Donkey Kong wants to make a 250 mL fruit smoothie. Being an ape, naturally he would like a large proportion of the smoothie to be bananas—at least 100 mL but not more than 150 mL. There should be at least 10 mL each of apples, cherries, and dates. However, for proper taste, there should be twice the amount of apples as cherries. The prices at the local supermarket are as follows:

Donkey Kong wants to spend as few coins as possible on his smoothie.

(a) What are the design variables in this optimization problem?

(b) Translate the written description of the problem into a formal optimization problem statement (see Chapter 9 in the textbook). Label each expression as an objective function, inequality constraint, equality constraint, or bound on a design variable.

(c) Using any strategy you wish, find the values of the design variables that minimize the objective function while satisfying the constraints (i.e., solve the optimization problem). How many coins will the smoothie cost? (If you decide to use Matlab, see the linprog function; if you prefer Python, see scipy.optimize. Does the answer returned by the software make sense?)

You are interviewing to work in a motion capture laboratory. During the job interview, you are presented with the 3-D marker convention shown in Figure 1. Since you have taken a biomechanics class, you feel confident in your ability to understand this marker convention. You quickly suspect that, for the shank, markers 3-5 are used to establish a Tracking Coordinate System (TCS) and that markers 1, 2, 6, and 7 are used to establish the Anatomic Coordinate System (ACS). You relay your suspicions to your interviewer, who is quite impressed with your ability. (NOTE: You have access to any computational tools during your interview).

1. You are told that markers 3-5 do form the TCS. Marker 4 is used as the origin. The x-axis is directed from marker 4 towards marker 3. The y-axis is directed anteriorly and is normal to the x-axis and the vector from marker 4 towards marker 5. The z-axis is normal to both the x-axis and the y- axis such that the axes form a right-handed basis. Using the values for the location the markers in the Global Coordinate System (GCS) shown in Figure 1, please derive the 4x4 homogeneous transformation matrix that expresses the  with respect to the GCS (  ) for the conditions shown in Figure 1.

2. You are once again correct and are told that markers 1,2,6, and 7 do form the ACS (at this point, you’re feeling good about the interview). You are told the following: “The origin for the shank ACS is the midpoint of markers 1 and 2. The z-axis is directed from the origin towards the midpoint of markers 6 and 7. The y-axis is directed anteriorly and is normal to the z-axis and a temp vector from marker 1 to marker 2. The x-axis normal to both the y-axis and z-axis and is directed medially.”  Using the values for the location the markers in the Global Coordinate System (GCS) shown in Figure 1, please derive the homogeneous transformation matrix that expresses the  with respect to the GCS (  ) for the conditions shown in Figure 1.

3. Using your solutions to parts 1 and 2, please derive the transformation matrix that expresses the  with respect to  ( ).

4. Which of the following transformation matrices are constant for the duration of an experiment (and therefore only need to be calculated once), and which vary with time?
   
                        
                        
   

5. You know that the markers used to establish the ACS are frequently removed from a subject during a motion analysis study. As a final test during your job interview, you are asked to find the knee flexion angle ( from question 7.2 above), the knee varus/valgus angle (), and the internal rotation angle () using a X-Y-Z Euler angle convention for some future instance in time (assume the thigh is the fixed frame “A” and that the shank is the moving frame “B”). Please provide your answer in degrees. To solve for the angles, please use your previous answers and the following additional information that was computed at time :

Figure 1. Reflective Markers on a Right Leg.