An Asymmetric Model for Quadrupedal Bounding in Place Romeo Orsolino 1 , Michele Focchi 1 , Darwin G. Caldwell 1 and Claudio Semini 1 Abstract— In this article we show our approach to determine the main gait parameters of a bounding gait for a quadruped robot. After introducing an asymmetric model that captures the relevant dynamics of quadrupeds we show how this can be employed in an optimization problem that computes a periodic limit cycle. The stability analysis shows that this solution is open loop unstable but can be made marginally stable by means of a state feedback of an augmented system. I. I NTRODUCTION The problem of legged robot locomotion consists in find- ing a series of footholds and joint torques which allow to perform a given motion and reach the final target. This study is typically done using simplified dynamic models and then mapping the result to the whole-body dynamics. For humanoids the most widely used simplified model, particularly suitable for highly dynamic motions, is the Spring Loaded Inverted Pendulum (SLIP). In the case of quadruped robots the applicability of this model is limited by their asymmetric structure even when restricted to the 2D sagittal plane. Considering an asymmetric mass distribution is fundamental to describe, for example, the pitch dynamics. A few studies, at the best of our knowledge, of an asymmetric SLIP model can be found in [1], [2], [3] and [4]. The following observations can be made regarding this asymmetry: • Static measurements on quadruped mammals, mainly dogs and horses, have shown that their Center of Mass (CoM) is always shifted towards the front of the body, resulting in an asymmetric structure. A consequence of this is that front limbs bear around the 60% of the animal’s weight in steady state locomotion [5], [6]. The same can happen on quadruped robots which, even if they have a symmetric skeleton, they might be equipped with exteroceptive sensors which are usually positioned in the front to acquire information of the environment in front of the robot. • Even in the presence of a perfectly symmetric quadruped, the kinematic limits and the manipulability properties of front and hind limbs do not allow them to push or pull their trunk with equal ease in any direction. • Biological observations have shown that load difference of each limb is tightly connected to their phase differ- ence in the gait cycle [7]. Following the above considerations we decided to employ an asymmetric simplified dynamic model which allows us 1 Department of Advanced Robotics, Istituto Italiano di Tecnologia, Via Morego, 30, 16163, Genova, Italy. email: {romeo.orsolino, michele.focchi, darwin.caldwell, claudio.semini}@iit.it to design a realistic controller that can be later easily mapped into the full dynamic model of HyQ, IIT’s hydraulic quadruped robot [8]. II. A MODEL FOR QUADRUPEDAL BOUNDING We designed a 2D dynamic model which could keep this inherent asymmetry into account as depicted in Fig. 1. The impulses J h and J f represent the integral of the ground reaction forces (GRFs) F h and F f applied to the trunk of the robot during the stance over one cycle by the hind and front legs respectively. Fig. 1: Snapshot of the simplified planar model of interaction between the trunk of a quadruped robot in the 2D sagittal plane and the ground. The red dot represents the geometric center of the trunk while the blue dots are the two hips. J h =  Tst 0 F h (t)dt J f =  Tst 0 F f (t)dt (1) The lever arms of the GRFs, l h and l f , are in general different from the quantity L/2 (L is the distance between the hind and front hips) and can be computed as a function of the trunk orientation, the foot contact point and the line of action of the GRFs, defined by the angle φ. Here we study the simplified case in which φ h = φ f = π/2 in such a way to obtain vertical impulses. The resulting equations of motion are: ¨ x =0 ¨ z = −g + F f m + F h m (2) ¨ θ = F f I l f − F h I l h where I is the inertia of the trunk computed as mr 2 , m is the mass and r is the radius of gyration with respect to the CoM [9]. In section III we will use a generalized state vector q =[x, ˙ x, z, ˙ z, θ, ˙ θ] T .