Coda AMPLIFIER 15.0 Uživatelská příručka stáhnout pdf (Strana 106)

Charnwood Dynamics Ltd. Coda cx1 User Guide – Advanced Topics III - 3

CX1 USER GUIDE - COMPLETE.doc 26/04/04

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(In either case the sequence described is that by which the segment arrives at its

orientation. Anyone wishing to ‘undo’ the rotations to return to the neutrally aligned

orientation must reverse carefully!)

Mathematical decomposition

Having settled for the ‘ZXY’ sequence we must calculate the compound rotation matrix:

zxy

= R

which looks like:

cosφ cosψ + sinθ sinφ sinψ sinθ sinφ cosψ − cosφ sinψ cosθ sinφ

zxy

= |

cosθ sinψ cosθ cosψ −sinθ

sinθ cosφ sinψ − sinφ cosψ sinθ cosφ cosψ + sinφ sinψ cosθ cosφ

If we use R

zxy

to form the image,U

zxy

, of distal EVB U

under this compound rotation we

obtain the same matrix again, the columns of which are the vectors u

```, u

``` and u

```

(re-oriented axial unit vectors), which are precisely the distal EVB vectors derived in the

segmental analysis along with u

, u

and u

, the proximal EVB vectors.

Conveniently, the 2

element of the 3

column of our matrix happens to be the scalar

product of proximal unit vector u

(= [0,1,0]

) with column (distal axis) vector u

```, i.e.

``` =

−sinθ

and we solve first, therefore, the X axis rotation angle θ as

θ = −sin

-1

{ u

```

Having found θ, its value may be plugged back into the equations for φ and ψ which

similarly arise from the scalar (or ‘dot’) products of proximal/distal axis unit vectors:

we obtain φ from the 1

element of the 3

column wherein

```

= cosθ sinφ

so that the flexion angle about the proximal Y axis is given by

φ = sin

-1

{ (u

```

) / cosθ }.

Likewise, from the 2

element of the 1

column:

```

= cosθ sinφ

so that the internal/external rotation angle about the Z axis is

ψ = sin

-1

{ (u

```

) / cosθ }.

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