Real-Time Gradient Waveform Design for Arbitrary $k$-Space Trajectories

Objective To develop a real-time method for designing gradient waveforms for arbitrary k-space trajectories that are time-optimal and hardware-compliant.Methods The gradient waveform is solved recursively under both the slew-rate and the trajectory constraints, which form a quadratic equation. The gradient constraint is enforced by thresholding the L2-norm of the gradient vectors. To ensure the existence of the solution, gradient magnitude is thresholded by the escape velocity. A Discrete-Time Forward and Backward Sweep strategy is then applied to further constrain the slew-rate. Trajectory and gradient reparameterization strategies are adopted to enhance the generality and preserve the sampling accuracy. The proposed method is compared with the conventional optimal control method across seven commonly adopted non-Cartesian trajectories. Imaging feasibility of the designed time-optimal gradient waveform was demonstrated by phantom and in vivo imaging experiments.Results The proposed method achieves a >89% reduction in computation time and a >98% reduction in slew-rate error simultaneously. The computation time of the proposed method is shorter than the gradient duration for all tested cases, validating the real-time capability of the proposed method.Conclusions The proposed method enables real-time and hardware-compliant gradient waveform design, achieving significant reductions in computation time and slew-rate overshoot compared to the previous method.Significance This is the first method achieving real-time gradient waveform design for arbitrary k-space trajectories.