Numerical Experiments

Chapter 5: First Order Delay Compensation for Nonlinear Control Methods . 77

5.4 Numerical Experiments

Table 5.1: Simulation parameters for two drones of different sizes.

Intel Aero Crazyflie2.0

mass 𝑚 2.40 kg 0.033 kg

inertia 𝐽_𝑥 3.60×10⁻³kg m² 1.57×10⁻⁵kg m² 𝐽_𝑦 3.60×10⁻³kg m² 1.66×10⁻⁵kg m² 𝐽_𝑧 7.20×10⁻³kg m² 2.93×10⁻⁵kg m²

arm length 30 cm 4.6 cm

thrust-to-weight 1.43 1.41

pos ctrl freq 50 Hz 100 Hz

pos ctrl delay Δ𝑓 20 ms 10 ms

att ctrl freq 250 Hz 500 Hz

ang ctrl freq 500 Hz 500 Hz

motor gain 𝚲𝜂 50 Hz 20 Hz

mixer delay Δ𝜂 2 ms 2 ms

5.4 Numerical Experiments

and a feedforward component:

f𝑑 =−g+ ¥p𝑑−K𝑃p˜ −K𝐷p¤˜ −K𝐼

∫

p˜𝑑 𝑡 . (5.18) The attitude determination is the same as in (5.15). And the attitude controller is a regulation PID feedback on𝝎:

𝝎𝑟 =−K𝑞q˜𝑣, 𝝎˜ =𝝎−𝝎𝑟, 𝝉𝑑 =−K𝑃,𝜔𝝎˜ −K𝐷 ,𝜔𝝎¤˜ −K𝐼 ,𝜔

∫ 𝝎˜𝑑 𝑡 .

(5.19)

The mixer without delay compensation technique is simply

u=B⁻¹w𝑑. (5.20)

Common commercial flight controllers such as BetaFlight and PX4 [23] deploy sim- ilar attitude control laws and mixers as (5.19) and (5.20).

Position Level Delay Compensation

Next, we use the position controller (5.9), the associated delay compensation (5.14) described in Sec. 5.3, and the regulation PID (5.19) for attitude control described above to showcase high-level compensation can drastically improve performance even low-level attitude control is only of a regulation type.

Differential Flatness

We can use nonlinear attitude controller described in (5.8) and (5.10) to achieve better tracking performance if desired angular rate𝝎𝑑 and angular acceleration𝝎¤𝑑

along our trajectoryp𝑑(𝑡)is calculated using (5.11) and (5.12) and differential flat- ness properties of multirotor dynamics. This formulation has been vastly used in aggressive trajectory tracking of multirotor drones to achieve state-the-art perfor- mances.

Mixer Level Delay Compensation

Built on top of differential flatness controller above, the actuation delays of the mo- tors can be compensated using (5.17).

Table 5.2: Summary of controllers definitions.

Abbrev. Name Position Control Attitude Control Mixer

Casc. PID (5.15) and (5.18) (5.19) (5.20)

Pos. DC (5.9), (5.14) and (5.15) (5.19) (5.20) Diff. Flat. (5.9) and (5.15) (5.10) (5.20)

Mixer DC (5.9) and (5.15) (5.10) (5.17)

Comb. DC (5.9), (5.14) and (5.15) (5.10) (5.17)

Combined Delay Compensation

Finally, the combined method where both the position level and the mixer level de- lay compensation are applied to differential flatness controller are included as the theoretical performance ceiling of the techniques proposed in this work.

Comparisons of Trajectory Tracking Performance

From Fig. 5.3 and Table 5.3, we see that all the control methods except the Cas- caded PID approach perform well tracking the xy-plane circular trajectory, with the combined delay compensation method performing best by a small margin. This is primarily because the desired thrust along the circle is constant, so methods that compensate for thrust delays do not offer an advantage on this particular trajectory.

The Intel Aero drone, which runs the position control on a separate flight computer, runs the position control loop at a slower frequency. This is one of the main rea- sons for degraded performance compared to the Crazyflie platform, which runs the position control on the flight controller.

In contrast to the circular trajectory, the figure-∞trajectory has a significant varia- tion in thrust along the desired trajectory. Consequently, in Fig. 5.4 and Table 5.3, we observe the performance improvement the delay compensation approaches we propose, which explicitly account for thrust delay and other system delays. Partic- ularly surprising is that the Position Compensation approach for the Crazyflie plat- form outperforms the differential flatness control with a significantly simpler attitude controller.

Combined Delay Compensation Technique on System with High Delay

As Fig. 5.3 and 5.4 as well as Table 5.3 show that the delay compensation at both position control and mixer levels improve tracking performance over baseline meth- ods. The combined delay compensation has about10%reduction in error for circu-

−2 0 2 x [m]

−1 0 1

y [m]

Intel Aero - 3s Circle

0.0 1.5 3.0 4.5 6.0

t [s]

0.0 0.5 1.0

pos. error [m]

Casc. PID Pos. DC Comb. DC

(a) Intel Aero3 sCircle

−2 0 2

x [m]

−1 0 1

y [m]

Crazyflie - 3s Circle

0.0 1.5 3.0 4.5 6.0

t [s]

0.0 0.5 1.0

pos. error [m]

Casc. PID Pos. DC Comb. DC

(b) Crazyflie2.0 3 sCircle

Figure 5.3: Simulation results tracking a fast circle trajectory. Intel Aero and Crazyflie2.0simulation tracking a2 mwide circle in the𝑥 𝑦-plane with a period of 3 s, starting from hover at the origin. Top plots show steady state trajectory, bottom plots show last6 sof flights.

−1.0 −0.5 0.0 0.5 1.0 x [m]

−0.5 0.0 0.5

z [m]

Intel Aero - 4s figure-∞

0 2 4 6 8

t [s]

0.0 0.2 0.4 0.6

pos. error [m]

Casc. PID Pos. DC Diff. Flat.

Mixer DC Comb. DC

(a) Intel Aero4 sFigure-∞

−1 0 1

x [m]

−0.5 0.0 0.5

z [m]

Crazyflie - 5s figure-∞

0.0 2.5 5.0 7.5 10.0 t [s]

0.0 0.2 0.4 0.6

pos. error [m]

Casc. PID Pos. DC Diff. Flat.

Mixer DC Comb. DC

(b) Crazyflie2.0 5 sFigure-∞

Figure 5.4:Simulation results tracking a fast Figure-∞trajectory. Intel Aero and Crazyflie2.0simulation tracking a Figure-∞in the𝑥 𝑧-plane, starting from hover at the origin. The periods are 4 s and 5 s, respectively. Top plots show steady state trajectory, bottom plots show last6 sof flights.

Table 5.3: Summary of trajectory tracking performance.

Intel Aero

Trajectory Controller kpk˜ _RMS[cm]

Circle (3 s)

Casc. PID 24.5

Pos. DC 7.10

Diff. Flat. 6.88 Mixer DC 6.91 Comb. DC 6.04

Figure-∞(4 s)

Cascaded PID 16.3

Pos. DC 7.29

Diff. Flat. 6.44 Mixer DC 5.51 Comb. DC 4.41 Crazyflie2.0

Circle (3 s)

Cascaded PID 21.9

Pos. DC 4.58

Diff. Flat. 3.42 Mixer DC 3.47 Comb. DC 3.04 Figure-∞(5 s) Cascaded PID 8.31

Pos. DC 3.07

Diff. Flat. 4.46 Mixer DC 2.64 Comb. DC 2.23

lar trajectories in both drones, and50% reduction for Figure-∞ trajectories. Such observation motivates us to investigate further its capabilities when trajectories be- come more aggressive, or similarly during high delay situations. We conducted the same trajectory tracking task as in Fig. 5.4 for Intel Aero comparing Diff. Flat. and Comb. DC method, at the same time varied motor delay and signal delay values for the simulations. As shown in Fig. 5.5, a drastic improvement of delay compensation controllers in RMS error can be observed for both high motor and/or high signal delays. A Diff. Flat. controller struggles as signal delays are over100 ms. It also indicates that the proposed method handles the mixed delay sources with ease, as combination of continuous and discrete delays persist in these simulations.

0 50 100 150 200 10¹

10²

2 × 10¹ 3 × 10¹ 4 × 10¹ 6 × 10¹

Motor Delay [ms]

Comb. DC

0.00 0.08 0.16 0.24

0 50 100 150 200

Pos. Computer Delay [ms]

10¹ 10²

2 × 10¹ 3 × 10¹ 4 × 10¹ 6 × 10¹

Motor Delay [ms]

Diff. Flat.

0.00 0.08 0.16 0.24

Figure 5.5: Heatmap showing performance degradation with increasing delays for the proposed delay compensation controller and the baseline differential flatness controller. Heat map plots of RMS position tracking error, in meters, ver- sus the position control loop computer delay and motor delay for the Intel Aero simulation tracking the4 sfigure-∞in Fig. 5.4. Top: proposed delay compensation control scheme. Bottom: baseline nonlinear controller based on differential flatness.