Johns Hopkins EN 600 445 - Homework Assignment 2 – 600.445 Fall 2003

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Homework Assignment 2 – 600.445 Fall 2003 Instructions and Score Sheet (hand in with answers) Name Email Other contact information (optional) Signature (required) I have followed the rules in completing this assignment _____________________________ Question Points Total Points Total 1.A 10 1.B 10 1.C 10 1.D 10 1.E 10 Question 1 Subtotal 50 2 20 3 20 4 20 Total 110 1. Remember that this is a graded homework assignment. 2. You are to work alone and are not to discuss the problems with anyone other than the TAs or the instructor. 3. It is otherwise open book, notes, and web. But you should cite any references you consult. 4. Please refer to the course organizational notes for a fuller listing of all the rules. I am not reciting them all here, but they are still in effect. 5. Unless I say otherwise in class, it is due before the start of class on the due date posted on the web. 6. Sign and hand in the score sheet as the first sheet of your assignment 7. Remember to include a sealable 8 ½ by 11 inch self-addressed envelope if you want your assignmentScenario Consider the pelvic osteotomy situation illustrated in Figure 1. Here we assume that a three locating pins (similar to those used in robodoc) have been inserted into the patient’s pelvis, and that a CT scan of the pelvis with the pins inserted has been produced. The patient has been placed onto the operating table. Also, a magnetic navigation system (here, the Northern Digital Aurora) is present in the room. Two surgical tools are available: • A probe/pointer device • An osteotome (essentially a fancy chisel) that will be used to cut the pelvis. 6 DOF Aurora sensors have been attached to the handle of each tool. The aurora is capable of determining the position and orientation of each sensor relative to the Aurora base unit. Initially, you can assume that the Aurora base unit will not move relative to the operating table and that the patient is also fixed on the operating table. Let: SFSTFPFPTFASFAPFTAFTFAFIFTIF Figure 1coordinate system associated with base unit coordinate system associated with operating table= coordinate system associated with pelvis (and CT of pelvis)= coordinate system associated with ATIS==FFFF sensor on osteotomecoordinate system associated with sensor on pointercoordinate system associated with the top of the 'th pinPkk==FF Further, assume that we have the following relationships: transformation from table to base coordinates (unknown) transformation from table to pelvis/CT coordinates (unknown)= desired position and orientation of osteotome tip relative to = cTATIIC IST==FFFFFAPoordinate transformation from osteotome sensor to osteotome tipcoordinate transformation from pointer sensor to pointer tipcoordinate transformation from base to osteotome sensor (measured)PTAS==FFF,,,coordinate transformation from base to pointer sensor (measured), coordinate transformation from CT to 'th pin coordinate systemdesired position and orientation of osteotome seIk Ik IkACk====FRpFGnsor relative to base (to compute) Question 1 Initially, we will assume that the registration process is performed as in “classic” Robodoc. I.e., the tip of the pointer probe is placed in a small dimple on the top of the pin. A. Write down an expression for ACF in terms of the other quantities defined above. I.e., give an expression that will give the position and orientation of the osteotome sensor with respect to the base unit that will place the osteotome tip into the desired pose ICF relative to IF . B. Assuming that all measurements are perfectly accurate, write down a series of equations that may be solved to determine ACF. (Hint: This is essentially a “registration” step. You may find it useful to consider combining TAF and TIF into a single (unknown) transformation AIF ). C. Now, assume that the registration step performed above is error free, but that the Aurora has developed an error, so that each time the pose (i.e., position and orientation) of the osteotome sensor is measured as ,AS jF , the actual position is *,, ,AS j AS j AS j=•∆FF F where ,AS j∆F is a small random error. Suppose that the surgeon (with hands of rock and nerves of steel) has placed the osteotome so that the measured value ,0ASF is exactly the desired value ACF . Give an expression giving the actual position and orientation *ICF of the osteotome tip relative to IF , in terms of ,0 ,0 ,0,AS AS AS∆=∆ ∆FRpG D. Using the approximations ,0 0,0 0()ASASskewpαε∆=+∆=RIGG Produce a linearized expression approximating ()1*ICIC IC−∆= •FF FE. Suppose that the Aurora began producing errors during the registration process so that each time the pose (i.e., position and orientation) of the probe sensor is measured as ,AP jF , the actual position is*,, ,=•∆AP j AP j AP jFF F where ,AP j∆F is a small random error. Assume that these errors are small enough so that you can use the approximations ,,()αε∆≈+∆=GGGAP j jAP j jskewRIp Assume, further, that you have been told the following information about these measurement errors: ,,,j,,,j,k , αεαα εεαεαγεη==≤≤GGjx jxjjy jyjz jzkjkk Produce a set of linearized constraints allowing you to bound the error in the placement of the osteotome. Assume that the error measuring the osteotome sensor is as in Part D with similar limits. (i.e., 0, 0,k and αγεη≤≤kk k) Question 2 Continuing the scenario from Question 1.A and B, assume that the Aurora has no measurement errors. Assume that the Aurora is located approximately 1 meter from the patient and that someone has bumped the Aurora base unit after registration, so that the new position and orientation of the Aurora base unit with respect to the operating table is given by *bump bump(),TA TAα=∆∆•FR pFGG where we know that bumpbump0.01 radians0.2 mmαε≤≤GG Suppose again that our rock solid, steely-nerved surgeon places the osteotome so that sensor reading ,0ASF is exactly the desired value ACF . Estimate the positional error introduced in the osteotome tip position. I.e., how far will the osteotome tip be from where it is supposed to be?Question 3 Suppose that we have modified the surgical procedure by affixing another Aurora sensor rigidly to the patient’s pelvis before the start of the procedure (but


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Johns Hopkins EN 600 445 - Homework Assignment 2 – 600.445 Fall 2003

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