Impedance Space Method: Time-Independent Parametric Ellipses for Robot Compliant Control

TL;DR

A novel 3D graphical representation for impedance control, called the impedance space, is proposed to foster the analysis of the dynamic behavior of robotic compliant controllers to overcome limitations of existing 2D graphical approaches.

Abstract

This paper proposes a novel 3D graphical representation for impedance control, called the impedance space, to foster the analysis of the dynamic behavior of robotic compliant controllers. The method overcomes limitations of existing 2D graphical approaches by incorporating mass, stiffness, and damping dynamics, and associates the impedance control parameters with linear transformations to plot a parametric 3D ellipse and its projections in 2D for a mass-spring-damper impedance under sinusoidal reference. Experimental evaluation demonstrates the effectiveness of the proposed representation for analysis of impedance control. The method applies to various compliant control topologies and can be extended to other model-based control approaches.

Figures (10)

• Figure 1: In \ref{['fig:impedance_block_diagram_1DoF']}, the desired impedance parameters compose the impedance controller blocks in blue. The blocks in orange represent the robot's dynamics that physically interact with the environment (green). The inner torque/force controller $C(s)$ is usually tuned to track with high-fidelity the reference force $f_{ref}$. The interaction force feedback shown by a dashed line highlights its dependency on inertia shaping; In \ref{['fig:impedance_block_diagram_robot']}, a physical equivalence for an impedance controller acting at the end-effector of a manipulator robot.
• Figure 2: Conventional 2D representation of the impedance stiffness and damping. Values of $k_d$ in and $d_d$ in . On the left side, on the $e(t) \times f_{int}(t)$ plane, plots for a pure spring, in blue, and a spring-damper, in orange. Note how the inclination of the main axis of the ellipse changes with respect to the line. On the right side, the reciprocal on the $\dot{e}(t) \times f_{int}(t)$ plane.
• Figure 3: The impedance ellipse binormal vector $\bm{b}$, normal to the trajectory plane of a mass-spring-damper impedance under sinusoidal input. The 3D elliptic trajectory is shown in light gray. The components of $\bm{b}$ are $b_1$ for the $e$-axis (red), $b_2$ for the $\dot{e}$-axis (green), and $b_3$ for the $f_{int}$-axis (blue). The left-hand side depicts the projection of the 3D graph on the damping plane, while the right-hand side shows the projection on the stiffness plane. Only in particular cases does the value of $\rho$, and $\varphi$ matches the slope of the 2D ellipse major axis ($\hat{\rho}$, $\hat{\varphi}$) in these planes.
• Figure 4: Visualization of the 3D theoretical impedance plots and its projections. Each column represents a specific combination of impedance parameters $k_d$ and $d_d$ ($m_d=0$ for all of them), while the rows show different projection of the 3D impedance plot for each case. The first row, in particular, shows the 3D impedance space. In \ref{['fig:ellipses_colored_0']}, where $k_d=0$ and $d_d=0$, only the original unit circle is plotted, from which the other columns derive. In \ref{['fig:ellipses_colored_D']}, $d_d \neq 0$ is added, and the original circle, in black, is elongated and then rotated by $\rho$ around the $e$ axis. An ellipse appears in the 3D plot, with a projection on the plane $\dot{e} \times f_{int}$ being a straight line with the slope given by $tan(\hat{\rho})=tan(\rho)=d_d$. In \ref{['fig:ellipses_colored_K']}, $k_d \neq 0$ is added to the original circle in \ref{['fig:ellipses_colored_0']}, which is elongated and then rotated by $\varphi$ around the $\dot{e}$ axis in this case. The 2D ellipse projection on $e \times f_{int}$ is also a straight line, with a slope given by $tan(\hat{\varphi})=tan(\varphi)=k_d$. Last but not least, in \ref{['fig:ellipses_colored_KD']}, the original circle is transformed into a 3D ellipse with $k_d \neq 0$ and $d_d \neq 0$. In this case, the 2D projections in both $e \times f_{int}$ and $\dot{e} \times f_{int}$ planes are also ellipses. It is important to highlight that, for these 2D projected ellipses, $tan(\hat{\varphi}) \neq k_d$ and $tan(\hat{\rho}) \neq d_d$. In particular, in the $e \times f_{int}$ plane it is possible to see how the inclination changes with respect to the pure stiffness case of \ref{['fig:ellipses_colored_K']}.
• Figure 5: Plot of the 2D projected ellipse in the $\dot{e} \times f_{int}$ plane, with a fixed $d_d=200$. Values of $k_d$ in . When a stiffness $k_d$ is added to the impedance controller, the slope of the ellipse major axis $tan(\hat{\rho})$ may significantly differ from $d_d$.