This paper deals with the control of a lower limb exoskeleton acting on the knee joint level. Classical L1 adaptive control law is proposed to ensure assistance-as-needed and resistive rehabilitation following a desired trajectory considered defined by a therapeutic doctor. This control law introduces a time lag within the desired trajectory tracking because of the presence of a filter in its structure. In order to mitigate this drawback, the classical L1 adaptive control is augmented by a nonlinear proportional control. The classical and augmented L1 adaptive control laws are tested in real-time using the exoskeleton EICOSI of LISSI-lab. Real-time experimental results highlight the utility of these control laws in assistance-as-needed and resistive rehabilitation. They also show the effectiveness of the augmented version of the L1 adaptive control.