An Experimental Study of Model-based Control for Planar Handed Shearing Auxetics Robots

Abstract

Parallel robots based on Handed Shearing Auxetics (HSAs) can implement complex motions using standard electric motors while maintaining the complete softness of the structure, thanks to specifically designed architected metamaterials. However, their control is especially challenging due to varying and coupled stiffness, shearing, non-affine terms in the actuation model, and underactuation. In this paper, we present a model-based control strategy for planar HSA robots enabling regulation in task space. We formulate equations of motion, show that they admit a collocated form, and design a P-satI-D feedback controller with compensation for elastic and gravitational forces. We experimentally identify and verify the proposed control strategy in closed loop.

Figures (10)

• Figure 1: Panel (a): Block scheme of the closed-loop system: we plan the steady-state behavior such that the end-effector matches the given desired position $p_\mathrm{ee}^\mathrm{d}$. The outputs of this planning are the steady-state actuation $\phi^\mathrm{ss}$ and a suitable end-effector orientation $\theta_\mathrm{ee}^\mathrm{d}$. After leveraging inverse kinematics to identify the desired and current configuration, $q$ is mapped into a collocated form where the inputs are decoupled. Finally, we use a P-satI-D feedback controller on the actuation coordinates $\varphi$. Panel (b): Visualization of the operational workspace of a planar HSA robot consisting of FPU rods. The colored area within the black dashed borders represents the positions the end-effector (visualized as a dot) can reach. The coloring denotes the mean magnitude of actuation (i.e., twisting of the rods). Furthermore, we plot three sample configurations: the unactuated straight configuration $q = [0, 0, 0]^\mathrm{T}$ (blue), maximum clockwise bending $q = [-11.2rad \per m, 0.08, 0.30]^\mathrm{T}$ (red), and maximum counter-clockwise bending $q = [11.2rad \per m, -0.08, 0.30]^\mathrm{T}$ (green).
• Figure 2: Experimental setup: the parallel robot consists of four HSA rods connected by a platform at their distal end. Four servo motors actuate the HSAs. We track the pose of the end-effector with a motion capture system by attaching reflective markers to the platform.
• Figure 3: Verification of the system model and the identified system parameters on an unseen trajectory with the HSA being randomly actuated through a GBN sequence: the solid line denotes the actual trajectory. In contrast, the dashed line visualizes the trajectory simulated with the system model. We report results for both FPU and EPU-based HSA.
• Figure 4: Step response of the baseline PID, P-satI-D (with gravity compensation), and P-satI-D + GC (with gravity cancellation) controllers on an FPU-based HSA robot.
• Figure 5: Experimental results for tracking a reference trajectory of eleven step functions with the baseline PID controller on an FPU-based HSA robot. Panel (a): End-effector position with the dotted and solid lines denoting the task-space reference and actual position, respectively. Panel (b): The planned (dotted) and the actual (solid) configuration. Panel (c): The planned (dotted) and the actual (solid) actuation coordinates of the collocated system. Panel(d): The saturated planar control inputs are visualized with solid lines, and the computed steady-state actuation with dotted lines.