# Virtual and Physical Prototyping of Reconfigurable Parallel Mechanisms with Single Actuation
## Appendix A Basics of Instantaneous Screw Theory  
The major idea of screw theory relies on [PERSON] theorem [32] (p. 26), by which any finite motion of a rigid body can be represented as a combination of rotation about some axis and translation parallel to it. This axis is called a screw axis. If the motion is infinitesimal, we can consider it instantaneous, and the screw axis becomes an instantaneous screw axis. This instantaneous screw motion can be characterized by three basic elements: the axis itself, the angular speed about this axis and the linear speed along it. We can combine these elements into a 6-D vector called (instantaneous) _screw_\(\mathbf{\upxi}\)[32] (p. 40):  
\[\mathbf{\upxi}=\omega\left[ \begin{array}{c}\mathbf{\upxi}\\ \mathbf{r}\times\mathbf{\upxi}\end{array} \right]+\upsilon\left[ \begin{array}{c} \mathbf{0}\\ \mathbf{\upxi}\end{array} \right], \tag{1}\]  
where \(\mathbf{\upxi}\) is a unit vector directed along the screw axis; \(\mathbf{r}\) is a vector pointed from the origin of the coordinate system to any point on the screw axis; \(\omega\) is an angular speed relative to the screw axis; and \(\upsilon\) is a linear speed about the screw axis. An alternative common notation for a screw implies using scalar parameters: magnitude \(s\) and pitch \(h=\upsilon/\omega\). In this case, a screw can be defined as follows [22] (p. 19):  
\[\mathbf{\upxi}=s\left[ \begin{array}{c}\mathbf{\upxi}\\ \mathbf{r}\times\mathbf{\upxi}+h\mathbf{\upxi}\end{array} \right],\text{ if }h\ \
eq\ \infty, \tag{2}\] \[\mathbf{\upxi}=s\left[ \begin{array}{c}\mathbf{0}\\ \mathbf{\upxi}\end{array} \right],\text{ if }h=\infty.\]  
Similarly, all the forces and moments acting on a rigid body can be reduced to a force directed along some axis and a moment about the axis. We can define screw \(\mathbf{\upxi}\) representing this resulting force and moment using relations similar to (1) and (2). To differ between these screws, terms _twist_ and _wrench_ are usually applied for \(\mathbf{\upxi}\) and \(\mathbf{\upxi}\) respectively. Twistsand wrenches of zero pitch (\(h=0\)) correspond to pure rotations and forces. Twists and wrenches of infinite pitch (\(h=\infty\)) correspond to pure translations and moments. Screws can be summed with each other and multiplied by a scalar--these properties allow screws to form vector spaces. Thus, the space of twists can define the possible velocities of the rigid body, while the space of wrenches represents all the forces and moments acting on it. If each twist \(\xi\) from the twist space and each wrench \(\zeta\) from the wrench space satisfy the following condition (22) (p. 24):  
\[\xi\ \circ\ \zeta= (\mathbf{P}\xi)^{\mathrm{T}}\zeta=0, \tag{10}\]  
where \(\mathbf{P}\) is a permutation matrix of the form:  
\[\mathbf{P}=\left[ \begin{array}{cc}\mathbf{0}_{3\times 3}&\mathbf{I}_{3\times 3} \\ \mathbf{I}_{3\times 3}&\mathbf{0}_{3\times 3}\end{array} \right], \tag{11}\]  
then the wrench space represents all forces and torques that constrain the body motion. Operation \"\(\circ\)\" is known as a _reciprocal product_, and screws \(\xi\), and \(\zeta\) that satisfy condition (10) are _reciprocal_. If two bodies are connected by a series of joints, we can associate unit twist \(\xi_{j}\) (\(s=1\) in expression (11)) \(j=1\ \dots\ m\) with each joint, where \(m\) is a number of joints. Then, twist system T comprising twists \(\xi_{1}\), \(\xi_{2}\), \(\dots\), \(\xi_{m}\) will define the space of possible velocities of one body relative to the other. The reciprocal wrench system W (each wrench of which is reciprocal to each twist of system T) corresponds to constraints between the bodies. For a parallel mechanism with \(n\) kinematic chains, each having \(m_{i}\) joints, \(i=1\ \dots\ n\), we can define twists systems T\({}_{i}\) of unit twists \(\xi_{ij},j=1\ \dots\ m_{i}\), associated with each chain. Wrench system W\({}_{i}\) reciprocal to T\({}_{i}\) corresponds to constraints that the \(i\)-th chain imposes on the motion of the output link.