Using Torsion Springs Linearly as Elastics
Torsion springs work similar to extension and compression springs in the sense they apply a linear force in relation to their angular extension. However, utilizing this angular extension in a linear manner may result in a non-linear relationship between extension and spring force.
Imagining that a string is attached to the two ends of a torsional spring and pulled apart, the forces are as shown:
The spring force and metal tension force components combine to oppose the outside pulling force. It is important to observe that as theta gets larger, the tension force of the metal does more opposing on the pull force than the spring force does. When the pull force reaches a large value and the spring extends all the way, the spring force will have no component acting against the pull force, and all the force from the pull will be opposed by the tension of the metal.
Knowing that the Spring force is equivalent to K(theta), we can calculate the overall spring force as a function of the linear extension (x), which is the factor that we will change in the situation that the spring is placed between a timing belt that is placed around a joint and motor pulley. In a linear relationship, the spring force is equivalent to kx, where k is the spring constant. In our case, we want k to be a function of x rather than a constant. Solving for the total opposing force created by the spring on the pull on the end, by both the tension in the metal and the spring force, we calculate the the linear force "constant" is:
This function shows us that torsional springs are a viable option that would be worthwhile to implement and test to create a variable stiffness actuator. We can observe that this actuator would have a large range of stiffness as it extends from a weak spring to a near infinite force applied by the tension of the metal of the spring itself.
When implementing, we will first want to confirm our calculation by collecting data and graphing it in comparison to our theoretical equation for the spring "constant." In order to prevent the spring to hang inverted because of it's off-center COG it would be beneficial to attach the torsion springs in pairs that are arranged to create what looks like a ladder. This method also ensures the the spring is being pulled linear from its two ends as the natural motion of the pair of springs attached in a ladder form is along a line. As the extension motor pulls the springs to their max extension, it will experience a large torque applied onto it, of a value that can only be calculated because in reality the spring cannot be held at full extension. Therefore we will need to calculate the max force of the springs when we pull the belt with the elastics to their "max" extension.