Ground-effect-controller

Supplementary material for the paper: Ground-Effect-Aware Modeling and Control for Multicopters.

The detailed data of the above video is in the Part.5. Control algorithm.

1. The quadrotor

1.1. Firmware for the flight controller

This firmware is included in a VMware virtual machine environment.

Downlink: Fireware with virtual environment

Downlink: Fireware

Password: fastlab@2024

1.2. Devices

Flight controller: CUAV V5

Lattice laser sensor: Laser

Onboard computer: Intel NUC

Motion capture system: NOKOV

1.3. CAD model

The CAD model of the quadrotor in this paper.

1.4. BOM list

Type	Name	Mass (g)	Quantity	Gross Mass (g)
Rack	Upper center plate	112.80	1	112.80
	Lower center plate	150.00	1	150.00
	Flight controller base PCB	14.00	1	14.00
	Flight controller core PCB	10.50	1	10.50
	Flight controller	41.30	1	41.30
Rack mounting	M3 Lockout nuts	0.40	20	8.00
	M3 Isolation column (6mm)	0.18	20	3.56
	M3 screw (16mm)	1.04	20	20.80
Undercarriage	Landing gear carbon clamp	4.90	4	19.60
	Landing gear carbon tube spacer	4.23	4	16.93
	Landing gear carbon tube	2.50	4	10.00
	Landing gear carbon tube sponge cylinder	3.25	4	13.00
	Landing gear carbon tube sponge round pad	0.40	4	1.59
	Motor	45.00	4	180.00
	Paddle (7 inches)	7.50	4	30.00
	M3 screw (20mm)	1.23	16	19.68
Onboard computer and battery	3D printed parts (on-board computer fixed)	56.50	1	56.50
	Intel NUC	494.20	1	494.20
	SSD	8.80	1	8.80
	RAM	8.20	2	16.40
	Battery 22.2V 1400mAh	233.80	1	233.80
	Reflector bracket	31.50	1	31.50
	Reflective (25mm)	6.52	5	32.60
Laser sensor and fixation	M3 screw (10mm)	0.77	8	6.16
	M3 Lockout nuts	0.40	8	3.20
	M3 screw (16mm)	1.04	4	4.16
	M3 Isolation column (6mm)	0.18	4	0.71
	M3 single pass aluminum column (12mm)	0.60	4	2.40
	M3 Lockout nuts	0.40	4	1.60
	Laser sensors fix carbon plates	10.60	1	10.60
	Laser sensor	8.30	1	8.30
Total				1562.70

2. The quadrotor platform for model validation

2.1. CAD model

The CAD model of the force measurement platform in this paper.

2.2. Platform data: The leveling torque

The following figure shows the data in the leveling torque experiment. The figure shows the relationship between leveling torque ${{\bf{\tau }}_G}$ and average rotor speed $\left| {{n_i}} \right|$. The black points are sensor data and the blue lines are model-fitting results.

2.3. Platform data: ROS bags and processing

To be uploaded.

3. Motor calibration and rotor speed control

3.1. Motor model

To control the rotors to the desired speeds, the rotors need to be modeled and calibrated.

The thrust $T_i$ and torque $M_i$ generated by a single rotor are:

$$T_i = k_T n_i^2,$$

$${M_i} = {k_I}{n^2} + {J_R}{{\dot n}_i}.$$

• $k_T$ is the thrust coefficient,
• $k_I$ is the torque coefficient,
• $J_R$ is the moment of inertia of the rotor,
• $i$ is the rotor number.

​ The following figure is the calibration of the (a)thrust and (b)torque model with the single motor platform in the paper. The static/dynamic modelmeans: without/with a differential term of rotor speed $\dot{n}$.

3.2. Throttol model of the flight controller

The rotor speeds are controlled through the throttle input ($t_i^{des} \in \left[ {0,1} \right]$), and the rotational speeds are fed back through BDhot. When the flight controller receives a throttle control signal (${t_c} \in [0,1]$), the rotor speed will be maintained at a roughly determined value. The relationship between the rotor speed and the throttle, after eliminating the influence of battery voltage, is usually a quadratic function:

$$n_{esc}(t_c) = c_2 t_c^2 + c_1 t_c + c_0.$$

We collect rotor speed and throttle data on the single-motor platform and calibrate the model. The data is illustrated in the following figure.

• (a) The relationship between rotor speed $n$ and throttle $t_c$.
• (b) The time series of motor speed, with the blue line representing the speed predicted using the throttle model.

It can be seen that the data demonstrates a strong alignment with the model.

3.3. Rotor speed control

The rotor speeds require closed-loop control; however, we do not implement this control for each motor individually. Instead, we apply closed-loop control to the combined acceleration generated by all rotors along ${\bf{z}}_B$. The control method for rotor speed is:

$$ t_i^{des} = t_i^{ref} + t_E, $$

$$ t_i^{ref} = n_{esc}^{ - 1}(n_i), $$

$$ t_E = K_P^T T_a^E + K_I^T \sum T_a^E, $$

$$ T_a^E = \left( {\sum\limits_i {{k_T}n{{_i^{des}}^2}} - \sum\limits_i {{k_T}{n_i}^2} } \right)/m $$

• $t_i^{ref}$ is the feedforward throttle obtained by $n_{esc}^{ - 1}$, the inverse function of throttol model,
• $t_E$ is the throttle from rotor speed error,
• $T_a^E$ is the acceleration error by all rotors,
• $K_P^T$, $K_I^T$ are the parameters of the Proportional-Integral controller.

4. Parameters

4.1. Parameter calibration

The results of parameter identification in the paper are shown in following table:

Symbol	Value	Name	Method
$k_T$	$4.0083 \times 10^{-8} , \text{N/rpm}^2$	Thrust coefficient	Single-rotor platform
	$3.7840 \times 10^{-8} , \text{N/rpm}^2$		Quadrotor platform
	$4.2958 \times 10^{-8} , \text{N/rpm}^2$		Real flight by hovering
$k_{TX}$	$4.678 \times 10^{-8} , \text{N/rpm}^2$	Torque by thrust coefficient (roll)	Quadrotor platform
$k_{TY}$	$3.588 \times 10^{-8} , \text{N/rpm}^2$	Torque by thrust coefficient (pitch)
$k_I$	$6.3859 \times 10^{-10} , \left( \text{N} \cdot \text{m} \right) / \text{rpm}^2$	Rotor torque coefficient	Single-rotor platform
$J_R$	$1.0556 \times 10^{-4} , \text{kg/m}^2$	Rotor inertia	Single-rotor platform
$g_1$	$1.804 \times 10^{-2}$	Ground effect coefficient	Quadrotor platform
$g_2$	$7.339 \times 10^{-3}$
$g_3$	$-3.365 \times 10^{-1}$
$g_4$	$4.126 \times 10^{-2}$
$g_5$	$6.494 \times 10^{-2}$
$c_2$	$-1.448471 \times 10^8$	Throttle curve parameter	Quadrotor platform
$c_1$	$5.228928 \times 10^8$
$c_0$	$1.033111 \times 10^8$
$d_x$	$0.3970 , \text{N/(m/s)}$	Rotor drag coefficient	Real flight
$d_y$	$0.3300 , \text{N/(m/s)}$
$m$	$1.696 , \text{kg}$	Mass of the quadrotor	Electronic scale
	$1.562 , \text{kg}$		Mechanical model
$I_x$	$0.00745220 , \text{kg/m}^2$	Inertia of the quadrotor	Mechanical model
$I_y$	$0.00792752 , \text{kg/m}^2$
$I_z$	$0.01249522 , \text{kg/m}^2$

4.2. Spearman's rank correlation

The following table shows the Spearman's rank correlation coefficient between variables.

• $\uparrow $ : Strong correlation;
• $\downarrow$ : Weak correlation.

Spearman coefficient	${{\bf{x}}_W}^ \top {{\bf{\tau }}_B}$	${{\bf{z}}_W}^ \top {{\bf{\tau }}_B}$	${{\bf{y}}_W}^ \top {{\bf{\tau }}_G}$	${{\bf{y}}_W}^ \top {{\bf{\tau }}_G}/\sin \delta$	${{\bf{f}}_{\bf{G}}}$
${\bf{N}_{base}}\left( 1 \right)$	-	-	$\uparrow -0.6640$	-	$\uparrow +0.4703$
${\bf{N}_{base}}\left( 2 \right)$	$\uparrow +0.8245$	-	-	-	-
${\bf{N}_{base}}\left( 3 \right)$	-	-	-	-	-
${\bf{N}_{base}}\left( 4 \right)$	-	$\uparrow +0.9251$	-	-	-
$h$	$\downarrow +0.0284$	$\downarrow +0.1781$	$\uparrow +0.3384$	-	$\uparrow -0.5465$
$\delta$	-	-	-	$\downarrow +0.0930$	$\downarrow +0.0054$

5. Control algorithm

The following figure shows the curves of Exp.~7($3m/s$, near-ground) in the paper. Every loop is well-controlled, including the rotor speed, thrust acceleration, body torque, etc.

6. Future work

The fluid simulation work is in progress.