SPARC: Spine with Prismatic and Revolute Compliance for Quadruped Robot

TL;DR

SPARC addresses the need for tunable, axis-specific spine compliance in quadruped robots by delivering a compact 3-DoF sagittal-plane spine with prismatic and revolute motion and programmable impedance. It combines three QDD actuators, a 1 kHz control board, and a PMU to implement an impedance control framework based on a planar RNEA model and a smooth Stribeck friction term, enabling accurate stiffness rendering and damping tuning. Bench experiments demonstrate stiff rendering from $k_x=300$ to $700$ N/m with relative error $\leq 1.5\%$, and dynamic tests show mass–spring–damper behavior across damping settings, with small phase deviations due to inertia coupling and friction effects. The work provides a portable, open-source platform for systematic studies of spine compliance in legged locomotion and lays the groundwork for future on-robot validation within a quadruped.

Abstract

We present SPARC, a compact, open-source 3-DoF sagittal-plane spine module that combines revolute (pitch) and prismatic (axial) motion with programmable task-space impedance for quadruped robots. The system integrates three torque-controlled actuators, a custom 1 kHz control board, and a protected power unit in a 1.26 kg package, enabling closed-loop stiffness and damping shaping along x, z, and theta. We develop an RNEA-based computed-acceleration controller with smooth Stribeck friction compensation to render spring-damper behavior without explicit inertia shaping. Bench experiments validate the approach. Quasi-static push-pull tests show linear force-displacement characteristics with commanded horizontal stiffness spanning 300-700 N/m and <= 1.5% relative error (R^2 >= 0.992, narrow 95% CIs). Dynamic displace-and-release trials confirm mass-spring-damper responses over multiple damping settings, with small, interpretable phase deviations due to configuration-dependent inertia and low-speed friction effects. A task-space PD controller produces roughly linear stiffness but with greater variability and coupling sensitivity. SPARC provides a portable platform for systematic studies of spine compliance in legged locomotion and will be released with complete hardware and firmware resources.

Figures (7)

• Figure 1: The compact SPARC system, including a 3-DoF sagittal-plane spine and its control module, is designed to mount between fore and hind body segments. Top: assembled prototype used in experiments. Bottom: structure of the spine unit (bottom).
• Figure 2: Hardware architecture of SPARC.
• Figure 3: Testbed of the SPARC system. One end of the spine unit is fixed, and the end effector is floating to avoid friction. A horizontal force $F_x$ is applied along the $x$–axis by pushing/pulling. The ruler is used to indicate the movement of the end effector and the true displacement used for analysis is obtained from encoder–based forward kinematics.
• Figure 4: Static response: measured $F_x$ vs. displacement for commanded $k_x\in\{300,400,500,600,700\}\,\mathrm{N/m}$. Dots: data; lines: OLS fits whose slopes recover stiffness near the command.
• Figure 5: Dynamic step response of the SPARC system, validating mass–spring–damper behavior. Each panel compares measured hardware response (solid) to the ideal theoretical response (dashed). Top row: $k=300\,\mathrm{N/m}$; bottom row: $k=500\,\mathrm{N/m}$. For each stiffness, four damping coefficients are shown ($b\in\{0,2,20,40\}\,\mathrm{N\cdot s/m}$). The solid line is the mean over 10 trials; the shaded region is $\pm1\sigma$.