In the SLIP model the body dynamics is described by a point mass. With this only leg force pointing to this point, mass can be described. During locomotion, the forces acting on the body are not necessarily directed to the center of mass (COM). For instance, in human walking, the stance leg forces point to a slightly above the COM. In order to describe such deviations of the leg force from the leg axis (from contact point at the ground to COM), the point mass needs to be replaced by an extended body, e.g. a rigid trunk as mentioned in EMSLIP. To study the control of a hopping robot, Poulakakis and Grizzle (2009) extended the SLIP model with a rigid upper body. They used the hybrid zero dynamics (HZD) approach to successfully control the system. Maus et al. (2010) applied the same extension (rigid upright trunk) to a bipedal SLIP model to implement the virtual pivot point (VPP) concept. This approach assumes leg forces to intersect at a fixed location above the body COM to keep postural balance like a Roly Ply toy (Fig. 1).

Figure 1. The Virtual pivot point (VPP) concept and virtual pendulum fo posture control from [Maus 2010].

With this model, both stable walking and running could be predicted. The predicted hip torques are similar to those observed in human walking. As a result, the inverted pendulum model of locomotion can be transferred to a periodic movement modeled by a regular virtual pendulum (VP) as shown in Fig. 1. VPP is an observation from human gait. However, it can be considered as a goal for control which results in VPPC (virtual pendulum posture control) for balancing upper body in TSLIP model for running or hopping [Sharbafi2013] and BTSLIP model for walking [Maus2010].

Figure 2. Three locomotor subcuntions control for stable gait using VPP concep.

As shown in Fig. 2, the TSLIP model with spring (like all SLIP-based models) for stance control, a swing leg adjustment e.g., VBLA for swing leg control and VPPC for balance can generate stable walking and running models. Here, we explain the formulation for VPPC. Considering TSLIP model, the required hip torque to direct the GRF such that it goes through a specific VPP is given as described in Fig. 3.

Figure 3. Formulation of VPPC for balancing upper body in TSLIP model.

In [Maus 2010], the VPP is considered to be on the trunk vertical axis (the line going from hip to CoM). In [Sharbafi 2013], angle between the VPP and the trunk orientation, called VPP angle is utilized to compensate perturbations effect for hopping in place. It was shown that adaptive VPP increases robustness against peruturbations. In this approach LQR (Linear quadratic regualator) is employed to find the desired VPP angle as the control input to return to hopping in place using the states at apex. This model is simulated in MATLAB as TSLIP_VPP_Model.

In this model, different following control approaches can be selected:

1. Stance control: fixed parameters of the spring or adaptation at midstance to reach a certain hopping heigth
2. Swing leg adjustment: Fixed angle, Peuker, VBLA
3. Balance control: fixed of adaptive VPP (with LQR or deadbeat controllers), HZD (hybrid zero dynamics)

In addition to the control features, the body parameters and environment parameters can be set. Finally, searching for stable solutions inside a range for searching parameters are possible.

Figure 4. GUI of the MATLAB code for the TSLIP model.

1. Derive the hip torque equation $\tau_{VPP}$ to generate VPP as shown in Fig. 3.
2. Develop the hip torque equations for BTSLIP model. (Hint: The single support of BTSLIP model will be similar to stance phase of TSLIP, and the double support will be given by repeating the hip torque equation of the single support for two legs)
3. Based on BTSLIP model given in EMSLIP and equations previous exercise, develop the walking model based on VPP. You can benefit from the model TSLIP_VPP_Model and extend it to a walking model.

Maus, H. M., Lipfert, S. W., Gross, M., Rummel, J., & Seyfarth, A. (2010). Upright human gait did not provide a major mechanical challenge for our ancestors. Nature Communications, 1, 70.

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.