Linear Approximation for Robot Arm Movement

p {margin: 0; padding: 0;} .ft00{font-size:20px;font-family:Arial;color:#000000;} .ft01{font-size:18px;font-family:ArialMT;color:#000000;} .ft02{font-size:18px;line-height:22px;font-family:ArialMT;color:#000000;} 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 thetaderivative 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 valueof pi over 4 plus delta theta. The result is approximately x of 1, pi over 4 plus the change inx, 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 betweenthe 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 Lminus delta theta.