Discovering Optimal Natural Gaits of Dissipative Systems via Virtual Energy Injection

TL;DR

The paper tackles energy efficiency in legged locomotion by offsetting dissipative losses with a virtual energy input parameterized by $\gamma$, enabling quasi-passive gait discovery. It then bridges to fully actuated gaits through a one-parameter input homotopy on $\varepsilon$, using direct collocation for root-search and a predictor–corrector continuation to arrive at energy-optimal actuation. The core contributions are a modular three-stage framework (virtual energy injection, root-search, and continuation), a formal treatment of passive/quasi-passive dynamics, and demonstrations on a Prismatic Monopod and a Sagittal Quadruped with series elastic actuation. The approach reduces reliance on large NLPs, scales to multi-legged systems, and provides a practical path to efficient elastic locomotion by exploiting natural dynamics while ensuring actuated realizability. Overall, the method offers a robust, computationally efficient route to design energy-efficient gaits for dissipative legged robots.

Abstract

Legged robots offer several advantages when navigating unstructured environments, but they often fall short of the efficiency achieved by wheeled robots. One promising strategy to improve their energy economy is to leverage their natural (unactuated) dynamics using elastic elements. This work explores that concept by designing energy-optimal control inputs through a unified, multi-stage framework. It starts with a novel energy injection technique to identify passive motion patterns by harnessing the system's natural dynamics. This enables the discovery of passive solutions even in systems with energy dissipation caused by factors such as friction or plastic collisions. Building on these passive solutions, we then employ a continuation approach to derive energy-optimal control inputs for the fully actuated, dissipative robotic system. The method is tested on simulated models to demonstrate its applicability in both single- and multi-legged robotic systems. This analysis provides valuable insights into the design and operation of elastic legged robots, offering pathways to improve their efficiency and adaptability by exploiting the natural system dynamics.

Figures (14)

• Figure 1: The presented approach utilizes three main concepts. We first apply virtual energy injection to obtain a quasi-passive model. A quasi-passive system is not supported by actuation but through a non-physical energy input scaled by one scalar parameter. Through root-search, we find a quasi-passive gait. The homotopic continuation allows the derivation of a fully actuated optimal gait. Overall, this approach leverages the natural dynamics of the robotic system, thereby avoiding the need to solve large NLPs with high-dimensional decision variables.
• Figure 2: Continuation on the first-order optimality. The coordinate system indicates the cost $c$ in the vertical and the search space $\mathcal{A}$ in the remaining dimensions. The continuation starts at the regular point $\boldsymbol{\psi}(\boldsymbol{\zeta}^*,\varepsilon = 0)$ -- a solution, which we call quasi-passive. From there, we move along a one-dimensional manifold (orange), where the first-order optimality condition $\mathbf{r}(\boldsymbol{\psi}^*) = \mathbf{0}$ is fulfilled. Note that we advance in the direction of increasing $\varepsilon$. The regular point $\boldsymbol{\psi}(\boldsymbol{\zeta}^*,\varepsilon = 1)$ marks the result with a locally optimal cost. The blue curve shows a path towards $\varepsilon = 1$, which is not optimal.
• Figure 3: The prismatic monopod model contains a main body with mass $m_\mathrm{B}$ and inertia $j_\mathrm{B}$, a leg with mass $m_\mathrm{L}$ and inertia $j_\mathrm{L}$ and a foot as point mass $m_\mathrm{F}$. There is a radial spring with stiffness $k_\mathrm{H}$ and damping $d_\mathrm{H}$ at the hip joint, as well as a linear spring with stiffness $k_\mathrm{L}$ and damping $d_\mathrm{L}$ between the leg and foot. The configuration of the system is described by $\mathbf{q} = [x\,z\,\phi\,\alpha\,l] ^{\top}$.
• Figure 4: The continuous states of the hopping forward gait of the prismatic monopod model, including virtual energy injection. The operating point constraint is set to the average speed $\Bar{\Dot{x}} = 0.3 \sqrt{l_0g}$. The gray area marks the stance phase. The main body swings over a stride and compensates the back and forward motion of the leg.
• Figure 5: Actuated hopping forward gait of the prismatic monopod as result of homotopic continuation. The cost function is $c = \boldsymbol{\xi} ^{\top} \boldsymbol{\xi}$. The gray area marks the stance phase. The continuous states and velocities are similar to the respective quasi-passive gait but show small differences, such as in the absolute angle of the main body. The plot on the bottom shows the actuation profile of the parallel elastic actuators for each grid point.