# Does it take energy to move something in a circle?

First, you are not equating the work done correctly. This is a good physics lesson. Please understand your equations before you use them. Blindly plugging in numbers will not work out. The equation you give is only true for motion in one dimension and with a constant force. Plugging in $0$ for displacement is not correct here. In general you need to look at infinitesimal displacements $\text d\mathbf x$ and calculate the work $\text d W=\mathbf F\cdot\text d\mathbf x$, then integrate (add up) the total work.

Now, I am assuming the ball starts and stops at rest. Therefore, the arm does work to increase the ball's speed, and then it does the same amount of negative work to bring it to rest. So the net work *is* $0$, but it is because the total change in kinetic energy is $0$ (since $W=\Delta K$), not because the displacement is $0$ around the circle.

Now, this is not the same thing as the robot using something like a battery. The robot (neglecting friction) has to apply forces to change the speed, and this requires power from the power supply. Just imagine yourself doing the action of the robot. You will need to exert effort to get the ball (and yourself) moving, and you will need to exert effort to get the ball (and yourself) to stop rotating.

The definition of work is energy = force * displacement * cos(theta)

This definition is only strictly accurate for movement on a straight line. In general, you need to solve an integral

$$E = \int_a^b\vec{F}\cdot d\vec{\ell}$$

Does the movement require energy?

Two reasons a real robot will require energy to do this operation:

Real robots have friction in their joints, so there will always be a force opposing the direction the robot is trying to move, and so the integral of force along the path of motion won't go to zero

If you are considering cases where the motion includes some components downward in relation to gravity and some components upward, not every robot will be designed to recover the energy gained by moving downward and use it to move the object back upward, so some energy will be used lifting the object on the upward part of the path.

If we can neglect external gravitational or electromagnetic fields then we only need to consider the kinetic energy of the ball. This is zero at the beginning of the motion and zero at the end, so the net change of energy in the ball is zero.

If we assume an ideal robot arm (no friction, perfect conductors, no air resistance etc.) then the energy that the robot arm puts into the ball to accelerate it at the start of the motion can be 100% recovered when the ball decelerates at the end of motion. So the net loss of energy from the robot arm is also zero.

In practice the robot arm will lose energy due to friction, resistive heating, air resistance etc.