Published online by Cambridge University Press: 12 September 2009
Equivalent closed-loop spherical mechanism
In the previous chapter, it was shown that any serial manipulator can be transformed into a closed-loop spatial mechanism by constructing a hypothetical closure link. This chapter will focus on the geometry of the new closed-loop mechanism.
A new closed-loop mechanism called the equivalent spherical mechanism will be formed from the original spatial closed-loop mechanism. The first step in creating the equivalent spherical mechanism is to give all the unit joint vectors, Si, which label revolute or cylindric joint axes, self-parallel translations so that they all meet in a common point O and so that they all point outward from O (see Figure 6.1). Thus the directions of the Si vectors are the same for the original spatial mechanism and the cointersecting arrangement.
Consider now that a unit sphere is drawn, centered at point O. The unit vectors Si will meet this sphere at a sequence of points, i = 1, 2, 3,…, and so forth, as shown in Figure 6.2. Links (arcs of great circles) can be drawn on the unit sphere joining adjacent points, 12, 23, 34,…, and so forth. For example, Figure 6.2 illustrates a spherical link joining points 1 and 2 such that the angle between S1 and S2 is α12, that is, the same angle as between S1 and S2 in the original spatial mechanism.
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.