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Soft exosuits are wearable devices that can provide assistance to individuals with impaired arm movement due to injuries or disabilities. The control of cable-driven soft exosuits has several challenges due to the complex nature of the system interaction and inherent nonlinearities. In this paper, we consider the problem of robust controller design for a soft exosuit considering actuator nonlinearities. This presented work discusses the design of a two-loop sliding mode controller for upper limb exosuits to compensate the effect of nonlinearities in the actuator, i.e., the DC motor and the cable-driven actuation unit. Our aim is to achieve the desired trajectory tracking in the presence of friction in the DC motor and backlash hysteresis in the Bowden cable, for which a two-loop control scheme is introduced. The outer loop generates a desired trajectory by compensating for the backlash in the Bowden cable. The inner loop controller is a robust position-tracking control that tackles the friction in the DC motor. Using Lyapunov stability analysis, we guarantee the convergence of tracking error to zero in the presence of disturbance. Simulation results show that the proposed control law can guarantee accurate tracking performance in the presence of unknown non-linearities.