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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., Comment: 23 pages, 9 figures, joint publication of the priority programme SPP 2353

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., Comment: 9 pages, 4 figures

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., Comment: submitted to 7th Annual Learning for Dynamics & Control Conference Research (L4DC 2025)

Trajectory optimization is an essential tool for generating efficient and 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 commonly used direct method. While the direct method provides considerable flexibility in its implementation, it is limited by its input space parameterization. In contrast, the indirect method improves accuracy by defining control inputs as functions of the system's states and costates. We tackle the convergence challenges associated with indirect shooting methods, particularly through the systematic development of gait libraries by utilizing numerical continuation methods. Our contributions include: (1) the formalization of a general periodic trajectory optimization problem that extends existing first-order necessary conditions for 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, including a novel approach for reconstructing Lagrange multipliers and costates from passive gaits; and (3) a comparative analysis of the indirect and direct shooting methods using a compass-gait walker as a case study, demonstrating the former's superior accuracy in generating optimal gaits. 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., Comment: submitted to the International Journal of Robotics Research (IJRR)

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.

Continuum robots have emerged as a promising technology in the medical field due to their potential of accessing deep sited locations of the human body with low surgical trauma. When deriving physics-based models for these robots, evaluating the models poses a significant challenge due to the difficulty in accurately measuring their intricate shapes. In this work, we present an optimization based 3D shape registration algorithm for estimation of the backbone shape of slender continuum robots as part of a pho togrammetric measurement. Our approach to estimating the backbones optimally matches a parametric three-dimensional curve to images of the robot. Since we incorporate an iterative closest point algorithm into our method, we do not need prior knowledge of the robots position within the respective images. In our experiments with artificial and real images of a concentric tube continuum robot, we found an average maximum deviation of the reconstruction from simulation data of 0.665 mm and 0.939 mm from manual measurements. These results show that our algorithm is well capable of producing high accuracy positional data from images of continuum robots., Comment: 11 pages, 8 figures, 2 algorithms, journal

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.

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)., Comment: 25 pages, 11 figures, under review

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.

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.

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., Comment: 8 pages, 7 figures. Paper under review

Classical turnpikes correspond to optimal steady states which are attractors of optimal control problems. In this paper, motivated by mechanical systems with symmetries, we generalize this concept to manifold turnpikes. Specifically, the necessary optimality conditions on a symmetry-induced manifold coincide with those of a reduced-order problem under certain conditions. We also propose sufficient conditions for the existence of manifold turnpikes based on a tailored notion of dissipativity with respect to manifolds. We show how the classical Legendre transformation between Euler-Lagrange and Hamilton formalisms can be extended to the adjoint variables. Finally, we draw upon the Kepler problem to illustrate our findings., Comment: 29 pages, 3 figures

Turnpikes have recently gained significant research interest in optimal control, since they allow for pivotal insights into the structure of solutions to optimal control problems. So far, mainly steady state solutions which serve as optimal operation points, are studied. This is in contrast to time-varying turnpikes, which are in the focus of this paper. More concretely, we analyze symmetry-induced velocity turnpikes, i.e. controlled relative equilibria, called trim primitives, which are optimal operation points regarding the given cost criterion. We characterize velocity turnpikes by means of dissipativity inequalities. Moreover, we study the equivalence between optimal control problems and steady-state problems via the corresponding necessary optimality conditions. An academic example is given for illustration.

The paper proposes first steps towards the formalization and characterization of time-varying turnpikes in optimal control of mechanical systems. We propose the concepts of velocity steady states, which can be considered as partial steady states, and hyperbolic velocity turnpike properties for analysis and control. We show for a specific example that, for all finite horizons, both the (essential part of the) optimal solution and the orbit of the time-varying turnpike correspond to (optimal) trim solutions. Hereby, the present paper appears to be the first to combine the concepts of trim primitives and time-varying turnpike properties., Comment: 14 pages, 3 figures

Symmetries, e.g. rotational and translational invariances for the class of mechanical systems, allow to characterize solution trajectories of nonlinear dynamical systems. Thus, the restriction to symmetry-induced dynamics, e.g. by using the concept of motion primitives, may be considered as a quantization of the system. Symmetry exploitation is well-established in both motion planning and control. However, the linkage between the respective techniques to optimal control is not yet fully explored. In this manuscript, we want to lay the foundation for the usage of symmetries in Model Predictive Control (MPC). To this end, we investigate a mobile robot example in detail where our contribution is twofold: Firstly, we establish asymptotic stability of a desired set point w.r.t. the MPC closed loop, which is also demonstrated numerically by using motion primitives applied to the parallel parking scenario. Secondly, if the optimization criterion is not consistent with the symmetry action, we provide guidelines to rigorously derive stability guarantees based on symmetry exploitation.

We present an optimization-based approach for trajectory planning and control of a maneuverable melting probe with a high number of binary control variables. The dynamics of the system are modeled by a set of ordinary differential equations with a priori knowledge of system parameters of the melting process. The original planning problem is handled as an optimal control problem. Then, optimal control is used for reference trajectory planning as well as in an MPC-like algorithm. Finally, to determine binary control variables, a MINLP fitting approach is presented. The proposed strategy has recently been tested during experiments on the Langenferner glacier. The data obtained is used for model improvement by means of automated parameter identification., Comment: to be published in Proceedings of the 2018 American Control Conference, 7 pages, 6 figures