KAAP686
Homework 2 (33 points)
1 (3 points): Which way do the unit vectors i, j, and k point, as they are described in part C below? Possible
answers are left, right, anterior, posterior, superior, inferior.
2 (15 points): Submit a hard copy of a Matlab script as described below.
3 (15 points): Submit hardcopy of at least two views of 3D plots of the five tracks: 3 markers and 2 hip joint
centers. Use different colors and symbols for each track, and supply a legend to indicate which track is which.
Plot symbols at each time point, connected by lines.
Write a Matlab script to:
A. Read in the data from one of the following marker data files.
1. File …\kaap686\reserve\pelvis_tracking\pelvis_markers_noheaders.txt has 10 columns of data.
The columns are: Frame number, left ASIS X, left ASIS Y, left ASIS Z, right ASIS X, right
ASIS Y, right ASIS Z, sacrum X, sacrum Y, sacrum Z. Frame rate= 100 Hz, units=mm, data is
in the global reference system, i.e. laboratory coordinates.
2. File …\kaap686\reserve\walk_data\gait_raw.txt has 39 columns of data. The columns are x,y,z
for: sacrum, left ASIS, thigh, knee, tibia, ankle, toe, right ASIS, thigh, knee, tibia, ankle, toe.
Frame rate= 50 Hz, units=mm, data is in the global reference system, i.e. laboratory coordinates.
B. Put the x,y,z data from L ASIS into one 3-column array called LASIS, and the x,y,z data from R ASIS
and sacrum into 3-column arrays RASIS and Sac.
C. Compute the right and left hip joint center locations, in global coordinates, at each instant of time (i.e. at
each frame) by using the following equations:
O=origin = midpoint of ASIS markers=(LASIS+RASIS)/2
Breadth = ASIS breadth=distance between RASIS and LASIS (always positive)
k=(RASIS-LASIS) / ||( RASIS-LASIS)|| = unit vector pointing from LASIS to RASIS
a=LASIS-Sac = vector pointing from sacrum to LASIS
b=RASIS-Sac = vector pointing from sacrum to RASIS
i= (a  b) / ||(a  b)||
j = (k  i) / ||(k  i)||
You can do the problem without using a rotation matrix, by using the following equations to locate the
hip joint centers. The equations are the same, whether you use row vectors or column vectors (as long
as you are consistent):
HCR = O + i*(0.24*Breadth) – j*(0.21*Breadth) + k*(.32*Breadth)
HCL = O + i*(0.24*Breadth) – j*(0.21*Breadth) – k*(.32*Breadth)
You can do the problem using a rotation matrix if you prefer. The correct equations depend on whether
you use row vectors or column vectors for marker and joint positions.
If you use column vectors, then i, j, and k are column vectors, and i’=transpose of i=row vector.
Likewise for j and j’ and k and k’. Then the equations are:
𝑖𝑥
𝐻𝐶𝑥
𝑂𝑥
[𝐻𝐶𝑦 ] = [𝑂𝑦 ] + [ 𝑗𝑥
𝑘𝑥
𝐻𝐶𝑧
𝑂𝑧
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𝑖𝑦
𝑗𝑦
𝑘𝑦
𝑖𝑧 +.24𝐵𝑟𝑒𝑎𝑑𝑡ℎ
𝑗𝑧 ] [−.21𝐵𝑟𝑒𝑎𝑑𝑡ℎ]
𝑘𝑧 ±.32𝐵𝑟𝑒𝑎𝑑𝑡ℎ
where + is for right and – is for left in the breadth vector. This can also be written as
R=[i’; j’; k’] (in Matlab, prime indicates transpose and semicolon separates rows in an array)
HCR = O + R*[+.24*ASISbreadth; -0.21*ASISBreadth; +.32*ASISBreadth)]
HCL = O + R*[+.24*ASISbreadth; -0.21*ASISBreadth; -.32*ASISBreadth)]
HCR, HCL, O, and the items in [ ] are column vectors. R is 3x3.
If you use row vectors, then i, j, and k are row vectors, and i’=transpose of i=column vector.
for j and j’ and k and k’. Then the equations are:
𝑖𝑥
𝐻𝐶
𝐻𝐶
𝐻𝐶
𝑂
𝑂
𝑂
𝑖
[ 𝑥
𝑦
𝑧] = [ 𝑥
𝑦
𝑧 ] + [+.24𝐵𝑟𝑒𝑎𝑑𝑡ℎ −.21𝐵𝑟𝑒𝑎𝑑𝑡ℎ ±.32𝐵𝑟𝑒𝑎𝑑𝑡ℎ] [ 𝑦
𝑖𝑧
where + is for right and – is for left in the breadth vector. This can also be written as
Likewise
𝑗𝑥
𝑗𝑦
𝑗𝑧
𝑘𝑥
𝑘𝑦 ]
𝑘𝑧
R=[i’, j’, k’] (in Matlab, prime indicate transpose and comma separates columns in an array)
Rhc = O + [+.24*ASISbreadth, -0.21*ASISBreadth, +.32*ASISBreadth)]*R
Lhc = O + [+.24*ASISbreadth, -0.21*ASISBreadth, -.32*ASISBreadth)]*R
HCR, HCL, O, and the items in [ ] are row vectors. R is 3x3.
D. Write a 7 column text file to disk. Columns should be t (in seconds, computed using correct frame rate
for the data set), X, Y, Z for R hip center ; X, Y, Z for L hip center. It should have as many rows as the
initial file has.
Thanks to Tom Kepple for the hip model and for the file pelvis_markers_noheaders.txt.
Two possible data sets were offered for the above assignment. Data set B (file pelvis_markers_noheaders.txt)
was collected on treadmill (approximate velocity 1.8 m/s), so there is no forward progression of pelvis during
the trial.
Two data sets are now described. Data set A could be used for spherical fitting homework problem. Data set B
is one of the options for the homework problem above.
A. BRPstand_full_data.txt and BRPwalk_data.txt
Stroke patient, 120 fps, units=mm.
+X=anterior, +Y=left, +Z=superior. X=0 just behind heels, Y=0 between feet, Z=0 on floor.
Markers use LPSIS & RPSIS instead of sacrum. Surprisingly, the medial knee and ankle markers ( LKNM,
LMAM, RKNM, RMAM) are present in the walking trial as well as the standing calibration trial, according to
the header info in the c3d file.
This data set could be useful as an example of spherical fitting to find hip joint center. The walking trial
would have to be used. There is no sacral marker, so this set cannot be used as a direct substitute for the
homework problem above.
BRPstand_full_data: 1.9 seconds
Markers
1-10
EARR
C7SP
11-20
RELB
RWRI
21-30
LTH2
LTH3
31-40
LFTL
LFTM
41-50
RSH3
RSH4
51-54
LMAM
RKNL
TRK1
LASI
LTH4
LFTT
RMAL
RKNM
TRK2
RASI
LSH1
RGTR
RHLT
RMAM
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TRK3
LPSI
LSH2
RTH1
RHLB
TRK4
RPSI
LSH3
RTH2
RFTL
LSHO
LILC
LSH4
RTH3
RFTM
LELB
RILC
LMAL
RTH4
RFTT
LWRI
LGTR
LHLT
RSH1
LKNL
RSHO
LTH1
LHLB
RSH2
LKNM
Columns: 162: x, y, z for each marker above.
37-48:
LASIx,y,z, RASIx,y,z, LPSIx,y,z, RPSIx,y,z
145-150:
LKNLx,y,z, LKNMx,y,z
154-159:
RKNLx,y,z, RKNMx,y,z
BRPwalk_data: 2.5 seconds; same markers as above minus EARR.
Markers
1-10
C7SP
TRK1
TRK2
TRK3
TRK4
LSHO
LELB
11-20
RWRI
LASI
RASI
LPSI
RPSI
LILC
RILC
21-30
LTH3
LTH4
LSH1
LSH2
LSH3
LSH4
LMAL
31-40
LFTM
LFTT
RGTR
RTH1
RTH2
RTH3
RTH4
41-50
RSH4
RMAL
RHLT
RHLB
RFTL
RFTM
RFTT
51-53
RKNL
RKNM
RMAM
Columns: 159: x, y, z for each marker above.
34-45:
LASIx,y,z, RASIx,y,z, LPSIx,y,z, RPSIx,y,z
142-147:
LKNLx,y,z, LKNMx,y,z
151-156: RKNLx,y,z, RKNMx,y,z
LWRI
LGTR
LHLT
RSH1
LKNL
RSHO
LTH1
LHLB
RSH2
LKNM
RELB
LTH2
LFTL
RSH3
LMAM
B. gait-raw.txt
Normal adult, 50 fps, units=mm, 142 frames=2.84 seconds of data.
This data could be used as a direct substitute for the data from Tom Kepple, since there are ASIS and sacral
marker data. Sacrum=columns 1-3, LASIS=columns 4-6, RASIS=columns 22-24.
Columns
1-9:
10-18:
19-27:
28-36:
37-39:
SACRx
LKNEx
LTOEx
RKNEx
RTOEx
SACRy
LKNEy
LTOEy
RKNEy
RTOEy
SACRz
LKNEz
LTOEz
RKNEz
RTOEz
LASIx
LTIBx
RASIx
RTIBx
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LASIy
LTIBy
RASIy
RTIBy
LASIz
LTIBz
RASIz
RTIBz
LTHIx
LANKx
RTHIx
RANKx
LTHIy
LANKy
RTHIy
RANKy
LTHIz
LANKz
RTHIz
RANKz