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Research – Nonlinear Mechanics
Design, modeling and control of modular tendon-driven elastic continuum mechanisms

Soft robotics

Taking insights from structural and continuum mechanics, this project aims to establish required modeling frameworks for a systematic design and control of soft robots.

Nature’s diversity has inspired scientists and engineers for centuries. The recent technological progress in material design and manufacturing processes has enabled a progressive transfer of nature’s conceptions to robotics and has led to the field of soft robotics. With a paradigm shift from rigid to soft, devices made out of highly deformable materials are developed in such a way that they intrinsically satisfy design criteria such as high flexibility, mechanical robustness, safe human-robot interaction, energy storage or shock absorbability. Soft robots are commonly actuated in two ways: by tendons with variable length routed along or within the body, or by pneumatic actuation which causes the system to deform by changing pressure level. Even though many soft robots have been tested, general modeling and control methods are still not available.

Models proposed for highly deformable bodies require a continuous description of the kinematics which results in systems with infinitely many degrees of freedom. Given that soft robots are composed of deformable bodies, the theory of rigid multi-body dynamics would not be suitable anymore. We will develop and apply nonlinear continuum models with appropriate finite element formulations. The discretization by finite elements describes the dynamics of soft robots eventually as finite degree of freedom systems. To design and control soft robots two inverse problems have to be solved.

(i) The static problem: what are the required control variables for the actuators for a given deformation of the system in space?

(ii) The dynamic problem: how do the actuators’ control variables have to change in time for a given motion of the system?

Journal articles

Harsch, J., Sailer, S., & Eugster, S. R. (2023). A total Lagrangian, objective and intrinsically locking-free Petrov--Galerkin SE(3) Cosserat rod finite element formulation. International Journal for Numerical Methods in Engineering, n/a(n/a), Article n/a. https://doi.org/10.1002/nme.7236
Abstract
Based on more than three decades of rod finite element theory, this publication combines the successful contributions found in the literature and eradicates the arising drawbacks like loss of objectivity, locking, path-dependence and redundant coordinates. Specifically, the idea of interpolating the nodal orientations using relative rotation vectors, proposed by Crisfield and Jelenic in 1999, is extended to the interpolation of nodal Euclidean transformation matrices with the aid of relative twists; a strategy that arises from the SE(3)-structure of the Cosserat rod kinematics. Applying a Petrov--Galerkin projection method, we propose a rod finite element formulation where the virtual displacements and rotations as well as the translational and angular velocities are interpolated instead of using the consistent variations and time-derivatives of the introduced interpolation formula. Properties such as the intrinsic absence of locking, preservation of objectivity after discretization and parameterization in terms of a minimal number of nodal unknowns are demonstrated by representative numerical examples in both statics and dynamics.
BibTeX
@article{Harsch2023, abstract = {Based on more than three decades of rod finite element theory, this publication combines the successful contributions found in the literature and eradicates the arising drawbacks like loss of objectivity, locking, path-dependence and redundant coordinates. Specifically, the idea of interpolating the nodal orientations using relative rotation vectors, proposed by Crisfield and Jelenic in 1999, is extended to the interpolation of nodal Euclidean transformation matrices with the aid of relative twists; a strategy that arises from the SE(3)-structure of the Cosserat rod kinematics. Applying a Petrov--Galerkin projection method, we propose a rod finite element formulation where the virtual displacements and rotations as well as the translational and angular velocities are interpolated instead of using the consistent variations and time-derivatives of the introduced interpolation formula. Properties such as the intrinsic absence of locking, preservation of objectivity after discretization and parameterization in terms of a minimal number of nodal unknowns are demonstrated by representative numerical examples in both statics and dynamics.}, author = {Harsch, Jonas and Sailer, Simon and Eugster, Simon R.}, doi = {https://doi.org/10.1002/nme.7236}, eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.7236}, file = {:Harsch2023.pdf:PDF}, journal = {International Journal for Numerical Methods in Engineering}, number = {n/a}, title = {A total Lagrangian, objective and intrinsically locking-free Petrov--Galerkin SE(3) Cosserat rod finite element formulation}, url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.7236}, volume = {n/a}, year = 2023 }
Link
https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.7236
Documents
- Harsch2023.pdf
Eugster, S. R., & Harsch, J. (2023). A family of total Lagrangian Petrov–Galerkin Cosserat rod finite element formulations. GAMM-Mitteilungen, n/a(n/a), Article n/a. https://doi.org/10.1002/gamm.202300008
Abstract
The standard in rod finite element formulations is the Bubnov–Galerkin projection method, where the test functions arise from a consistent variation of the ansatz functions. This approach becomes increasingly complex when highly nonlinear ansatz functions are chosen to approximate the rod's centerline and cross-section orientations. Using a Petrov–Galerkin projection method, we propose a whole family of rod finite element formulations where the nodal generalized virtual displacements and generalized velocities are interpolated instead of using the consistent variations and time derivatives of the ansatz functions. This approach leads to a significant simplification of the expressions in the discrete virtual work functionals. In addition, independent strategies can be chosen for interpolating the nodal centerline points and cross-section orientations. We discuss three objective interpolation strategies and give an in-depth analysis concerning locking and convergence behavior for the whole family of rod finite element formulations.
BibTeX
@article{Eugster2023a, abstract = {The standard in rod finite element formulations is the Bubnov–Galerkin projection method, where the test functions arise from a consistent variation of the ansatz functions. This approach becomes increasingly complex when highly nonlinear ansatz functions are chosen to approximate the rod's centerline and cross-section orientations. Using a Petrov–Galerkin projection method, we propose a whole family of rod finite element formulations where the nodal generalized virtual displacements and generalized velocities are interpolated instead of using the consistent variations and time derivatives of the ansatz functions. This approach leads to a significant simplification of the expressions in the discrete virtual work functionals. In addition, independent strategies can be chosen for interpolating the nodal centerline points and cross-section orientations. We discuss three objective interpolation strategies and give an in-depth analysis concerning locking and convergence behavior for the whole family of rod finite element formulations.}, author = {Eugster, Simon R. and Harsch, Jonas}, doi = {https://doi.org/10.1002/gamm.202300008}, eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/gamm.202300008}, journal = {GAMM-Mitteilungen}, number = {n/a}, pages = {e202300008}, title = {A family of total Lagrangian Petrov–Galerkin Cosserat rod finite element formulations}, url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/gamm.202300008}, volume = {n/a}, year = 2023 }
Link
https://onlinelibrary.wiley.com/doi/abs/10.1002/gamm.202300008
Documents
- Eugster2023a.pdf
Harsch, J., Ganzosch, G., Barchiesi, E., Ciallella, A., & Eugster, S. R. (2022). Experimental analysis, discrete modeling and parameter optimization of SLS-printed bi-pantographic structures. Mathematics and Mechanics of Solids, 27(10), Article 10. https://doi.org/10.1177/10812865221107623
Abstract
Making use of experimental data for bias extension, shearing, and point-load tests in large deformation regime for rectangular and square bi-pantographic specimens, we perform a numerical identification to fit the a priori parameters of a planar discrete spring model. The main objective of the work is to develop an automatized optimization process based on the Nelder�Mead simplex algorithm for identifying the constitutive parameters of discrete modeling of bi-pantographic structures, as well as assessing its descriptiveness and predictive capacity. The analysis allows to conclude that there exists a single set of parameters for the adopted discrete modeling such that it is descriptive and predictive for several different tests and for a wide range of deformations.
BibTeX
@article{Harsch2022a, abstract = {Making use of experimental data for bias extension, shearing, and point-load tests in large deformation regime for rectangular and square bi-pantographic specimens, we perform a numerical identification to fit the a priori parameters of a planar discrete spring model. The main objective of the work is to develop an automatized optimization process based on the Nelder�Mead simplex algorithm for identifying the constitutive parameters of discrete modeling of bi-pantographic structures, as well as assessing its descriptiveness and predictive capacity. The analysis allows to conclude that there exists a single set of parameters for the adopted discrete modeling such that it is descriptive and predictive for several different tests and for a wide range of deformations.}, author = {Harsch, Jonas and Ganzosch, Gregor and Barchiesi, Emilio and Ciallella, Alessandro and Eugster, Simon R}, doi = {10.1177/10812865221107623}, eprint = {https://doi.org/10.1177/10812865221107623}, file = {:Harsch2022a.pdf:PDF}, journal = {Mathematics and Mechanics of Solids}, number = 10, pages = {2201-2217}, title = {Experimental analysis, discrete modeling and parameter optimization of SLS-printed bi-pantographic structures}, url = {https://doi.org/10.1177/10812865221107623}, volume = 27, year = 2022 }
Link
https://doi.org/10.1177/10812865221107623
Documents
- Harsch2022a.pdf
Eugster, S. R., Harsch, J., Herrmann, M., Capobianco, G., Bartholdt, M., & Wiese, M. (2022). Soft pneumatic actuator model based on a pressure-dependent spatial nonlinear rod theory. IEEE Robotics and Automation Letters, 7(2), Article 2. https://ieeexplore.ieee.org/document/9691781
Abstract
To describe the mechanical behavior of a soft pneumatic actuator, we present a nonlinear rod theory, which recognizes the chamber pressurization as contribution to the resultant contact forces and couples. Accordingly, the pressure actuation can be considered as part of the rod’s constitutive laws, which relate the contact interactions with the rod’s kinematics and the chamber pressures. The theory allows for nonlinear constitutive laws, which are capable of describing strengthening or softening behavior of largely deformable materials. The theory was applied to a three-chamber actuator to predict its centerline as well as the cross-section orientations in static equilibrium. The resulting governing equations were spatially discretized using beam finite elements. We evaluated four different actuator models that are contained in the presented theory regarding describability and predictability of the end effector position. Existing rod models with linear elastic material laws can describe the tip-displacement with an averaged deviation of 9.1 mm in the training set and 11.4 mm in the test set. With a deviation of 2.8 mm and 4.9 mm in the training and test set, respectively, the most enhanced model, for which the pressure chamber radii increase with increasing pressure, is more than 2 times more accurate.
BibTeX
@article{Eugster2022c, abstract = {To describe the mechanical behavior of a soft pneumatic actuator, we present a nonlinear rod theory, which recognizes the chamber pressurization as contribution to the resultant contact forces and couples. Accordingly, the pressure actuation can be considered as part of the rod’s constitutive laws, which relate the contact interactions with the rod’s kinematics and the chamber pressures. The theory allows for nonlinear constitutive laws, which are capable of describing strengthening or softening behavior of largely deformable materials. The theory was applied to a three-chamber actuator to predict its centerline as well as the cross-section orientations in static equilibrium. The resulting governing equations were spatially discretized using beam finite elements. We evaluated four different actuator models that are contained in the presented theory regarding describability and predictability of the end effector position. Existing rod models with linear elastic material laws can describe the tip-displacement with an averaged deviation of 9.1 mm in the training set and 11.4 mm in the test set. With a deviation of 2.8 mm and 4.9 mm in the training and test set, respectively, the most enhanced model, for which the pressure chamber radii increase with increasing pressure, is more than 2 times more accurate.}, author = {Eugster, S. R. and Harsch, J. and Herrmann, M. and Capobianco, G. and Bartholdt, M. and Wiese, M.}, journal = {IEEE Robotics and Automation Letters}, number = 2, pages = {1-8}, title = {Soft pneumatic actuator model based on a pressure-dependent spatial nonlinear rod theory}, url = {https://ieeexplore.ieee.org/document/9691781}, volume = 7, year = 2022 }
Link
https://ieeexplore.ieee.org/document/9691781
Documents
- Eugster2022c.pdf
Harsch, J., Capobianco, G., & Eugster, S. R. (2021). Finite element formulations for constrained spatial nonlinear beam theories. Mathematics and Mechanics of Solids, 26(12), Article 12. https://doi.org/10.1177/10812865211000790
Abstract
A new director-based finite element formulation for geometrically exact beams is proposed by weak enforcement of the orthonormality constraints of the directors. In addition to an improved numerical performance, this formulation enables the development of two more beam theories by adding further constraints. Thus, the paper presents a complete intrinsic spatial nonlinear theory of three kinematically different beams which can undergo large displacements and which can have precurved reference configurations. Moreover, the hyperelastic constitutive laws allow for elastic finite strain material behavior of the beams. Furthermore, the numerical discretization using concepts of isogeometric analysis is highlighted in all clarity. Finally, all presented models are numerically validated using exclusive analytical solutions, existing finite element formulations, and a complex dynamical real-world example.
BibTeX
@article{Harsch2021a, abstract = {A new director-based finite element formulation for geometrically exact beams is proposed by weak enforcement of the orthonormality constraints of the directors. In addition to an improved numerical performance, this formulation enables the development of two more beam theories by adding further constraints. Thus, the paper presents a complete intrinsic spatial nonlinear theory of three kinematically different beams which can undergo large displacements and which can have precurved reference configurations. Moreover, the hyperelastic constitutive laws allow for elastic finite strain material behavior of the beams. Furthermore, the numerical discretization using concepts of isogeometric analysis is highlighted in all clarity. Finally, all presented models are numerically validated using exclusive analytical solutions, existing finite element formulations, and a complex dynamical real-world example.}, author = {Harsch, Jonas and Capobianco, Giuseppe and Eugster, Simon R}, doi = {10.1177/10812865211000790}, eprint = {https://doi.org/10.1177/10812865211000790}, file = {:Harsch2021a.pdf:PDF}, journal = {Mathematics and Mechanics of Solids}, number = 12, pages = {1838-1863}, title = {Finite element formulations for constrained spatial nonlinear beam theories}, url = {https://doi.org/10.1177/10812865211000790}, volume = 26, year = 2021 }
Link
https://doi.org/10.1177/10812865211000790
Documents
- Harsch2021a.pdf
Capobianco, G., Harsch, J., Eugster, S. R., & Leine, R. I. (2021). A nonsmooth generalized-alpha method for mechanical systems with frictional contact. International Journal for Numerical Methods in Engineering, 122(22), Article 22. https://doi.org/10.1002/nme.6801
- BibTeX
BibTeX
@article{Capobianco&Harsch&Eugster&Leine2021, author = {Capobianco, G. and Harsch, J. and Eugster, S. R. and Leine, R. I.}, doi = {https://doi.org/10.1002/nme.6801}, journal = {International Journal for Numerical Methods in Engineering}, number = 22, pages = {6497--6526}, title = {A nonsmooth generalized-alpha method for mechanical systems with frictional contact}, volume = 122, year = 2021 }
Barchiesi, E., Harsch, J., Ganzosch, G., & Eugster, S. R. (2020). Discrete versus homogenized continuum modeling in finite deformation bias extension test of bi-pantographic fabrics. Continuum Mechanics and Thermodynamics. https://doi.org/10.1007/s00161-020-00917-w
Abstract
A 2D-continuum model describing finite deformations in plane of discrete bi-pantographic fabrics has been recently obtained by applying an asymptotic procedure based on a set of local generalized coordinates. Rectangular bi-pantographic prototypes were additively manufactured by selective laser sintering using polyamide as raw material. Displacement-controlled bias extension tests were performed on such specimens for total elastic deformations up to ca. 25\%. Experimental force measurements, complemented by discrete displacement measurements obtained by local digital image correlation, were used to fit the continuum model. In the present paper, a global and minimal set of generalized coordinates, alternative to the one used for the homogenization, is introduced for the discrete model. The mechanical constitutive parameters appearing in the discrete model are then found by means of collected experimental data. Finally, a comparison between experiments, the discrete and the continuum model is presented. It is concluded that (a) the discrete model and the experimental data are in excellent agreement, and that (b) the continuum retains the relevant phenomenology of the discrete system even for a rather low number of cells.
BibTeX
@article{Barchiesi2020, abstract = {A 2D-continuum model describing finite deformations in plane of discrete bi-pantographic fabrics has been recently obtained by applying an asymptotic procedure based on a set of local generalized coordinates. Rectangular bi-pantographic prototypes were additively manufactured by selective laser sintering using polyamide as raw material. Displacement-controlled bias extension tests were performed on such specimens for total elastic deformations up to ca. 25{\%}. Experimental force measurements, complemented by discrete displacement measurements obtained by local digital image correlation, were used to fit the continuum model. In the present paper, a global and minimal set of generalized coordinates, alternative to the one used for the homogenization, is introduced for the discrete model. The mechanical constitutive parameters appearing in the discrete model are then found by means of collected experimental data. Finally, a comparison between experiments, the discrete and the continuum model is presented. It is concluded that (a) the discrete model and the experimental data are in excellent agreement, and that (b) the continuum retains the relevant phenomenology of the discrete system even for a rather low number of cells.}, author = {Barchiesi, E. and Harsch, J. and Ganzosch, G. and Eugster, S. R.}, day = 12, doi = {10.1007/s00161-020-00917-w}, file = {:Barchiesi2020.pdf:PDF}, issn = {1432-0959}, journal = {Continuum Mechanics and Thermodynamics}, month = {09}, title = {Discrete versus homogenized continuum modeling in finite deformation bias extension test of bi-pantographic fabrics}, url = {https://doi.org/10.1007/s00161-020-00917-w}, year = 2020 }
Link
https://doi.org/10.1007/s00161-020-00917-w
Documents
- Barchiesi2020.pdf

Proceedings

Harsch, J., Capobianco, G., & Eugster, S. R. (2021). Dynamic simulation of the Wilberforce pendulum using constrained spatial nonlinear beam finite elements. PAMM, 21(1), Article 1. https://doi.org/10.1002/pamm.202100110
Abstract
More than 100 years ago, Lionel Robert Wilberforce did investigations On the Vibrations of a Loaded Spiral Spring 1. The spring was clamped at its upper side and on the other side, perpendicular to the spring axis, a steel cylinder was attached. Four screws with adjustable nuts were symmetrically attached around the cylinder in order to change its moment of inertia (Fig. 1). In this paper the Wilberforce pendulum is modeled by a rigid body attached to a constrained spatial nonlinear Timoshenko beam, discretized with B-spline shape functions. As shown by a numerical experiment, the presented model is capable of reproducing the characteristic pendulum motion.
BibTeX
@article{Harsch2021, abstract = {More than 100 years ago, Lionel Robert Wilberforce did investigations On the Vibrations of a Loaded Spiral Spring [1]. The spring was clamped at its upper side and on the other side, perpendicular to the spring axis, a steel cylinder was attached. Four screws with adjustable nuts were symmetrically attached around the cylinder in order to change its moment of inertia (Fig. 1). In this paper the Wilberforce pendulum is modeled by a rigid body attached to a constrained spatial nonlinear Timoshenko beam, discretized with B-spline shape functions. As shown by a numerical experiment, the presented model is capable of reproducing the characteristic pendulum motion.}, author = {Harsch, Jonas and Capobianco, Giuseppe and Eugster, Simon R.}, doi = {https://doi.org/10.1002/pamm.202100110}, eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/pamm.202100110}, file = {:Harsch2021.pdf:PDF}, journal = {PAMM}, number = 1, pages = {e202100110}, title = {Dynamic simulation of the Wilberforce pendulum using constrained spatial nonlinear beam finite elements}, url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/pamm.202100110}, volume = 21, year = 2021 }
Link
https://onlinelibrary.wiley.com/doi/abs/10.1002/pamm.202100110
Documents
- Harsch2021.pdf
Capobianco, G., Harsch, J., Eugster, S. R., & Leine, R. I. (2021). Simulating mechanical systems with frictional contacts using a nonsmooth generalized-alpha method. Proceedings in Applied Mathematics and Mechanics (PAMM), 21, e202100141:1--3. https://doi.org/10.1002/pamm.202100141
Abstract
In this paper, we introduce a nonsmooth generalized-alpha method for the simulation of mechanical systems with frictional contact. In many engineering applications, such systems are composed of rigid and flexible bodies, which are interconnected by joints and can come into contact with each other or their surroundings. Prominent examples are automotive, wind turbine, and robotic systems. It is known from structural mechanics applications, that generalized-alpha schemes perform well for flexible multibody systems without contacts. This motivated the development of nonsmooth generalized-alpha methods for the simulation of mechanical systems with frictional contacts 2, 3, 5. Typically, the Gear-Gupta-Leimkuhler approach is used to stabilize the unilateral constraint, such that numerical penetration of the contact bodies can be avoided - a big issue of the most popular time-stepping schemes such as Moreau's scheme. The nonsmooth generalized-alpha method presented in this paper is derived in 2 and in contrast to 3,5 accounts for set-valued Coulomb-type friction on both velocity and acceleration level. Finally, we validate the method using a guided flexible hopper as a benchmark mechanical system.
BibTeX
@inproceedings{Capobianco&Harsch&Eugster&Leine2021PAMM, abstract = {In this paper, we introduce a nonsmooth generalized-alpha method for the simulation of mechanical systems with frictional contact. In many engineering applications, such systems are composed of rigid and flexible bodies, which are interconnected by joints and can come into contact with each other or their surroundings. Prominent examples are automotive, wind turbine, and robotic systems. It is known from structural mechanics applications, that generalized-alpha schemes perform well for flexible multibody systems without contacts. This motivated the development of nonsmooth generalized-alpha methods for the simulation of mechanical systems with frictional contacts [2, 3, 5]. Typically, the Gear-Gupta-Leimkuhler approach is used to stabilize the unilateral constraint, such that numerical penetration of the contact bodies can be avoided - a big issue of the most popular time-stepping schemes such as Moreau's scheme. The nonsmooth generalized-alpha method presented in this paper is derived in [2] and in contrast to [3,5] accounts for set-valued Coulomb-type friction on both velocity and acceleration level. Finally, we validate the method using a guided flexible hopper as a benchmark mechanical system.}, author = {Capobianco, G. and Harsch, J. and Eugster, S. R. and Leine, R. I.}, booktitle = {Proceedings in Applied Mathematics and Mechanics (PAMM)}, doi = {https://doi.org/10.1002/pamm.202100141}, pages = {e202100141:1--3}, title = {Simulating mechanical systems with frictional contacts using a nonsmooth generalized-alpha method}, volume = 21, year = 2021 }
Documents
- Capobianco_x_Harsch_x_Eugster_x_Leine-Simulating_mechanical_systems_with_frictional_contact_using_a_nonsmooth_generalized-alpha_method.pdf

This research forms part of the project “Design, modeling and control of modular tendon-driven elastic continuum mechanisms”, which is conducted in cooperation with the Institute of Robotics and Mechatronics at DLR, Oberfaffenhofen, and is supported by the German Science Foundation (DFG) under project number 405032572. The project is part of the Priority Programm 2100 “Soft Material Robotic Systems”.

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Institute of Robotics and Mechatronics

SPP 2100

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