Free Software to remove shake on mp4 videos

I'm sure someone will come along with a better answer.

There are 2 aspects to shake: (1) blurry photos because the camera sensor cant adjust to the speed of movement. (2) The pictures are fine, but the camera wobbles.

for (1), the game is already over.

for (2) where the picture is mostly sharp, there is a nasty, free, time consuming way.
Convert the pictures to jpg I used vlc, yes it was fiddly, no I can not remember the exact method, but was more than one setting - think 3 in 2 different locations

For the jpg choose a reference point - say a persons left eye pupil. Note the coordinates on the screen where that pupil always has to be and make sure the pupil on all the successive shots match those coordinates too. The surroundings will possibly have to move, but that pupil always be in that same coordinates, by adding or cutting borders, but the end product of each jpg has to be the same dimensions even it if they are all larger than the original, so that the jpg can be converted back to film, the photos overlay each other. In some situations you will loose some information around the borders through croping and gain white areas due to border addition.

assemble all the photos use something like ffmpeg -r 60 -f image2 -s 1920x1080 -i pic%04d.png -vcodec libx264 -crf 25 -pix_fmt yuv420p test.mp4 - you will have to google this command to suit your needs

If all the frames are still present and you have not added too many or removed to many, the audio of the original will still be in sync, so import both new and old films into VSCD and turn off the viewing feature on the old so to only play the sound track. You will also most likely have to trim the film borders to a smaller size to remove the unwanted re-positioned borders

In the grand scheme of things, this is probably what you are after, and the full body camera ones are more expensive
https://vimeo.com/149806262

[Edit2]
The easy way in the PRO version of vsdc - about £20