Authors: Christiane Sommer, Vladyslav Usenko, David Schubert, Nikolaus Demmel, Daniel Cremers Description: Continuous-time trajectory representation has recently gained popularity for tasks where the fusion of high-frame-rate sensors and multiple unsynchronized devices is required. Lie group cumulative B-splines are a popular way of representing continuous trajectories without singularities. They have been used in near real-time SLAM and odometry systems with IMU, LiDAR, regular, RGB-D and event cameras, as well as for offline calibration. These applications require efficient computation of time derivatives (velocity, acceleration), but all prior works rely on a computationally suboptimal formulation. In this work we present an alternative derivation of time derivatives based on recurrence relations that needs O(k) instead of O(k^2) matrix operations (for a spline of order k) and results in simple and elegant expressions. While producing the same result, the proposed approach significantly speeds up the trajectory optimization and allows for computing simple analytic derivatives with respect to spline knots. The results presented in this paper pave the way for incorporating continuous-time trajectory representations into more applications where real-time performance is required.