Screw theory

[4] 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;[1] 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.

This is in part because of the relationship between screws and dual quaternions which have been used to interpolate rigid-body motions.

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.

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 torque, on the other hand, is a pure moment that is not bound to a line in space and is an infinite pitch screw.

Now, introduce the ordered pair of real numbers â = (a, b), called a dual scalar.

These definitions allow the following results: A common example of a screw is the wrench associated with a force acting on a rigid body.

The vector V = v + d × ω is the velocity of the point in the body that corresponds with the origin of the fixed frame.

There are two important special cases: (i) when d is constant, that is v = 0, then the twist is a pure rotation about a line, then the twist is and (ii) when [Ω] = 0, that is the body does not rotate but only slides in the direction v, then the twist is a pure slide given by For a revolute joint, let the axis of rotation pass through the point q and be directed along the vector ω, then the twist for the joint is given by, For a prismatic joint, let the vector v pointing define the direction of the slide, then the twist for the joint is given by, The coordinate transformations for screws are easily understood by beginning with the coordinate transformations of the Plücker vector of line, which in turn are obtained from the transformations of the coordinate of points on the line.

Consider the movement of a rigid body defined by the parameterized 4x4 homogeneous transform, This notation does not distinguish between P = (X, Y, Z, 1), and P = (X, Y, Z), which is hopefully clear in context.

The solution is the matrix exponential This formulation can be generalized such that given an initial configuration g(0) in SE(n), and a twist ξ in se(n), the homogeneous transformation to a new location and orientation can be computed with the formula, where θ represents the parameters of the transformation.

The combination of a translation with a rotation effected by a screw displacement can be illustrated with the exponential mapping.

Consider the set of forces F1, F2 ... Fn act on the points X1, X2 ... Xn in a rigid body.

The work by the forces over the displacement δri=viδt of each point is given by Define the velocities of each point in terms of the twist of the moving body to obtain Expand this equation and collect coefficients of ω and v to obtain Introduce the twist of the moving body and the wrench acting on it given by then work takes the form The 6×6 matrix [Π] is used to simplify the calculation of work using screws, so that where and [I] is the 3×3 identity matrix.

In the study of robotic systems the components of the twist are often transposed to eliminate the need for the 6×6 matrix [Π] in the calculation of work.

[1] In this case the twist is defined to be so the calculation of work takes the form In this case, if then the wrench W is reciprocal to the twist T. The mathematical framework was developed by Sir Robert Stawell Ball in 1876 for application in kinematics and statics of mechanisms (rigid body mechanics).

[4] Felix Klein saw screw theory as an application of elliptic geometry and his Erlangen Program.

The use of a symmetric matrix for a von Staudt conic and metric, applied to screws, has been described by Harvey Lipkin.

[12] Other prominent contributors include Julius Plücker, W. K. Clifford, F. M. Dimentberg, Kenneth H. Hunt, J. R.

[13] The homography idea in transformation geometry was advanced by Sophus Lie more than a century ago.

Even earlier, William Rowan Hamilton displayed the versor form of unit quaternions as exp(a r)= cos a + r sin a.

The idea is also in Euler's formula parametrizing the unit circle in the complex plane.

William Kingdon Clifford initiated the use of dual quaternions for kinematics, followed by Aleksandr Kotelnikov, Eduard Study (Geometrie der Dynamen), and Wilhelm Blaschke.

[14] In 1940, Julian Coolidge described the use of dual quaternions for screw displacements on page 261 of A History of Geometrical Methods.

[15] Coolidge based his description simply on the tools Hamilton had used for real quaternions.