Frame of References and Newton's Law of Motion

A Brief Review of Newton's Laws of Motion

Let's review certain basic concepts of motion, namely Newton's first two Laws of Motion, which are presumably as basic and fundamental as any natural law can be:

(1) The Law of Inertia: A body which has no force acting on it will move with uniform motion (that is, with constant speed and direction).

(2) The Force Law: If a force acts on a body, it will not move uniformly, but will be accelerated in the direction of the force at a rate proportional to the force, and inversely proportional to its inertia, or mass.

Now, these two laws seem very simple and obvious, and perfectly reasonable and correct. So much so, that if we see an object which is moving uniformly, we presume that it must not have any force (or at least, any net force) acting on it; whereas if we see an object which is accelerating, we presume it must have some force acting on it, in the direction of its acceleration. The strange thing is, that it is not only very easy, but actually more normal than not, for Newton's Laws of Motion to be wrong. For we often find ourselves in a situation in which bodies appear to be accelerating under the influence of some force, even though no such force is actually acting on them.

Inertial Frames of Reference

To understand how such a statement could possibly be true, we need to discuss frames of reference. A frame of reference is simply that portion of the world around us, which we use to measure the motion of moving bodies. For all practical purposes, the world around us appears to be at rest, and insofar as that statement is true, then any motion we measure relative to our surroundings is correctly observed, and if a motion appears uniform, it must truly be uniform, and if the motion appears nonuniform, then it must truly be nonuniform.

But suppose that instead of using the world around us, we use some particular portion of the world, such as a railway car, which is moving relative to the rest of the world. As the car moves along its tracks, the motion of whatever object we are observing will not be measured correctly, but will have an error equal to the motion of the railway car. So wouldn't that affect our observations of the moving object? Of course it would; but as long as the motion of the railway car is absolutely uniform and unchanging, whatever error we make in observing the moving object will be absolutely constant and unchanging, as well; so if the object has uniform motion in the everyday world that is our normal frame of reference, then it will appear to have a different but still absolutely uniform motion in the frame of reference represented by the railway car. And if the moving object is moving non uniformly, because there is a force acting on it, in the everyday world, then it will also be moving non uniformly, by exactly the same amount, in the frame of reference of the moving railway car.

This concept, that a uniformly moving frame of reference, such as the railway car, should not change the laws of motion, was actually first proposed, albeit in a slightly different form, decades prior to Newton's stating his Laws, by Galileo Galileo. Galileo proposed that in all frames of reference which are moving uniformly relative to each other, the laws of nature must be the same. This statement encompasses not only Newton's Laws, but all the laws of nature, and is the basis of what we called Galilean Relativity.

Non-Inertial Reference Frames

But how in the world could you possibly find yourself in such a situation that the Law of Inertia would appear to be wrong? Very easily; for remember, in discussing the uniformly moving railway car, it was stated to be moving uniformly, so that any observations of a moving object would have an error, relative to the everyday world, but an error that was as constant as the motion of the railway car, so that uniform motion still appeared uniform, and nonuniform motion still appeared nonuniform.

But what if the motion of the railway car is not constant? Then the error introduced into measurements of the moving object would be changing with time, which would make a constant motion look nonuniform, and therefore accelerated, which would suggest that a force is acting on the moving object, which is obviously incorrect, if it is really moving with constant motion. In other words, if your frame of reference has a non-uniform, or accelerated motion, then the Law of Inertia will appear to be wrong, and you must be in a non-inertial frame of reference. So, although all frames of reference which are moving uniformly relative to an inertial reference frame are also inertial reference frames, all frames of reference which are moving non-uniformly (are accelerated) relative to an inertial reference frame are non-inertial reference frames.

 Frame-of-reference for combining rotations

When we are modelling rotation it is important to consider the frame of reference. In the following we will use the example of an aeroplane, to a person on the ground the aeroplane might be turning right, but to a person in the aeroplane the ground will appear to be turning left. So the orientation of the plane relative to the ground is minus the orientation of the ground relative to the plane.
But the situation gets more complicated when we are trying to work out the result of a sequence of rotations, I can think of two possible ways to combine two rotations. In the first case we used the fixed global co-ordinates for everything, in the second case we specify the second rotation in the local co-ordinates of the aeroplane, which have been rotated by the first rotation.
Note: the following assumes:

• A rectangular (Cartesian) co-ordinate system.
• A right hand convention for direction of rotations.

The following examples show two possible ways to combine rotations:

Global frame-of-reference

always defines rotations in global coordinates, in this case x=right, y=up, z=toward viewer. In this case the matrix or quaternion representations of the individual rotations can be combined by multipling in the following order:
[resulting transform] = [second transform] * [first rotation]
To see why this is we could multiply vector Vin by matrix [A] to produce an intermediate vector Vmid, we then multipy this by matrix [B]. If we sustitute [A]Vin for Vmid we can see that the overall matrix = [B][A].

Local frame-of-reference

Defines rotations relative to local coordinates, in other words the coordinate system rotates with the object. In this case the matrix or quaternion representations of the individual rotations can be combined by multipling in the following order:
[resulting transform] = [first rotation] * [second transform]
The easiest way to work in this situation is, instead of imagining ourselves looking at the object from outside and calculating its coodinates, instead we imagine ourselves sitting on the object and measure the coordinates of external points. This means that instead of working in terms of the objects position transform, we are working in terms of its inverse (which is the transpose for rotations as explained here). We can now calculate the result of consecutive transforms by multipling the individual transforms as we did before. Normally when we multiply two matricies, we calculate an element by combinig the corresponding row in the first matrix by the corresponding column in the second matrix. However since the matricies are transposed this is equivilant to reversing the order of matricies from the global case. This forms an isomorphism as shown here:

Which to use

The Global frame of reference is easiest for us to understand and it makes sense when we are moving objects by hand one step at a time. However, for real world objects, such as aeroplane or a robot arm the first rotation tends to alter the coordinate reference for susequent rotations. Therefore the local frame of reference might be used more often.

Example

Here are a set of rotations in global and local for comparison:

Global Frame-of-Reference	Local Frame-of-Reference
Initial Orientation:	Initial Orientation:
First rotate about y (in global frame) through +90 degrees to give: 0 0 1 0 1 0 -1 0 0	First rotate about y (in local frame which is the same as global frame at this stage) through +90 degrees to give: 0 0 1 0 1 0 -1 0 0 Note that the local co-ordinates are rotated with the aeroplane
Now rotate about z through +90 degrees. resulting matrix 0 -1 0 0 0 1 -1 0 0 = rotate +90 about absolute z (this rotation) 0 -1 0 1 0 0 0 0 1 * rotate +90 about y (first rotation) 0 0 1 0 1 0 -1 0 0	Now rotate about z in the updated local co-ordinates through +90 degrees. resulting matrix 0 0 1 1 0 0 0 1 0 = rotate +90 about y (first rotation) 0 0 1 0 1 0 -1 0 0 * rotate +90 about local z (this rotation) 0 -1 0 1 0 0 0 0 1
Now rotate about y through -90 degrees. resulting matrix 1 0 0 0 0 1 0 -1 0 = rotate -90 about absolute y (this rotation) 0 0 -1 0 1 0 1 0 0 * result from above 0 -1 0 0 0 1 -1 0 0 So the final result is equivalent to rotating about the x axis (even though none of the individual rotations were about the x axis)	Now rotate about y in the updated local co-ordinates through -90 degrees. resulting matrix 1 0 0 0 0 -1 0 1 0 = result from above 0 0 1 1 0 0 0 1 0 * rotate -90 about absolute y (this rotation) 0 0 -1 0 1 0 1 0 0

References

• http://cseligman.com/text/physics/inertial.htm