arXiv subjects

Kathrin Flaßkamp

Publications and source records attributed to Kathrin Flaßkamp.

At least 19 recordsLinked to original sources

arXiv2026

Reconstruction of continuum robots by marker-free shape registration of image data using a kinematic model

Continuum robots are slender, flexible manipulators that navigate confined, curved workspaces and are gaining traction in aerospace, inspection, automation, and minimally invasive medical applications. Predicting their shape from physics-based models alone remains challenging, making accurate measurement of the deformed backbone essential for model validation and reference-data acquisition. We present an optimization-based shape-registration algorithm that fits a parametric three-dimensional curve directly to image observations within a photogrammetric pipeline, targeting marker-free measurement rather than sensing under occlusion. By matching reconstruction points to robot pixels, the method requires no prior knowledge of the robot's location in each image. Across most configurations, the estimated backbone deviates from ground truth by less than 1 mm (0.67% of the robot's length). On real concentric-tube continuum robots, the reconstruction agrees with ten discrete manual photogrammetric measurements over 18 configurations, while replacing the manual procedure with an automated pipeline that runs in roughly 0.5 s per configuration.

cs.RO↗

arXiv2026

Structure-Preserving Gaussian Processes Via Discrete Euler-Lagrange Equations

In this paper, we propose Lagrangian Gaussian Processes (LGPs) for probabilistic and data-efficient learning of dynamics via discrete forced Euler-Lagrange equations. Importantly, the geometric structure of the Lagrange-d'Alembert principle, which governs the motion of dynamical systems, is preserved by construction in the absence of external forces. This allows learning physically consistent models that overcome erroneous drift in the system's energy, thereby providing stable long-term predictions. At the core of our approach lie linear operators for Gaussian process conditioning, constructed from discrete forced Euler-Lagrange equations and variational discretization schemes. Thereby and unlike prior work, the method enables learning dynamics from discrete position snapshots, i.e., without access to a system's velocities or momenta. This is particularly relevant for a large class of practical scenarios where only position measurements are available, for instance, in motion capture or visual servoing applications. We demonstrate the data-efficiency and generalization capabilities of the LGPs in various synthetic and real-world case studies, including a real-world soft robot with hysteresis. The experimental results underscore that the LGPs learn physically consistent dynamics with uncertainty quantification solely from sparse positional data and enable stable long-term predictions.

cs.LG↗

arXiv2026

A data-based image representation for continuous-time LTI systems

We derive a numerically stable method to compute an image representation of an unknown linear system only from data, leveraging a continuous-time version of Willems et al.'s fundamental lemma. To this end, we use derivatives approximated by algebraic differentiators. Our novel image representation avoids solving differential-algebraic equations and significantly reduces computational complexity by eliminating redundant degrees of freedom corresponding to the number of unknown quantities to be identified. Simulation results confirm the effectiveness of the proposed approach, even in the presence of severe measurement disturbances.

math.OC↗

arXiv2026

Individual Minima-Informed Multi-Objective Model Predictive Control for Fixed Point Stabilization

Multi-objective model predictive control (MOMPC) for fixed point stabilization requires an automated a priori decision-making (DM) mechanism to translate a high-level preference into a single solution. To this aim, we introduce an approach called individual minima-informed DM. This class of methods can be implemented through two sequential optimizations, regardless of the number of objectives, thereby improving the real-time capability of MOMPC. These methods operate on Pareto fronts (PFs) and leverage the individual minima (IM), which are characteristic Pareto-optimal points. By this, we aim to facilitate mapping a high-level preference to a point on the PF. Several approaches exist to guarantee the closed-loop stability of an MOMPC scheme. This work builds upon an approach known from the literature, which combines a quasi-infinite horizon scheme with an additional descent condition on the costs. Assuming that the terminal ingredients of the quasi-infinite horizon approach are fixed, then the size of a PF or the DM space is determined solely by the descent condition. This paper examines both the IM-informed DM methods and their integration into a stabilizing MOMPC scheme. The main contributions are twofold. First, we propose and systematically analyze six variants of IM-informed DM methods, including two novel methods, designed to facilitate the translation of a high-level preference to a point on the PF. Second, to retain the largest possible DM space for these methods, we show that they can be embedded into an MOMPC framework while preserving closed-loop stability under a descent condition that is less restrictive than in the literature. We further present a practical method for constructing the required terminal ingredients. A numerical case study confirms the closed-loop stability of the proposed framework and illustrates the potential benefit of adapting the preference online.

math.OC↗

arXiv2026

The Indirect Method for Generating Libraries of Optimal Periodic Trajectories and Its Application to Economical Bipedal Walking

Trajectory optimization is an essential tool for generating efficient, dynamically consistent gaits in legged locomotion. This paper explores the indirect method of trajectory optimization, emphasizing its application in creating optimal periodic gaits for legged systems and contrasting it with the more common direct method. While the direct method provides flexibility in implementation, it is limited by its need for an input space parameterization. In contrast, the indirect method improves accuracy by computing the control input from states and costates obtained along the optimal trajectory. In this work, we tackle the convergence challenges associated with indirect shooting methods by utilizing numerical continuation methods. This is particularly useful for the systematic development of gait libraries. Our contributions include: (1) the formalization of a general periodic trajectory optimization problem that extends existing first-order necessary conditions to a broader range of cost functions and operating conditions; (2) a methodology for efficiently generating libraries of optimal trajectories (gaits) utilizing a single shooting approach combined with numerical continuation methods; (3) a novel approach for reconstructing Lagrange multipliers and costates from passive gaits; (4) a comparative analysis of the indirect and direct shooting methods using a compass-gait walker as a case study, demonstrating the improved accuracy of the indirect method in generating optimal gaits; and (5) demonstrating applicability to the more complex legged robot RABBIT, with ten dynamic states and four inputs. The findings underscore the potential of the indirect method for generating families of optimal gaits, thereby advancing the field of trajectory optimization in legged robotics.

cs.RO↗

arXiv2026

Non-Extreme Individual Minima for Improved Pareto Front Sampling Efficiency and Decision-Making

In multi-objective optimization, the set of optimal trade-offs -- the Pareto front -- often contains regions that are extremely steep or flat. The Pareto optimal points in these regions are typically of limited interest for decision-making, as the marginal rate of substitution is extreme: a marginal improvement in one objective necessitates a significant deterioration in at least one other objective. These unfavorable trade-offs frequently occur near the individual minima, where single objectives attain their minimum values without considering the remaining criteria. To address this, we propose the concept of \emph{non-extreme individual minima} that relies on the notion of $L$-practical proper efficiency. These points can serve as a less sensitive replacement for \emph{standard} individual minima in subsequent related methods. Specifically, they allow for a more practical restriction of the Pareto front sampling within a refined utopia-nadir hyperbox, provide a meaningful basis for image space normalization, and can enhance decision-making techniques, such as knee-point methods, by focusing on regions with acceptable trade-offs. We provide a computationally efficient algorithm to determine these non-extreme individual minima by solving at most $2n_J$ standard weighted-sum scalarizations, where $n_J$ is the number of objectives. To ensure robustness across varying objective scales, the method incorporates an integrated image space normalization strategy. Numerical examples, specifically a convex academic case and a non-convex real-world application, demonstrate that the method successfully excludes practically irrelevant regions in the image space.

math.OC↗

arXiv2025

Optimality-Informed Neural Networks for Solving Parametric Optimization Problems

Many engineering tasks require solving families of nonlinear constrained optimization problems, parametrized in setting-specific variables. This is computationally demanding, particularly, if solutions have to be computed across strongly varying parameter values, e.g., in real-time control or for model-based design. Thus, we propose to learn the mapping from parameters to the primal optimal solutions and to their corresponding duals using neural networks, giving a dense estimation in contrast to gridded approaches. Our approach, Optimality-informed Neural Networks (OptINNs), combines (i) a KKT-residual loss that penalizes violations of the first-order optimality conditions under standard constraint qualifications assumptions, and (ii) problem-specific output activations that enforce simple inequality constraints (e.g., box-type/positivity) by construction. This design reduces data requirements, allows the prediction of dual variables, and improves feasibility and closeness to optimality compared to penalty-only training. Taking quadratic penalties as a baseline, since this approach has been previously proposed for the considered problem class in literature, our method simplifies hyperparameter tuning and attains tighter adherence to optimality conditions. We evaluate OptINNs on different nonlinear optimization problems ranging from low to high dimensions. On small problems, OptINNs match a quadratic-penalty baseline in primal accuracy while additionally predicting dual variables with low error. On larger problems, OptINNs achieve lower constraint violations and lower primal error compared to neural networks based on the quadratic-penalty method. These results suggest that embedding feasibility and optimality into the network architecture and loss can make learning-based surrogates more accurate, feasible, and data-efficient for parametric optimization.

math.OC↗

arXiv2025

Efficient Collision-Avoidance Constraints for Ellipsoidal Obstacles in Optimal Control: Application to Path-Following MPC and UAVs

This article proposes a modular optimal control framework for local three-dimensional ellipsoidal obstacle avoidance, exemplarily applied to model predictive path-following control. Static as well as moving obstacles are considered. Central to the approach is a computationally efficient and continuously differentiable condition for detecting collisions with ellipsoidal obstacles. A novel two-stage optimization approach mitigates numerical issues arising from the structure of the resulting optimal control problem. The effectiveness of the approach is demonstrated through simulations and real-world experiments with the Crazyflie quadrotor. This represents the first hardware demonstration of an MPC controller of this kind for UAVs in a three-dimensional task.

eess.SY↗

arXiv2025

Neural network-enhanced integrators for simulating ordinary differential equations

Numerous applications necessitate the computation of numerical solutions to differential equations across a wide range of initial conditions and system parameters, which feeds the demand for efficient yet accurate numerical integration methods.This study proposes a neural network (NN) enhancement of classical numerical integrators. NNs are trained to learn integration errors, which are then used as additive correction terms in numerical schemes. The performance of these enhanced integrators is compared with well-established methods through numerical studies, with a particular emphasis on computational efficiency. Analytical properties are examined in terms of local errors and backward error analysis. Embedded Runge-Kutta schemes are then employed to develop enhanced integrators that mitigate generalization risk, ensuring that the neural network's evaluation in previously unseen regions of the state space does not destabilize the integrator. It is guaranteed that the enhanced integrators perform at least as well as the desired classical Runge-Kutta schemes. The effectiveness of the proposed approaches is demonstrated through extensive numerical studies using a realistic model of a wind turbine, with parameters derived from the established simulation framework OpenFast.

math.NA↗

arXiv2024

The impact of AI on engineering design procedures for dynamical systems

Artificial intelligence (AI) is driving transformative changes across numerous fields, revolutionizing conventional processes and creating new opportunities for innovation. The development of mechatronic systems is undergoing a similar transformation. Over the past decade, modeling, simulation, and optimization techniques have become integral to the design process, paving the way for the adoption of AI-based methods. In this paper, we examine the potential for integrating AI into the engineering design process, using the V-model from the VDI guideline 2206, considered the state-of-the-art in product design, as a foundation. We identify and classify AI methods based on their suitability for specific stages within the engineering product design workflow. Furthermore, we present a series of application examples where AI-assisted design has been successfully implemented by the authors. These examples, drawn from research projects within the DFG Priority Program \emph{SPP~2353: Daring More Intelligence - Design Assistants in Mechanics and Dynamics}, showcase a diverse range of applications across mechanics and mechatronics, including areas such as acoustics and robotics.

eess.SY↗

arXiv2024

Efficient Avoidance of Ellipsoidal Obstacles with Model Predictive Control for Mobile Robots and Vehicles

In real-world applications of mobile robots, collision avoidance is of critical importance. Typically, global motion planning in constrained environments is addressed through high-level control schemes. However, additionally integrating local collision avoidance into robot motion control offers significant advantages. For instance, it reduces the reliance on heuristics and conservatism that can arise from a two-stage approach separating local collision avoidance and control. Moreover, using model predictive control (MPC), a robot's full potential can be harnessed by considering jointly local collision avoidance, the robot's dynamics, and actuation constraints. In this context, the present paper focuses on obstacle avoidance for wheeled mobile robots, where both the robot's and obstacles' occupied volumes are modeled as ellipsoids. To this end, a computationally efficient overlap test, that works for arbitrary ellipsoids, is conducted and novelly integrated into the MPC framework. We propose a particularly efficient implementation tailored to robots moving in the plane. The functionality of the proposed obstacle-avoiding MPC is demonstrated for two exemplary types of kinematics by means of simulations. A hardware experiment using a real-world wheeled mobile robot shows transferability to reality and real-time applicability. The general computational approach to ellipsoidal obstacle avoidance can also be applied to other robotic systems and vehicles as well as three-dimensional scenarios.

cs.RO↗

arXiv2024

Globally convergent homotopies for discrete-time optimal control

Homotopy methods are attractive due to their capability of solving difficult optimisation and optimal control problems. The underlying idea is to construct a homotopy, which may be considered as a continuous (zero) curve between the difficult original problem and a related, comparatively easy one. Then, the solution of the easier one is continuously perturbed along the zero curve towards the sought-after solution of the original problem. We propose a methodology for the systematic construction of such zero curves for discrete-time optimal control problems drawing upon the theory of globally convergent homotopies for nonlinear programs. The proposed framework ensures that for almost every initial guess at a solution there exists a suitable homotopy path that is, in addition, numerically convenient to track. We demonstrate the results by solving optimal path planning problems for a linear system and the nonlinear nonholonomic car (Dubins' vehicle).

math.OC↗

arXiv2024

Koopman Based Trajectory Optimization with Mixed Boundaries

Trajectory optimization is a widely used tool in the design and control of dynamical systems. Typically, not only nonlinear dynamics, but also couplings of the initial and final condition through implicit boundary constraints render the optimization problem non-convex. This paper investigates how the Koopman operator framework can be utilized to solve trajectory optimization problems in a (partially) convex fashion. While the Koopman operator has already been successfully employed in model predictive control, the challenge of addressing mixed boundary constraints within the Koopman framework has remained an open question. We first address this issue by explaining why a complete convexification of the problem is not possible. Secondly, we propose a method where we transform the trajectory optimization problem into a bilevel problem in which we are then able to convexify the high-dimensional lower-level problem. This separation yields a low-dimensional upper-level problem, which could be exploited in global optimization algorithms. Lastly, we demonstrate the effectiveness of the method on two example systems: the mathematical pendulum and the compass-gait walker.

math.OC↗

arXiv2024

Trim turnpikes for optimal control problems with symmetries

Motivated by mechanical systems with symmetries, we focus on optimal control problems possessing symmetries. Following recent works, which generalized the classical concept of static turnpike to manifold turnpike, we extend the exponential turnpike property to the exponential trim turnpike for control systems with symmetries induced by abelian or non-abelian groups. Our analysis is mainly based on the geometric reduction of control systems with symmetries. More concretely, we first reduce the control system on the quotient space and state the turnpike theorem for the reduced problem. Then we use the group properties to obtain the trim turnpike theorem for the full problem. Finally, we illustrate our results on the Kepler problem and the Rigid body problem.

math.OC↗

arXiv2024

Optimization-based motion primitive automata for autonomous driving

Trajectory planning for autonomous cars can be addressed by primitive-based methods, which encode nonlinear dynamical system behavior into automata. In this paper, we focus on optimal trajectory planning. Since, typically, multiple criteria have to be taken into account, multiobjective optimization problems have to be solved. For the resulting Pareto-optimal motion primitives, we introduce a universal automaton, which can be reduced or reconfigured according to prioritized criteria during planning. We evaluate a corresponding multi-vehicle planning scenario with both simulations and laboratory experiments.

eess.SY↗

arXiv2023

Hamiltonian Neural Networks with Automatic Symmetry Detection

Recently, Hamiltonian neural networks (HNN) have been introduced to incorporate prior physical knowledge when learning the dynamical equations of Hamiltonian systems. Hereby, the symplectic system structure is preserved despite the data-driven modeling approach. However, preserving symmetries requires additional attention. In this research, we enhance HNN with a Lie algebra framework to detect and embed symmetries in the neural network. This approach allows to simultaneously learn the symmetry group action and the total energy of the system. As illustrating examples, a pendulum on a cart and a two-body problem from astrodynamics are considered.

cs.LG↗

arXiv2022

Multi-Objective Model-Predictive Control for Dielectric Elastomer Wave Harvesters

This contribution deals with multi-objective model-predictive control (MPC) of a wave energy converter (WEC) device concept, which can harvest energy from sea waves using a dielectric elastomer generator (DEG) power take-off system. We aim to maximise the extracted energy through control while minimising the accumulated damage to the DEG. With reference to system operation in stochastic waves, we first generate ground truth solutions by solving an optimal control problem, and we analyse the MPC performance to determine a prediction horizon that trades off accuracy and efficiency for computation. Fixed weights in the MPC scheme can produce unpredictable costs for variable sea condition, meaning the average rate of cost accumulation can vary vastly. To steer this cost growth, we propose a heuristic to adapt the algorithm by changing the weighting of the cost functions using for fulfilling the long-time goal of accumulating a small enough damage in a fixed time. A simulated case-study is presented in order to evaluate the performance of the proposed MPC framework and the weight-adaptation algorithm. The proposed heuristic proves to be able to limit the amount of accumulated damage while remaining close to (or even improving) the energy yield obtained with a comparable fixed-weight MPC.

eess.SY↗

arXiv2022

Path Planning for Concentric Tube Robots: a Toolchain with Application to Stereotactic Neurosurgery

We present a toolchain for solving path planning problems for concentric tube robots through obstacle fields. First, ellipsoidal sets representing the target area and obstacles are constructed from labelled point clouds. Then, the nonlinear and highly nonconvex optimal control problem is solved by introducing a homotopy on the obstacle positions where at one extreme of the parameter the obstacles are removed from the operating space, and at the other extreme they are located at their intended positions. We present a detailed example (with more than a thousand obstacles) from stereotactic neurosurgery with real-world data obtained from labelled MPRI scans.

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