Screw theory

Screw theory
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Screw_theory".

The pitch of a pure screw relates rotation about an axis to translation along that axis.

Screw theory was developed by Sir Robert Stawell Ball in 1876, for application in kinematics and statics of mechanisms (rigid body mechanics). It is a way to express displacements, velocities, forces and torques in three dimensional space, combining both rotational and translational parts. Recently screw theory has regained importance and has become an important tool in robot mechanics, mechanical design, computational geometry and multi-body dynamics.

Fundamental theorems include Poinsot's theorem (Louis Poinsot, 1806) and Chasles' theorem (Michel Chasles, 1832). Other prominent contributors include Julius Plücker, W. K. Clifford, F. M. Dimentberg, Kenneth H. Hunt, J. R. Phillips.

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1 Basic concepts
2 Twists as general displacements
3 Calculating Twists
- 3.1 Revolute Joints
- 3.2 Prismatic Joints
4 Calculus of screws
5 See also
6 References
7 External links

Basic concepts

The basic terms associated with screw theory are screw, twist and wrench.

Screw

In the sense of rigid body motion, a screw is a way of describing a displacement. It can be thought of as a rotation about an axis and a translation along that same axis. Any general displacement can be described by a screw, and there are methods of converting between screws and other representations of displacements, such as homographic transformations.

In rigid body dynamics, velocities of a rigid body and the forces and torques acting upon it can be represented by the concept of a screw. The first kind of screw is called a twist, and represents the velocity of a body by the direction of its linear velocity, its angular velocity about the axis of translation, and the relationship between the two, called the pitch. The second kind of screw is called a wrench, and it relates the force and torque acting on a body in a similar way.

Apart from the internal force that keeps the body together this motion does not require a force to be maintained, provided that the direction is a principal axis of the body.

In general, a three dimensional motion can be defined using a screw with a given direction and pitch. Four parameters are required to fully define a screw motion, the 3 components of a direction vector and the angle rotated about that line. In contrast, the traditional method of characterizing 3-D motion using Euler Angles requires 6 parameters, 3 rotation angles and a 3x1 translation vector.

A pure screw is simply a geometric concept which describes a helix. A screw with zero pitch looks like a circle. A screw with infinite pitch looks like a straight line, but is not well defined.

Any motion along a screw can be decomposed into a rotation about an axis followed by a translation along that axis. Any general displacement of a rigid body can therefore be described by a screw.

Twist

Twists represent velocity of a body. For example, if you were climbing up a spiral staircase at a constant speed, your velocity would be easily described by a twist. A twist contains 6 quantities. Three linear and three angular. Another way of decomposing a twist is by 4 line coordines (see Plücker coordinates), 1 scalar pitch value and 1 twist magnitude.

Wrench

Wrenches represent forces and torques. One way to conceptualize this is to consider someone who is fastening two wooden boards together with a metal screw. The person turns the screw (applies a torque), which then experiences a net force along its axis of rotation.

Transformations

Twists

The velocities of each particle within a rigid body define a helical field called the velocity twist. To move representation from point A to point B, one must account for the rotation of the body such that:

In screw notation velocity twists transform with a 6x6 transformation matrix

$\hat{v_B} = \begin{bmatrix} \vec v_B \\ \vec \omega \end{bmatrix} = \begin{bmatrix} 1 & - \vec r_{AB} \times \\ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} \vec v_A \\ \vec \omega \end{bmatrix}$

Where:

$\vec v_A$ denotes the linear velocity at point A

$\vec v_B$ denotes the linear velocity at point B

$\vec \omega$ denotes the angular velocity of the rigid body

$\vec r_{AB} \times$ denotes the 3x3 cross product matrix (see: Cross product)

Wrenches

Similarly the equipolent moments expressed at each location within a rigid body define a helical field called the force wrench. To move representation from point A to point B, once must account for the forces on the body such that:

In screw notation force wrenches transform with a 6x6 transformation matrix

$\hat{f_B} = \begin{bmatrix} \vec f \\ \vec \tau_B \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ - \vec r_{AB} \times & 1 \end{bmatrix} \cdot \begin{bmatrix} \vec f \\ \vec \tau_A \end{bmatrix}$

Where:

$\vec \tau_A$ denotes the equipollent (link: wikibooks.org ) moment at point A

$\vec \tau_B$ denotes the equipollent (link: wikibooks.org ) moment at point B

$\vec f$ denotes the total force applied to the rigid body

$\vec r_{AB} \times$ denotes the 3x3 cross product matrix (see: Cross product)

Twists as general displacements

Given an initial configuration $g\left(0\right) \in SE\left(n\right)$ , and a twist $\xi \in R^n$ , the homogeneous transformation to a new location and orientation can be computed with the following formula:

$g\left(\theta\right) = \exp(\hat{\xi}\theta) g\left(0\right)$

where $θ$ represents the parameters of the transformation.

Calculating Twists

Twists can be easily calculated for certain common robotic joints.

Revolute Joints

For a revolute joint, given the axis of revolution $\omega \in R^3$ and a point $q \in R^3$ on that axis, the twist for the joint can be calculated with the following forumula:

$\xi = \begin{bmatrix} q \times \omega \\ \omega \end{bmatrix}$

Prismatic Joints

For a prismatic joint, given a vector $v \in R^3$ pointing in the direction of translation, the twist for the joint can be calculated with the following formula:

$\xi = \begin{bmatrix} v \\ 0 \end{bmatrix}$

Calculus of screws

The science of screw mapping has been advanced by the use of dual quaternions developed by W.K. Clifford, Eduard Study, F.M. Dimentberg, and more recently by A.T. Yang (see reference). In brief, multiplications of dual numbers correspond to shear mapping, and inner automorphisms by unit quaternions model rotations about an axis; the synthesis of these operations in the dual quaternions displays the screw mapping through a ring multiplication. The transformation of equation (24) on page 271 of Yang's 1974 essay is an example of the application of a projectivity in inversive ring geometry.

References

Ball, R. S. (1876). The theory of screws: A study in the dynamics of a rigid body. Hodges, Foster.
William Kingdon Clifford (1873), "Preliminary Sketch of Biquaternions", Paper XX, Mathematical Papers, p.381.
A.T. Yang (1974) "Calculus of Screws" in Basic Questions of Design Theory, William R. Spillers, editor,Elsevier, pages 266 to 281.
Roy Featherstone (1987). Robot Dynamics Algorithms. Springer. ISBN 0898382300.