User:Prof McCarthy/screwtheory

Screw theory refers to the algebra and calculus of pairs of vectors, such as forces and moments and angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies.

The conceptual framework was developed by Sir Robert Stawell Ball in 1876 for application in kinematics and statics of mechanisms (rigid body mechanics).

The value of screw theory derives from the central role that the geometry of lines plays in three dimensional mechanics, where lines form the screw axes of spatial movement and the lines of action of forces. The pair of vectors that form the Plucker 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.

A remarkable result of screw theory is that geometric calculations for points using vectors have parallel geometric calculations for lines obtained by replacing vectors with screws. This is termed the transfer principle.

Screw theory notes that all rigid-body motion can be represented as rotation about an axis along with translation along the same axis; this axis need not be coincident with the object or particle undergoing displacement. In this framework, screw theory expresses displacements, velocities, forces, and torques in three dimensional space.

Recently screw theory has regained importance and has become 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.

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.



Basic concepts
Here we describe screw, twist, and wrench in general terms, and save the mathematics for the following sections.

Screw
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 displacement. This is known as Chasles' theorem. The six parameters that define a screw displacement are the four independent components of the Plucker 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 give by three Euler Angles that define the rotation and the three components of the translation vector.

The points in a body undergoing a constant screw 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.

Twist
A twist represents 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.

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 Plucker 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.

Algebra of Screws
Let a screw be an ordered pair S=(S, 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 dual scalars. Let the addition and subtraction of these numbers be componentwise, and define multiplication as âĉ=(a, b)(c, d) = (ac, ad+bc). The multiplication of a screw S=(S, V) by the dual scalar â=(a, b) is computed componentwise to be,
 * $$ \hat{a}\mathsf{S} = (a, b)(\mathbf{S}, \mathbf{V}) = (a\mathbf{S}, a\mathbf{V}+b\mathbf{S}).$$

Finally, introduce the dot and cross products of screws by the formulas:
 * S•T = (S, V)•(T, W) = (S•T, S•W+V•T) and
 * S×T = (S, V)×(T, W) = (S×T, S×W+V×T).

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
 * sinẑ = sinφ + dcosφ and cosẑ = cosφ - dsinφ.

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

These definitions allow the following results:
 * Unit screws satisfy the relation |S|=√(S•S)=1 and are the Plucker coordinates of a line;
 * S•T=|S||T|cosẑ, where ẑ=(φ, d) defines the angle φ between the axes of S and T around their common normal and d is the distance between these axes along the common normal. ẑ=(φ, d) is said to define the dual angle between the lines that form the axes of S and T;
 * S×T = |S||T|sinẑN, where N is 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.

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 Fi i=1,...,n, acting on a rigid body is simply the sum of the individual wrenches Wi, that is
 * R = Σin Wi = Σin (Fi, Pi×Fi).

Notice that the case of two equal but opposite forces F and -F acting at points A and B respectively, yields the resultant R=(F-F, A×F - B× F) = (0, (A-B)×F). This shows that screws of the form M=(0, M) 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, D(t)=([A(f)],d(f)), where [A] is a rotation matrix and d is a translation vector. This causes a point p that is fixed in moving body to trace a curve P(t) in the fixed frame given by,

\mathbf{P}(t) = [A(t)]\mathbf{p} + \mathbf{d}(t). $$

The velocity of P is

\mathbf{V}_P(t) = \left[\frac{dA(t)}{dt}\right]\mathbf{p} + \mathbf{v}(t), $$ where v is velocity of the origin of the moving frame, that is dd/dt. Now substitute p= [AT](P-d) into this equation to obtain,

\mathbf{V}_P(t) = [\Omega]\mathbf{P} + \mathbf{v} - [\Omega]\mathbf{d}\quad\mbox{or}\quad\mathbf{V}_P(t) = \mathbf{\omega}\times\mathbf{P} + \mathbf{v} + \mathbf{d}\times\mathbf{\omega}, $$ where [Ω]=[dA/dt][AT] is the angular velocity matrix and ω is the angular velocity vector.

The screw T=(ω, v + d×ω) 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: (i) when d is constant, that is v=0, then the twist is a pure rotation about the line L=(ω, d×ω), 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 L=(0, v).

Screws and spatial displacements
The coordinate transformations for screws are easily understood by beginning with the coordinate transformations of the Plucker 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=([A], d), where [A] 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 Plucker coordinates,
 * $$ \mathsf{q}=(\mathbf{q}-\mathbf{p}, \mathbf{p}\times(\mathbf{q}-\mathbf{p})),$$

then in the fixed frame we have the transformed point coordinates P=[A]p+d and Q=[A]q+d, which yield.
 * $$\mathsf{Q}=(\mathbf{Q}-\mathbf{P}, \mathbf{P}\times(\mathbf{Q}-\mathbf{P})) = ([A](\mathbf{q}-\mathbf{p}), [A](\mathbf{p}\times(\mathbf{q}-\mathbf{p})) + \mathbf{d}\times[A](\mathbf{q}-\mathbf{p}))$$

Thus, a spatial displacement defines a transformation for Plucker coordinates of lines given by

\begin{Bmatrix} \mathbf{Q}-\mathbf{P} \\ \mathbf{P}\times(\mathbf{Q}-\mathbf{P}) \end{Bmatrix} = \begin{bmatrix} A & 0 \\ DA & A \end{bmatrix} \begin{Bmatrix} \mathbf{q}-\mathbf{p} \\ \mathbf{p}\times(\mathbf{q}-\mathbf{p}) \end{Bmatrix}. $$ The matrix [D] is the skew symmetric matrix that performs the cross product operation, that is [D]y=d×y.

The 6x6 matrix [Â]=([A], [DA]) obtained from the spatial displacement D=([A], d) operates on a screw s=(s.v) to obtain:
 * $$\mathsf{S} = [\hat{A}]\mathsf{s}, \quad\text{or}\quad (\mathbf{S}, \mathbf{V}) = ([A], [DA])(\mathbf{s}, \mathbf{v})  = ([A]\mathbf{s}, [A]\mathbf{v}+[DA]\mathbf{s}).$$

The dual matrix [Â]=([A], [DA]) has determinant 1 and is called a dual orthogonal matrix.

Twists and spatial displacements
Consider the movement of a rigid body defined by the parameterized 4x4 homogeneous transform,
 * $$ \textbf{P}(t)=[T(t)]\textbf{p} =

\begin{Bmatrix} \textbf{P} \\ 1\end{Bmatrix}=\begin{bmatrix} A(t) & \textbf{d}(t) \\ 0 & 1\end{bmatrix} \begin{Bmatrix} \textbf{p} \\ 1\end{Bmatrix}.$$ This notation does not distinguish between P = (X, Y, Z, 1), and P = (X, Y, Z), 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,
 * $$ \textbf{V}_P = [\dot{T}(t)]\textbf{p} =

\begin{Bmatrix} \textbf{V}_P \\ 0\end{Bmatrix} = \begin{bmatrix} \dot{A}(t) & \dot{\textbf{d}}(t) \\ 0 & 0 \end{bmatrix} \begin{Bmatrix} \textbf{p} \\ 1\end{Bmatrix}.$$ 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(t), that is
 * $$\textbf{V}_P=[\dot{T}(t)][T(t)]^{-1}\textbf{P}(t) = [S]\textbf{P},$$

where
 * $$[S] = \begin{bmatrix} \Omega & -\Omega\textbf{d} + \dot{\textbf{d}} \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} \Omega & \mathbf{d}\times\omega+ \mathbf{V}_O \\ 0 & 0 \end{bmatrix}.$$

Recall that [Ω] is the angular velocity matrix. The matrix [S] is an element of the Lie algebra se(3) of the Lie group SE(3) of homogeneous transforms. The components of [S] are the components of the twist screw, and for this reason [S] is also often called a twist.

From the definition of the matrix [S], we can formulate the ordinary differential equation,
 * $$[\dot{T}(t)] = [S][T(t)],$$

and ask for the movement [T(t)] that has a constant twist matrix [S]. The solution is the matrix exponential
 * $$[\dot{T}(t)] = e^{[S]t}.$$

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 $$\theta$$ represents the parameters of the transformation.

Important formulas in kinematics define the velocity and acceleration of points in a moving body as they trace trajectories in three dimensional space. This is particularly important for the center of mass of a body, which is used to derive equations of motion using either Newton's second law or Lagrange's equations.

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 formula:

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

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:

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

Screws by reflection
In transformation geometry, the elemental concept of transformation is the reflection (mathematics). In planar transformations a translation is obtained by reflection in parallel lines, and rotation is obtained by reflection in a pair of intersecting lines. To produce a screw transformation from similar concepts one must use planes in space: the parallel planes must be perpendicular to the screw axis, which is the line of intersection of the intersecting planes that generate the rotation of the screw. Thus four reflections in planes effect a screw transformation. The tradition of inversive geometry borrows some of the ideas of projective geometry and provides a language of transformation that does not depend on analytic geometry.

Work of a forces acting on a rigid body
Consider the set of forces F1, F2 ... Fn act on the points X1, X2 ... Xn in a rigid body. The trajectories of Xi, i=1,...,n  are defined by the movement of the rigid body with rotation [A(t)] and the translation d(t) of a reference point in the body, given by
 * $$ \mathbf{X}_i(t)= [A(t)]\mathbf{x}_i + \mathbf{d}(t)\quad i=1,\ldots, n, $$

where xi are coordinates in the moving body.

The velocity of each point Xi is
 * $$\mathbf{V}_i = \vec{\omega}\times(\mathbf{X}_i-\mathbf{d}) + \mathbf{v},$$

where ω is the angular velocity vector and v is the derivative of d(t).

The work by the forces over the displacement δri=viδt of each point is given by
 * $$ \delta W = \mathbf{F}_1\cdot\mathbf{V}_1\delta t+\mathbf{F}_2\cdot\mathbf{V}_2\delta t + \ldots + \mathbf{F}_n\cdot\mathbf{V}_n\delta t.$$

Define the velocities of each point in terms of the twist of the moving body to obtain
 * $$ \delta W = \sum_{i=1}^n \mathbf{F}_i\cdot (\vec{\omega}\times(\mathbf{X}_i -\mathbf{d}) + \mathbf{v})\delta t. $$

Expand this equation and collect coefficients of ω and v to obtain
 * $$ \delta W = (\sum_{i=1}^n \mathbf{F}_i)\cdot\mathbf{v}\delta t + (\sum_{i=1}^n \mathbf{X}_i \times\mathbf{F}_i)\cdot\vec{\omega}\delta t  + (\sum_{i=1}^n \mathbf{F}_i)\cdot\mathbf{d}\times \vec{\omega}\delta t =  (\sum_{i=1}^n \mathbf{F}_i)\cdot(\mathbf{v}+\mathbf{d}\times \vec{\omega}) \delta t + (\sum_{i=1}^n \mathbf{X}_i \times\mathbf{F}_i)\cdot\vec{\omega}\delta t .$$

Introduce the twist of the moving body and the wrench acting on it given by
 * $$ \mathsf{T} = (\vec{\omega},\mathbf{d}\times \vec{\omega} +\mathbf{v})=(\mathbf{T},\mathbf{T}^\circ),\quad\mathsf{W} = \sum_{i=1}^n (\mathbf{F}_i, \mathbf{X}_i \times\mathbf{F}_i)=(\mathbf{W},\mathbf{W}^\circ), $$

then work takes the form
 * $$\delta W = (\mathbf{W}\cdot\mathbf{T}^\circ + \mathbf{W}^\circ \cdot\mathbf{T})\delta t.$$

The 6x6 matrix [Π] is used to simply the calculation of work using screws, so that
 * $$\delta W = (\mathbf{W}\cdot\mathbf{T}^\circ + \mathbf{W}^\circ \cdot\mathbf{T})\delta t = \mathsf{W}[\Pi]\mathsf{T}\delta t,$$

where
 * $$ [\Pi] =\begin{bmatrix} 0 & I \\ I & 0 \end{bmatrix},$$

and [I] is the 3x3 identity matrix.

Reciprocal screws
If the virtual work of a wrench on a twist is zero, then the forces and torque of the wrench are constraint forces relative to the twist. The wrench and twist are said to be reciprocal, that is if
 * $$\delta W =\mathsf{W}[\Pi]\mathsf{T}\delta t = 0,$$

then the screws W and T are reciprocal.

Twists in robotics
In the study of robotic systems the components of the twist are often transposed to eliminate the need for the 6x6 matrix [Π] in the calculation of work. In this case the twist is defined to be
 * $$\check{\mathsf{T}} = (\mathbf{d}\times \vec{\omega} +\mathbf{v},\vec{\omega})$$

so the calculation of work takes the form
 * $$\delta W =\mathsf{W}\cdot\check{\mathsf{T}}\delta t.$$

In this case, if
 * $$\delta W =\mathsf{W}\cdot\check{\mathsf{T}}\delta t= 0,$$

then the wrench W is reciprocal to the twist T.