## Transformation of loads

The loads (forces and moments) were measured in the implant-base coordinate system x’ y’ z’ (IBS). A force vector in the __IBS__ is __F__’ = (Fx’, Fy’, Fz’), a moment vector is __M__’ = (Mx’, My’, Mz’). For the implant “HIP JOINT” the IBS was fixed to the left bone, for “HIP JOINT III”, “KNEE” and “SHOULDER” to the right bone. For subjects with implants at the opposite joint, the loads were first mirrored to the other side. Then they were mostly transformed to the bone-based system x y z (BBS). A force vector in the __BBS__ is __F__ = (Fx, Fy, Fz), a moment vector is __M__ = (Mx, My, Mz). Only for the knee joint and the spinal implants they were left in the implant-based system x’ y’ z’.

The example shows the coordinate systems IBS and BBS of an implant of type “HIP JOINT III”. Definitions of the coordinate systems of the other implants are described in the manual under the caption “Implants”.

Relative to the __BBS__ the implant is rotated three times in the order 1, 2, 3. The three rotation angles α1, α2, α3 are stated relative to the axes of the __BBS__! Their order and values are shown in the window “Info Patient” of the OrthoLoad videos:

In example 1 (implant type “HIP JOINT III”, right-sided implant) the three rotations and their order 1 2 3 are:

- clockwise by angle α
_{1}= α_{x }= +2° around axis +x - clockwise by angle α
_{2}= α_{y }= -2° around axis +y - clockwise by angle α
_{3}= α_{z }= -15° around axis +z

For this right-sided implant “HIP JOINT III”, a negative angle α_{z} indicates an anteversion of the implant neck.

In example 2 (implant type “HIP JOINT III”, right-sided implant) the three rotations and their order 1 2 3 are:

- clockwise by angle α
_{1}= α_{x }= +17° around axis +z - clockwise by angle α
_{2}= α_{y }= +8° around axis +x - clockwise by angle α
_{3}= α_{z }= 0° around axis +y

For this left-sided implant an anteversion is indicated by a positive angle α_{z}.

For transforming a force __from__ the __IBS__ x’ y’ z’ __to__ the __BBS__ x y z, three transformations have to be performed, using the transformation matrices __T___{1} (α_{1}), __T___{2}(α_{2}), __T___{3}(α_{3}). The complete transformation matrix __T__’ is then:

__T__‘ = __T___{3} (α_{3}) * __T___{2}(α_{2}) * __T___{1}(α_{1}) for __F__ = __T__‘ * __F__’ (note the inverse order 3 2 1 of matrices!)

For transforming the loads from the __BBS__ x y z __to__ the __IBS__ x’ y’ z’, this calculation has to be performed in the __reverse__ order and with __ne__g__ative__ angles:

__T__ = __T___{1} (-α_{1}) * __T___{2}(-α_{2}) * __T___{3}(-α_{3}) for __F__’ = __T__ * __F__

Rotations around the axes x y z are performed by these matrices:

For example 1 the complete transformation matrices T’, T are therefore:

__T__‘ = __T___{z}(-15°) * __T___{y} (-2°) * __T___{x}(2°) for __F__ = __T__‘ *__F__’

__T__ = __T___{x} (-2°) * __T___{y}(+2°) * __T___{z}(+15°) for __F__’ = __T__ *__F__

For example 2 they are:

T’ = T_{y} (0°) * T_{x}(8°) * T_{z}(17°) for F = T‘ * F‘

T = T_{z} (-17°) * T_{x}(-8°) * T_{y}(0°) for F‘ = T * F

Transformations of the moments M and M’ are performed separately in an analogue way.

**Measuring Units % Body Weight and Newton**

In the OrthoLoad videos, the loads are mostly reported in %BW (percent of body weight) for the forces and %BW*m for the moments, except for the spine implants were they are stated in Newton. The subject’s body weight in Newton is stated in the window “Info Patient” (examples see above). To transform loads from %BW/ %BW*m to N / Nm, the forces / moments have to be multiplied by 1% of the body weight, in example 1 by 7.809 and in example 2 by 6.9.