## Abstract

Research into legged robotics is primarily motivated by the prospects of building machines that are able to navigate in challenging and complex environments that are predominantly non-flat. In this context, control of contact forces is fundamental to ensure stable contacts and equilibrium of the robot. In this paper we propose a planning/control framework for quasi-static walking of quadrupedal robots, implemented for a demanding application in which regulation of ground reaction forces is crucial. Experimental results demonstrate that our 75-kg quadruped robot is able to walk inside two high-slope (\(50^\circ \)) V-shaped walls; an achievement that to the authors’ best knowledge has never been presented before. The robot distributes its weight among the stance legs so as to optimize user-defined criteria. We compute joint torques that result in no foot slippage, fulfillment of the unilateral constraints of the contact forces and minimization of the actuators effort. The presented study is an experimental validation of the effectiveness and robustness of QP-based force distributions methods for quasi-static locomotion on challenging terrain.

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## Notes

In the following we keep using \(x_{com}\) even if in the implementation we actually used \(x_{com-base}\).

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## Acknowledgments

This research has been funded by the Fondazione Istituto Italiano di Tecnologia.

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## Appendix

### Appendix

### 1.1 Intuitive justification of foot placement

This section explains our choices regarding foot positioning for quadrupedal walking on v-shaped terrain. We show that, when the robot stands on three feet, having an *acute* support triangle is convenient for maintaining the robot in equilibrium. We know that the robot is in equilibrium when the net external force and moment (about any point) acting on it are zero. We define a reference frame \(O_1\) located at foot 1 (see Fig. 10), with the axis \(z_1\) aligned with gravity and the axis \(x_1\) pointing towards foot 2 (which we assume to be approximately aligned with foot 1). At the equilibrium, the net moment \(m\in \mathbb {R}^3\) about \(z_1\) has to be zero, that is:

where \(P_{xy}\in \mathbb {R}^{3\times 3}\) projects onto the \(x_1y_1\) plane, \(P_z\in \mathbb {R}^{3\times 3}\) projects onto the \(z_1\) axis, \(f_2 (f_3)\in \mathbb {R}^3\) is the GRF at the foot 2 (3), and \(p_{12}, p_{13} \in \mathbb {R}^3\) are the lever arms from foot 1 to foot 2 and 3, respectively. The first term of (17) always generates a positive moment about \(z_1\) because of the unilaterality constraints, i.e., \(f_{2y}>0\). To have equilibrium then we need \(f_3\) (i.e., the second term) to generate a negative moment about \(z_1\). In other words \((P_{xy}f_3)\) must lie in the right halfspace delimited by the line passing through feet 1 and 3. Similarly, computing the net moment about \(z_2\) (i.e., the *z* axis of the frame \(O_2\)), we can infer that to have equilibrium \((P_{xy}f_3)\) must lie in the left halfspace delimited by the line passing through feet 2 and 3. This implies that \((P_{xy}f_3)\) must lie—not only inside the friction cone, but also—inside the *support cone*, that is the cone originating in \(O_3\) and delimitated by two sides of the support triangle (green cone in Fig. 10). We can then state that having an *acute* support triangle leaves more freedom in the choice of \(f_3\) because it results in a bigger area of intersection between the friction cone and the support cone. If \(p_3\) gets too close to \(p_1\) or \(p_2\), a part of the friction cone of \(f_3\) stops intersecting the support cone, leaving less freedom for the choice of \(f_3\) (e.g. red support triangle in Fig. 10).

Taking advantage of these insights we planned contact configurations that generate acute support triangles. A gait sequence that satisfies this requirement is *RH*, *RF*, *LH*, *LF*, in which we set an initial offset positions for the feet along the *x* direction (see Fig.10 (bottom)).

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Focchi, M., del Prete, A., Havoutis, I. *et al.* High-slope terrain locomotion for torque-controlled quadruped robots.
*Auton Robot* **41**, 259–272 (2017). https://doi.org/10.1007/s10514-016-9573-1

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DOI: https://doi.org/10.1007/s10514-016-9573-1