Robot Arm Solution
Now let us begin our linear approximation. We will think of x as a function of L and theta. We will write x of L, theta as L plus root 2 cosine of theta. I can then take the derivative of this function, so the L derivative is just 1. And the theta derivative is equal to minus root 2 times sine of theta. I can approximate the value of x at 1 plus delta L by looking at what happens to the value of pi over 4 plus delta theta. The result is approximately x of 1, pi over 4 plus the change in ×, which is x sub L of 1, pi over 4 delta L plus x sub theta of 1, pi over 4 delta theta. This is the change in x, which we labeled delta x in our graph. This is the change between the initial situation when L was 1, and theta was pi over 4, and the new situation. The formula for delta x is here. Let’s just plug in. What is x sub L of 1 comma pi over 4? Well, x of L is always 1. So there we have 1. And what’s x sub theta of 1 comma pi over 4? Plugging in. It happens to be the minus 4. Now we’ve got this equation- delta x is delta L minus delta theta. x(L, O)=L+sqrt2cosθ X_L=1 X_θ=-sqrt2sinθ x(1+ΔL,pi/4+Δθ)~~x(1,pi/4)+X_θ(1,pi/4)ΔL+X(1,pi/4)Δθ }Δx Δx=ΔL-Δθ
