  This work investigates an application-driven co-design problem where the
motion and motors of a six degrees of freedom robotic manipulator are optimized
simultaneously, and the application is characterized by a set of tasks. Unlike
the state-of-the-art which selects motors from a product catalogue and performs
co-design for a single task, this work designs the motor geometry as well as
motion for a specific application. Contributions are made towards solving the
proposed co-design problem in a computationally-efficient manner. First, a
two-step process is proposed, where multiple motor designs are identified by
optimizing motions and motors for multiple tasks one by one, and then are
reconciled to determine the final motor design. Second, magnetic equivalent
circuit modeling is exploited to establish the analytic mapping from motor
design parameters to dynamic models and objective functions to facilitate the
subsequent differentiable simulation. Third, a direct-collocation-based
differentiable simulator of motor and robotic arm dynamics is developed to
balance the computational complexity and numerical stability. Simulation
verifies that higher performance for a specific application can be achieved
with the multi-task method, compared to several benchmark co-design methods.
