HOMEWORK #3 BME 473 ~ Applied Biomechanics Due during Week #10
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1 HOMEWORK #3 BME 473 ~ Applied Biomechanic Due during Week #1 1. We dicued different angle et convention in cla. One common convention i a Bod-fied X-Y-Z rotation equence. With thi convention, the B frame i firt rotated about b b angle θ, then rotated about b b angle θ, and finall rotated about b b angle b θ. Thi rotation equence reult in the following rotation matri (where c = coθ, = inθ, etc.). A R B BodXYZ cc = c + c cc + c c + c c + c c c c Uing the eample from cla a a primer, and uing the atan2 function, pleae derive the formula necear to etract the value of angle θ, θ, and θ from the above rotation matri. For implicit, ou ma aume that no angle equal either or 9. Pleae epre our anwer in term of element of the rotation matri ( r, r, etc.) 2. You are interviewing to work in a motion capture laborator. During the job interview, ou are preented with the 3-D marker convention hown in Figure 1. Since ou have taken BME 473, ou feel confident in our abilit to undertand thi marker convention. You quickl upect that, for the hank, marker 3-5 are ued to etablih a Segment Coordinate Frame (Shank) and that marker 1,2,6, and 7 are ued to etablih the Skeletal Coordinate Frame (Tibia). You rela our upicion to our interviewer, who i quite impreed with our abilit. a. You are told that marker 3-5 do form the Shank Coordinate Frame. Marker 4 i ued a the origin. The -ai i directed from marker 4 toward marker 3. The -ai i normal to the -ai and the vector from marker 4 toward marker 5. The -ai i normal to both the -ai and the -ai. Uing the value for the location the marker in the Lab Coordinate Frame hown in Figure 1, pleae derive the 44 tranformation matri that epree the Shank Coordinate Frame (defined uing marker 3-5) with repect to the Lab Coordinate Frame for the condition hown in Figure 1. b. You are once again correct and are told that marker 1, 2, 6, and 7 do form the Tibia Coordinate Frame (at thi point, ou re feeling good about the interview). You are told the following: The origin for the Tibia Coordinate Frame i the midpoint of marker 1 and 2. The -ai i directed from the origin toward the midpoint of marker 6 and 7. The -ai i directed
2 anteriorl and i normal to the -ai and a temp vector from marker 1 to marker 2. The -ai normal to both the -ai and -ai and i directed mediall (Note: numerou lab ue thi convention.) Uing the value for the location the marker in the Lab Coordinate Frame hown in Figure 1, pleae derive the tranformation matri that epree the Tibia Coordinate Frame (defined uing marker 1, 2, 6, and 7) with repect to the Lab Coordinate Frame for the condition hown in Figure 1. c. Uing our olution to part a and b, pleae derive the tranformation matri that epree the Tibia Coordinate Frame (defined uing marker 1,2,6, and 7) with repect to Shank Coordinate Frame (defined uing marker 3-5). d. You know that the marker ued to etablih the Tibia Coordinate Frame (defined uing marker 1, 2, 6, and 7) are frequentl removed from a ubject during a motion anali tud. A a final tet during our job interview, ou are aked to find the knee fleion angle ( θ from quetion 1), the knee varu/valgu angle ( θ ), and the internal rotation angle ( θ ) uing a Bodfied X-Y-Z rotation equence for ome future intance in time (aume the Femur i the fied frame A and that the Tibia i the moving frame B ). Pleae provide our anwer in degree. To olve for the angle, pleae ue our anwer to part c and the following additional information: Lab Shank T = Lab Thigh T = Thigh Femur T =
3 Figure 1: Reflective Maker on a Right Leg
4 3. A common problem encountered b the elderl i lo of balance, which ometime reult in falling. One biomechanical hpothei ugget that falling in the elderl reult from a lack of ufficient mucle trength. To tud the iue of trength, uppoe we aked a number of elderl women to rie from a chair in our motion lab. Our aim i to etimate the net joint torque for the major joint of the lower etremit during thi tak. Once we calculate the joint torque ued to tand up from a chair, we can compare them to the maimum joint torque that our ubject can generate (maimum joint torque are a meaure of mucle trength). Thi comparion will how what percentage of our ubject maimum trength mut be ued to rie from a chair. Suppoe we ued high-peed video camera, together with marker poitioned judicioul over variou bod landmark to etimate the time hitor of the abolute angular diplacement of the hank, thigh, and trunk (i.e., θ1, θ2, θ3 in Figure 2). Thee data were ubequentl filtered and numericall differentiated to obtain the abolute angular velocitie and acceleration of each of thee bod egment over time. Auming that the it-to-tand activit can be characteried b predominantl planar (2-D) motion of all the bod egment, a imple threeegment, planar linkage i ued to model the human keleton (Figure 2). Further aume that the bod egment are connected b frictionle ingle degree-offreedom revolute (i.e., hinge) joint. In term of kinematic data onl: a. Pleae derive complete analtical epreion for the net joint torque eerted at the ankle (T1), knee (T2), and hip (T3) during it-to-tand motion. Aume that the foot remain flat on the floor. b. Pleae derive complete analtical epreion for the ground reaction force eerted in the horiontal (fore-aft) and vertical direction during the it-totand motion. Again aume that the foot remain flat on the floor and that it i male. Note that if ou could meaure Fg and Fg that ou could calculate T1 without an kinematic data. c. Would ou conider thi problem to be invere or direct dnamic? Wh? d. What i the phical ignificance of the interegmental force (e.g., F1)? Are the calculated interegmental force equal to the true bone-on-bone force? Wh or wh not?
5 Figure 2: Planar 3 egment model
6 4. Uing our preliminar data from problem 3, we were able to land a mall grant and have ued the mone to purchae a force plate. With our improved eperimental et-up we can now meaure the horiontal (fore-aft) and vertical ground reaction force during the it-to-tand motion. Alo, our video tem remain operational, and we hall continue to record the time hitor of the angular diplacement of the hank, thigh, and trunk. On the bai of the three egment, planar model given in Figure 2, ue the kinematic and force-plate data now available to: a. Derive complete analtical epreion for the net joint torque eerted at the ankle (T1), knee (T2), and hip (T3) during the it-to-tand motion. Again, aume that the foot remain flat on the floor. b. B comparing the term in our equation for T1, T2, and T3 derived in problem 4a and 3a, dicu the advantage to being able to accuratel etimate the ground reaction force and relevant kinematic parameter when calculating joint torque.
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