Virtual model control (VMC)

Overview

Virtual model control (VMC) designs a controller from imaginary mechanical parts: springs, dampers and, in some designs, masses. Each part is attached to a point on the robot and produces a virtual force there. Every force is mapped to joint torques through the transpose of that point's Jacobian, and the torques from all parts are summed 2610.05695 §2.1 [arxiv:2610.05695 eq. 1]. The controller never computes a joint-space target, so it needs no inverse kinematics 2610.05695 §3.

The rule in eq. 1 is shared by every variant. What differs is the layout of the virtual parts. The paper credits Pratt et al. (2001) with introducing VMC for walking bipedal robots 2610.05695 §5; that paper is not in this wiki. The one card so far, Virtual model control for compliant reaching under uncertainties, applies the method to a humanoid arm.

Variants

Simple reaching

A spring and a damper join the end effector to a fixed target. The force is a PD law in end-effector space, and orientation uses rotational springs and dampers in the same way 2610.05695 §2.2 2610.05695 eq. 2. On hardware the paper compares 15 and 45 N/m stiffness at 1.5 Ns/m damping. The stiffer run gave smaller error and faster motion. At 15 N/m the spring force did not overcome friction at the end of the reach 2610.05695 §4.1 fig. 4. Card: Virtual model control for compliant reaching under uncertainties.

Geometric reaching

The end effector is joined by a spring and a damper to a virtual L-shaped rigid body. That body has its own inertia, rotational spring and damper, and its geometry sets the position the end effector is pulled toward 2610.05695 §2.2. The card gives no values for the body's parameters and reports no result for this design.

Moving-reference reaching

The target moves along a straight line from start to goal. A scalar ramps from zero to one in finite time, and the spring law is applied to the moving target 2610.05695 §2.2. The paper says this limits the force needed over long distances 2610.05695 §2.2. Compared with simple reaching, the card reports lower peak force and less lateral deviation at 0.1 and 0.5 m/s, with a final error of about 5 cm 2610.05695 §4.1 fig. 5.

Force-driven reaching

A virtual cart with mass and viscous friction is attached to the end effector through a spring and a damper. A constant force pulls the cart along a straight path. If the end effector is blocked, the cart stops too, so the robot force is bounded by that pull 2610.05695 §2.2. The cart's mass and friction are not stated, and the card reports no result for this design.

Obstacle avoidance

Each obstacle is joined to the end effector by a repulsive, nonlinear spring and damper. Stiffness and damping grow as the separation shrinks, and the forces from several obstacles add up 2610.05695 §2.3 2610.05695 §4.2. The paper uses one piecewise stiffness law. It falls to zero at 20 cm, and the damping law is not stated 2610.05695 §4.2. The card tests one 10 cm cylinder, two 6 cm cylinders, and a non-convex pair. The first two were avoided. With the non-convex pair the end effector fluctuated near a local minimum of the obstacle springs. All three results are qualitative 2610.05695 §4.2 fig. 6.

When to use which

Implementation sketch

The sketch sums the torques of several virtual parts, as eq. 1 states. The force law for each part comes from the design chosen above. Only the spring-damper law eq. 2 is written out in the card, so the other laws appear as placeholders.

# One control cycle with several virtual parts  [eq. 1] [§3]
# Input:  q, dq (joint state), parts (virtual springs, dampers, masses)
# Output: tau (joint torques)

def vmc_step(q, dq, parts):
    # Read each attachment point and its Jacobian from the joint state  [§3]
    points, jacobians = forward_kinematics(q)
    # Update each part's state: spring extension, damper velocity  [§3]
    states = part_states(parts, points, dq)
    # Virtual force of each part, e.g. k x + c dx for a spring-damper  [eq. 2]
    forces = [part_force(p, s) for p, s in zip(parts, states)]
    # Sum the Jacobian-transposed forces into joint torques  [eq. 1]
    tau = sum(J.T @ f for J, f in zip(jacobians, forces))
    return tau

Conversion to motor current is a hardware step. On the Dynamixel motors the paper uses a linear approximation read from the motor's performance graph 2610.05695 §3. It is not included here. Gravity compensation is not stated 2610.05695 §3, so check whether the target robot needs it.

Contradictions

No disagreements between papers, since the page has one card. Within that card:

• The 5 cm error of the moving reference is attributed to low stiffness, "as in the previous case". The previous case used 15 N/m, while the moving-reference run used 45 N/m 2610.05695 §4.1. The stated cause does not clearly match the run it is attached to.
• The damping term for a moving target is ambiguous. eq. 2 uses the error rate, and the text does not say whether the target's velocity enters 2610.05695 §2.2.
• The obstacle damping is written with two different arguments in §2.3, and no damping law is given. The distance unit in the piecewise stiffness law is not stated. Centimetres fit the piece boundaries, which is an inference 2610.05695 §4.2.

Papers

• Virtual model control for compliant reaching under uncertainties: six VMC designs for reaching, moving reference and obstacle avoidance, with a contact test on a humanoid arm; results are author-reported.