Optimal Virtual Model Control for Robotics: Design and Tuning of Passivity-Based Controllers

TL;DR

This work presents a principled framework for designing and tuning passivity-based robotic controllers via virtual mechanisms. By modeling the controller as a passive virtual structure interconnected with the robot, and optimizing its parameters through differentiable simulations of rigid-body dynamics, the approach achieves stable, task-oriented impedance while accommodating nonlinear dynamics. Key contributions include a formalization of virtual forces/elements, a tractable min–max optimization combining $ ext{L}_2$ and $ ext{L}_{inity}$ costs, and the use of algorithmic differentiation to tune controllers for reaching tasks and complex surgical robotics. Experimental validation on a 7-DOF laparoscopic setup and a Franka Emika arm demonstrates stable sim-to-real transfer and substantial performance gains under disturbances and tracking requirements. The framework offers interpretable, physically grounded control design with scalable tuning capabilities for high-stakes robotic applications.

Abstract

Passivity-based control is a cornerstone of control theory and an established design approach in robotics. Its strength is based on the passivity theorem, which provides a powerful interconnection framework for robotics. However, the design of passivity-based controllers and their optimal tuning remain challenging. We propose here an intuitive design approach for fully actuated robots, where the control action is determined by a `virtual-mechanism' as in classical virtual model control. The result is a robot whose controlled behavior can be understood in terms of physics. We achieve optimal tuning by applying algorithmic differentiation to ODE simulations of the rigid body dynamics. Overall, this leads to a flexible design and optimization approach: stability is proven by passivity of the virtual mechanism, while performance is obtained by optimization using algorithmic differentiation.

Figures (19)

• Figure 1: The virtual mechanism for laparoscopic surgery. The motion of the virtual 'instrument', in red, is constrained to a pre-defined remote centre of motion (RCM) through prismatic and rotatory joints. The virtual instrument is 'attached' to the robot through springs and dampers. Springs are shown in a stretched configuration, at equilibrium there would be no distance between the virtual instrument and the surgical tool.
• Figure 2: Virtual forces ${\bm{F}}_1$ and ${\bm{F}}_2$ acting on points ${\bm{z}}_1$ and ${\bm{z}}_2$.
• Figure 3: Components symbols. A) Spring. B) Damper. C) Inerter. D) Revolute joint. E) Prismatic joint.
• Figure 4: The nonlinear spring force ${\bm{F}} = \sigma \tanh(k|{\bm{z}}|/\sigma) \frac{{\bm{z}}}{|{\bm{z}}|}$ for different values of the parameters $k$ and $\sigma$.
• Figure 5: Virtual model controller block diagram. $({\bm{F}}_e, {\bm{z}}_e)$ is an external port. $r$ represent external inputs to the controller. This is a general structure for any torque controller robot providing joint position and velocity feedback. In the case of the virtual instrument controller of Fig. \ref{['fig:VirtualInstrument']}, described in detail in Section \ref{['sec:SurgeryVM']}, the 'virtual structure' comprises of the rotating joints, and the mass/inertia of the virtual instrument itself, while the springs and dampers between the virtual instrument and robot are 'interface components'.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4