600.445/645 Fall 2015 1 of 10

Homework Assignment 2 – 600.445/645 Fall 2015 (Circle One)

Instructions and Score Sheet (hand in with answers) Name Name

Email Email

Other contact information (optional) Other contact information (optional)

Signature (required) I/We have followed the rules in completing this assignment _____________________________

Signature (required) I/We have followed the rules in completing this assignment _____________________________

Note: 445 students may answer 645 questions for extra credit, but max total grade will still be 100

Question Points (445) Points (645) Totals

1A 1 1

1B 4 4

1C 5 5 10

2A 5 5

2B 5 5

2C 10 5

2D 10 10

2E 10 10

2F 0 5 40

3A 5 5

3B 10 5

3C 5 5

3D 10 10

3E 10 10

3F 10 10

3G 0 5 50

Total 100

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1. Remember that this is a graded homework assignment. It is the functional equivalent of a take-home exam.

2. You are to work alone or in teams of two 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 assignment

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Question 1 Suppose that we have F

* = FΔFR =ΔFLF , where

F = [R,!p]

ΔFL = [ΔRL,Δ!pL ]≈ [I+ sk( !αL ),

!εL ]

ΔFR = [ΔRR,Δ!pR ]≈ [I+ sk( !αR ),

!εR ]

A. Give expressions for ΔFL and ΔFR in terms of F, ΔFR , and ΔFL , avoiding tautologies like ΔFL =ΔFL .

B. Give expressions for ΔRL , ΔRR , Δ!pL , and Δ

!pR in terms of the other

quantities while avoiding tautologies. C. Give simplified expressions for the linearized error variables

!αL ,

!αR ,

!εL ,

and !εR in terms of the other quantities while avoiding tautologies.

600.445/645 Fall 2015 4 of 10

Question 2

FAB = [Rot(!xC,θ ),

!pAB ]

A"

B"

C"

FAC

FBC

!xC

!yC

!zC

Consider the figure above. Here you can think of A and B as being some sort of pose sensors and C as being a target. The position and orientation of the two sensors is given by FAB = [Rot(

!xC,θ),

!pAB ]. I.e., the relative orientation

of A and B is given by a rotation about the !x axis of C. FAC and FBC are the

positions and orientations returned by the measurement sensors. Of course, there is some error, so that FAC

* = FACΔFAC , where ΔFAC ≈ [I+ sk( !αAC ),!εAC ]

and similarly for FBC* and FAB

* . Suppose that we have some information

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about the uncertainties. In particular, the sensors may be less accurate in “depth” than they are laterally. This is expressed component wise as

!εAC ≤ [η,η,ν ]T (i.e., εAC,x ≤ η, εAC,y ≤ η, εAC,z ≤ν), and η≤ν!εBC ≤ [η,η,ν ]T

Otherwise, we know that the errors are “small”. Clearly, the two measurements FAC and FBC can be combined to produce a new estimated value FAC

est for FAC* . This new estimate will have an associated error

FAC* = FAC

estΔFACest , where ΔFAC

est ≈ [I+ sk( !αACest ),!εAC

est ].

Note: The questions below mostly emphasize estimating accuracy, based on simple linearized models of error propagation. A. Write down the components of FAC

est = [RACest ,!pAC

est ] based on the kinematic calculation FAC

est = FABC = FABFBC .

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• Notational Hint: Please express this as FACest = FABC = FABFBC and

then give expressions for RABC and !pABC . We are adding this

notation to the problem to simplify the grading and also to help clarify the question and approach.

B. Write down the components of ΔFABCest = [ΔRABC

est ,Δ!pABC

est ] based on the kinematic calculation FAC

est = FABC = FABFBC .

C. Write down the linearized approximation components [!αABC,

!εABC ]

ΔFABC ≈ [I+ sk( !αABC ),!εABC ].

D. Since we also know the measurement accuracy associated with ΔFAC , we have additional information. All this information can be combined to produce an improved estimate FAC

est of C relative to A with

ΔFACest ≈ [I+ sk( !αAC

est ),!εAC

est ]. Write down a linearized set of constraints saying all that can be estimated about the value of

!εAC

est . Note that this will include what you know about the measurement uncertainties, together with the results of Question 2C. In your final answer, please expand out the results of Question 2C, so that the variables

!αABC and

!εABC do not appear, though you should probably include them in showing how you got to the final answer.

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E. Suppose now that we have the following additional information:

!εAB ≤ εAB

max, αAB ≤αABmax , αBC ≤αBC

max , αAC ≤αACmax

Further, suppose that we have the following values:

εABmax = 0.5 mm, αAB

max = αACmax = αBC

max = 0.001 radians

η = 0.5 mm, ν = 5 mm

θ = 90!, "pAB = [0,−500 mm, -500 mm]T

"pAC = [0,0, -500 mm]T

Estimate how accurately you can compute an estimate for FACest .

F. (600.645 only) Under the same accuracy assumptions as above, what is the smallest value of θ for which you can guarantee that

!εAC

est ≤3mm?

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Question 3

FAB = [Rot(!xC,θ ),

!pAB ]

A"

B"

C"

FAC FBC

!xC

!yC

!zC

!gi

!hii

!dii

!ptip

FAD

Consider now the situation in the figure above, which adds to that used for

Question 2. The target body C has been attached to a pointer, and a calibration has been performed to compute the position

!ptip of the pointer tip

relative to the coordinate system of C. In addition a fiducial structure D has been attached to the patient’s body. This fiducial object comprises a number of point fiducials at locations

!di relative to the coordinate system of D. The

sensor A senses these point fiducials and reports that the location of the i'th fiducial is at location

!gi relative to the coordinate system of A. Similarly,

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sensor B reports that that the location of the i'th fiducial is at location !hi

relative to the coordinate system of B. There is, of course, some error:

!ptip

* =!ptip +

!εtip where

!εtip ≤ψ

!di

* =!di +

!εD,i where

!εD,i ≤ δ

!gi

* =!gi +

!εA,i where

!εA,i ≤ [η,η,ν ]T

!hi

* =!hi +

!εB,i where

!εB,i ≤ [η,η,ν ]T

Assume that you have software that computes FACest , with

FACest * = FAC

estΔFACest and ΔFAC

est ≈ [I+ sk( !αACest ),!εAC

est ]. Further, we have the following nominal values:

!d1 = [0,50,0]T

!g1 = [0,50,−650]T

!d2 = [50,0,0]T

!g2 = [50,0,−650]T

!d3 = [0,−50,0]T

!g3 = [0,−50,−650]T

!d3 = [−50,0,0]T

!g4 = [−50,0,−650]T

A. Since we have redundant information, the A and B sensor values may be

combined to produce an estimate !gi

est of the position of the i'th fiducial

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relative to sensor A. Under the assumptions of Questions 2A-2E, how accurately can you determine

!gi

est ? I.e., how can you constrain !εi

est , where

!gi

est * =!gi

est +!εi

est ?

B. Describe a method for computing FAD from the !gi

est .

C. Assuming that your algorithm for computing FAD is correct, give a formula for computing the position

!pDt of the pointer tip relative to the coordinate

system of D. D. Assume now that (due to our superior analysis and possible invocation of

an oracle) we can assume that !εi

est ≤µ. How accurately can you determine FAD? I.e., can you express some constraints or limits on

ΔFAD = [I+ sk( !αAD ),!εAD ]?

E. Write a formula for the error !εDt in

!pDt , under the assumptions of

Questions 3A-3D. F. Suppose that we know that FAC

est = [I,[0,0,500]T ], !ptip = [0,0,−100]T ,

µ = 2 mm , and ψ= δ = 0.1 mm. Estimate numerical limits on !εDt .

G. (645 only) Under the assumptions above, what value of µ would be required in order to ensure that

!εDt ≤ 2 mm?