In this section, we present the two main template models which are SLIP (Spring Loaded Inverted Pendulum) and IP (Inverted Pendulum) models.

Complex simulation models are often directly related to the structure of the human body (body segments corresponding to bones, muscles, tendons and other soft tissues). In contrary, the design of conceptual simplified models highly depend on mechanical intuition like in the inverted pendulum (IP) model (Cavagna et al. 1963), the lateral leg spring (LLS) model (Schmitt and Holmes, 2000) or the spring-loaded inverted pendulum (SLIP) model (Blickhan, 1989; McMahon and Cheng, 1990). These models are focusing on describing the axial leg function as a simple telescopic leg spring, with either a constant leg length during stance (IP model) or a leg force proportional to the amount of leg compression (LLS or SLIP model). The assumption of spring-like leg function can be found in approximation experimentally both in animals [Blickhan93] and humans (Lipfert 2009) during steady state locomotion. However, there are also clear deviations in the locomotion dynamics that are not well described by these simple models.

The key limitations of both the IP model and the SLIP model as the most common template models for legged locomotion are summarized in Extended models (Model zoo). Corresponding model extensions that are suitable to overcome these limitations are also presented. It is important to note that we only select elementary extensions of the model, however, also combinations of the model extensions are possible to consider, like XT-SLIP (Sharbafi et al. 2013a) which is an extended SLIP model with trunk (T-SLIP), and added leg mass (M-SLIP) or the ballistic walking model presented of Mochon and McMahon (1980). Model extensions can address either mechanics or control of the system. Another class of model extensions comprises muscles (e.g. single-joint and two-joint muscles with muscle fiber-tendon dynamics) and neural circuits (e.g. sensory feedback pathways) describing muscle stimulation and integration of sensory signals. A sophisticated extension of the SLIP model including muscles, reflex pathways and segmented legs is the gait model of Geyer and Herr (Geyer and Herr, 2010), which originates on the neuro-muscular model introduced by Geyer et al., (2003) or in nmF model presented in mc3.

One of the most recognized models is the spring-loaded inverted pendulum (SLIP) which consists of a point mass atop a massless spring. This model provides a good description of human gaits, such as walking, hopping and running. Despite its high level of abstraction, it supported and inspired the development of successful legged robots and was used as explicit targets for control, over the years.

Since the information about developing SLIP model is already described in http://wiki.ifs-tud.de/biomechanik/modellierung, here we briefly explain this model. Fig. 1 shows the SLIP model and the related equations for hopping in place. In these equations, the only variable is the CoM (Center of Mass) height and a periodic movement is adjustable by the initial dropping height ($y_0$), leg stiffness ($k$), rest length ($l_0$) and the body mass ($m$). This model includes two phases: stance; when the leg is in contact with the ground, and flight; in which there is no contact with the ground. In SLIP, the stance phase is modeled by a spring-mass system, while the flight is modeled by a ballistic free motion. Switch from stance to flight happens when the leg reaches its rest length which is called takeoff. Touchdown is the moment that the leg hits the ground and flight phase turns to stance phase.

Figure 1. SLIP model for hopping in place.

One level extension of the SLIP model for hopping in place will be forward hopping or running model. As the legs are considered to be massless, the model for running and forward hopping are the same. The SLIP model can precisely predict hopping and running. Fig. 2 shows the model and the related equations. In this model, the angle of attack can be considered as a control parameter for swing leg adjustment. The simplest method is using a fixed angle of attack which can result in a stable gait with a small region of attraction. In the other words, this method is very sensitive to the desired angle of attack. In SLIP model and locomotor subfunctions we describe more about swing leg control in SLIP model.

Figure 2. SLIP Running model.

There is a strong belief that unlike running, the stance leg behavior is not spring like in walking. However, Geyer et al presented a BSLIP (Bipedal SLIP) model to predict human walking (Geyer et al. 2006). They showed that BSLIP is better than inverted pendulum in predicting some features of walking such as CoM motion and GRF (ground reaction force) patterns. Fig. 3, presents the BSLIP and the related equations. In this model, the single support phase (in which one leg is in contact with the ground) is similar to the stance phase of SLIP model for running. The new model is given for the double support, in which two legs are in contact with the ground. In this phase, two springs are considered between the point mass at CoM and the ground. There is no flight phase and takeoff and touchdown conditions are similar to SLIP model.

Figure 3. BSLIP model for running.

In running the leg force length curves represent almost a linear relationship that can be modeled by a spring. In contrary, there are many studies in which human/animal walking is modeled with an inverted pendulum (e.g. [Alexander76], [Mochon80]). See Fig. 4 demonstrating human walking, the concept of inverted pendulum and the passive walker robot built by McGeer in 1990 using the similar principles of locomotion. In this model, which is also called simplest model (Garcia 1998), the initial condition (speed) of the CoM generates the forward movement as an inverted pendulum motion. However, energy is lost at touchdown when the impact occurs. So to balance energy, we should introduce source of energy injection. The common methods for energy injection in IP are using gravity, by moving on a sloped terrain, adding (impulsive) pushoff, or adding an actuator between two legs. In passive dynamic walker the first method is selected. In this approach finding the precise value for slope which add required energy to compensate impacts are very critical. We have implemented this model in WEBOT (a simulator with physic engine), shown in the following video.

As can be seen this model is very sensitive to uncertainties, initial conditions and disturbances. Many videos of building such a robot can be found in youtube. In the following video, a motorized robot is developed by Steve Collins which is working based on PDW (passive dynamic walking) motion principles, but on flat terrain. Unlike McGeer robot this one has knees whici can also walk on inclined terrain without actuation.

These walking mechanisms and robots have exhibited similar walking behavior and are sometimes constructed in ways that are mechanically analogous to an inverted pendulum. More specifically, the overall motions of the trunk, stance leg, and swing leg of walking humans, animals, mechanisms and the inverted pendulum model share many similarities: During the stance phase of human walking, when a single leg is on the ground, the body tends to rise and then fall as it pivots about the foot. This is similar to the way an inverted pendulum moves about its pivot. However, the inverted pendulum model has instantaneous double support which is totally different from human walking. Though the concept of a bipedal inverted pendulum is dramatically simple when compared to walking humans or other animals, it can nonetheless help us understand and predict many aspects of walking, including potentially universal and physiologically-independent principles of locomotion.

Figure 4. The concept of the inverted pendulum model in walking from human to robot.

In Fig. 5, the simplest walking model (another name for IP) which was presented by Garcia in 1998 is demonstrated with the related motion equations. In this figure, the single support continuous phase is represented by differential equations while the instantaneous double support (discrete phase) is given by the difference equations. As can be seen, the ground slope plays a key role in the motion dynamics and stability of the system. The phase plot shows a hybrid limit cycle which includes two continuous and two discrete phases. This cycle illustrates a complete stride comprising two steps. In (Goswami 1999) the effects of the slope as a parameter for bifurcation and even turning to chaotic behavior are analyzed.

Figure 5. The simplest model (Inverted pendulum) of walking.

Legged locomotion even represented by template models are complex, nonlinear and hybrid. Therefore, stability analysis is complicated and sometimes impractical. In simulations, one of the easiest methods is the step-to-fall measure meaning the number of steps that the model can take before falling is considered as a stability measure. In the ideal case, this number should approach infinity but in practice, we need to determine a limited integer number to stop simulation when the number of steps reaches this value. In many studies, 50 or 25 are considered as this threshold. It means that if the model does not fall for 50 steps it is sufficient to call the system stable. Simulation time before falling (time-to-fall) can be also considered as a measure of stability. These methods are based on the concept of metastability or beyond stability from physics. For example, if you build a robot that can walk for 1 hour an then falls, it is sufficent for performing the intended tasks. For stabilizing this system, one option could be resetting the states before reaching the failure time (e.g., 50 min in this case). So stopping and resetting the robot states to the initial values every 50 minutes guarantees the stability of the system for any duration.

The main drawback of this method for investigating stability is that not falling for a while is not equal to performing the task in a correct manner. For example, if the desired gait is walking at $1~m/s$ and the model stops after two steps or converges to walk in place (with average speed $0~m/s$), this solution should not be considered as a stable solution with respect to the control goal though it is stable based on control engineering definition. One method to solve this problem is using Poincar\'e return map analysis. In this method, a Poincar\'e section is defined to simplify the complex system by detecting the intersection of the system phase trajectories with this surface. For example, in running, assume that the Poincar\'e section is the apex in which the vertical speed of CoM becomes zero and the second derivative is negative ($\dot{y}=0$ and $\ddot{y}<0$). Linearizing the system around this section simplifies the hybrid system to a discrete 1D system and turns the asymptotically stable limit cycle to an asymptotically stable fixed point resulted from the Poincar\'e return map as follows:

$y_{apex}(i+1)=\rho(y_{apex}(i))$.

With this formulation, first the fixed point ($y_{apex}^*$) should be found in which $y_{apex}^*=\rho(y_{apex}^*)$. Then with linearization of the model around this point and finding the eigenvalues, the stability can be investigated. If the eigenvalues are inside the unit circle the system is asymptotically stable if there is one eigenvalue on the unit circle an the rest are inside it, the system is neutrally stable and if one or more eigenvalues are outside the unit circle the system is unstable. Figure 6 shows the concept schamitcally by drawing $y_{apex}(i+1)$ versus $y_{apex}(i)$ and also showing the results for running with SLIP model from [Seyfarth02]. In this figure, the intersection point of this phase trajectory with the straight line $y_{apex}(i+1)=y_{apex}(i)$ (black line) gives the fixed point and the slope of the curve at this point represents the eigenvalue. Further description about implementation of this example in MATLAB can be found in LM 4 Anwendung.

Figure 6. Stability analysis using Poincar\'e return map and its application on running with SLIP from [Seyfarth02]. Arrows show how to proceed when starting from an initial condition ($y_{apex}(0)$).

1. Using the SLIP model equations for vertical hopping, find the required relation between the mass and spring properties and the initial conditions that generate stable response meaning not hitting the ground.
2. Assume the stability condition of the previous exercise is satisfied, find three different behavioral regimes (Hint: three regimes comprise one fixed point and two cyclic motion)
3. Use the SLIP model presented in Spring-mass model, simulate the model with different initial conditions and verify your response in the previous exercise.
4. Use the MATLAB files in Return map to analyze stability of the responsesusing the Poincar\'e return map analysis.
5. What is the definition of a limit cycle and hybrid limit cycle found with inverted pendulum model in passive dynamic walking?
6. Why the passive dynamic walker is so sensitive to the initial condition and parameter tuning as can be seen in the video of simulation with WEBOT.

[Alexander76] Alexander, R. McN. (1976). Mechanics of bipedal locomotion. Perspectives in experimental biology, 1, 493-504.

[Blickhan93] Blickhan, R., & Full, R. J. (1993). Similarity in multilegged locomotion: bouncing like a monopode. Journal of Comparative Physiology A, 173(5), 509-517.

[Garcia98] Garcia M, Chatterjee A, Ruina A, Coleman M. (1998), The Simplest Walking Model: Stability, Complexity, and Scaling. ASME. Journal of Biomechanical Engineering;120(2):281-288.

[Cavagna63] Cavagna, G., Saibene, F., & Margaria, R. (1963). External work in walking. Journal of Applied Physiology, 18, 1–9.

[Mochon80] Mochon, S., & McMahon, T. A. (1980). Ballistic walking. Journal of biomechanics, 13(1), 49-57.

[Seyfarth02] Seyfarth, A., Geyer, H., Günther, M., & Blickhan, R. (2002). A movement criterion for running. Journal of biomechanics, 35(5), 649-655.

[Sharbafi13a] Sharbafi, M. A., Maufroy, C., Ahmadabadi, M. N., Yazdanpanah, M. J., & Seyfarth, A. (2013a). Robust hopping based on virtual pendulum posture control. Bioinspiration & biomimetics, 8(3), 036002.

[Sharbafi13b] Sharbafi, M. A., Ahmadabadi, M. N., Yazdanpanah, M. J., Mohammadinejad, A., & Seyfarth, A., (2013b) “Compliant hip function simplifies control for hopping and running,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).