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

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

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\}

.

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.

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.

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.

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.

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}

).

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
$l$	$0.275\text{\,}\mathrm{m}$	$R_{p}$	$0.114\text{\,}\mathrm{m}$	$m$	$3.039\text{\,}\mathrm{k}\mathrm{g}$
$I_{xx}$	$0.063$	$I_{yy}$	$0.062$	$I_{zz}$	$0.095$
$N_{p}$	$4$	$\alpha_{\rm limit}$	$\pm 225\text{\,}\mathrm{\SIUnitSymbolDegree}$	$t_{\rm servo}$	$0.048\text{\,}\mathrm{s}$
$k_{q}/k_{t}$	$0.0165\text{\,}\mathrm{m}$	$f_{t,\epsilon}$	$1.5\text{\,}\mathrm{N}$	$\lambda$	$90\text{\,}\mathrm{\SIUnitSymbolDegree}$
Integral Term & Wrench Estimator, Sec. V

TABLE IV: Model & Control Parameters

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.

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.

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

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.

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.

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.

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.

Figure 31

Figure 32

Figure 33

Figure 34

Figure 35

Figure 36

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

Abstract

Figures & tables

Appendix

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