Humanoid SRL: Augmenting Humanoids with Supernumerary Limbs for Coordinated Loco-Manipulation

Fukang Liu; Feiyang Wu; Paul Barsa; Jaehwi Jang; Akshat Jain; Zhigen Zhao; Yipu Chen; Zhaoyuan Gu

Figure 1: Configurations and real world tasks. The humanoid platform is equipped with a supernumerary

4

-DoF arm in neck-mounted and back-mounted embodiments. (A–G) Photographs of both configurations in distinct SRL poses. (H–T) Task snapshots illustrating the diversity of behaviors enabled by the tri-arm morphology, including reach-augmented manipulation, coordinated manipulation, dynamic loco-manipulation, and human–robot collaborative whole-body behaviors.

Figure 2: Morphological augmentation of workspace and support. ( A ) Neck-mounted and back-mounted SRL embodiments. ( B ) Workspace augmentation in body-frame. ( i ) Reachable workspace overlays for the native arms only, native arms with neck-mounted SRL, and native arms with back-mounted SRL. ( ii ) Directional reach comparison in the body frame (F: forward; R: rearward; LL: left lateral; LR: right lateral; U: upward; D: downward). ( iii ) Workspace decomposition into single-arm volume, dual-arm co-reach, and tri-arm co-reach. ( C ) Translational manipulability isotropy across nominal standing frame slices. The neck-mounted SRL enhances manipulability in frontal and overhead regions, whereas the back-mounted SRL improves performance in the rear, lateral, and lower regions. ( D ) Projected CoM distributions for sampled SRL configurations and payloads of

0-4

kg, compared with the foot-contact polygons of the selected standing configurations.

Figure 3: Support and active stability augmentation with the SRL. ( A, B ) CAREN perturbation experiments with and without back-mounted SRL support against the external frame. ( C ) Forward leaning while the native arms carry a payload and the SRL presses against a wall. ( D, E ) CoM responses with SRL support under repeated

10

cm perturbations and without support under increasing perturbation amplitudes, including balance loss at

7.5

cm. The

10

cm amplitude is the experimental safety limit. ( F ) Analysis of wall bracing. ( i ) Global-frame CoM and SRL-endpoint trajectories relative to the feet midpoint. The feet-only support polygon and projected contact hull are shown separately from conditional static-equilibrium regions computed with unilateral contacts, wall and floor friction coefficients

\mu=1

, and wall-normal-force caps of

5\%

,

10\%

, and

20\%

of modeled weight

W

. ( ii ) Forward CoM position versus time, showing a peak excursion of

16.1

cm beyond the feet-only forward boundary.

Figure 4: Reach-augmented manipulation with the SRL. (A–D) High-shelf retrieval, shelf transfer without torso reorientation, native-arm-supported reach-around, and concurrent rear/front manipulation. (E–H) Corresponding reference-based analyses. (i) Body-frame signed distance to the sampled native-arm workspace; positive values indicate points outside, and shading indicates voxel-size sensitivity (

15

–

25

mm). (ii) World-frame SRL and torso heights (E) ; initial-heading-frame SRL trajectory between shelves (F) ; projected CoM trajectory and conditional static-support regions (G) ; and instantaneous body-frame SRL and right-hand positions (H) .

Figure 5: Coordinated augmentation through tri-arm manipulation. ( A ) Shelf transfer, in which the native arms lift a box from a low shelf and the SRL assists in final placement on a high shelf. ( B ) Coordinated loading, in which the native arms lift a container while the SRL retrieves an object from a top shelf and places it into the container. ( C ) Stacked-box stabilization, in which the native arms lift a multi-layer box stack while the SRL provides an additional stabilizing contact. ( D )( i ) The vertical motion of the SRL and native left hand in the instantaneous body frame during during shelf transfer. ( D )( ii ) The full object path to the final high-shelf placement. ( E ) The heights of the SRL and native left hand in the instantaneous body frame during coordinated loading. ( F ) Contact schematic and Ferrari–Canny sensitivity analysis for stacked-box stabilization, showing that the added SRL contact increases wrench support and enlarges the region of positive

\epsilon

values under a frictional contact model.

Figure 6: Dynamic and collaborative whole-body augmentation with tri-arm coordination. ( A ) Collaborative table transport with a human partner, where the native arms share the table load while the SRL stabilizes a loose object during lifting and walking. ( B ) Box transport through a weighted door, with the native arms maintaining the carried object while the neck-mounted SRL pushes the door during locomotion. ( C ) Door opening with the back-mounted SRL, which engages the handle from behind while the native arms continue carrying a box.

Figure 7: Hierarchical framework for supernumerary humanoid loco-manipulation. Trajectory optimization generates dynamically feasible whole-body references that coordinate the native limbs and SRL under contact and actuation constraints. Learn-to-Teach combines privileged teacher supervision and shared experience to train a student policy for reference tracking using noisy proprioceptive observations. The student policy transfers zero-shot to hardware, enabling coordinated manipulation, support, and locomotion through whole-body feedback control. Vision-guided references extend the same control framework to autonomous task execution.

Figure S1 : Exploded view of the supernumerary arm. . The shoulder comprises three co-located joints, with the actuators housed in high-infill 3D-printed brackets. The elbow link consists of two parallel aluminum plates that provide structural rigidity and accommodate a dual-supported bearing arrangement and belt-tensioning mechanism. The elbow is actuated remotely through a

3{:}1

belt transmission. The hollow aluminum forearm provides structural stiffness and internal cable routing, with an interface for different end-effectors. The assembly is shown with a parallel-jaw gripper.

Property	Specification
Degrees of freedom / arm length	$4$ / $83.0$ cm
Upper arm / forearm length	$37.3$ / $45.7$ cm
Total mounted mass	$4.7$ kg
Continuous / peak payload	$2$ / $4$ kg
Gripper mass	$0.5$ kg
Battery / approximate runtime	$36$ V, $8$ Ah / $\sim 65$ min

Table S1 : SRL system and integration specifications

Joint	Motion	Actuator	External ratio	Continuous torque	Peak torque	Joint range
J1	Shoulder yaw	RS-02	$1{:}1$	$\pm 6$ Nm	$\pm 17$ Nm	$[-180^{\circ},180^{\circ}]$
J2	Shoulder pitch	RS-03	$1{:}1$	$\pm 20$ Nm	$\pm 60$ Nm	$[-100^{\circ},100^{\circ}]$
J3	Shoulder roll	RS-02	$1{:}1$	$\pm 6$ Nm	$\pm 17$ Nm	$[-90^{\circ},90^{\circ}]$
J4	Elbow	RS-02	$3{:}1$	$\pm 6$ Nm	$\pm 17$ Nm	$[-125^{\circ},50^{\circ}]$

Table S2 : Joint and actuator specifications of the SRL

Component	Material	Fabrication	Detail
Main shoulder bracket	PLA	3D printed	Actuator housing
Primary elbow plate	AL-6061	Waterjet	Elbow structure
Secondary elbow plate	AL-6061	Waterjet	Elbow structure
Elbow shells	PLA	3D printed	Elbow enclosure
Forearm tube	AL-6061	OTS	Internal cable routing
Wrist shaft	AL-7075	Machined	Keyed interface

Table S3 : Mechanical components and fabrication of the SRL

Figure S2 : SRL tracking under different payloads. (A, B) Encoder-derived paths with a

5

lb payload, projected onto the

xy

plane for the square (

z=0.30

m) and the

xz

plane for the circle (

y=0.50

m). The square has

0.30

m sides, and the circle has a

0.30

m radius; black dashed curves indicate commanded reference trajectories. Colors denote mean reference speeds of approximately

0.10

,

0.20

, and

0.30

m/s. (C, D) Time-weighted 3D position RMSE versus payload for the square and circle trajectories, respectively. Each point summarizes approximately ten analyzed cycles from a single recording.

Figure S3 : Multi-view visualization of the augmented workspace. Reachable workspaces for the neck-mounted (top row) and back-mounted (bottom row) configurations, displayed from isometric, front, side, and top views. Translucent volumes represent the reachable workspaces of the native arms and the SRLs, illustrating their spatial overlap.

Setting	Value or implementation
Solver	Crocoddyl FDDP
Time integration	Symplectic (semi-implicit) Euler
Time step	$\Delta t=0.005$ s
Control parameterization	Piecewise-constant joint torques
Maximum iterations	$200$ per phase
Solver stopping threshold	$10^{-8}$

Table S4 : Trajectory-optimization settings for the neck-mounted example

Input	Symbol	Dimension	Student	Teacher
Base Linear Velocity	$\mathbf{\dot{p}}_{t}^{\rm b}$	$3$	✓	✓
Base Angular Velocity	$\omega^{\rm b}_{t}$	$3$	✓	✓
Projected Gravity	$\mathbf{g}_{t}$	$3$	✓	✓
History Motor Joint Position	$\mathbf{q}^{\rm j}_{\rm hist}$	$200$	✓	✓
Motor Joint Velocity	$\mathbf{\dot{q}}^{\rm j}_{t}$	$20$	✓	✓
Ref. End-effector Position	$\hat{\mathbf{{p}}}^{\rm ee}_{t}$	$15$	✓	✓

Table S5 : Observation spaces.

Category	Term	Expression	Weight
Task Reward	Motor Joint Position	$\exp(-5\\|\mathbf{{\hat{q}}}^{\rm j}_{t}-\mathbf{{{q}}}^{\rm j}_{t}\\|_{2}^{2})$	$3$
	Base Position	$\exp(-20\\|\mathbf{\hat{p}}^{\rm b}_{t}-\mathbf{{p}}^{\rm b}_{t}\\|_{2}^{2})$	$3$
	Base Orientation	$\exp(-50\\|\hat{\boldsymbol{\theta}}^{\rm b}_{t}-{\boldsymbol{\theta}}^{\rm b}_{t}\\|_{2}^{2})$	$3$
	Base Linear Velocity	$\exp(-2\\|\mathbf{\hat{\dot{p}}}_{t}^{\rm b}-\mathbf{{\dot{p}}}^{\rm b}_{t}\\|_{2}^{2})$	$3$
	Base Angular Velocity	$\exp(-0.5\\|\hat{\boldsymbol{\omega}}^{\rm b}_{t}-{\boldsymbol{\omega}}^{\rm b}_{t}\\|_{2}^{2})$	$3$
	End-effector Position	$\exp(-20\\|\hat{\mathbf{{p}}}^{\rm ee}_{t}-{\mathbf{{p}}}^{\rm ee}_{t}\\|_{2}^{2})$	$3$

Table S6 : Reward Components and Weights.

Category	Parameter	Type	Range / Std
Observation	Ref. End-effector Position (m)	Additive Gaussian	$\sigma=0.04$
	Motor Joint Position (rad)	Additive Gaussian	$\sigma=0.035$
	Joint Velocity (rad/s)	Additive Gaussian	$\sigma=0.03$
	Base Linear Velocity (m/s)	Additive Gaussian	$\sigma=0.03$
	Base Angular Velocity (rad/s)	Additive Gaussian	$\sigma=0.03$
	Projected gravity direction (unitless)	Gaussian; unit norm	$\sigma=0.015$

Table S7 : Domain Randomization Parameters.

Parameter	Value
PPO clipping parameter $\epsilon$	$0.3$
Discount factor $\gamma$	$0.97$
Learning rate	$5\times 10^{-5}$
Entropy coefficient	$1\times 10^{-2}$
Rollout length	$20$
Number of minibatches	$32$

Table S8 : RL policy training hyperparameters.

			$\Delta$ vs. Asym-PPO (%)
Metric	Asym-PPO	L2T (ours)	Mean	Max
Episodic return (teacher $=1$ ) $\uparrow$	0.916	0.964	+5.3	+9.1
End-effector MPKPE (mm) $\downarrow$	124.1	113.3	+11.2	+32.6
Joint-position RMSE (rad) $\downarrow$	0.053	0.048	+8.6	+35.0

Table S9 : Comparison of whole-body controller performance. Both policies use onboard observations. Values are task averages, with episodic return normalized by the privileged teacher for each task. Relative improvements are computed per task and then averaged; positive values indicate higher return or lower tracking error. MPKPE measures end-effector position error in the robot’s base frame. Results use one training seed per task.

Figure S4 : Distribution of controller performance improvements. Relative improvements of the L2T student over Asym-PPO in episodic return, end-effector tracking, and joint-position tracking. Each dot represents an evaluated task, and horizontal bars indicate averages across tasks. Positive values indicate higher return or lower tracking error for L2T; negative values favor Asym-PPO.

Type	Signal	Dim.	Rate	Shelf	Box	Door
Observation	Torso / SRL end-effector RGB-D	$240\times 424\times 4$ each	15 Hz	✓	✓	✓
	Motor positions	$24$	200 Hz	✓	✓	✓
	Gripper state (last command)	$1$	200 Hz	✓
	Policy mode (last command)	$1$	200 Hz			✓
Action	End-effector positions (body frame)	$3\times 5$	50 Hz	✓	✓	✓
	Base pose (body frame)	$6$	50 Hz	✓	✓

Table S10 : Diffusion-policy observation and action specifications. Sensor rates are logging rates; training data are resampled to

50

Hz, with each sample paired with the most recent images. Check marks indicate signals used by each task-specific policy: top-shelf retrieval (Shelf), stacked-box stabilization (Box), and object transport during door interaction (Door).

Category	Parameter	Value
Architecture	Diffusion backbone	1D conv. U-Net (LeRobot)
	U-Net channels	$(512,1024,2048)$ (shelf, box); $(128,256,512)$ (door)
	Kernel size / GroupNorm groups	$5$ / $8$
	Timestep embedding dim.	$128$
	Conditioning	FiLM (scale and shift)
	Visual encoder	ResNet-18, GroupNorm, shared across cameras

Table S11 : Diffusion-policy architecture and training parameters

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