Trajectory adaptation of biomimetic equilibrium point for stable locomotion of a large-size hexapod robot


This paper proposes a control scheme inspired by the biological equilibrium point hypothesis (EPH) to enhance the motion stability of large-size legged robots. To achieve stable walking performances of a large-size hexapod robot on different outdoor terrains, we established a compliant-leg model and developed an approach for adapting the trajectory of the equilibrium point via contact force optimization. The compliant-leg model represents well the physical property between motion state of the robot legs and the contact forces. The adaptation approach modifies the trajectory of the equilibrium point from the force equilibrium of the system, and deformation counteraction. Several real field experiments of a large-size hexapod robot walking on different terrains were carried out to validate the effectiveness and feasibility of the control scheme, which demonstrated that the biologically inspired EPH can be applied to design a simple linear controller for a large-size, heavy-duty hexapod robot to improve the stability and adaptability of the motion in unknown outdoor environments.

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The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: authors gratefully thank the support of the Natural Science Foundation of China (No. 61773139), the Shenzhen Science and Technology Program (Nos. KQTD2016 11215134654) and the Shenzhen special fund for future industrial development (No. JCYJ20160425150757025).

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Correspondence to Wei Guo.

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Appendix A

To calibrate the stiffness coefficient of the robot leg, namely \({K_{iz}}\) in Eq. (2), a simple method was used.

First, we initialized the robot posture with six legs supporting the robot body on a rigid flat terrain, and set a desired body altitude d. Then we employed a plumb line with one end fixed on the robot body, and the other end perpendicular to the ground. By measuring the length of the plumb line, we obtained the actual body altitude \(d'\) under the gravity of the robot. At last, we measured the vertical foot force \({{}^C{F_{iz}}^\prime }\) of leg i by reading the feedback data from the 3D contact-force sensor equipped. Based on \({{}^C{F_{iz}}^\prime }\) and the body altitude deviation \(d-d'\), \({K_{iz}}\) was obtained, as shown in Eq. (24). The schematic diagram of the calibration process is shown in Fig. 22.

$$\begin{aligned} K_{iz} = \frac{{}^C{F_{iz}}^\prime }{d-d'} \end{aligned}$$
Fig. 22

The schematic diagram of the calibration process

Appendix B

In the authors’ previous work (Zha et al. 2019), the macro terrain can be nearly abstracted as a support plane which is constructed with all the support feet. The general equation of this plane can be expressed in Eq. (25), which has the same form as the Eq. (1) in reference (Zha et al. 2019).

$$\begin{aligned} Ax + By + Dz + 1 = 0 \end{aligned}$$

The coefficients of Eq. (25), namely A, B, and D can be computed using the least square method, as shown in Eq. (26).

$$\begin{aligned} \left[ {\begin{array}{*{20}{c}} {\mathrm{A}}\\ {\mathrm{B}}\\ {\mathrm{D}} \end{array}} \right] = {\left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{}{{a_{12}}}&{}{{a_{13}}}\\ {{a_{21}}}&{}{{a_{22}}}&{}{{a_{23}}}\\ {{a_{31}}}&{}{{a_{32}}}&{}{{a_{33}}} \end{array}} \right] ^{ - 1}}\left[ {\begin{array}{*{20}{c}} { - \sum {{}^C{P_{ix}}^\prime } }\\ { - \sum {{}^C{P_{iy}}^\prime } }\\ { - \sum {{}^C{P_{iz}}^ * } } \end{array}} \right] \end{aligned}$$


$$\begin{aligned} \left\{ \begin{array}{l} {a_{11}} = \sum {{}^C{P_{ix}}{{^\prime }^2}} \\ {a_{22}} = \sum {{}^C{P_{iy}}{{^\prime }^2}} \\ {a_{33}} = \sum {{}^C{P_{iz}}{{^ * }^2}} \\ {a_{12}} = {a_{21}} = \sum {{}^C{P_{ix}}^\prime \cdot {}^C{P_{iy}}^\prime } \\ {a_{13}} = {a_{31}} = \sum {{}^C{P_{ix}}^\prime \cdot {}^C{P_{iz}}^ * } \\ {a_{23}} = {a_{32}} = \sum {{}^C{P_{iy}}^\prime \cdot {}^C{P_{iz}}^ * } \\ {}^C{P_{iz}}^ * = {}^C{P_{iz}}^\prime + {{{}^C{F_{iz}}^\prime } / {{K_{iz}}}} \end{array} \right. \end{aligned}$$

where \({}^C{P_{ix}}^\prime \), \({}^C{P_{iy}}^\prime \), and \({}^C{P_{iz}}^\prime \) represent the computation-based foot positions of the support leg i along the x, y and z directions in the robot body coordinate C, respectively. \({}^C{P_{iz}}^ *\) represents the actual foot position of the support leg i along the z direction in the robot body coordinate C. \({}^C{F_{iz}}^\prime \) represents the actual vertical foot force of the support leg i achieved from the feedback data of the 3D contact force sensor. \(K_{iz}\) represents the vertical stiffness coefficient of the support leg i.

It is well to be noticed that during the computation of Eq. (26), the three foot positions along different directions are not obtained through the same way. Just as discussed in Sect. 2.2, the tangential deformation of the support leg i is too small and can be neglected. Therefore, \({}^C{P_{ix}}^\prime \) and \({}^C{P_{iy}}^\prime \) which are obtained through the forward kinematics computation can be used precisely to represent the actual foot positions (the kinematics modeling process of the large-size hexapod robot employed in this paper can be found in the authors’ previous work (Chen et al. 2019)). But for the vertical foot position, because of the obvious vertical deformation of leg i, the forward kinematics computation along the z direction is not accurate enough. In other words, \({}^C{P_{iz}}^\prime \) obtained directly through the forward kinematics computation cannot be used directly to represent the actual foot position. Therefore, \({}^C{P_{iz}}^ *\) with the vertical deformation considered is used to represent the actual vertical foot position.

Based on the A, B, and D computed from Eq. (26), the perpendicular distance from the origin of the body coordinate C to the support plane, namely the actual body height \(d'\) during hexapod locomotion, can be obtained through the computation of Eq. (28).

$$\begin{aligned} d' = \frac{1}{{\sqrt{{A^2} + {B^2} + {D^2}} }} \end{aligned}$$

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Chen, C., Zha, F., Guo, W. et al. Trajectory adaptation of biomimetic equilibrium point for stable locomotion of a large-size hexapod robot. Auton Robot 45, 155–174 (2021).

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  • Equilibrium point hypothesis
  • Compliant-leg model
  • Contact force optimization
  • Deformation counteraction
  • Large-size hexapod robot