Randomized surfing

I have been thinking about random movements, but I was not able to devise an algorithm which have a linear displacement grow with time. Every time I put, roughly speaking, a choice of stop, back, forward. I see that bot perform diffusive/brownian type motion which as math teaches has displacement grows as sqrt of time. So the simplest of all guns: head on has the highest chance of success. From other hand it is very unlikely, that such bot will reach the ends of the guess factor ranges. Which yet again make it easy target due to limited ranges in a GF metric.

My current bot has such randomness buil in, affected by the danger map, yet it still likes to hang in one place, which makes it an easy target to GF guns (I think) and may be even DC guns.