A piezoelectric model based multi-objective optimization of robot gripper design

Abstract

The field of robotics is evolving at a very high pace and with its increasing applicability in varied fields, the need to incorporate optimization analysis in robot system design is becoming more prominent. The present work deals with the optimization of the design of a 7-link gripper. As actuators play a crucial role in functioning of the gripper, the actuation system (piezoelectric (PZ), in this case) is also taken into consideration while performing the optimization study. A minimalistic model of PZ actuator, consisting different series and parallel assembly arrangements for both mechanical and electrical parts of the PZ actuators, is proposed. To include the effects of connector spring, the relationship of force with actuator displacement is replaced by the relation between force and the displacement of point of actuation at the physical system. The design optimization problem of the gripper is a non-linear, multi modal optimization problem, which was originally formulated by Osyczka (2002). In the original work, however, the actuator was a ‘constant output-force actuator model’ providing a constant output without describing the internal structure. Thus, the actuator model was not integrated in the optimization study. Four different cases of the PZ modelling have been solved using multi-objective evolutionary algorithm (MOEA). Relationship between force and actuator displacement is obtained using each set of non-dominated solutions. These relationships can provide a better insight to the end user to select the appropriate voltage and gripper design for specific application.

Appendices

Appendix A: Problem formulation

1.1 A.1 Design variables

In the optimization process, seven design variables have been considered (same as original study (Osyczka 2002)), consists of link lengths, offsets and joint angle: x=(a, b, c, e, f, l, δ)^T, where a, b, c denote the link lengths, e, f, l denote link offsets and the joint angle between elements b and c is δ. A sketch of the gripper design is shown in Fig. 1.

1.2 A. 2 Problem formulation

The multi-objective problem for the optimization study can be formulated by integrating the the actuator modelling part with the original problem formulation. In this section, the problem formulation is discussed in detail.

1.2.1 A.2.1 Force analysis

In a two dimensional mechanism, bending of the link attached to actuator is avoided as the actuator can undergo translational motion to adjust the stresses. Hence, this link can be treated as a truss element. The force balance on link1 is as shown in Fig. 36.

Fig. 36

Free Body Diagram (FBD) of link1 of robot gripper. The actuator force, P, can be divided into two equal forces acting separately on point A and F (point F is shown in Fig. 1, which is a mirror image of point A). RR is the reaction force at point B

The structure is in static equilibrium, therefore equating horizontal forces to obtain

$$\begin{array}{@{}rcl@{}} \frac{P}{2} &=& RR\times\cos(\alpha). \end{array} $$

(35)

where RR is the reaction force on link a and the actuating force applied by the actuator on the gripper is given by P.

Rearranging above equation

$$\begin{array}{@{}rcl@{}} RR &=& \frac{P}{2\times\cos\alpha}. \end{array} $$

(36)

In Fig. 37, link 2 and 3 are shown with point C hinged. Taking moment equilibrium at C

$$\begin{array}{@{}rcl@{}} \sum M_{xC} &=& 0, \end{array} $$

(37)

$$\begin{array}{@{}rcl@{}} RR\times\sin(\alpha+\beta)\times b &=& F_{k}\times c, \end{array} $$

(38)

$$\begin{array}{@{}rcl@{}} F_{k} & = & \frac{RR\times\sin(\alpha+\beta)\times b}{c}, \end{array} $$

(39)

$$\begin{array}{@{}rcl@{}} F_{k} &=& \frac{P\times b\sin(\alpha+\beta)}{2\times c\cos\alpha}. \end{array} $$

(40)

Fig. 37

Free Body Diagram (FBD) of link 2 of robot gripper

1.2.2 A.2.2 Link geometry analysis

From Pythagoras theorem, in Δ ACD (Fig. 38), we get

$$\begin{array}{@{}rcl@{}} g^{2} &=& (l-z)^{2}+e^{2}, \\ g &=& \sqrt{(l-z)^{2}+e^{2}}. \end{array} $$

Using cosine law in Δ ABC

$$\begin{array}{@{}rcl@{}} \cos(\alpha - \phi) &=& \left( \frac{a^{2}+g^{2}-b^{2}}{2\times a \times g}\right). \end{array} $$

Fig. 38

Geometrical construction for the gripper mechanism. In Δ ACD, g is the hypotenuse distance between point A and point C and ϕ is the angle between AC and AD

Solving the above equation for α, we get,

$$\begin{array}{@{}rcl@{}} \alpha &=& \arccos (\frac{a^{2}+g^{2}-b^{2}}{2 \times a\times g})+\phi. \end{array} $$

Again, from cosine law in Δ ABC, for angle (β+ϕ)

$$\begin{array}{@{}rcl@{}} \cos(\beta + \phi) &=& \left( \frac{b^{2}+g^{2}-a^{2}}{2 \times b \times g}\right). \end{array} $$

Solving the above equation for β, we get,

$$\begin{array}{@{}rcl@{}} \beta &=& \arccos \left( \frac{b^{2}+g^{2}-a^{2}}{2 \times b \times g}\right)-\phi. \end{array} $$

Also, from Δ ACD we can get

$$\begin{array}{@{}rcl@{}} \phi &=& \arctan\left( \frac{e}{l-z}\right). \end{array} $$

1.3 A.3 Constraints

The gripper configuration is physically constrained at various points, for obtaining the required movement. These physical restrictions can be represented in the formulation as the problem constraints.These formulated constraints are multi-modal and non-linear in nature. The formulated constraints for the study are discussed in detail as following:

• 1. At the maximum actuator displacement, the distance between both ends of the gripper should be less than minimal dimension of the object, for proper gripping.

  $$\begin{array}{@{}rcl@{}} g_{1}(\mathbf x) &=& Y_{\min}-y(\mathbf x,Z_{\max})\geq 0. \end{array} $$

  (41)

  in the above equation, y(x,z)=2×[e+f+c× sin(β+δ)] denotes the distance between two ends of the gripper and Y _min is the minimal dimension of the object to be gripped. The parameter Z _max corresponds to the maximum actuator displacement.

• 2. The distance between gripper ends for maximum actuator displacement (Z _{m a x} ) should be greater than zero:

  $$\begin{array}{@{}rcl@{}} g_{2}(\mathbf x) &=& y(\mathbf x,Z_{\max})\geq0. \end{array} $$

  (42)

• 3. When the actuator displacement is zero, the distance between two ends of the gripper should be greater than the maximum dimension object to be gripped.

  $$\begin{array}{@{}rcl@{}} g_{3}(\mathbf x) &=& y(\mathbf x,0)-Y_{\max}\geq 0. \end{array} $$

  (43)

  where Y _max denotes the maximum dimension of the object to be gripped.

• 4. The maximum range of the displacement of the gripping ends of the gripper should be greater than or equal to the distance between the gripping ends corresponding to zero actuator displacement:

  $$\begin{array}{@{}rcl@{}} g_{4}(\mathbf x) &=& Y_{G}-y(\mathbf x,0)\geq 0. \end{array} $$

  (44)

  where Y _G is the maximum displacement that gripper ends can attain.

• 5. Geometric constraints for the gripper mechanism can be given as:

  $$\begin{array}{@{}rcl@{}} g_{5}(\mathbf x) &=& (a+b)^{2}-l^{2}-e^{2}\geq0. \end{array} $$

  (45)

  The geometric interpretation of constraint g ₅(x) is shown in Fig. 39.

  $$\begin{array}{@{}rcl@{}} g_{6}(\mathbf x) &=& (l-Z_{\max})^{2}+(a-e)^{2}-b^{2}\geq 0. \end{array} $$

  (46)

  Fig. 39

  

  Geometric illustration of constraint g ₅(x) for robot gripper design

  The geometric interpretation of constraint g ₆(x) can be seen from Fig. 40.

  $$\begin{array}{@{}rcl@{}} g_{7}(\mathbf x) &=& l-Z_{\max}\geq0. \end{array} $$

  (47)

• 6. Minimum force to grip the object should be greater than or equal to chosen limiting gripping force:

  $$ g_{8}(\mathbf x) = \min_{z} F_{k}(\mathbf x,z)-FG\geq 0, $$

  (48)

  where FG is the assumed minimal griping force.

Fig. 40

Geometric illustration of constraint g ₆(x)

1.4 A.4 Objective functions

The objective functions for an optimized gripper design, have to be formulated based on link geometry analysis. The formulated functions used in this optimization study are as follows:

• 1. For any gripper mechanism, the most crucial aspect is to ensure a steady firm grip on the object to be gripped. Hence, the first objective function must be formulated in a way such that this requirement is addressed. We have assumed the difference between the maximum and minimum value of gripping force that will be applied on the object during the whole operation, as our first objective function.

  $$ F_{1}(\mathbf x) = \max_{z} F_{k}(\mathbf x,z)- \min_{z} F_{k}(\mathbf x,z). $$

  (49)

• 2. One of the most desirable characteristic in any mechanism, is to have a low energy consumption. In a gripper mechanism, lower power consumption can be ensured by having a higher force transformation ratio. Hence, the second objective function for the present study is formulated as to maximize the force transformation ratio of the mechanism. Force transformation ratio in the initial study was defined as the ratio between the applied actuating force P and the resulting minimum gripping force at the tip of link c (Osyczka (2002)):

  $$ F_{2}(\mathbf x) = \frac{P}{\min_{z} F_{k}(\mathbf x,z)}. $$

  (50)

  However, as actuator modelling is taken in consideration in the present study, the actuator force P is no longer a constant and varies with actuator displacement. The second modified objective can be defined as

  $$ F_{2}(\mathbf x) = \max_{z}\left( \frac{P(\mathbf x,z)}{F_{k}(\mathbf x,z)}\right). $$

  (51)

Appendix B: Previous results

The results of the innovization study done by Datta and Deb (2011) are presented in this appendix. For better understanding and interpretation of the results, corresponding link lengths, link offsets and joint angle are also shown along with the plots. Figure 41 shows the relationships between link lengths a,b and offsets e,l with force transformation ratio (F ₂). It is clear from the figure that a and b must be fixed at 250 mm, e must be 100 mm where as l should be 0 mm.

Fig. 41

Variation of link length a,b and offsets e,l with force transformation ratio

Link length c varies with F ₂ as a straight line with slope = 243.6 and intercept = 0, as shown in Fig. 42. Figures 43 and 44 shows that f must be fixed at 37 mm and δ should be 1.72 radian.

Fig. 42

Variation of offset c with force transformation ratio

Fig. 43

Variation of offset f with force transformation ratio

Fig. 44

Variation of joint angle δ with force transformation ratio