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Control saturation is an important limitation in practical control systems and it is well known that performance degradation or instability may result if this limitation is not effectively addressed. Using ideas from the gradient projection method in nonlinear programming, we propose a new anti-windup scheme for multi-input, multi-output nonlinear dynamic controllers. The key idea is to project the controller state update law onto the tangent plane of the active saturation constraints. To do this, we first extend the gradient projection method to the continuous-time case that can accommodate multiple nonlinear constraints. This is then used for anti-windup compensation, resulting in a hybrid controller that switches its state update law over arbitrary combinations of saturating controls. Simulations on a nonlinear two-link robot driven by an adaptive sliding mode controller illustrates its effectiveness and limitations.