Size plot and legend position programmatically for publication

Preparing a plot for a publication is far from trivial as I've learned over the past couple of days, so I was hoping for some help. There are three things I would like help with:

1. Size plot acccording to journal requirements
2. Programmatically position the legend outside of the plot (in the 'southoutside' position, centre aligned) without the plot obscuring the legend. Note if I use 'southoutside', the legend does not centre align, hence why I have not used it below
3. Print image in accordance with sizing requirements

For reference, I am seeking to size the figure 84 mm x 84 mm to satisfy the requirements for Springer Guidelines for Illustrations

Here is a much simplified version of some code to demonstrate my issue:

% Inputs
x_pos = [2.5,5,7.5];
depth_0 = [0,0,0];
depth_nom = [-0.0038 -0.0035 -0.004];

set(0,'DefaultFigureWindowStyle','normal')
set(0,'defaulttextinterpreter','latex')
set(0,'DefaultLegendInterpreter','latex')
set(0,'defaultAxesTickLabelInterpreter','latex');


h(1) = figure;

% Plot
plot(x_pos,depth_0,'marker','o','MarkerFaceColor','r','MarkerEdgeColor','r','MarkerSize',1)
hold on
plot(x_pos,depth_nom,'marker','o','MarkerSize',1)

% Figure properties
h(1).Units = 'centimeters';
h(1).Position(3) = 8.4;
h(1).Position(4) = h(1).Position(4); % aspect ratio 1:1
FontSize = 12;
set(h(1).Children, 'FontSize', FontSize);
axis square
axis tight
xlim([0 10])
xticks([0:2.5:10])
ylim([-0.025 0.025])
yticks(-0.025:.025:.025)
ylabel('$\eta$ [m]')
xlabel('$x$ [m]')
legendWidth = .3; legendHeight = .3; legendX0 = .5-(legendWidth/2); legendY0 = .0001; % legend sizing (normalised)
set(gca,'LooseInset',max(get(gca,'TightInset'), 0.02)) % remove unnecessary white space

% Get the object handles
haxis = get(h(1),'CurrentAxes');
set(haxis,  'Units', 'centimeters')            
% Set the figure size and position
pos = get(h(1), 'Position');
outerpos = get(haxis, 'OuterPosition');
set(haxis, 'OuterPosition',[0, 0, outerpos(3), outerpos(4)]);
set(h(1), 'Position', [pos(1), pos(2), outerpos(3), outerpos(4)]);
% Paper position in the eps
set(h(1), 'PaperPositionMode', 'auto');

% Legend
leg = legend('Zero flow','$U_{0}$','FontSize',FontSize,'NumColumns',2,'AutoUpdate','off');
legend boxoff
set(leg, 'Position', [legendX0, legendY0, legendWidth,legendHeight])

exportgraphics(figure(1), 'test.pdf', 'ContentType', 'vector');

My preference is to do this programmatically rather than manually sizing and positioning the figure properties.

When I insert the printed image into my Latex document, it is categorically not 84 mm wide to fit a single column.

I have tried a few of the functions on File Exchange ('Export Fig' - doesn't print pdfs in 2021b; 'Pro Plot' - doesn't deal with legends outside of the plot; fig- the exact figure size and no white space' - didn't resolve my issues)

I'd be very grateful for any help. Thank you.

[dpb]

You're writing the legend on top of the xlabel position for starters by not letting the axes resize to make room for it.

If I just use

hLg=legend('Zero flow','$U_{0}$','FontSize',FontSize,'NumColumns',2,'Location','southoutside');

it looks pretty good to me and centered on the axes of the figure. Personally, I think that would look better than trying to center it on the whole figure.

I know there have been issues with printing to size for since forever; I've never had the need, thankfully, my publication days were too far in the past to submit electronically.

[Daniel Rowe]

Thanks for your response. To be clear, the issue was slightly conflated if the screen was docked and I manually adjusted the legend position, which led to the plot auto resizing. Anyway, since then, I discovered that by changing the 'DataAspectRatioMode' attribute to 'manual', the auto-resizing is switched off, furthermore, I gave up with the legend outside of the plot and reverted to 'Location','best' as this seems more robust.