A new frontier of assistive devices aims at designing exoskeletons based on fabric and flexible materials for applications where kinematic transparency is the primary requirement. Bowden-cable transmission is the widely employed solution in most of the aforementioned applications due to advantages in durability, lightweight, safety, and flexibility. The major advantages of soft assistive devices driven by bowden-cable transmissions can be identified in the superior ergonomics and wearability, allowing users to freely move and allocating the actuation stages far from the end-effector. However, control accuracy in bowden-cable transmission presents some intrinsic limitation due to nonlinearities such as static and dynamic friction, occurring between the cables and the bowden sheaths, and backlash hysteresis. Friction and backlash effects are known to be related to the curvature of the flexible sheath, which is not directly measurable and can vary during human motion. In this paper we describe our new wearable exosuit for upper limb assistance and in particular we introduce a mathematical model for backlash hysteresis compensation. The implementation of a nonlinear adaptive controller is described in detail and experimentally tested on the proposed design as a backlash compensation strategy: results report that the adaptive controller improves the accuracy in position tracking (ie R M S E in trajectory tracking≈ 1∘) by compensating for time-varying backlash and continuously updating the model parameters. The backlash hysteresis model and the proposed control scheme are validated first on a custom-designed test bench and then applied to control the soft exoskeleton worn by a subject affected by bilateral brachial plexus injury.