Soft robots leverage material compliance to achieve safe interaction and adaptive motion. While such compliance is a core concept of soft robotics, practical robotic systems often require additional stiff parts, such as rigid actuators or linkages connected by joints. Despite their practical relevance, many existing optimization studies consider only fully deformable bodies. In this work, we propose an automated design framework for soft robots in a broader sense, allowing the simultaneous optimization of soft body shape, skeletal layout, and time-dependent actuation for locomotion tasks. The robot consists of a soft body externally supported by skeletal truss structures and driven by solenoid actuators. The soft body is simulated using the material point method [1], while the truss network and the actuators are represented by extended position-based dynamics [2] to capture constrained motion and internal load transfer in a differentiable manner. The soft body shape is represented by a density field, and the skeletal structure is represented using a ground-structure approach. The voltage inputs to the solenoid actuators are optimized following a time-dependent design formulation in a prior study [3]. We apply the proposed framework to the design of a 150-mm-scale robot for a locomotion task. The optimized result features a soft body reinforced by self-weight-supporting skeletal structures, enabling stable and periodic walking motion. Furthermore, the optimized design is fabricated, and its locomotion performance is demonstrated in a real-world environment. [1] Hu, Y., Fang, Y., Ge, Z., Qu, Z., Zhu, Y., Pradhana, A., & Jiang, C. (2018). A moving least squares material point method with displacement discontinuity and two-way rigid body coupling. ACM Transactions on Graphics (TOG), 37(4), 1-14. [2] Macklin, M., Müller, M., & Chentanez, N. (2016, October). XPBD: position-based simulation of compliant constrained dynamics. In Proceedings of the 9th International Conference on Motion in Games (pp. 49-54). [3] Yuhn, C., Sato, Y., Kobayashi, H., Kawamoto, A., & Nomura, T. (2023). 4D topology optimization: Integrated optimization of the structure and self-actuation of soft bodies for dynamic motions. Computer Methods in Applied Mechanics and Engineering, 414, 116187.