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

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.


Introduction
e 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. ere were also some other recent works implementing the PID controller on positioning control of servo motors with different tuning methods [3,4]. e 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]. is 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. is 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][12][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].
is 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][17][18].
is 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][20][21][22][23][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][26][27].
e remainder of this article is organized as follows: e mathematical model of the linear servo cart system is established in Section 2, while Section 3 outlines the IDC design for linear positional control of the linear servo cart system. e 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.

Mathematical Model of the Linear Servo
Cart System Figure 1 shows the linear servo cart system. e 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 where M, v c , and B eq are the mass of the cart, linear velocity of the cart, and the equivalent viscous damping coefficient, respectively, and F aj is the armature inertial force due to motor rotation acting on the cart which can be defined as where η g is the efficiency of the gear box, K g is the gear ratio, and τ aj is the armature inertial torque which can be expressed as e angular velocity of the motor shaft can be translated into linear velocity of the cart with the following equation: By substituting (3) and (4), (2) can be rewritten as 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: where and the actuator gain is Note that η m is the efficiency of the motor, K g , k t , and k m are the gear ratio, motor torque constant, and back-emf constant, respectively, and r mp and R m is the radius of the motor pinion and the motor resistance.
Finally, the equivalent inertia term can be expressed as

Design of IDC Control
e dynamics of the linear servo cart system can be expressed by rearranging (6) as follows: where F � −J −1 eq B eq v c and G � J −1 eq A m . It is known that the velocity of the cart can be obtained by taking the derivative of its linear position: 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, x c from its desired position, x cd is defined as Similarly, the error function for linear velocity can be written as e 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. e equation takes the following form: where c 1 , c 2 , and c 3 are coefficients that allow the constraint differential equations as in (14) and (15) to achieve uniform asymptotic stability. erefore, they must be selected carefully as suggested in [15]. e following can be obtained by taking the derivative from (12) and (13): Taking further derivative on (16) yields 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: where 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 where A + is the MPGI of A given as Note that λ is the null control and P is the null projection, which can be expressed as Note that A has a dimension of 2 × 1, in which the P in (24) will be zero when the property of pseudo-inverse A + A � 1. is will make the null control not useful, and therefore, we will not consider it in our control design.

IDC Singularity Avoidance
e 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 as u and can be defined as e homogeneous part of equation (25) is asymptotically stable, while c in the forcing phrase is a positive real-valued constant. e dynamically scaled generalized inverse (DSGI) is formulated as . (26) us, we can update the IDC-based control input voltage by the following expression: Finally, we can update (10) by the following expression:

Numerical Simulations
To evaluate performances of the designed controller, we performed numerical simulations on a linear servo cart model having parameters as given in Table 1. e proposed IDC was having dynamic gains of n 1 � 2 and n 2 � 0.01 and tracking gains of c 1 � 50, c 2 � 21000, and c 3 � 0.1.

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 Figure 2. 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.

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 Figure 4 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 K p � 274.6159 and velocity gains K v � 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. is simulation results also suggest that IDC is a robust controller.   results are as shown in Figure 6. We can see from the simulation results that IDC has the ability to track the linear servo cart's speed smoothly without overshooting. e squared error norm measured from both controllers also suggests that the IDC perform slightly better than the LC controller as shown in Figure 7.

Conclusions
An IDC has been successfully designed to control linear position and speed of a linear servo cart system. e 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. rough 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.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.