A Physical Model of Hidden Space Dynamics
HBR 180898

One way to think of the smooth dynamics in our model is in terms of a physical system, where for each frame in a segment, the value of the dynamic state at that time is connected to the segment target by a spring, and is also pulled towards its predecessor and successor values by other springs.

Fig 1. Physical "spring" model of coarticulation.

Fig 1 shows our coarticulation model expressed in terms of springs and beads. The positions of a sequence of massless beads represents the state of the hidden dynamic parameters at different times, and these beads are connected to a target position depending on which segment they are in by massless springs. Other springs connect adjacent beads (the beads are constrained to move in the vertical plane). In summary, there are two functions for two different kinds of springs here:

The way in which the target pliancy previously described fits into this model, is to set the stiffness of the springs connected to the respective target position. The second set of springs between adjacent beads are all set to the same stiffness. As it is the pliancy ratios, and not the absolute pliancies that matter, the pliancies of these springs can be set to unity.

The solution to the steady-state positions of the beads arrives when the forces on all of the beads are equal to zero. The forces acting on any one bead in the chain is the sum of the force applied by the spring attached to the target and the forces applied by the two springs attached to the two adjacent beads:

where s(i) is the target index which applies at time i.

The reason why we have used massless beads and springs here is because we are only interested in the steady-state condition of this system of springs, so by making the beads/springs massless, the system always exists in this condition.

For the electrical engineers, the same problem can also be expressed in terms of resistors and voltages (Fig 2). The targets and hidden dynamic states are now expressed as voltages, and the springs can be replaced with resistors.

Fig 2. Electrical analogue of spring system.

The voltage at each node between the top row of resistors represents the values in the hidden dynamic trajectory.