Researchers at Beijing Institute of Technology have now tackled this measurement gap head-on. In a study published in Space: Science & Technology, the team proposed an error-state extended Kalman filtering method built directly on multibody integrators, allowing the dynamics of a flexible cable net to be predicted and corrected without the cumbersome model transformations that conventional filtering demands. The work addresses a problem that has long frustrated the multibody dynamics community: the cable-net system is inherently a high-dimensional, nonlinear, strongly rigid-flexible coupled constrained multibody system, and its governing equations take the form of index-3 differential-algebraic equations, or DAEs. These equations mix differential equations with algebraic constraints, and traditional approaches force analysts to convert them into ordinary differential equations and then linearize them before a Kalman filter can touch them—a process that is complex, lacks generality, and often degrades accuracy.

The numerical obstacles run deeper still. Direct integration of index-3 DAEs is notoriously stiff, and conventional numerical integrators converge slowly on such problems. The Jacobian matrix becomes severely ill-conditioned when large time steps are used—its condition number, in fact, grows with the cube of the inverse of the time step—and the algebraic constraints that hold the system together are prone to violation, a phenomenon known as constraint drift. For a state estimator that must run fast enough to guide a capture in real time, these are not academic inconveniences; they are showstoppers. The Beijing Institute of Technology team therefore set out to develop a filtering method that requires no model transformation, can use the multibody DAE directly for state prediction, and can maintain constraint satisfaction throughout the estimation process.

The foundation of the new approach is a carefully constructed multibody model of the recovery system itself. The cable-net recovery sequence unfolds in three phases—following, entering, and capturing—during which trolleys drive the cable motion to track the rocket’s landing point. Controlling the position of the capture container center depends on accurate cable dynamic information, which is exactly what the model must supply. Each arresting cable is discretized into variable-length cable elements using the Arbitrary Lagrangian–Eulerian formulation of the Absolute Nodal Coordinate Formulation, a technique well suited to cables whose length changes as they are paid out and reeled in. The complete system comprises four cables, each driven at both ends by trolleys and subjected to distributed loads such as plume impingement, and the resulting dynamic equations are high-dimensional, nonlinear, and strongly coupled between rigid and flexible modes.

To tame the index-3 DAEs, the researchers applied two complementary remedies. First, they performed a scaling transformation on the equations with respect to dimensionless time and the magnitude of the constraint equations. Second, they combined this with an augmented Lagrangian formulation for index reduction. The combined effect is striking: the Jacobian matrix condition number and the error propagation magnitude become independent of the time step, and the numerical behavior of the constrained system reaches stability comparable to that of ordinary differential equations. This means the integrator no longer punishes the user for choosing the larger time steps that real-time operation demands—a property that proves decisive later in the validation.

On top of this stabilized model sits the filtering framework itself. The method adopts an indirect filtering strategy that decouples state prediction from error covariance update. In the prediction stage, one of two multibody integrators—the generalized-α method or the backward differentiation formula (BDF)—performs temporal integration of the scaled, index-reduced DAEs, outputting generalized coordinates and velocities in a single step, with no conversion to ODEs and no linearization of the original multibody system. Error-state covariance propagation employs a simplified state transition matrix to keep the computational burden manageable. In the correction stage, the posterior estimate of the error state is computed from 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 at every step.

The measurement function is designed with operational flexibility in mind. In single-cable mode, the filter ingests the three-dimensional position and velocity of a target point on one arresting cable. In four-cable mode, it measures the average of the target points on all four cables—effectively the position and velocity of the capture container center, the quantity that ultimately matters for steering the net onto the descending rocket. This dual-mode design allows the same filtering machinery to serve both local cable monitoring and global capture-point control.

Numerical simulations put the method through its paces, systematically examining the effects of integrator type, time step, augmented Lagrangian parameters, and the number of cable elements. The results are telling. In a baseline error-model simulation, the estimate deviates from the true value by 1.1 meters—beyond the allowable capture tolerance of 1 meter—whereas both error-state EKF implementations significantly outperform it. With a small time step of 1 millisecond, the displacement root-mean-square errors are 0.0066 meters for the generalized-α integrator and 0.026 meters for BDF, while the velocity RMSEs come in at 0.2951 and 0.2927 meters per second, respectively. The generalized-α method proves superior in both displacement accuracy and numerical stability: it remains stable as the time step increases, whereas BDF diverges beyond 1 millisecond. Most importantly for practical deployment, when the time step grows to 5 milliseconds, the ratio of CPU time to real time for the generalized-α method drops below 1, demonstrating genuine real-time operational capability.

Scaling up to the complete four-cable-net system, the team compared two estimation strategies. A four-channel decentralized estimation based on the single-cable model achieves accuracy comparable to centralized estimation built on the full-system model, while significantly improving computational efficiency. This finding has clear engineering significance: instead of paying the price of a monolithic high-dimensional model, operators can run parallel single-cable estimators and recover essentially the same capture-relevant accuracy at a fraction of the computational cost.

The implications extend beyond a single recovery architecture. By showing that a Kalman filter can operate directly on multibody DAEs—using stabilized integrators, index reduction, and constraint-violation correction to sidestep the classical DAE-to-ODE detour—the study offers an effective solution for state estimation in flexible cable-net recovery systems and technical support for the broader theoretical development of state estimation in complex flexible multibody systems. As reusable launch vehicles grow heavier and launch cadences accelerate, the ability to sense, in real time, the invisible dynamics of a flexible capture net may well determine whether cable-net recovery matures from an appealing concept into the workhorse of routine, low-cost spaceflight.

Subject of Research: Error-state Kalman filtering with multibody integrators for state estimation of flexible cable-net recovery systems for reusable launch vehicles

Article Title: An error-state kalman filter with multibody integrator for state estimation of flexible cable-net systems

Article References: An error-state kalman filter with multibody integrator for state estimation of flexible cable-net systems. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: Kalman filtering, error-state estimation, multibody dynamics, differential-algebraic equations, flexible cable-net, reusable launch vehicles, generalized-alpha method, backward differentiation formula, Absolute Nodal Coordinate Formulation, constraint violation correction, real-time estimation, lidar measurements