Inverse Kinimeatics

We were first going to use the Jacobian and task space control to solve for the joint angles, however once we did a little research we discovered I would be rather simple to do inverse kinematics. The figure define all variable used to solve the inverse kinematics.

The total height the ball is launched from above the mouth of the cup is defined as

Tocalculate the first joint angle, θ₁, that finds the correct bearing it is a simple inverse tangent relationship shown below.

The range joint angle, θ₂, is a little bit more difficult to calculate. We first calculated the angle of maximum range. This calculation uses ‘z’ calculated above, but since ‘z’ is a function of ‘θ₂‘, preformed an iterative search to determine the angle of maximum range. The variable ‘v’ is the velocity of the ball leaving the barrel, this value was determined experimentally.

We next calculated the range of the shot and the maximum range the shooter is able to achieve at the current height.

If the range exceeded the shooters maximum range, the angle of maximum range was used to take the shot. If not then the equation below was used to calculate the angle required to make the shot.

Before we sent the angles to the servos we first removed some angle biases that were experimentally determined. The bearing bais was -4deg and the range bias was 2deg.