9.2.1 Nonlinear 3DOF Model

To obtain the nonlinear 3‐DOF motion model of a helicopter, we first introduce its hardware components. Photos of a single helicopter are shown in Figures 9.3–9.5. Two DC motors are used to drive the propellers, which in turn generate control forces to regulate the 3‐DOF motions of the helicopter. These motors are defined as the front and the back motors respectively. The body frame is suspended from an instrumented joint, which connects the centre of the body frame and the end of a long arm, and is free to pitch about its centre. The arm is installed on the base through a 2‐DOF instrumented joint, which allows the helicopter body to elevate and travel. A counterweight is located at the other end of the arm such that the effective mass of the helicopter can be adjusted to a suitable value. In addition, an optional ADS can be installed on the arm. This includes an ADS motor that can drive an adjustable weight moving along the arm. The ADS motor is controlled independently, and therefore can act as an external disturbance or model uncertainty. The elevation motion can be generated by applying a positive voltage to each motor, and positive pitch by applying greater voltage to the front motor. The travel motion is the result of tilted thrust vectors when the body pitches. The three attitude angles are measured by encoders mounted on the instrumented joints. Therefore, pitch, elevation, and travel are the three degrees of freedom from which the 3DOF‐Heli gets its name.

A convenient way to describe the motions of the 3DOF‐Heli is through a body‐frame defined along the axes of the equipment, centralized around the baseline rotational pivot, as shown in Figure 9.4. For a fleet of 3DOF‐Helis, the nomenclature associated with the th 3DOF‐Heli, , is as follows:

• is the elevation angle;
• is the elevation rate;
• is the pitch angle;
• is the pitch rate;
• is the travel angle;
• is the travel rate;
• , , and are the moments of inertia about the elevation axis, pitch axis, and travel axis, respectively;
• is the force constant of the motor–propeller combination;
• is the distance from the elevation axis to the centre of the helicopter body;
• is the distance from the pitch axis to either motor;
• is the effective mass of the helicopter body;
• is the gravitational acceleration constant;
• , , and denote disturbances acting on the elevation channel, pitch channel and travel axis, respectively;
• and are the voltages applied to the front motor and back motor, respectively;
• and denote the sum and the difference of and , respectively.

The differential equation of motion for the the laboratory experimental setup, which can be derived using the Lagranges formalism, is written in the following well‐known form (see examples in the literature [2, 3] and Chapter 6):

(9.1)

where the inertia matrix , the Coriolis matrix , the gravity vector , the disturbance input , the generalized forces , and the state variable vector .

For simplicity, the generalized inertia matrix is often reduced to a diagonal matrix with constant entries. To be specific, the mathematical model of the form (1) for the th 3DOF‐Heli reads as follows:

(9.2)

(3)

(9.4)

where

(9.5)

(9.6)

and:

• Equation (9.2) describes the elevation dynamics: Due to the presence of the counterweight, the second‐order equation of motion accounts for the pendulum‐like oscillatory characteristic, where is the vertical lift thrust component, is the restorative spring torque, is the torque generated because the pivot points of the elevation and pitch motions are not exactly in the helicopter's body, and is the unknown aerodynamic and mechanical damping torque.
• Equation (9.3) describes the pitch dynamics: Here, the second‐order equation of motion accounts for the lightly‐damped oscillatory characteristics: is the rotor torque, the restorative spring torque, and is the rotor damping.
• Equation (9.4) describes the travel dynamics: Here, is the propulsive thrust component resulting from the pitch motion with a nonzero ; is the unknown aerodynamic drag as well as some amount of mechanical friction at the hinge.

Consider the equations of motion given in (9.10)–(9.12) with relationship equations (9.6). The electric voltage controls collectively the speed of the two propellers, and a positive produces a positive lifting moment. The electric voltage results in differential changes in the two rotor speeds, and a positive produces a positive pitch moment.

Let

(9.7)

(9.8)

(9.9)

Then Equations (9.2)–(9.4) are reduced to:

(9.10)

(9.11)

(9.12)

In Equations (9.10)–(9.12), the aerodynamic drag, the mechanical friction, and the rotor damping are not explicitly considered. Instead, we take into account both model uncertainties and friction effects, by lumping them together as the additive input disturbances , , and .

9.2.2 2DOF Model for Elevation and Pitch Control

Assuming travel motion can be achieved by highly precise pitch tracking, we simplify the 3‐DOF attitude dynamics to 2‐DOFs, namely elevation and pitch. The simplified model describing the elevation and pitch motions of the th helicopter ( ) is then:

(9.13)

(9.14)

where the pitch angle, , is limited to the range mechanically, and the nominal values of the involved parameters are as presented in Table 9.1.

Parameter	Value	Parameter	Value
	0.1188 N/V		1.034 kg
	0.660 m		0.045 kg
	0.178 m		9.81 m/
	0.094 kg

Equations (9.13)–(9.14) can be written in the following compact form:

(9.15)

where

(9.16)

(9.17)

(9.18)

Since pitch angle is mechanically limited to the range in all experiments, the inertia matrix for each is a positive‐definite matrix.

Equation (9.15) can be formulated in matrix format as:

(9.19)

where , , , , , and are vectors, and is a diagonal inertia matrix. These vectors have the following expressions:

Since , , for any , Equation 9.19 is equivalent to:

(9.20)

where

(9.21)

(9.22)

We use to denote the desired angular‐position trajectories for the four helicopters, and the desired angular‐velocity trajectories. For helicopters, 2‐D mutual angular‐position synchronization means that:

(9.23)

and 2‐D zero‐error trajectory tracking in angular position means

(9.24)

Correspondingly, the 2‐D mutual angular‐velocity synchronization means

(9.25)

and the 2‐D zero‐error trajectory tracking in angular velocity means

(9.26)

In the context of this application, synchronization error is used to identify how the attitude trajectories of each 3‐DOF helicopter converge with respect to each other.

Let:

(9.27)

(9.28)

(9.29)

Then, MVS (9.15) can be modelled by the vector equation (5.4) or the following two decoupled scalar equations:

(9.30)

where with ( ) is the control input vector to be determined, and with ( ) is the normalized disturbance input vector. This further implies that the control concepts and approaches introduced in Chapter 6 are applicable in designing for the MVS considered.

Once or has been determined, are computed according to Equation (9.27). Therefore, in each of the experiments below, the design of or is the main focus: we will introduce several control experiments where objectives (9.23)–(9.26) are achieved in an asymptotic manner or an approximately asymptotic manner. We start by introducing an experiment in which the GSE‐based synchronized tracking controller developed in Section is applied and implemented.

9.3.1 Objective

The experiments are conducted on a platform consisting of three 3‐DOF helicopters, which are labelled Heli. I, Heli. II and Heli. III, respectively, as shown in Figure 9.6. The model parameters for control design are given in Table 9.2.

Parameter/gain	Heli. I	Heli. II	Heli. III
, kg m	1.044	1.030	1.017
, kg m	0.0455	0.0455	0.0455
Mass , kg	0.142 1	0.128	0.113
, N/Volt	0.625	0.625	0.625
, m	0.648	0.648	0.648
, m	0.178	0.178	0.178
ADS system	Yes	No	No
Feedback gains, [ , ]	[30.0 15.0]	[30.0 15.0]	[30.0 15.0]
Sync. feedback gains,	1.0	1.0	1.0
Sync. coupling gains,	1.0	1.0	1.0

Value measured when ADS is at the farthest position from propellers, 0.170 for the middle position in the slide bar, 0.213 for the nearest position from propellers.

The objectives of this experiment are:

• to design a controller using the approach proposed in Section for the elevation axis, to achieve synchronized trajectory tracking asymptotically; that is, to achieve (9.23)–(9.26) asymptotically;
• to implement the controller and compare its performance to that of a controller without GSE feedback (obtained by setting and , , in controller (5.142));
• to verify the robustness of the proposed controller with respect to the disturbances resulting from activated ADSs.

9.3.2 Initial Conditions and Desired Trajectories

A quintic polynomial trajectory in (9.31) is designed for the attitude motion of the elevation:

(9.31)

where the coefficients can be determined from the position, velocity, and acceleration requirements at both boundaries. If the desired maneuver is from 0 to in 8 s, the coefficients are , , and . This trajectory guarantees that , , and .

9.3.3 Control Strategies

As an initial‐stage experimental investigation, we applied the proposed synchronization controller to only one axis of the laboratory helicopters, i.e. the elevation axis, and use the Quanser‐provided LQR controller for the other axes (pitch and travel). The synchronization process takes place along the same axis (attitude motion), but not among different axes of a single vehicle. From the dynamic equations (9.13) and (9.14), it can be seen that, despite the fact that the elevation motion and pitch motion are uncoupled, the elevation motion does not influence the pitch motion, and is completely controllable with respect to input since is mechanically limited to the range . Thus it is reasonable to apply the proposed synchronization controller to the elevation axis only and a different controller to the pitch axis.

The control law (5.142) is applied to control the elevation motion, with control parameters as given in Table 9.2. For comparison purposes between the different synchronization errors, we implement and verify the following three control strategies.

• Control Strategy I: The synchronization error and the coupled attitude error are chosen with given as in (3.44).
• Control Strategy II: The synchronization error and the coupled attitude error are chosen with as given in (3.45);
• Control Strategy III: Apply a simple variation of controller (5.142) without synchronization strategy, obtained by setting and in (5.142).

For all the above strategies, the synchronization error defined with given in (3.44) is employed as the uniform synchronization error for performance evaluation.

9.3.4 Disturbance Condition

The ADS on Heli. I is activated by a square wave command (see Figure 9.7) so as to initiate a performance disturbance to the control system. Here, 0 denotes the farthest position from the propellers, and 0.26 the nearest position). In this way, the effective mass of Heli. I will vary between 0.142 kg and 0.213 kg. If there is a parametric disturbance, such as the mass disturbance in our case, asymptotic convergence cannot be guaranteed, although the system may still be ISS.

Strategy	Heli. I ( )	Heli. II ( )	Heli. III ( )
	3.136	1.968	1.978
No	1.257	0.290	0.378
	1.319	0.264	1.319
	2.466	2.217	2.206
I	0.677	0.466	0.466
	0.527	0.264	0.440
	2.385	2.261	2.259
II	0.641	0.641	0.641
	0.506	0.440	0.352

The aim of introducing the mass disturbance in the experiments is to verify the robustness of the control strategies. We will inspect how these 3DOF‐Helis respond without and with synchronization strategies, and compare the synchronization performance of different control strategies under parametric disturbance.

9.3.5 Experimental Results

Figure 9.10 shows the experimental results of elevation tracking control of three 3DOF‐Helis without a synchronization strategy (by setting and ). Figures 9.8–9.9 are the experimental results using synchronization strategy I and II, respectively. The maximal elevation tracking and synchronization errors are listed in Table 9.3. For each experiment, the three values are the maximal elevation trajectory tracking errors during maneuver (from 0 to 20 s), after maneuver (maneuver completed, after 20 s in our experiments), and the maximal synchronization error during the whole experiment, respectively.

It can be seen from Figure 9.10 that the elevation trajectory tracking error of Heli. I is large because of the ADS. Table 9.3 shows that the maximal elevation trajectory tracking errors during and after the maneuver are and , respectively. As there is no interconnection between these 3‐DOF helicopters, the elevation tracking motion of Helis II and III do not react to the tracking motion of Heli. I due to ADS. Thus, asymptotic convergence of Helis II and III has been achieved. For this reason, the synchronization errors between them are large and the maximal value is . However, the elevation trajectory tracking and synchronization errors can be markedly reduced using the proposed synchronization strategies. For example, the maximal synchronization error is reduced to using Control Strategy I and to using Control Strategy II.

Strategy II is expected to produce better performance than Control Strategies I and III because it uses more neighbor information in each individual controller. However, the better performance is obtained at the cost of a greater online computational burden, less reliability, and more control effort, as can be seen from Figures 9.9–9.10 for control voltages for all the above cases.

9.3.6 Summary

In this experiment, a model‐based synchronized trajectory tracking control strategy for multiple 3‐DOF helicopters is presented and verified. With the proposed synchronization controller, both the attitude trajectory tracking errors and the attitude synchronization errors can achieve asymptotic convergence. The introduction of the generalized synchronization concept allows more space for designing different synchronization control strategies.

Experimental results conducted on the three 3DOF‐Heli setup demonstrate the effectiveness of the proposed synchronization controller. The investigation indicates that better performance can be realized, but at the cost of computational burden, poorer reliability, and control effort. A tradeoff between synchronization performance and implementation costs should be made before choosing the synchronization strategy for a real system.

Current and future work in this area includes development of:

• an adaptive synchronization controller for systems with parametric variation
• new synchronization strategies.

9.4.1 Objective

The objectives of the experiment are:

• to design a robust controller by applying the UDE‐based approach proposed in Section for both elevation and pitch channels;
• to implement the controller, compare its performance under different UDE parameters, and evaluate the effect of UDE parameter on synchronization and tracking errors;
• to verify the robustness of the controller to disturbances from the activated ADS.

9.4.2 Initial Conditions and Desired Trajectories

The initial states of the four helicopters are specified as follows:

(9.32)

Without loss of generality, the following non‐constant desired trajectories are adopted for experiments:

(9.33)

where all angles are given in degrees.

9.4.3 Control Strategies

Consider the directed communication topology graph shown in Figure 9.11. It is seen that only H1 has access to the desired trajectories, and Condition 3 is satisfied with , and all other entries of and are 0. Then, it follows that:

(9.34)

(9.35)

Consider the controller (5.19) with (5.20) and (5.32) for the motion synchronization of both elevation and pitch channels. Use and to denote the two entries of controller component , and to denote the two entries of disturbance estimate vector , and to denote the two entries of estimation error vector ; that is, , , and , where .

For the parameters of and , we choose

(9.36)

which results in

(9.37)

(9.38)

(9.39)

(9.40)

where the subscripts and are added to the symbols introduced in Section , to distinguish the two channels.

For comparison purposes, four experimental cases are studied. The same nominal controllers and with parameters as in (9.36) are applied for each case. The differences between these cases are:

• For Case 4, only and are applied, which corresponds to the situation in Section and is equivalent to applying the controller (5.19) with , and , to the four helicopters.
• For Case 2, UDEs (5.32) with parameters and are additionally applied for each helicopter.
• For Case 3, UDEs (5.32) with smaller parameters of and are applied for each helicopter, and none of the equipped ADSs are activated.
• For Case 4, the same controllers as for Case 3 are applied, but the ADSs of helicopters 2 and 4 are activated.

In addition, for the controller design of each helicopter, the model parameters given in Table 9.1 are also used in this experiment, which is different from the situation considered in the above Experiment 1 (where the differences among the inertial parameters of the involved helicopters are explicitly considered). This implies that larger modeling errors are neglected in constructing the controller for Experiment 2, since, as shown in Table 9.2, ADSs on the helicopters lead to obvious uncertainties in the inertial parameters and .

9.4.4 Experimental Results and Discussions

9.4.5 Summary

To show the performance differences more explicitly, the maximum magnitudes of the tracking errors for the four cases are summarized and compared in Tables 9.4 and 9.5; the responses over the same time interval are considered in each case. From these tables, it is easy to see that both the tracking and synchronization accuracy improves if UDEs with smaller UDE parameters are used.

Case	H1 ( )	H2 ( )	H3 ( )	H4 ( )
1
2
3
4

Case	H1 ( )	H2 ( )	H3 ( )	H4 ( )
1
2
3
4

In this experiment, the UDE‐based robust synchronized tracking control of four 3‐DOF helicopters has been verified under the condition that only a small subset of the helicopters has access to the desired attitude information. This robust controller is obtained by augmenting a distributed tracker with a continuous disturbance estimator that produces a signal to compensate for the effect of disturbances involved in the actual dynamics. Experimental results demonstrate that both the tracking and synchronization accuracy are greatly improved by UDEs with suitable parameters, and that the smaller the UDE parameter is, the smaller the tracking errors and synchronization errors are.

9.5.1 Objective

The objectives of the experiment are:

• to design a robust controller by applying the sliding‐mode control approach proposed in Section for both elevation and pitch channels;
• to implement the controllers with or without disturbance compensation, and compare the performance;
• to check the control performance with respect to exogenous disturbances.

9.5.2 Initial Conditions and Desired Trajectories

The initial angular positions of the involved four helicopters are:

(9.41)

(9.42)

(9.43)

(9.44)

The initial angular velocities of all the helicopters is zero; that is, for each . The reference trajectory of angular position is set to:

(9.45)

(9.46)

so all the helicopters are expected to move sinusoidally in both channels, with amplitudes of and and frequencies of 0.15 Hz and 0.1 Hz for the elevation channel and pitch channel, respectively.

9.5.3 Control Strategies

Consider the same information‐exchange graph as shown in Figure 9.11 for this experiment. We apply and implement the controllers (5.79), (5.94), and (5.96) for the motion synchronization of both elevation and pitch channels.

The same nominal inertial parameters used in Experiment 2 are also used to construct the controller for the helicopters in this experiment, despite the fact that the differences among the parameters of the four helicopters are obvious. This in turn implies that the results obtained in this experiments also demonstrate the system's robustness with respect to the uncertainties in inertial parameters. Furthermore, by the definition of given by (5.18), one can see that the disturbance compensation term reflects not only the effect of the disturbances and uncertainties of vehicle itself, but also that of its neighbors.

The controller gains for each helicopter are the same and given by:

(9.47)

These are tuned manually according to the response of the experiment, and the other parameters , , , are accordingly determined using (5.116)–(5.119).

In addition, in order to attenuate chattering caused by the discontinuous signum function, we use standard saturation functions in the control law. The slope of the saturation functions is chosen as 1000.

9.5.4 Experimental Results and Discussions

For comparison purposes, three experimental cases are designed and implemented in this experiment, corresponding to different disturbance conditions or disturbance compensation strategies.

• Case 1: the ADSs on helicopters 2 and 4 are turned off, and the control component is used for disturbance compensation for each helicopter.
• Case 2: the ADSs on helicopters 2 and 4 are turned on, and the control component is used for disturbance compensation for each helicopter.
• Case 3: the ADSs on helicopters 2 and 4 are turned off (as in Case 1), and the control component is removed from the implemented controller for each helicopter; in other words, the disturbance is not compensated for any helicopter.

9.5.5 Summary

In these experiments, the decentralized output‐feedback synchronized tracking control strategy, as developed in Section , was implemented and verified on a platform consisting of four 3‐DOF helicopters with only angular position measurements available. In particular, the effect of the disturbance compensation term included in controller (5.79) was evaluated.