This part and also the two other parts EM and EMSLIP are basically a summarized version of subchapter 3.6 of a book chapter recently published which is edited by Maziar Sharbafi and Andre Seyfarth titled: Bioinspired legged locomotion. The book can be found here and a complete description of these models can be found in the last subchapter of the third chapter of this book.

This lecture includes two 45 min sessions.

In the basic inverted pendulum models a point mass sits on top of a rigid leg focusing on CoM movement and energetics. In the simplest model (resulting in the passive dynamic walker), no controller is consiederd for different locomotor subfunctions. In this model all subfuncitons are results of passive dynamics of the system e.g., the swing leg motion is based on pendulum movement. One step further is moving on flat ground instead of sloped one which may add the hip torque (between two legs) for swing leg adjustment or the stance leg control may be basically concentrated on pre-impact pushoff. However, the balance control is completely ignored in this model. According to the focus of the IP model which is on energetics, there are not several extensions considering upper body. In general, the IP model was extended in the following directions :

• stance leg: (a) adding leg mass, inertia and damping, (b) adaptation of leg parameters during motion and (c ) increasing number of segments
• swing leg: (a) addition of one or more legs (b) increasing number of segments in swing leg, (c ) adding leg mass
• Further extensions: (a) PDW with upperbody (b) 3d IP model.

The inverted pendulum model is developed to explain walking. Because of the rigidity of the stance leg, no flight phase exists to represent running or hopping. Therefore, the minimum number of legs in this model is 2. Addition of number of legs for this model is not common because it is usually applied to understand human gaits or developing passive dynamic robots except for increasing stability in 3d e.g., McGeer passive walker (McGeer 1990). However, rimless wheel model can be considered as an extension of inverted pendulum model with adding more legs, which are coupled with a fixed angle between limbs (McGeer 1990).

The original IP model does not consider displacements in axial leg direction during stance and thus forces the CoM to move on a circular arch. Introducing a prismatic joint in stance leg converts the IP into a SLIP model if the generated force is proportional to the leg length. In that respect SLIP can be considered as an extension of IP with additional leg spring.

Figure 1. Comparison of LIPM and IP regarding 6 determinants of gaits.

For a long time, many studies in walking were related to the CoM movement described by the inverted pendulum paradigm and the six determinants of gait (Kuo 2007). Based on minimization of CoM displacement, the six determinants of gait theory (Saunders, Inman, & Eberhart, 1953) results in no vertical CoM excursion in walking. In 1991, the linear inverted pendulum model was introduced by Kajita & Tani (1991), in which the leg force is determined to compensate gravity resulting in zero vertical acceleration. The ground reaction force can only act along the leg axis (CoP-CoM line) and the vertical element of the leg force should be equal to the body weight ($Mg$). According to parameters shown in Fig. 4c, the required leg force ($F_l$) to achieve the CoM height ($h_0$) when the horizontal distance between CoM and CoP equals $x$ is computed as follows.

$F_l=\frac{Mg}{h_0}l=\frac{Mg \sqrt{x^2+h_0^2 }}{h_0}$

Following this approach, leg force is predicted to increase with leg lengthening, which is opposite to experimental findings (Lipfert et al., 2012) and the concept of spring-like leg function. The LIPM model was used to develop capture point concept (Pratt et al. 2006) as a method for leg adjustment to reach zero forward speed at vertical leg configuration within one step. Some versions of this model consider a rotation around the CoM by using an upper body (e.g., Kajita & Tani 1991 with a constant angular velocity) or a flywheel with torque control (Pratt et al. 2006), as shown in Fig. 4.

Figure 2. Linear Inverted pendulum model formulation from [Kajita 91].

Figure 3. LIP + Flywheel model from [Pratt 2006].

1. In the IP model, the swing leg is moving forward to take the step, while in rimless wheel model (below figure), the next leg which is coming from above the stance leg with backward motion. Can this phenomenon increase stability? Is there any relation between this model and swing leg retraction paper [Seyfarth03]? What is the different of this swing leg adjustment approach with fixed angle of attack?

1. Which model can better represent human CoM motion, SLIP, IP or LIP?
2. Could you derive the dynamics presented in Fig. 4?

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

Here we focus on an important method of foot placement as a control strategy for the second locomotor subfunction. This method is developed based on the LIP model. This control approach is based on zeroing the CoM velocity by determining an appropriate foot position for the swing leg (see Fig. 4). This idea of was presented within the capture point concept in walking [Pratt2006a]. For a bipedal system, capture state is defined as the state with zero kinetic energy level. By placing the foot (CoP) on a capture point P, the controlled motion dynamics moves the states to reach the capture state. The set of all capture points is called capture region. These concepts can be extended to n-step capture point and n-step capture region using leg swinging and recursive definition [Pratt2006a]. If n approaches ∞ the n-step walking can be achieved because the capture region converges to the area on the ground that the foot can be placed at without falling. For implementation on robots, usually simple models like inverted pendulum [Pratt2006a] or linear inverted pendulum [Pratt2006b] are utilized to find the capture point analytically.

Figure 4. Capture point concept from [Pratt 2006].

In Fig. 5, the formulation of capture point for the LIP model is give. As aforementioned, this formulation can be extended to $n$ step and generate stable walking by converging $n$ to $\inf$. This method was used to control the humanoid robots in walking.

Figure 5. Formulation of capture point for LIP model.

The “pure” IP model with massless legs (Alexander 1976), (Hemami & Golliday 1977) (Wisse et al. 2006) is rarely utilized in walking analysis. Addition of a mass to the legs can simplify control (e.g., based on passive swing leg movement) and also makes the model more realistic. The resulting model was called “the simplest walking model” (Garcia et al. 1998) or the “compass gait model” (Goswami et al. 1996). The compass gait concept was already pointed out by (Borelli 1680) in his famous book „De Motu Animalium“. This popular model is able to represent walking without the need for active control of the swing leg. The stability of the predicted gait was well analyzed in (Goswami et al. 1996).

Different leg mass locations are considered, like the leg’s CoM position (at about the center of the leg) like in the passive dynamic walking model (McGeer1990) or small masses at tip toes (Garcia et al. 1998). A very similar model compared to the simplest walking model is the Acrobot model (Westervelt et al. 2007). In this model the mass is distributed along the leg and not concentrated at the hip. In general, addition of leg mass i) can simplify control, ii) enable passive walking down a shallow slope, iii) permits describing leg swinging (another locomotion sub-function) but at the same time it (iv) requires control e.g. of hip torques when walking on flat terrain to stabilize the gait and to compensate for energy losses (energy management).

There are few extended models of IP with an upper body. In most of such extended IP models, traditional control engineering methods are used for keeping the torso upright (McGeer 1988) (Grizzle et al. 2001) (Gregg & Spong 2009). In [Wisse 2004], a rigid trunk is added to the simplest walking model. The upper body is confined to the bisector of the angle between two legs. With this kinematic constraint, the model has only two degrees of freedom. In this model, the body mass is split to 4 smaller masses which are ordered decreasingly as: one big mass (about 60% of the body mass at COM), one smaller mass (about 25% of the body mass) at pelvis, and two small masses at feet (less than 9% of the body mass at each foot). In addition, a rotational spring is considered between the two legs to support swing leg movement. This model achieves robust walking against $8\%$ disturbances of the initial conditions. Fig. 6 shows this model with a table representing the parameters to mimic human body parameters.

Figure 6. Inverted pendulum model with additional trunk. The figure is adopted from [Wisse 2004].

The focus of conceptual model based gait analyses is on 2d motion in sagittal plane. Moving from side to side as they with the lateral placement rotating about the vertical (yaw) axis at the ankles are the two observations in biological legged locomotion. One of the first attempts for such extensions was 3d model of passive dynamic walking (Fig. 6c) incorporating both roll and yaw rotation (McGeer 1993). However, they found that the model couldn’t stably walk without control. Representing the theoretical stability of a walking machine that rocks side to side without yaw motion, Kuo could stabilize the passively unstable system by a simple control scheme inheriting much of the passive behavior (Kuo1999). In (Zijlstra & Hof 1997) a 3d inverted pendulum model was utilized to explain human walking in 3d space with a sinusoidal left-right movement of CoM. Using such a 3d compass gait model, Gregg & Spong 2009) extended the planar walking into directional 3d dynamic walking (e.g., moving on a circle) by controlled reduction approach. Other extensions like 3d LIPM (Kajita et al. 2001), 3d IP+torso (Gregg & Spong 2009), the generalized 3d IP (Sakka et al. 2010) and 3-segmented IP based model with small actuation at ankle (Wisse et al. 2001) are instances of studies to build an anthropomorphic 3d model for stable walking based on inverted pendulum model. Recently, 3LP, a linear 3D linear IP-based model including torso and swing dynamics is presented by Faraji & Ijspeert (2016) to represent all three sub-functions of legged locomotion with a IP based model. In addition, they could predict nonlinear speed frequency relationship as one optimality trends in human walking.

Figure 7. Linear inverted pendulum model concept in 3d.

1. Can you derive the equation for the capture point in one step $x_capture=\dot{x}\frac{\sqrt{y_H}}{g}$ from Fig. 6.
2. How can the IP model with upper body explained in Fig. 6 address the locomotor subfunctions. Consiering the regular IP model, approximate the upper body angle during motion. How far will it be from upright trunk?
3. Assume the model of Fig. 6 and replace the constraint of having the upper body in the midway of the legs with two rotational springs betweenthe legs and trunk. What is your explanation of this model? How can you comparethese models?
4. How difficult is extending the IP model from 2d to 3d? See [Kuo99] to respond this question.

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