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

Xinyuan Luo; Chunyuan Yang; Boyuan Chen; Xianyi Cheng

cs.ROSep 30, 2026

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

Authors: Xinyuan Luo, Chunyuan Yang, Boyuan Chen, Xianyi Cheng

Organizations: Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708, USA

Abstract

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.

Figures & tables

Method	Direction	Angular	Tunable	Gen. Track.	Scope	Loco.
Gentle [ 5 ]	$\times$	$\times$	100–300 †	$\checkmark$	Upper body	$\times$
CHIP [ 7 ]	$\times$	$\times$	$1/k:~0$ – $0.05^{\ddagger}$	$\checkmark$	EE	$\times$
SoftMimic [ 4 ]	$\times$	$\times$	40–1000	$\times$	EE	$\times$
CEER [ 6 ]	$\times$	$\times$	$\times$	$\checkmark$	EE	$\times$
LAC [ 11 ]	$\times$	$\checkmark$	10–500	$\checkmark$	Arms + torso	$\times$
Ours	$\checkmark$	$\times$	100–600	$\checkmark$	EE + root	$\checkmark$

TABLE I: Comparison of humanoid compliance methods.

Fig. 2: System overview. Stage 1 trains a stiff low-level whole-body tracking policy with an EE–root command interface. In Stage 2, the low-level policy is held fixed while high-level residual policies learn EE or root command corrections that produce the desired response to external forces under commanded stiffness or a selected root mode. At deployment, an analytical compliance-space gate composes 10 EE compliance experts according to the commanded Cartesian stiffness, while a root compliance policy is selected through mode switching. The low-level policy remains active throughout, enabling online adjustment of EE stiffness and root mode switching.

Policy	Nom. Err. (m) $\downarrow$	Median (N/m)	P95 (N/m)
3-kp stiff (30 N)	$0.0456\pm 0.0140$	621.4	1793.0
CEER [ 6 ]	$0.1385\pm 0.1657$	103.3	285.0
SONIC [ 1 ]	$0.0955\pm 0.0482$	192.3	342.4
3-kp stiff (40 N)	$0.0524\pm 0.0148$	544.7	961.8
3-kp stiff (50 N)	$0.0528\pm 0.0134$	672.1	3426.0
3-kp stiff (60 N)	$0.0830\pm 0.0269$	406.5	1088.8

TABLE II: Low-level tracking-policy and training-force characterization. Apparent stiffness values describe natural behavior and are not errors relative to a target.

Fig. 3: Natural apparent stiffness of representative low-level policies. Positive and negative directions are combined for each axis. Thin lines show P5–P95, thick lines show the interquartile range, and dots show the median. Dotted lines indicate $200$ and $600~\mathrm{N/m}$ .

Policy	EE Comp. Err. (m) $\downarrow$	Nom. EE Err. (m) $\downarrow$	App. Stiffness (N/m)
Analytical MoE	$\mathbf{0.023\pm 0.016}$	$0.049\pm 0.045$	$219.6\pm 49.2$
Fixed-stiffness HL	$0.028\pm 0.014$	$0.050\pm 0.040$	$241.5\pm 55.7$
Fixed-stiffness E2E	$0.082\pm 0.034$	$0.100\pm 0.046$	$316.6\pm 512.0$

TABLE III: Fixed-stiffness compliance evaluation at

200~\mathrm{N/m}

Fig. 4: Commanded versus apparent stiffness over the 16-configuration 100–600 N/m evaluation, shown separately for the $x$ , $y$ , and $z$ axes. Curves show the median response, shaded regions show the interquartile range, and the dashed line denotes ideal tracking. For readability, the unusually broad stiffness-conditioned E2E IQR band is omitted only in the $x$ panel.

	Accuracy			Trend
Method	Compliance Matrix Err. $\downarrow$	EE Comp. Err. (m) $\downarrow$	Stiffness MAPE $\downarrow$	Slope $\rightarrow 1$	$R^{2}$ $\uparrow$
Analytical MoE	$\mathbf{0.278}$	$\mathbf{0.0349}$	$\mathbf{0.249}$	$\mathbf{0.764}$	0.779
Stiff.-cond. HL	0.677	0.0566	1.325	0.136	0.040
Stiff.-cond. E2E	1.064	0.0884	5.208	3.430	0.003
Oracle-force LL	0.304	0.0459	0.256	0.485	$\mathbf{0.792}$

TABLE IV: Full-range directional EE compliance over 16 stiffness configurations (ideal trend slope and

R^{2}

1

Mode	Root drift (m) $\downarrow$	Velocity error (m/s) $\downarrow$	Apparent $B$ median / target
3-kp LL only	0.3627	0.0955	–
Resistance	0.3103	0.0987	–
$B=60$	1.7806	0.1143	57.0 / 60
$B=200$	0.7163	0.1031	201.4 / 200

TABLE V: Root-compliance evaluation using the 3-kp root-residual action space without yaw control.

Mode	3-kp	3-kp+yaw	5-kp	5-kp+yaw
Resistance	0.3103	0.2032	0.4964	$\mathbf{0.1574}$
$B=60$	0.1143	0.1217	0.1094	$\mathbf{0.1073}$
$B=200$	0.1031	$\mathbf{0.0678}$	0.0904	0.0700

TABLE VI: Root-policy action-space ablation. Resistance is evaluated by planar root drift (m), and damping by velocity-response error (m/s). All variants include root-command residuals; 5-kp adds foot-position residuals, and +yaw adds a root-yaw residual.

Fig. 5: Directional EE compliance tasks: writing “8” (a–c), straight-line writing under disturbance (d–f), and writing on a tilted surface (g–l). Tunable EE-stiffness demonstrations: force-gauge pulling (m–o) and yoga-ball grasping (p–q).

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CEER: Compliant End-Effector and Root Control as a Unified Interface for Hierarchical Humanoid Loco-Manipulation

Xinyuan Luo, Xingrui Chen, Xunjian Yin, Hongxuan Wu+5

Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708, USA

Humanoid Loco-Manipulation End-Effector Pose Whole-Body Control Long-Horizon Manipulation

Open paper AskPDF arXiv

Humanoid robots have achieved impressive locomotion performance, yet contact-rich and long-horizon manipulation remains a major bottleneck. Manipulation is inherently contact-rich and demands compliant whole-body control for stable interaction, while its diversity and long-horizon nature favor modular, planner-compatible interfaces over joint-space tracking. We propose CEER, a compliant end-effector-root (EE-root) control abstraction for modular humanoid loco-manipulation within a hierarchical planning framework. CEER enables compliance-aware whole-body control in an interpretable task space defined by root motion commands and end-effector pose targets, and supports plug-and-play integration with heterogeneous high-level planners. A teacher-student framework is adopted to distill a general motion-tracking controller into a low-level policy that consumes only EE-root commands. We further construct a hierarchical system that integrates heterogeneous planners and task modules through the EE-root interface, enabling diverse manipulation tasks without retraining the underlying whole-body policy. Experiments in simulation and on hardware demonstrate 3.3 cm end-effector tracking accuracy with substantially reduced jerk compared to baselines, stable contact-rich manipulation under teleoperation, and up to 70% success in simulated single-object loco-manipulation tasks within a room-scale environment. These results indicate that compliant EE-root control provides a practical abstraction for humanoid loco-manipulation, enabling modular and scalable integration of diverse skills.

Abstract

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