LOInK: Learned Optimal Inverse Kinematics via Structured Neural Surrogate Models

Michael Somerfield; Damian Abood; Ruigang Wang; Ian R. Manchester

cs.ROSep 18, 2026

LOInK: Learned Optimal Inverse Kinematics via Structured Neural Surrogate Models

Authors: Michael Somerfield, Damian Abood, Ruigang Wang, Ian R. Manchester

Organizations: Australian Robotic Inspection and Asset Management (ARIAM) Hub, the Australian Centre for Robotics (ACFR), and the School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Australia

Abstract

We introduce Learned Optimal Inverse Kinematics (LOInK), a method to generate approximately optimal solutions to an inverse kinematics problem. When trained on data consisting of sampled configurations and associated task variables and a given cost function, LOInK learns a bi-Lipschitz invertible mapping from configuration space to a decoupled task/latent space, and moreover, the latent space is structured so as to place cost-minimizing solutions at the origin. This enables efficient sampling of cost-minimizing solutions via a network-inversion algorithm based on operator splitting. We demonstrate the proposed approach on three problems: an illustrative three degree-of-freedom manipulator problem; a quadrupedal climbing robot for which LOInK can generate near-optimal solutions on average 31 times faster and up to 100 times faster than a constrained optimization approach; and a simulated soft actuator as a purely data-driven example, in which LOInK can explicitly generate high-quality solutions, unlike existing generative approaches that require diverse sampling and evaluation of candidate solutions.

Figures & tables

Fig. 1: The main idea of LOInK is to learn an invertible map $\mathcal{G}$ (a bi-Lipschitz diffeomorphism) between configuration space and a decoupled task/latent-space coordinate system. Moreover, on the latent space, the origin $z=0$ corresponds to the cost-minimizing solution. Then, for a given task variable $y$ , the optimal configuration can be evaluated as $x^{\star}=\mathcal{G}^{-1}([y;0])$ .

Fig. 2: LOInK model structure and training process, where we have highlighted the decoupled output

[y\;z]^{T}\in\mathbb{R}^{n}

and the separate loss functions for each output space. In the task space we impose a fitting loss between dataset tasks

y_{i}\in\mathbb{R}^{p}

and network output

\hat{y}_{i}:=\hat{y}(x_{i})

whilst the latent space fitting loss structures the latent space so that the mapping

\hat{J}(z_{i},\hat{y}_{i})

defined in ( 10 ) approximates the auxiliary costs

J_{i}

from the dataset

\mathcal{D}

Fig. 3: Demonstrative example of the planar 3-DoF manipulator, where the task space is the horizontal position. At each of three task space locations, there is a particular minimal cost. In the learned latent space $z_{1},z_{2}$ , cost increases radially from the origin.

Fig. 4: Configuration velocities resulting from end-effector trajectory tracking with different values of $\mu$ . We observe that both solutions effectively track the desired task trajectory $y$ (red), while the larger value for $\mu$ produces smoother velocity profiles, as expected c.f. Sec IV-D .

Fig. 5: Left: Magneto robot with center of mass and gravity direction. Right: End-effector force $f$ (red) resulting from adhesion force $\sigma$ (blue), frictional normal force $f_{n}$ (green) and tangential force $f_{t}$ (magenta).

Fig. 6: Comparison between a cost minimizing solution (blue) for Magneto with required adhesion force $J_{f}=1.004$ N and a non-minimizing solution (red) with $J_{f}=13.047$ N. Note the straight upper limbs of the optimal solution. The direction of gravity is shown with the blue arrow.

Solver	Time [ms]	EE error [mm]	Cost ( $J$ )
LOInK	5.47 (5.31–5.78)	38.9 (17.47–76.5)	0.237 (0.08–0.46)
IPOPT	171.1 (77.6–617.7)	15.6 (7.24–30.4)	0.227 (0.08–0.43)

TABLE I: Comparisons between LOInK and optimization method IPOPT [mean (min-max)].

Fig. 7: Example postures for the simulated three-segment planar HSA robot, showing configurations that can be achieved on the platform through its actuation ranges.

Fig. 8: Histogram comparing positional error of the task space position across 2000 end-effector positions.

Fig. 9: Histogram showing the ratio of actuation efforts versus an optimization-based reference, where our solutions are obtained at

z=0

for a sample of 2000 random end-effector targets.

Fig. 10: Costs of the distribution obtained by sampling

z\sim\mathcal{N}(0,cI)

with varying

c

Solver	Time [ms]	EE error (mm)	Cost ( $J$ )
LOInK	2.66 (2.66-2.67)	0.44 (0.0022-35.049)	0.17 (0.0003-0.59)
IKFlow	4.51 (4.43-4.56)	1.42 (0.0037-43.80)	0.38 (0.01-0.71)

TABLE II: Comparison with IKFlow for the Soft Robot Manipulator: Solve times, EE error, and cost [mean (min-max)]

Inverse Kinematics

Inverse kinematics maps desired robot or articulated-body motions to joint configurations while handling constraints, redundancy, and multiple solutions.

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MorphIK: Morphology-Conditioned Neural Inverse Kinematics for Unknown Robots

Lennart Clasmeier, Jan Gerrit Habekost, Cornelius Weber, Stefan Wermter

Knowledge Technology Group, Department of Informatics, University of Hamburg, Hamburg, Germany

Cross-Embodiment Robot Learning Inverse Kinematics Robot Kinematics

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Neural models can learn to generate various solutions to the inverse kinematics problem from data, but are usually limited to a single robot. We present MorphIK, a flow-matching model that solves inverse kinematics for revolute-joint-based kinematic chains it has never seen during training. The model uses a transformer architecture to encode the robot's morphology along with the target pose. This encoding then conditions a flow-matching head that generates poses from noise. Trained on purely synthetic data from procedurally generated robots, the model reaches a precision of about 5 cm on unseen real-world robots with 6 to 9 Degrees of Freedom. For higher precision, the model serves as an excellent Prior for further optimization algorithms, reducing error to less than 1 cm after a single step of Damped Least Squares optimization and to sub-1 mm error after 3 steps in most cases. Building on flow matching's generative capabilities to produce highly diverse outputs, our model can efficiently sample the robot's null space, providing a wide variety of configurations for the same pose. Thus, overall, MorphIK allows learning and generalizing neural inverse kinematics for a multitude of known and unknown robots.

Abstract

Inverse Kinematics

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