Implicit Euler Discrete-Time Set-Valued Admittance Control for Impact-Contact Force Control

TL;DR

This work tackles instability in admittance control during impact with unknown environmental stiffness by introducing a two-set-valued feedback framework: an outer first-order sliding-mode loop constrains actuator torque via a normal-cone formulation, and an inner Multivariable Sliding Twisting Algorithm (MSTA) loop ensures robust motion against differentiable impact forces $f_c$. The method is discretized with an implicit-Euler approach to preserve the set-valued nature in real-time, yielding a practical implementation that couples two interdependent inclusions for $\tau$ and $u_s$. Theoretical stability analyses for contact and no-contact scenarios, plus discrete-time convergence results under perturbed disturbances, support strong robustness claims. Simulation and hardware experiments across one- and two-DoF manipulators and a linear-motor testbed demonstrate improved force-tracking, reduced overshoot, and resilience to torque saturation and unknown stiffness compared with leading methods. The approach promises safer, more adaptable interaction control for robots operating in contact-rich, uncertain environments.

Abstract

Admittance control is a commonly used strategy for regulating robotic systems, such as quadruped and humanoid robots, allowing them to respond compliantly to contact forces during interactions with their environments. However, it can lead to instability and unsafe behaviors like snapping back and overshooting due to torque saturation from impacts with unknown stiffness environments. This paper introduces a novel admittance controller that ensures stable force control after impacting unknown stiffness environments by leveraging the differentiability of impact-contact forces. The controller is mathematically represented by a differential algebraic inclusion (DAI) comprising two interdependent set-valued loops. The first loop employs set-valued first-order sliding mode control (SMC) to limit input torque post-impact. The second loop utilizes the multivariable super-twisting algorithm (MSTA) to mitigate unstable motion caused by impact forces when interacting with unknown stiffness environments. Implementing this proposed admittance control in digital settings presents challenges due to the interconnected structure of the two set-valued loops, unlike implicit Euler discretization methods for set-valued SMCs. To facilitate implementation, this paper offers a new algorithm for implicit Euler discretization of the DAI. Simulation and experimental results demonstrate that the proposed admittance controller outperforms state-of-the-art methods.

Figures (8)

• Figure 1: Illustration of the proposed admittance controller with an inner loop of MSTA (Multivariable Sliding Twisting Algorithm) and an outer loop of normal cone $\mathcal{N}_{\mathcal{F}}$ that is equivalent to the inversed map of a first-order SMC.
• Figure 2: Illustration of one DoF manipulator is accelerated by $f_d$ to impact with unknown environments in terms of stiffness. The parameters are $m_1=5$kg, $l=0.5$m, friction coefficient $\mu=0.1$ and torque saturation $F=3$Nm;
• Figure 3: The simulation results of impact-contact scenario in Fig. \ref{['fig_manipulator_one-DOF']} with an environment of stiffness $k_s=2\times 10^3$N/m, $f_d=-2$N. The proposed algorithm \ref{['equ:kikuuwe_adm_proposed']} denoted as "Prop", the work in Tahara_2021_ICRA denoted as "Tahara", and the work Kikuuwe_TRO_2019 denoted by "Kiku" are implemented and compared with time-stepping size $h=0.001$s; (a) The angular position $q$ of the manipulator and its proxy position $q_x$; (b) The force command $f_d$ and actual contact force $f_c$; (c) The input actuation torque $\tau$ and saturation levels $\tau_{\max}=F$, $\tau_{\min}=-F$;
• Figure 4: Illustration of two-link planar manipulator is accelerated by $f_d$ to impact with unknown environments in terms of stiffness. The parameters are $m_1=6$kg, $m_2=9$kg,$l_1=0.4$m, $l_2=0.6$m, $J_1=0.32$kg$\cdot$m$^2$, $J_2=1.08$kg$\cdot$m$^2$, friction coefficient $\mu=0.1$ and torque saturations $F_1=3$Nm and $F_2=4$Nm;
• Figure 5: The simulation results of impact-contact scenario in Fig. \ref{['fig_manipulator_2-DOF']} with an environment of stiffness $k_s=2\times 10^3$N/m, $f_d=[0,-2]^T$N. The proposed algorithm \ref{['equ:kikuuwe_adm_proposed']} denoted as "Prop", and the work Kikuuwe_TRO_2019 denoted by "Kiku" are implemented and compared with time-stepping size $h=0.001$s; (a) The position $y$ of the end-effector of manipulator and its proxy position $y_p$ along the $y$-axis direction in the Cartesian space; (b) The force command $f_d$ and actual interaction force $\bar{f}_c$ along the $x$-axis (friction) and $y$-axis (contact force), i.e., $\bar{f}_{c,x}$ and $\bar{f}_{c,y}$ in the Cartesian space; (c) The input actuation torque $\tau$ of two joints and saturation levels $\tau_{1,\max}=F_1$, $\tau_{1,\min}=-F_1$, $\tau_{2,\max}=F_2$, $\tau_{2,\min}-F_2$;