Screw theory is the algebraic calculation of pairs of vectors, such as angular and linear velocity, or forces and moments, that arise in the kinematics and dynamics of rigid bodies.^[1][2]

Screw theory provides a mathematical formulation for the geometry of lines which is central to rigid body dynamics, where lines form the screw axes of spatial movement and the lines of action of forces. The pair of vectors that form the Plücker coordinates of a line define a unit screw, and general screws are obtained by multiplication by a pair of real numbers and addition of vectors.^[3]

Important theorems of screw theory include: The Transfer Principle proves that geometric calculations for points using vectors have parallel geometric calculations for lines obtained by replacing vectors with screws.^[4] Chasles' theorem proves that any change between two rigid object poses can be performed by a single screw. Poinsot's theorem proves that rotations about a rigid object's major and minor -- but not intermediate -- axes are stable.

Screw theory is an important tool in robot mechanics,^[5][6][7][8] mechanical design, computational geometry and multibody dynamics. This is in part because of the relationship between screws and dual quaternions which have been used to interpolate rigid-body motions.^[9] Based on screw theory, an efficient approach has also been developed for the type synthesis of parallel mechanisms (parallel manipulators or parallel robots).^[10]

Basic concepts

A spatial displacement of a rigid body can be defined by a rotation about a line and a translation along the same line, called a screw motion. This is known as Chasles' theorem. The six parameters that define a screw motion are the four independent components of the Plücker vector that defines the screw axis, together with the rotation angle about and linear slide along this line, and form a pair of vectors called a screw. For comparison, the six parameters that define a spatial displacement can also be given by three Euler angles that define the rotation and the three components of the translation vector.

Screw

A screw is a six-dimensional vector constructed from a pair of three-dimensional vectors, such as forces and torques and linear and angular velocity, that arise in the study of spatial rigid body movement. The components of the screw define the Plücker coordinates of a line in space and the magnitudes of the vector along the line and moment about this line.

Twist

A twist is a screw used to represent the velocity of a rigid body as an angular velocity around an axis and a linear velocity along this axis. All points in the body have the same component of the velocity along the axis, however the greater the distance from the axis the greater the velocity in the plane perpendicular to this axis. Thus, the helicoidal field formed by the velocity vectors in a moving rigid body flattens out the further the points are radially from the twist axis.

The points in a body undergoing a constant twist motion trace helices in the fixed frame. If this screw motion has zero pitch then the trajectories trace circles, and the movement is a pure rotation. If the screw motion has infinite pitch then the trajectories are all straight lines in the same direction.

Wrench

The force and torque vectors that arise in applying Newton's laws to a rigid body can be assembled into a screw called a wrench. A force has a point of application and a line of action, therefore it defines the Plücker coordinates of a line in space and has zero pitch. A torque, on the other hand, is a pure moment that is not bound to a line in space and is an infinite pitch screw. The ratio of these two magnitudes defines the pitch of the screw.

Algebra of screws

Let a screw be an ordered pair

${\displaystyle {\mathsf {S}}=(\mathbf {S} ,\mathbf {V} ),}$

where S and V are three-dimensional real vectors. The sum and difference of these ordered pairs are computed componentwise. Screws are often called dual vectors.

Now, introduce the ordered pair of real numbers â = (a, b) called a dual scalar. Let the addition and subtraction of these numbers be componentwise, and define multiplication as

The multiplication of a screw S = (S, V) by the dual scalar â = (a, b) is computed componentwise to be,

Finally, introduce the dot and cross products of screws by the formulas:

which is a dual scalar, and which is a screw. The dot and cross products of screws satisfy the identities of vector algebra, and allow computations that directly parallel computations in the algebra of vectors.

Let the dual scalar ẑ = (φ, d) define a dual angle, then the infinite series definitions of sine and cosine yield the relations

which are also dual scalars. In general, the function of a dual variable is defined to be f(ẑ) = (f(φ), df′(φ)), where df′(φ) is the derivative of f(φ).

These definitions allow the following results:

• Unit screws are Plücker coordinates of a line and satisfy the relation
  $${\displaystyle |{\mathsf {S}}|={\sqrt {{\mathsf {S}}\cdot {\mathsf {S}}}}=1;}$$

  
• Let ẑ = (φ, d) be the dual angle, where φ is the angle between the axes of S and T around their common normal, and d is the distance between these axes along the common normal, then
  $${\displaystyle {\mathsf {S}}\cdot {\mathsf {T}}=\left|{\mathsf {S}}\right|\left|{\mathsf {T}}\right|\cos {\hat {z}};}$$

  
• Let N be the unit screw that defines the common normal to the axes of S and T, and ẑ = (φ, d) is the dual angle between these axes, then
  $${\displaystyle {\mathsf {S}}\times {\mathsf {T}}=\left|{\mathsf {S}}\right|\left|{\mathsf {T}}\right|\sin {\hat {z}}{\mathsf {N}}.}$$

  

Wrench

A common example of a screw is the wrench associated with a force acting on a rigid body. Let P be the point of application of the force F and let P be the vector locating this point in a fixed frame. The wrench W = (F, P×F) is a screw. The resultant force and moment obtained from all the forces F_i, i = 1,...,n, acting on a rigid body is simply the sum of the individual wrenches W_i, that is

${\displaystyle {\mathsf {R}}=\sum _{i=1}^{n}{\mathsf {W}}_{i}=\sum _{i=1}^{n}(\mathbf {F} _{i},\mathbf {P} _{i}\times \mathbf {F} _{i}).}$

Notice that the case of two equal but opposite forces F and −F acting at points A and B respectively, yields the resultant

${\displaystyle \mathsf{R}=(\mathbf{F}-<!--\; -\; -->\mathbf{F},\mathbf{A}\times <!--\; \times \; -->\mathbf{F}-<!--\; -\; -->\mathbf{B}\times <!--\; \times \; -->\mathbf{F}}$Zdroj:https://en.wikipedia.org?pojem=Screw_theory
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