Design and Development of an Adaptive Robotic Gripper

Abstract

In this paper, the design and development of an adaptive gripper are presented. Adaptive grippers are useful for grasping objects of varied geometric shapes by wrapping fingers around the object. The finger closing sequence in adaptive grippers may lead to ejection of the object from the gripper due to any unbalanced grasping force and such grasp failure is common for lightweight objects. Designing of the proposed gripper is focused on ensuring a stable grasp on a wide variety of objects, especially, lightweight objects (e.g., empty plastic bottles). The proposed actuation mechanism is based on movable pulleys and tendon wires which ensure that once a link stops moving, the other links continue to move and wrap around the object. Further, optimisation is used to improve the design of the adaptive gripper and the optimised gripper has been developed using 3D printing. Finally, validation is done by executing object grasping on common household objects using an industrial robot fitted with the developed gripper.

Appendix

1.1 A. Kinematics of the Gripper

See Fig. 26

Fig. 26

Kinematics schematic with parameter annotations and frame assignments for the two-finger gripper

The kinematics for the fingers is determined using the Denavit-Hartenberg (DH) notation. With the convention of DH frame assignments, the reference frames are attached to the links of the fingers, where the $Z$–axis of link frame coincides with the joint axis as shown in Fig. 26. The frames are denoted as [${X}_{k}$ ${Y}_{k}$ ${Z}_{k}$], where the subscript $\{\mathrm{k}\}$ is the link number starting with $\{k=0\}$ for the fixed knucle link. The DH parameters for both the fingers are given in Table

Table 2 DH parameters for both fingers of the two-finger gripper

2, where the four DH parameters denoted as ${\mathrm{\alpha }}_{\mathrm{k}-1}$, ${\mathrm{a}}_{\mathrm{k}-1}$, ${\mathrm{d}}_{\mathrm{k}}$ and ${\uptheta }_{\mathrm{k}}$ are for the interconnection between link $k-1$ and link$k$.

The forward kinematics relationship between frame $\{0\}$ and the fingertip frame {3} is determined by using the DH parameters given in Table 2 as follows.

$${}_{3}{}^{0}T=\left[\begin{array}{cc}\begin{array}{cc}{r}_{11}& {r}_{12}\\ {r}_{21}& {r}_{22}\end{array}& \begin{array}{cc}{ r}_{13}& { p}_{x}\\ {r}_{23}& {p}_{y}\end{array}\\ \begin{array}{cc}{r}_{31}& {r}_{32}\\ 0& 0\end{array}& \begin{array}{cc}{ r}_{33} & {p}_{z}\\ 0& 1\end{array}\end{array}\right]$$

(29)

where, the matrix elements for the finger (F1) are as follows

$${p}_{x}= \mathrm{cos}\left({q}_{11}+{q}_{12}\right){L}_{3}-\mathrm{sin}({q}_{11}+{q}_{12}){D}_{3}+\mathrm{cos}\left({q}_{11}\right){L}_{1}+{L}_{0}$$

$${p}_{y}= 0$$

$${p}_{z}= \mathrm{sin}\left({q}_{11}+{q}_{12}\right){L}_{3}+\mathrm{cos}({q}_{11}+{q}_{12}){D}_{3}+\mathrm{sin}\left({q}_{11}\right){L}_{1}$$

$${r}_{11}= \mathrm{cos}({q}_{11}+{q}_{12})$$

$${r}_{13}= -\mathrm{sin}({q}_{11}+{q}_{12})$$

$${r}_{31}= \mathrm{sin}({q}_{11}+{q}_{12})$$

$${r}_{33}= \mathrm{cos}({q}_{11}+{q}_{12})$$

$${r}_{22}=1$$

$${r}_{12}={r}_{21}={r}_{23}={r}_{32}=0$$

1.2 B. Grasp Quality Metric

Here, a brief formulation is given to find the grasp quality while more details can be best found in [37]. The contact friction is modelled by approximating the friction cones with eight-sided pyramids having a unit length and a half angle of ${tan}^{-1}{\mu }_{s}$, where ${\mu }_{s}$ is the static friction coefficient as shown in Fig. 

Fig. 27

Approximation of the friction cone with an eight-sided pyramid

27.

Then the contact force ${\varvec{f}}$ transfer to the object through contact is the linear combination of the vectors used to approximate the eight sides of the pyramid.

$${\varvec{f}}={\sum }_{j}^{m}{\alpha }_{j}{{\varvec{f}}}_{j}$$

(30)

where, ${\alpha }_{j}>0 , {\sum }_{j}^{m}{\alpha }_{j}=1$ and number of sides of the pyramid $m=8$

The contact wrench can be defined as the six-dimensional vectors formed by contact forces and torques at the contact point and is given as follows.

$${{\varvec{w}}}_{i,j}=\left(\begin{array}{c}{{\varvec{f}}}_{i,j}\\ \lambda ({{\varvec{d}}}_{i}\times {{\varvec{f}}}_{i,j})\end{array}\right)$$

(31)

where, ${{\varvec{f}}}_{i,j}$ is the $j$-th component of the friction cone at the $i$-th point of contact. ${{\varvec{d}}}_{i}$ is the distance vector from torque origin to the $i$-th point of contact. The scalar $\lambda$ enforces the constraint $\Vert {\varvec{\tau}}\Vert \le \Vert {\varvec{f}}\Vert$.

Now, assembling all the contact wrenches gives the convex hull or the polygon as follows.

$$W=ConvexHull\left({\bigcup }_{i}^{n}\{{{\varvec{w}}}_{i,1}, {{\varvec{w}}}_{i,2},\dots ..,{{\varvec{w}}}_{i,m}\}\right)$$

(32)

The volume $v$ of the hull gives an invariant measure of grasp quality subjected to the origin of the wrench space lies within the convex hull. This quality measure is used as a basis for the optimisation of the link dimensions.

1.3 C. Design of Position-Based Impedance Controller

Grasping task involves a physical interaction between the finger and the object. Impedance control is preferred for a task that involves physical contact with the environment. Position-based impedance control can regulate positions and contact forces by regulating the dynamic behaviour of the system as a whole. The independent control of the two joints is not possible due to underactuated nature of the system. Here, the joints of the finger are controlled in joint space. The impedance controller is designed on top of a PID block as shown in Fig. 

Fig. 28

Block diagram of the position-based impedance controller for the gripper

28.

Let ${\mathrm{q}}_{\mathrm{d}}\in {\mathrm{R}}^{\mathrm{n}}$ be the desired joint displacements. To achieve the desired joint trajectories and required torque for generating contact forces, the desired joint displacements are modified to a new set of joint displacement ${\mathrm{q}}_{\mathrm{s}}\in {\mathrm{R}}^{4}$. These modified displacements and actual joint displacements are used to compute displacement errors. The following PID feedback control law is used to generate the actuating torques.

$$\uptau ={\mathrm{K}}_{\mathrm{p}}\left({\mathrm{q}}_{\mathrm{s}}-\mathrm{q}\right)-{\mathrm{K}}_{\mathrm{v}}\dot{\mathrm{q}}+{\mathrm{K}}_{\mathrm{i}}\int \left({\mathrm{q}}_{\mathrm{s}}-\mathrm{q}\right)\mathrm{d\tau }$$

(33)

where${\mathrm{K}}_{\mathrm{p}}$, ${\mathrm{K}}_{\mathrm{v}}$, ${\mathrm{K}}_{\mathrm{i}}$ are proportional, derivative and integral constants, respectively.

The modified joint displacements and the required joint displacements are related as follows.

$${\mathrm{q}}_{\mathrm{s}}={\mathrm{q}}_{\mathrm{d}}+{\uptau }_{\mathrm{c}}/{\mathrm{K}}_{\mathrm{s}}$$

(34)

where ${\mathrm{K}}_{\mathrm{s}}$ is the impedance stiffness.