Screw theory
Screw theory is the algebraic calculation of pairs of vectors, also known as dual vectors – such as angular and linear velocity, or forces and moments – that arise in the kinematics and dynamics of rigid bodies.
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.
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;
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, 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. Based on screw theory, an efficient approach has also been developed for the type synthesis of parallel mechanisms.
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 . 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 pairwhere and 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, called a dual scalar. Let the addition and subtraction of these numbers be componentwise, and define multiplication as
The multiplication of a screw by the dual scalar 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 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, where ′ is the derivative of .
These definitions allow the following results:
- Unit screws are Plücker coordinates of a line and satisfy the relation
- Let 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
- Let N be the unit screw that defines the common normal to the axes of S and T, and is the dual angle between these axes, then
Wrench
Notice that the case of two equal but opposite forces F and −F acting at points A and B respectively, yields the resultant
This shows that screws of the form
can be interpreted as pure moments.
Twist
In order to define the twist of a rigid body, we must consider its movement defined by the parameterized set of spatial displacements,, where is a rotation matrix and d is a translation vector. This causes a point p that is fixed in moving body coordinates to trace a curve P in the fixed frame given byThe velocity of P is
where v is velocity of the origin of the moving frame, that is dd/dt. Now substitute p = into this equation to obtain,
where = is the angular velocity matrix and ω is the angular velocity vector.
The screw
is the twist of the moving 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: when d is constant, that is v = 0, then the twist is a pure rotation about a line, then the twist is
and when = 0, that is the body does not rotate but only slides in the direction v, then the twist is a pure slide given by
Revolute joints
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,Prismatic joints
For a prismatic joint, let the vector v pointing define the direction of the slide, then the twist for the joint is given by,Coordinate transformation of screws
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.Let the displacement of a body be defined by D = , where is the rotation matrix and d is the translation vector. Consider the line in the body defined by the two points p and q, which has the Plücker coordinates,
then in the fixed frame we have the transformed point coordinates P = p + d and Q = q + d, which yield.
Thus, a spatial displacement defines a transformation for Plücker coordinates of lines given by
The matrix is the skew-symmetric matrix that performs the cross product operation, that is y = d × y.
The 6×6 matrix obtained from the spatial displacement D = can be assembled into the dual matrix
which operates on a screw s = to obtain,
The dual matrix = has determinant 1 and is called a dual orthogonal matrix.
Twists as elements of a Lie algebra
Consider the movement of a rigid body defined by the parameterized 4x4 homogeneous transform,This notation does not distinguish between P =, and P =, which is hopefully clear in context.
The velocity of this movement is defined by computing the velocity of the trajectories of the points in the body,
The dot denotes the derivative with respect to time, and because p is constant its derivative is zero.
Substitute the inverse transform for p into the velocity equation to obtain the velocity of P by operating on its trajectory P, that is
where
Recall that is the angular velocity matrix. The matrix is an element of the Lie algebra se of the Lie group SE of homogeneous transforms. The components of are the components of the twist screw, and for this reason is also often called a twist.
From the definition of the matrix , we can formulate the ordinary differential equation,
and ask for the movement that has a constant twist matrix . The solution is the matrix exponential
This formulation can be generalized such that given an initial configuration g in SE, and a twist ξ in se, the homogeneous transformation to a new location and orientation can be computed with the formula,
where θ represents the parameters of the transformation.