The robot comprises a chassis (1) with a first set of legs (2a,2b) with suction pads (6) on tilting supports (7) mounted in a linearly displaceable manner with regard to the chassis and a second set of legs (4) with suction pads (17) mounted in a rotary manner with regard to the chassis. Certain control means are adapted in order to control the functioning of said first and second sets of legs in a coordinated manner for displacing the chassis in any direction on the work surface (S). It includes support feet (9) and a system (25,10,11) for levelling the chassis in relation to a curvature of the work surface. Fitted on the chassis is a work unit (44) with a head for working on the work surface via an opening (12) in the chassis. It includes means for displacing said head along three axes and pivoting the head on two axes with respect to the chassis.