Modular Robot Control with Motor Primitives

TL;DR

The paper introduces a modular robot control framework built on motor primitives, combining Elementary Dynamic Actions (EDA) and Dynamic Movement Primitives (DMP) within a Norton-equivalent network to form independent, stable modules. It defines four basic modules (joint-space, task-space position, and two orientations using SO(3) and H1) and proves independence via superposition of virtual trajectories and impedances, plus closure of stability through passivity with energy-based analysis. The approach enables IK-free task-space control, seamless handling of kinematic singularities and redundancy, and energy-efficient interaction with environments, validated by planar and KUKA iiwa14 experiments and modular imitation learning. The work positions motor primitives as a practical inductive bias for learning and a framework to account for biological motor behavior while offering a constructive tool for robust robotic manipulation.

Abstract

Despite a slow neuromuscular system, humans easily outperform modern robot technology, especially in physical contact tasks. How is this possible? Biological evidence indicates that motor control of biological systems is achieved by a modular organization of motor primitives, which are fundamental building blocks of motor behavior. Inspired by neuro-motor control research, the idea of using simpler building blocks has been successfully used in robotics. Nevertheless, a comprehensive formulation of modularity for robot control remains to be established. In this paper, we introduce a modular framework for robot control using motor primitives. We present two essential requirements to achieve modular robot control: independence of modules and closure of stability. We describe key control modules and demonstrate that a wide range of complex robotic behaviors can be generated from this small set of modules and their combinations. The presented modular control framework demonstrates several beneficial properties for robot control, including task-space control without solving Inverse Kinematics, addressing the problems of kinematic singularity and kinematic redundancy, and preserving passivity for contact and physical interactions. Further advantages include exploiting kinematic singularity to maintain high external load with low torque compensation, as well as controlling the robot beyond its end-effector, extending even to external objects. Both simulation and actual robot experiments are presented to validate the effectiveness of our modular framework. We conclude that modularity may be an effective constructive framework for achieving robotic behaviors comparable to human-level performance.

Figures (9)

• Figure 1: (A) The three Elementary Dynamic Actions (EDA). Submovements (orange box) and oscillations (blue box) correspond to kinematic primitives and mechanical impedances (green box) manage physical interaction. (B) Elements of EDA combined using a Norton equivalent network model. The virtual trajectory $\mathbf{x}_0(t)$ (yellow box) consists of submovements (orange box) and/or oscillations (blue box), and mechanical impedances $\mathbf{Z}$ (green box) govern the dynamics of physical interaction. The Norton equivalent network model provides an effective framework to combine the two distinct domains in robotics: the information domain (left) and physical domain (right). Figure modified from hogan2013generalhogan2017physical.
• Figure 2: (A) A definition of a module, which consist of a pair of mechanical impedance $\mathbf{Z}$ and the virtual trajectory $\mathbf{x}_0$ to which the impedance is connected nah2024robot. For the virtual trajectory, a combination of discrete and/or rhythmic movements is used. (B) The major modules used for robot control: a module for joint-space ($\mathbf{Z}_q, \mathbf{q}_0$) (Section \ref{['subsec:module_for_joint_space']}), a module for task-space position ($\mathbf{Z}_p, \mathbf{p}_0$) (Section \ref{['subsec:module_for_task_space_position']}), a module for task-space orientation, $(\mathbf{Z}_r, \mathbf{R}_0)$, which could either use spatial rotation matrices (Section \ref{['subsec:module_for_task_space_orientation_SO3']}) or unit quaternions (Section \ref{['subsec:module_for_task_space_orientation_H1']}). Note that the conversion between spatial rotation matrices and unit quaternions can be conducted (Appendix \ref{['appx_sec:quat_so3_conversion']}).
• Figure 3: A two degrees-of-freedom planar robotic manipulator controlled using Equation \ref{['eq:two_modules_superimposed']}, and its configuration manifold which is a $T^2$ torus. (A–C) (respectively (D-F)) The robot passing through kinematic singularity (i.e., the straight-arm posture) to change from right-hand (respectively left-hand) to left-hand (respectively right-hand) configuration. In the planar robot diagram, red markers depict $\mathbf{p}_0$ for task-space position impedance $\mathbf{Z}_p$ (Section \ref{['subsec:module_for_task_space_position']}). In (B, C) and (E, F), the purple robot configurations depict the virtual left-hand $\mathbf{q}_{0,L} = [0.2\pi, 0.6\pi]$ and right-hand $\mathbf{q}_{0,R} = [0.8\pi, -0.6\pi]$ configurations, respectively. On the torus, the potential energy $U_q + U_p \circ \mathbf{h}_p$ is plotted, where red indicates lower values. Circle markers depict the robot’s current configuration; diamond markers depict the singular configuration (i.e., the straight-arm posture). In (B, C) and (E, F), purple star markers depict the virtual joint configurations $\mathbf{q}_{0,L}$ and $\mathbf{q}_{0,R}$, respectively. Parameters of the impedance modules: $\mathbf{K}_p=60\mathbb{I}_2$, $\mathbf{B}_p=20\mathbb{I}_2$, $\mathbf{K}_q=2\mathbb{I}_2$. Code script used for MuJoCo simulation: 2DOF_singularity.py. MATLAB script used for visualization: main_2DOF_singularity.m.
• Figure 4: KUKA iiwa14 robotic manipulator controlled using Equation \ref{['eq:three_modules_superimposed']}. (A,B) Using the three control modules, the whole robot's workspace can be utilized. For the experiment, iiwa14_singularity1 KUKA application was used. (C,D) Using the three control modules, the robot can seamlessly pass through singular configuration to change between "up-hand" and "down-hand" configurations. The purple robots depict the virtual (C) down-hand $\mathbf{q}_{0,D} =[-0.06, 0.81,0.31,-1.52,-0.09,-0.66, 0.00]$rad and (D) up-hand $\mathbf{q}_{0,U} = [0.06, 2.12, -0.28, 1.15, 0.20, 0.53, 0.24]$rad configurations, respectively. For the experiment, iiwa14_singularity2 was used. The experimental data was visualized in MATLAB using main_iiwa14_singularity_visualize.m.
• Figure 5: Analysis and quantification of kinematic singularity of the KUKA LBR iiwa14 robotic manipulator. (A) The robot’s workspace regions that were inaccessible using conventional methods khatib1987unifiedchiaverini1997singularity. Dots depict workspace locations where the singular value of matrix $\bm{\Lambda}^{-1}(\mathbf{q})$ is less than or equal to 0.03 lachner2020influence, where $\bm{\Lambda}^{-1}(\mathbf{q}) = \mathbf{J}(\mathbf{q})\mathbf{M}^{-1}(\mathbf{q})\mathbf{J}^{\top}(\mathbf{q}) \in \mathbb{R}^{6\times6}$. Based on this threshold value, 30% of the robot's workspace is unavailable. For visualization, points that meet the threshold but were less than 0.1m apart were excluded. A Delaunay triangulation algorithm was applied to tessellate the remaining points. (B) Using the proposed modular approach, which allows the robot to seamlessly go in and out of singular configuration, the entire workspace of the robot becomes accessible. MATLAB script used for computation and visualization: main_iiwa14_singularity_quantify.m. For the analysis and quantification, the first and last joints of the iiwa14 were fixed at zero. For each of the remaining five joints, 30 equally spaced sample points were generated between their respective minimum and maximum joint limits. The percentage of singular configurations was calculated by the ratio of sample points meeting the threshold to the total number of sampled points.