  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 $\mathcal{O}(k)$ instead of $\mathcal{O}(k^2)$ 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.
