Fast implicit approximation of the Minkowski sum of two general ellipsoids with a superellipsoid

This study proposes a computational method for approximating the Minkowski sum of two ellipsoids in three-dimensional space using a superellipsoid. Computing the Minkowski sum using high-precision mesh generation is expensive. Therefore, methods for approximating the Minkowski sum using functional representations have been proposed. In particular, numerous approaches use ellipsoids for approximation but suffer from accuracy limitations. The shape matrix and exponents of the superellipsoid are derived through a closed-form computation with constant time complexity of O ( 1 ) , independent of any boundary sampling resolution. Moreover, because the superellipsoid is an implicit function, the inside–outside testing of points is straightforward and requires only the evaluation of an inequality. In our experiments, we evaluate various parameter combinations for two general ellipsoids. The results demonstrate that the superellipsoid approximates the Minkowski sum with small error. Additionally, a comparison with an existing method that approximates the Minkowski sum of rotational ellipsoids using superellipsoids shows comparable accuracy, while the proposed method generalizes the approach to arbitrary ellipsoids. Regarding execution time, the method exhibits consistently fast and stable performance, as it relies solely on closed-form computation. The theoretical framework of this method has the potential to be extended to N -dimensional spaces, and is expected to find applications not only in rigid body simulations in three dimensions but also in computational tasks, such as grasp stability evaluation for robotics in six dimensions.