Review of the geometry of screw Axes

by Nasser Abbasi




Definitions, Important formulas, and terminology

Planar displacement

General motion of a body in 2D that includes rotation and translation. We use $[A]$ for the rotation matrix, the vector $\QTR{bf}{d}$ for the translation and the matrix $[T]$ for displacement. Hence we write MATH

Spatial displacement

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.

Homogeneous Transformation

Recall that the transformation that defines spatial displacement, which is given by MATH is non-linear due to the presence of the translation term $\QTR{bf}{d}$. 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 $1$ and rewrite the transformation, now calling it as $T$ which maps $X$ to $x$ as

MATH

Or in full component form

MATH

Or in short form

MATH

The rotation axis

The set of points that remain fixed during the rotation defined by $[A]$. We use Rodrigues vector $\QTR{bf}{b}$ to define the rotation axis.

A Pole or a Fixed point

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 $[I-A]$ which for 3D is not possible since $A$ has an eigenvalue of 1.

The following diagram is an example of a pole under planar displacement.


pole2d.png

For spatial displacement (3D), one condition, when satisfied will result in a fixed point. This condition occurs when the translation vector $\QTR{bf}{d}$ is perpendicular to the rotation vector $\QTR{bf}{b}$. The pole in this case is given by MATH

Not only is this point $\QTR{bf}{c}$ fixed, by any point on the line MATH is also fixed. Where $t$ is a parameter and $\QTR{bf}{S}$ is a unit vector. Hence we can a fixed line under such a spatial displacement, and this line is called the rotation axis.

Rotational displacement

This is a special case of spatial displacement when the translation vector $\QTR{bf}{d}$ is perpendicular to the rotation axis $\QTR{bf}{b.}$ To make the difference more clear, the following diagram is an illustration of spatial displacement which is not rotational displacement, and one which is.


rotational_disp.png

Rodrigues vector $\QTR{bf}{b}$

Defines the rotation axis under $[A]$. Using $\QTR{bf}{S}$ as the unit vector along $\QTR{bf}{b}$, then MATH where $k$ is the length of vector $\QTR{bf}{b}$ given by MATH where $\theta$ is the rotation angle around the rotation axis. Hence we can write MATH

We see that the MATH

Hence at $\theta=\pm180^{0}$ , MATH which goes to infinity. A plot of the function MATH is below showing the discontinuities at $\pm\frac{\pi}{2}$


tan.png

Screw

Screw is defined as a pair of vectors MATH such that MATH and MATH. The pitch of the screw $p_{\omega }$ is defined as MATH

Plücker coordinates of a line

Given a line defined by a parametric equation MATH, where $C$ is a reference point and $S$ is a unit vector along the line, the Plücker coordinates of this line is given by MATH.

MATH represents the moment of the line around the origin of the reference frame. The following diagram helps to illustrates this.


plucker_coordinates.png

screw defined in terms of the plücker coordinates of its axis

We said above that a screw is defined as a pair of vectors MATH. Given the plücker coordinates of the screw line MATH, then we can write the pair of vectors that defines the screw associated with this screw line as follows

MATH

Where $p_{\omega}$ is the screw pitch and MATH

Important relations between Cayley's Matrix $\left[ B\right] $

These are some important formulas put here for quick reference.

MATH

See (4A) below for derivation of the above.

Where MATH

and also

MATH

WhereMATH

From $[B]$ we obtain a row vector and call it MATH

(This is rodrigues vector)

From $\left[ S\right] $ we obtain a row vector and call it

MATH

(This is unit vector along the screw axis).

We also have

MATH

And in terms of the vectors MATH the above becomes

MATH

Introduction to the rotation matrix in 3D

We seek to derive an expression for the rotation matrix in 3D. Consider a point $x_{1}$ in 3D being acted upon by a rotation matrix $A$. Let the final coordinates of this point be $x_{2}$. We consider the position vectors of these points, so we will designate these points by their position vectors $\QTR{bf}{x}_{1}$ and $\QTR{bf}{x}_{2}$ from now on. The following diagram illustrate this.


cayley.png

Hence we have that MATH

Since MATH we can write MATH

The geometric meaning of the above last equation is shown in the following diagram


cayley2.png

To introduce the rotation matrix $A$ into the equations, we can write MATH as MATH

Hence MATH

Similarly, we obtain

MATH

From (3) we obtain

MATH

Substitute the above expression for $\QTR{bf}{x}_{1}$ into (2) we obtain

MATH

We now call the matrix MATH as $B$

MATH

We can also write from above the following

MATH

Now rewrite (3A) as

MATH

What is the geometric meaning of $\left[ B\right] $? we see that it is an operator that acts on vector MATH to produce the vector MATH, however from the diagram above we see that the vector MATH is perpendicular to MATH and scaled down.

Hence

MATH

This implied that $B$ is skew-symmetric and have the form given by

MATH

Matrix $B$ can be written as a column vector called MATH such that for any vector $y$ we have MATH

The vector $\QTR{bf}{b}$ is called Rodrigues vector.

Let the vector MATH then we write

MATH

Where $\QTR{bf}{b}$ is some vector perpendicular to $\QTR{bf}{y}$ such that its cross product with $y$ results in MATH

The vector $\QTR{bf}{b}$ is the vector that defines the rotation axis. The length of this vector is $k$ and a unit vector along $\QTR{bf}{b}$ is called $\QTR{bf}{S}$, Hence we can write MATH

Now we solve for $B.$

We will use equation (3B) to find the form of $B$ for different rotations.

Consider for example the 3D rotation around the $x$ axis given by MATH

From (3B) we obtain


bx.png

For example for $\theta=45^{0}$ the $B$ matrix is

MATH

Hence we see that $b_{z}=0$, $b_{y}=0$, $b_{x}=-0.41421$ henceMATH

Hence geometrically, Rodrigues vector is along the $x$ axis and in the positive direction as illustrated in the following diagram.


cayley4.png

The size of the $\QTR{bf}{b}$ vector depends on the angle or rotation $\theta$. The Rodrigues vector will be largest when $\theta$ is almost $180^{0}$ and smallest when $\theta$ is zero.

Screw displacement

We now derive an expression for the invariant under spatial displacement, which is the screw axis.

Consider the following general spatial displacement


screwaxis.png

We start be decomposing the translation vector $\QTR{bf}{d}$ into 2 components: One parallel ($\QTR{bf}{d}_{2})$ and one perpendicular MATH to the rotation axis of $[A]$ as follows (notice that $\QTR{bf}{d}^{\ast}$ is perpendicular to the $x-axis$ which is where the rotation occurs around)


screwaxis2.png

Recall from earlier that the vector $\QTR{bf}{S}$ is a unit vector along the rotation axis of $[A]$ in the direction of $\QTR{bf}{b}$. We found earlier that MATH. Hence we can write that MATH where MATH

Let us redraw the above diagram putting all of these symbols to make the discussion more clear.


screwaxis3.png

Now since MATH

Then the spatial displacement operator $T$ can be written as follows

MATH

Hence the spatial displacement is MATH

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
screwaxis4.png

Derivation for expression for finding reference point for screw axis

Rotational displacement have a fixed point given by

MATH

The derivation of the above equation is as follows.

Since we seek a fixed point $\QTR{bf}{C}$, then we write

MATH

Using Cayley's formula, derived above in equation (3C), reproduced below

MATH

and substitute for $A$ in (4), we obtain

MATH

Multiply both sides by $[I-B]$

MATH

But by definition MATH

Hence (4B) becomes

MATH

Take the cross product of both sides w.r.t. $\QTR{bf}{b}$ we obtain

MATH

To simplify the above, use the relation

MATH

Apply the above relation on the RHS of (4C), hence (4C) can be rewritten as

MATH

But the vector $\QTR{bf}{C}$ is perpendicular to $\QTR{bf}{b}$ hence MATH and the above simplifies to

MATH

Now we continue to derive an expression for the screw axis.

We now consider a line $L$ that passed through this point $\QTR{bf}{C}$ and is parallel to the rotation axis of $[A]$ (in other words, along the same direction as the vector $\QTR{bf}{S}$). Any point along this line remain fixed relative to the rotational displacement MATH part of the spatial displacement.

In addition, since the translation part of the spatial displacement, and given by MATH is a translation in the same direction and slides along the vector $kS$ as $k$ changes, then this line will also remain fixed relative to the translation part as well.

Hence we conclude that the line $L$ will remain fixed relative to the overall spatial displacement $T$.

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 $\QTR{bf}{C}$ first. Let take an example similar to the above diagrams, where say $\theta=30^{0},$ MATH, MATH , henceMATH hence MATH

On the above diagram we now can draw the screw axis using the above coordinates for the point $\QTR{bf}{C}$


screwaxis5.png

It is important to note that it is the line given by MATH (the screw axis) which is fixed under the spatial displacement, and not any one single point on this line.

The Screw Matrix

We now derive a new expression for spatial displacement using the screw axis line, which we denote as$\ \QTR{it}{S}$, the angle of rotation $\theta$ and the amount of slide $k$ 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 MATH 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

  1. The screw axis line which we call $\QTR{it}{S}$ with the plucker coordinates MATH

  2. The angle or rotation $\theta$

  3. The amount of slide $k$

This is in addition to the mathematical object we examined earlier which is

Since

MATH

Since $C$ is a fixed point under the translation by $\QTR{bf}{d}^{\ast}$ hence we write

MATH

Hence MATH

Substitute the above into (5) we obtain

MATH

But the spatial displacement $T$ is defined as

MATH

Hence using (5A) the above becomes

MATH

Using the notation of $\theta$ for angle of rotation and the slide $k$ and $\QTR{Large}{S}$ to denote the screw axis, we can write (5B) as

MATH

So now (5C) is an expression for the spatial operator $T$ in terms of $\QTR{bf}{S,}k\ $and $\theta$

Recall that MATH

Where MATH

We call MATH the screw Matrix.

The following diagram helps to illustrate this.


screwmatrix.png

The spatial displacement of screws

So far we have discussed spatial displacements applied to points. We showed two Matrices can be used to accomplish this. The homogeneous transformation matrix MATH and the screw matrix MATH MATH where MATH

We now show a matrix $[\hat{T}]$ 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


screw_spatial.png

Since MATH operates on the Plücker coordinates of a line, then we write

MATH

To processed further, we now assume a point $\QTR{bf}{q}$ on the line $x$ such that $\QTR{bf}{x}=$ MATH as illustrated below
screw_spatial_2.png
Hence we can now write, in the new coordinates

MATH

But

MATH

And

MATH

Since $[A]$ is a rotation matrix, then

MATH

Where $[D]$ is a skew-symmetric matrix defined such that MATH

Hence (6D) can be written as

MATH

By substitution of (6E) and (6C) into RHS of (6B) we obtain

MATH

But $\QTR{bf}{q-p=x}$

Hence the above becomes

MATH

Now substitute $\QTR{bf}{q=p+x}$ in the above we obtain

MATH

Hence

MATH

The we can writeMATH

We now analyze the spatial displacement of the screw axis under $[\hat{T}]$

Recall that a screw axis Plücker coordinates are written as

MATH

Where $\QTR{bf}{S}$ is the unit vector in along the axis, $\QTR{bf}{p}$ is the fixed reference point on the axis and $p_{\omega}$ is the screw pitch and MATH where $\QTR{bf}{W}$ is the Rodrigues vector. To make things more clear, we illustrate these quantities in the following diagram


screwaxis7.png

We now perform spatial displacement on the screw axis using its Plücker coordinates MATH

Rewrite the above as general plücker coordinates MATH then the spatial displacement of this general line is as seen above in (6B) becomes

MATH

We need seek to evaluate the above coordinates for the screw axis given in (6C)

In other words, given MATH and MATH we need to find MATH and $[D][A]\QTR{bf}{w}$

The first plücker coordinate $\omega\QTR{bf}{S}$ transforms easily as MATH

But MATH is just the Rodrigues vector in the new coordinates system which we call lower case $\QTR{bf}{w}$ hence

MATH

Now we need to transform the second plücker coordinate MATH

With the help of the $[D]$ matrix which can be used to rewrite the cross product of 2 vectors as $[D]$ times one the 2 vectors, we can write

MATH

But MATH hence the above becomes

MATH

And since $\omega$ is a scalar, we can write the above as

MATH

Now we need to compute MATH which is MATH

MATH

Hence

MATH

The screw axis of a displacement

Here we show that the screw axis is invariant of the $6\times 6$ transformation matrix $[\hat{T}]$ derived in the last section.

Gives the screw axis line $S$ defined by its plucker coordinates MATH we need to show the following

MATH

(1) can be written as

MATH

Now, if we can find solution to the above others than $S=0$ then we have showed that (1) is valid. Equation (1) can be written asMATH

But

MATH

Hence we obtain

MATH

Hence we obtain 2 equations

MATH

From the first equation we obtain MATH substitute into the second equation

MATH

Introduce MATH hence the above becomes

MATH

And from the cayley's formula for $A$

MATH

Then (3) becomes

MATH

Hence

MATH

But we know that MATH where $\theta $ is the rotation angle, and MATH hence (4) becomes

MATH

Hence MATH

Hence we showed that a non zero solution for (2) exist given by MATH where $V$ is given in (5). This shows that (1) is valid which is what we wanted to show.

Hence the screw axis $S$ is invariant of the $6\times 6$ transformation matrix $[\hat{T}]$.

References

  1. Geometric Design Of Linkages. By Professor J.Michael McCarthy. Springer publication.

  2. Introduction to Theoretical Kinematics. By Professor J.Michael McCarthy

  3. Class notes, MAE245. Theoretical Kinematics spring 2004. UCI. Professor J.Michael McCarthy