cs.ROOct 10, 2026

Modeling and Control of a Rigid-soft Hybrid-link Humanoid Robot

Authors: Zewen He, Taiki Ishigaki, Ko Yamamoto

Organizations: Department of Mechano-informatics, The University of Tokyo, 7-3-1 Hongo Bunkyo-ku, 113-8656, Tokyo, Japan

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Abstract

Physical flexibility plays a significant role not only in human motions but also in robot motions. For example, a humanoid robot with passive and flexible elements, which can be regarded as a rigid--soft hybrid-link system, has the advantages of storing energy and mitigating impact in a dynamic motion. However, the motion design and control of a hybrid-link system are challenging to achieve and have only been discussed in the literature. In this study, we propose the framework of inverse kinematics and compliance optimization-based control for a hybrid-link humanoid system that represents the continuous deformation of a flexible structure using the piece-wise constant strain (PCS) model. The proposed method was validated using forward dynamics simulations for standing, jumping, and walking motions. Then, we built an experimental setup using a biped humanoid robot with leaf-spring type prosthesis leg, in which the prosthesis deformation was measured by a motion capture system. Using this setup, we experimentally validated the proposed method for standing balance control in both double- and single-support cases.

Figures & tables

Figure 1: Overview of hybrid Hydra , which consists of a hydraulically-driven humanoid robot, Hydra Ko et al. (2021) , wearing a leaf-spring type prosthesis at each leg: (a) its appearance, (b) hybrid-link model with PCS model of the prosthesis legs, (c) the detail of the PCS, and (d) experimental setup including motion capture system for measuring the deformation.
Figure 2: (a) Schematic illustration of the PCS model Renda et al. (2018) . A beam or rod structure is divided into a finite number of segments . The six-dimensional strain is assumed to be spatially constant. (b) Illustration of a hybrid-link system. The blue parts represent the soft-part modeled using the PCS model, and the white parts represent rigid-part modeled using the rigid-body multi-link system. The base-link of the system is not fixed to the environment. An external force is considered to be acting on the end-effector of the system.
Figure 3: Equilibrium between the gravity and elastic force in the hybrid-link system. The soft-part deforms due to the gravitational force, as indicated by the blue parts.
Figure 4: (a) Pitch rotation angle plotted to represent the connecting link attitude. (b) Attitude of the connecting link between rigid and soft parts. A larger angle represents larger PCS forward bending deformation. The step duration is 1 s and the double-support period of each step marked by gray regions, is 0.4 s. In this walking motion, the robot first steps forward with its right leg. In each double-support period, the whole-body COG shifts from one leg to the other.
Figure 5: (a) Illustration of the double-support case. The left foot is considered as a fixed base, and the right foot has an equation constraint on its velocity. (b) Relationship between the different spaces of a hybrid system comprising a closed kinematic chain. The rectangles in the figure represent the velocity spaces. The solid straight arrows show the transformation direction between different velocities. The hollow curved arrows represent the transformation relationship between different space compliance.
Figure 6: Control framework of a hybrid-link humanoid robot.
 COG gain αi\alpha_{i}yyzz
PD/1001.0
DCC1.0, 1.0, 10001002 y\sqrt{y}
Table 1: Parameter settings in double-support balance control simulations. ( yy and zz are the null-space stiffness and damping values, respectively)
 COG gain αi\alpha_{i}yyzz
Take-off phase1.0, 1.0, 10003000.2*2 y\sqrt{y}
Flight phase (PD)/5005.0
Touch-down phase1.0, 1.0, 10001002 y\sqrt{y}
Table 2: Parameter settings in Vertical hopping control simulations. ( yy and zz are the null-space stiffness and damping values, respectively)
 COGwaistfootfootnull-
 gain αi\alpha_{i}att.pos.att.space
single-0.8, 0.8,k=50,k=3000,k=200,yy =60
support1000d=0.8d=1d=1zz =10
double-0.8, 0.8,\\\yy =20
support1000   zz =8
Table 3: Parameter settings in bipedal walking control simulations. ( yy and zz are the null-space stiffness and damping values, respectively)
Figure 7: Simulation results of double support balancing. An external impact (1,000 N ×\times 0.02 s) was added to the body link in the xx direction at 0.2 s.
Figure 8: Snapshots of DCC optimization simulation during vertical hopping and landing (single cycle). The green line shows the ground reaction force on each vertex of the contact plane.
Figure 9: Joint torques in the vertical hopping and landing motion. The torque of the left-side joints is similar to that of the right-side joints. The gray region shows the flight phase.
Figure 10: COG position in the zz -axis of a twice vertical hopping motion on hybrid Hydra. The dashed black line is the reference trajectory generated by the 3D ZMP model and the magenta line denotes the control result of the simulation. The gray regions show the flight phases in the simulation.
Figure 11: Simulation results of COG (actual and reference) and ZMP of the hybrid walking motion when \mbox{\boldmathK}_{S}=\mbox{diag}\{50\} .
Figure 12: Actual (forward dynamics simulation) and reference (hybrid IK) PCS strains of right leg during walking motion.
Figure 13: Snapshots of walking motion control achieved using the proposed method. The first 2 steps are shown. A total of 9 steps were simulated.
Figure 14: Software architecture of the hybrid Hydra.
COG gain αi\alpha_{i}knee task gainyyzz
0.5, 1.0, 1000K=800, D=5.0100.2
Table 4: Parameter setting of hybrid Hydra for the double-support standing experiment
Figure 15: Compliance test with DCC optimization. The red dashed arrows illustrate the motion tendency of the robot. It can be seen that at 1.35 s after being impacted, the robot bends its body along the curved arrow direction to counteract the impact.
Figure 16: Screenshots of the push recovery experiment applying DCC control. The solid red line shows the external force applied to the robot by a stick. The red dashed lines represent the motion tendency of the robot. After being subjected to an external impact, the robot tilts backward, reaching the maximum COG displacement at approximately 1 s. Simultaneously, during this process, the robot bends its body link to minimize the COG error, which also demonstrates its whole-body compliance. Subsequently, the robot gradually returned to its COG equilibrium position at approximately 3.2 s. During this time, the robot straightened its body and achieved a balanced posture.
Figure 17: (a) COG trajectories in xx directions of the push recovery experiment applying DCC control. In this experiment, three external impacts were applied in the −x-x direction. From the figures, it can be observed that each time an impact occurs, the COG shifts in the −x-x direction, reaches its maximum displacement, and then returns to its initial state after approximately 1 to 2 s later. (b) Waist joint angle in yy direction (pitch). The measured waist angle increased after impact each time and recovered after several seconds.
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