**Locomotion in Natural and Artificial Systems****Compliant and Variable-Impedance Robotics****Modeling and**Simulation of Constrained Dynamical Systems**Non-linear Oscillators for Energy Harvesting Applications****Energy Efficient Compliant Actuator Designs**

**Locomotion in Natural and Artificial Systems: **The goal of this research is to discover the *physical reasons* and control principles that govern movements of humans and animals and to utilize these findings to *build next generation robotic devices*. We have focused on the control aspect of this problem by developing an *impedance controller* that is capable of generating *autonomous self-organized locomotion. * Experiments on an *anthropomorphic biped robot* demonstrate one particular implementation of this control approach.

**D. J. Braun**, J. E. Mitchell, M. Goldfarb, Actuated Dynamic Walking in a Seven-Link Biped Robot,*IEEE/ASME Trans**ac**tions on Mechatronics*, vol. 17, no.1, pp. 147-156, 2012.**D. J. Braun**, M. Goldfarb, A Control Approach for Actuated Dynamics Walking in Biped Robots,*IEEE Transactions on Robotics*, vol. 25, no. 6, pp. 1292-1303, 2009.

**Compliant and Variable-Impedance Robotics****: **In this research we investigate *how impedance control strategies emerge from first principles of optimality, and how these strategies can be applied to variable impedance actuation*. We utilize *computational optimal control* to devise controllers that exploit the intrinsic compliance and the natural dynamics of the system to achieve better performance. Experiments on the **state-of-the-art variable stiffness DLR Hand-Arm System developed at the German Aerospace Center** demonstrate the benefits.

**D. J. Braun**, F. Petit, F. Huber, S. Haddadin, P. van der Smagt, A. Albu-Schäffer and S. Vijayakumar, Robots Driven by Compliant Actuators: Optimal Control under Actuation Constraints,*IEEE Trans**ac**tions on Robotics*, vol. 29, no. 5, pp. 1085-1101, 2013.**D. J. Braun**, M. Howard, and S. Vijayakumar, “Exploiting variable stiffness in explosive movement tasks,”*in 2011 Robotics: Science and Systems*, (Los Angeles, California, USA), June/July 2011.

**Modeling and Simulation of Constrained Dynamical Systems: **Simulation of constrained dynamical systems, modeled with *differential algebraic equations (DAEs),* is required in many engineering disciplines. However unlike ordinary differential equations (ODEs), that can be effectively integrated with explicit numerical methods (e.g., Runge-Kutta),* integration of DAEs often requires sophisticated implicit integrators enhanced with iterative constraint stabilization.* In this research, we developed an explicit numerical integration scheme which is easy to implement and can be used to accurately simulate constrained dynamical systems modeled with differential algebraic equations. *The basic assumption during the development of the integration scheme is that the computational environment is non-ideal* i.e., the numerical solution is error contaminated, such that, neither the kinematic constraints, nor the energy-type conservation law, can be exactly satisfied. Using this assumption, a systematic derivation is carried out which resulted in an equation of motion that directly incorporates the correction terms required for precise satisfaction of the kinematic constraints and prevents energy drift along the numerical solution. Importantly, the kinematic constraint correction terms and the energy correction terms are derived to be independent of each other by taking the geometry of the constrained dynamics rigorously into account. This idea enables *decoupled constraint and energy correction, which leads to significant improvement in accuracy over long time integrations*.

*A simple case study example:*

**D. J. Braun**and M. Goldfarb, Simulation of Constrained Mechanical Systems – Part I: An Equation of Motion,*ASME Journal of Appli**ed**Mechanics*, vol. 79, issue 4, 041017, 2012.**D. J. Braun**, M. Goldfarb, Elimination of Constrained Drift in the Numerical Simulation of Constrained Dynamical Systems,*Computer Methods in Applied Mechanics and Engineering*, vol. 198, no. 37-40, pp. 3151-3160, 2009.

**Non-linear oscillators for Energy Harvesting Applications: **Vibration energy harvesting devices are built to convert kinetic energy into useful electricity. Most of these devices are linear oscillators, capable of large gain amplification at resonant excitation, but have limited bandwidth under more typical, possibly random ambient vibrations. The goal of this project was to develop a non-linear oscillator for adaptive energy harvesting from low-frequency natural vibrations. The ambition was to uncover a physical phenomenon for autonomous energy harvesting applications.

*Working principle of a oscillatory energy accumulator:*

**D.J. Braun**, A. Sutas and S. Vijayakumar, Self-tuning Bistable Parametric Oscillator: Near-optimal Amplitude Maximization without Model Information,*Physical Review E*, 95, 012201, 2017.**D.J. Braun**, Optimal parametric feedback excitation of nonlinear oscillators,*Physical Review Letters*, 116, 044102, 2016.

**Energy Efficient Compliant Actuator Designs: **While robotic systems are able to achieve previously unprecedented agility, robustness and adaptability, they operate with particularly low efficiency. Energy efficient compliant actuation is the missing ingredient and the key enabler of next-generation robotic systems: prosthetic devices, exoskeletons, domestic robots, rescue robots. The goal of this project is to develop a computational approach for computer-aided design of efficient compliant actuators. This approach will be used to develop a new generation wearable autonomous exoskeleton for human augmentation and targeted biomedical applications.

- V. Chalvet and
**D.J. Braun**, Algorithmic Design of Low Power Variable Stiffness Mechanisms,*IEEE Transactions on Robotics,*2017. - V. Chalvet and
**D.J. Braun**, Criterion for the Design of Low Power Variable Stiffness Mechanisms,*IEEE Transactions on Robotics,*vol. 32, no. 4, pp. 1002–1010, Aug. 2017. - A. Dahiya and
**D.J. Braun**, Efficiently Tuneable Positive-Negative Stiffness Actuator,*IEEE International Conference on Robotics and Automation,*Singapore, pp. 1235-1240, May 2017.