International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. B. ch. 3.3, pp. 372-373   | 1 | 2 |

Section 3.3.1.3.10. The three-axis joystick

R. Diamonda*

aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England
Correspondence e-mail: rd10@cam.ac.uk

3.3.1.3.10. The three-axis joystick

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The three-axis joystick is a device which depends on compound transformations for its exploitation. As it is usually mounted it consists of a vertical shaft, mounted at its lower end, which can rotate about its own length (the Y axis of display space, Section 3.3.1.3.1[link]), its angular setting, φ, being measured by a shaft encoder in its mounting. At the top of this shaft is a knee-joint coupling to a second shaft. The first angle φ is set to zero when the second shaft is in some selected direction, e.g. normal to the screen and towards the viewer, and goes positive if the second shaft is moved clockwise when seen from above. The knee joint itself contains a shaft encoder, providing an angle, ψ, which may be set to zero when the second shaft is horizontal and goes positive when its free end is raised. A knob on the tip of the second shaft can then rotate about an axis along the second shaft, driving a third shaft encoder providing an angle θ. The device may then be used to produce a rotation of the object on the screen about an axis parallel to the second shaft through an angle given by the knob. The necessary transformation is then [\eqalign{ {\bi R} &= \pmatrix{\cos \varphi &0 &-\sin \varphi\cr 0 &1 &0\cr \sin \varphi &0 &\cos \varphi\cr} \pmatrix{1 &0 &0\cr 0 &\cos \psi &\sin \psi\cr 0 &-\sin \psi &\cos \psi\cr}\cr &\quad \times \pmatrix{\cos \theta &-\sin \theta &0\cr \sin \theta &\cos \theta &0\cr 0 &0 &1\cr} \pmatrix{1 &0 &0\cr 0 &\cos \psi &-\sin \psi\cr 0 &\sin \psi &\cos \psi\cr}\cr &\quad \times \pmatrix{\cos \varphi &0 &\sin \varphi\cr 0 &1 &0\cr -\sin \varphi &0 &\cos \varphi\cr}}] which is [\eqalign{ &\left(\matrix{c^{2}\psi s^{2}\varphi + (1 - c^{2}\psi s^{2}\varphi) c\theta &-s\psi c\psi s\varphi (1 - c\theta) - c\psi c\varphi s\theta\cr \noalign{\vskip3pt} -s\psi c\psi s\varphi (1 - c\theta) + c\psi c\varphi s\theta &s^{2}\psi + c^{2}\psi c\theta\cr \noalign{\vskip3pt} -c^{2}\psi s\varphi c\varphi (1 - c\theta) - s\psi s\theta &s\psi c\psi c\varphi (1 - c\theta) - c\psi s\varphi s\theta\cr}\right.\cr \noalign{\vskip5pt} &\left.\matrix{\phantom{-s\psi c\psi s\varphi (1 - c\theta) + c\psi c\varphi s\theta s^{2}\psi + c^{2}} &-c^{2}\psi s\varphi c\varphi (1 - c\theta) + s\psi s\theta\cr \noalign{\vskip3pt} &s\psi c\psi c\varphi (1 - c\theta) + c\psi s\varphi s\theta\cr \noalign{\vskip3pt} &c^{2}\psi c^{2}\varphi + (1 - c^{2}\psi c^{2}\varphi) c\theta\cr}\right)}] in which cos and sin are abbreviated to c and s, which is the standard form with [l = -\cos \psi \sin \varphi], [m = \sin \psi], [n = \cos \psi \cos \varphi].








































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