# Notes on [Customizable Three-Dimensional Printed Origami Soft Robotic Joint With Effective Behavior Shaping for Safe Interactions](https://ieeexplore.ieee.org/abstract/document/8481372/keywords#keywords)

#### Author:- [Uddesh Tople](https://github.com/uddeshtople)

###### Tags : `soft robotics` `SoRA` `origami` `chamber`

## Brief Outline
A Soft robotic joint with rotary actuator was proposed in this paper. Proposed approach facilitates a fully customizable joint design towards the desired force capability and motion range.

The above image shows the design of pneumatic soft Robotic rotary actuator.

## Geometric design
The proposed Soft Origami Rotary Actuator (SoRA) joint consists of two antagonistic airtight origami chambers,

The maximum and minimum central angles formed are :-

$\theta_{min}$ = $\frac{2nt}{r_1}$
$\theta_{max}$ = $\frac{2na}{r_2}$ = $\frac{2nc}{r_1}$

From the figure above we get :-
$r_2$ - $r_1$ = $b$
$\frac{r_1}{r_2}$ = $\frac{c}{a}$

From above 2 equations,
$r_1$ = $\frac{bc}{(a-c)}$ $\&$ $r_2$ = $\frac{ab}{(a-c)}$

Therefore,
$\theta_{min}$ = $\frac{2nt(a-c)}{bc}$ $\&$ $\theta_{max}$ = $\frac{2n(a-c)}{b}$

---

### Features added in design to improve air-tightness and performance


**Motion constraint of origami chamber:-** With a large external load opposing the actuation torque, radial bulging occurs. Rigid pins with bearings were added to connect the chamber to the rigid skeleton, as bulging constraints.

**Connection buckles:-** To ensure bidirectional force interaction between chamber and rigid skeleton.

**Pneumatic connection to chamber:-** To ensure air-tightness and oppose air inlet at high working pressure which may cause structure distortion.

## Modelling

### Actuator Modelling
The pressure difference between the two origami chambers causes rotation of the rigid plate with torque T.

Generated Torque in terms of torque generated by chamber1 & chamber2 is given by:-
$T$ = $T_1$ + $T_2$
$T_1$ = $-k \alpha$ $+$ $\int_{r_1}^{r_2}(P_1-P_2)hrdr$
$T_2$ = $-k \alpha$ $-$ $\int_{r_1}^{r_2}(P_2-P_0)hrdr$

$k$ is inherent stiffness constant.
$P_o$ is ambient temperature

Subsituting
$T$ = $-2k \alpha + \frac{h({r_2}^2-{r_1}^2)}{2}$ $(P_1 - P_2)$

Joint stiffness constant $K$ is given by:-
$K=\frac{dT}{d \alpha}= 2k - \frac{h({r_2}^2-{r_1}^2)}{2}$ $\frac{\partial(P_1-P_2)}{\partial\alpha}$

Assuming Adiabatic Ideal gas:-
$P_1 = m_1 RT_a/V_1$
$P_2 = m_2 RT_a/V_2$

$V_1 = \theta_1h({r_2}^2-{r_1}^2)$
$V_2 = \theta_2h({r_2}^2-{r_1}^2)$

$\theta_1 = \pi + \alpha$
$\theta_2 = \pi - \alpha$

Combining above 3 equations, we get:-
$\frac{\partial P_1}{\partial \alpha} =\frac{\partial P_1}{\partial V_1} \frac{\partial V_1}{\partial \alpha} = -\gamma\frac{P_1}{\pi + \alpha}$
$\frac{\partial P_2}{\partial \alpha} =\frac{\partial P_2}{\partial V_2} \frac{\partial V_2}{\partial \alpha} = -\gamma\frac{P_2}{\pi - \alpha}$
where $\gamma$ = 1.4 for air

subsituting it in $K$
$K = 2k+\gamma\frac{h({r_1}^2-{r_0}^2)}{2(\pi^2-\alpha^2)}[\pi(P_1+P_2)+\alpha(P_2-P_1)]$

## Control and Behaviour Shaping


* One rotary encoder and two pressure sensor is used for feedback of joint angle, velocity and pressure with kalman filters.
* Proportional valve is provided at each chamber for high-quality air flow.
* A low level proportional control for internal chamber adopted for high level stiffness and angle control with PID feedback.
* The outer loop time constant is much larger than that of the inner loop as angle rotation in joint take time to accumulate.
* Without external disturbances, PID control is sufficient for following desired trajectories, But under fast impacts, PID control will not respond fast sufficiently.
* Therefore, joint will follow its passive stiffness set by the preset pressures and respond spontaneously.
* Under slow disturbances, the joint will be subject to both PID and passive stiffness.

*combining passive stiffness presetting and active PID cascade control, the proposed SoRA joint exhibits distinctive characteristics for impact response and disturbance rejection.*

## Conclusion
The proposed SoRA joints are not only strong, precise and responsive but they can also be PID-controlled along with passive stifness to follow fast-changing desired trajectories.