Abstract

Motion planning for robotic systems traditionally discretizes time into fixed waypoints, then solves for joint or task-space configurations at each node. Continuous-time trajectory optimization (CTTO) instead parameterizes the entire trajectory as a smooth function, typically a Gaussian process or a polynomial spline, and optimizes over the function’s hyperparameters directly. This allows exact enforcement of kinematic and dynamic constraints at any query time, not just at discrete nodes, and produces smoother, more energy-efficient motions. Deploying CTTO on embedded platforms, where compute is constrained and real-time deadlines are hard, introduces a distinct set of engineering challenges that are now being systematically addressed.

Continuous-Time Representations

The two dominant representations in current literature are Gaussian Process (GP) motion planning and polynomial spline parameterization. GP-based methods, pioneered by GPMP2 from the Robot Learning Lab at Georgia Tech, represent trajectory distributions as GP priors and optimize the posterior under obstacle and kinematic constraints. The GP prior encodes smoothness implicitly through kernel choice, typically a Matern or white-noise-on-acceleration kernel, avoiding explicit smoothness penalties in the objective.

Polynomial spline methods, including the B-spline and piecewise-polynomial approaches widely used in MAV (micro aerial vehicle) research from groups at UPenn GRASP Lab and MIT CSAIL, offer closed-form derivatives and straightforward constraint enforcement at arbitrary arc-length positions. Minimum-snap trajectories for quadrotors remain one of the most successful examples of this class, with industrial adoption in drone delivery platforms.

Embedding CTTO in Real-Time Loops

The core challenge is that CTTO solves a nonlinear optimization problem, typically via gradient-based solvers such as GPMP2’s Gaussian belief propagation or the iSAM2 factor graph solver from GTSAM. On a desktop workstation with a modern CPU and 20-50 ms solve budgets, this is manageable. On embedded systems, such as ARM Cortex-A78 class processors or the Zynq UltraScale+ MPSoC used in some industrial controller boards, solve time must shrink by an order of magnitude.

Two strategies have emerged. First, warm-starting: the optimizer initializes from the previous trajectory, which for slowly changing environments yields convergence in far fewer iterations than cold-start. Second, problem reduction through active-set trimming, keeping only obstacle constraints within a proximity threshold active in the current solve and discarding the rest. Together these techniques reduce median solve time for a 6-DOF arm re-plan from roughly 80 ms to 12-18 ms on Cortex-A78 hardware, as reported in 2025 papers from the Robotics and Autonomous Systems Group at Imperial College London.

Hardware accelerators offer a complementary path. Small matrix-matrix multiply operations dominate the inner loop of GP belief propagation, and FPGA-implemented fixed-point matrix units can execute these at clock-rate speeds unavailable to the general-purpose CPU. Xilinx-based implementations have demonstrated 4x speedups on the covariance update step with modest FPGA resource utilization, making this a viable path for production embedded controllers.

Practical Deployment

Integrating CTTO into a production robot requires handling edge cases that clean benchmarks omit: abrupt environment changes that invalidate the warm start, communication jitter from distributed sensor nodes, and worst-case solve time guarantees required by safety certifiers. Mixed-criticality system designs, where a fast reactive controller handles immediate collision avoidance while CTTO runs asynchronously at lower priority, are the current practical standard. The ROS 2 real-time executor and the Orocos RTT framework both provide the scheduling primitives to implement this architecture.

Conclusion

Continuous-time trajectory optimization is moving from a research technique to a deployable component in embedded robot controllers. Warm-starting, active-set reduction, and selective hardware acceleration have brought solve times within reach of real-time budgets on arm-class hardware. As tooling matures and safety certification frameworks adapt, CTTO is well-positioned to replace fixed-horizon discretized planners in the next generation of industrial manipulators and mobile platforms.