Creating biomimetic systems can be dated back to the 1950s, where master arms used for remote handling of radioactive materials have been developed (Carignan, 2000, p. 2). These interface systems are called bio-mimetic, because they are designed to mimic the sensations of a human operator (Fisch et al., 2007, p. 1f.). Already existing interfaces for human senses of sight and hearing are well developed, but within the last decades these systems have been extended to a third human sense, which is called haptics. By employing new materials and drive components, the mass and friction in these devices have been reduced, making the current generation of haptic devices adequate for many applications (Carignan, 2000, p. 2). With further development of force-torque sensors, such as strain gauge and state feedback mechanisms (e.g. accelerometer) have been incorporated in haptic devices.

Within the scope of this work, the haptic device controller (hardware) is coupled with a SLIP model (section 2.1). Differentiations are made between simple and complex running models. Complex modelling approaches include more specified variables describing human locomotion into detail, which requires more computational power. In order to avoid confusion by complexity and to make the model executable on devices with smaller computational capacity, the model is kept simple including only necessary variables (4.1. Main Variables).

In this section a commonly used model in the biomechanics of human movement (esp. running) is described, which is called spring loaded inverted pendulum model (SLIP). The SLIP model is inspired by the observation that the musculoskeletal system shows elastic behaviour while jumping or running. Some species of animals (e.g. kangaroo, ostrich, horses) can store up to 70% of the kinetic body energy at the time of landing in specific elastic elements. This stored energy can be used for the take-off (Alexander, 1976), which contributes to an efficient and energy saving movement. Thus, developing a model based on elastic elements (i.e. springs) is an obvious idea.

In general, any physical model should have a close relationship to reality, based on assumptions (elasticity) and observable effects, such as center of mass and ground reaction forces.

Simply said, the SLIP model consist of a point mass located at the center of mass (CoM), which is working as a pivot during the ground contact phase. When the spring returns to its initial state length, the flight phase begins. This simple approach of describing the dynamics of running remains relevant after more than 25 years of research.

Figure 1: A simple visualisation of the SLIP model

Regarding Fig. 1 the SLIP model can be described by three variables:

• Center of Mass (= point mass)
• Foot position
• Leg function (comparable to a spring)

In the years 1989-1990 Blickhan, McMahon and Cheng mentioned and described the spring-loaded inverted pendulum model (SLIP; Fig. 1). The SLIP model is inspired by the observation that the musculoskeletal system during running and jumping operates elastically. According to Arslan (2008, p. 8), the spring loaded inverted pendulum (SLIP) is the simplest and fundamental template to analyse dynamical locomotion. It was originally used to model running and jumping. There have even been several attempts to extend the model to walking, which is naturally less stable than running (Rossi, 2015, pp. 7ff.). Hence, to reduce complexity of the human musculoskeletal system and to increase stability, the reductive SLIP model is used. According to Hutter et al. (2010) this model is simple, self-stabilising and accurate in predicting the point mass trajectory (Rossi, 2015, p. 7).

In contrast to walking, both feet are off the ground at some point in time. Respectively, there are two phases (Fig. 2) in one-legged running:

• Flight phase
• Stance Phase

Within the flight phase, the only force acting on the SLIP running model is the gravitation by the earth. According to Arslan, (2008, pp. 8f.), this phase can be further subdivided (Table 1).

Table 1: Subdivision into phases of running

While running, a recurring change between flight and contact phases takes place. In the flight phase, the point mass trajectory is similar to a ballistic curve. When the point mass reached the highest point of the ballistic trajectory (Apex), the vertical velocity changes from positive to negative, this means that the point mass is descending. During stance, the terms compression and decompression (Fig. 2) play an important role. The leg compression is part of stance phase after the touchdown, here, the rate of leg length change is negative until minimum spring length is reached, i.e. maximum compression of the spring. In this case, the point mass reaches its minimum height, which means that the vertical velocity is positive and the point mass is ascending. Maximum compression of the spring leg is reached, when the center of mass is above the foot. A lot of elastic energy is stored and waiting to be released. The release of energy takes place in the decompression phase, which is a subperiod of stance phase as well. The rate of leg length change is positive and the stored elastic energy of the spring decreases, until the take-off, where the spring returns to its initial state length. The next ground contact is made by the opposite leg.

Figure 2: Subdivision into phases of running (in accordance to Arslan, 2008, p. 9)

Transitional events should be considered while running (Arslan, 2008, pp. 10f.), e.g. maximum height or minimum leg length are reached during these events:

1. Apex
2. Touchdown
3. Bottom
4. Take-off

In the following, the upper mentioned events (1-4) will be described in detail. The biomechanical expression Apex means, the maximum height of the point mass trajectory by the SLIP model is reached. In order to check this event, vertical velocity can be used, i.e. $𝑣_y = 0$. The touchdown event occurs during the descent phase. In this case, the current leg length (𝐿) is equal to the touchdown leg length ($𝐿_{TD}$), i.e. 𝐿 = 𝐿_{TD} $𝑣_y < 0$. At the beginning of stance phase, the touchdown leg length $𝐿_{TD}$ is equal to the leg rest length $𝐿_0$. During the simulation, the leg compression is calculated, hence, the current leg length 𝐿, gets updated ($𝐿 < 𝐿_0$). Reaching the lowest point of the point mass (center of mass) trajectory is called bottom event. This transitional event occurs between the compression and decompression subphases. During this event, the spring potential energy reaches its maximum, whereas the leg is at minimum length. Consequently, the rate of change of leg length should be 𝐿 = 0. The take-off describes the transition from stance phase to flight phase and can be expressed as $𝐿 = 𝐿_{TO} ∧ 𝑣_y > 0$. Here, the current leg length (𝐿) is equal to the take-off leg length ($𝐿_{TO}$) and the vertical velocity ($𝑣_y$) increases.

Stable locomotion is dependent on the initial state and input parameters. As described in 2.1, the SLIP model is described by a point mass 𝑚 that is attached to a massless prismatic spring with its resting length $𝐿_0$ and stiffness constant 𝑘 (Hutter et al., 2010, p. 4935). Both, the stiffness constant and the rest length, model the leg, which is carrying the whole body weight. The forces produced by the leg result in measurable ground reaction forces. As the SLIP runner is part of a simulation with a predefined starting parameter set and as the surface conditions don’t change over time, the result remains the same. To test the model for stability, a success criterion is needed. In contrast to previous works (Seyfarth, Geyer, Günther & Blickhan, 2002), where a SLIP model has been developed under stable running conditions, i.e. the initial configuration set is predefined and optimized for the simulation, this work primarily deals with unstable configuration settings. Hence, the goal of stable running is achieved, if the computed SLIP model reaches at minimum 5 more steps after the regular fall down. Generally speaking, the SLIP model may fall down, if no external forces or stiffness changes were applied by haptic device control.

Within the scope of this work, the interaction interface between the human wrist and the virtual SLIP model in the scenario of running play a major role. Since the goal of this work is the coupling of a running SLIP model with a robotic hardware device, a question might come up concerning the control mechanism of said device. Here, the combination of science and technology, with a well developed feedback design criterion, plays a major role (Moussette, 2012). Research knowledge on haptic device control has already been adopted for various purposes, e.g. Melendez-Calderon et al. (2011, p. 1) from Imperial College London developed a dual-wrist robotic interface, called Hi5, which allows the implementation of computer-controlled dynamic conditions and record interaction forces and EMG signals of both partners.

As it can be seen in Fig. 3, the term haptics deals with three main disciplines, i.e. science, technology and biomedical engineering. Science is important, since haptic feedback is perceived by skin receptors. Across synaptic gaps, neural information flows from physiological membranes to the somatosensory cortex. Apart from physiological science, knowledge about the cognitive system of the human brain is necessary, in order to understand the meaning of haptic interaction.

Based on (neuro-)scientific results, technological approaches are made and innovative robotic systems could be created. Hence, technology symbolizes practical implementation of scientific facts and biomechanical models. In the context of rehabilitation and assistance in everyday life, technology is responsible for the creation of devices and robotic systems, which are helping people in the execution of individual activities.

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