This paper presents a continuous-time radar-inertial and lidar-inertial odometry approach using a Gaussian process motion prior. The method improves computational efficiency by using a sparse prior, reducing the complexity of preintegration and interpolation. A white-noise-on-acceleration motion prior is used, treating the gyroscope as a direct measurement while preintegrating accelerometer measurements to form relative velocity factors. The odometry is implemented using sliding-window batch trajectory estimation, achieving real-time performance. The approach is efficient and demonstrates improved radar odometry performance by 43% with the inclusion of an IMU. The paper also introduces a Gaussian process motion prior for continuous-time state estimation, enabling efficient batch trajectory estimation with linear computational complexity. The method is applied to both radar-inertial and lidar-inertial odometry, demonstrating robustness under adverse weather conditions. The approach is compared across different datasets and weather conditions, showing improved performance in challenging scenarios. The paper also discusses the use of Gaussian processes for continuous-time state estimation, including the use of a white-noise-on-acceleration prior and the benefits of using a sparse kernel. The method is applied to both radar and lidar odometry, demonstrating the effectiveness of continuous-time state estimation in handling motion distortion and improving accuracy. The paper also presents experimental results on three datasets, including KITTI-raw, the Newer College Dataset, and the Boreas dataset, showing the effectiveness of the approach in real-world scenarios. The results demonstrate the robustness of the method in challenging environments and the effectiveness of the Gaussian process motion prior in improving odometry performance. The paper concludes that the proposed approach provides a robust and efficient solution for continuous-time radar-inertial and lidar-inertial odometry.This paper presents a continuous-time radar-inertial and lidar-inertial odometry approach using a Gaussian process motion prior. The method improves computational efficiency by using a sparse prior, reducing the complexity of preintegration and interpolation. A white-noise-on-acceleration motion prior is used, treating the gyroscope as a direct measurement while preintegrating accelerometer measurements to form relative velocity factors. The odometry is implemented using sliding-window batch trajectory estimation, achieving real-time performance. The approach is efficient and demonstrates improved radar odometry performance by 43% with the inclusion of an IMU. The paper also introduces a Gaussian process motion prior for continuous-time state estimation, enabling efficient batch trajectory estimation with linear computational complexity. The method is applied to both radar-inertial and lidar-inertial odometry, demonstrating robustness under adverse weather conditions. The approach is compared across different datasets and weather conditions, showing improved performance in challenging scenarios. The paper also discusses the use of Gaussian processes for continuous-time state estimation, including the use of a white-noise-on-acceleration prior and the benefits of using a sparse kernel. The method is applied to both radar and lidar odometry, demonstrating the effectiveness of continuous-time state estimation in handling motion distortion and improving accuracy. The paper also presents experimental results on three datasets, including KITTI-raw, the Newer College Dataset, and the Boreas dataset, showing the effectiveness of the approach in real-world scenarios. The results demonstrate the robustness of the method in challenging environments and the effectiveness of the Gaussian process motion prior in improving odometry performance. The paper concludes that the proposed approach provides a robust and efficient solution for continuous-time radar-inertial and lidar-inertial odometry.