phyphox Forums - Experiments and Ideas

Measured IMU position inside the iPhone 17

Tue, 27 Oct 2026 15:17:22 +0000

Hi all,

I wanted to know where exactly the accelerometer/gyroscope (IMU) sits inside my iPhone 17.

Apple does not publish this, and the estimates I found online turned out to be wrong. So I measured it with phyphox and a turntable. Sharing the method and result in case it is useful to others.

Device / software

iPhone 17 (iPhone18,3), iOS 26.6.1
phyphox 1.2.1, experiment "Acceleration with g" + "Gyroscope", 100 Hz, exported as .xlsx

Setup
The phone lies flat in a pocket made of pins on a wooden turntable, with its housing center about 120 mm from the axis and the long side tangential. Three runs were done in the same pocket:

display up (reference)
display down, flipped about the long (y) axis
display down, flipped about the short (x) axis

Each run: spin up by hand, then let it coast down.
(photo of the setup attached)

Method
Rigid-body model in the device frame (Apple/phyphox axes: x right, y up, z out of the display; phyphox accelerometer = specific force, i.e. +9.81 m/s² in z when lying display-up):

Code:
ax = wz^2 * lx + dwz/dt * ly + bx
ay = wz^2 * ly - dwz/dt * lx + by

(lx, ly) is the vector from the IMU to the rotation axis. bx and by absorb offset and tilt.

Linear least squares per run, using the same smoothing filter on data and regressors.
Time shift between gyro and accelerometer estimated by grid search: consistently +7 to +8 ms.
Confidence intervals from a moving-block bootstrap, because the residuals are autocorrelated.
The runs are combined via l_k = S_k*c - p, where p is the IMU position relative to the housing center, c the housing center -> axis vector, and S_k the flip of run k. Because the phone is flipped about both axes, p follows from the data without relying on the measured 120 mm or on the alignment angle.

Consistency check: the lever arm from the centripetal term (wz^2) and from the tangential term (dwz/dt) agree to about 1 mm in every run.

Result (origin = geometric center of the housing 149.6 x 71.5 mm, looking at the display, 95% CI):

x = -29.0 mm [-30.2, -27.9] (left half, about 7 mm from the left edge)
y = -0.1 mm [-1.4, +1.0] (practically at mid-height)
z: not observable with this method

(plot attached)

Caveats

Single device. I assumed about 1 mm repositioning tolerance in the pocket, and this dominates the CI. The pure statistical error is about 0.1 mm.
Sensor scale-factor errors are not included. A 1 % gyro scale error would shift the lever arm by about 2 %.
The data put the axis at about 116 mm from the housing center, not the 120 mm I measured with a ruler. Probably the turntable runs slightly eccentric. The result for p does not depend on this.
phyphox detail (from the iOS source): the accelerometer is raw CMAccelerometerData x (-9.81), the gyroscope is CMDeviceMotion.rotationRate (bias-corrected), not raw CMGyroData.

I'd be happy if someone could repeat this with another iPhone 17 or other models. I'll also share my Octave script and my measurement data.

Best regards,
Prototype

Code:
% Estimation of the IMU position (iPhone 17, phyphox) from several turntable runs
%
% Rigid-body model for a rotation about the device z-axis (phone frame, phyphox
% convention: specific force, +x right, +y up, +z out of the display). With
% l = (lx, ly) being the vector from the IMU to the rotation axis:
%   ax = wz²*lx + dwz/dt*ly + bx
%   ay = wz²*ly - dwz/dt*lx + by
% (bx, by: sensor offset + gravity leakage due to tilt)
%
% Combination of the runs: the phone sits in the same pocket in every run, only
% its orientation changes. With c = vector housing center -> rotation axis in the
% phone frame of the reference run (display up, see turntable.png) and p = IMU
% position relative to the housing center:
%   l_k = S_k*c - p
% S_k maps the reference placement into the phone frame of run k:
%   display up   (reference)            S = [ 1 0; 0  1]
%   display down, flipped about y-axis  S = [-1 0; 0  1]
%   display down, flipped about x-axis  S = [ 1 0; 0 -1]
%   display up,   rotated by 180°       S = [-1 0; 0 -1]
% Components of c that the runs cannot resolve are taken from the setup prior
% (R, phi). With the runs 'up' + 'down about y' this applies to the y-direction.

% clean up
clear all;
close all;
clc;

% settings
runs = struct( ...
  'file', {'Translation und Rotation 2026-09-29 16-10-10.xlsx', ...
           'Translation und Rotation 2026-10-26 15-35-40.xlsx', ...
           'Translation und Rotation 2026-10-27 13-36-20.xlsx'}, ...
  'S',    {[1 0; 0 1], [-1 0; 0 1], [1 0; 0 -1]}, ...
  'up',   {true, false, false});  % display up? (checked against az)
R      = 120;    % prior: distance housing center -> rotation axis (mm)
sigR   = 1;      % uncertainty of R (mm, 1 sigma)
phi    = 0;      % prior: angle of the rotation axis seen from the housing center (deg)
sigphi = 2;      % uncertainty of phi (deg, 1 sigma)
sigpos = 1;      % repositioning tolerance of the phone in the pocket (mm, 1 sigma)
Tsm    = 0.10;   % smoothing window (s)
taus   = -0.05 : 0.001 : 0.05;  % search grid for the gyro time shift (s)
Lblk   = 1.0;    % block length for the moving-block bootstrap (s)
B      = 2000;   % number of bootstrap samples
conf   = 0.95;   % confidence level
m2mm   = 1000;
rand('seed', 42);
randn('seed', 42);

% load a phyphox export (cached as .mat next to the .xlsx)
function [ACC, GYRO] = load_run(file)
  [~, name] = fileparts(file);
  cache = [name, '.mat'];
  if exist(cache, 'file')
    load(cache);
  else
    pkg load io;
    pkg load windows;
    ACC  = xlsread(file, 'Accelerometer');  % t (s), ax, ay, az (m/s²)
    GYRO = xlsread(file, 'Gyroscope');      % t (s), wx, wy, wz (rad/s)
    save('-v7', cache, 'ACC', 'GYRO');
  end
end

% build the (smoothed) linear model for a given gyro time shift tau
function [Y, X] = build_model(ta, ax, ay, tw, wz, tau, nsm, keep)
  w  = interp1(tw + tau, wz, ta, 'linear', 'extrap');
  dw = gradient(w, ta);
  % the same linear filter is applied to data and regressors -> model stays exact
  f  = @(v) movmean(v, nsm);
  W2 = f(w.^2)(keep);
  A  = f(dw)(keep);
  ax = f(ax)(keep);
  ay = f(ay)(keep);
  o  = ones(size(W2));
  z  = zeros(size(W2));
  Y  = [ax; ay];
  X  = [W2,  A, o, z;
        -A, W2, z, o];
end

% fit a single run: time shift, least squares, moving-block bootstrap
function r = fit_run(ACC, GYRO, Tsm, taus, Lblk, B)
  ta  = ACC(:, 1) - ACC(1, 1);
  tw  = GYRO(:, 1) - ACC(1, 1);
  dt  = median(diff(ta));
  nsm = max(1, round(Tsm/dt));
  % only use samples covered by the gyro, minus the filter edges
  keep = ta > tw(1) + max(abs(taus)) & ta < tw(end) - max(abs(taus));
  keep(1 : nsm) = false;
  keep(end - nsm + 1 : end) = false;
  n = sum(keep);

  ssr = zeros(size(taus));
  for k = 1 : numel(taus)
    [Y, X] = build_model(ta, ACC(:, 2), ACC(:, 3), tw, GYRO(:, 4), taus(k), nsm, keep);
    ssr(k) = sum((Y - X*(X\Y)).^2);
  end
  [~, kbest] = min(ssr);
  r.tau = taus(kbest);
  r.ssr = ssr;

  [Y, X] = build_model(ta, ACC(:, 2), ACC(:, 3), tw, GYRO(:, 4), r.tau, nsm, keep);
  th = X\Y;
  r.th  = th;
  r.rms = sqrt(mean((Y - X*th).^2));
  r.lx_rad = X(1:n, 1)\(Y(1:n) - th(3) - X(1:n, 2)*th(2));              % ax ~ wz²*lx
  r.lx_tan = X(n+1:end, 1)\(Y(n+1:end) - th(4) - X(n+1:end, 2)*th(2));  % ay ~ -dwz/dt*lx
  r.t = ta(keep);
  r.Y = Y;
  r.Ym = X*th;
  r.n = n;
  r.az = mean(ACC(:, 4));

  % residuals are strongly autocorrelated -> resample blocks
  L   = max(1, round(Lblk/dt));
  nb  = ceil(n/L);
  thb = zeros(B, numel(th));
  for b = 1 : B
    s   = randi(n - L + 1, nb, 1);
    idx = (s + (0 : L - 1))';
    idx = idx(1 : n)';
    thb(b, :) = ([X(idx, :); X(n + idx, :)]\[Y(idx); Y(n + idx)])';
  end
  r.thb = thb;
end

% combine runs: weighted least squares for u = [c; p] incl. setup prior on c
function u = combine(l, S, c0, Wc, sigpos)
  M = [];
  y = [];
  for k = 1 : numel(l)
    M = [M; [S{k}, -eye(2)]/sigpos];
    y = [y; l{k}/sigpos];
  end
  M = [M; [Wc, zeros(2)]];
  y = [y; Wc*c0];
  u = M\y;
end

% 1) fit every run
nr = numel(runs);
for k = 1 : nr
  [ACC, GYRO] = load_run(runs(k).file);
  res(k) = fit_run(ACC, GYRO, Tsm, taus, Lblk, B);
  if (res(k).az > 0) != runs(k).up
    warning('run %d: az = %.2f m/s² does not match up = %d', k, res(k).az, runs(k).up);
  end
end

% 2) combine the runs (nominal + Monte Carlo over bootstrap, prior and repositioning)
c0  = R*[cosd(phi); sind(phi)];
Ec  = [cosd(phi), -sind(phi); sind(phi), cosd(phi)];   % radial/tangential directions
Sc  = Ec*diag([sigR, R*sigphi*pi/180].^2)*Ec';          % prior covariance of c
Wc  = chol(inv(Sc));
S   = {runs.S};
l   = arrayfun(@(r) r.th(1:2)*m2mm, res, 'UniformOutput', false);
u   = combine(l, S, c0, Wc, sigpos);
ub  = zeros(B, 4);
for b = 1 : B
  lb = arrayfun(@(r) r.thb(b, 1:2)'*m2mm + sigpos*randn(2, 1), res, 'UniformOutput', false);
  ub(b, :) = combine(lb, S, c0 + chol(Sc)'*randn(2, 1), Wc, sigpos)';
end
c  = u(1:2);
p  = u(3:4);
pb = ub(:, 3:4);
% how much does each coordinate of p depend on the prior? (1 = only prior, 0 = only data)
J  = zeros(2);
for j = 1 : 2
  dc = zeros(2, 1);
  dc(j) = 1;
  du = combine(l, S, c0 + dc, Wc, sigpos) - u;
  J(:, j) = du(3:4);
end

% report
q  = [(1 - conf)/2, (1 + conf)/2];
ci = @(v) quantile(v, q);
printf('Single runs (vector IMU -> rotation axis in device coordinates, %2.0f%% CI, block bootstrap):\n', conf*100);
for k = 1 : nr
  r = res(k);
  printf('\n  Run %d: %s\n', k, runs(k).file);
  printf('    display %s (az = %+.2f m/s²), tau = %+.0f ms, RMS = %.3f m/s²\n', ...
         {'down', 'up  '}{runs(k).up + 1}, r.az, r.tau*1000, r.rms);
  printf('    lx = %8.2f mm  [%8.2f, %8.2f]   (radial %.2f / tangential %.2f)\n', ...
         r.th(1)*m2mm, ci(r.thb(:, 1)*m2mm), r.lx_rad*m2mm, r.lx_tan*m2mm);
  printf('    ly = %8.2f mm  [%8.2f, %8.2f]\n', r.th(2)*m2mm, ci(r.thb(:, 2)*m2mm));
  printf('    bx = %6.3f m/s², by = %6.3f m/s²\n', r.th(3), r.th(4));
end
printf('\nConsistency: residuals of the combined equations l_k = S_k*c - p (mm):\n');
for k = 1 : nr
  printf('  Run %d: [%+6.2f, %+6.2f]\n', k, S{k}*c - p - l{k});
end
printf('\nRotation axis seen from the housing center (reference placement): c = [%.2f, %.2f] mm\n', c);
printf('  => R = %.2f mm (prior %.1f +- %.1f), phi = %.2f deg (prior %.1f +- %.1f)\n', ...
       norm(c), R, sigR, atan2d(c(2), c(1)), phi, sigphi);
printf('\nIMU position relative to the housing center (x right, y up, looking at the display):\n');
printf('  x = %7.2f mm   [%7.2f, %7.2f]   share of prior R/phi: %3.0f %%\n', p(1), ci(pb(:, 1)), 100*norm(J(1, :)));
printf('  y = %7.2f mm   [%7.2f, %7.2f]   share of prior R/phi: %3.0f %%\n', p(2), ci(pb(:, 2)), 100*norm(J(2, :)));

% visualization
figure(1);
for k = 1 : nr
  r = res(k);
  subplot(nr, 2, 2*k - 1);
  plot(r.t, r.Y(1:r.n), '-', r.t, r.Ym(1:r.n), '--');
  grid on;
  ylabel(sprintf('run %d: a_x (m/s²)', k));
  subplot(nr, 2, 2*k);
  plot(r.t, r.Y(r.n+1:end), '-', r.t, r.Ym(r.n+1:end), '--');
  grid on;
  ylabel(sprintf('run %d: a_y (m/s²)', k));
end
xlabel('t (s)');
legend('measurement (smoothed)', 'model');

figure(2, 'position', [100 100 800 700]);
hold on;
rectangle('Position', [-71.5/2, -149.6/2, 71.5, 149.6], 'Curvature', 0.15);
plot(pb(:, 1), pb(:, 2), '.', 'Color', [0.7 0.7 0.9]);
plot(p(1), p(2), 'r+', 'MarkerSize', 14, 'LineWidth', 2);
plot(0, 0, 'k+');
axis equal;
axis([-60 60 -80 80]);
grid on;
xlabel('x (mm)');
ylabel('y (mm)');
title('iPhone 17: IMU position (looking at the display)');
legend('housing 71.5 x 149.6 mm', 'bootstrap / Monte Carlo', 'estimate', 'housing center', ...
       'location', 'eastoutside');
text(p(1) + 4, p(2) + 22, sprintf('x = %.1f mm [%.1f, %.1f]\ny = %.1f mm [%.1f, %.1f]\n(%2.0f%% CI)', ...
     p(1), ci(pb(:, 1)), p(2), ci(pb(:, 2)), conf*100));
print(2, 'imu_position_estimate.png', '-dpng', '-r120');

  turntable.png (Size: 441.58 KB / Downloads: 6)

  imu_position_estimate.png (Size: 28.67 KB / Downloads: 5)

  Translation und Rotation 2026-09-29 16-10-10.xlsx (Size: 237.56 KB / Downloads: 4)

  Translation und Rotation 2026-10-26 15-35-40.xlsx (Size: 143.61 KB / Downloads: 5)

  Translation und Rotation 2026-10-27 13-36-20.xlsx (Size: 141.27 KB / Downloads: 7)

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Capture d'écran 2025-07-27 221829.png (Size: 58.62 KB / Downloads: 545)

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