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Causality issues in orientation control of an under-actuated drill machine.

1. INTRODUCTION

Under actuated systems are almost two to three decades old but with innovative ideas and latest trends this technological field offers new challenges to scientists and engineers for further research and experimentation. The ideas are not limited only for exploring new & innovative and versatile means of machining but field is open and rather more importantly to carry out further improvements including addressing of practical problems, improve efficiency, reduce wastages in terms of materials, time and maintenance cost etc, to carryout performance and error analysis in existing and already developed techniques [3], [4], [5], [6], [7], [8], [9], [10], [11], [12].

The paper is sequel to [1] in which modelling of a two axis drill machine was presented. The drill bit is free to move along X & Y axis as shown in figure 1, while it spins or rotates about its z-axis. The movement in X & Y axis are executed or implemented by electromagnetic windings on the stator and permanent magnets mounted on the rotor which in this case is drill bit itself. The magnetic field produced by the poles of stator winding interacts with that of permanent magnets mounted on the drill bit produces a torque which in turns is used for changing the drill orientation. The stator poles are excited at a particular phase in every revolution. Due to the mounting and weight limitations a single pair of electromagnets was used and therefore the desired orientation is achieved only by using single actuation thereby reducing the dual axis actuation to single for a two degree of freedom control. Hence the overall system reduces to under-actuated system as the degrees of freedom are greater than the number of control inputs.

[FIGURE 1 OMITTED]

The solution provided by [2] is based on the unlimited magnitude of the actuating signal. In physical system such unlimited amplitude is not available as it leads to saturation. On the other hand if a constraint is put on the maximum amplitude of control signal then this will pose serious limitation on the control input which restricts the overall movement of the drill and leads to stability issues too. The proposed remedy to this limitation is the use of a Pulse Width modulation technique [13], [14], [15], in which a pulse of varying width having fixed amplitude is used. This resulted in overcoming the control effort limitation as well as provides a smooth control for precise movement of the drill bit [2]. However in case of pulse width modulating control signal, the challenge of causality remains. Here a novel technique is introduced to avoid system becomes non-casual.

The remainder of the paper is organized as follows: section II covers the system description followed by modeling in section III. Section IV covers the discrete time equivalent model and Section V covers the causality issues followed by conclusions and references.

2. SYSTEM DESCRIPTION

The plant is a small drill machine and its construction is shown in the figure 1[2]. It spins or rotates about its z-axis while the up & down and right & left movements are about x-axis and y-axis respectively. Due to the mounting limitations mainly because of its miniature size only a single pair of magnets is used. As the drill bit is rotating therefore both axis can be excited but only one axis can be excited at one time, thus converting the fully actuated drill machine into an under actuated system.

Detailed description of drill machine was explained in detail [2]. The stator is a cylindrical body comprising of single phase winding while the rotor is a permanent magnet mounted on a spherical joint which is allowing its motion about the three axes. The two frames of reference one that is attached with the rotor (X', Y', Z') and the other with the stator (X, Y, Z) are shown in the Figure 2. When the stator coil is energized at specific roll instants a torque about the X' axis of the rotor is generated. This results the rotation of drill bit about its Y' axis. As the rotor is spinning at [omega] radians per second therefore due to precession phenomenon the applied torque in one axis causes a motion in the axis which is perpendicular to itself and to the spin axis as well.

3. MATHEMATICAL MODELING

A spinning drill bit with a high angular rate about z-axis resembles two degrees of freedom gyroscope, therefore following model of gyroscope [16] is used :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[[theta].sub.x] and [[theta].sub.y] are the angular position of the bit about X & Y axes respectively while [[tau].sub.x] and [[tau].sub.y] are the applied torques in stator frame of reference. Due to single pair of pole is mounted on the rotor, the applied torque available in stator frame is given by (2)

[[tau].sub.x] = [k.sub.i][tau] cos[omega]t [[tau].sub.y] = [k.sub.i][tau] sin[omega]t (2)

The (1) will be modified as (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

The combined moment of inertia of the drill bit and permanent magnet along X & Y axes is assumed to be same because of axis symmetry and is denoted by J and [J.sub.z] about Z axis respectively. Whereas b and [K.sub.i] represents the coefficient of friction and torque constant. H being the angular momentum is given by the following relationship:

H = ([J.sub.z] - J)[omega] (4)

To represent (3) in state space form, [[theta].sub.x] and [[theta].sub.y] are represented as [x.sub.3] and [x.sub.4] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as [x.sub.1] and [x.sub.2]. The applied torques [[tau].sub.x] and [[tau].sub.y] are represented as [u.sub.1] and [u.sub.2]. Finally the state space representation in stator frame of reference is given in (5).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The (5) can be written in matrix form (6), the output vector y(t) is formed by [x.sub.3] and [x.sub.4] states and the input u(t) is given by (7).

[??](t) = A x(t) + B u(t) y(t) = C x(t) (6)

u(t) = [tau] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The output vector y comprises of angular positions [[theta].sub.x] and [[theta].sub.y] in stator frame of reference.

4. EQUIVALENT MODEL IN DISCRETE TIME

Figure 3 depicts one actuation cycle of period T applied for one complete revolution of drill bit. Where [DELTA] defines the shift of rectangular pulse center from the start of revolution. It is actually the representation of phase of actuation. Whereas a is the fixed amplitude of the pulse. Pw defines the width of rectangular pulse and varies according to the requirement of the control signal magnitude on the lines of Pulse width modulation concept. This marks the difference in approach from the [2] where a pulse with fixed width having varying amplitude was used for the actuation.

Using the input signal u(t) characteristics of figure (4) a closed form is built by considering one complete revolution of the drill bit from time KT to (K+1)T. This model is developed by dividing the signal into three separate intervals i.e., interval I, II and III. It must be noted that during interval I and III the system is un-actuated.

4.1. Interval I : i.e., from KT [right arrow] (KT + [DELTA] - Pw/2) the states at (KT + [DELTA] - Pw/2) using [17] are given by (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

4.2. Interval II : i.e., from (KT + [DELTA] - Pw/2) [right arrow] (KT + [DELTA] - Pw/2) using result of (9) we have (10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

4.3. Interval III : i.e., from (KT + [DELTA] - Pw/2) [right arrow] x((K + 1)T) we have (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Substituting values from (10) into (11) we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

After further simplifying (12) we get:

x((K + 1)T) [congruent to] [e.sup.AT] x (KT) + [e.sup.A(T-[DELTA])] (Pw + [A.sup.2]P[w.sup.3]/24 + [A.sup.4]P[w.sup.5]/1920)BU (12)

[FIGURE 4 OMITTED]

Thus (12) describe the states at time (T+1) for a specific [P.sub.w] and [DELTA]. This was also verified from a simulation results as shown in figure 4, when same input signal is given to the drill bit model (8) and its derived solution (12).

5. CAUSALITY ISSUES

One of the major drawbacks in using the concept of pulse width modulated control signal for this application is that the system actuation becomes non-causal. A system is said to be causal system if its output depends on present and past inputs only. Therefore the output of casual system depends on present and past inputs, i.e., y(n) is a function of x(n), x(n-1), x(n- 2), x(n-3). Mathematically it can be represented as

y(n) = x(n) + x(n-2) (13)

Similarly a system whose present response depends on future values of the inputs is called as a non-causal system. Mathematically it can be represented as

Y(n) = x(n) + x(n+1) (14)

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

A novel concept of sliding adjustment is introduced which effectively do not allow actuation to become non-causal. The three possible cases are graphically shown in Figure 5, 6 and 7. Their mathematical representation is shown in Table 1. The center of the pulse "delta" has to be determined at the end of every rotation of the shaft and the actuation may become non-causal for some values of A as shown in Figure-5 & Figure-7. An arbitrary threshold zone "th" is introduced so that the actuation should not fall within this zone. The observation is always at the start of zone which slides to deny the actuation to fall within "th" zone. As clear from the Figure-5 & Figure-7 that when the reference is changed for (i+1) rotation then the next observation is so placed as to avoid the system becomes non-causal and align the center of actuation with the reference.

6. CONCLUSION

A closed form in discrete time domain is presented for orientation control of an under-actuated drill machine. PWM approach was used to overcome the practical limitation of controlling signal. Sliding adjustment is effectively introduced to avoid system becomes non- casual. The same is supported by simulation results (Figure 8). The derived model is also verified by comparing it with actual model through simulations. The derived model is useful for designing the feedback control system for orientation control. An pulse equivalent area based technique [18] has already been developed for orientation control of similar model under discussion since the typical control techniques [19, 20, 21] are not applicable for this class of systems. However techniques like back propagation neural network [22] can be applied for orientation control of similar systems.

7. REFERENCES

[1.] Atif, A., Malik, M.B., "Modeling and Simulation for Orientation Control of an Under-actuated Drill Machine", SDIWC, The Second International Conference on Electrical and Electronic Engineering, Telecommunication Engineering, and Mechatronics (EEETEM2016) conference, 2016, DOI: 10.13140/RG.2.1.3289.5767

[2.] Malik, M.B., Malik, F.M., and Munawar, K., "Orientation control of a 3-d under-actuated drill machine based on discrete-time equivalent model", International Journal Robotics and Automation, Vol. 27, Issue 4, DOI:10.2316/Journal.206.2012.4.206-3324, 2012.

[3.] He, J., Gao, F., Zhang, D., " Design and performance analysis of a novel parallel servo press with redundant actuation", International Journal of Mechanics and Materials in Design, Vol. 10, Issue 2, pp 145-163, 2014.

[4.] Criens, C., Willems, F., Steinbuch, M., "Under actuated air path control of diesel engines for low emissions and high efficiency", Proceedings of the FISITA 2012 World Automotive Congress, Vol. 189, pp 725-737, 2013.

[5.] Dong, W., Gu, G.Y., Zhu, X., Ding, H., "Solving the boundary value problem of an under-actuated quadrotor with subspace stabilization approach", Journal of Intelligent & Robotic Systems, DOI: 10.1007/s10846-014-0161-3, 2014.

[6.] Liu, Y., and Yu. H., "A survey of underactuated mechanical systems", Control Theory Appl., IET, Vol. 7, pp. 921-935, 2013.

[7.] Pucci, D., Romano, F., and Nori.F., "Collocated adaptive control of underactuated mechanical systems", IEEE trans on Robotics, Vol. 31, Issue. 6, pp. 1527-1536, 2015

[8.] Khan, Q.S., and Akmeliawati, R., " Robust Control of Underactuated Systems: Higher Order Integral Sliding Mode Approach", Mathematical Problems in Engineering, Vol. 2016, 11 pages, DOI:10.1155/2016/5641478, 2016

[9.] Baker Al-Bahri, Zainab Al-Qurashi., "Practical neural controller for robotic manipulator", vol 2, no. 1, International Journal of Digital Information and Wireless Communications (IJDIWC), jan-2012.

[10.] Maalouf, D., Chemori, A., and Creuze. V., " L1 Adaptive Depth and Pitch Control of an Underwater Vehicle with Real-Time Experiments", Ocean Engineering (Elsevier), DOI: 10.1016/j.oceaneng.2015.02.002, pp. 66-77, 2015.

[11.] Bara E., Aydin Y.k, "Robust nonlinear composite adaptive control of quadrotor", vol 4, no. 2, International Journal of Digital Information and Wireless Communications (IJDIWC), 2014

[12.] Matthias S., Peter B., "Ultra-Low Complexity Control Mechanisms for Sensor Networks and Robotic Swarms", vol 3, no. 1, International Journal on New Computer Architectures and Their Applications (IJNCAA), 2013

[13.] Barr, Michael. "Pulse Width Modulation," Embedded Systems Programming, September 2001, pp. 103-104.

[14.] T. Sakamoto and N. Hori, "New PWM schemes based on the principle of equivalent areas," IEEE International Symposium on Industrial Electronics, L'Aquila, Italy, pp. 505-510, 2002.

[15.] T. Sakamoto, N. Hori and T. Ueno, "Closed-loop control using a low-frequency PWM signal," Proc. IEEE Methods and Models in Automation and Robotics, Miedzyzdroje, Poland, pp. 627-632, 2003.

[16.] Savet, P.H., "Gyroscopes theory & design with applications to instrumentation, guidance and control", McGraw-Hill, 1961.

[17.] W. J. Rugh, "Linear System Theory", 2nd Edition, Prentice Hall, 1996.

[18.] A. Ali and M. B. Malik, "Pulse Equivalent Area based Orientation of a Class of Under-Actuated System under Constraints", Under review in Mehran University Research Journal Of Engineering & Technology.

[19.] C. T. Chen, Linear Systems: Theory and Design. Saunders College Publishing, third edition, 1984.

[20.] G. F. Franklin, D. Powell, and M. L. Workman, "Digital Control of Dynamical Systems", Addison-Wesley, 1996

[21.] Katsuhiko Ogata., "Modern Control Engineering" 5th Edition, Prentice Hall, 2009

[22.] M. Z. Rehman 1, N. M. Nawi, "Improving the Accuracy of Gradient Descent Back Propagation Algorithm (GDAM) on Classification Problems", vol 1, no. 4, International Journal on New Computer Architectures and Their Applications (IJNCAA), 2011

Atif Ali * and Mohammad Bilal Malik

Department of Electrical Engineering, College of Electrical and Mechanical Engineering, National University of Sciences and Technology (NUST), Islamabad, Pakistan

* Email : atif@ceme.nust.edu.pk
Table 1 Three possible cases for Sliding Adjustment

CASE   Condition               ith Rotation       (i+1)th Rotation

I      if {[DELTA] -           [MATHEMATICAL      [MATHEMATICAL
         [absolute               EXPRESSION NOT
         value of (Pw/2)]}       REPRODUCIBLE       EXPRESSION NOT
         [less than or equal     IN ASCII]          REPRODUCIBLE
         to] th                                     IN ASCII]
II     if {[DELTA] -           [MATHEMATICAL
         [absolute               EXPRESSION NOT
         value of (Pw/2)]}       REPRODUCIBLE
         [greater than or        IN ASCII]
         equal to] [T.sub.s]
III    otherwise               [MATHEMATICAL
                                 EXPRESSION NOT
                                 REPRODUCIBLE
                                 IN ASCII]
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Author:Ali, Atif; Malik, Mohammad Bilal
Publication:International Journal of New Computer Architectures and Their Applications
Article Type:Report
Date:Apr 1, 2016
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