# Air Muscle Actuator (G)

Linear actuator with force characteristics of biological muscle

• 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

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Gas opening for pressurizing the actuator. Connect this port to the pressure source for the actuator. The pressure at this port controls the force generated by the actuator.

One of two end caps at which the actuator force is applied. Connect this port to the fixture on which the actuator is to be mounted or to the load that the actuator is to move.

One of two end caps at which the actuator force is applied. Connect this port to the fixture on which the actuator is to be mounted or to the load that the actuator is to move.

Thermal boundary condition between the gas volume of the actuator and its surroundings. Connect this port to blocks in the Thermal domain to capture heat transfer by conduction, convection, or radiation, for example, or to insulate the actuator from the environment.

## Parameters

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Distance between the end caps of the actuator in the relaxed state, when it is both unloaded and unpressurized. Inflating the actuator during simulation causes it to widen and simultaneously to shorten. The actuator is shorter than its braids, which is to say that its initial length should be less than the Unstretched braid length parameter.

Length of a braid as measured with its turns straightened and any loads removed. If the braids are elastic, their (common) length will vary with during simulation pressure and load. As the braids are wound about the longitudinal axis of the actuator, their length is always greater than the Initial actuator length parameter.

Number of turns that a braid makes about the longitudinal axis of the actuator from one end cap to the second.

Choice of whether to model the stretching of a braid. The default option, `Inelastic braids`, treats the braids as flexible but inextensible. This is the assumption implicit in the original Chou-Hannaford equation. The length of a braid is then constant during simulation.

The alternative option, `Elastic braids`, allows the braids to extend with actuator load and pressure. The length of the braid is then treated as variable during simulation. This change adds a correction to the Chou-Hannaford equation for force.

Sum of the thicknesses of the braided shell and of the bladder inside it. A finite thickness adds to the Chou-Hannaford equation for force a correction term. (In the original Chou-Hannaford equation, the shell and bladder are assumed to be infinitely thin.)

Area normal to flow at the gas port of the actuator.

Number of braids in the shell surrounding the bladder. As the braids are each a bundle of intertwined strands, the number of braids is not generally the same as the number of strands.

#### Dependencies

This parameter is active when the Braid stretching option is set to ```Elastic Braids```.

Number of strands in a braid. The product of this number and the Total number of braids parameter gives the total number of strands in the shell.

#### Dependencies

This parameter is active when the Braid stretching option is set to ```Elastic Braids```.

Average diameter of a strand throughout its length.

#### Dependencies

This parameter is active when the Braid stretching option is set to ```Elastic Braids```.

Young's modulus of elasticity for the material of the strands. This parameter determines the degree to which the strands extend under pressure.

#### Dependencies

This parameter is active when the Braid stretching option is set to ```Elastic Braids```.

Pressure condition in the immediate surroundings of the actuator. The gauge pressure in the actuator is obtained relative to this pressure. Select `Atmospheric pressure` to use the atmospheric value specified in the Two-Phase Fluid Properties (2P) block or ```Specified pressure``` to use a different pressure.

Absolute pressure in the immediate surroundings of the actuator. The actuator contracts when the gauge pressure at port A (measured against this value) is greater than zero.

#### Dependencies

This parameter is active when the Environment pressure specification option is set to ```Specified pressure```.

Stiffness coefficient for the contact force at the limits of the actuator's range of motion. The contact force keeps the actuator from extending or contracting beyond its physical limits. The contact force is modeled using a spring-damper system. The spring element adds a restorative force that pulls the end caps back once they reach either limit of motion.

For more information on the force model, see the Translational Hard Stop block.

Damping coefficient for the contact force at the limits of the actuator's range of motion. The contact force keeps the actuator from extending or contracting beyond its physical limits. This force is modeled using a spring-damper system. The damper adds a viscous force to the end caps once they reach a limit of motion, causing them to slow to a stop.

For more information on the force model, see the Translational Hard Stop.

## Version History

Introduced in R2018b