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

\textbf{Objective: }To develop a real-time method for designing gradient waveforms for arbitrary $k$-space trajectories that are time-optimal and hardware-compliant. \textbf{Methods: }The gradient waveform is solved recursively under both the slew-rate and the trajectory constraints. The gradient constraint is enforced by thresholding the $\ell_2$-norm of the next gradient vector. The constraints form a quadratic equation. To ensure the existence of the solution, a novel Discrete-Time Forward and Backward Sweep (DTFBS) strategy is proposed. To ensure the existence of the trajectory derivatives, the trajectory function is reparameterized as a piecewise cubic polynomial function with $C^2$ continuity. To ensure trajectory fidelity, the output gradient waveform is reparameterized by the finite difference of the trajectory samples. Simulation experiments across seven commonly adopted non-Cartesian trajectories were conducted to validate generality, time-optimality, real-time capability, slew-rate accuracy, and improvements over prior work. Imaging feasibility of the designed time-optimal gradient waveform was validated in phantom and in vivo experiments. \textbf{Results: }The proposed method achieves a $>89\%$ reduction in computation time and simultaneously reduces slew-rate overshoot by $>98\%$ compared to the prior method across all involved trajectories. 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. \textbf{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. \textbf{Significance: }This is the first method achieving real-time gradient waveform design for arbitrary $k$-space trajectories.