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Planck LFI Alignment Analysis approach

Planck LFI Alignment Analysis approach. y A1. y A3. y A2. y A4. y. y L2. y L1. x. x L1. x L2. x A3. x A4. x A2. x A1. z A4. z A1. z A3. z A2. z. z L1. z L2. y L4. y L3. x L4. x L3. z L4. z L3. General structure analysis. 1 - Feed Horn

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Planck LFI Alignment Analysis approach

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  1. Planck LFI AlignmentAnalysis approach

  2. yA1 yA3 yA2 yA4 y yL2 yL1 x xL1 xL2 xA3 xA4 xA2 xA1 zA4 zA1 zA3 zA2 z zL1 zL2 yL4 yL3 xL4 xL3 zL4 zL3 General structure analysis 1 - Feed Horn 2 - Frame top (OMT for 70 GHz FH) 3 - Main Frame 4 - Exapode 4 3 2 1 PC Local reference frame Coupling reference frame Planck LFI - Feed Horn alignment

  3. Dimensional tolerances Form tolerances Thermal distortions (elements 3 and 4) Matching modalities. Changes of position and orientation of each local coupling reference frame. Changes of position and orientation of each localreference frame. Uncertainty contributions Planck LFI - Feed Horn alignment

  4. P yA OA xA yB zA OB xB zB Matrix approach The orientation of each reference frame and the position of its origin can be described in any reference frame by one 4x4 matrix, keeping into account rotations and translations at the same time. A  B Planck LFI - Feed Horn alignment

  5. yA2 yA4 yA3 yA1 y 4 yL2 yL1 x 3 xL1 xL2 xA4 xA2 xA1 xA3 2 zA3 zA1 zA2 zA4 z zL2 zL1 1 yL4 yL3 xL4 PC xL3 zL4 zL3 ML1 MA1 ML2 MA2 ML3 MA3 ML4 MA4 Position vector of the PC in the L1 ref. frame ML1 describes the L1 ref. frame in the A1 ref. frame MA1 describes the A1 ref frame in the L2 ref frame ML2 describes the L2 ref. frame in the A2 ref. frame MA2 describes the A2 ref. frame in the L3 ref. frame ... Planck LFI - Feed Horn alignment

  6. yA4 yA1 yA3 yA2 4 yL2 yL1 3 xL1 xL2 xA3 xA2 xA4 xA1 2 zA1 zA2 zA3 zA4 zL2 zL1 1 yL4 yL3 xL4 PC xL3 zL4 zL3 Position vector of the PC in the absolute ref. frame y x z ML1 MA1 ML2 MA2 ML3 MA3 ML4 MA4 Position vector of the PC in the L1 ref. frame M FH orientation (angles given following any chosen rule) Position of the PC (x,y,z) Planck LFI - Feed Horn alignment

  7. y y P P z x x x z Tolerances effects FH 44 GHz Local frame L1: - origin in correspondence of the P point - x axis correspondent to the FH real axis - y axis passing from the centers of the pins locations Planck LFI - Feed Horn alignment

  8. Coupling frame A1: - origin in the theoretical P point (on the FH) - x axis correspondent to the FH theoretical axis - y axis passing from the centers of the pins locations Planck LFI - Feed Horn alignment

  9. y y y P P P z x x x x z z Contributions 1 3 2 1 - translation in x of the L1 origin 2 - rotations around A1 z & y axis 3 - pins location in the yz plane: translations in x e z of the L1 origin and rotation around the A1 x axis (general dimensional tolerance) Planck LFI - Feed Horn alignment

  10. The shown uncertainty causes produce a modification of the orientation of the L1 ref. frame as regards the A1 ref. frame (2 & 3) and a translation of the L1 origin in the A1 ref. frame (1). There are further uncertainty contributions relating the uncertainty connected to the theoretical position of the PC: they influence the components of the position vector for the PC in the L1 ref. frame. Planck LFI - Feed Horn alignment

  11. Numerical simulation • Using the Monte-Carlo simulation technique, for eachuncertainty contribution, a series of N values is generated within the range of values defined by the considered tolerance (Uniform or normal distribution). • A series of N vectors, representing the N different situations is found, in consequence of the considered tolerances. • For each situation (vector), the rotation-translations matrixes are computed. Planck LFI - Feed Horn alignment

  12. Three angles, defining the orientation of the feed horn, are extracted from the total M rotation-translations matrix. • The position of the PC in the absolute ref. frame is obtained by multiplying the M matrix per the VPC1 vector. • The three found rotations and the the three found translations give the FH space location. Planck LFI - Feed Horn alignment

  13. The six degrees of freedom (position of the PC, orientation of the FH axis) of the FH are considered as random variables. • For each random variable a series of N possible numerical values has been obtained by the previously explained method. • Starting from the N values (constituting a sample), available for each variable, a proper probability distribution can be computed for this (its parameters are estimated). So a mean value and a standard deviation are calculated for the six degrees of freedom. Planck LFI - Feed Horn alignment

  14. Qualitative results (single FH dof) Number of simulations N=1500 Planck LFI - Feed Horn alignment

  15. Simulation tool A dedicated numerical analisys tool has been designed in MS VBA for Excel environment. On request the simulation tool can be shown. Planck LFI - Feed Horn alignment

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