# Efficient Derivative Computation for Cumulative B-Splines on Lie Groups

arXiv: https://arxiv.org/abs/1911.08860 · HTML: https://mlsanity.com/paper/1911.08860/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Nov 2019.

## Built on by (most cited first, 28 of 28)

- [CMax-SLAM: Event-based Rotational-Motion Bundle Adjustment and SLAM System using Contrast Maximization](https://mlsanity.com/paper/2403.08119/) (2024)
- [MCD: Diverse Large-Scale Multi-Campus Dataset for Robot Perception](https://mlsanity.com/paper/2403.11496/) (2024)
- [Dynamic Event Camera Calibration](https://mlsanity.com/paper/2107.06749/) (2021)
- [Asynchronous Multi-View SLAM](https://mlsanity.com/paper/2101.06562/) (2021)
- [Continuous-Time vs. Discrete-Time Vision-based SLAM: A Comparative Study](https://mlsanity.com/paper/2202.08894/) (2022)
- [Real-time Multi-Adaptive-Resolution-Surfel 6D LiDAR Odometry using Continuous-time Trajectory Optimization](https://mlsanity.com/paper/2105.02010/) (2021)
- [Event-Based Feature Tracking in Continuous Time with Sliding Window Optimization](https://mlsanity.com/paper/2107.04536/) (2021)
- [Traj-LO: In Defense of LiDAR-Only Odometry Using an Effective Continuous-Time Trajectory](https://mlsanity.com/paper/2309.13842/) (2023)
- [EF-Calib: Spatiotemporal Calibration of Event- and Frame-Based Cameras Using Continuous-Time Trajectories](https://mlsanity.com/paper/2405.17278/) (2024)
- [Continuous-Time Spatiotemporal Calibration of a Rolling Shutter Camera-IMU System](https://mlsanity.com/paper/2108.07200/) (2021)
- [Event-based Photometric Bundle Adjustment](https://mlsanity.com/paper/2412.14111/) (2024)
- [Forecasting Continuous Non-Conservative Dynamical Systems in SO(3)](https://mlsanity.com/paper/2508.07775/) (2025)
- [MITI: SLAM Benchmark for Laparoscopic Surgery](https://mlsanity.com/paper/2202.11496/) (2022)
- [Traj-LIO: A Resilient Multi-LiDAR Multi-IMU State Estimator Through Sparse Gaussian Process](https://mlsanity.com/paper/2402.09189/) (2024)
- [Continuous-Time SO(3) Forecasting with Savitzky--Golay Neural Controlled Differential Equations](https://mlsanity.com/paper/2506.06780/) (2025)
- [Hyperion -- A fast, versatile symbolic Gaussian Belief Propagation framework for Continuous-Time SLAM](https://mlsanity.com/paper/2407.07074/) (2024)
- [Jacobian Computation for Cumulative B-Splines on SE(3) and Application to Continuous-Time Object Tracking](https://mlsanity.com/paper/2201.10602/) (2022)
- [Constrained Visual-Inertial Localization With Application And Benchmark in Laparoscopic Surgery](https://mlsanity.com/paper/2202.11075/) (2022)
- [A Tutorial on Uniform B-Spline](https://mlsanity.com/paper/2309.15477/) (2023)
- [Joint Spatial-Temporal Calibration for Camera and Global Pose Sensor](https://mlsanity.com/paper/2403.00976/) (2024)
- [Mitigating Heterogeneity among Factor Tensors via Lie Group Manifolds for Tensor Decomposition Based Temporal Knowledge Graph Embedding](https://mlsanity.com/paper/2404.09155/) (2024)
- [LiDAR-based Registration against Georeferenced Models for Globally Consistent Allocentric Maps](https://mlsanity.com/paper/2412.02533/) (2024)
- [HDVIO2.0: Wind and Disturbance Estimation with Hybrid Dynamics VIO](https://mlsanity.com/paper/2504.00969/) (2025)
- [Temporal and Rotational Calibration for Event-Centric Multi-Sensor Systems](https://mlsanity.com/paper/2508.12564/) (2025)
- [eKalibr-Inertial: Continuous-Time Spatiotemporal Calibration for Event-Based Visual-Inertial Systems](https://mlsanity.com/paper/2509.05923/) (2025)
- [Unleashing the Power of Discrete-Time State Representation: Ultrafast Target-based IMU-Camera Spatial-Temporal Calibration](https://mlsanity.com/paper/2509.12846/) (2025)
- [Scanline-Aware Animatable Gaussian Avatars from Rolling-Shutter Videos](https://mlsanity.com/paper/2608.17314/) (2026)
- [LieSpline-DP: Lie-Group B-Spline Diffusion Policy for Smooth Robot Manipulation](https://mlsanity.com/paper/2609.15162/) (2026)

## Abstract (arXiv)

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(k2) 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.