Effector-Centric NMPC of Tiltable-Multirotors for Offset-Free Omnidirectional Aerial Manipulation

Jinjie Li; Yicheng Chen; Johannes Kübel; Haokun Liu; Junichiro Sugihara; Moju Zhao

cs.ROAug 18, 2026

Effector-Centric NMPC of Tiltable-Multirotors for Offset-Free Omnidirectional Aerial Manipulation

Authors: Jinjie Li, Yicheng Chen, Johannes Kübel, Haokun Liu, Junichiro Sugihara, Moju Zhao

Organizations: DRAGON Lab, Department of Mechanical Engineering, The University of Tokyo, Tokyo 113-8656, Japan

Abstract

Aerial manipulation extends robotic operations to previously inaccessible aerial environments. Unlike arm-equipped aerial systems, tiltable-multirotors can directly generate six-degree-of-freedom wrenches through their flight bases, enabling both efficient movement and omnidirectional operation by tilting the thrust direction. This work presents a design analysis and a wrench-based control framework for tiltable-multirotors in aerial manipulation. We show that a four-rotor tiltable configuration provides a balance between interference-free propeller sizing and hovering efficiency across different attitudes, and its null-space redundancy is crucial for traversing singular configurations under physical constraints. We further show that an upward end-effector placement yields a favorable trade-off between geometric clearance and available wrench. To address disturbances, we propose a dual strategy consisting of a modified integral term for model error and an acceleration-based estimator for external wrenches. Building on these insights, we develop an effector-centric nonlinear model predictive control (NMPC) framework that integrates design choices, singularity handling, and disturbance compensation into a unified formulation. The proposed framework runs fully onboard at 100 Hz on a custom-built tiltable-quadrotor. Real-world experiments, including a 90-deg step cartwheel rotation, whiteboard pushing, and continuous 360-deg valve turning, demonstrate the feasibility of wrench-based omnidirectional manipulation with singularity traversal on a one-DoF-per-arm tiltable-quadrotor.

Figures & tables

Fig. 1: Representative experiments included in this article : (a)–(c) demonstrate effector-centric omnidirectional flight; (d) presents external wrench estimation and compensation, which enables the whiteboard-pushing task in (e); and (f) shows continuous turning of a vertically installed valve.

Ref.	Control	Servo Dyn.	Sing. Handling	Wrench Est.	EE- Centric
[ 38 ]	Feedback Linearization	No	No	No	No
[ 35 ]	Switching PID	No	No	No	No
[ 21 ]	IQTO	No	No	No	No
[ 24 ]	Switching PID	No	No	Yes	No
[ 28 ]	NMPC	Yes	No	No	No
[ 31 ]	Impedance	No	No	Yes	No

TABLE I: Comparison with representative works on tiltable-quadrotors

Fig. 2: Simplified block diagram of the proposed effector-centric NMPC framework. The task planner generates predefined trajectories or teleoperation pose commands. Secs. II and III provide the basis for the main algorithms in Secs. IV – VI , whose results are implemented in Sec. VII . Note that the allocation is performed before the controller to provide the reference. The entire framework is validated in Sec. VIII .

Fig. 3: Diagram of a tiltable-multirotor with the ENU world frame

\left\{\mathcal{W}\right\}

(X East, Y North, Z Up), the FLU body frame

\left\{\mathcal{B}\right\}

(X Forward, Y Left, Z Up), and the tool frame

\left\{\mathcal{T}\right\}

for end-effectors. The

i

th rotor

\left\{\mathcal{R}_{i}\right\}

rotates around the X-axis (red) of

\left\{\mathcal{E}_{i}\right\}

Panel of Figure 5 — Fig. 4: Rotor interference and feasible propeller sizes of tiltable-multirotors. The central region of the platform remains unaffected by rotor airflow for any tilting angle, making it a potentially suitable location for mounting an end-effector. Fewer rotors allow a larger feasible propeller size.

Fig. 5: Hovering TWR over different attitudes with rotor numbers from 3 to 6, where a higher value indicates lower efficiency. The trirotor adopts a T-shaped layout, whereas the other configurations distribute the rotors evenly. Red dots mark representative attitudes. As the rotor number increases, the required total thrust becomes more uniformly distributed over the attitude space.

Panel of Figure 7 — Fig. 6: Geometric analysis of the required operation distance and feasible rod radius. Vertical mounting ( $\lambda=90^{\circ}$ ) provides a short operation distance and a large allowable rod radius within the considered geometry , making it suitable for large-wrench tasks.

Panel of Figure 8 — Fig. 7: Separated force and torque envelopes in the body frame at $5\text{\,}\mathrm{\SIUnitSymbolDegree}$ resolution, neglecting gravity. Directions with smaller attainable wrench values correspond to larger internal force and hence lower efficiency. This trend remains unchanged when gravity is included.

Panel of Figure 9 — Fig. 8: Coupled wrench envelope for applying the wrench to a vertical surface. Vertical mounting ( $\lambda=90^{\circ}$ ) provides a larger envelope than horizontal mounting ( $\lambda=0^{\circ}$ ), and the $+$ configuration further enlarges the envelope. The $+$ configuration at other mounting angles may cause collisions with the propellers and is therefore omitted.

Panel of Figure 10 — Fig. 9: Separated force and torque envelopes in the world frame for different mounting angles $\lambda$ . The line connecting the origin to any point on the envelope indicates the pointing direction of the end-effector. The force envelope is clearly shifted downward by gravity, whereas the torque envelope remains symmetric about the horizontal plane. Vertical mounting ( $\lambda=90^{\circ}$ ) provides larger envelopes than horizontal mounting ( $\lambda=0^{\circ}$ ).

Panel of Figure 11 — Fig. 10: Illustration of singularity test motion (a) and the heuristic term (b). In (b), if $f_{ir}\geq f_{t,\rm thresh}$ , the angle is unchanged as in vector 1). If $f_{ir}<f_{t,\rm thresh}$ as in vectors 2) and 3), the heuristic term modifies them into the $+Z$ plane, avoiding motor-cable winding. When $f_{ir}=0$ , $\alpha_{ir}\leftarrow$ 0\text{,}\mathrm{\SIUnitSymbolDegree}$$ .

Fig. 11: Comparison of different allocation methods when traversing singular points. PInv: pseudo-inverse allocation. For the tiltable-trirotor test, rotor 3 is always shut down. The tiltable-quadrotor outperforms the tiltable-trirotor by exploiting the null space to pass singular points without winding the cables.

Method	Time [ms]	Method	Time [ms]
PInv (SVD)	0.103	PInv (RtIn)	0.101
LSTSQ	0.181	WtPInv (Single)	0.103
Constrained QP	18.780	PInv (SVD) + QP	14.815
PInv (SVD) + Alloc.	0.310	PInv (SVD) for Tri	0.195

TABLE II: Calculation Time of Different Allocation Methods

Fig. 12: Diagram of the proposed framework for handling model error and external wrench in NMPC. All low-pass filters

H(s)

use the same parameters to introduce identical delay. In practice, the estimates

{}^{W}\hat{\bm{f}}_{de}

and

{}^{B}\hat{\bm{\tau}}_{de}

are further filtered and processed by a gate function ( 27 ) to enhance flight stability.

Fig. 13: Simulated comparison of reference-error-based [ 28 ] ( top ) and prediction-error-based ( bottom ) integral terms when tracking

{}^{W}p_{z}=

2\text{,}\mathrm{m}$$ . The former continues to accumulate until the reference is reached (

t=

4\text{\,}\mathrm{s}

) and exhibits overshoot, whereas the latter is independent of the reference error and remains well behaved.

Fig. 14: Frequency-response comparison of the wrench estimator and the integral term. The estimator captures external wrenches, while the integral term compensates for near-static residuals and responds weakly to captured wrenches, indicating their complementary rather than competing roles.

Symbol	Definition	Symbol	Definition
${{}^{W}\overline{\bm{p}}_{k}}$	${{}^{W}\bm{p}_{T_{o}}}-{{}^{W}\bm{p}_{T_{o},r}}$	${{}^{W}\overline{\bm{v}}_{T_{o},k}}$	${{}^{W}\bm{v}_{T_{o}}}-{{}^{W}\bm{v}_{T_{o},r}}$
${{}^{W}_{T}\overline{\bm{q}}_{k}}$	$\mathcal{V}\left({{}^{W}_{T}\bm{q}}^{*}_{r}\circ{{}^{W}_{T}\bm{q}}\right)$	${{}^{T}\overline{\bm{\omega}}_{k}}$	${{}^{T}\bm{\omega}}-{{}^{T}\bm{\omega}_{r}}$
${\overline{\bm{\alpha}}_{k}}$	${\bm{\alpha}}-{\bm{\alpha}_{c,r}}$	${{}^{W}\overline{\bm{f}}_{de,k}}$	$-{{}^{W}\bm{f}_{de}}-\bm{0}$
${{}^{B}\overline{\bm{\tau}}_{de,k}}$	$-{{}^{B}\bm{\tau}_{de}}-\bm{0}$	${\overline{\bm{f}}_{c,k}}$	$\bm{f}_{c}-\bm{f}_{c,r}$
${\overline{\bm{\alpha}}_{c,k}}$	${\bm{\alpha}_{c}-\bm{\alpha}}$ [ 28 ]

TABLE III: Definition of Error Terms in Effector-Centric NMPC

Fig. 15: Experimental comparison between CoG-centric and EE-centric formulations during a

0\text{\,}\mathrm{\SIUnitSymbolDegree}

–

180\text{\,}\mathrm{\SIUnitSymbolDegree}

pitch rotation. In this representative tracking task, the EE-centric formulation shows lower RMSE in the X direction (

0.053\text{\,}\mathrm{m}

vs.

0.068\text{\,}\mathrm{m}

) and pitch angle (

4.837\text{\,}\mathrm{\SIUnitSymbolDegree}

vs.

5.039\text{\,}\mathrm{\SIUnitSymbolDegree}

Fig. 16: Snapshot of the self-designed omnidirectional tiltable-quadrotor, Beetle-Art-Omni , together with images of its tilting module at various angles.

Fig. 17: Information flow of our system, where normal rectangles represent hardware and rounded rectangles represent software.

Model, Sec. II & Design, Sec. III & Reference, Sec. IV
Param.	Value	Param.	Value	Param.	Value
$l$	$0.275\text{\,}\mathrm{m}$	$R_{p}$	$0.114\text{\,}\mathrm{m}$	$m$	$3.039\text{\,}\mathrm{k}\mathrm{g}$
$I_{xx}$	$0.063$	$I_{yy}$	$0.062$	$I_{zz}$	$0.095$
$N_{p}$	$4$	$\alpha_{\rm limit}$	$\pm 225\text{\,}\mathrm{\SIUnitSymbolDegree}$	$t_{\rm servo}$	$0.048\text{\,}\mathrm{s}$
$k_{q}/k_{t}$	$0.0165\text{\,}\mathrm{m}$	$f_{t,\epsilon}$	$1.5\text{\,}\mathrm{N}$
Integral Term & Wrench Estimator, Sec. V

TABLE IV: Model & Control Parameters

Panel of Figure 22 — Fig. 18: Rotational flight about the pitch and roll axes. The RMSEs are: $p_{x}$ : $0.0478\text{\,}\mathrm{m}$ , $p_{y}$ : $0.1199\text{\,}\mathrm{m}$ , $p_{z}$ : $0.0467\text{\,}\mathrm{m}$ ; roll: $6.3287\text{\,}\mathrm{\SIUnitSymbolDegree}$ , pitch: $3.7819\text{\,}\mathrm{\SIUnitSymbolDegree}$ , yaw: $3.9590\text{\,}\mathrm{\SIUnitSymbolDegree}$ , where Euler angle errors are converted from quaternion errors for readability. The CoG curve denotes the change of the CoG point during this flight. Complete entry/exit data are retained to present the full process.

Panel of Figure 23 — Fig. 19: Continuous vertical cartwheel flight around the end-effector. The RMSEs are: $p_{x}$ : $0.0514\text{\,}\mathrm{m}$ , $p_{y}$ : $0.1616\text{\,}\mathrm{m}$ , $p_{z}$ : $0.1110\text{\,}\mathrm{m}$ ; roll: $5.4626\text{\,}\mathrm{\SIUnitSymbolDegree}$ , pitch: $7.4434\text{\,}\mathrm{\SIUnitSymbolDegree}$ , yaw: $2.3075\text{\,}\mathrm{\SIUnitSymbolDegree}$ . Complete entry/exit data are retained to present the full experimental process.

Panel of Figure 24 — Fig. 20: Discrete vertical cartwheel flight to evaluate singularity points. The RMSEs are: $p_{x}$ : $0.0784\text{\,}\mathrm{m}$ , $p_{y}$ : $0.0875\text{\,}\mathrm{m}$ , $p_{z}$ : $0.0867\text{\,}\mathrm{m}$ ; roll: $7.7260\text{\,}\mathrm{\SIUnitSymbolDegree}$ , pitch: $8.1508\text{\,}\mathrm{\SIUnitSymbolDegree}$ , yaw: $6.5117\text{\,}\mathrm{\SIUnitSymbolDegree}$ . The transient peaks in the attitude error are caused by the step changes of the body-Z angle reference.

Fig. 21: Open-loop wrench-estimation data. The offsets of the wrench sensor and the estimator are removed using the average over the first $3\text{\,}\mathrm{s}$ . Compared with the filtered ground truth (gt) over the initial $25\text{\,}\mathrm{s}$ , the force RMSE is X: $0.234\text{\,}\mathrm{N}$ , Y: $0.264\text{\,}\mathrm{N}$ , Z: $0.424\text{\,}\mathrm{N}$ ; the torque RMSE is X: $0.087\text{\,}\mathrm{N}\text{$\!\cdot\!$}\mathrm{m}$ , Y: $0.090\text{\,}\mathrm{N}\text{$\!\cdot\!$}\mathrm{m}$ , Z: $0.034\text{\,}\mathrm{N}\text{$\!\cdot\!$}\mathrm{m}$ . The yellow region denotes the period when the protection frame, rather than the end-effector, is poked, highlighting the estimator’s ability to sense external wrenches applied away from the EE. — Fig. 21: Open-loop wrench-estimation data. The offsets of the wrench sensor and the estimator are removed using the average over the first $3\text{\,}\mathrm{s}$ . Compared with the filtered ground truth (gt) over the initial $25\text{\,}\mathrm{s}$ , the force RMSE is X: $0.234\text{\,}\mathrm{N}$ , Y: $0.264\text{\,}\mathrm{N}$ , Z: $0.424\text{\,}\mathrm{N}$ ; the torque RMSE is X: $0.087\text{\,}\mathrm{N}\text{$ !\cdot! $}\mathrm{m}$ , Y: $0.090\text{\,}\mathrm{N}\text{$ !\cdot! $}\mathrm{m}$ , Z: $0.034\text{\,}\mathrm{N}\text{$ !\cdot! $}\mathrm{m}$ . The yellow region denotes the period when the protection frame, rather than the end-effector, is poked, highlighting the estimator’s ability to sense external wrenches applied away from the EE.

Fig. 22: Snapshots of closed-loop external wrench compensation tests: (a) force and (b) torque.

Panel of Figure 27 — Fig. 23: Flight data without and with wrench compensation corresponding to Fig. 22 a. Without compensation, the robot is easily pushed far away, and the tracking controller generates a maximum force of about $14\text{\,}\mathrm{N}$ . With compensation, the robot remains near the starting point and produces up to $30\text{\,}\mathrm{N}$ . Some integral terms (e.g., ${{}^{W}f_{dm,y}}$ and ${{}^{B}\tau_{dm,y}}$ , ( 19 )) continue to increase, since model error is not fully compensated.

Panel of Figure 28 — Fig. 24: Flight data without and with wrench compensation corresponding to Fig. 22 b. Without compensation, torque induces a position change, whereas with compensation this effect is eliminated. Some integral terms (e.g., ${{}^{W}f_{dm,y}}$ , ${{}^{B}\tau_{dm,y}}$ , ( 19 )) keep increasing, as model error is not fully compensated.

Panel of Figure 29 — Fig. 25: An operator pushes the side of a whiteboard via teleoperation. With wrench compensation activated, the robot’s force accumulates until it exceeds static friction, causing the board to slide. The external-wrench estimate is absent during some intervals (e.g., $24\text{\,}\mathrm{s}$ to $30\text{\,}\mathrm{s}$ ) because the estimator enters the STOP and CALIB states.

Panel of Figure 30 — Fig. 26: Shared-control valve-turning experiment. The yellow and gray regions indicate valve engagement and commanded continuous rotation, respectively. The RMSEs are: $p_{x}$ : $0.108\text{\,}\mathrm{m}$ , $p_{y}$ : $0.082\text{\,}\mathrm{m}$ , $p_{z}$ : $0.028\text{\,}\mathrm{m}$ ; roll: $6.120\text{\,}\mathrm{\SIUnitSymbolDegree}$ , pitch: $6.930\text{\,}\mathrm{\SIUnitSymbolDegree}$ , yaw: $2.925\text{\,}\mathrm{\SIUnitSymbolDegree}$ . The external wrench is shown only for information and is not used for compensation.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

\bm{A}\;=\;\begin{pmatrix}\frac{p_{E_{1},y}}{\rho_{1}}&0&\cdots&\frac{p_{E_{N_{p}},y}}{\rho_{N_{p}}}&0\\[3.0pt] -\,\frac{p_{E_{1},x}}{\rho_{1}}&0&\cdots&-\,\frac{p_{E_{N_{p}},x}}{\rho_{N_{p}}}&0\\[5.0pt] 0&1&\cdots&0&1\\[2.0pt] -\,\frac{\gamma_{1}\,p_{E_{1},y}}{\rho_{1}}\;+\;\frac{p_{E_{1},x}\,p_{E_{1},z}}{\rho_{1}}&p_{E_{1},y}&\cdots&-\,\frac{\gamma_{N_{p}}\,p_{E_{N_{p}},y}}{\rho_{N_{p}}}\;+\;\frac{p_{E_{N_{p}},x}\,p_{E_{N_{p}},z}}{\rho_{N_{p}}}&p_{E_{N_{p}},y}\\[3.0pt] \frac{\gamma_{1}\,p_{E_{1},x}}{\rho_{1}}\;+\;\frac{p_{E_{1},y}\,p_{E_{1},z}}{\rho_{1}}&-\,p_{E_{1},x}&\cdots&\frac{\gamma_{N_{p}}\,p_{E_{N_{p}},x}}{\rho_{N_{p}}}\;+\;\frac{p_{E_{N_{p}},y}\,p_{E_{N_{p}},z}}{\rho_{N_{p}}}&-\,p_{E_{N_{p}},x}\\[3.0pt] -\,\rho_{1}&-\,\gamma_{1}&\cdots&-\,\rho_{N_{p}}&-\,\gamma_{N_{p}}\end{pmatrix}

(28)

Appendix

Table 37

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