## Lower body fixed values

The Newington-Gage model is used to define the positions of the hip joint centers in the pelvis segment.

If the InterAsis distance has not been entered in the subject measurements, this is calculated as the mean distance between the LASI and RASI markers, for each frame in the trial for which there is a valid position for each marker.

If the Asis to Trocanter distances have not been entered, they are calculated from the left and right leg lengths using the formula:

AsisTrocDist = 0.1288 * LegLength – 48.56

This is done independently for each leg.

The value C is then calculated from the mean leg length:

C = MeanLegLength*0.115 – 15.3, aa is half the InterAsis distance, and mm the marker radius.

These are used to then calculate the offset vectors for the two hip joint centers (LHJC and RHJC) as follows:

X = C*cos(theta)*sin(beta) – (AsisTrocDist + mm) * cos(beta)

Y = -(C*sin(theta) – aa)

Z = -C*cos(theta)*cos(beta) – (AsisTrocDist + mm) * sin(beta)

where theta is taken as 0.5 radians, and beta as 0.314 radians.

For the right joint center, the Y offset is negated (since Y is in the lateral direction for the pelvis embedded coordinate system).

The position of the top of the lumbar vertebra 5 (the reference point for Dempster data) is then estimated as

(LHJC + RHJC)/2 + (0.0, 0.0, 0.828) * Length(LHJC – RHJC)

where the value 0.828 is a ratio of the distance from the hip joint center level to the top of the lumbar 5 compared to distance between the hip joint centers on the pelvis mesh.

Knee and ankle offsets are then calculated by adding half the measured joint width and marker diameter to give the distance from the center point of the marker to the joint center.

The general direction of the subject walking in the global coordinate system is then found by looking at the first and last valid position of the LASI marker. The X displacement is compared to the Y displacement. If the X displacement is bigger, the subject is deemed to have been walking along the X axis either positively or negatively, depending on the sign of the X offset. Otherwise, the Y axis is chosen. These directions are used to define a coordinate system matrix (similar to a segment definition) denoted the ProgressionFrame. Note that it's assumed that the Z axis is always vertical, and that the subject is walking along one of these axes, and not diagonally, for example.

If the distance between the first and last frame of the LASI marker is less than a threshold of 800mm however, the progression frame is calculated using the direction the pelvis is facing during the middle of the trial. This direction is calculated as a mean over 10% of the frames of the complete trial. Within these frames, only those which have data for all the pelvis markers are used. For each such frame, the rear pelvis position is calculated from either the SACR marker directly, or the center point of the LPSI and RPSI markers. The front of the pelvis is calculated as the center point between the LASI and RASI markers. The pelvis direction is calculated as the direction vector from the rear position to the front. This direction is then used in place of the LASI displacement, as described above, and compared to the laboratory X and Y axes to choose the Progression Frame.

## Pelvis

First the pelvis segment coordinate system is defined from the waist markers. The origin is taken as the midpoint of the two asis markers. The dominant axis, taken as the Y axis, is the direction from the right asis marker to the left asis marker. The secondary direction is taken as the direction from the sacrum marker to the right asis marker. If there is no sacrum marker trajectory, the posterior markers are used. If both are visible, the mean is used. If just one is visible, then that one is used. The Z direction is generally upwards, perpendicular to this plane, and the X axis generally forwards.

The position and scale of the pelvis is thus determined by the two asis markers, since they determine the origin of the coronal orientation of the pelvis. The posterior sacral markers (or psis markers) determine only the anterior tilt of the pelvis. Their actual distance behind the asis markers and lateral position is immaterial, allowing a sacral wand marker to be used, for example.

If the asis markers are also used to calculate the inter asis distance, they are therefore also used to determine the lateral positions of the hip joint centers within the pelvis segment. It is important for these to be as accurate as possible, since they affect the determination of the femur segments, and thus influence both the hip angles, and also the knee joint angles.

## Knee Alignment Device (KAD)

For the model to determine the knee and ankle joint centers, the markers must be very carefully positioned, and it is the responsibility of clinical staff to use their anatomical knowledge to position markers such that the model is able to make as good an approximation to the joint centers as possible.

The dynamic model uses the Thigh and Shank wand markers to define the plane of containing the joint centers, and one method of marker placement is to carefully position these markers to align with your judgment of where the joint centers are.

Alternatively, the Knee Alignment Device (KAD) may be used. This must be placed on the patient during the static trial to indicate the plane of the knee joint center. Then the model calculates the relative angle of the Thigh wand marker, and this angle is used in the dynamic trial to determine the joint center without the KAD. This technique relies on the accurate placement of the KAD, rather than the accurate placement of the wand marker.

## Knee joint center

### Static knee joint center calculation

If a KAD is being used in the static model, firstly a virtual KNE marker is determined by finding the point that is equidistant from the three KAD markers, such that the directions from the point to the three markers are mutually perpendicular.

For the right knee, the markers RKAX, RKD1, RKD2 must be labeled in a clockwise direction, and for the left knee, the markers LKAX, LKD1, LKD2 must be labeled anti-clockwise. That is, if the two KD markers are positioned anteriorly, the upper marker should be KD1.

The joint center KJC is then determined using the chord function with the HJC, KNE and KAX. The HJC-KJC and KJC-KNE lines will be perpendicular, and the KJC-KNE line has a length equal to the knee offset (KO).

The thigh marker rotation offset () is then calculated by projecting its position on to a plane perpendicular to the HJC-KJC line.

If a KAD is not being used in a static trial, then processing proceeds exactly as for a dynamic trial.

Note that for static trials without a KAD, the anterior-posterior position of the KJC is determined by the position of the THI wand marker, and the value of wand offset value that is entered (if you do not enter a value, a value of zero is assumed). Correct determination of the KJC (and the AJC) is very important, especially for the kinetic calculations. In the clinic, you have to assess which method of marker positioning gives the best estimate of the KJC: using a KAD or using the THI marker.

### Dynamic knee joint center calculation

In the dynamic model, the KJC is determined using the modified chord function, from the global position of the HJC, the thigh wand marker (THI), and the knee marker (KNE), together with the knee offset (KO), and thigh wand angle offset () from the subject measurements.

KJC is found such that the KNE marker is at a distance of KO from the KJC, in a direction perpendicular to the line from the HJC to KJC. It is also found such that the angle between the KJC-KNE line and the KJC-THI line, projected onto a plane perpendicular to the HJC-KJC line, is the same as the thigh wand offset angle.

There is only one position for the KJC that satisfies these two conditions.

## Femur

The femur origin is taken as the knee joint center. The primary Z axis is taken from the knee joint center (KJC) to the hip joint center (HJC). The secondary axis is taken parallel to the line from the knee joint center to the knee marker (or virtual knee marker, for static KAD trials). This in fact directly gives the direction of the Y axis. For both the left and the right femur, the Y axis is directed towards the left of the subject. The X axis for both femura is hence directed forwards from the knee.

Note that in a static trial although a KAD determines the plane in which the knee joint center lies, it does not directly determine the lateral orientation of the "knee axis" which is implicitly defined as the Y axis of the femur segment. The lateral orientation is defined by the vertical orientation of the Z axis (the line joining the hip and knee joint centers). The Y axis may pass either above or below the KNE marker.

## Ankle joint center

The ankle joint center is determined in a similar manner to the knee joint center (see Knee joint center).

### Static ankle joint center calculation

In static trials with a KAD, the KAX marker is used to define the plane of the knee axis, and the plane of the ankle axis is assumed to be parallel to this. A value for Tibial Torsion can be entered, and the plane of the in which the Ankle joint center lies will be rotated by this amount relative to the plane containing the KAX marker.

Thus the AJC is found using the modified chord function, such that it has a distance equal to the ankle offset from the ANK marker (AO), and such that the ANK-AJC line forms an angle equal to the Tibial Torsion with the projection of the KAX-AJC line into the plane perpendicular to the KJC-AJC line. Note that Tibial Torsion is thus considered as an external rotation of the ankle axis relative to the knee axis.

The shank marker rotation offset is then calculated by projecting its position onto the same plane. Note that this value takes into account the value of the tibial torsion, and in general, you would expect it to be slightly less than the value for Tibial Torsion, if the TIB wand marker is conventionally placed.

### Dynamic ankle joint center calculation

In the dynamic trial, and static trials without a KAD, the ankle joint center is calculated from the knee joint center, shank wand marker and ankle marker with the ankle offset and shank rotation offset using the modified chord function. Thus the ankle joint center is at a distance of ankle offset from the ankle marker, and the angle between the KJC-AJC-ANK plane and the KJC-AJC-TIB plane is equal to the tibia rotation offset.

## Tibia

### Tortioned tibia

The tibial rotation offset as determined by the static trial already takes into account the tibial torsion. Thus a Tortioned Tibia is defined with an origin at the AJC, the Z Axis in the direction from the AJC to the KJC, the Y axis leftwards along the line between the AJC and ANK marker, and the X axis generally forwards. This is representative of the distal end of the tibia.

### Untortioned tibia

A second tibia is also generated representing the tibia before tibial torsion is applied, by rotating the X and Y axes of the Tortioned Tibia round the Z axis by the negative of the tibial torsion (i.e. externally for +ve values). This represents the proximal end, and is used to calculate the knee joint angles.

## Foot

### Static foot calculation

The heel marker is used in the static trial, and the model effectively makes two segments. For both segments, the AJC is used as the origin.

The main foot segment is constructed using the TOE-HEE line as the primary axis. If the settings for the model have the foot flat check box selected (ie you have selected **Left Foot** and/or **Right Foot** in the **Assume Horizontal** properties for the **Process Static Plug-in Gait Model** pipeline operation), then HEE is moved vertically (along the global Z axis) to be at the same height as TOE. This line is taken as the Z axis, running forwards along the length of the foot. The direction of the Y axis from the untortioned tibia is used to define the secondary Y axis. The X axis thus points down, and the Y axis to the left.

A second foot segment is constructed, using the TOE-AJC as the primary axis, and again the Y axis of the untortioned tibia to define the perpendicular X axis and the foot Y axis (the 'uncorrected' foot).

The Static offset angles (Plantar Flexion offset and Rotation offset) are then calculated from the 'YXZ' Cardan Angles between the two segments (rotating from the 'uncorrected' segment to the heel marker based foot segment). This calculation is performed for each frame in the static trial, and the mean angles calculated. The static plantar-flexion offset is taken from the rotation round the Y axis, and the rotation offset is the angle round the X axis. The angle round the Z axis is ignored.

### Dynamic foot calculation

In the dynamic trial, the foot is calculated in the same way as for the 'uncorrected' foot. The resulting segment is then rotated first round the Y axis by the Plantar Flexion offset. Then the resulting segment is rotated around its X axis by the rotation offset.