Exercises 22: Optimization

In these exercises we'll take a close look at inverse kinematics, through the lens of continuous optimization. In fact, you'll implement a scheme like this one "for real" in A4. Hopefully these exercises will build up your confidence in understanding the concepts behind the A4 tasks, and provide a bit deeper understanding of what's going on, how to improve your IK solver---and why things might fail!

Build a Kinematic Chain Consider a linear kinematic chain connecting three vertices p0, p1, p2, and angles of rotation theta0, theta1, as depicted above. Ultimately, this chain will try to "reach" for a given goal point x, i.e., it will try to make the final vertex close to x. For now, just give expressions for the locations of q0, q1, and q1 as functions of the original positions p0, p1, p2 and the current angles theta0, theta1.

Define the Energy

Write an expression for an energy Phi of the angles theta0 and theta1 that is minimized when the chain is as close as possible to reaching the goal. You may also use other quantities in this [removed]such as the points p or q), as long as the dependence of q on the angles theta0 and theta1 is clearly indicated.

Give an expression for the gradient of the energy Phi with respect to the angles theta0 and theta1.