# Linear Positional and Speed Control of Servo Carts Using Inverse Dynamic Control

Authors: Ibrahim M. Mehedi; Mohd Heidir Mohd Shah; Rahtul Jannat
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Dynamic inverse- (DI-) based control technique has been utilized in many applications and proven to be effective. Recently, the inverse dynamic control (IDC), an expansion to the classical DI technique, has been trending with implementation in many areas. It has been proved that IDC is capable of overcoming some limitations in DI-based techniques, particularly in cancellation of useful nonlinearities. This paper extends the implementation of IDC on the positional and speed control of the linear servo cart system. Simulation results further proves that IDC is an effective and robust controller evidently when comparing it with the proportional velocity and lead compensator controller.


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## Body

## 1. Introduction

The linear servo cart system is used widely in industries especially in manufacturing and automation process. Some of the application example includes pick-and-place system, tool-feeding system in machining process, and indexing of operations like drilling, stamping, and embossing. It became a popular choice in the linear-positioning tool, thanks to the combination of strengths from its actuator and ease of positioning control from servo units. Due to that, various types of controllers have been proposed to improve the positional and speed control of servo systems including the ever-present PID controller and its variances. Although broadly accepted in industries, the traditional PID controllers are known for their poor performance under the influence of disturbances. To tackle this issue, the authors in [1] introduce two-degree-of-freedom PID positional controller where the position reference responses and disturbance responses were allowed to be designed independently, while the authors in [2] combine the application of position interpolation method and modified incremental PID. There were also some other recent works implementing the PID controller on positioning control of servo motors with different tuning methods [3, 4].The functionality of the linear servo cart system involves movement of the mechanical component which can lead to friction resulting in the introduction of highly nonlinear disturbance to the control output. One of the popular methods used to deal with nonlinearity is sliding mode controllers (SMCs). However, a conventional SMC with switching control action suffers a major drawback in the form of chattering. To overcome this weakness, a boundary layer around the switching surface [5] and an integral SMC with switching gains [6] were proposed.Another common way of dealing with nonlinearities is by employing nonlinear dynamic inversion- (NDI-) based technique as shown in [7]. This type of controller is designed by enforcing a stable linear error dynamics intuitively. It is widely applied especially in aerospace fields such as unmanned combat aerial vehicles (UCAV) [8], pitch axis autopilots [9], and quadrotors [10]. Despite the effectiveness shown by NDI in the presence of disturbances and noises, there is a big limitation in the implementation of such controllers as they are only applicable to the system where the model is known accurately. This can be solved using incremental NDI (INDI), an expansion to the classical NDI which only requires a small part of a model to be known. Some implementation of INDI can be found in [11–13].Inverse dynamic control (IDC) is another expansion of the NDI family where control by inversion of dynamic inverse constraints is achieved through the Moore–Penrose generalized inverse [14]. This technique overcomes many limitations of the classical NDI especially in the cancellation of useful nonlinearities, robustness concerns, and computational challenges arising with square matrix inversion [15]. IDC has been implemented in many applications [16–18]. This paper extends the design and implementation of IDC on the positional and speed control of the linear servo cart system. In this paper, a generalized inverse control technique is demonstrated for Quanser’s linear servo cart system in response to other recent advances in modern control techniques, such as adaptive fuzzy control, fractional order control, and inverse control [19–24]. Moreover, by using sensor technology and new materials to optimize concrete maintenance, we can collect necessary data to justify the proposed control law [25–27].The remainder of this article is organized as follows: The mathematical model of the linear servo cart system is established in Section2, while Section 3 outlines the IDC design for linear positional control of the linear servo cart system. The technique used to avoid singularity is explained in Section 4, and simulation setup and the results are discussed in Section 5 before this paper is concluded in Section 6.


## 2. Mathematical Model of the Linear Servo Cart System

Figure1 shows the linear servo cart system. The relationship between the force applied to the cart by the DC motor and resultant motion of the cart can be derived by applying Newton’s second law of motion and D’Alembert’s principle to the system, as in(1)Mv˙ct+Fajt=Fct−Beqvct,where M, vc, and Beq are the mass of the cart, linear velocity of the cart, and the equivalent viscous damping coefficient, respectively, and Faj is the armature inertial force due to motor rotation acting on the cart which can be defined as(2)Faj=ηgKgτajrmp,where ηg is the efficiency of the gear box, Kg is the gear ratio, and τaj is the armature inertial torque which can be expressed as(3)τaj=Jmω˙mt.Figure 1
Linear servo cart system.The angular velocity of the motor shaft can be translated into linear velocity of the cart with the following equation:(4)ωmt=Kgvctrmp.By substituting (3) and (4), (2) can be rewritten as(5)Faj=ηgKg2Jmv˙ctrmp2.With that, the force in (1) can now be expressed in terms of the linear velocity of the cart and by considering both the electrical parts and the equation of motion:(6)Jeqv˙ct+Beqvct=AmVmt,where(7)Beq=ηgkg2ηmktkm+Bcrmp2Rmrmp2Rm,and the actuator gain is(8)Am=ηgkgηmktrmpRm.Note thatηm is the efficiency of the motor, Kg, kt, and km are the gear ratio, motor torque constant, and back-emf constant, respectively, and rmp and Rm is the radius of the motor pinion and the motor resistance.Finally, the equivalent inertia term can be expressed as(9)Jeq=Mc+ηeqkg2Jmrmp2.


## 3. Design of IDC Control

The dynamics of the linear servo cart system can be expressed by rearranging (6) as follows:(10)v˙c=F+GVm,where F=−Jeq−1Beqvc and G=Jeq−1Am. It is known that the velocity of the cart can be obtained by taking the derivative of its linear position:(11)x˙c=vc.In order to track the linear position of the cart precisely, an error function in the form of squared error function of the actual position,xc from its desired position, xcd is defined as(12)ξxc=n1xc−xcd2=n1exc2.Similarly, the error function for linear velocity can be written as(13)ξvc=n2vc−vcd2=n2evc2.The constraint linear time-varying ordinary differential equation is established based on the deviation functions. Note that the differential orders are corresponding to the relative degree of the deviation functions. The equation takes the following form:(14)ξ¨xc+c1tξ˙xc+c2tξxc=0,(15)ξ˙vc+c3tξvc=0,where c1, c2, and c3 are coefficients that allow the constraint differential equations as in (14) and (15) to achieve uniform asymptotic stability. Therefore, they must be selected carefully as suggested in [15].The following can be obtained by taking the derivative from (12) and (13):(16)ξ˙xc=2n1exce˙xc,(17)ξ˙vc=2n2evce˙vc+2n2evcF+GVm−v˙cd.Taking further derivative on (16) yields(18)ξ¨xc=2n1e˙xc2+2n1excF+GVm−v˙cd.By putting the time derivatives of the equations in the constraint dynamics as in (14) and (15), equations (16)–(18) can be transformed into their algebraic form shown as follows:(19)AVm=B,where(20)A=A1A2=2n1excG2n2evcG,(21)B=B1B2=2n1excv˙cd−2n1e˙xc2−2n1excF−2c1n1exce˙xc−c2n1exc22n2evcv˙cd−2n2evcF−c3n2evc2.Due to the underdetermined nature of the algebraic expression, (19) can have an infinite number of solutions. Using generalized inversion by the Greville method, solutions in the equation are parameterized as(22)Vm=A+B+pλ,where A+ is the MPGI of A given as(23)A+=ATATA′.Note thatλ is the null control and P is the null projection, which can be expressed as(24)P=1−A+A.Note thatA has a dimension of 2×1, in which the P in (24) will be zero when the property of pseudo-inverse A+A=1. This will make the null control not useful, and therefore, we will not consider it in our control design.


## 4. IDC Singularity Avoidance

The main trouble with generalized inversion techniques is the singularity which is caused by discontinuities in the MPGI matrix function and eventually leads to the structure to go unbounded. Such happens when the inverted matrix tends to switch its rank.To overcome this problem, we introduce a dynamic scaling factor which is expanded within MPGI. We denote the scaling factor asu and can be defined as(25)u˙t=−ut+γexct2+evct2,u0>0.The homogeneous part of equation (25) is asymptotically stable, while γ in the forcing phrase is a positive real-valued constant. The dynamically scaled generalized inverse (DSGI) is formulated as(26)A∗=ATATA+ut.Thus, we can update the IDC-based control input voltage by the following expression:(27)Vm∗=A∗B.Finally, we can update (10) by the following expression:(28)v˙c=F+GA∗B.


## 5. Numerical Simulations

To evaluate performances of the designed controller, we performed numerical simulations on a linear servo cart model having parameters as given in Table1. The proposed IDC was having dynamic gains of n1=2 and n2=0.01 and tracking gains of c1=50, c2=21000, and c3=0.1.Table 1
Parameters of the linear servo cart system.
ParametersValuesUnitsMotor armature resistance,Rm2.6ΩRotor moment of inertia,Jm3.9×10−7Kgm2Motor current torque constant,kt7.7×10−3Nm/AMotor efficiency,ηm1—Motor back-emf constant,km7.7×10−3V/rad/sGear ratio,kg3.71—Gear efficiency,ηg1—Mass of the cart,M0.57kgMotor pinion radius,rmp6.4mmEquivalent viscous damping coefficient,Beq4.3—


### 5.1. Linear Position Tracking

In the first simulation, we set the desired motion profile to move the cart 100 mm from its initial position and back continuously in a period of 5 seconds with acceleration and deceleration time set at 15% from the period as shown in Figure2. By using the nominal linear servo cart system parameters, we showed that IDC is able to track the position of the cart according to the profile set accurately as suggested by the actual motion profile in Figure 2 and the squared error norm in Figure 3.Figure 2
Position tracking of the linear servo cart system using IDC.Figure 3
Squared error norm for position tracking.


### 5.2. Robust Analysis for Sinusoidal Position Tracking

We evaluate robustness of our designed controller by setting a 0.2 Hz sinusoidal input reference as shown in Figure4 having 150 mm of maximum amplitude with the numerical value of the linear servo cart system changed to 20% from the one in the previous simulation. For comparison, the system was also simulated using a proportional-velocity controller with proportional gains Kp=274.6159 and velocity gains Kv=5.5272. It was obvious from the squared error norm in Figure 5 that IDC outperforms the PV controller in linear position tracking of the linear servo cart system. This simulation results also suggest that IDC is a robust controller.Figure 4
Sinusoidal position tracking of the linear servo cart system.Figure 5
Squared error norm for sinusoidal position tracking.


### 5.3. Linear Speed Tracking

To evaluate performance of IDC on tracking the speed of the linear servo cart system, we simulate the scenario by giving a 0.25 Hz square wave set point with an amplitude of 0.2 m/s. Another simulation is done using a lead compensator (LC) for comparison, and the results are as shown in Figure6. We can see from the simulation results that IDC has the ability to track the linear servo cart’s speed smoothly without overshooting. The squared error norm measured from both controllers also suggests that the IDC perform slightly better than the LC controller as shown in Figure 7.Figure 6
Speed tracking of the linear servo cart system.Figure 7
Squared error norm for speed tracking.


## 5.1. Linear Position Tracking

In the first simulation, we set the desired motion profile to move the cart 100 mm from its initial position and back continuously in a period of 5 seconds with acceleration and deceleration time set at 15% from the period as shown in Figure2. By using the nominal linear servo cart system parameters, we showed that IDC is able to track the position of the cart according to the profile set accurately as suggested by the actual motion profile in Figure 2 and the squared error norm in Figure 3.Figure 2
Position tracking of the linear servo cart system using IDC.Figure 3
Squared error norm for position tracking.


## 5.2. Robust Analysis for Sinusoidal Position Tracking

We evaluate robustness of our designed controller by setting a 0.2 Hz sinusoidal input reference as shown in Figure4 having 150 mm of maximum amplitude with the numerical value of the linear servo cart system changed to 20% from the one in the previous simulation. For comparison, the system was also simulated using a proportional-velocity controller with proportional gains Kp=274.6159 and velocity gains Kv=5.5272. It was obvious from the squared error norm in Figure 5 that IDC outperforms the PV controller in linear position tracking of the linear servo cart system. This simulation results also suggest that IDC is a robust controller.Figure 4
Sinusoidal position tracking of the linear servo cart system.Figure 5
Squared error norm for sinusoidal position tracking.


## 5.3. Linear Speed Tracking

To evaluate performance of IDC on tracking the speed of the linear servo cart system, we simulate the scenario by giving a 0.25 Hz square wave set point with an amplitude of 0.2 m/s. Another simulation is done using a lead compensator (LC) for comparison, and the results are as shown in Figure6. We can see from the simulation results that IDC has the ability to track the linear servo cart’s speed smoothly without overshooting. The squared error norm measured from both controllers also suggests that the IDC perform slightly better than the LC controller as shown in Figure 7.Figure 6
Speed tracking of the linear servo cart system.Figure 7
Squared error norm for speed tracking.


## 6. Conclusions

An IDC has been successfully designed to control linear position and speed of a linear servo cart system. The controller design has been discussed in detail, particularly in setting up dynamic constraints in the form of constraint differential equations. We have shown that the control law of stable linear position and speed tracking is achieved by inverting the constraint differential equations using MPGI. Through simulations, we have demonstrated that the IDC is an effective and robust controller with performance of tracking linear position and speed of the linear servo cart system, surpassing the performance of PV and LC controllers.



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