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Precise SE(3) End-Effector Tracking in Whole-Body Humanoid Control

Joohwan Seo, Xiaofeng Guo, Jinkun Cao, Roberto Horowitz, Rocky Duan, Guanya Shi, Koushil Sreenath

arXiv:2610.09479v1cs.ROcs.SYeess.SY

TL;DR

Precise end-effector pose tracking during humanoid whole-body motion must contend with base oscillations, gravity, dynamic coupling, and locomotion disturbances. ResGAC combines geometric task-space feedback with residual reinforcement learning and uses a ground-attached heading frame to limit pelvis-motion propagation to the target. Across four real-world tracking trajectories it reports lower position and orientation errors than baselines, and it achieves higher peg-in-hole success than SONIC.

  • Problem

    Humanoid manipulation requires accurate end-effector translation and rotation during locomotion and balance despite base oscillations, gravity, whole-body dynamics, and foot impacts.

    Source: §I · p. 1

  • Method

    ResGAC combines floating-base geometric admittance control for nominal arm targets with residual reinforcement learning in a shared joint-position action space, using a ground-attached heading frame to limit pelvis-motion propagation.

    Source: §I · p. 2

  • Results

    ResGAC achieves the lowest position and orientation errors on all four tracking trajectories and succeeds in 18/20 peg-in-hole trials (90%), compared with 10/20 (50%) for SONIC-Teleop-v1.1.

    Source: B. Manipulator Tracking Performance · p. 6, D. Peg-in-Hole Task · p. 7

  • Takeaways & Limitations

    The ground-attached heading frame prevents pelvis roll, pitch, and vertical motion from propagating to the desired end-effector pose while retaining planar locomotion.

    Source: §I · p. 2, §V · p. 8

Abstract

from arXiv · show

Precise end-effector tracking during humanoid whole-body motion is challenging due to floating-base oscillations, gravity, dynamic coupling, and locomotion-induced disturbances. We propose ResGAC, a whole-body humanoid controller for precise end-effector pose tracking that combines geometric admittance control (GAC) with residual reinforcement learning. GAC provides structured $\SE$ task-space feedback and generates nominal arm joint-position targets, while residual RL compensates for unmodeled dynamics and coordinates locomotion and balance in the shared joint-position action space. The left-invariant geometric formulation allows the same GAC law to be used across manipulation reference frames. This enables the use of a ground-attached heading frame that preserves planar locomotion while removing pelvis roll, pitch, and heave from the manipulation reference, thereby reducing reference-induced end-effector motion during locomotion. ResGAC is validated on a real Unitree G1 humanoid. Across four standing end-effector tracking benchmarks, ResGAC consistently outperforms representative baselines, including SONIC, achieving lower translational and rotational errors. Real-world experiments further demonstrate reduced propagation of pelvis motion to the desired end-effector pose using the proposed ground-attached heading frame. ResGAC achieves $90\%$ success in a standing peg-in-hole task compared with $50\%$ for SONIC, and accurate world-frame $\SE$ end-effector pose tracking during lower-body motion. Experimental videos are included in the supplementary material and are also available on the project website: https://resgac.github.io/ResGAC-website/.

I. INTRODUCTION

ResGAC combines geometric SE(3) feedback with residual reinforcement learning for precise humanoid end-effector tracking during locomotion and balance. Real-world evaluation reports better tracking than representative baselines, while a ground-attached heading frame limits pelvis-motion propagation to the target.

I. INTRODUCTION

  • Precise humanoid manipulation requires translational and rotational end-effector accuracy despite base oscillations, gravity, whole-body dynamics, and foot impacts.Prior work has explored task-space interfaces and stabilization, but high-precision SE(3) tracking during humanoid whole-body motion remains comparatively under-emphasized.Source: §I · pp. 1–2
  • ResGAC combines floating-base geometric admittance control (GAC), which generates nominal joint-position targets from task-space feedback, with residual RL in the shared joint-position action space.The learned correction compensates for unmodeled dynamics and coordinates locomotion and balance.Source: §I · p. 2
  • A left-invariant GAC law supports different manipulation reference frames, including a ground-attached heading frame that removes pelvis roll, pitch, and heave while preserving planar locomotion.This frame is introduced to prevent those pelvis motions from propagating to the desired end-effector pose.Source: §I · p. 2
  • ResGAC consistently outperforms representative baselines, including SONIC, across real-world end-effector tracking, walking, peg-in-hole, and world-frame tracking evaluations.The framework is validated on a Unitree G1 humanoid.Source: §I · p. 2

A. Lie Group and Frame Notation

  • The formulation distinguishes world, manipulation-reference, and end-effector frames, representing relative end-effector pose in SE(3) and relative velocity as a body twist.It also introduces Lie-group notation, including hat and vee maps between vectors and skew-symmetric matrices.Source: A. Lie Group and Frame Notation · p. 2

B. Problem Formulation

The problem is to track dual-arm end-effector poses while the humanoid executes locomotion and maintains balance. The command combines desired walking motion, pelvis height, and reference-frame end-effector poses.

B. Problem Formulation

  • The objective is accurate tracking of commanded end-effector position and orientation while executing locomotion and maintaining balance.The command represents left and right desired poses in the selected manipulation reference frame.Source: B. Problem Formulation · p. 3
  • The humanoid uses joint-position targets tracked by low-level PD controllers, so task-space feedback is mapped to joint targets before residual corrections are applied.With zero residual, the nominal controller is recovered; the learned action provides a bounded correction to the nominal command.Source: B. Problem Formulation · p. 3
  • A pelvis-frame command transmits locomotion-induced roll, pitch, and vertical oscillations to the desired end-effector trajectory, motivating a reference that retains planar motion without that propagation.A fixed relative pose can still produce a moving world-frame target when the reference frame moves.Source: B. Problem Formulation · p. 3

III. METHOD

The method combines floating-base GAC with residual RL: GAC supplies nominal arm targets, while the policy corrects those targets and controls the remaining whole-body joints. The relative Jacobian is constructed to express end-effector motion with respect to a moving manipulation frame.

III. METHOD

  • GAC generates nominal arm joint-position targets, while residual RL corrects the arm targets and controls the remaining whole-body joints.The two commands are combined within the whole-body controller.Source: §III · p. 3

1) Calculation of Jacobian Matrix:

  • The relative Jacobian and velocity are constructed for the end effector with respect to the moving manipulation reference frame using world-aligned quantities transformed into the end-effector body frame.This construction addresses interfaces that primarily provide frame quantities relative to the world rather than relative to each other.Source: 1) Calculation of Jacobian Matrix · p. 3
  • The full relative Jacobian includes joints contributing to end-effector motion, but GAC is applied only through arm-joint columns, leaving the other joints to RL.The full Jacobian is used to evaluate relative end-effector velocity.Source: 1) Calculation of Jacobian Matrix · p. 3

2) Geometric Admittance Control with Null-Space Handling:

GAC acts as a geometric task-space tracking controller that converts SE(3) pose error into nominal joint-position targets. Its left-invariant feedback supports changes of manipulation reference frame, and the resulting command is combined with residual policy output.

2) Geometric Admittance Control with Null-Space Handling:

  • GAC is used as a task-space tracking controller, not as a wrench-driven compliant controller, and generates joint-position targets from SE(3) pose error.The command is interpretable as a differential inverse-kinematics step.Source: 2) Geometric Admittance Control with… · p. 3
  • ResGAC also supplies pose-tracking error and relative end-effector velocity to the residual policy as an approximately Markovian task-space tracking state under the approximate closed-loop dynamics.The resulting joint-position target is combined with the residual policy.Source: 2) Geometric Admittance Control with… · pp. 3–4
  • The left-invariant geometric feedback is unchanged by a common change of reference coordinates, allowing the same law across manipulation frames.Its translational and rotational stiffness matrices are positive definite.Source: 2) Geometric Admittance Control with… · p. 4
  • The desired body-frame velocity is discretized and mapped to joint space with a damped pseudoinverse, using null-space regulation for the redundant 7-DoF arm.The nominal arm configuration and null-space gain define the posture regulation.Source: 2) Geometric Admittance Control with… · p. 4

B. Residual Reinforcement Learning

ResGAC combines structured GAC arm feedback with residual RL that corrects arm commands and controls other joints for locomotion and balance. The actor observes the command and task state, while privileged simulation quantities are reserved for the critic.

C. Selection of Manipulation Reference Frame

The manipulation reference frame determines which floating-base motions reach the end-effector command. The ground-attached heading frame preserves planar motion while excluding pelvis roll, pitch, and heave, although physical disturbances still require learned compensation.

D. Training of ResGAC

The training reward jointly targets end-effector tracking and locomotion while regularizing undesirable whole-body behavior. Gait-cycle drift penalties distinguish unintended net motion from periodic locomotion oscillations.

1) Reward Function:

  • The reward combines position terms based on squared-L2 and L1 errors with an orientation term defined by intrinsic SO(3) trace error.Source: 1) Reward Function · p. 5
  • Residual magnitude and mean-action acceleration are penalized, while locomotion uses periodic gait phases offset by π and gait-cycle averages.Source: 1) Reward Function · p. 5
  • Three drift penalties target unintended translation during turning, yaw during linear commands, and lateral velocity, using gait-cycle averages to avoid penalizing periodic oscillations.Source: 1) Reward Function · p. 5

2) Curriculum:

ResGAC training progresses from fixed-target standing to broader task-space commands and locomotion. The trajectories are generated in task space from feasible arm configurations rather than retargeted whole-body human-motion data.

2) Curriculum:

  • Training begins with fixed-target standing so the residual policy can cooperate with nominal GAC while maintaining balance.Source: 2) Curriculum · p. 5
  • Later stages introduce trajectory variation and locomotion, tighten tracking, and broaden task-space commands for teleoperation and Mime-man.Source: 2) Curriculum · p. 5
  • End-effector trajectories are generated directly in task space from feasible arm configurations without retargeted whole-body human-motion data.Source: 2) Curriculum · p. 5

3) Symmetry Augmentation and Regularization:

ResGAC combines symmetry-regularized learning with geometric task-space feedback and residual learning, then evaluates tracking on real humanoid hardware. It achieves lower errors than the baselines across four standing trajectories.

3) Symmetry Augmentation and Regularization:

  • Sagittal-reflection augmentation mirrors replay transitions, while actor equivariance and critic invariance regularize the learned policy toward symmetric behavior.The symmetry terms are added to the corresponding RL objectives.Source: 3) Symmetry Augmentation and… · p. 5

IV. EXPERIMENTS

  • The real-world Unitree G1 evaluation covers pose trajectory tracking, arm stabilization while walking, peg-in-hole manipulation, and world-frame end-effector tracking.The standing tracking trajectories are held out from training; baselines use pelvis-frame commands when supported, except SONIC-Teleop-v1.1, which uses its heading-frame representation.Source: §IV · p. 6, B. Manipulator Tracking Performance · p. 6

A. Implementation Details

  • Training uses 4096 parallel IsaacSim environments on one NVIDIA L40S GPU, and the six-stage curriculum takes approximately 16 hours.GAC and residual policy run at 50 Hz; the G1 low-level PD controllers track the resulting joint-position commands.Source: A. Implementation Details · p. 6

B. Manipulator Tracking Performance

  • ResGAC converges faster and has lower steady-state position and orientation errors than E2E-RL and SONIC-Teleop-v1.1 after setpoint changes.Figure 2 aggregates 25 command-transition intervals; solid curves show mean errors and shaded regions show one standard deviation.Source: Figure 2 · p. 6, B. Manipulator Tracking Performance · p. 6
  • The standing comparison reports mean ± standard deviation over temporal samples and both arms, with only steady-state samples included for setpoint tracking.Evaluation trajectories are held out from training, although training covers their task-space ranges and motion classes.Source: B. Manipulator Tracking Performance · p. 6
  • ResGAC yields the lowest errors on all four trajectories, with 6.5/6.6 mm position and 1.37/1.67° orientation errors on setpoint/pick-and-place, 54–67% lower position and approximately 72% lower orientation error than the best baseline.Table III reports standing hardware results; setpoint values are steady-state samples after command transitions.Source: B. Manipulator Tracking Performance · p. 6, Table III · p. 7

C. End-effector Stabilization via Frame Selection

During walking arm-holding, the ground-attached heading frame reduces some pelvis-motion transmission to the end effector compared with the pelvis frame. The effect differs by motion component.

C. End-effector Stabilization via Frame Selection

  • The transmission ratio compares end-effector and pelvis motion standard deviations for roll, pitch, and z; a lower ratio indicates weaker pelvis-motion transmission.The walking comparison uses ResGAC policies commanding a fixed end-effector pose in either the pelvis frame or H0.Source: C. End-effector Stabilization via… · p. 7
  • With H0, roll transmission falls from 0.958 to 0.790 and EEF MAE from 1.11° to 0.81°; z EEF MAE falls from 22.5 mm to 12.6 mm and transmission from 1.068 to 0.884.Pelvis-motion statistics remain comparable between the policies; pitch transmission is similar (0.954 and 0.964) because the lever-like arm configuration primarily transmits locomotion-induced impacts in pitch.Source: C. End-effector Stabilization via… · p. 7

D. Peg-in-Hole Task

In a standing peg-in-hole task, ResGAC succeeds more often than SONIC-Teleop-v1.1. In the Mime-man demo, ResGAC tracks world-frame end-effector poses more closely while the humanoid moves its body.

D. Peg-in-Hole Task

  • ResGAC succeeds in 18/20 peg-in-hole trials (90%), compared with 10/20 (50%) for SONIC-Teleop-v1.1.The task uses a 10-mm diametral clearance and an externally estimated target affected by marker-to-hole registration uncertainty.Source: D. Peg-in-Hole Task · p. 7

E. World-frame Tracking: Mime-Man Demo

  • World-frame commands are converted into H0 at each control step, and a causal inverse-dynamics compensator is applied to reduce closed-loop tracking lag.The compensator approximates tracking dynamics as a second-order system; reference derivatives are computed numerically for causality.Source: E. World-frame Tracking: Mime-Man Demo · pp. 7–8
  • In the Mime-man demo, ResGAC achieves world-frame position and orientation MAEs of 8.8 mm and 2.69°, versus 30.0 mm and 13.75° for E2E-RL.The comparison uses the same H0 reference formulation, global wrapper, and world-to-H0 command conversion for both controllers.Source: E. World-frame Tracking: Mime-Man Demo · pp. 7–8, Figure 3 · p. 8

V. CONCLUSION

  • The authors conclude that ResGAC supports precise SE(3) task-space control for whole-body humanoid loco-manipulation.The conclusion cites improved translational and rotational tracking, stabilization through frame selection, peg-in-hole success, and world-frame tracking during body motion.Source: §V · p. 8
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