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Optimization-based dynamic motion planning and control for quadruped robots

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Fast trajectory planning and control frameworks improve locomotion robustness against disturbances and uncertainties. In this paper, dynamic motions are optimized under the constraint of a decoupled spring-loaded inverted pendulum model. Subsequently, a hierarchical Quadratic Programming whole-body controller is employed to execute the planned trajectories while ensuring compliance with all physical feasibility constraints. Both the motion planner and the whole-body controller operate within the same high-frequency control loop. Furthermore, the unified whole-body controller governs all gait phases, including flight phases. The proposed algorithms are evaluated through simulation and real experiments, showcasing dynamic gaits such as hopping, twist jumping, and trotting in challenging environments. The algorithms demonstrate resilience against external disturbances and environmental uncertainties.

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The authors would like to thank Ke Wang from Imperial College London for the trajectory generation of the 90\(^\circ \) twist jumping, and Dr. Wolfgang Merkt from the University of Oxford for the experimental tests of the proposed algorithms on the ANYmal Boxy robot at Oxford.


This work was supported in part by Grants EPSRC UK RAI Hubs NCNR (EP/R02572X/1), ORCA (EP/R026173/1) and EU Horizon 2020 THING (ICT-27-2017 780883), the Liaoning and Guangdong Basic and Applied Basic Research Foundation [2023JH2/101600033] [2022A1515140156], the Fundamental Research Funds for the Central Universities [82232025], the State Key Laboratory of Robotics and Systems funding [SKLRS-2023-KF-16]. Dalian Science and Technology Innovation Funding (2023JJ12GX017).

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All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by GX. The first draft of the manuscript was written by GX and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Guiyang Xin.

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1.1 Dimension reduction

Equation (20) can be directly split into two equations

$$\begin{aligned}{} & {} \textbf{M}_f\ddot{\textbf{q}}+\textbf{h}_f=\textbf{J}_{c,f}^\top \varvec{\lambda } \end{aligned}$$
$$\begin{aligned}{} & {} \textbf{M}_a\ddot{\textbf{q}}+\textbf{h}_a=\varvec{\tau }+\textbf{J}_{c,a}^\top \varvec{\lambda } \end{aligned}$$

where \(\textbf{M}_f\in \mathbb {R}^{6\times (6+n)}\), \(\textbf{h}_f \in \mathbb {R}^6\) and \(\textbf{J}_{c,f} \in \mathbb {R}^{3\,m \times 6}\) denote the floating base components of the dynamic matrices while \(\textbf{M}_a\in \mathbb {R}^{n\times (6+n)}\), \(\textbf{h}_a \in \mathbb {R}^n\) and \(\textbf{J}_{c,a} \in \mathbb {R}^{3\,m \times n}\) represent the actuated components. Apparently \(\varvec{\tau }\) is determined by Eq. (39) as follows

$$\begin{aligned} \varvec{\tau }=\textbf{M}_a\ddot{\textbf{q}}+\textbf{h}_a-\textbf{J}_{c,a}^\top \varvec{\lambda } \end{aligned}$$

Therefore, the decision variable can be reduced to \(\textbf{y}=\begin{bmatrix}\ddot{\textbf{q}}^\top&\varvec{\lambda }^\top \end{bmatrix}^\top \), and the physical feasibility constraint Eq. (28) becomes

$$\begin{aligned} \begin{bmatrix} -\textbf{h}_f \\ dot{\textbf{J}}_{c}\dot{\textbf{q}} \\ -\mathbf {\infty } \\ \varvec{\tau }_{\textrm{min}}-\textbf{h}_a\end{bmatrix} \le \begin{bmatrix}\textbf{M}_f &{} -\textbf{J}_{c,f}^\top \\ \textbf{J}_c &{} \textbf{0} \\ \textbf{0} &{} \textbf{C}_{\textrm{in}} \\ \textbf{M}_a &{} -\textbf{J}_{c,a}^\top \end{bmatrix}\textbf{y} \le \begin{bmatrix} -\textbf{h}_f \\ dot{\textbf{J}}_c\dot{\textbf{q}} \\ \textbf{0} \\ \varvec{\tau }_{\textrm{max}}-\textbf{h}_a\end{bmatrix} \end{aligned}$$

The dimensions of the cost functions also need to be modified using the reduced decision variable.

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Xin, G., Mistry, M. Optimization-based dynamic motion planning and control for quadruped robots. Nonlinear Dyn 112, 7043–7056 (2024).

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