Off-Axis Compliant RCM Joint with Near-Isotropic Stiffness and Minimal Parasitic Error

Federico Mariano; Elena De Momi; Giovanni Berselli; Jovana Jovanova; Just L. Herder; Leonardo S. Mattos

cs.ROMar 30, 2026

Off-Axis Compliant RCM Joint with Near-Isotropic Stiffness and Minimal Parasitic Error

Authors: Federico Mariano, Elena De Momi, Giovanni Berselli, Jovana Jovanova, Just L. Herder, Leonardo S. Mattos

Organizations: Biomedical Robotics Lab, Department of Advanced Robotics, Istituto Italiano di Tecnologia, Genoa, Italy. · Department of Electronics, Information and Bioengineering, Politecnico di Milano, Milan, Italy · DIME, Università di Genova, Genoa, Italy · Department of Maritime and Transport Technology, Delft University of Technology, Delft, The Netherlands. ·

Abstract

This paper presents an off-axis, monolithic compliant Remote Center of Motion (RCM) joint for neuroendoscopic manipulation, combining near-isotropic stiffness with minimal parasitic motion. Based on the Tetra II concept, the end-effector is placed outside the tetrahedral flexure to improve line of sight, facilitate sterilization, and allow rapid tool release. Design proceeds in two stages: mobility panels are sized with a compliance-based isotropy objective, then constraining panels are synthesized through finite-element feasibility exploration to trade stiffness isotropy against RCM drift. The joint is modeled with beam elements and validated via detailed finite-element analyses, including fatigue-bounded stress constraints. A PA12 prototype is fabricated by selective laser sintering and characterized on a benchtop: a 2 N radial load is applied at the end-effector while a 6-DOF electromagnetic sensor records pose. The selected configuration produces a stiffness-ellipse principal axis ratio (PAR) of 1.37 and a parasitic-to-useful rotation ratio (PRR) of 0.63%. Under a 4.5° commanded rotation, the predicted RCM drift remains sub-millimetric (0.015-0.172 mm). Fatigue analysis predicts a usable rotational workspace of 12.1°-34.4° depending on direction. Experiments reproduce the simulated directional stiffness trend with typical deviations of 6-30%, demonstrating a compact, fabrication-ready RCM module for constrained surgical access.

Figures & tables

Panel of Figure 1 — Fig. 1 : (a) Representation of a double parallelogram RCM mechanism. The mutual parallelism of A, B, C, and D forces the prolongation of the EE to pass through the RCM - (b) Representation of a RCM mechanism with circular guides.

Fig. 2 : View of dual compliant joint composed of two identical joint connected in series. This joint can move the EE around the RCM along the pitch and roll directions. [ 17 ]

Fig. 3 : Isometric view of Tetra II joint composed of nine flexural walls ( $p_{1},...,p_{9}$ ). The blue panels, obtained connected the nodes $N1,N2,N3$ and $N4$ , are responsible for joint motion and are therefore the ones to be modified to have isotropic stiffness behavior. The red panels block the joint’s rotation, constraining the tool from following circular rotations around the RCM.

Panel of Figure 4 — Fig. 4 : Scheme of the line of sight of the surgeon over the entry area. The green cone represents the visible area while the black cone represents the non visible area. In (a) the joint has the EE located at the center of tetrahedral structure, while in (b) the joint has the EE located at the side of tetrahedral structure. The EE is represented with the red line.

Fig. 5 : Scheme of the three panels responsible for the joint movement. Highlighted are nodes $N1,N2,N3$ and $N4$ , along with the outputs of the optimization $L_{1},L_{2},L_{3},t_{1},t_{2},t_{3},\theta_{1}$ and $\theta_{2}$ .

Parameter	Type	Interval
$L_{ref}$	Length	$20\,\mathrm{mm}\leq L_{ref}\leq 100\,\mathrm{mm}$
$t_{ref}$	Thickness	$1\,\mathrm{mm}\leq t_{ref}\leq 6\,\mathrm{mm}$
$t_{\triangle}$	Thickness	$1\,\mathrm{mm}\leq t_{\triangle}\leq 6\,\mathrm{mm}$
$H$	Height	$20\,\mathrm{mm}\leq H\leq 100\,\mathrm{mm}$
$\alpha$	Angle	$10^{\circ}\leq\alpha\leq 70^{\circ}$

TABLE I : Parameter ranges

Panel of Figure 7 — Fig. 6 : Representation of the joint from frontal view in (a) , lateral view in (b) and isometric view in (c) . In blue, the structure is responsible for the movement of the end effector, and in red, the constraining structure locks the RCM during the movement. Highlighted are the nodes $N1,N2,N3$ and $N4$ with the angle $\alpha$ and all the thickness $t1,t2,t3$ and $t_{\triangle}$ .

Fig. 7 : Workflow of the study: MATLAB identifies candidate geometries that satisfy the isotropic-stiffness condition for the mobility panels. A parametric FEM model is then evaluated and the metrics $IsoErr$ and $ParErr$ are computed from the output displacements. A feasibility analysis selects admissible designs and returns Solutions 1–4. These candidates are re-simulated with FEM, and the best configuration is chosen using the Principal-Axis Ratio $PAR$ , the Parasitic-to-useful Rotation Ratio $PRR$ , and the composite index $J$ .

Fig. 8 : Experimental setup used to validate the joint in real-life conditions. A mass producing a force of $2N$ is attached at point N4, while an Aurora microsensor, operating within the Aurora electromagnetic field, is used to measure the movement of point N4.

Panel of Figure 10 — Fig. 9 : (a) 3D model of the final compliant joint. (b) Polar graph showing

Parameter	Definition	Value / Ratio
$L_{1}$	reference length	$L_{ref}$
$L_{2}/L_{1}$	—	3.12
$L_{3}/L_{1}$	—	2.00
$t_{2}$	reference thickness	$t_{ref}$
$t_{1}/t_{2}$	—	1.03
$t_{3}/t_{2}$	—	2.01

TABLE II : Parameters of the isotropic stiffness solution.

Case	$L_{ref}$ [mm]	$t_{ref}$ [mm]	PAR
1	$20$	$1$	1.12
2	$40$	$1$	1.06
3	$40$	$2$	1.10

TABLE III : Verification of isotropic stiffness via ANSYS for three geometric scalings.

	Conf. 1	Conf. 2	Conf. 3	Conf. 4
$t_{ref}$ [mm]	1.59	1.15	1.33	1.83
H [mm]	97.56	84.01	95.98	67.59
$L_{ref}$ [mm]	64.10	70.84	97.86	99.76
$t_{\triangle}$ [mm]	3.35	3.55	3.42	1.67
$\alpha$ [ ∘ ]	34.40	46.33	42.68	26.61
IsoErr	0.12	0.22	0.27	0.52

TABLE IV : configuration obtained from the feasibility analysis

	Conf. 1	Conf. 2	Conf. 3	Conf. 4
PAR	1.17	1.29	1.37	1.87
PRR	0.032	0.015	0.0063	0.0041
J	0.0055	0.0043	0.0023	0.0036

TABLE V : configuration obtained from the feasibility analysis

Panel of Figure 15 — Fig. 10 : Graphs of two feasibility metrics, IsoErr (left column, in blue) and ParErr (right column, in red) versus the five inputs. ( a-b ) $L_{ref}$ , ( c-d ) $H$ , ( e-f ) $t_{ref}$ , ( g-h ) $t_{\triangle}$ , and ( i-j ) $\alpha$ . In each graph, the shown variable sweeps its range while the other four are fixed at best configuration obtained after the optimization ( $L_{ref}=97.86mm$ , $H=95.98mm$ , $t_{ref}=1.33$ , $t_{\triangle}=3.42mm$ and $\alpha=42.68^{\circ}$ .

$\theta_{f}$ [ ∘ ]	Stiffness [N/m]	Parasitic error [mm]
0	285.27	0.066
10	301.02	0.092
20	318.18	0.117
30	334.22	0.141
40	345.61	0.159
50	349.08	0.170

TABLE VI : Stiffness and parasitic error for a 4.5 ∘ rotation.

Fig. 11 : In blue, the rotational workspace of the joint around $\theta_{f}$ . - In red is the minimum workspace for superimposing the EE over the surgical trajectory.

Angle [ ∘ ]	Exp. mean $\pm$ SD	Sim.	% Error
	Stiffness [N/m]
0	374.05 $\pm$ 6.14	285.27	23.73
30	479.45 $\pm$ 30.22	334.22	30.29
60	423.41 $\pm$ 17.67	343.48	18.88
90	333.21 $\pm$ 28.82	297.02	10.86
120	289.33 $\pm$ 4.04	261.08	9.76

TABLE VIII : Comparison between experimental and simulated stiffness values, with associated percentage error.

Metric	Value
MAE [N/m]	63.19
RMSE [N/m]	73.92
MAPE [%]	16.88
Bias [N/m]	60.15

TABLE IX : Global error metrics for the comparison between experimental and simulated stiffness values.

Mechanism	Description	Key features
Tetra II [ 15 ]	A compliant RCM joint based on a tetrahedral flexure architecture, which maintains a fixed remote center of motion while steering the EE.	The EE is located on the central axis of the tetrahedral structure, which limits surgical usability in the operating room.
Off-axis compliant mechanism (unoptimized) [ 17 ]	The EE holder is shifted from the center of the tetrahedral structure to a lateral (off-axis) position to improve surgical accessibility.	Stiffness is highly anisotropic ( $PAR=21.63$ ) and the peak parasitic error is relatively high ( $1.10~mm$ ).
Dual-joint compliant mechanism [ 17 ]	Two identical off-axis joints are connected in series to reduce parasitic RCM displacement and improve stiffness isotropy.	Stiffness anisotropy is reduced (PAR = 1.55) with a lower peak parasitic error (0.051 mm).
Off-axis compliant joint (optimized)	Starting from the Tetra II architecture, the geometry is optimized, via feasibility analysis, to reduce parasitic motion while maintaining near-isotropic stiffness.	Near-isotropic stiffness (PAR = 1.37) with minimal peak parasitic error (0.172 mm).

TABLE X : Comparison between compliant RCM joint variants and their main performance indicators.

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General-purpose manipulators can position ophthalmic instruments under a software Remote Centre of Motion (RCM). However, at some reachable configurations that are nonsingular for the tip task alone, the RCM and tip-tracking objectives conflict and the servo oscillates. We formalise this failure as a constraint-induced task singularity: the RCM-projected tip Jacobian loses rank while the unprojected tip Jacobian stays well conditioned. Thus, the effective tip authority collapses although the target remains reachable. We present a high-precision RCM servo architecture with built-in singularity recovery: for each commanded target, an authority-guided planner selects a well-conditioned arm configuration that reaches it and plans an RCM-consistent route, executed by a fixed hierarchical-QP controller. On a 200-scenario kinematic benchmark, the complete nominal-and-recovery protocol reaches 128/128 near-singular targets and 71/72 near-joint-limit targets (100% and 99%, respectively), with the software-RCM residual below 0.142 mm. On a physical 6-DOF arm it executes across four RCM entry points, with a worst-case kinematically reconstructed RCM residual of 178 μm and a median tip-regulation error of 32μm, within the 200μm control requirement adopted for the ophthalmic task.

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