Modeling

Long Solenoid Approximation

Each right-handed helix of the Smart Braid is connected to an adjacent left-handed helix. In this configuration, the current moves up the actuator along one hand of helix and down the actuator along the other hand. The current always circles the axis of the actuator in the same direction like a solenoid. This allows us to use the Long Solenoid equation to  predict the change in inductance.

As McKibben Muscles contract, their length, l, decreases while their cross-sectional area, A, increases. The total number of times that the current circles the axis, N, remains constant. When using rubber tubes and air for the inside of the solenoid, the magnetic permeability of the core can be assumed to remain constant at the level of vacuum (4*π x10^-7).

Neumann Formula

Each right-handed helix of the Smart Braid is connected to an adjacent left-handed helix. In this configuration, the current moves up the actuator along one hand of helix and down the actuator along the other hand. The current always circles the axis of the actuator in the same direction like a solenoid. This allows us to use the Long Solenoid equation to  predict the change in inductance.

As McKibben Muscles contract, their length, l, decreases while their cross-sectional area, A, increases. The total number of times that the current circles the axis, N, remains constant. When using rubber tubes and air for the inside of the solenoid, the magnetic permeability of the core can be assumed to remain constant at the level of vacuum (4*π x10^-7).

Stress and Resistance Calculations

Smart Braids need to be designed to bear the high-stresses of pressurized actuation and have low resistance. The wires that make up the braid also need to be able to withstand repeated bending and cyclic stress. For both methods of inductance measurement that we present, high circuit resistance (and relatively low inductance values) can make the measurement of inductance difficult. For these reasons, the ideal wire material would probably be thinly-insulated, stranded aluminum wire. Aluminum has both high conductivity and high tensile strength.  Due to the limited commercial availability of aluminum wire, however, we use copper wire in our experiments.

The stress in the wires can be calculated using an equation presented by Davis and Caldwell.  

P is the pressure in the actuator, D is the diameter, L is the length, n is the number of turns each helix makes around the long axis, N is the total number of helices, and A is the cross-sectional area of the wire.

Both of the inductance measurement techniques we present require that resistance of the wires be somewhat limited. This point will be discussed further in the "Inductance Measurement Techniques" subsection of "Fabrication."