C. David Remy

Iskandar Khemakhem, Dominik Tschemernjak, Maximilian Raff et al.

In the fields of robotics and biomechanics, the integration of elastic elements such as springs and tendons in legged systems has long been recognized for enabling energy-efficient locomotion. Yet, a significant challenge persists: designing a robotic leg that perform consistently across diverse operating conditions, especially varying average forward speeds. It remains unclear whether, for such a range of operating conditions, the stiffness of the elastic elements needs to be varied or if a similar performance can be obtained by changing the motion and actuation while keeping the stiffness fixed. This work explores the influence of the leg stiffness on the energy efficiency of a monopedal robot through an extensive parametric study of its periodic hopping motion. To this end, we formulate an optimal control problem parameterized by average forward speed and leg stiffness, solving it numerically using direct collocation. Our findings indicate that, compared to the use of a fixed stiffness, employing variable stiffness in legged systems improves energy efficiency by 20 % maximally and by 6.8 % on average across a range of speeds.

Maximilian Raff, Kathrin Flaßkamp, C. David Remy

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.

Kristin M. de Payrebrune, Kathrin Flaßkamp, Tom Ströhla et al.

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.

Maximilian Raff, Nelson Rosa, C. David Remy

In this work, we present a nonlinear dynamics perspective on generating and connecting gaits for energetically conservative models of legged systems. In particular, we show that the set of conservative gaits constitutes a connected space of locally defined 1D submanifolds in the gait space. These manifolds are coordinate-free parameterized by energy level. We present algorithms for identifying such families of gaits through the use of numerical continuation methods, generating sets and bifurcation points. To this end, we also introduce several details for the numerical implementation. Most importantly, we establish the necessary condition for the Delassus' matrix to preserve energy across impacts. An important application of our work is with simple models of legged locomotion that are often able to capture the complexity of legged locomotion with just a few degrees of freedom and a small number of physical parameters. We demonstrate the efficacy of our framework on a one-legged hopper with four degrees of freedom.

Kevin Green, Nils Smit-Anseeuw, Rodney Gleason et al.

This paper describes the design and control of a support and recovery system for use with planar legged robots. The system operates in three modes. First, it can be operated in a fully transparent mode where no forces are applied to the robot. In this mode, the system follows the robot closely to be able to quickly catch the robot if needed. Second, it can provide a vertical supportive force to assist a robot during operation. Third, it can catch the robot and pull it away from the ground after a failure to avoid falls and the associated damages. In this mode, the system automatically resets the robot after a trial allowing for multiple consecutive trials to be run without manual intervention. The supportive forces are applied to the robot through an actuated cable and pulley system that uses series elastic actuation with a unidirectional spring to enable truly transparent operation. The nonlinear nature of this system necessitates careful design of controllers to ensure predictable, safe behaviors. In this paper we introduce the mechatronic design of the recovery system, develop suitable controllers, and evaluate the system's performance on the bipedal robot RAMone.

Daniel Bruder, Xun Fu, R. Brent Gillespie et al.

Controlling soft continuum manipulator arms is difficult due to their infinite degrees of freedom, nonlinear material properties, and large deflections under loading. This paper presents a data-driven approach to identifying soft manipulator models that enables consistent control under variable loading conditions. This is achieved by incorporating loads into a linear Koopman operator model as states and estimating their values online via an observer within the control loop. Using this approach, real-time, fully autonomous control of a pneumatically actuated soft continuum manipulator is achieved. In several trajectory following experiments, this controller is shown to be more accurate and precise than controllers based on models that are unable to explicitly account for loading. The manipulator also successfully performs pick and place of objects with unknown mass, demonstrating the efficacy of this approach in executing real-world manipulation tasks.

Nils Smit-Anseeuw, C. David Remy, Ram Vasudevan

Safety guarantees are valuable in the control of walking robots, as falling can be both dangerous and costly. Unfortunately, set-based tools for generating safety guarantees (such as sums-of-squares optimization) are typically restricted to simplified, low-dimensional models of walking robots. For more complex models, methods based on hybrid zero dynamics can ensure the local stability of a pre-specified limit cycle, but provide limited guarantees. This paper combines the benefits of both approaches by using sums-of-squares optimization on a hybrid zero dynamics manifold to generate a guaranteed safe set for a 10-dimensional walking robot model. Along with this set, this paper describes how to generate a controller that maintains safety by modifying the manifold parameters when on the edge of the safe set. The proposed approach, which is applied to a bipedal Rabbit model, provides a roadmap for applying sums-of-squares verification techniques to high dimensional systems. This opens the door for a broad set of tools that can generate safety guarantees and regulating controllers for complex walking robot models.

Daniel Bruder, Brent Gillespie, C. David Remy et al.

Controlling soft robots with precision is a challenge due in large part to the difficulty of constructing models that are amenable to model-based control design techniques. Koopman Operator Theory offers a way to construct explicit linear dynamical models of soft robots and to control them using established model-based linear control methods. This method is data-driven, yet unlike other data-driven models such as neural networks, it yields an explicit control-oriented linear model rather than just a "black-box" input-output mapping. This work describes this Koopman-based system identification method and its application to model predictive controller design. A model and MPC controller of a pneumatic soft robot arm was constructed via the method, and its performance was evaluated over several trajectory following tasks in the real-world. On all of the tasks, the Koopman-based MPC controller outperformed a benchmark MPC controller based on a linear state-space model of the same system.

Daniel Bruder, C. David Remy, Ram Vasudevan

Soft robots are challenging to model due in large part to the nonlinear properties of soft materials. Fortunately, this softness makes it possible to safely observe their behavior under random control inputs, making them amenable to large-scale data collection and system identification. This paper implements and evaluates a system identification method based on Koopman operator theory in which models of nonlinear dynamical systems are constructed via linear regression of observed data by exploiting the fact that every nonlinear system has a linear representation in the infinite-dimensional space of real-valued functions called observables. The approach does not suffer from some of the shortcomings of other nonlinear system identification methods, which typically require the manual tuning of training parameters and have limited convergence guarantees. A dynamic model of a pneumatic soft robot arm is constructed via this method, and used to predict the behavior of the real system. The total normalized-root-mean-square error (NRMSE) of its predictions is lower than that of several other identified models including a neural network, NLARX, nonlinear Hammerstein-Wiener, and linear state space model.

Daniel Bruder, Audrey Sedal, Ram Vasudevan et al.

The compliant structure of soft robotic systems enables a variety of novel capabilities in comparison to traditional rigid-bodied robots. A subclass of soft fluid-driven actuators known as fiber reinforced elastomeric enclosures (FREEs) is particularly well suited as actuators for these types of systems. FREEs are inherently soft and can impart spatial forces without imposing a rigid structure. Furthermore, they can be configured to produce a large variety of force and moment combinations. In this paper we explore the potential of combining multiple differently configured FREEs in parallel to achieve fully controllable multi-dimensional soft actuation. To this end, we propose a novel methodology to represent and calculate the generalized forces generated by soft actuators as a function of their internal pressure. This methodology relies on the notion of a state dependent fluid Jacobian that yields a linear expression for force. We employ this concept to construct the set of all possible forces that can be generated by a soft system in a given state. This force zonotope can be used to inform the design and control of parallel combinations of soft actuators. The approach is verified experimentally with the parallel combination of three carefully designed actuators constrained to a 2DOF testing rig. The force predictions matched measured values with a root-mean-square error of less than 1.5 N force and 8 x 10^(-3)Nm moment, demonstrating the utility of the presented methodology.

Audrow Nash, Yu-Ming Chen, Nils Smit-Anseeuw et al.

In this paper, we optimize over the control parameter space of our planar-bipedal robot, RAMone, for stable and energetically economical walking at various speeds. We formulate this task as an episodic reinforcement learning problem and use Covariance Matrix Adaptation. The parameters we are interested in modifying include gains from our Hybrid Zero Dynamics style controller and from RAMone's low-level motor controllers.