Optimizing Design and Control of Running Robots Abstracted as Torque Driven Spring Loaded Inverted Pendulum (TD-SLIP)

TL;DR

The paper addresses co-design of sub-500 g legged runners by abstracting motion with a torque-driven damped SLIP (TD-SLIP) and solving a mixed-integer optimization using MDPSO. It introduces a full TD-SLIP formulation, a Castigliano-based leg stiffness model, a 5th-order open-loop voltage control, and symmetry/repeatability constraints to drive feasible gait design. Two case studies optimize either the touchdown angle repeatability ($\theta_{Diff}=|\theta_0-\theta_{TD2}|$ with Gaussian noise $\mathcal{N}(0, \epsilon^2)$, $\epsilon=1.29^{\circ}$) or the actuation energy ($F=\int V_a i_a dt$), yielding designs with relative stiffness $k_{rel}=\frac{k_0 l_0}{mg}$ around 12–16 that are robust to noise. Validation with 100 noisy trials shows the energy-optimized design achieves comparable stability with lower energy and more gait cycles, demonstrating practical design guidance for micro- to milli-robot runners.

Abstract

Legged locomotion shows promise for running in complex, unstructured environments. Designing such legged robots requires considering heterogeneous, multi-domain constraints and variables, from mechanical hardware and geometry choices to controller profiles. However, very few formal or systematic (as opposed to ad hoc) design formulations and frameworks exist to identify feasible and robust running platforms, especially at the small (sub 500 g) scale. This critical gap in running legged robot design is addressed here by abstracting the motion of legged robots through a torque-driven spring-loaded inverted pendulum (TD-SLIP) model, and deriving constraints that result in stable cyclic forward locomotion in the presence of system noise. Synthetic noise is added to the initial state in candidate design evaluation to simulate accumulated errors in an open-loop control. The design space was defined in terms of morphological parameters, such as the leg properties and system mass, actuator selection, and an open loop voltage profile. These attributes were optimized with a well-known particle swarm optimization solver that can handle mixed-discrete variables. Two separate case studies minimized the difference in touchdown angle from stride to stride and the actuation energy, respectively. Both cases resulted in legged robot designs with relatively repeatable and stable dynamics, while presenting distinct geometry and controller profile choices.

Figures (8)

• Figure 1: A schematic torque-driven damped spring-loaded inverted pendulum (TD-SLIP) model, with a hip torque and damper parallel to the spring, showing one cycle of stance and flight. Cartesian coordinates $x$, $y$ are used as global coordinates and $\theta$, $\zeta$ are used for locally defining the leg position during stance.
• Figure 2: Sketch of the leg geometry used for simulation. The leg is represented as a semicircle with a rectangular cross-section.
• Figure 3: Panel (a) shows how the trajectory of the center of mass of a single stable stance phase evolves with time. The $x$ and $y$ coordinates were scaled by $l_0$. Panel (b) shows the scaled voltage profile required by a motor to match the torque evaluated at the hip and rotational speed during the stance phase shown in panel (a).
• Figure 4: Renderings of optimized designs. Red: case study 1 -- minimizing $\theta_{\text{Diff}}$; and Blue: case study 2 -- minimizing $F$. The sphere diameter is indicative of the relative system mass.
• Figure 5: Cost function and constraint convergence history for case study 1 where variation in touchdown angle was minimized. Feasible results are highlighted in green.