KAAP686
Homework 4
(70 points)
1. (4 points) u = 1.5i – 0.5j + k, v = i + j + 2k .
a. (2) Find the length and direction of u x v
b. (2) Find the length and direction of v x u.
2. (2 points) In figure at right, |F| = 30 N and |PQ|=20 cm. What
is the magnitude of the torque on the bolt at P? (Don’t forget
units.)
3. (6 points) On the second row (blank row) of diagrams below, draw the vectors u = i+j,
v = i - j, and w = u x v as vectors starting at the origin. Use a small circle with an x in it to
denote a vector straight into the page. Use a small circle with a dot in the middle to denote a
vector straight out of the page. If two vectors overlap, indicate clearly on the diagram. Use a
different color (consistently) to show each vector on all 4 diagrams below. The first row
below shows examples, for vectors r, g, and b, whose coordinates are r=(1,0,0.5), g=(0,1,0),
b=(0.5,1,1.5). The plots don’t have to be exactly correct, but get the general idea correct.
Note that on each of the four diagrams, you should draw and label all three vectors: u, v, and
w.
Y
g
b
Z
b
r
r
X
Y
Z
r
r
g
g
Y
Z
X
Z
b
b
X
X
Y
Z
Z
Y
g
X
X
The next questions refer to the locations of markers at jugular notch (incisura jugularis, IJ)
xiphoid process (PX), spines of C7 and T8 (C7, T8), and the left glenohumeral center (GHL).
The coordinates of these points in the global (laboratory) reference frame are IJ=(200;150;300);
PX=(202.5;131;300.5); C7=(196;155;298); T8=(196.5;125;297.5); GHL=(209;152;282). (Data
in cm, from spreadsheet …/excel/upper_limb_coords.xls.) Give all answers in the global
reference frame coordinate system.
Y
4. (2 points) What is the location of the midpoint of IJ and C7?
5. (2 points) What is the location of the midpoint of PX and T8?
6. (2 points) What is the vector FROM midpoint of PX and T8 TO midpoint of IJ and C7?
7. (4 points) What is the unit vector nyt from midpoint of PX and T8 to midpoint of IJ and C7
(i.e. same direction as #6)? This is the +Yt axis direction, i.e. Y axis of the thorax, as
recommended by the International Shoulder Group (Wu et al. (2005), J Biomech
38: 981-992.)
8. (4 points) What is the unit vector nzt found by crossing nyt onto the vector from IJ to C7?
(Don’t forget that we want the unit vector, and be sure to get the sign right for vector from IJ
to C7.) This is the +Zt axis direction, i.e. Z axis of the thorax, as recommended by the ISG
(Wu et al., 2005). (This similar but not identical to the +Xt direction described by Jim Richards in his
presentation on shoulder angle measurements,
www.udel.edu/HNES/faculty/richards/BluePresentation_files/frame.htm, retrieved 20090312. Jim Richards’s
thoracic axis definitions in that presentation are more similar to those of FCT van den Helm,
www.fbw.vu.nl/research/Lijn_A4/shoulder/isg/proposal/protocol.html, retrieved 20090313, but not quite the
same.)
9. (2 points) Does the +Zt axis point to the subject’s right or left?
10. (4 points) What is an equation of the plane perpendicular to nzt, passing through IJ? This is
the midsagittal plane. The ISG recommends using IJ as the origin of the thoracic coordinate
system.
11. (4 points) What is the unit vector nxt which is found by crossing nyt onto nzt ?
(nxt = nyt × nzt.) This is the +Xt axis recommended by the ISG.
12. (2 points) Does the +Xt axis point anteriorly or posteriorly?
13. (4 points) Use your previous answers (#7, #11) to show that nzt × nxt = nyt .
14. (6 points) What are the coordinates of the reflection, through the midsagittal plane, of the
left glenohumeral rotation center? Remember that you already know the unit normal to the
plane (#8) and a point in the plane (IJ). (Check: The reflection of the left GH center should
be about where the right GH center is located.)
15. (30 points) Compute and plot Euler and helical angles for simulated shoulder rotation data.
The data is in file shouldersimdata2.txt. +X points right, +Y points forward, and +Z points
up. Compute Euler angles two ways: for rotation about X then Y then Z, and for rotation
about Y then X then Z. As you will see, they do not always give the same answers. The
marker data file has 22 columns: time, plus x,y,z for 7 markers. The markers are labeled in
header lines in the files. From the markers, compute i,j,k for the thorax, and i,j,k for the
humerus, using the Int. Shoulder Group recommendations for defining axes from markers
(Wu et al., J Biomech 2005), as explained in file matrices_rotations_2.(docx or pdf), which is
posted on the web site under "Notes".
When complete, you will have angle ax=flexion/extension versus time, computed 3 ways:
EulerXYZ, EulerYXZ, and helical. Plot all three versions of ax versus time on a single plot
to facilitate comparison. You will also have angle ay=ADduction/ABduction, computed the
same 3 ways. Plot all three versions of ay on a single graph. You will also have
angle az=internal/external rotation, computed the same 3 ways. Plot all three versions of az on
a single graph.
(15 points) Hard copy of the three plots described above. See shoulderangleHW.jpg for an
example of what this could look like.
(15 points) Hard copy of your code.
You can do it any way you like. The following files are in the homework section of course
web site. The first four files are ones you need to complete, if you are using Matlab. The
remaining files are complete, and are provided to help complete the assignment.
shoulder_angle_estimation_HW.m
Main file which needs filling in. You will probably
want to call the function files euleranglesXYZ_HW.m, euleranglesYXZ_HW.m, and
helicalangles_HW.m.
euleranglesXYZ_HW.m
Function file that returns the Euler angles ax,ay,az, given
the matrices Rprox and Rdist. You need to fill in this incomplete function, which is based on
equations 38-41 from the notes. I've already inserted a line of code for equation 38, so you
only need to add code to implement equations 39-41.
euleranglesYXZ_HW.m
Function file that returns the Euler angles ax,ay,az, given
the matrices Rprox and Rdist. You need to fill in this incomplete function. You'll need
Equations 47-49 from the notes.
helicalangles_HW.m
Function file that returns the helical angles ax,ay,az, given
the matrices Rprox and Rdist. You need to fill in this incomplete function, which is based on
equations 6, 19, 22-26. I’ve already inserted code for equation 6, so you only need to add
code to implement equations 19 and 22-26. You may find it useful to call function
rot_helixmtx.m, which is already complete (see below). If you call it, make sure it is in an
accessible directory.
shouldersimdata2.txt Marker data for trunk and humerus (simulated).
shoulderangleHW.jpg Screenshot showing my results for this assignment. Hand in a
similar plot, on paper, and a printout of your code (.VI or .m).
rot_helixmtx.m
Function file that returns rotation matrix for a helical rotation,
given the values for axis n and angle phi. This function is complete. You may find it useful to
call this function from helicalangles_HW.m.