This lecture includes two 45 min sessions.
First, we explain the IP model and its properties, then we describe its abilities in predicting locomotor subfunctions.

In Fig. 1, the model with related equations are presented. AS explained in TM, this model includes a continuous single support and a discrete model for the instantaneous double support. The body mass ($M$)is mainly concentrated at the hip and two small masses ($m$) are attached to the foot (end of the legs). These are ignorable compared to the body mass ($m<<M$). The reason for attaching these small masses is benefiting from the pendulum-like motion of the swing leg to passively complete the step on the slope.

Figure 1. The Inverted Pendulum (simplest) model and related equations

Defining $\beta=\frac{m}{M}$, this value can be set to zero in the dynamics of the CoM motion represented by the angle of the stance leg with respect to the line perpendicular to the surface (vertical direction) as shown in Fig. 1. For the swing leg dynamics $\beta$ can be removed by multiplying the equation to $\frac{1}{\beta}$. Thus, after rescaling time by $\sqrt{\frac{l}{g}}$, the equations of the motion for IP model can be given as follows:

\begin{equation} \ddot{\theta}=sin(\theta-\gamma)\\ \end{equation}

\begin{equation} \ddot{\varphi}=\dot{\theta}^2sin(\varphi)+cos(\theta-\gamma)sin(\varphi)+sin(\theta-\gamma) \end{equation}

Touchdown happens when the angle between two legs ($\varphi$) is twice the stance leg angle $\theta$. At touchdown, the leg angles are relabeled and the angular speeds are computed using conservation of angular momentum about the stance foot contact point and the hip hinge. This equation shows that the magnitude of the CoM velocity vector decreases at impact. This results in energy loss that will be compensated by the potential energy from the gravitational force on the sloped ground. Later we discuss other methods to inject energy on flat terrain.

Figure 2. Simulation result of IP for stable walking on slope, (right) angles vs. time and (left) The hybrid limit cycle.

Fig. 2 shows the simulation results of the IP model presented above. Switching between stance and swing leg at foot strike occurs together with an impulsive change in angular speed while the configuration is considered to be fixed. Red and blue curves in the left figure shows how the stance and swing leg angles are switching. In the right figure the phase portraits are depicted for the stance and swing leg angles ($\theta$ and $\theta-\varphi$). The limit cycle is generated comprising the continuous motion of two legs and the discrete mapping between them in a stride (2 steps).

There are extensions of IP model which will be described in Extended models (Model zoo). Here we just explain the compass gait model which is very similar to the simplest model whit minor changes. In the compass gait model, the leg masses ($m$) are considered at the center of mass of the legs and not at foot. More importantly, they are comparable with the body mass $M$. Thus, the system dynamics will be more complex and the simplification made above are not valid. Fig. 3 summarize the differences between these two models.

Figure 3. Compass gait and IP

Using the compass model which is closer to the physical models (e.g., passive dynamic walkers), Goswami has performed analyses on the effect of the ground slope on the behavior of this walking model. He showed that by increasing the slope, bifurcation happens meaning period doubling in this case. As can be seen in Fig. 4., at the moment of the first bifurcation at $\gamma=0.015$, one step periodic motion turns to 2 step periodic walking. This periodic doubling happens again before $\gamma=0.018$ and continues afterward. Finally, at about $\gamma=0.019$, chaos happens and after that, no stable solution is found.

Figure 4. Bifurcation and chaos in walking with compass gait model on the sloped terrain (from Goswami 1996). The bifurcation parameter is the ground slope.

<note tip>session 2: (45 min)</note>

As the stance leg is rigid in IP model, there is no motion in the axial leg direction and the leg force is also determined by the gravitational force. However, for moving on the flat ground two approaches are considered. First one is impulsive pushoff just before touchdown and the second one is considering hip torque between two legs to inject energy into the system. Here e describe the first method which relates to the stance leg control. In other words, generating leg force by spring during whole single support is replaced by an impulsive force at (infinitely short time before) touchdown. The equations of the IP model in existance of the impulsive pushoff are described below

Figure 5. Pushoff as a stance control for IP model.

With this model the mount of energy which is lost at impact should be compensated by pushoff. As a result a limit cycle can be generated for walking. There are some analysis which shows that this method is more efficient than the second method of exerting hip torque continously (similar to energy suppliance by the gravity in PDW). We explain about this idea in the following. Andy Ruina explains this idea in the following video.

The swing leg motion in IP is given by the pendulum motion with moving pivot point. For this, it has a physical representation which may better predict human swing leg movement than pure SLIP model with a fixed angle of attack. Ther are also attempts to attach a physical swing leg to the SLIP model which resulted in improving human swing leg motion prediction in walking and running [Kneusel 2005, Mohammadinejad2015, Sharbafi2017]. In general representing the swing leg by a passive pendulum seems to be a useful method for understanding human foot placement. Linear inverted pendulum (LIPM) is an extended IP which is mainly used for defining capture point as a foot placement strategy. This model will be described in Extended models (Model zoo).

As aforementioned, the second approach for injecting energy to IP model is exerting hip torque. This affects the swing leg motion. For example in [Kuo 2002], a rotational spring is considered between the legs. Any kind of hip torque that generates the same amount of energy that the model loses at touchdown can yield a periodic walking. In the following, we compare this approach and pushoff regarding their ability in reducing energy consumption.

Similar to SLIP model, without trunk, balancing cannot be addressed with this template model. In Extended models (Model zoo) we present some extensions of this model with an additional trunk. However, there are not several studies on such extended models and the main usage of the IP model is explaining the energy consumption (minimization) and understanding what humans minimize during walking.

As two extremes of energy injection method, the lost energy at impact need to be compensated by either pushoff or hip torque. In Fig. 6, different scenarios are compared schematically. In the left figure, pushoff generates energy exactly equal what is lost at impact. As can be seen in this figure the preimpact velocity vector ($\bar{v}_{com}^-$) is changed by the pushoff force $\hat{F}_trail$such that after impact the magnitude of the velocity ($\bar{v}_{com}^+$) will be the same as ($\bar{b}_{com}^-$). In this scenario, no hip torque exists and $W^(+)$ should be equal to $W^(-)$. If the pushoff in less than this amount the (shown in the middle figure), the post-impact velocity is smaller that the preimpact velocity ($\bar{v}_{com}^+$<$\bar{v}_{com}^-$). Here the remained amount of energy should be generated by the hip torque (energy injection). Finally in the right figure, the pushoff is too much resulting in $\bar{v}_{com}^+$>$\bar{v}_{com}^-$. In this case, the hip torque should work against the motion like a brake (energy absorption). From this figure it looks that the first case consumes the minimum energy.

Figure 6. Energy management in IP model

Writing the equations, the aformentioned idea of having optimal motion just by sufficient impulsive pushoff before impact can be proved [Kuo 2002]. This can be easily computed by differentiating from $W_{total}$ computed in Fig. 7. Note that in the figure the absolute value of the hip energy should be added to the pushoff energy to find the total energy because negative energy is also costly and cannot reduce energy consumption.

Figure 7. Optimality of the pre-touchdown impulsive pushoff in walking.

Using optimal control, Bhounsule et al. built Cornell Ranger which can walk about 65km without recharging the batteries. The following video shows the performance of this robot [Bhounsule14].

1. Can you derive the impact equations of the inverted pendulum model using conservation of angular momentum about the stance foot and the hip hinge?
2. How do you explain the relation between the right and left leg angles regarding Fig. 2?
3. Using the equations of the simplest model of walking, simulate walking on the slope and draw the angles and the limit cycle like Fig. 2.
4. Using the model generatedin the previous exercise, change the ground slope and see how it affects the periodicity of the responses and how bifurcation and chaos ocurr.
5. Using the IP model with hip torque and pushoff and prove that the most optimal control strategy is generating sufficient pushoff without hip torque.

[Bhounsule14] Bhounsule, P. A., Cortell, J., Grewal, A., Hendriksen, B., Karssen, J. D., Paul, C., & Ruina, A. (2014). Low-bandwidth reflex-based control for lower power walking: 65 km on a single battery charge. The International Journal of Robotics Research, 33(10), 1305-1321.

[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.

[Goswami96] Goswami, A., Thuilot, B., & Espiau, B. (1996). Compass-like biped robot part I: Stability and bifurcation of passive gaits (Doctoral dissertation, INRIA).

[Kuo 2002] Kuo, A. D. (2002). Energetics of actively powered locomotion using the simplest walking model. Journal of biomechanical engineering, 124(1), 113-120.

[Kuo 2007] Kuo, A. D. (2007). The six determinants of gait and the inverted pendulum analogy: A dynamic walking perspective. Human movement science, 26(4), 617-656.