# Air Muscle Actuator (G)

**Libraries:**

Simscape /
Fluids /
Gas /
Actuators

## Description

The Air Muscle Actuator (G) block models a linear actuator popular in robotics for its
characteristics reminiscent of biological muscle. The actuator comprises an expandable
bladder in a braided shell. When the bladder is pressurized, the pair widens and
simultaneously shortens, producing at their end caps a contractile force. The bladder is
pressurized at Gas port **A**; the force is exerted at Mechanical
Translational ports **R** and **C**.

**Air Muscle in Relaxed State**

Air muscles are often installed in pairs—one muscle serving as agonist, the other as antagonist. Pairs of this sort are common in the human body, where the biceps (in the arm) accompanies the triceps, and the quadriceps (in the leg) accompanies the hamstrings. The muscles attach at one end to a joint, but at an offset so as to produce a torque. When the net torque is other than zero, and if loading conditions allow, the joint rotates.

### Actuator Force

The mass and energy balances of the actuator are as described for the Translational Mechanical Converter (G) block. The actuator force, however, is based on the standard Chou-Hannaford equation (with two corrections made for the simplifying assumptions of the original model). In its original form, the Chou-Hannaford equation gives:

$${F}_{\text{C-H}}=\frac{\pi {D}_{\text{M}}^{2}P}{4}\left[3{\left(\frac{L}{l}\right)}^{2}-1\right],$$

where:

*F*is the contractile force exerted by the actuator on its ends. The subscript`C-H`

denotes the theoretical value of the original Chou-Hannaford model.*D*is the diameter of the bladder and shell assembly. The subscript`M`

denotes its maximum theoretical value—that in which the braids of the shell are at right angles to its longitudinal axis.*P*is the gauge pressure in the bladder (measured against the environment external to the actuator).*L*is the length of the actuator (the distance between the mechanical ports**R**and**C**.*l*is the natural length of a braid (before it is stretched in a pressurized bladder). The braids, as they are wound about the longitudinal axis of the actuator, are always longer than the actuator itself).

The maximum theoretical diameter of the actuator is defined as:

$${D}_{\text{M}}=\frac{l}{n\pi},$$

where *n* is the number of turns that a braid
makes about the longitudinal axis of the actuator.

Implicit in the Chou-Hannaford equation are the assumptions of infinitely thin
bladder and shell and of inelastic braids incapable of stretching. Both assumptions
can lower the accuracy of the model and are corrected for in this block. The
correction for the stretching in a braid replaces the constant length
*l* with the variable length
*l*^{*}:

$${l}^{*}=\frac{Cl+\sqrt{{\left(Cl\right)}^{2}+12{L}^{2}\left(C+1\right)}}{2\left(C+1\right)}+\frac{2nP{D}^{2}}{Ed},$$

where *l* is the natural length of a braid used
in the original Chou-Hannaford equation and:

*C*is a correction term for braid stretching.*E*is Young's modulus of elasticity for the material of the braids.*d*is the diameter of a strand in a braid (each braid being a bundle of tightly intertwined strands).

The correction term for braid stretching is defined as:

$$C=\frac{{n}^{2}{\pi}^{2}E{d}^{2}N}{PlL},$$

where *N* is the total number strands in the
braided shell. The correction for the thickness of the bladder and shell adds to the
total actuator force the factor:

$${F}_{\text{T}}=\pi P\left[t\left(2D-\frac{{D}_{\text{M}}^{2}}{D}\right)-{t}^{2}\right]$$

where *t* is the aggregate thickness of the
bladder and shell and the subscript `T`

denotes the correction for
thickness. The total actuator force is:

$$F={F}_{\text{H-F}}+{F}_{\text{T}},$$

where the strand length used in the calculation of the
Chou-Hannaford term is the variable
*l*^{*}. This force is counteracted at the
limits of extension and contraction by translational hard stops. These are modeled
as described for the Translational Hard Stop
block.

### Modeling Assumptions

There is no flow resistance between the gas entrance (port

**A**) and the interior of the actuator.There is no thermal resistance between the actuator wall (port

**H**) and the gas that it encloses.The actuator is hermetic and does not leak.

The effects of friction and inertia are ignored.

The bladder and shell are perfectly cylindrical no matter their inflation level.

The longitudinal elasticity of the bladder is ignored.

## Ports

### Conserving

## Parameters

## Extended Capabilities

## Version History

**Introduced in R2018b**