Optimum dynamic design of flexible robotic manipulator

Abstract

A closed form solution to Optimum Dynamic Design (ODD) of a flexible link Robotic Manipulator is derived considering the link to be the Euler–Bernoulli beam. The Finite Element Model (FEM) for dynamic equation is solved to maximize the stiffness and minimize the mass in the formulation of ODD procedure. The optimum pre-assigned parameters of link end deflections and the maximum payload capacity form the constraint inputs. The redundant progressive series formed as an objective function is solved by Optimal Interface Theory, briefed in the Appendix. The philosophy of Interface Theory lies in the observation that with the progression, the grip improves and the grasp diminishes under the loads of the physical world problems. For the theoretical analysis, the manipulator link is considered to be a revolute jointed with two degrees of freedom (DOF)—a deflection and a rotation at each node. Based on the requirements of better productivity and higher precision in high speed industrial robots, the platform for the maximization of stiffness, minimization of mass has been prepared to demonstrate the ODD procedure. In the ODD problem, the deflection, velocity and acceleration vectors act as grip coefficients and the stiffness, damping and the mass are considered as the grasp variables.

Appendix: Brief of interface theory

The linear redundant equation, in the general form with ‘n’ number of terms is given by

$$ \sum {a_{i} X_{i} = b} $$

(A1)

The attempt here is to find X _i , the Interface Variables, given a _i —the coefficients and b—the output. To make the task of derivation simple, (A1) is assumed with three terms as,

$$ a_{1} X_{1} + a{}_{2}X_{2} + a_{3} X_{3} = b $$

(A2)

Equation A2 after decoupling is written in the form

$$ a_{1} X_{1} + V_{1} = 0 $$

(A3)

where $V_{1} = a_{2} X{}_{2} + a_{3} X_{3} - b$ is the decoupled segment.

Equation A3 is an indeterminate equation that has infinite number of roots In the first step, the handle roots are assumed as (X ₁, V ₁) = (1, −a ₁)

The solutions to (A3) is by the process of parameterization. The infinite roots are (X ₁, V ₁) = (h, −a ₁ h)

Parameterization is the treatment of proportionating the handle roots. by an adaptor (variable) known as Interface Adaptor (h).

Hence

$$ a_{2} X_{2} + V_{2} = - a_{1} h $$

(A4)

where $V_{2} = a_{3} X_{3} - b$

Solution to (A4) is $\left( {X_{2} ,V_{2} } \right) = \left[ {\left( {1 - {\frac{{a_{1} }}{{a_{2} }}}} \right)h, - a_{2} h} \right]$

Now,

$$ a_{3} X_{3} + V_{3} = - a_{2} h $$

(A5)

The Eqs. (A3), (A4) and (A5) are the segments of Eq. (A2). If they were individually different, then the Interface Adapter can have any or different values. In this case the adapter is same for all the segments, which is kept as variable till the final segmentation and finally estimated to be the ratio of output (b) and the coefficient of the last term.

Solution to Eq. A5 is $\left( {X_{3} ,V_{3} } \right) = \left[ {\left( {1 - {\frac{{a_{2} }}{{a_{3} }}}} \right)h, - a_{3} h} \right]$

But $V_{3} = - b = - a_{3} h$

Hence the Interface Adapter is

$$ h = {\frac{b}{{a_{3} }}} $$

In the general Eq. A1, the solutions are given by $X_{1} = {\frac{b}{{a_{n} }}}$

$X_{i + 1} = \left( {1 - {\frac{{a_{i} }}{{a_{i + 1} }}}} \right) \cdot \left( {{\frac{b}{{a_{n} }}}} \right)\;{\text{for }}i = 1\;{\text{to}}\;\left( {n - 1} \right).$

It may be noted that the maximization of variables occurs when a _i are arranged in the ascending order i.e., when a _i  < a _i+1. With this arrangement it may be observed that X _i get solved to take values in the descending order i.e., X _i  > X _i+1. In the present case, the optimal solution implies to positive and real roots. For all ‘X _i ’ to be positive and real, b > 0, $\left( {1 - {\frac{{a_{i} }}{{a_{i + 1} }}}} \right) > 0$ and a _n is the highest positive coefficient in the series. This is possible when the series with positive output (b) is arranged such that, a _i  < a _i+1.

This is the condition for optimality. This is used to solve the equation of motion to arrive at ODD. By this process it may be observed that X _i  > X _i+1, which indicates that an optimal balance is struck between grasp and grip variables. The same concept is accommodated to derive the closed form solution in ODD.

Hence it is arrived at the conclusion that in this optimization theory, with the progression the grip improves and the grasp diminishes.