is rotating (that's the omega x A). With that in mind, the first term on the RHS of (2) leads to the first term on the RHS of (3). The second term in (2) gives the second term and one of the Wi x P(o)i terms. The other Wi x P(o)i comes from the third term in (2), which also gives the fourth term in (3) by direct differentiation and the final term as omega x A. Any standard dynamics text (Greenwood, Meirovitch, etc) will have a much more detailed discussion of this. > Apologies for the way that I have done sub/super > scripts. Ditto. -- ------------------------------------------------------------------------------ Carl Gooch | Why am I inside at a keyboard when gooch@leland.stanford.edu | I could be outside riding bike? ------------------------------------------------------------------------------
puchm@cutmcvax.cs.curtin.edu.au (RichardPuchmayer) (02/15/91)
Dear netters,
The following question refers to :
Bayliss C. McInnis and Cheng-Kang Frank Liu (1986) "Kinematics
and Dynamics in Robotics: A Tutorial Based Upon Classical
Concepts of Vectorial Mechanics",IEEE Journal of Robotics
and Automation. Vol:RA-2, No:4, December 1986.
If you don't have this article please ignore this posting.
The authors differentiate equation (2) on page 181 to get
equation (3).
(2):
Vp = Vi + P(o)i/i + Wi X Pi
to get
(3):
Ap = Ai + P(oo)i/i + 2Wi X P(o)i + W(dot)i X Pi +
Wi X ( Wi X Pi) ^
|
|
I only get one of these. ie. a one instead of a two.
Please, someone, explain.
How is this done, or is this a typo ?
If it is a typo, what is the correct answer.
PS: I hope this makes sense.
Apologies for the way that I have done sub/super
scripts.
Much thanks in advance.
Richard.
--
/-------------------------------------------+---------------------------------\
| Some of us are poets, some of us are not. | puchm@cutmcvax.cs.curtin.edu.au |
| Richard Puchmayer, Masters Student at | Sorry but I |
| Curtin University of Western Australia | don't know any other addresses |
+-------------------------------------------+---------------------------------+
| I know nothing, so can hold no opinions for myself or others.....:-) |
\-----------------------------------------------------------------------------/gerry@frc2.frc.ri.cmu.edu (Gerry Roston) (02/23/91)
Taking the derivatives of rotating vectors, etc, is always somewhat confusing. However, if you want to see a good systematic way of doing this, I strongly recommend "Dynamics" by Kane and Levinson. Also see their book "Spacecraft Dynamics" for a similar treatment of orientations. -- Gerry Roston (gerry@cs.cmu.edu) | Society is produced by our wants, Field Robotics Center, | and government by our wickedness;... Carnegie Mellon University | Pittsburgh, PA, 15213 | Thomas Paine, "Common Sense", Jan 1776 (412) 268-6557 |