In this chapter, the results of the simulation experiments are presented. The goal to be achieved within this work is to examine a simple SLIP model while running under unstable condition sets. Therefore, certain initial parameters have been changed at the beginning of a specified simulation set of trials, e.g. trials with a different set of initial horizontal velocity values (𝑣R) have been executed. Following values have been used for the simulation experiment: $v_x$ = 2.0 𝑚/𝑠, $v_x$ = 2.5 𝑚/𝑠, $v_x$ = 3.0 𝑚/𝑠, $v_x$ = 3.5 𝑚/𝑠, $v_x$ = 4.0 𝑚/𝑠. Since the haptic control design (chapter 3) within this work is based on the human-in-the-loop (HITL) approach, the subject is able to modify certain parameter values of the SLIP model in real-time. Beside the manipulation of the virtual stiffness factor 𝑘 (initial value: 22,500 𝑁/𝑚), the HITL is able to apply an external force (𝐹_𝑒𝑥𝑡) to the point mass in real-time (5.4 Enabling Haptics).

In chapter 6.1 (General trials), the simulation results are illustrated (Fig. 17, 21) in a 𝑥-/𝑦-diagramm showing the point mass trajectory over the achieved running distance. The sections 6.2-6.4 show simulation results of SLIP running under unstable initial conditions while using haptic device control.

Within this section, experimental trials were performed, where no haptic input is introduced by the user, i.e. haptic device control is disabled during these experimental trials. There was neither a change in the stiffness, nor has an external force been added during stance phase to the point mass.

A distinction is made between two changing (initial) variables. As previously mentioned, the simulation has been performed by using different start values for the horizontal velocity 𝑣R and the vertical CoM position 𝑦/. Both scenarios, are explained in the next sections.

The following Fig. 17 shows that the initial velocity 𝑣R has a major impact on the behaviour of the SLIP model. Starting the simulation with an initial velocity of 5 𝑚/𝑠, the SLIP model immediately returns into a stable state after the first touchdown. The Apex height does not vary with regards to the steps made. By decreasing the initial velocity value 𝑣R, it can be seen, that the system gets unstable and the Apex height increases significantly with every step made during the simulation, until the SLIP model falls down. The slower the running model starts; the less steps can be achieved. This is due to minor kinetic energy provided to the system.

Figure 17: Changing initial velocity conditions without haptic input by the user

Without haptic influence (i.e. adding an external force or changing the spring stiffness parameter) from the user’s side, the SLIP model falls down for various initial height conditions. It should be noted that only the height condition has changed, all other start parameters are set to their respective default values. Taking the horizontal velocity as an example, the initial value is set to 5 𝑚/𝑠.

Figure 18: Changing initial height conditions without haptic input by the user

Regarding Fig. 18, it should be noted that the optimum height is 1 𝑚. In this particular case, the foot of the SLIP model doesn’t touch the ground yet, even if the 𝑦-position of the CoM is set to 1 𝑚 as well. This is due to the angle of attack, which is set to 70° by default. Furthermore, the impact of the first touchdown rises with increasing initial height conditions. The explanation is obvious, since during flight phase only gravitational force is acting on the point mass. Neglecting aerodynamic frictions, the vertical point mass velocity decreases continuously, until the first ground contact takes place, thus intensifying the impact.

From this point onwards, the current horizontal velocity, as well as the spring stiffness are decisive factors for the further curve trajectory of the CoM. More information on the importance and function of model parameters can be found in chapter 4.1 (Main Variables).

In the following, experimental results with an external force application to the point mass are shown. Based on the current angle 𝛼 between leg axis and the ground line, the externally applied force is split into a horizontal and a vertical component. In total, 5 different constellations, with regards to changing velocity 𝑣R, are examined. Since the SLIP model is very stable for an initial velocity of 5 𝑚/𝑠, this case is left out. It is more relevant to see, how the SLIP model behaves, when decreasing horizontal velocity at the beginning of the simulation.

Furthermore, it can be seen, that with an increasing number of attempts, the SLIP model improves, but it still falls down after a certain amount of steps (i.e. 6 max. steps). Regarding Fig. 19 (4th trial, 5th trial), the running model was able to achieve a distance of approximately 4.5 𝑚 within a time range of 3 𝑠. Since this is the first series of trials (in total 5 trials), a major improvement with regards to the achieved running distance is noticed. All experimental trials are performed on a single subject. Therefore, it might be interesting to compare Sensimotor learning results with more than one person.

Figure 19: Haptic force application – 5 trials with 𝑣x = 2.0 𝑚/𝑠

By increasing initial velocity 𝑣R to 2.5𝑚/𝑠 (Fig. 20), the running distance is approximately twice as big as in the previous case (𝑣R = 2.0 𝑚/𝑠). Also the amount of steps increased from 6 to 9 steps. Unfortunately, the system crashed during the 4th trial. The CoM 𝑦-trajectory, as well as the change in Apex height is similar to the curve characteristics of the green trajectory (5th trial). Again the 3rd trial is the worst one regarding the achieved running distance. It should be noticed, that the concept of sensimotor learning is applying. Thus, the subject was able to respond more smoothly to the impacts and ground reaction forces occurring during stance phase.

Figure 20: Haptic force application – 5 trials with 𝑣x = 2.5 𝑚/𝑠

Adding another 0.5 𝑚/𝑠 to the initial velocity 𝑣x (Fig. 21), the systems seems to get stable. With regards to the upper subplot, it can be seen, that the SLIP runner is running forwards during the first 4 simulation seconds. Afterwards, the system remains stable, but the SLIP model is running backwards due to the rapidly decreasing angle of attack during the touchdown phase. The model doesn’t fall down, because an external force is applied continuously from the touchdown to the take-off. Even if the angle of attack $𝛼_0$ is set to 70° (4.1 Main Variables), the model isn’t able to run forwards. 14 steps are counted in Fig. 21, but only the first 9 steps might be treated as valid values. In real world scenarios, it is impossible to touch down with an angle of 70° (without kinetic energy in horizontal 𝑥-direction) and therefore to remain in the same position while jumping up and down. The 4th trial is comparable to the previous case (𝑣R = 2.5 𝑚/𝑠). Differences are noticed in fluctuations of the Apex height change, which seems to be more stable for an initial velocity 𝑣R = 3.0 𝑚/𝑠.

Based on these two experimental cases, a hypothesis may be formulated stating that increasing Apex height changes might cause unstable point mass trajectories during running.

Figure 21: Haptic force application – 5 trials with 𝑣x = 3.0 𝑚/𝑠

The results for running with an initial horizontal velocity of 𝑣R = 4.0 𝑚/𝑠 seem to be more constant, than in previous trials. It should be noted, that the deviation between the smallest (3rd trial, yellow line) and the greatest (4th trial, purple line) running distance is approximately 2.5 𝑚 (Fig. 22). Within this series of trials, the SLIP model reached more than 10 𝑚 of running distance twice. Comparing this to previous results (𝑣x = 2.0 𝑚/𝑠 and 𝑣x = 3.0 𝑚/𝑠), where the maximum running distance is about 4-5 𝑚, a big difference is noted. The series with an initial velocity state of 𝑣x = 2.5 𝑚/𝑠 (Fig. 20), where a running distance of 9 𝑚 has been reached, forms the only exception. This result can be treated as an outlier, because within all other trials a significantly smaller running distance (1 − 4 𝑚) has been achieved.

The SLIP model shows a more stable locomotion pattern with an increased start velocity. If the initial velocity 𝑣x gets too high (i.e. significantly higher than 5.0 𝑚/𝑠), the system returns back to an unstable state as well.

As part of this last series of trials (𝑣x = 4.0 𝑚/𝑠), all 5 attempts show great results. According to the principles of trial and error, the user learned how to apply external forces with each running cycle. Of course, the good results are also due to the increased start velocity 𝑣R. Nevertheless, the SLIP model still falls down after at least 6 steps. It would be interesting to see, how the sensimotor learning progress looks like after more than only 5 trials. Maybe a trained user is able to achieve a greater running distance or to stabilize the locomotion pattern, that no big Apex height changes are noticed. This and many more ideas can be read in chapter 8 (Future Work).

Figure 22: Haptic force application – 5 trials with 𝑣x = 4.0 𝑚/𝑠

The following results show the CoM trajectory (related to the achieved running distance) with respect to a – in real-time – changing spring stiffness parameter during stance phase. This time, no external force is applied to the point mass; only internal forces act on the CoM, which are produced by a massless prismatic spring. In the first series of trials (with 𝑣x = 2.0 𝑚/𝑠; Fig. 23), a maximum running distance of 5 𝑚 is reached by the SLIP model. After the first attempts, the user roughly learned how to manipulate the spring stiffness parameter with respect to stabilizing the locomotion pattern. Hence, the last trial (number 5) has the longest simulation time. In this attempt the SLIP model didn’t fall down. However, the 1st trial appears to be more successful, even though the running distance of approximately 4 𝑚 is 1 𝑚 less than in the 5th trial. This is due to the smooth CoM trajectory (blue line) in trial number 1, where no big Apex height changes occur, unlike in the CoM trajectory (green line) in trial number 5.

Figure 23: Haptic stiffness manipulation – 5 trials with 𝑣x = 2.0 𝑚/𝑠

Regarding Fig. 24, a difference is noted by increasing the initial horizontal velocity to 𝑣x = 2.5 𝑚/𝑠. The SLIP running model falls down immediately after the first step has been made. Thereafter, considering the last attempt (purple line), more than 30 𝑚 running distance are achieved by the model. Comparing this results to the prior experimental series of trials (𝑣x = 2.0 𝑚/𝑠), there are certain similarities. This can be seen as an effect of sensimotor learning. Comparing outcomes to the previous setup, where external forces have been applied to the point mass with a maximum running distance of roughly 10 𝑚, results show a triplication of the distance accomplished by the SLIP model. Under previous initial conditions, where 𝑣x = 2.0 𝑚/𝑠, all trials terminate with a SLIP model jumping and falling backwards (see upper subplot). In this setup (𝑣x = 2.5 𝑚/𝑠) – apart from trial number 2 – the SLIP runner falls down when running/jumping forwards. Considering the purple line (4th trial) in Fig. 24, after approximately 4 𝑠 of simulation time, a minimum change of the Apex is noticed. After 5.5 𝑠, the CoM trajectory (lower subplot) of the SLIP runner shows growing fluctuations between Apex and touchdown position of the CoM. It is obvious, that the spring stiffness has changed drastically, due to the fluctuations.

Figure 24: Haptic stiffness manipulation – 4 trials with 𝑣x = 2.5 𝑚/𝑠

According to Fig. 25, it can be assumed that the SLIP model is going to fall down within the next simulation seconds. After two unsuccessful trials, the following three attempts show roughly normal CoM curve trajectories, despite the big Apex height fluctuations. These fluctuations are apparently the reason for the future fall-down of the SLIP model. Nevertheless, a running distance of over 20 𝑚, has been achieved three times. It would be interesting to find out, how sensimotor training can affect the user customized input fed to the haptic device robot. Basically speaking, it is possible to achieve a totally stable running pattern with an initial horizontal velocity 𝑣x = 3.0 𝑚/𝑠.

Figure 25: Haptic stiffness manipulation – 5 trials with 𝑣x = 3.0 𝑚/𝑠

According to the scenario with an initial horizontal velocity 𝑣x of 4.0 𝑚/𝑠, within the first trial, a little more than 10 𝑚 distance has been reached. Afterwards the SLIP model falls back and the simulation terminates, if the CoM 𝑦-value is below 0. A stable locomotion is reached in the 4th trial. After increasing changes in Apex height (first 3 𝑠, Fig. 26), the user figured out an appropriate strategy how to stabilize the running pattern.

A proper solution might be to decrease spring stiffness during the compression phase. Before take-off, the spring stiffness should be increased. There are many possible approaches, e.g. increasing the stiffness immediately before touchdown and reducing it while the spring length is minimized. This opens a new field of research considering the control of the real-time changing spring stiffness value during stance phase while running. Further interesting topics can be found out in chapter 8 (Future Work).

Figure 26: Haptic stiffness manipulation – 5 trials with 𝑣x = 4.0 𝑚/𝑠

In the following experimental trials with spring stiffness manipulation and different initial state height ($𝑦_0$) of the point mass are examined and evaluated.

As it can be seen in Fig. 27, first two trials are almost stable and no big Apex height change is noticed. In each of the five trials, the SLIP model is running forwards (upper subplot) with an achieved running distance of almost 40 𝑚, reached in the first two trials. Within the last three trials a maximum running distance of almost 15 𝑚 has been reached. Thus, while the first attempts more than 15 steps have been made in the first attempts, but afterwards the system falls down after approximately 3 steps, due to inappropriate haptic stiffness regulation at the beginning of the simulation (i.e. first touchdown).

Figure 27: Haptic stiffness manipulation – 5 trials with CoM height 𝑦0 = 1.2 𝑚

The same hypothesis applies, when using an initial CoM height of 1.5 𝑚. The first impact within the 3rd trial is very smooth compared to the first attempt, where the SLIP model falls down after the third step. 10 steps are made in the following trial, then the model falls down as well. Before the fall-down (Fig. 28, red line), the curve characteristics between the red and the yellow line are very similar. In one scenario the SLIP running model achieves a stable state.

Figure 28: Haptic stiffness manipulation – 3 trials with CoM height y0 = 1.5 :

Since the main topic of this thesis is haptic device control with regards to velocity changes in a SLIP model, the various running scenarios with changing initial CoM height are only described marginally. The idea is to point out further investigation opportunities when modifying another initial state parameter (i.e. CoM height) within the SLIP model. More information on this topic can be found in the following section.

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