Right.
So it lacks torque and is too nose heavy. And probably needs some reprogramming.
If I turn left or right on ANY floor it turn exactly the same as if its on a smooth pavement.
You need your chair drastically improving. Quite easy to do. Big low impedance batteries, properly programmed controller with correctly set motor compensation. Because you should not be able to tell easily what the front wheels are on or which way they are pointing. The chair should just go where its told, when its told in linear fashion. And possibly thicker battery cables and a controller with enough Amps... Also small batteries wont work well as they suffer voltage drop under load reducing torque.
http://www.wheelchairdriver.com/gopro/puncture.mp4 steers fine turning on the spot, reversing, on carpet, grass, whatever. I cant tell its got a flat tyre. Because its programmed correctly, has rearward CG position, and big powerful controller/batteries. Tyre was flat for about 10 days... Too busy to fix it! A rubber mat wouldnt and shouldnt make any difference. If I put it against a wall and try to turn the drive tyres screech and leave marks and smoke...