State constrained optimal control of a ball pitching robot

• Offline optimal control of an underactuated two-link pitching robot
• Integral approximations of time-global constraints
• Adjoint method is used to perform gradient computation.
• A gradient-based optimization algorithm computes the optimal motor torque profile.
• Numerical experiments illustrate the effectiveness of the proposed approach.

We present a method for offline optimal control of a two-link ball pitching robot with the aim of throwing a ball as far as possible. The pitching robot is connected to a motor via a non-linear torsional spring at the shoulder joint. The elbow joint is passive and loaded with a linear torsional spring. We model the system based on an Euler–Lagrange formulation. Constraints on the motor torque and power as well as the angular velocity of the motor shaft are included in the model. By using an interior point method with gradients supplied by a discrete adjoint method, we numerically solve the resulting constrained control problem of finding the optimal piecewise constant motor torque profile and release position. Numerical experiments illustrate the effectiveness of our strategy as well as the effect of the constraints on the objective. In our experiments, the optimal motor torque gives rise to motions comprising an initial backswing; a transition, where the elbow spring accumulates potential energy; and finally a fast acceleration phase leading up the ball release.