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CEER2: Directional and Tunable End-Effector and Root Compliance for Humanoid Loco-Manipulation

Xinyuan Luo, Chunyuan Yang, Boyuan Chen, Xianyi Cheng

arXiv:2609.38709v1cs.RO

TL;DR

Humanoid motion tracking alone does not specify how the robot should respond to contact while preserving task motion. This work combines tunable directional end-effector compliance and selectable root responses through hierarchical control. Simulation and real-world experiments demonstrate compliance behaviors, while the implementation remains constrained by its underlying tracking policy.

  • Problem

    Existing compliant humanoid controllers do not jointly specify directional, tunable end-effector compliance and root response, although isotropic softening can sacrifice task-relevant accuracy.

    Source: §I · pp. 1–2

  • Method

    A hierarchical controller modulates a fixed whole-body tracking policy, with an analytical compliance-space mixture of experts composing end-effector responses and root policies selected by mode.

    Source: §I · p. 2, 1) Analytical Compliance-Space MoE · p. 4

  • Results

    The analytical mixture of experts achieves a compliance-matrix error of 0.278, EE compliance-tracking error of 0.0349 m, and stiffness MAPE of 0.249, and experiments demonstrate distinct compliance configurations in loco-manipulation.

    Source: 2) Full-Range Directional EE Compliance · p. 6, §V · p. 8

  • Takeaways & Limitations

    The framework supports online end-effector stiffness adjustment and root-mode switching, with experiments showing compliant manipulation, payload adaptation, and collaborative physical interaction.

    Source: §V · p. 8

    Achievable compliance is bounded by the stiff tracking policy, including insufficient x-direction stiffness and limited locomotion that constrain end-effector range and root interaction performance.

    Source: §V · p. 8

Abstract

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Humanoids are increasingly capable of tracking complex whole-body motions, but physical interaction introduces a different challenge. When a robot makes contact with a person or the environment, it needs to respond to external forces while preserving the motion needed for the task. This response can vary across directions in the end-effectors and on the body. For example, an end effector may need to accommodate contact force in one direction while maintaining motion accuracy in another, while the robot body may resist an external force or move with it. We present a compliance framework for humanoid loco-manipulation that combines directional and tunable end-effector (EE) compliance with selectable root compliance for external force rejection or force following. A hierarchical reinforcement learning controller modulates a fixed whole-body tracking policy through high-level EE and root commands, while interaction forces are estimated from proprioceptive history. Our simulation and real-world experiments on a humanoid demonstrate directional stiffness control, online stiffness adjustment, distinct root compliance, compliant manipulation, and collaborative carrying.

I. INTRODUCTION

Humanoid tracking policies still need a way to respond to contact forces without sacrificing task motion. This work combines directional, tunable end-effector compliance with selectable root responses, controlled through a hierarchical policy.

I. INTRODUCTION

  • Directional stiffness accommodates contact in one direction while preserving task-relevant accuracy in others, unlike isotropic softening that can sacrifice that accuracy.Writing is one motivating example: low stiffness normal to the surface and high in-plane stiffness.Source: §I · p. 2
  • The framework combines independently adjustable end-effector stiffness with root resistance and damping modes for concurrent physical interaction.End-effector stiffness is tunable over 100–600 N/m.Source: §I · p. 2
  • A fixed, stiff whole-body tracking policy is modulated by high-level residual policies, with an analytical compliance-space mixture of experts composing end-effector responses.The framework also uses a force estimator and teacher–student learning with privileged force observations.Source: §I · p. 2
  • Real-world manipulation and loco-manipulation tasks validate directional and tunable end-effector compliance together with root compliance in physical interaction.The comparison table is identified as covering humanoid compliance methods, but no comparison values are supplied here.Source: §I · p. 2, Table I · p. 2

A. Compliance Control for Humanoid Robots

Prior humanoid work addresses several aspects of compliance, force awareness, and contact structure, while this method specifies directional end-effector and root responses. Its method uses a fixed tracking backbone with high-level compliance policies and defines end-effector compliance through force-dependent displacement.

A. Compliance Control for Humanoid Robots

B. Force- and Tactile-Aware Humanoid Interaction

  • Force- and tactile-aware approaches provide interaction information, but sensing or estimating contact does not itself specify distinct compliance behaviors.The cited approaches include force-adaptive control, tactile sensing, and proprioceptive contact or wrench estimation.Source: B. Force- and Tactile-Aware Humanoid… · p. 2

C. Contact-Aware Humanoid Loco-Manipulation

  • Contact-aware methods preserve or command contact relationships through retargeting, contact labels or schedules, and geometry-aware features.These methods address contact structure and where or when contacts occur, rather than specifying a compliance response.Source: C. Contact-Aware Humanoid… · p. 3

III. METHOD

  • A fixed, sufficiently stiff tracking policy serves as the backbone, while high-level policies realize softer behaviors through command residuals and root-mode switching.Separate high-level policies are trained for different compliance settings.Source: §III · p. 3

A. Directional and Tunable EE Compliance

  • Directional end-effector compliance is defined by desired Cartesian displacement under external force, with commanded stiffness specified along Cartesian axes.The method applies this definition to each end effector.Source: A. Directional and Tunable EE Compliance · p. 3

B. Root Resistance and Damping Modes

Root compliance is defined by how the root moves in response to external forces. The framework offers resistance and two force-following damping modes.

B. Root Resistance and Damping Modes

  • The root offers one resistance mode that rejects external forces and two force-following modes, with damping coefficients of 60 and 200 N s/m.These are designated low- and high-damping modes, respectively.Source: B. Root Resistance and Damping Modes · p. 3
  • In damping mode, the target root velocity responds to the net force while retaining the nominal planar velocity command.The specified relationship applies damping rather than stiffness: the robot keeps moving while pushed or pulled instead of stopping at an offset equilibrium.Source: B. Root Resistance and Damping Modes · p. 3

C. Stage 1: Low-Level Whole-Body Tracking Policy Training

A stiff whole-body tracking policy provides the backbone for high-level compliance control. High-level policies add command residuals to its EE–root targets, with training objectives shaping the desired responses and penalizing large or nonsmooth residuals.

C. Stage 1: Low-Level Whole-Body Tracking Policy Training

  • The low-level controller tracks robot-state and EE–root targets with coordinated joint actions while maintaining stability.It is trained on motion tracking under upper-body disturbance forces, with references unchanged.Source: C. Stage 1: Low-Level Whole-Body… · p. 3

D. Stage 2: High-Level Residual Policy Training

1) Residual Actions and Training Objectives:

  • EE residual actions are six Cartesian position offsets, passed through tanh, scaled by expert-specific per-axis limits, and added to nominal EE commands.The 100 N/m soft-axis experts use a 0.35 m limit; other experts use 0.15 m.Source: 1) Residual Actions and Training… · p. 3
  • The training objective includes task errors and rewards, while also penalizing large and temporally nonsmooth command residuals.D and R denote damping and resistance, and σ_m > 0 is the error scale for objective m.Source: 1) Residual Actions and Training… · p. 4

2) EE Compliance Experts:

The high-level policies use stiffness-specific EE experts and separate root-compliance policies, trained through teacher–student learning. At deployment, the system composes EE experts by commanded stiffness and selects a root policy by mode.

2) EE Compliance Experts:

  • Ten EE experts cover isotropic 600 N/m and directional 100, 200, and 400 N/m stiffness settings for each Cartesian axis.Each expert changes EE position while leaving nominal EE orientation to the low-level command; separate experts address the similar responses observed when one policy spans stiffness settings.Source: 2) EE Compliance Experts · p. 4

3) Root Compliance Policies:

  • Three separate root policies provide resistance and damping responses, with damping targets set at B = 60 or 200 N s/m.The resistance policy minimizes planar root displacement; the damping policies track force-dependent velocities.Source: 3) Root Compliance Policies · p. 4

4) Teacher–Student Training:

  • Teacher–student training uses privileged force and simulation-state observations for the teacher, while estimators learn from deployable observation histories.A force-estimation head predicts hand forces for EE policies, and net force plus a body-location label for root policies; deployment uses estimated latent and force features.Source: 4) Teacher–Student Training · p. 4

E. Deployment-Time Policy Composition

  • At deployment, an analytical compliance-space mixture composes EE experts by commanded Cartesian stiffness, while a root-mode command selects resistance or damping.This supports online stiffness adjustment, root-mode switching, and combined EE–root compliance through the fixed low-level policy.Source: E. Deployment-Time Policy Composition · p. 4

1) Analytical Compliance-Space MoE:

The analytical MoE interpolates EE-expert outputs in compliance space, while root compliance uses mode-based policy selection. Because the residuals modify separate command components, EE and root compliance can operate together.

1) Analytical Compliance-Space MoE:

  • Neighboring EE experts are interpolated linearly in compliance space because ideal displacement is force multiplied by compliance, where C_d = 1/K_d.The MoE composes experts independently on each Cartesian axis, and its weights are determined directly by commanded compliance without learned parameters.Source: 1) Analytical Compliance-Space MoE · p. 4

2) Root Policy Selection:

  • The root-mode command selects one of three Stage 2 policies rather than interpolating them, because root response depends on foot–ground contacts and support conditions..Source: 2) Root Policy Selection · pp. 4–5

3) Combined EE–Root Control:

  • EE and root compliance can run simultaneously because their residuals modify different components of the nominal EE–root command.No priority rule or policy reset is required.Source: 3) Combined EE–Root Control · p. 5

IV. EXPERIMENTS

  • The experiments assess commanded directional EE stiffness and root modes, full-range control accuracy, and real-world loco-manipulation with combined EE–root compliance.The protocol first evaluates the EE hierarchy, full-range compliance, and workspace consistency, then root modes and action spaces, before real-world demonstrations.Source: §IV · p. 5

A. Experimental Setup and Evaluation Protocol

The evaluation uses simulated perturbations and real-robot experiments, with compliance measured through tracking error and force-to-displacement apparent stiffness. Low-level characterization selects the 30 N 3-kp policy as the backbone for the main experiments.

A. Experimental Setup and Evaluation Protocol

  • Each EE compliance configuration is tested in 60 trials spanning two hands, three axes, two force directions, and five force magnitudes.Forces ramp for 0.5 s and are held for 2.0 s; measurements average the final 0.5 s.Source: A. Experimental Setup and Evaluation… · p. 5
  • Apparent stiffness is estimated as applied force divided by resulting EE displacement along each Cartesian axis.Compliance-tracking error measures distance from the actual EE position to the compliant target; nominal tracking error measures distance to the nominal command.Source: A. Experimental Setup and Evaluation… · p. 5

B. Low-Level Policy Characterization

  • The 30 N 3-kp policy has the lowest nominal error and median apparent stiffness near 600 N/m, so it is used in the main experiments.Increasing training force from 30 to 60 N does not monotonically increase apparent stiffness and degrades nominal tracking, most clearly at 60 N.Source: B. Low-Level Policy Characterization · p. 5, Table II · p. 5
  • CEER is naturally more compliant than the other representative policies, SONIC has an intermediate response, and their apparent-stiffness distributions differ substantially.The broad uncommanded distributions motivate explicit high-level compliance control.Source: B. Low-Level Policy Characterization · p. 5, Figure 3 · p. 5
  • The 100–600 N/m command range uses 600 N/m near the low-level policy’s overall median stiffness and a workspace-limited 100 N/m lower bound.At 100 N/m and 30 N force, the required displacement is 0.30 m; the soft-axis residual limit is 0.35 m.Source: B. Low-Level Policy Characterization · pp. 5–6

C. End-Effector Compliance Evaluation

The hierarchy improves fixed-target EE compliance, while the analytical MoE performs best across the full directional stiffness range. Its specialized experts help preserve distinct commanded responses where stiffness-conditioned policies compress the range.

1) Fixed-Target Hierarchy Ablation:

  • 0.023±0.016 m and 0.028±0.014 m EE compliance-tracking errors for analytical MoE and fixed-stiffness HL outperform 0.082±0.034 m for fixed-stiffness E2E.The two hierarchical methods also achieve apparent stiffness closer to the 200 N/m target.Source: 1) Fixed-Target Hierarchy Ablation · p. 6

2) Full-Range Directional EE Compliance:

  • Table IV reports the analytical MoE achieves the lowest compliance-matrix error (0.278), EE compliance-tracking error (0.0349 m), and stiffness MAPE (0.249), and has the slope closest to 1.Source: 2) Full-Range Directional EE Compliance · p. 6
  • A single stiffness-conditioned policy compresses command responses, whereas the analytical MoE composes specialized behaviors; even the oracle-force baseline trails the MoE in slope and tracking error.The oracle-force low-level baseline assumes ideal Cartesian position control, an assumption contradicted by finite apparent stiffness and coupled whole-body dynamics.Source: 2) Full-Range Directional EE Compliance · p. 6

3) Workspace Consistency:

Workspace consistency was tested across seven paired EE poses using the full 16-configuration protocol. Component errors were similar in mean across axes, with the largest posewise mean at raised-hand z+.

3) Workspace Consistency:

  • The full 16-configuration protocol was applied at seven paired EE poses, totaling 6,720 trials.The poses comprise the center and 0.05 m offsets along each positive and negative Cartesian axis.Source: 3) Workspace Consistency · p. 6
  • Fig. 4 plots commanded against apparent stiffness separately for x, y, and z; curves show medians, shaded regions show interquartile ranges, and the dashed line is ideal tracking.The stiffness-conditioned E2E interquartile band is omitted only in the x panel for readability.Source: Figure 4 · p. 7
  • 24.8, 22.1, and 25.8 mm are the mean x-, y-, and z-component errors across tested poses; the largest posewise mean component error is 43.7 mm at raised-hand z+.The authors suggest reduced joint-limit margin or kinematic redundancy with elevated arms may explain the largest error; left x and right z responses are generally less accurate, respectively.Source: 3) Workspace Consistency · p. 7

D. Root Compliance Evaluation

The root policies produce distinct resistance and damping responses, while additional foot and yaw control helps resistance more than damping. The paper also reports simulation-to-real transfer for EE and root compliance tasks.

D. Root Compliance Evaluation

  • Resistance moves the root least, B = 60 damping follows the pull most, and B = 200 gives an intermediate response; median apparent damping closely matches both commanded values.The evaluation applies 5, 10, 20, and 30 N forces in four horizontal directions at the torso, shoulders, wrists, and hands.Source: D. Root Compliance Evaluation · p. 7
  • Foot-position and yaw residuals substantially improve resistance through active turning and stepping, but their damping improvements are smaller and configuration-dependent, leaving contact- and gait-dependent damping unresolved.Table VI compares planar root drift for resistance and velocity-response error for damping; 5-kp adds foot-position residuals and +yaw adds a root-yaw residual.Source: D. Root Compliance Evaluation · p. 7, Table VI · p. 7

E. Real-World Evaluation of EE and Root Compliance

  • All policies were transferred directly from simulation to real-world tests of tunable stiffness, directional EE compliance, and combined EE–root compliance, including online EE-stiffness adjustment.Because manual force and displacement measurements provide only approximate stiffness estimates, the evaluation focuses primarily on task-level performance.Source: E. Real-World Evaluation of EE and… · p. 7

1) Tunable EE Compliance and Online Stiffness Adjustment:

Real-world tests demonstrate tunable, directional, and combined EE–root compliance in manipulation and collaborative carrying. The authors conclude that these configurations support safer contact, more accurate manipulation, payload adaptation, and physical interaction, while noting limits inherited from the low-level tracker.

1) Tunable EE Compliance and Online Stiffness Adjustment:

  • Under K_y = 100, 200, and 600 N/m, force-gauge pulls yield apparent stiffness values of 214.6, 338.6, and 551 N/m, respectively.EE stiffness is adjusted online without restarting the low-level policy; increasing K_z also reduces vertical displacement under the same 3 kg payload.Source: 1) Tunable EE Compliance and Online… · p. 7

2) Directional EE Compliance:

  • Directional stiffness supports writing: low surface-normal and high tangential stiffness preserve contact and improve trajectory accuracy, while isotropically high stiffness broke the pen holder in another trial.Isotropically soft control often loses contact when writing “8”; increased stiffness along a perpendicular disturbance direction visibly reduces trajectory protrusion.Source: 2) Directional EE Compliance · p. 7

3) Combined Directional EE and Root Compliance:

  • In collaborative carrying with a 3 kg payload, directional EE compliance plus B = 60 root damping completes the task, whereas isotropically low EE stiffness lets the box slip.The EE is compliant along the contact y-axis and stiff along x and z, while the root follows human interaction forces.Source: 3) Combined Directional EE and Root… · pp. 7–8

V. CONCLUSION

  • The conclusion attributes safer contact, more accurate manipulation, payload adaptation, and collaborative physical interaction to different compliance configurations demonstrated in simulation and real-world experiments.The framework combines directional and tunable EE compliance with root damping and resistance.Source: §V · p. 8
  • Achievable compliance is bounded by the stiff low-level tracker: insufficient x-direction stiffness limits EE stiffness range, and limited locomotion constrains root interaction performance.Physical interaction is also often rough, with heavy steps and no deliberate foot-placement adaptation.Source: §V · p. 8

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