^{1}

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This paper presents a model of a target tracking system assembled in a moving body. The system is modeled in time domain as a nonlinear system, which includes dry friction, backlash in gear transmission, control input tensions saturation, and armature current saturation. Time delays usually present in digital controllers are also included, and independent control channels are used for each motor. Their inputs are the targets angular errors with respect to the system axial axis and the outputs are control tensions for the motors. Since backlash in gear transmission may reduce the systems accuracy, its effects should be compensated. For that, backlash compensation blocks are added in the controllers. Each section of this paper contains a literature survey of recent works dealing with the issues discussed in this article.

In many military and civilian applications we can find equipments assembled into moving bodies, which must follow a target. They need to be isolated from the rotational motion of the body where they are mounted, called body 0 in this work (see Figure

Target tracking system assembled in a moving body (0).

In the present work, the controller described in Gruzman et al. [

In this work, actuators are permanent magnet DC motors controlled by armature voltage. Sensor errors and noise are not considered, and the motion of body 0 is prescribed. Control signals (voltages) saturation and armature current limiters are included in the model.

Many authors, as Masten [

a sophisticated time domain target tracking system model,

a backlash compensation block that is added to the controllers previously developed for systems without backlash in Gruzman et al. [

The equations of motion of the device can be obtained by Lagrange formulation [

Orientation coordinates of body 0 with respect to inertial frame (

Backlash model (between rotor 1 and body 1).

It can be seen in Figure

Writinng the Lagrange equations for the device, one has

The system’s Lagrangian is given by the sum of bodies 1, 2 and rotor 1 and rotor 2 Lagrangians, each containing the potential and kinetic energy of the bodies, which are assumed to be rigid in this work. The right-hand side terms of (

The viscous friction torques at body 1, body 2, rotor 1, and rotor 2 axes are, respectively, given by

The electromotive torque in a rotor of a permanent magnet DC motor is given by (

_{b}

By a similar way, (

Armature current limiters are usually employed to avoid damages to the motors and circuits. They keep the current between a maximum value (

The torque between the rotor and the body driven by that rotor (body 1 or body 2) is affected by backlash present at the transmission. When the body is transversing the backlash gap, this torque will be equal to zero. The approach adopted in Nordin and Gutman [

The gear reduction ratio is given by

The torque at the shaft without inertia is

The backlash angle is _{b}

A new state variable is introduced in this model (

For the set rotor 2-body 2, the equations are

The backlash angle is

Dry friction torques may be found in the system; therefore they should be included in the model. There are many numerical models for dry friction available in literature [

The dynamic equations of the system (obtained from the Lagrange equations) in a matrix form is given by (

Calling a vector

In the following, we will concentrate our analysis in the generalized coordinate

If

If it is equal to zero and the modulus of the sum of all torques (

It should be stressed that it is not convenient to consider

Consider that at a certain instant during the numerical resolution of the equations system (

As previously discussed, a numerical damping term should be added to dry friction static torque because body 1 is in stick regime. An expression proposed in [

The resultant system has four first-order ODEs, given by (

The components of the vector

The components of the vector

with data provided by the sensors, at an instant

the ODEs are integrated from

at instant (

the integration process continues from (

then, the process is repeated until the final simulation instant is reached.

The controllers need to provide the adequate voltages to the motors to keep the axial axis pointed to the target. This means that the angular errors of azimuth (

It will be assumed that the TAES provides the targets centroid coordinates in the image plane, as shown in Figure

Angular errors of azimuth and elevation.

In the inner loops PI controllers are used. The inputs are the errors comprising the desired absolute angular speeds minus the absolute angular speeds measured with the rate gyros. If digital controllers are used, the control signals provided to the pan and tilt motors are given, respectively, by the following equations:

_{P}_{I}

If the absolute angular speed takes a long time to reach the desired value, the sum in (

In the outer loop, the TAES provides information about the targets elevation and azimuth angular errors, as shown in Figure

Pan motor general control structure.

The inputs for the FLC used at the pan motor control are the angular azimuth error (_{2}

Both FLCs use the input and output membership functions shown in Figure

Rule-base for the FLC with three inputs (

NL | NM | NS | Z | PS | PM | PL | |||

NL | Z | Z | Z | Z | Z | PS | PM | ||

NM | Z | Z | Z | Z | Z | PS | PM | ||

NS | Z | Z | Z | Z | PS | PM | PM | ||

Z | Z | Z | Z | Z | PM | PM | PL | ||

PS | Z | Z | Z | PS | PM | PL | PL | ||

PM | Z | Z | Z | PS | PL | PL | PL | ||

PL | Z | Z | Z | PM | PL | PL | PL | ||

NL | NL | NL | NL | NM | Z | PS | PM | ||

NM | NL | NL | NL | NS | Z | PS | PM | ||

NS | NL | NL | NM | NS | PS | PM | PM | ||

Z | NL | NM | NM | Z | PM | PM | PL | ||

PS | NM | NM | NS | PS | PM | PL | PL | ||

PM | NM | NS | Z | PS | PL | PL | PL | ||

PL | NM | NS | Z | PM | PL | PL | PL | ||

NL | NL | NL | NL | NM | Z | Z | Z | ||

NM | NL | NL | NL | NS | Z | Z | Z | ||

NS | NL | NL | NM | NS | Z | Z | Z | ||

Z | NL | NM | NM | Z | Z | Z | Z | ||

PS | NM | NM | NS | Z | Z | Z | Z | ||

PM | NM | NS | Z | Z | Z | Z | Z | ||

PL | NM | NS | Z | Z | Z | Z | Z |

Membership functions for the FLC with three inputs (

The membership functions and rule-base presented were adjusted through simulations of systems with outer loop time delay (

The controller described in previous sections is projected for systems without backlash. Its performance will be significantly worse if large backlash angles are present in gear transmission, since backlash causes delays, oscillations, and inaccuracy. To avoid backlash, some designers, as Arambel et al. [

The backlash compensation block is added in the stabilization loop, according to Figure

Backlash compensation controller in the stabilization loop.

Two tension saturation blocks should be used: the first before the backlash compensation block and the second after it. If there is no tension saturation block before the backlash compensation controller, the compensation tension may be in some cases ignored, as in the following example.

A target tracking system is supplied by a battery that provides tensions between

The idea behind the backlash compensation approach used in this work can be explained with the schematic draw of Figure

Schematic of a drive system with backlash and the compensation approach.

In order to have the desired relative motion between

The theory presented for the linear system of Figure

Correspondent generalized coordinates.

Variable | Rotor 1 – body 1 set | Rotor 2 – body 2 set |
---|---|---|

Since in most of the practical applications many parameters of the system may be unknown, the backlash compensation block developed in this work consists of a controller that does not require a model of the system. Besides, it differs from other backlash compensation techniques that do not use a model of the system, because they provide compensation to the inputs instead of the output of the main controller, as demonstrated in Figure

Schematic of traditional backlash compensation and the new approach.

Traditional approach

New approach

Some examples of these backlash compensation techniques can be found in Kim et al. [

The flexibility is an advantage of the backlash compensation approach proposed in this work, when compared to the traditional techniques, since it is not designed for a specific type of main controller. In fact it can be a PID, FLC, state feedback controller, LQR, and so forth. Besides, it can be implemented for main controllers working with different feedback variables as joint angles, joint speeds, absolute angular speeds, absolute angular accelerations, and so forth. This is a desired factor for backlash compensation controllers used in gimbaled tracking systems, because not every system has a controller that pursues to keep body 2 absolute angular speeds in desired values. Several examples of gimbaled tracking systems where the main controller pursues to keep the joint angles in desired values can be found in the works presented by Skoglar [

The driven axes and rotors angular positions and velocities are required for the backlash compensation controller proposed in this work. It is assumed that sensors are assembled after the rotors and the axes in order to measure the generalized coordinates (_{m}_{m}

The total backlash angles in each gear are required for the compensation block proposed. Therefore, it assumed in this work that this parameter is known. Simple experiments can be done for a good estimation of the total backlash angle in a transmission train: the rotor is kept fixed and the driven axis is turned manually until no further motion is allowed, and the angular displacement corresponds to the total backlash angle. In addition, the total backlash can be estimated online by adaptative controllers [

The strategy to compensate the backlash nonlinearity consists of a heuristic approach, as presented in Figure

Pan motor stabilization loop with backlash compensation.

In Figure

The FLC backlash compensation block for the pan and tilt motor control use the input and output membership functions shown in Figure

Rule-base for the FLC with three inputs (

PM | PS | Z | NS | NM | |||

PL | PH | PH | PH | PM | PM | ||

PM | PH | PM | PM | PM | PS | ||

PS | PH | PS | PS | PS | PS | ||

Z | PM | PS | PS | PS | Z | ||

NS | PM | PS | PS | Z | NM | ||

NM | PS | PS | Z | NM | NH | ||

NL | PS | Z | NM | NH | NH | ||

PL | Z | Z | Z | Z | Z | ||

PM | Z | Z | Z | Z | Z | ||

PS | Z | Z | Z | Z | Z | ||

Z | Z | Z | Z | Z | Z | ||

NS | Z | Z | Z | Z | Z | ||

NM | Z | Z | Z | Z | Z | ||

NL | Z | Z | Z | Z | Z | ||

PL | PH | PH | PM | Z | NS | ||

PM | PH | PM | Z | NS | NS | ||

PS | PM | Z | NS | NS | NM | ||

Z | Z | NS | NS | NS | NM | ||

NS | NS | NS | NS | NS | NH | ||

NM | NS | NM | NM | NM | NH | ||

NL | NM | NM | NH | NH | NH |

Membership functions for the FLC with three inputs (

The membership functions and rule-base for the backlash compensation controller are chosen to reduce delays, oscillations, and inaccuracies, by providing motors overdrives such that backlash region is quickly, but with smooth reengagement, transversed, according to the strategy presented in Figure

In this section two different cases will be simulated. In the first the target will move in a circle five meters ahead body 0, that is fixed in inertial frame. In the second case, body 0 and the target will move. The objective of the controllers is to keep the angular errors of azimuth and elevation equal to zero. The curves in black are the angular errors of a system without backlash compensation controllers and

The following data were used in all simulation.

Gravity acceleration (^{2}. Initial conditions are

Resultant coefficient of rigidity in the drive trains (

Body 1 mass (^{2}/rad s. Dry torques friction at bodies 1 and 2 axes is disregarded. Position vector of the center of the Cardan suspension (point

Bodies 1 and 2 inertia tensors, expressed in coordinates of body 1 and 2 frames, are respectively,

Rotors 1 and 2 inertia tensors, expressed in coordinates of body 1 and 2 frames, are respectively,

Viscous friction coefficients at the rotors axes are ^{2}/rad s. Dry friction dynamic torques at the rotors axes are

Tracking loop (FLC) is the membership functions and rule-base presented in Section

Stabilization loop (PI control): the following gains were obtained with Ziegler-Nichols tuning in [

Backlash compensation block (FLC): membership functions and rule-base presented in Section

Outer loop time delay (

The results presented in Figure

According to the results presented in Figure

The results presented in Figure

Root mean square (rms) of elevation and azimuth angular errors.

Simulation | Correspondent curve in Figure | rms | rms |
---|---|---|---|

black | 0.0338 rad | 0.0525 rad | |

red | 0.2465 rad | 0.2656 rad | |

green | 0.0766 rad | 0.1 rad |

Angular errors of elevation (a) and azimuth (b).

Angular errors of elevation (a) and azimuth (b).

Once again, by the results presented in Figure

Root mean square (rms) of elevation and azimuth angular errors.

Simulation | Correspondent curve in Figure | rms | rms |
---|---|---|---|

black | 0.0183 rad | 0.0696 rad | |

red | 0.1051 rad | 0.2122 rad | |

green | 0.0539 rad | 0.1173 rad |

This paper presented the modeling and control of a target tracking system assembled into a moving body. The system is modeled in time domain, instead of the most traditional frequency domain approach. This allowed a more realistic model of the system, since the inertial parts of the equations of motion were not linearized and other nonlinearities such as dry friction, backlash in gear transmission, control tensions saturation, and armature current saturation were included in the model. Different time delays in each control loop were also included in the model. The presentation of the model was done in several steps, each of them containing a literature survey of recent works where similar problems were analyzed. Initially, the generalized coordinates used in the model were defined and the Lagrange equations for the system were presented. Despite the assumption that the system is composed of rigid bodies, the Lagrange equations allowed to consider flexibility in gear transmissions that were included in the nonconservative generalized torques of the equation. In this torques, the motors electromotive force, viscous friction, backlash, and dry friction were also included. The model does not include body flexibility and aerodynamic effects, which may increase inaccuracy in the system performance. Such effects should appear more drastically in the cases that the TAES is large and more flexible as an antenna. Further developments of the model might be pointed toward this direction.

Next, a methodology was presented for the numerical integration of the equations of motion simultaneously with the update of the control signals with time delays. For finding of the control signals, independent controllers, with two loops each, were used for each motor. The outer loop, that contains a fuzzy logic controller, used information about the targets angular errors of azimuth and elevation and calculates the desired absolute angular speeds for body 2, to point the axial axis to the target. The inner loop, that is faster, uses information from the other sensors in order to keep body 2 angular absolute speeds in the desired values. A PI controller with antiwindup action is used in this loop. Tuning the gains of PI controller through Ziegler-Nichols method requires obtaining an absolute angular velocity response of the device with its actual parameters. This step could be avoided if other controller design techniques are used, such as robust control.

Afterwards, special attention was given to the problems caused by backlash in gear transmission. To reduce its effects a backlash compensation block was included in the stabilization loop. This compensation block consisted of a fuzzy logic controller whose inputs are the tension obtained in the PI controller and the relative angular position and velocity between the driven axis and the rotor (divided by the gear ratio). The output is an extra tension that is added to the tension calculated in the PI controller, assuring that the backlash region is quickly transverse, but with smooth reengagement. The backlash compensation controller can be used in combination with different principal controllers, which in this work corresponds to the inner loop PI controller. It provides compensation signal to the output of the main controller, instead of changing its inputs, as usual in traditional backlash compensation techniques. By last, results of numerical simulation were presented. It is shown the improvement in performance of a system that has large backlash in gear transmission, but whose controller has been previously tuned to work without backlash. A suggestion for future work is the inclusion of noise and sensors inaccuracy in the model.

The authors gratefully acknowledge the support of CNPq.