Report writing at UCI during work on my MSc in Mechanical Engineering. 2006
General motion of a body in 2D that includes rotation and translation. We use for the rotation matrix, the vector for the translation and the matrix for displacement. Hence we write
General motion of a body in 2D that includes rotation and translation. Another name for this is rigid displacement. Distance between points in a body remain unchanged before and after spatial displacement.
Recall that the transformation that deﬁnes spatial displacement, which is given by is non-linear due to the presence of the translation term . It is more convenient to be able to work with linear transformation, therefore we add a fourth component to the position vector which is always and rewrite the transformation, now calling it as which maps to as
Or in full component form
Or in short form
The set of points that remain ﬁxed during the rotation deﬁned by . We use Rodrigues vector to deﬁne the rotation axis.
A pole is point that remains ﬁxed during planar or spatial displacement. Planar displacement (2D) have a pole, but 3D spatial displacement do not have a pole in general, since the requirement for a pole is to have an inverse for which for 3D is not possible since has an eigenvalue of 1.
The following diagram is an example of a pole under planar displacement.
For spatial displacement (3D), one condition, when satisﬁed will result in a ﬁxed point. This condition occurs when the translation vector is perpendicular to the rotation vector . The pole in this case is given by
Not only is this point ﬁxed, by any point on the line is also ﬁxed. Where is a parameter and is a unit vector. Hence we can a ﬁxed line under such a spatial displacement, and this line is called the rotation axis.
This is a special case of spatial displacement when the translation vector is perpendicular to the rotation axis To make the diﬀerence more clear, the following diagram is an illustration of spatial displacement which is not rotational displacement, and one which is.
Deﬁnes the rotation axis under . Using as the unit vector along , then where is the length of vector given by where is the rotation angle around the rotation axis. Hence we can write
We see that the
Hence at , which goes to inﬁnity. A plot of the function is below showing the discontinuities at
Screw is deﬁned as a pair of vectors such that and . The pitch of the screw is deﬁned as
Given a line deﬁned by a parametric equation , where is a reference point and is a unit vector along the line, the Plücker coordinates of this line is given by represents the moment of the line around the origin of the reference frame. The following diagram helps to illustrates this.
We said above that a screw is deﬁned as a pair of vectors . Given the plücker coordinates of the screw line , then we can write the pair of vectors that deﬁnes the screw associated with this screw line as follows
Where is the screw pitch and
These are some important formulas put here for quick reference.
See (4A) below for derivation of the above where
From we obtain a row vector and call it
(This is rodrigues vector) From we obtain a row vector and call it
(This is unit vector along the screw axis). We also have
And in terms of the vectors the above becomes
We seek to derive an expression for the rotation matrix in 3D. Consider a point in 3D being acted upon by a rotation matrix . Let the ﬁnal coordinates of this point be . We consider the position vectors of these points, so we will designate these points by their position vectors and from now on. The following diagram illustrate this.
Hence we have that
Since we can write
The geometric meaning of the above last equation is shown in the following diagram
To introduce the rotation matrix into the equations, we can write as
Similarly, we obtain
From (3) we obtain
Substitute the above expression for into (2) we obtain
We now call the matrix as
We can also write from above the following
Now rewrite (3A) as
What is the geometric meaning of ? we see that it is an operator that acts on vector to produce the vector , however from the diagram above we see that the vector is perpendicular to and scaled down.
This implied that is skew-symmetric and have the form given by
Matrix can be written as a column vector called such that for any vector we have
The vector is called Rodrigues vector.
Let the vector then we write
Where is some vector perpendicular to such that its cross product with results in
The vector is the vector that deﬁnes the rotation axis. The length of this vector is and a unit vector along is called , Hence we can write
Now we solve for
We will use equation (3B) to ﬁnd the form of for diﬀerent rotations.
Consider for example the 3D rotation around the axis given by
From (3B) we obtain
For example for the matrix is
Hence we see that , , hence
Hence geometrically, Rodrigues vector is along the axis and in the positive direction as illustrated in the following diagram.
The size of the vector depends on the angle or rotation . The Rodrigues vector will be largest when is almost and smallest when is zero.
We now derive an expression for the invariant under spatial displacement, which is the screw axis.
Consider the following general spatial displacement
We start be decomposing the translation vector into 2 components: One parallel ( and one perpendicular to the rotation axis of as follows (notice that is perpendicular to the which is where the rotation occurs around)
Recall from earlier that the vector is a unit vector along the rotation axis of in the direction of . We found earlier that . Hence we can write that where
Let us redraw the above diagram putting all of these symbols to make the discussion more clear.
Then the spatial displacement operator can be written as follows
Hence the spatial displacement is
Hence we see that the spatial displacement can be viewed as rotational displacement followed by pure translation. Recall from above that rotational displacement is a special type of spatial displacement where the translation part is perpendicular to the rotation axis. We can represent the above equation geometrically as follows
Rotational displacement have a ﬁxed point given by
The derivation of the above equation is as follows.
Since we seek a ﬁxed point , then we write
Using Cayley's formula, derived above in equation (3C), reproduced below
and substitute for in (4), we obtain
Multiply both sides by
But by deﬁnition
Hence (4B) becomes
Take the cross product of both sides w.r.t. we obtain
To simplify the above, use the relation
Apply the above relation on the RHS of (4C), hence (4C) can be rewritten as
But the vector is perpendicular to hence and the above simpliﬁes to
Now we continue to derive an expression for the screw axis.
We now consider a line that passed through this point and is parallel to the rotation axis of (in other words, along the same direction as the vector ). Any point along this line remain ﬁxed relative to the rotational displacement part of the spatial displacement.
In addition, since the translation part of the spatial displacement, and given by is a translation in the same direction and slides along the vector as changes, then this line will also remain ﬁxed relative to the translation part as well.
Hence we conclude that the line will remain ﬁxed relative to the overall spatial displacement .
This line is called the screw axis. And this type of decomposing the spatial displacement into rotational displacement followed by pure translation is called the screw displacement.
How to geometrically ﬁnd the screw axis? Let us ﬁnd the point ﬁrst. Let take an example similar to the above diagrams, where say , , hence
On the above diagram we now can draw the screw axis using the above coordinates for the point
It is important to note that it is the line given by (the screw axis) which is ﬁxed under the spatial displacement, and not any one single point on this line.
We now derive a new expression for spatial displacement using the screw axis line, which we denote as, the angle of rotation and the amount of slide along the screw axis.
The screw matrix is a new mathematical operator that we can use to denote spatial displacement between 2 diﬀerent reference frames. Earlier we showed that we can use the homogeneous transformation operator to denote spatial displacement, and now we seek to obtain a new expression for a spatial displacement operator which is a function of the following 3 parameters
This is in addition to the mathematical object we examined earlier which is
Since is a ﬁxed point under the translation by hence we write
Substitute the above into (5) we obtain
But the spatial displacement is deﬁned as
Hence using (5A) the above becomes
Using the notation of for angle of rotation and the slide and to denote the screw axis, we can write (5B) as
So now (5C) is an expression for the spatial operator in terms of and
We call the screw Matrix.
The following diagram helps to illustrate this.
So far we have discussed spatial displacements applied to points. We showed two Matrices can be used to accomplish this. The homogeneous transformation matrix and the screw matrix where
We now show a matrix which is used for the spatial displacement of a line and not just a point. This is based on using the plücker coordinates of a line to represent the line. Geometrically this is illustrated in the following diagram
Since operates on the Plücker coordinates of a line, then we write
To processed further, we now assume a point on the line such that as illustrated below
Hence we can now write, in the new coordinates
Since is a rotation matrix, then
Where is a skew-symmetric matrix deﬁned such that
Hence (6D) can be written as
By substitution of (6E) and (6C) into RHS of (6B) we obtain
Hence the above becomes
Now substitute in the above we obtain
The we can write
We now analyze the spatial displacement of the screw axis under
Recall that a screw axis Plücker coordinates are written as
Where is the unit vector in along the axis, is the ﬁxed reference point on the axis and is the screw pitch and where is the Rodrigues vector. To make things more clear, we illustrate these quantities in the following diagram
We now perform spatial displacement on the screw axis using its Plücker coordinates
Rewrite the above as general plücker coordinates then the spatial displacement of this general line is as seen above in (6B) becomes
We need seek to evaluate the above coordinates for the screw axis given in (6C)
In other words, given and we need to ﬁnd and
The ﬁrst plücker coordinate transforms easily as
But is just the Rodrigues vector in the new coordinates system which we call lower case hence
Now we need to transform the second plücker coordinate
With the help of the matrix which can be used to rewrite the cross product of 2 vectors as times one the 2 vectors, we can write
But hence the above becomes
And since is a scalar, we can write the above as
Now we need to compute which is
Here we show that the screw axis is invariant of the transformation matrix derived in the last section.
Gives the screw axis line deﬁned by its plucker coordinates we need to show the following
(1) can be written as
Now, if we can ﬁnd solution to the above others than then we have showed that (1) is valid. Equation (1) can be written as
Hence we obtain
Hence we obtain 2 equations
From the ﬁrst equation we obtain substitute into the second equation
Introduce hence the above becomes
And from the cayley's formula for
Then (3) becomes
But we know that where is the rotation angle, and hence (4) becomes
Hence we showed that a non zero solution for (2) exist given by where is given in (5). This shows that (1) is valid which is what we wanted to show.
Hence the screw axis is invariant of the transformation matrix .