1416:, i.e relative position vectors, i.e. directed distances from a point A to a point B) fixed to the body. I called these directed axes a local CS, but you are right: theoretically, they don't need to pass through an origin (common intersection), and don't need to have units to measure coordinates, so they don't have to be a CS. However, this is just splitting hair. I have never heard an engineer referring to "a basis set codirectional with a set of directed axes fixed to the airplane" (nor to "a basis set fixed to the airplane"). That's too long and unnecessarily complex. As far as I know, they just refer, much more simply, to "a CS fixed to the airplane", or "local CS". And this CS is also used to measure the time-invariant (local) position of points of interest such as the vertices of a mesh describing the shape of the external surface of the airplane.
401:
points whose position relative to each other is fixed. A reference frame is kinematically indistinguishable from a rigid body. It has no inherent origin, coordinate axes, or basis vectors. A set of basis vectors can be fixed arbitrarily to a reference frame or rigid body, describing a set of directions in which orientations, velocities, and accelerations can be expressed. Finally, linear position requires a coordinate system or a set of coordinate axes, which requires a set of basis vectors and an origin, an arbitrary point fixed in the reference frame. A given reference frame can have any number of sets of basis vectors and coordinate systems attached to it, none of which is "preferred" or inherent.
101:
1568:
doesn't make any sense, because all points on a rigid body generally have different velocity at the same time (except along the axis of rotation). To state "the velocity of body B" is meaningless without specifying the reference point, so truly the velocity is only defined for points. In colloquial language, this point is typically assumed to be the center of mass or the centroid, such as when referring to a car driving, a person running, or a ball flying through the air and spinning rapidly. This does not mean that it is valid to say that a rigid body as a velocity.
423:
expressed in different bases. Putting the rotation matrix into the general equation creates a very restrictive definition, and leaves the reader unprepared to handle any situation in which not all of his vectors are available in the desired basis. A huge amount of vector analysis can be performed correctly while completely ignorant of bases, and this type of analysis is much simpler than it would be if one had to drag multiple direction cosines through all the equations. The matrices are clumsy to manipulate relative to the vectors.
1511:
and the order of the new paragraphs is not optimized. Wouldn't it be better for the readers to remove our latest edits, put them in a separate temporary page, and lock the article until you manage to obtain a completely new and publishable version of the article? I mean that an article, if possible, should not appear as a "work in progress". If it has reached a sound structure, that structure should not be substituted with another, unless the new one is equally sound, or at least reasonably sound.
1788:
define the placement in space of the whole rigid body. In other words, I agree with you that a basis which rotates with the body is enough to define the orientation of planes or axes that are fixed to the body, but these planes or axes, even together with a reference point, are not enough to describe the whole body. To measure the local position of all other particles, you need a coordinate system. And only if you have local positions, you can transform them to the global coordinate system.
31:
667:, or basis vectors, if you like). And this is typically the first step of the algorithm to compute an orientation matrix, or an orientation vector, or a set of Euler angles. And, if these axes are also the axes of a CS, in which you know the (local) position of other important particles of the object (for instance, the tail and tip of the ariplane, the point of application of external forces such as engine thrust, etc.) then you can complete the task.
1495:, which is a self-guided computer course in dynamics. There is certainly more to be done, like perhaps to add mathematical definitions of angular velocity and angular acceleration, or to give examples of how to use the many abstract equations I have just added. I also want to improve the equation about "spatial acceleration" to make basis- and coordinate system-independent like the others, but it is not something I am familiar with.
91:
64:
22:
318:
more general than just the position and velocity of a body relative to the "world". They can be defined relative to any arbitrary point of reference and any arbitrary reference frame. If the article needs two sets of basis vectors and/or a coordinate system for comparison, it should be introduced when it is first used, after the definitions of position and velocity.
1029:
583:
perfectly correct. I mean that too many people (mainly engineers) forget that the orientation is a position (notice that I am an engineer as well). Since we (I agree with you) don't like to define rigid body position using the concepts of (hypothetic) translation and rotation from reference position, I suggest another approach. See if you like my next edit, please.
264:
high mathematics. Even in introductory physics courses the distinction is often clearly explained (see for instance the widely used textbook by
Reisnick and Holliday: Physics, Part 1, John Wiley and Sons). According to this distinction, the displacement is (even etymologically) a change in placement, i.e. a change in position, i.e. a simplified representation of a
397:
certain contexts, I think this is awkward and unnecessarily ambiguous. I don't think it is necessarily bad to define a new word that would encompass both without the ambiguity. (We use "state" to refer to the four quantities of position, velocity, orientation, and angular velocity when speaking in 6dof, or just position and velocity when speaking in 3dof.)
1467:
directions with no origin. It is possible to describe an infinite number of
Cartesian coordinate systems using a single set of basis vectors, where the basis vectors are based on geometric features of the body (not axes). There is plenty of analysis that can be performed using only basis vectors and points without bothering to define any coordinate axes.
362:"pose" to indicate position + orientation, but this is just because they didn't realize that the orientation is a position. And it is a neologism used only by a very small group of people. I am sure that no serious physics textbook will ever adopt this horrible terminology. I believe that Lagrange used the expression "
393:"You incorrectly stated that there's no orientation vector that can be differentiated. There is an orientation vector and it can be differentiated. Its derivative is not the angular velocity, but it can be differentiated." -- Hopefully my recent edit has sufficiently clarified this point to your satisfaction.
687:
Perhaps I do not use the same definition of "axis" as you do, but in my experience an axis has a fixed physical location because it is required to pass through the origin. This is distinct from a basis vector, which has no spatial location, but only describes a direction. Therefore you don't need a
357:
You can mention in a footnote, if you like, that some people use the word "position" as a synonym of "linear position", and that they use "orientation" to indicate angular position. But I showed you that this is a questionable terminology, not compatible with the terminology that the whole world uses
268:
motion. A displacement occurs in a measurable interval of time, a position exists at a given time, with no reference whatsoever to past or future instants. In other words, the tail and tip of a position vector are two points that exist at the same time (origin and particle), while the tail and tip of
1756:
by introducing extraneous "coordinate systems" everywhere. The way it was written before was better. For example, you can use any arbitrary point in a rigid body for the purpose of defining position vectors, but now the article falsely claims that this point has to be the origin of some coordinate
1530:
By redoing the kinematical formulas I have not actually removed any important content from the article, and I haven't really changed the structure. I replaced a paragraph followed by formulas with a presentation of basically the same information in an interlocking manner, with better explanation of
1510:
What you are trying to do is useful, but the article's structure was sound before your edits. Now parts are missing (we wrote that position and velocity are vector quantities, we did not write this for accelerations; you inserted addition theorems: why only for angular velocity and linear position?)
1457:
The phrase "a basis set fixed to the airplane" is no longer or more complex than "a CS fixed to the airplane". It is true, engineers don't typically say "a basis set codirectional with a set of directed axes fixed to the airplane", but that's because it is unnecessary to specify directed axes. And
582:
Both expressions are correct, but too many people forget the first one. It is not too hard to use 3 or 4 words that have been used for centuries. And if I write "the position of this body is represented by a linear position vector and an orientation matrix" everybody should consider this terminology
434:
basis by the appropriate application of rotation matrices. Vectors can even be expressed in bases that are not fixed to the reference frame the vector is associated with. Once the vector analysis is done, the necessary input vectors can easily be expressed in the correct bases via simple relations
280:
I also disagree about deleting the description of the two coordinate systems L and G. The rest of the article refer to these systems. The reader needs to know that it is impossible to define the orientation of a rigid body without defining two coordinate systems. For instance, how can you state that
263:
I disagree about using the word displacement as a synonym of position. It is used with this meaning by some authors, especially in elementary physics textbooks, but it is more advisable, when possible, to take into account the distinction between the two concepts. Notice that this distinction is not
1778:
I have reviewed our previous discussion. I perfectly understand your point, however I think you did not consider that defining the linear and angular position of a coordinate system (or, if you like, the linear position of a reference point, and the angular position of a basis set) is not enough to
1711:
Refs 4,5,6,7 are either no longer available online or incorrectly cited. Is this a print textbook, an online course, or something else entirely? My interest is partly selfish - it looks like a rather useful reference and I would love to find it :). If it's an online course or textbook, a link would
1567:
And a final point: I disagree with your revert of my edit where I stated that "the velocity of a rigid body is undefined, it is only defined for points fixed to the body." You reverted it to state that the velocity of a rigid body is the velocity of any arbitrary reference point on the body. This
422:
For this reason it is unnecessary to introduce rotation matrices (the A(t)'s) into the general equations that define the fundamental relations of position, velocity, and acceleration of a point on a rigid body. It only confuses the issue by making people ignorant of the fact that vectors often are
339:
The position of a rigid body includes the orientation. This is absolutely correct. In physics, linear and angular are the keywords that distinguish the two parts of kinematics. There is a position, a velocity and an acceleration both in angular and in linear kinematics. You say (even for the motion
1614:
This article contradicts the following from the article on angular velocity: "In general, the angular velocity in an n-dimensional space is the time derivative of the angular displacement tensor which is a second rank skew-symmetric tensor". I'm assuming this quote is accurate, and this article is
1451:
This appears to agree with my definition of basis vectors, because in your parenthetical statement you are saying that they need not be codirectional with axes, they only have to be based on geometric features of the body such as line segments connecting points fixed in the body. I would add that
396:
In my common experience (aerospace engineering), the word "position" means only linear position, and "orientation" means only orientation. There is not a general term in my technical vocabulary to describe the two together, though I believe there should be. While "position" may encompass both in
317:
I didn't read the rest of the article, so pI was unaware that it is necessary to establish two coordinate systems (or sets of basis vectors) for the purposes of the examples. However, I do not believe that this should be a part of the descriptions of position and velocity. Those concepts are far
1550:
The general formula for the addition of angular accelerations does not take on the same simple form as the addition theorem for angular velocity; it contains cross products of the angular velocities. I can add it if you believe it would be appropriate, but it actually didn't occur to me to do so
598:
I do like your edit. There is not much I would change in it other than some personal wording preferences, which is not worth the effort. Next up, let's tackle those kinematic equations by getting rid of the direction cosine matrices from the equations and making them basis-independent, and then
207:
I agree, and more importantly since in relativity one can see that the whole rigid body concept is an ideal approximation that is incongruent with the speed of light limit. Of course without leaving out the fact/concept of rigid motion in relativity (note that rigid motion redirects here), and to
226:
I generalized the section about position a little better than before, and removed its dependence on a pair of reference frames. I think it is easier to understand without weighing it down by presenting the definition simultaneously with an example which never gets finished. I also added a short
1787:
To reconstruct the position of these particles, a translation and a rotation are not enough. You also need the local position of all the particles, relative to a "local coordinate system". This is the reason why I believe that defining a local coordinate system is always required if you want to
400:
There is no disagreement here that two "coordinate systems" (your terminology) are necessary to define the quantities we are interested in. In light of that, I believe it is important not to conflate coordinate systems, reference frames, and basis vectors. A reference frame is merely a set of
274:
Of course, you can imagine a hypothetical displacement that brings the particle from the arbitrary origin of a coordinate system (where the particle has never been) to the point where the particle is located at a given time. But still there is a difference between this hypothetical displacement
305:
I understand that there is such a thing as a rotation vector. I was trying to emphasize the fact that there is no such thing as an orientation vector that is the angular analog to position, because the natural inclination of many readers will be to assume "hey, I can integrate velocity to get
1466:
as well as many of my colleagues here at
Lockheed Martin, who use this terminology and make the same distinction between basis vectors and CSs. It is not splitting hairs. A CS has a single, specific origin and it measures distance along its axes. Basis vectors are merely a set of reference
888:
1295:. As far as I know, your distinction between axis and versor (basis vector) is correct and useful. Good point. Thank you for reminding me. That allows me to better understand your distinction between basis set and CS (08:18, 28 August 2008). Here are my two cents about this distinction:
1783:
rigid body. The latter is actually defined as the position of all the particles which the object is composed of (or at least the main points which define the shape of the object, for instance the vertices of a cube, or the points which approximately define the external surface of an
309:
As for position and displacement, I agree with your statement. I shouldn't have equated them. I was struggling to come up with a word for "translational position" that was not merely the word "position", since this article attempts to use position as an umbrella term for position
366:" to indicate both the linear and angular position (and Lagrange's terminology, together with his equations of motion, are frequently used in the scientific community). This is the only correct way to indicate unambiguously both the linear and angular position, as far as I know.
698:
Now I will give an example of why reference frames and basis vectors (and by extension, coordinate systems) are not equivalent. Imagine two reference frames (say inertial frame N and rigid body B), each with a set of three mutually perpendicular unit vectors fixed to it,
1361:
as "parallel to the chord line of the wing, positive forward", I have not given it a physical location based on any particular points on the body. I have only defined its direction and specified that it is fixed to the body. I can create a second basis vector
1330:" a RF or rigid body, you give to basis vectors a physical meaning that they don't have. Only axes or CSs can be fixed to a RF or rigid body, because they are defined using a set of points fixed to the RF or rigid body, one of which is typically the origin (see
1562:
in form, it would not be appropriate to call them that, because they are not truly addition theorems. The velocity formula resembles it visually, but the superscripts are not analogous. The present titles are much more descriptive of the function of these
617:
OK, give me some time. I am busy right now, and I have to read again your previous comment, trying to understand exactly what you mean (notice that there was a discussion about the distinction between reference frames (RF) and coordinate systems (CS) in
378:, if you like), and that should be made clear. I will adjust my example to stress this concept again: how can you state that an airplane is horizontal, without first defining both its longitudinal (coordinate) axis, and the horizontal (coordinate) axis?
238:
as an "angular position vector" that you could take the derivative of. If this article needs a better definition of angular velocity than the one I put in it, I can provide one, but it will be more technical and perhaps not appropriate for this topic.
1247:
There are several methods to carry out the calculations to the final answer, but all of them require expressing vectors in bases that are not fixed to the reference frame in which the vector is defined. In other words, merely to say "the velocity of
1408:"Fixed in direction" means nothing. Since the body is free to rotate, their direction is time-varying. My point is that you cannot use the expression "fixed to the rigid body". The ony way to describe the behaviour of those basis vectors,
1024:{\displaystyle _{\text{N}}\mathbf {v} _{P}{\text{ }}={\text{ }}_{\text{N}}\mathbf {v} _{Q}{\text{ }}+{\text{ }}_{\text{B}}\mathbf {v} _{P}{\text{ }}+{\text{ }}_{\text{N}}\mathbf {\omega } _{\text{B}}\times \mathbf {r} _{QP}}
369:
I agree that the two coordinate systems should not be defined as global (G) and local (L). You are right: any coordinate system is ok to define the position of another coordinate system. But you need two coordinate systems
1252:
in N" fully describes a specific physical quantity, but it does not fully define the measure numbers of your vector, since that velocity can be expressed in terms of any arbitrarily selected set of basis vectors fixed in
419:. Vectors are vectors independent of their basis, and two vectors can be added, subtracted, cross-multiplied, dot-multiplied, or magnituded (if that's a word) algebraically without concerning oneself about the basis.
1526:
Ok, starting a separate temporary article may be the proper way to do things until we get all the issues hammered out. I don't know how to do that, however, so I would appreciate it if you took care of the set up.
275:(which is more properly called a position) and a real displacement. And I suggest to use a terminology which reflects and respects this distinction, as well the etymological interpretation of the word "displacement".
1728:
I added those references. It is a textbook that is only available in print, despite its title. The same authors have published several fantastic textbooks that are available for free from
Cornell University here:
1458:
even if you have never heard an engineer say "a basis set fixed to the airplane", I have. I use this terminology all the time. I can show you textbooks, give you the names of several professors and lecturers at
1303:
Any basis set is just a collection of numbers (a matrix if you like) which have no physical meaning. This is also true for any other vector. The tail of a vector is always supposed to be at point (0,0,0) (see
192:
I think that the problems of defining a relativistic version of the rigid body should be mentioned on this page. Several paradoxes in relativity theory stem from this issue (e.g., the barn and pole-vaulter).
333:
You incorrectly stated that there's no orientation vector that can be differentiated. There is an orientation vector and it can be differentiated. Its derivative is not the angular velocity, but it can be
1299:
Two axes are enough to define a
Cartesian CS. Indeed, their intersection is the origin, and the third axis can be easily found according to the right-hand or left-hand rule (depending on your choose of
1180:
1837:
497:
1370:
as completing the right-handed set and it points "down" relative to the belly of the aircraft. These vectors don't have an origin and are not fixed in location; they are only fixed in direction.
35:
1763:
We had a long discussion several years ago on this very talk page where you and I discussed a similar set of concerns. Please review it and let me know what you think about my objection.
1128:
1076:
869:
693:
I read most of your long discussion with Brews Ohare over on the other page. I generally agree with him that a coordinate system and a reference frame are definitely not equivalent ideas.
1390:
I think we are in agreement that basis vectors have no fixed location. I think we disagree in that I don't believe the direction of a basis vector must be linked to a coordinate axis.
1842:
1857:
688:
set of axes to describe the relative orientation of two reference frames, only two sets of basis vectors, or one set of basis vectors and three points in the second frame as you said.
1752:
I wanted to revert all five of Paulo.dL's most recent edits, but realized that would be inflammatory. The overall effect of Your edits, Paulo, is to make the paragraphs you touched
1827:
314:
orientation. Do you have a better word to use in this case? Or perhaps we should not use position to also refer to orientation, and keep it strictly for "translational position"?
644:
MarcusMaximus, I have not read with attention your previous comment (I will), but I can post a short reflexion in the meantime. We don't actually need two coordinate systems (CS)
1357:. Sorry for jumping from rockets to airplanes, but an airplane provides much more obvious physical features to use as references than a rocket does. If I define a basis vector
1226:
415:
I also have a problem with the section of the article containing all those equations. It is only asking for trouble to use vectors algebraically and to tie them to a specific
829:
One common example where it is necessary to express vectors in bases not attached to their native reference frames is calculating the inertial velocity of the center of mass
254:
I agree about the fact that the angular velocity is not the derivative of the orientation vector. However, I disagree that an orientation vector does not exist. It does (see
169:
74:
258:; as well as a rotation matrix, a rotation vector can be used to represent an orientation; in this case, you refer to them as "orientation matrix" and "orientation vector").
354:
There's no single word to indicate both linear and angular velocity, except for "velocity" alone. The same is true for acceleration, and the same must be true for position.
1434:"Fixed in direction" is certainly not meaningless; in context it should be apparent that I meant "fixed in direction relative to the rigid body (the airplane)".
157:
1491:
with my editing; I couldn't stand those clunky kinematical equations the way they were before. I borrowed heavily from a textbook by Kane and
Levinson called
290:
article. If you like to add a better definition, you are welcome. Actually, it is possible to compute it in several (more or less intuitive) different ways.
1452:
you can specify them as normal to a plane, among other definitions. These geometric features are not necessarily axes, and do not form a coordinate system.
1832:
1257:
reference frame, not necessarily in N. This should establish definitively that reference frames are not equivalent to vector bases or coordinate systems.
426:
The fundamental equations should stand on their own, basis-less. The concept of vector bases should be introduced separately, to ingrain the concept that
1822:
1862:
1852:
1732:
The source I cited draws nearly all its material from those books. It would probably be a good idea for me to go through and change the citations.
147:
100:
1847:
1554:
The general formulas for the addition of linear velocities and accelerations of two different points are there on the page; they are the
1817:
1713:
1616:
123:
555:
And take away the word displacement, as we already agreed. I do not agree that there is an urgent need to use a single word to say
1147:
1791:
However, I do not have time to explain this in the article, so I partially reverted my edits. See if you like the new text.
1760:
One improvement you made was to remove references to the meaningless concept of angular position of points and particles.
212:. I took the liberty of adding the Relativity wikiproject banner although I'm not a member of such group. Did I do wrong?
114:
69:
438:
1867:
44:
1650:
1635:
358:
for velocity and acceleration. By the way, I recently discovered that a few bioengineers decided to use the horrible
306:
position, so that must mean I can integrate angular velocity to get orientation", which is, of course, not the case.
1638:. In short, we all agree that unbounded lines or planes (in general N-D spaces) can be validly described as rigid
1543:. Notice that I didn't remove the formula based on screw theory, but I think it needs to be generalized as well.
872:
1546:
I didn't include "addition theorems" for angular acceleration, linear velocity, or linear acceleration because:
1095:
1043:
836:
652:", plus the position of at least three particles of the rigid object. The observer cannot measure without a CS.
21:
1730:
1717:
1620:
1768:
1737:
1697:
1600:
1573:
1500:
1472:
1395:
1375:
1331:
1262:
619:
604:
520:
406:
323:
244:
648:
the linear and angular position of a rigid body. We need only one of them, that is the CS attached to the "
363:
1595:. Shouldn't these two pages be merged, or at least move most of the content of this page to the other?
1591:
content. There is little on this page that is not rigid body dynamics-related; possibly the section on
1448:, i.e relative position vectors, i.e. directed distances from a point A to a point B) fixed to the body.
50:
1199:
1588:
122:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
1798:
1764:
1733:
1693:
1658:
1596:
1569:
1516:
1496:
1468:
1421:
1391:
1371:
1343:
1313:
1280:
1258:
672:
627:
600:
588:
516:
402:
383:
319:
295:
240:
194:
1539:
were in the article before, but in a less general form. There previously were no equations for
824:, all of which might be useful in analysis, and none of which required a set of coordinate axes.
1275:
You wrote a lot of intersting statements. I will discuss them one at a time, as soon as I can.
1182:, the angular velocity of the rocket in inertial space (in N) expressed in a body-fixed basis (
1689:
1673:
1636:
Talk:Orientation (geometry)#Physical objects, bodies, figures, segments, spaces, planes, lines
1488:
1130:, the velocity of the mass center in the body frame (in B) expressed in the body fixed basis (
209:
1305:
1078:, the inertial velocity (in N) of the IMU expressed in a basis fixed to the inertial frame (
287:
198:
1677:
507:
in each respective subscripted basis, and R is the direction cosine matrix expressing the
255:
227:
section on velocity and angular velocity and gave the important characteristics of each.
1228:, the position vector from the IMU to the mass center expressed in the body-fixed basis (
1681:
213:
106:
1444:, is to say that they are just codirectional with directed axes (or directed straight-
1412:, is to say that they are just codirectional with directed axes (or directed straight-
1811:
1794:
1654:
1512:
1417:
1339:
1309:
1276:
668:
623:
584:
541:(or simply position) to indicate the position of a particle (center of mass, usually)
379:
291:
281:
an airplane is horizontal, without first defining its longitudinal (coordinate) axis?
534:
I believe that we need to introduce the terminology used in physics books. I mean:
1445:
1413:
269:
a displacement vector indicate the position of the particle at two different times.
1685:
90:
63:
655:
However, typically we use the three or more selected particles to define three
286:
By the way, the angular velocity of a rigid body is explained in detail in the
1592:
1366:
as "normal to the plane of symmetry and directed positive rightward". Define
234:
the time derivative of "angular position". Angular velocity is a vector, but
96:
1646:? In other words, does it make sense to speak of a "body" of infinite size?
359:
1463:
1459:
649:
1779:
define the "placement in space" (or position, in general terms) of the
119:
1649:
IMPORTANT NOTE: If you are interested, please post your contribution
664:
503:
is a 3x1 column matrix containing the measure numbers of the vector
1688:. I think that page needed. Does anyone mind if I start it back at
727:
of body B has nonzero velocity is both the B frame and the N frame.
1802:
1772:
1741:
1721:
1701:
1684:, but none that get into the specific mathy details of SE(3) as a
1662:
1624:
1604:
1577:
1520:
1504:
1476:
1425:
1399:
1379:
1347:
1317:
1284:
1266:
676:
631:
608:
592:
524:
410:
387:
327:
299:
248:
230:
One thing I changed was the definition of angular velocity: it is
216:
202:
1326:. When you, in your previous comments, refer to "basis vectors
1440:
The ony way to describe the behaviour of those basis vectors,
820:, giving us four options for an expression of the velocity of
15:
1442:
if you want them to represent the orientation of a rigid body
1410:
if you want them to represent the orientation of a rigid body
736:
can be expressed in two different reference frames (N or B),
1587:
It occurs to me that this page consists almost entirely of
1634:
An interesting discussion about terminology is ongoing on
1175:{\displaystyle _{\text{N}}\mathbf {\omega } _{\text{B}}}
682:
I agree with the first paragraph of your comment above.
1676:
which was a stub and now redirects here. There's also
663:
axes attached to the rigid body (represented by three
442:
1838:
Knowledge level-5 vital articles in
Physical sciences
1551:
because it wasn't featured in the source I was using.
1202:
1150:
1098:
1046:
891:
839:
441:
492:{\displaystyle \scriptstyle {x_{b}=_{b}R_{a}*x_{a}}}
118:, a collaborative effort to improve the coverage of
1680:, which barely mentions SE(3). There is a page for
1630:
Valid or invalid generalizations of the term "body"
1220:
1174:
1122:
1070:
1023:
863:
491:
551:to indicate the orientation of a local basis set.
1843:Start-Class vital articles in Physical sciences
871:based on the inertial velocity measured at its
1858:Start-Class physics articles of Mid-importance
1334:). On the contrary, basis vectors can only be
1038:The inputs to this calculation are typically:
349:(linear) acceleration and angular acceleration
1828:Knowledge vital articles in Physical sciences
599:introducing the concept of bases separately.
515:) before doing the last plug-and-chug steps.
8:
1123:{\displaystyle _{\text{B}}\mathbf {v} _{P}}
1071:{\displaystyle _{\text{N}}\mathbf {v} _{Q}}
864:{\displaystyle _{\text{N}}\mathbf {v} _{P}}
19:
1558:formulas. Since they do not resemble the
1537:acceleration of two points on a rigid body
58:
1209:
1204:
1201:
1166:
1161:
1154:
1149:
1114:
1109:
1102:
1097:
1062:
1057:
1050:
1045:
1012:
1007:
997:
992:
985:
981:
973:
967:
962:
955:
951:
943:
937:
932:
925:
921:
913:
907:
902:
895:
890:
855:
850:
843:
838:
723:, respectively. Suppose the mass center
481:
468:
458:
448:
443:
440:
1483:Kinematical equations, basis-independent
1748:Coordinate system, schmoordinate system
796:in N can also be expressed in terms of
60:
1533:velocity of two points on a rigid body
1353:I don't agree with your statements in
346:(linear) velocity and angular velocity
343:(linear) position and angular position
1560:addition theorem for angular velocity
530:Point 1 - Linear and angular position
7:
764:). To be explicit, the velocity of
112:This article is within the scope of
49:It is of interest to the following
1833:Start-Class level-5 vital articles
1642:. Can they be also validly called
768:in B can be expressed in terms of
14:
1531:each equation. The formulas for
1386:After rereading your comments in
1221:{\displaystyle \mathbf {r} _{QP}}
1823:Knowledge level-5 vital articles
1556:one point moving on a rigid body
1541:one point moving on a rigid body
1205:
1110:
1058:
1008:
963:
933:
903:
851:
640:Part 2 - Local coordinate system
99:
89:
62:
29:
20:
1863:Start-Class relativity articles
1853:Mid-importance physics articles
1583:Merge with Rigid Body Dynamics?
152:This article has been rated as
1605:04:38, 11 September 2008 (UTC)
1:
1625:20:44, 16 February 2010 (UTC)
1615:not. Can anyone verify this?
1578:01:44, 5 September 2008 (UTC)
1521:17:46, 4 September 2008 (UTC)
1505:08:34, 3 September 2008 (UTC)
1477:02:11, 5 September 2008 (UTC)
1426:16:24, 4 September 2008 (UTC)
1400:19:38, 2 September 2008 (UTC)
1380:19:26, 2 September 2008 (UTC)
1348:12:09, 2 September 2008 (UTC)
1318:12:08, 2 September 2008 (UTC)
1285:11:53, 2 September 2008 (UTC)
1267:09:46, 2 September 2008 (UTC)
167:This article is supported by
132:Knowledge:WikiProject Physics
126:and see a list of open tasks.
1848:Start-Class physics articles
1668:New page: Rigid-body motion?
1663:11:55, 15 January 2011 (UTC)
135:Template:WikiProject Physics
677:10:03, 30 August 2008 (UTC)
632:09:31, 30 August 2008 (UTC)
609:08:32, 30 August 2008 (UTC)
593:17:26, 28 August 2008 (UTC)
525:08:42, 28 August 2008 (UTC)
430:vector can be expressed in
411:08:18, 28 August 2008 (UTC)
388:18:03, 27 August 2008 (UTC)
328:16:35, 27 August 2008 (UTC)
300:14:39, 27 August 2008 (UTC)
249:10:38, 27 August 2008 (UTC)
1884:
1818:Start-Class vital articles
511:basis vectors in terms of
158:project's importance scale
1803:17:02, 11 July 2011 (UTC)
217:00:54, 9 April 2008 (UTC)
203:18:26, 7 April 2008 (UTC)
170:the relativity task force
166:
151:
84:
57:
1773:20:41, 8 July 2011 (UTC)
1742:05:56, 8 June 2011 (UTC)
1722:21:28, 7 June 2011 (UTC)
1702:00:27, 25 May 2011 (UTC)
1672:There was a page called
1338:the axes of a local CS.
740:in two different bases (
1388:Fixed and codirectional
1355:Fixed and codirectional
1332:talk:Frame of reference
1324:Fixed and codirectional
620:Talk:Frame of reference
340:of a single particle):
1692:with a DaB link here?
1450:
1222:
1176:
1124:
1072:
1025:
865:
792:, and the velocity of
493:
236:there is no such thing
1640:collections of points
1438:
1223:
1177:
1125:
1073:
1026:
866:
494:
222:Position and velocity
36:level-5 vital article
1712:be useful. Thanks.
1200:
1148:
1096:
1044:
889:
875:fixed in B at point
837:
439:
364:generalized position
1868:Relativity articles
1589:rigid body dynamics
879:. The equation is:
115:WikiProject Physics
1336:codirectional with
1218:
1172:
1120:
1068:
1021:
861:
489:
488:
45:content assessment
1690:rigid body motion
1674:rigid body motion
1169:
1157:
1105:
1053:
1000:
988:
983:
975:
958:
953:
945:
928:
923:
915:
898:
846:
210:Ehrenfest paradox
185:
184:
181:
180:
177:
176:
1875:
1610:angular velocity
1306:vector (spatial)
1293:Basis set and CS
1227:
1225:
1224:
1219:
1217:
1216:
1208:
1181:
1179:
1178:
1173:
1171:
1170:
1167:
1165:
1159:
1158:
1155:
1129:
1127:
1126:
1121:
1119:
1118:
1113:
1107:
1106:
1103:
1077:
1075:
1074:
1069:
1067:
1066:
1061:
1055:
1054:
1051:
1030:
1028:
1027:
1022:
1020:
1019:
1011:
1002:
1001:
998:
996:
990:
989:
986:
984:
982:
976:
974:
972:
971:
966:
960:
959:
956:
954:
952:
946:
944:
942:
941:
936:
930:
929:
926:
924:
922:
916:
914:
912:
911:
906:
900:
899:
896:
870:
868:
867:
862:
860:
859:
854:
848:
847:
844:
732:The velocity of
545:angular position
498:
496:
495:
490:
487:
486:
485:
473:
472:
463:
462:
453:
452:
288:angular velocity
140:
139:
138:physics articles
136:
133:
130:
109:
104:
103:
93:
86:
85:
80:
77:
66:
59:
42:
33:
32:
25:
24:
16:
1883:
1882:
1878:
1877:
1876:
1874:
1873:
1872:
1808:
1807:
1750:
1709:
1694:—Ben FrantzDale
1678:Euclidean group
1670:
1632:
1612:
1585:
1493:Online Dynamics
1485:
1203:
1198:
1197:
1160:
1151:
1146:
1145:
1108:
1099:
1094:
1093:
1056:
1047:
1042:
1041:
1006:
991:
980:
961:
950:
931:
920:
901:
892:
887:
886:
849:
840:
835:
834:
642:
539:linear position
532:
477:
464:
454:
444:
437:
436:
377:
373:
334:differentiated.
256:rotation vector
224:
190:
137:
134:
131:
128:
127:
105:
98:
78:
72:
43:on Knowledge's
40:
30:
12:
11:
5:
1881:
1879:
1871:
1870:
1865:
1860:
1855:
1850:
1845:
1840:
1835:
1830:
1825:
1820:
1810:
1809:
1806:
1805:
1792:
1789:
1785:
1749:
1746:
1745:
1744:
1708:
1705:
1682:rotation group
1669:
1666:
1631:
1628:
1611:
1608:
1584:
1581:
1565:
1564:
1552:
1524:
1523:
1484:
1481:
1480:
1479:
1454:
1453:
1435:
1431:
1430:
1429:
1428:
1403:
1402:
1383:
1382:
1321:
1320:
1301:
1290:
1289:
1288:
1287:
1270:
1269:
1244:
1243:
1242:
1241:
1215:
1212:
1207:
1195:
1164:
1153:
1143:
1117:
1112:
1101:
1091:
1065:
1060:
1049:
1035:
1034:
1033:
1032:
1018:
1015:
1010:
1005:
995:
979:
970:
965:
949:
940:
935:
919:
910:
905:
894:
881:
880:
858:
853:
842:
826:
825:
729:
728:
695:
694:
690:
689:
684:
683:
641:
638:
637:
636:
635:
634:
612:
611:
580:
579:
568:
553:
552:
542:
531:
528:
484:
480:
476:
471:
467:
461:
457:
451:
447:
391:
390:
375:
371:
367:
355:
352:
351:
350:
347:
344:
336:
335:
303:
302:
283:
282:
277:
276:
271:
270:
260:
259:
223:
220:
189:
186:
183:
182:
179:
178:
175:
174:
165:
162:
161:
154:Mid-importance
150:
144:
143:
141:
124:the discussion
111:
110:
107:Physics portal
94:
82:
81:
79:Mid‑importance
67:
55:
54:
48:
26:
13:
10:
9:
6:
4:
3:
2:
1880:
1869:
1866:
1864:
1861:
1859:
1856:
1854:
1851:
1849:
1846:
1844:
1841:
1839:
1836:
1834:
1831:
1829:
1826:
1824:
1821:
1819:
1816:
1815:
1813:
1804:
1800:
1796:
1793:
1790:
1786:
1782:
1777:
1776:
1775:
1774:
1770:
1766:
1765:MarcusMaximus
1761:
1758:
1755:
1747:
1743:
1739:
1735:
1734:MarcusMaximus
1731:
1727:
1726:
1725:
1723:
1719:
1715:
1714:128.32.240.58
1706:
1704:
1703:
1699:
1695:
1691:
1687:
1683:
1679:
1675:
1667:
1665:
1664:
1660:
1656:
1652:
1647:
1645:
1641:
1637:
1629:
1627:
1626:
1622:
1618:
1617:146.6.200.225
1609:
1607:
1606:
1602:
1598:
1597:MarcusMaximus
1594:
1590:
1582:
1580:
1579:
1575:
1571:
1570:MarcusMaximus
1561:
1557:
1553:
1549:
1548:
1547:
1544:
1542:
1538:
1534:
1528:
1522:
1518:
1514:
1509:
1508:
1507:
1506:
1502:
1498:
1497:MarcusMaximus
1494:
1490:
1482:
1478:
1474:
1470:
1469:MarcusMaximus
1465:
1461:
1456:
1455:
1449:
1447:
1446:line segments
1443:
1436:
1433:
1432:
1427:
1423:
1419:
1415:
1414:line segments
1411:
1407:
1406:
1405:
1404:
1401:
1397:
1393:
1392:MarcusMaximus
1389:
1385:
1384:
1381:
1377:
1373:
1372:MarcusMaximus
1369:
1365:
1360:
1356:
1352:
1351:
1350:
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1319:
1315:
1311:
1307:
1302:
1298:
1297:
1296:
1294:
1286:
1282:
1278:
1274:
1273:
1272:
1271:
1268:
1264:
1260:
1259:MarcusMaximus
1256:
1251:
1246:
1245:
1239:
1235:
1231:
1213:
1210:
1196:
1193:
1189:
1185:
1162:
1152:
1144:
1141:
1137:
1133:
1115:
1100:
1092:
1089:
1085:
1081:
1063:
1048:
1040:
1039:
1037:
1036:
1016:
1013:
1003:
993:
977:
968:
947:
938:
917:
908:
893:
885:
884:
883:
882:
878:
874:
856:
841:
832:
828:
827:
823:
819:
815:
811:
807:
803:
799:
795:
791:
787:
783:
779:
775:
771:
767:
763:
759:
755:
751:
747:
743:
739:
735:
731:
730:
726:
722:
718:
714:
710:
706:
702:
697:
696:
692:
691:
686:
685:
681:
680:
679:
678:
674:
670:
666:
662:
658:
653:
651:
647:
639:
633:
629:
625:
621:
616:
615:
614:
613:
610:
606:
602:
601:MarcusMaximus
597:
596:
595:
594:
590:
586:
577:
573:
569:
566:
562:
558:
557:
556:
550:
546:
543:
540:
537:
536:
535:
529:
527:
526:
522:
518:
517:MarcusMaximus
514:
510:
506:
502:
482:
478:
474:
469:
465:
459:
455:
449:
445:
433:
429:
424:
420:
418:
413:
412:
408:
404:
403:MarcusMaximus
398:
394:
389:
385:
381:
368:
365:
361:
356:
353:
348:
345:
342:
341:
338:
337:
332:
331:
330:
329:
325:
321:
320:MarcusMaximus
315:
313:
307:
301:
297:
293:
289:
285:
284:
279:
278:
273:
272:
267:
262:
261:
257:
253:
252:
251:
250:
246:
242:
241:MarcusMaximus
237:
233:
228:
221:
219:
218:
215:
211:
205:
204:
200:
196:
187:
172:
171:
164:
163:
159:
155:
149:
146:
145:
142:
125:
121:
117:
116:
108:
102:
97:
95:
92:
88:
87:
83:
76:
71:
68:
65:
61:
56:
52:
46:
38:
37:
27:
23:
18:
17:
1780:
1762:
1759:
1754:less general
1753:
1751:
1710:
1671:
1648:
1643:
1639:
1633:
1613:
1586:
1566:
1559:
1555:
1545:
1540:
1536:
1532:
1529:
1525:
1492:
1486:
1441:
1439:
1409:
1387:
1367:
1363:
1358:
1354:
1335:
1327:
1323:
1322:
1300:handedness).
1292:
1291:
1254:
1249:
1237:
1233:
1229:
1191:
1187:
1183:
1139:
1135:
1131:
1087:
1083:
1079:
876:
833:of a rocket
830:
821:
817:
813:
809:
805:
801:
797:
793:
789:
785:
781:
777:
773:
769:
765:
761:
757:
753:
749:
745:
741:
737:
733:
724:
720:
716:
712:
708:
704:
700:
660:
656:
654:
645:
643:
581:
575:
571:
567:position" or
564:
560:
554:
548:
544:
538:
533:
512:
508:
504:
500:
431:
427:
425:
421:
416:
414:
399:
395:
392:
316:
311:
308:
304:
265:
235:
231:
229:
225:
206:
191:
168:
153:
113:
51:WikiProjects
34:
1707:References?
1686:Lie algebra
576:orientation
549:orientation
41:Start-class
1812:Categories
1784:airplane).
1563:equations.
657:orthogonal
646:to measure
188:Relativity
75:Relativity
1437:You said:
360:neologism
214:JunCTionS
39:is rated
1795:Paolo.dL
1757:system.
1655:Paolo.dL
1593:geometry
1513:Paolo.dL
1464:UC-Davis
1460:Stanford
1418:Paolo.dL
1340:Paolo.dL
1328:fixed to
1310:Paolo.dL
1277:Paolo.dL
669:Paolo.dL
661:directed
650:observer
624:Paolo.dL
585:Paolo.dL
572:position
499:, where
380:Paolo.dL
292:Paolo.dL
208:mention
665:versors
565:angular
156:on the
129:Physics
120:Physics
70:Physics
1644:bodies
1487:I was
1194:), and
711:, and
561:linear
374:and CS
195:Ty8inf
47:scale.
1781:whole
1651:there
808:, or
780:, or
752:, or
417:basis
28:This
1799:talk
1769:talk
1738:talk
1718:talk
1698:talk
1659:talk
1621:talk
1601:talk
1574:talk
1535:and
1517:talk
1501:talk
1489:bold
1473:talk
1462:and
1422:talk
1396:talk
1376:talk
1344:talk
1314:talk
1281:talk
1263:talk
673:talk
659:and
628:talk
605:talk
589:talk
574:and
563:and
521:talk
407:talk
384:talk
324:talk
296:talk
266:real
245:talk
199:talk
1724:AJ
1308:).
1255:any
873:IMU
738:and
622:).
547:or
432:any
428:any
370:(CS
312:and
232:not
148:Mid
1814::
1801:)
1771:)
1740:)
1720:)
1700:)
1661:)
1653:.
1623:)
1603:)
1576:)
1519:)
1503:)
1475:)
1424:)
1398:)
1378:)
1368:B3
1364:B2
1359:B1
1346:)
1316:)
1283:)
1265:)
1240:).
1238:B3
1236:,
1234:B2
1232:,
1230:B1
1192:B3
1190:,
1188:B2
1186:,
1184:B1
1163:ω
1142:),
1140:B3
1138:,
1136:B2
1134:,
1132:B1
1090:),
1088:N3
1086:,
1084:N2
1082:,
1080:N1
1004:×
994:ω
818:B3
816:,
814:B2
812:,
810:B1
806:N3
804:,
802:N2
800:,
798:N1
790:B3
788:,
786:B2
784:,
782:B1
778:N3
776:,
774:N2
772:,
770:N1
762:B3
760:,
758:B2
756:,
754:B1
750:N3
748:,
746:N2
744:,
742:N1
721:B3
719:,
717:B2
715:,
713:B1
709:N3
707:,
705:N2
703:,
701:N1
675:)
630:)
607:)
591:)
578:".
523:)
475:∗
409:)
386:)
326:)
298:)
247:)
201:)
73::
1797:(
1767:(
1736:(
1716:(
1696:(
1657:(
1619:(
1599:(
1572:(
1515:(
1499:(
1471:(
1420:(
1394:(
1374:(
1342:(
1312:(
1279:(
1261:(
1250:P
1214:P
1211:Q
1206:r
1168:B
1156:N
1116:P
1111:v
1104:B
1064:Q
1059:v
1052:N
1031:.
1017:P
1014:Q
1009:r
999:B
987:N
978:+
969:P
964:v
957:B
948:+
939:Q
934:v
927:N
918:=
909:P
904:v
897:N
877:Q
857:P
852:v
845:N
831:P
822:P
794:P
766:P
734:P
725:P
671:(
626:(
603:(
587:(
570:"
559:"
519:(
513:a
509:b
505:x
501:x
483:a
479:x
470:a
466:R
460:b
456:=
450:b
446:x
435:(
405:(
382:(
376:2
372:1
322:(
294:(
243:(
197:(
173:.
160:.
53::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
