With heavy reusable launch vehicles becoming a key technology for low-cost and high-frequency space exploration, cable-net recovery schemes have attracted considerable attention due to their strong adaptability to landing dispersion and attitude deviations. However, under multiple sources of interference—including engine plume impingement, trolley motion, and flexible vibrations—the dynamic state information of the flexible cable-net recovery system is difficult to acquire directly, severely constraining the accuracy of closed-loop control and capture reliability. Although Kalman filtering combined with multibody dynamics models can achieve state estimation for complex systems, the flexible cable-net system is inherently a high-dimensional, nonlinear, strongly rigid-flexible coupled constrained multibody system, whose dynamic equations are formulated as index-3 differential-algebraic equations (DAEs). Conventional methods require the conversion of DAEs into ordinary differential equations (ODEs) followed by linearization, a process that is complex and lacks generality; conventional numerical integration methods converge slowly when dealing with such stiff problems, with the Jacobian matrix becoming severely ill-conditioned under large time steps and constraints being prone to violation. Therefore, developing a filtering method that does not require model transformation, can directly utilize the multibody DAE for state prediction, and can maintain constraint satisfaction has become a key technical challenge for state estimation in flexible cable-net recovery.
In a recent study published in Space: Science & Technology , researchers from Beijing Institute of Technology proposed an error-state extended Kalman filtering method based on multibody integrators. The study adopts an indirect filtering strategy that decouples state prediction from error covariance update—directly employing two types of multibody integrators, the generalized-α method and the backward differentiation formula (BDF), for one-step state prediction without requiring model conversion or linearization of the original multibody system. By applying scaling transformation and augmented Lagrangian formulation for index reduction, the Jacobian condition number is improved, and the numerical behavior reaches stability comparable to that of ODEs; additionally, constraint-violation correction is applied to the estimated states to ensure that the states always satisfy the system constraints. Numerical simulation results demonstrate that under a small time step of 1 ms, the root-mean-square errors of displacement estimation corresponding to the generalized-α and BDF integrators are 0.0066 m and 0.026 m, respectively, with the generalized-α method outperforming BDF in both displacement accuracy and numerical stability; when the time step is increased to 5 ms, the ratio of CPU time to real time for the generalized-α method drops below 1, indicating real-time operational capability. In the complete four-cable-net system, the four-channel decentralized estimation strategy based on the single-cable model achieves accuracy comparable to that of the full-system centralized estimation, while significantly improving computational efficiency. This research provides an effective solution for state estimation in flexible cable-net recovery systems and offers technical support for the theoretical development of state estimation for complex flexible multibody systems.
First, this paper focuses on the state estimation problem in the cable-net recovery system for heavy reusable launch vehicles and establishes a multibody dynamics model for the flexible cable-net system. The cable-net recovery scheme captures the landing rocket via a ground-based arresting cable-net, offering greater adaptability to landing position and attitude dispersions compared to landing-leg recovery methods. However, during the recovery process, multiple sources of interference—including engine plume impingement, trolley motion, and flexible vibrations—render the dynamic state of the arresting cables difficult to measure directly, severely constraining the accuracy of closed-loop control. As illustrated in Fig. 1, the cable-net recovery process comprises three phases: following, entering, and capturing; the trolley drives the cable motion to track the rocket landing point, and the control of the capture container center position relies on accurate cable dynamic information. The system is modeled as a flexible multibody system, with each arresting cable discretized into variable-length cable elements using the Arbitrary Lagrangian–Eulerian (ALE) formulation of the Absolute Nodal Coordinate Formulation (ANCF). The complete cable-net system consists of four cables, and its dynamic equations are formulated as high-dimensional, nonlinear, strongly rigid–flexible coupled index-3 differential-algebraic equations (DAEs). Fig. 2 presents the multibody model structure of the four-cable-net system, where each cable is driven at both ends by trolleys and subjected to distributed loads such as plume impingement. The direct numerical integration of index-3 DAEs faces severe numerical difficulties, including a Jacobian matrix condition number that grows with the cube of the inverse of the time step and constraint drift. To address these issues, the study performs scaling transformation on the DAEs with respect to dimensionless time and constraint equation magnitude, and combines this with an augmented Lagrangian formulation for index reduction. This approach renders the Jacobian matrix condition number and error propagation magnitude independent of the time step, achieving numerical behavior comparable to that of ordinary differential equations (ODEs).
Second, this paper proposes an error-state extended Kalman filtering method based on multibody integrators, employing the generalized-α method and the backward differentiation formula (BDF) as the two multibody integrators for constructing the prediction stage, respectively. As illustrated in Fig. 3, the framework adopts an indirect filtering strategy that decouples state prediction from error covariance update, directly utilizing the multibody integrators for one-step state prediction without requiring model conversion or linearization of the original multibody system, thereby circumventing the complex DAE-to-ODE transformation inherent in conventional Kalman filtering. In the prediction stage, the multibody integrators perform temporal integration of the DAEs after scaling and index reduction, outputting the generalized coordinates and generalized velocities. The error-state covariance propagation employs a simplified state transition matrix to enhance computational efficiency. In the correction stage, the posterior estimate of the error state is computed based on lidar measurements, and the estimated state is projected back onto the constraint manifold through constraint-violation correction, ensuring that both position and velocity estimates consistently satisfy the system constraints. The measurement function is divided into single-cable mode and four-cable mode depending on the measurement scheme: the single-cable mode measures the three-dimensional position and velocity of a target point on a single cable; the four-cable mode measures the average values of the target points on the four cables, i.e., the position and velocity of the capture container center.
Finally, this paper validates the effectiveness of the proposed method through numerical simulations, and systematically analyzes the effects of integrator type, time step, augmented Lagrangian parameters, and the number of cable elements on estimation accuracy and computational efficiency. Based on the estimation results of the lateral displacement and velocity of the target point on a single arresting cable, the error-model simulation deviates from the true value by 1.1 m, exceeding the allowable capture tolerance of 1 m, whereas the estimates from both error-state EKF implementations significantly outperform the error-model simulation. The displacement root-mean-square errors (RMSEs) are 0.0066 m (generalized-α) and 0.026 m (BDF), and the velocity RMSEs are 0.2951 m/s and 0.2927 m/s, respectively, validating the effectiveness of the filtering method. Figs. 4 and 5 present the RMSE and the ratio of CPU time to real time under different time steps, respectively; the generalized-α method remains stable as the time step increases, whereas BDF diverges beyond 1 ms. At a time step of 5 ms, the CPU-time-to-real-time ratio of the generalized-α method drops below 1, demonstrating real-time operational capability. In the state estimation of the complete four-cable-net system, Figs. 6 and 7 compare two estimation strategies: the four-channel decentralized estimation based on the single-cable model achieves accuracy comparable to that of the centralized estimation based on the full-system model, while significantly improving computational efficiency. This research provides an effective solution for state estimation in flexible cable-net recovery systems and offers technical support for the theoretical development of state estimation for complex flexible multibody systems.
Space: Science & Technology
An Error-State Kalman Filter with Multibody Integrator for State Estimation of Flexible Cable-Net Systems
2-Jul-2026