Generate trajectory between 2 points to achieve a desired momentum

Question

I need to plan path for robotic arm(having a tennis racquet as end-effector) whose aim is to hit a tennis ball in a specific direction(to return ball to the opponent). Now I have an algorithm which moves the arm to the position(in the desired position and orientation of the racquet) where ball is expected to be(in a linear path shown in image below) . But the racquet just touches the ball because there is no momentum. Due to this, the ball falls nearby and does not reaches the opponent.

In the image below:

• Red Dot -> Starting position of end effector.
• Dark Blue Dot -> End position of end effector.
• Pink Line -> The desired orientation(face of the racquet) to hit the ball.
• Dotted Light Blue Line -> Desired ball trajectory after being hit by the racquet.
• Green Line -> Current trajectory.

I need to find a curvy trajectory(how humans hit the tennis ball, how they generate a swing) between 2 given points to achieve a desired momentum at the end point so that the ball can reach the opponent. Refer the desired trajectory image below.

In the image below:

• Red Dot -> Starting position of end effector.
• Dark Blue Dot -> End position of end effector.
• Pink Line -> The desired orientation(face of the racquet) to hit the ball.
• Dotted Light Blue Line -> Desired ball trajectory after being hit by the racquet.
• Green Line -> Desired trajectory.

In this image, the green trajectory is expected to generate a better momentum in the desired direction(the light blue line).

QUESTION:

• How to generate trajectory between 2 points to achieve a desired(or maximum) momentum at the end point(second point)?

Answer 1

There are a few options,

1. put it "all in the wrist", move the wrist fulcrum in the correct position and orientation to strike the ball with the racket and then time the wrist to swing at the correct time to hit the ball. You can pick other joints further up the arm to be the primary swing fulcrum and perhaps add some momentum from the other joints. But that gets trickier as you need to ensure the contact point isn't in line with the swing joint axis and the contact is not near the limits of the swing joint's movement range.

2. building off of that once you solved the IK for the contact position find the direction for all the joints that moves the racket back from the contact point, given that when you move the arm make every joint approach the contact position from that direction holding back the final move forward until it's time to swing.

Answer 2

The telemetry problem should be handled by a learning algorithm. I'd start by nesting a couple different approaches. For any one shot, I'd predict a set of contact points, create a small set of racket state vectors at each contact point, and then create a big set of trajectories. I'd bin the trajectories according to the contact location and state vector conditions. Then I'd run a post swat trajectory prediction on each, choose one trajectory using some parametrized strategy, take the swing, and compare the results to the prediction. You can now use a set of genetic learning algorithms to sort the mechanical issues of getting hits in the middle of the racket (requires a module for ball state vector prediction and racket state vector prediction), to improve the telemetry prediction issues of trajectories when you do hit the middle of the racket, and to improve the effectiveness of the shot selection strategy.

And after trolling though some papers, that seems to be a verified approach to the problem now - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9185249/