Application of neural network training in forward kinematics simulation for a novel modular hybrid manipulator with experimental validation

Abstract

This contribution addresses forward kinematic solution of modular hybrid manipulator which includes two similar Stewart mechanisms in serial form, known as 2-(6UPS) manipulator. First, using geometrical and vectorial analysis, mathematical model of kinematic analysis for 2-(6UPS) is extracted. Due to highly nonlinear characteristic of kinematic model of 2-(6UPS), which is related to complicated configuration of manipulator, forward kinematic solutions of mechanism is so difficult. Therefore, we proposed artificial neural network based on wavelet analysis to resolve forward kinematics of 2-(6UPS). Also, we used wavelet neural network (WNN) to approximate specific trajectory in circle oscillates in z direction and spiral with semi-cardioid base paths for mid and upper platforms movement respectively. Comparison between the results of proposed network and closed form solution of kinematics for 2-(6UPS) shows proper performance of proposed network in $<$1 % error. Also, WNN results are verified by experimental results which are obtained by image processing.

Appendix

In this paper, we specified two different trajectories for centre point of moving (mid and upper) platforms. The mentioned trajectories are included equations for displacement of $(x, y ,z ,\alpha , \beta , \gamma )$ in time (t) for centre point of moving platforms.

• Mid plate:

  Trajectory of mid platform is oscillating circle as follows:

  $$\begin{aligned} x^{\mathrm{m}}(t)= & {} 200 \sin \left( \frac{\pi }{6.3}t\right) \cos \left( \frac{\pi }{6.3}t\right) \end{aligned}$$

  (31)
  $$\begin{aligned} y^{\mathrm{m}}(t)= & {} 200 \sin ^{2}\left( \frac{\pi }{6.3}t\right) \end{aligned}$$

  (32)
  $$\begin{aligned} z^{\mathrm{m}}(t)= & {} 20 \sin (4t) \end{aligned}$$

  (33)
  $$\begin{aligned} \alpha ^{\mathrm{m}}(t)= & {} \beta ^{\mathrm{m}}(t)=\gamma ^{\mathrm{m}}(t)=0 \end{aligned}$$

  (34)

• Upper plate:

  Trajectory of upper platform is spiral with semi-cardioid base as follows:

  $$\begin{aligned} x^{\mathrm{u}}(t)= & {} 150\sin \left( \frac{t}{2}\right) \cos \left( \frac{\pi }{2} \cos \left( \frac{t}{2}\right) \right) \end{aligned}$$

  (35)
  $$\begin{aligned} x^{\mathrm{u}}\left( t\right)= & {} 150\sin \left( \frac{t}{2}\right) \sin \left( \frac{\pi }{2} \cos \left( \frac{t}{2}\right) \right) \end{aligned}$$

  (36)
  $$\begin{aligned} x^{\mathrm{u}}(t)= & {} 15t \end{aligned}$$

  (37)
  $$\begin{aligned} \alpha ^{\mathrm{u}}(t)= & {} 0 \end{aligned}$$

  (38)
  $$\begin{aligned} \beta ^{\mathrm{u}}(t)= & {} -10 \sin (t) \end{aligned}$$

  (39)
  $$\begin{aligned} \gamma ^{\mathrm{u}}(t)= & {} 5 \sin (t/2) \end{aligned}$$

  (40)