you will need some form of image stabilization when you are filming and moving at the same time.
There is several ways to do this
1: EIS or Electronic Image stabilization.
This is made by camera, what happen is the camera pick a smaller area on the CMOS image chip, and as this move around on the rest of the CMOS due to shaking the camera track this and put it in the middle.
So to get EIS of say 1080P footage the CMOS capturing that footage have to be a little larger than what is needed to capture 1080 so camera can track that 1080p as it move around on the rest of the CMOS pixels.
2: OIS or Optical Image stabilization.
This is done in the lens, and so i am not sure if its suited to action camera use as there is some moving parts.
Another solution is to buy a steadycam gimbal, this is a gimbal with motors on it and some hardware that allways keep your camera steady and level.
Steadycam gimbals some in all sizes and prices, from pro solutions for big cameras to small ones for little actioncameras.
Shooting 1080 at 60 fps means a more smoothe footage not least of faster moving things, i use it too as it allow me to slow down footage with 50% and still have pretty nice smoothe running "slow motion" footage.
As you are capturing 60 frames every second and not only 30, this mean for 60 FPS shooting you got to have a shorter exposure timing and as such the chance of sharper footage.
But this is really only in play in low light situations where cameras will go as slow in exposure it can to make bright footage, and so if you film 60 FPS the exposure timing have to be at least 1/60 of a second for each frame, otherwise you dont have enuff time every second to capture 60 frames.
This in turn mean filming in 1080 30 FPS the camera can drop down to 1/30 second exposure timing, leaving each frame 2 X longer time to expose than filming 60 FPS.
As long as there is plenty of light the camera will use much faster exposure timings, so in full sunshine the camera spend more time waiting on capturing the next of 60 freame every second, here we are takling houndreds or thousinds of a second exposure time.
This can change as the camera can allso regulate ISO values, this is simply said the light sensetivety of the camera, a really high ISO number = more light sensetive ( film in old cameras ) or sensetivety of the CMOS in modern digital cameras.
