Task-space impedance control of a parallel Delta robot using dual quaternions and a neural network

Abstract

The Delta robot, widely used in fast pick-and-place applications with pure position control, is a parallel kinematic chain with three rotational inputs resulting in three pure translations at the end-effector. This paper proposes a complete task-space impedance control with inverse dynamics to give this robot compliant behavior, enabling it to be used in tasks involving physical interaction. For that purpose, the well-known usage of dual quaternion algebra for kinematics modeling is novelly integrated with a neural network to compose a compact representation for the forward kinematics function, that is singularity-free and suitable for real-time calculation. This network computes the forward kinematics more than 150 times faster than a numeric equation solving algorithm, with an average estimation error of less than 0.5 mm. The proposed algorithm is implemented in a rigid body simulator, and the performance of the complete system is analyzed.

Appendix: Intermediate angles for the Delta robot

This appendix describes how to compute the two intermediate angles for each articulated arm of the Delta robot, $\theta _{2i}$ and $\theta _{3i}$, as shown in Fig. 7. Therefore, 6 angles must be computed, starting with the definition of the Cartesian coordinates of each connection point ($c_i$) between the end-effector and the parallelogram arms, relative to the frame of the corresponding rotational joint.

$$\begin{aligned} c_{1x}= & {} x + h - r \end{aligned}$$

(38)

$$\begin{aligned} c_{1y}= & {} y \end{aligned}$$

(39)

$$\begin{aligned} c_{2x}= & {} - \dfrac{x}{2} + \dfrac{\sqrt{3}}{2} y + h - r \end{aligned}$$

(40)

$$\begin{aligned} c_{2y}= & {} - \dfrac{\sqrt{3}}{2} x - \dfrac{y}{2} \end{aligned}$$

(41)

$$\begin{aligned} c_{3x}= & {} - \dfrac{x}{2} - \dfrac{\sqrt{3}}{2} y + h - r \end{aligned}$$

(42)

$$\begin{aligned} c_{3y}= & {} \dfrac{\sqrt{3}}{2} x - \dfrac{y}{2} \end{aligned}$$

(43)

Fig. 7

Intermediate angles of the Delta robot, shown on left-side and front views. The position of the connection point $c_i$ between the end-effector and the parallelogram arm is also highlighted

Intermediate angles are then computed with the following two relations. The default contradomain of $\cos ^{-1}$, from 0 to $\pi$, is compatible with the definition depicted in Fig. 7.

$$\begin{aligned} \theta _{3i}= & {} \cos ^{-1} \left( \dfrac{c_{iy}}{b} \right) \end{aligned}$$

(44)

$$\begin{aligned} \theta _{2i}= & {} \cos ^{-1} \left( \dfrac{c_{ix}^2 + c_{iy}^2 + z^2 - a^2 - b^2}{2ab \sin {\theta _{3i}}} \right) \end{aligned}$$

(45)

After computing $\theta _{2i}$ and $\theta _{3i}$ for each articulated arm, it is possible to compute both intermediate Jacobian matrices, as described in Sect. 3.2 of the main manuscript.