Can virtual model control enable compliant reaching under uncertainties and obstacles?

A controller built from virtual springs and dampers

Virtual model control (VMC) does not compute a trajectory and then track it with stiff joint control. Instead, the designer attaches a 'virtual' mechanism — springs, dampers, and sometimes a small cart with mass — between the robot's end effector and a target, an obstacle, or a moving reference point. The forces from that mechanism are converted into joint torques through the robot's Jacobian, so planning and control happen in the same step.

The paper's argument is that this makes behavior predictable: the robot moves as the designed mechanical interaction implies. It also makes the design process intuitive, because adding a task feature means adding or reshaping a virtual component rather than re-deriving a planner. The cost is stated up front by the authors: motion accuracy and trajectory optimality are sacrificed for controlled sensitivity to the environment.

Six virtual models cover reaching and avoidance

The paper proposes six ways to design the virtual model. Simple reaching attaches a spring and damper from the end effector to the goal, which the authors note resembles proportional-derivative control in end-effector space. Geometric reaching connects the end effector to a virtual L-shaped structure with mass that rotates, producing constrained motion such as a circular path. Moving-reference reaching pulls the end effector toward a target that travels along a straight line, regulated by a parameter that goes from zero to one in finite time.

Force-driven reaching attaches a virtual cart to the end effector through spring-damper components and pulls the cart with a constant force. Because the cart also stops when the end effector is obstructed, the total force the robot exerts is bounded by the cart's force, and the desired motion resumes when the obstruction is removed. Obstacle and multi-obstacle avoidance add repulsive nonlinear springs between the end effector and each known obstacle, with stiffness that grows as distance shrinks.

Stiffer springs reach faster but with less margin

In the obstacle-free test, two simple-reaching runs used the same damping of 1.5 Ns/m but spring stiffness of 15 N/m and 45 N/m. The stiffer spring produced a smaller error and faster motion. With 15 N/m, the spring force in the final part of the trajectory was not enough to overcome friction, leaving roughly a 5 cm error from the target.

The same 5 cm shortfall appeared in moving-reference reaching at 45 N/m stiffness, again attributed to low stiffness at the end of the reach. The authors also report that during transients the end effector deviated along the y-axis, which they trace to interaction among the virtual spring, arm inertia, and inaccuracy in converting torque commands to motor current. Moving-reference reaching limited high forces and fast motion and reduced that y-axis deviation, because lower speeds diminish the effect of imprecise torque-current conversion and inertia.

Known obstacles are handled by force balance, not re-planning

With a 10 cm cylindrical obstacle, the robot followed the straight reference until the repulsive spring pushed it away, then accelerated toward the goal once clear. The repulsive stiffness was defined piecewise by distance to the nearest obstacle point: −46r+250 N/m below 5 cm, −3.6r+38 N/m from 5 to 10 cm, −0.2r+4 N/m from 10 to 20 cm, and zero beyond 20 cm.

Two 6 cm cylinders were avoided by attaching a repulsive spring to each. The authors emphasize that, unlike path-planning algorithms, adding a second obstacle did not significantly increase computation because no pre-planning or re-planning is required. The harder case was two partially overlapping obstacles forming a nonconvex shape: trajectories fluctuated near local minima of the elastic potential field before avoiding them. The authors present this as a limitation of the design and call for an improved virtual model controller.

Unknown obstructions pause motion and then release it

When a human hand blocked the end effector around x = 40 cm during simple reaching, the end effector stopped temporarily and resumed as soon as it was released; the authors report the interaction had no effect on the final error. The measured difference from the inverse-kinematics baseline was in contact force: the VMC controller applied up to 4.53 N, compared with 35.77 N for joint-space PID based on inverse kinematics.

In a second scenario, an acrylic plate of unknown presence and position partially blocked a moving-reference reach at 0.5 m/s. The end effector was held in place for about 0.5 seconds after contact, then slid along the plate and reached the goal. The authors explain this as the virtual spring extending until it exerted enough force to overcome friction and slide. They note the physical perturbation resembles the virtual perturbation produced by the known-obstacle springs.

One humanoid arm, tuned by trial and error

Validation used the Sciurus17 humanoid, a lightweight low-cost platform with 17 degrees of freedom in total: seven per arm, two for the head, and one for the torso. The experiments used a force-controlled 8-DoF arm configuration. Dynamixel motors provide current control, and torque-to-current conversion is approximated as a linear equation from the motor's performance graph. Control relies on forward kinematics and Jacobians; inverse kinematics is not needed to drive the robot.

The authors are explicit that the paper only scratches the surface of VMC design. Choosing which virtual mechanisms to use and tuning their mechanical parameters was done by trial and error. They list optimal mechanism design and parameter tuning, task prioritization in complex settings, the precision-compliance trade-off, and adaptive virtual mechanisms as open directions. The nonconvex-obstacle case is the clearest in-paper demonstration that composing springs can create local minima that a better design would need to address.

Choose VMC when contact safety outweighs precision

The paper's own comparison sets the priority: if a reaching task must tolerate unexpected contact, VMC's bounded interaction force and automatic resumption matter more than hitting the goal exactly. The 4.53 N versus 35.77 N gap under human obstruction, and the roughly 5 cm errors traced to insufficient spring force against friction, are two sides of the same design choice.

The scenario split is equally specific. For known obstacles, repulsive springs avoid re-planning cost but can stall near local minima when obstacles form nonconvex shapes. For unknown obstacles, the moving-reference and force-driven mechanisms let the robot stop, slide, and recover without sensing the obstacle in advance. Readers evaluating this approach should therefore ask which of those two regimes their task resembles, and whether a few centimeters of steady-state error is acceptable in exchange for contact forces an order of magnitude lower.

Sources used in this answer