Daniel F. B. Haeufle

Hill-type muscle model with serial damping and eccentric force–velocity relation

Hill-type muscle models are commonly used in biomechanical simulations to predict passive and active muscle forces. Here, a model is presented which consists of four elements: a contractile element with force-length and force-velocity relations for concentric and eccentric contractions, a parallel elastic element, a series elastic element, and a serial damping element. With this, it combines previously published effects relevant for muscular contraction, i.e. serial damping and eccentric force-velocity relation. The model is exemplarily applied to arm movements. The more realistic representation of the eccentric force-velocity relation results in human-like elbow-joint flexion. The model is provided as ready to use Matlab and Simulink code.

The role of intrinsic muscle properties for stable hopping—stability is achieved by the force–velocity relation

A reductionist approach was presented to investigate which level of detail of the physiological muscle is required for stable locomotion. Periodic movements of a simplified one-dimensional hopping model with a Hill-type muscle (one contractile element, neither serial nor parallel elastic elements) were analyzed. Force-length and force-velocity relations of the muscle were varied in three levels of approximation (constant, linear and Hill-shaped nonlinear) resulting in nine different hopping models of different complexity. Stability of these models was evaluated by return map analysis and the performance by the maximum hopping height. The simplest model (constant force-length and constant force-velocity relations) outperformed all others in the maximum hopping height but was unstable. Stable hopping was achieved with linear and Hill-shaped nonlinear characteristic of the force-velocity relation. The characteristics of the force-length relation marginally influenced hopping stability. The results of this approach indicate that the intrinsic properties of the contractile element are responsible for stabilization of periodic movements. This connotes that (a) complex movements like legged locomotion could benefit from stabilizing effects of muscle properties, and (b) technical systems could benefit from the emerging stability when implementing biological characteristics into artificial muscles.

Integration of intrinsic muscle properties, feed-forward and feedback signals for generating and stabilizing hopping

It was hypothesized that a tight integration of feed-forward and feedback-driven muscle activation with the characteristic intrinsic muscle properties is a key feature of locomotion in challenging environments. In this simulation study it was investigated whether a combination of feed-forward and feedback signals improves hopping stability compared with those simulations with one individual type of activation. In a reduced one-dimensional hopping model with a Hill-type muscle (one contractile element, neither serial nor parallel elastic elements), the level of detail of the muscles force–length–velocity relation and the type of activation generation (feed-forward, feedback and combination of both) were varied to test their influence on periodic hopping. The stability of the hopping patterns was evaluated by return map analysis. It was found that the combination of feed-forward and proprioceptive feedback improved hopping stability. Furthermore, the nonlinear Hill-type representation of intrinsic muscle properties led to a faster reduction of perturbations than a linear approximation, independent of the type of activation. The results emphasize the ability of organisms to exploit the stabilizing properties of intrinsic muscle characteristics.

Energy management that generates terrain following versus apex-preserving hopping in man and machine

While hopping, 12 subjects experienced a sudden step down of 5 or 10 cm. Results revealed that the hopping style was “terrain following”. It means that the subjects pursued to keep the distance between maximum hopping height (apex) and ground profile constant. The spring-loaded inverse pendulum (SLIP) model, however, which is currently considered as template for stable legged locomotion would predict apex-preserving hopping, by which the absolute maximal hopping height is kept constant regardless of changes of the ground level. To get more insight into the physics of hopping, we outlined two concepts of energy management: “constant energy supply”, by which in each bounce—regardless of perturbations—the same amount of mechanical energy is injected, and “lost energy supply”, by which the mechanical energy that is going to be dissipated in the current cycle is assessed and replenished. When tested by simulations and on a robot testbed capable of hopping, constant energy supply generated stable and robust terrain following hopping, whereas lost energy supply led to something like apex-preserving hopping, which, however, lacks stability as well as robustness. Comparing simulated and machine hopping with human hopping suggests that constant energy supply has a good chance to be used by humans to generate hopping.

Can Quick Release Experiments Reveal the Muscle Structure? A Bionic Approach

The goal of this study was to understand the macroscopic mechanical structure and function of biological muscle with respect to its dynamic role in the contraction. A recently published muscle model, deriving the hyperbolic force-velocity relation from first-order mechanical principles, predicts different force-velocity operating points for different load situations. With a new approach, this model could be simplified and thus, transferred into a numerical simulation and a hardware experiment. Two types of quick release experiments were performed in simulation and with the hardware setup, which represent two extreme cases of the contraction dynamics: against a constant force (isotonic) and against an inertial mass. Both experiments revealed hyperbolic or hyperbolic-like force-velocity relations. Interestingly, the analytical model not only predicts these extreme cases, but also additionally all contraction states in between. It was possible to validate these predictions with the numerical model and the hardware experiment. These results prove that the origin of the hyperbolic force-velocity relation can be mechanically explained on a macroscopic level by the dynamical interaction of three mechanical elements. The implications for the interpretation of biological muscle experiments and the realization of muscle-like bionic actuators are discussed.

Nature as an engineer: one simple concept of a bio-inspired functional artificial muscle

The biological muscle is a powerful, flexible and versatile actuator. Its intrinsic characteristics determine the way how movements are generated and controlled. Robotic and prosthetic applications expect to profit from relying on bio-inspired actuators which exhibit natural (muscle-like) characteristics. As of today, when constructing a technical actuator, it is not possible to copy the exact molecular structure of a biological muscle. Alternatively, the question may be put how its characteristics can be realized with known mechanical components. Recently, a mechanical construct for an artificial muscle was proposed, which exhibits hyperbolic force-velocity characteristics. In this paper, we promote the constructing concept which is made by substantiating the mechanical design of biological muscle by a simple model, proving the feasibility of its real-world implementation, and checking their output both for mutual consistency and agreement with biological measurements. In particular, the relations of force, enthalpy rate and mechanical efficiency versus contraction velocity of both the constructs technical implementation and its numerical model were determined in quick-release experiments. All model predictions for these relations and the hardware results are now in good agreement with the biological literature. We conclude that the construct represents a mechanical concept of natural actuation, which is suitable for laying down some useful suggestions when designing bio-inspired actuators.

Proof of concept: Model based bionic muscle with hyperbolic force-velocity relation

Recently, the hyperbolic Hill-type force-velocity relation was derived from basic physical components. It was shown that a contractile element CE consisting of a mechanical energy source active element AE, a parallel damper element PDE, and a serial element SE exhibits operating points with hyperbolic force-velocity dependency. In this paper, a technical proof of this concept was presented. AE and PDE were implemented as electric motors, SE as a mechanical spring. The force-velocity relation of this artificial CE was determined in quick release experiments. The CE exhibited hyperbolic force-velocity dependency. This proof of concept can be seen as a well-founded starting point for the development of Hill-type artificial muscles.

Evaluating Morphological Computation in Muscle and DC-Motor Driven Models of Hopping Movements

In the context of embodied artificial intelligence, morphological computation refers to processes which are conducted by the body (and environment) that otherwise would have to be performed by the brain. Exploiting environmental and morphological properties is an important feature of embodied systems. The main reason is that it allows to significantly reduce the controller complexity. An important aspect of morphological computation is that it cannot be assigned to an embodied system per se, but that it is, as we show, behavior- and state-dependent. In this work, we evaluate two different measures of morphological computation that can be applied in robotic systems and in computer simulations of biological movement. As an example, these measures were evaluated on muscle and DC-motor driven hopping models. We show that a state-dependent analysis of the hopping behaviors provides additional insights that cannot be gained from the averaged measures alone. This work includes algorithms and computer code for the measures.

Proof of concept of an artificial muscle: Theoretical model, numerical model, and hardware experiment

Recently, the hyperbolic Hill-type force-velocity relation was derived from basic physical components. It was shown that a contractile element CE consisting of a mechanical energy source (active element AE), a parallel damper element (PDE), and a serial element (SE) exhibits operating points with hyperbolic force-velocity dependency. In this paper, the contraction dynamics of this CE concept were analyzed in a numerical simulation of quick release experiments against different loads. A hyperbolic force-velocity relation was found. The results correspond to measurements of the contraction dynamics of a technical prototype. Deviations from the theoretical prediction could partly be explained by the low stiffness of the SE, which was modeled analog to the metal spring in the hardware prototype. The numerical model and hardware prototype together, are a proof of this CE concept and can be seen as a well-founded starting point for the development of Hill-type artificial muscles. This opens up new vistas for the technical realization of natural movements with rehabilitation devices.

External control strategies for self-propelled particles: optimizing navigational efficiency in the presence of limited resources

We experimentally and numerically study the dependence of different navigation strategies regarding the effectivity of an active particle to reach a predefined target area. As the only control parameter, we vary the particles propulsion velocity depending on its position and orientation relative to the target site. By introducing different figures of merit, e.g., the time to target or the total consumed propulsion energy, we are able to quantify and compare the efficiency of different strategies. Our results suggest that each strategy to navigate towards a target has its strengths and weaknesses, and none of them outperforms the other in all regards. Accordingly, the choice of an ideal navigation strategy will strongly depend on the specific conditions and the figure of merit which should be optimized.