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Modeling and Control of a Rigid-soft Hybrid-link Humanoid Robot
Zewen He, Taiki Ishigaki, Ko Yamamoto
TL;DR
Humanoid control with passive flexible elements remains challenging because nonlinear deformation and gravity complicate motion design and balance. This paper combines hybrid inverse kinematics with compliance optimization for a PCS-modeled humanoid. Simulations and experiments validate the approach, including reported jumping, walking, and balance outcomes.
Problem
Model-based motion design and control for humanoids with passive flexible elements remain rarely investigated because their continuous nonlinear deformation is complex and can disturb balance under gravity.
Source: §1 · p. 3Method
The paper combines inverse kinematics that accounts for elastic deformation with compliance optimization for a hybrid-link humanoid modeled using piece-wise constant strain.
Source: §1 · p. 4, §2.1 · p. 6Results
The proposed dynamically consistent compliance optimization improved balance control in simulation; jumping height was 3 times higher than the reference trajectory without consideration of elasticity, and walking COG error in the x-direction stayed within 0.16 m.
Source: §7 · p. 29Takeaways & Limitations
The authors conclude that passive elasticity contributed to energy saving and enhanced locomotion capabilities, based on comparisons of joint torques between hybrid and rigid robots.
Source: §7 · p. 29Choosing a smaller gravity-term weight stabilizes numerical calculation but can increase COG error, which the authors observed becoming large enough for the COG to almost reach the ground.
Source: §3.2 · p. 11
Abstract
from arXiv · showhide
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.
1. Introduction
Control of humanoids with passive flexible elements remains challenging because their nonlinear deformation complicates motion design, while gravity-induced leg deformation can threaten balance. This study develops hybrid-link inverse kinematics and compliance optimization, then validates the approach through simulations and robot experiments.
1. Introduction
- Passive flexible elements can store energy and absorb impact, but their nonlinear continuous deformation makes model-based motion design and control challenging.Gravity-induced deformation can also disturb balance, requiring control to account for interactions between actuated joints and passive elements.Source: §1 · p. 3
- The study proposes hybrid inverse kinematics that balances elastic restoring force against gravity, alongside compliance optimization that accounts for passive viscoelasticity.The controller directly optimizes compliance matrices and uses force/torque control because those matrices can have nonzero off-diagonal elements.Source: §1 · pp. 3–4
- The proposed schemes were validated through simulations and experiments of a humanoid equipped with prostheses, with the study targeting energy-saving capabilities during dynamic motion.Simulations covered standing, hopping, and walking; the experimental platform validated balance control.Source: §1 · p. 4
- Beyond the earlier conference paper's compliance optimization and standing/hopping simulations, this study adds hybrid inverse kinematics, walking simulation, and experimental validation through flexible-deformation measurement.The earlier work is identified as He et al. (2024).Source: §1 · pp. 4–5
2. Rigid-soft Hybrid-link System
The system represents flexible structures with a piece-wise constant strain model and combines them with rigid-body multi-link components. Its floating-base dynamics include passive viscoelasticity, which the actuated rigid part must account for when maintaining system stability.
2.1. Piece-wise Constant Strain Model Renda et al. (2018)
- The PCS model discretizes a soft continuum into segments with constant six-dimensional strain, representing its configuration along the central axis.For inextensible Kirchhoff-Love segments, the model uses three angular-strain degrees of freedom rather than six.Source: §2.1 · pp. 5–6
2.2. Dynamics of a Hybrid-link System with Floating-base Ishigaki and Yamamoto (2021)
- The hybrid-link system integrates a PCS-modeled flexible structure with a rigid-body multi-link system, and the soft part can be located anywhere in the system.The base link is floating, and the system's generalized coordinates include the base, rigid-part, and soft-part degrees of freedom.Source: §2.2 · pp. 6–7, Figure 2 · p. 7
- The soft part has passive viscoelasticity without actuation, so control must generate rigid-part inputs while accounting for its elastic effects and system stability.The soft-part torque is expressed using stiffness and viscosity matrices and deformation relative to its undeformed coordinates.Source: §2.2 · p. 8
3. Inverse Kinematics for a Hybrid-link Humanoid Robot
Hybrid IK chooses active joint coordinates and passive soft-part deformation together, accounting for gravitational and elastic effects. Its gravity term has a physical force-balance interpretation, and the walking formulation incorporates reference COG acceleration.
3.1. Hybrid IK that minimizes Potential Energies
- Hybrid IK jointly calculates passive deformation and active joint coordinates by minimizing elastic and gravitational potential energy while meeting task-space position references.The formulation assumes a contact foot and can be solved numerically using the Levenberg–Marquardt method.Source: §3.1 · pp. 8–10
3.2. Physical Interpretation of Gravity Term in Cost Function
- At an optimal solution with zero COG error, the soft-part condition expresses equilibrium among elastic force, gravity, and gravity-induced moment.With residual COG error, the weighting matrix W acts as virtual stiffness; choosing W too small relative to KS can permit substantial error, while large W can destabilize optimization.Source: §3.2 · pp. 10–11
3.3. Hybrid IK for Bipedal Walking
- The walking formulation incorporates reference COG acceleration, and simulated prosthesis deformation shifts between support legs during double support.In single support, the support-leg deformation remains near 0.06 rad while the swing-leg deformation stays near its initial 0.02 rad; a transition deformation spike is attributed to ground reaction force.Source: §3.3 · pp. 11–13
4. Control of Hybrid-link Humanoid based on Compliance Optimization
The section motivates optimizing active compliance directly because the QP feasible solution space is limited in hybrid-link systems. It develops compliance mappings and optimization for single- and double-support control.
4.1. Feasible Solution Space in QP-based Method for Contact Force Distribution
- In the tested single- and double-support cases, Eq. (26) is overdetermined, limiting feasible contact-force solutions in the QP method.The stated matrix dimensions yield more equations than variables.Source: §4.1 · p. 14
4.2. Motion Control as Compliance Optimization
- The hybrid stiffness matrix combines active and passive stiffness blocks with an interference block that feeds passive-position error into active control.The lower-left block is zero because the method has no feedback term determining passive torque from passive coordinates.Source: §4.2 · p. 15
4.3. Compliance Optimization in Double-support State
- Double-support foot contact constrains joint velocity, reducing the joint-space dimension before compliance is mapped between joint, double-support, and task spaces.The method solves the double-support-to-task-space mapping before the joint-to-double-support mapping.Source: §4.3 · p. 17
4.3.1. Solving Eq. (42)
- Dynamically-consistent compliance optimization weights the double-support-space solution using inertia and uses a dynamically consistent inverse of its Jacobian.The method interprets the resulting feedback terms as task-space and null-space terms.Source: §4.3.1 · p. 18
4.3.2. Solving Eq. (41)
- The method sets the reference stiffness off-diagonal block to zero to avoid control instability, then solves the two compliance optimizations in one procedure.The combined procedure substitutes the solution for Eq. (42) into the formulation for Eq. (41).Source: §4.3.2 · p. 19
5. Validations Through Simulations
Forward-dynamics simulations evaluated standing balance, vertical hopping, and bipedal walking under the proposed control. The simulations maintained balance after a disturbance, produced repeated hops, and completed walking with reported COG and strain errors.
5.1. Simulation Setting
- Forward-dynamics simulations covered double-support balance, vertical hopping, and bipedal walking, using empirically chosen KS = 50E and DS = 0.02E rather than values estimated from an actual prosthesis.The control framework generates the COG trajectory using a 3D COG-ZMP model and sets the feed-forward torque to gravity compensation.Source: §5.1 · p. 19, Figure 6 · p. 20
5.2. Simulations of Standing Balance Control
- After a 1,000 N impact over 0.02 s, DCC maintained standing balance with smooth ZMP convergence, while the PD-controlled robot fell.The impact was applied to the body link in the x direction at 0.2 s.Source: §5.2 · p. 20, Figure 7 · p. 21
5.3. Simulations of Vertical Hopping
- The hopping simulation reached 0.11 m at 0.68 and 1.95 s versus the 0.03 m reference at 0.57 and 1.87 s; take-off peak waist, hip, and ankle torques were smaller than in the original model.The authors attribute the higher jump to elastic energy stored in the prosthesis during take-off.Source: §5.3 · p. 21, Figure 8 · p. 22, Figure 9 · p. 23
5.4. Simulations of Bipedal Walking
- Walking ended with about +0.1 m COG error in x and 5.6% RMS error between reference and actual PCS strains, supporting feasibility of the proposed methods.The walking simulation applied the motion generated by hybrid inverse kinematics as its reference.Source: §5.4 · p. 22
6. Experimental Validation of Balance Control
The experimental setup measured leaf-spring prosthesis deformation on the hybrid Hydra with motion capture. The experiments demonstrated whole-body compliance and push recovery under DCC control in standing balance tests.
6.1. Experimental Setup
- The setup measured prosthesis deformation with optical markers and motion capture; PCS inverse kinematics ran at about 0.7 ms per iteration, with compliance updates at about 300 Hz.The controller used three inextensible Kirchhoff-Love segments per prosthesis leg and stiffness and viscosity values chosen to closely match estimated prosthesis values.Source: §6.1 · pp. 23–25
6.2. Whole-body Compliance Experiment
- With DCC, the robot countered a slow push by rotating its body link in the COG task-space null-space, demonstrating whole-body compliance.The experiment set the COG feedback gain in the x direction to 0.5.Source: §6.2 · p. 26, Figure 15 · p. 27
6.3. Push Recovery Experiment
- Under a larger impact and higher COG gain, DCC produced body and waist bending, returned measured COG values to their initial values, and kept x displacement within 0.5 cm.The experiment used a COG gain of 5.0 in the x direction; the robot did not fall.Source: §6.3 · pp. 26–27, Figure 16 · p. 28
7. Conclusion
The conclusion presents hybrid inverse kinematics and dynamically consistent compliance optimization for a hybrid-link humanoid. Simulations and experiments report balance, jumping, walking, and standing-control results.
7. Conclusion
- Hybrid IK optimizes generalized coordinates to satisfy elastic-gravity force equilibrium, explicitly accounting for passive elastic deformation due to gravity.The optimization considers elastic and gravity potential energy and uses the Levenberg-Marquardt method.Source: §7 · p. 28
- The proposed compliance framework optimizes active compliance given passive compliance, and dynamically consistent optimization showed balance-control advantages in simulation.The control results were also validated for jumping and walking motions.Source: §7 · pp. 28–29
- After each of three −x impacts under DCC control, COG returned to its initial state approximately 1 to 2 s later.Waist pitch increased after each impact and recovered after several seconds.Source: Figure 17 · p. 29
- Jumping height was 3 times higher than the reference trajectory without consideration of elasticity; walking COG error in x stayed within 0.16 m.The RMS error of PCS strain between hybrid IK and control results was 5.6%.Source: §7 · p. 29
- The hybrid-link robot’s COG displacement in x was controlled within 0.5 cm during double-support balance experiments.The experimental platform combined a humanoid robot with deformation measurement based on motion capture.Source: §7 · p. 29
Appendix A. Hybrid IK in Double-support State
The appendix describes the double-support hybrid IK optimization as a constrained version of the corresponding problem, with an additional foot constraint.
Appendix A. Hybrid IK in Double-support State
- The double-support optimization adds constraints on the foot to an optimization similar to Eq. (12).This extension addresses the double-support state.Source: Appendix A · p. 30
- The appendix defines a Lagrangian L(θ) to discuss the necessary condition of the double-support problem.The Lagrange multiplier λ is specified as belonging to R6.Source: Appendix A · p. 30
- The Lagrange multiplier λ can be regarded as the contact force on the additional foot, and the bottom two rows of Eq. (A.5) express force equilibrium.That equilibrium includes elastic forces from two soft parts, gravity, and the contact force.Source: Appendix A · p. 30
Appendix B. Derivation of Eq. (19) and Eq. (20)
The appendix derives Eqs. (19) and (20) by differentiating Eq. (18), representing the Jacobian matrices, and rewriting the gradient condition.
Appendix B. Derivation of Eq. (19) and Eq. (20)
- The derivation begins by taking the derivative of Eq. (18).Source: Appendix B · p. 30
- The appendix represents the Jacobian matrices used in the derivation.Source: Appendix B · p. 31
- The gradient condition ∇gT(θ∗) = 0 is rewritten using Eq. (B.1) as Eqs. (19) and (20).Source: Appendix B · p. 31
Appendix C. Derivation of DCC Optimization
The appendix derives the general solution of the dynamically consistent compliance optimization and relates it to a previously given formulation.
Appendix C. Derivation of DCC Optimization
- The appendix presents a general solution to the compliance optimization problem.Source: Appendix C · p. 31
- The general solution is rewritten using the mapping given by Eq. (C.2).Source: Appendix C · p. 31
- The resulting formulation is identified as the general solution of dynamically consistent compliance formulation Yamamoto et al. (2021), given by J2C2 = CJ T†.Source: Appendix C · p. 31