E19 numerical methods for engineering applications spring

Question

E19: Numerical Methods for Engineering Applications Spring 2016 - HOMEWORK 8

1. Jacobian-based IK

The end effector position of one of the legs of the LittleDog quadruped robot (diagrammed above) is determined by the forward kinematics (FK) equation

Given a desired position p_d = (x_d, y_d, z_d) of the end effector, we want to solve the inverse kinematics (IK) problem to determine the joint angles θ such that p(θ) = p_d.

To answer the questions below, it will be useful to examine and modify the ld_leg.py code from the course website.

a. Why is IK generally more complicated than FK?

b. Although we can sometimes solve the IK problem analytically (for instance, for the LittleDog leg kinematics), state some reasons why we might prefer to implement it numerically.

c. A singular configuration of a robotic manipulator is one in which the rank of J drops below n. Show that the LittleDog leg is singular at θ = 0.

d. Explain why singular configurations must be avoided when using Gauss-Newton to solve the IK problem.

e. Extend ld_leg.py to use the Gauss-Newton algorithm to solve for the joint angles θ to achieve a desired position of p_d = (1, 2.5, -6).

2. Lagrange Multipliers

You have been tasked with building an aluminum pressure vesel in the shape of a capped cylinder, as shown below:

The vessel consists of a cylindrical section of length l and radius r, capped by two hemispheres of radius r. The material costs for the hemispheres are $200 per square meter for the endcaps, and $100 per square meter for the cylinder. Use the Lagrange multiplier technique to minimize the cost of building the pressure vessel subject to the constraint that it encloses a volume of exactly 10π cubic meters.

a. Write down the objective function f(l, r) to be minimized, in terms of l and r. It should be equal to the surface area of each type of section, weighted by the appropriate cost.

b. Write down the constraint function h(l, r) to set equal to zero. It should be the total volume of all sections minus the desired volume.

c. Use the Lagrange multiplier method to compute the gradient of the function L(l, r, λ) = f(l, r) + λh(l, r) and solve for ∇L = 0.

You may find it useful to write a program to help you solve this problem, or at least compute gradients for part c above. The sympy package can be very helpful in computing derivatives, see ld_leg.py for examples. If you do write a program to help you, please turn it in online as well.

Attachment:- Assignment.rar