Conceptualization, Y.Y. and J.S.; methodology, Y.Y., W.C., S.H. and Q.L.; software, Y.Y., J.S., W.C. and Q.L.; validation, Y.Y.; formal analysis, Y.Y. and J.S.; investigation, Y.Y. and J.S.; resources, M.Z.; data curation, Y.Y., J.S. and Y.G.; writing—original draft preparation, Y.Y., J.S., S.H. and Y.G.; writing—review and editing, M.Z., Y.Y., J.S., S.H., Y.G., W.C., X.C. and X.L.; visualization, Y.Y. and Y.G.; supervision, M.Z. and X.L.; project administration, M.Z.; funding acquisition, M.Z. All authors have read and agreed to the published version of the manuscript.

Figure 1. Three-Particle Model.

Figure 1. Three-Particle Model.

Figure 2. Overall control framework.

Figure 2. Overall control framework.

Figure 3. Schematic diagram of collision experiment.

Figure 3. Schematic diagram of collision experiment.

Figure 4. CoM position of the robot under the frontal impact of a 9 kg rigid ball.

Figure 4. CoM position of the robot under the frontal impact of a 9 kg rigid ball.

Figure 5. CoM position of the robot under the frontal impact of an 11 kg rigid ball.

Figure 5. CoM position of the robot under the frontal impact of an 11 kg rigid ball.

Figure 6. Swing foot position of the robot under the frontal impact of a 9 kg rigid ball.

Figure 6. Swing foot position of the robot under the frontal impact of a 9 kg rigid ball.

Figure 7. Swing foot position of the robot under the frontal impact of an 11 kg rigid ball.

Figure 7. Swing foot position of the robot under the frontal impact of an 11 kg rigid ball.

Figure 8. Screenshots of the motions generated by our method in the frontal impact test.

Figure 8. Screenshots of the motions generated by our method in the frontal impact test.

Figure 9. Centroidal pitch angular momentum of the robot under the frontal impact of a 9 kg rigid ball.

Figure 9. Centroidal pitch angular momentum of the robot under the frontal impact of a 9 kg rigid ball.

Figure 10. Centroidal pitch angular momentum of the robot under the frontal impact of an 11 kg rigid ball.

Figure 10. Centroidal pitch angular momentum of the robot under the frontal impact of an 11 kg rigid ball.

Figure 11. Computation time cost of TP-MPC in the frontal impact test. (a) 9 kg ball; (b) 11 kg ball.

Figure 11. Computation time cost of TP-MPC in the frontal impact test. (a) 9 kg ball; (b) 11 kg ball.

Figure 12. CoM position of the robot under the side impact of a 2.7 kg rigid ball.

Figure 12. CoM position of the robot under the side impact of a 2.7 kg rigid ball.

Figure 13. CoM position of the robot under the side impact of a 3.2 kg rigid ball.

Figure 13. CoM position of the robot under the side impact of a 3.2 kg rigid ball.

Figure 14. Swing foot position of the robot under the side impact of a 2.7 kg rigid ball.

Figure 14. Swing foot position of the robot under the side impact of a 2.7 kg rigid ball.

Figure 15. Swing foot position of the robot under the side impact of a 3.2 kg rigid ball.

Figure 15. Swing foot position of the robot under the side impact of a 3.2 kg rigid ball.

Figure 16. Centroidal roll angular momentum of the robot under the side impact of a 2.7 kg rigid ball.

Figure 16. Centroidal roll angular momentum of the robot under the side impact of a 2.7 kg rigid ball.

Figure 17. Centroidal roll angular momentum of the robot under the side impact of a 3.2 kg rigid ball.

Figure 17. Centroidal roll angular momentum of the robot under the side impact of a 3.2 kg rigid ball.

Figure 18. Screenshots of the motions generated by our method in the side impact test.

Figure 18. Screenshots of the motions generated by our method in the side impact test.

Figure 19. Computation time cost of TP-MPC in the side impact test. (a) 2.7 kg ball; (b) 3.2 kg ball.

Figure 19. Computation time cost of TP-MPC in the side impact test. (a) 2.7 kg ball; (b) 3.2 kg ball.

Table 1. Tasks, Constraints, and Priority Setting of Whole-Body Control for Single Leg Balance Standing.

Table 1. Tasks, Constraints, and Priority Setting of Whole-Body Control for Single Leg Balance Standing.

PriorityTasksTasks DimensionConstraintsConstraints Dimension1Floating Base Dynamics6Joint Torque102Linear Momentum6ZMP9Torso PostureFriction Cone3Foot Position & Posture16 Contact Wrench