Kinematic-Time Profiling in Mechatronic Systems

Lift Report 2018; 44(4): 1-11

Kinematic-Time Profiling in Mechatronic Systems Lutfi Al-Sharif Professor Mechatronics Engineering Department The University of Jordan, Amman 11942, Jordan This series of articles is based on lecture notes used by the author in the course entitled: “Advanced Drive Systems” that was delivered at the Mechatronics Engineering Department at The University of Jordan (Amman, Jordan) to undergraduate Mechatronics Engineering students in the period between 2006 and 2018. Abstract Planning the kinematics of the journey in mechatronic systems is very critical for the success of these systems. In the majority of cases this involves planning the velocity against time. However, in some cases, the planning is based on displacement against time. The use of a single quintic polynomial for displacement as used in robotic manipulator arms is reviewed. A practical numerical example is given and solved in detail. In elevator systems, it is customary to use a blended polynomial approach. A practical numerical example is given. A comparison is then carried out for a journey of 600 m and a duration of 50 seconds between the single polynomial and the blended polynomial. The paper concludes with an introduction to the concept of cascade control as used in mechatronic systems. The two-loop cascade control structure requires the velocity time profile as an input reference. The three-loop cascade control structure requires the displacement-time profile as an input reference. 1. INTRODUCTION Speed profiling is used in all motion control mechatronic systems. It is defined as the planning of the speed against time during the journey in order to optimize one or more of the following parameters: 1. 2. 3. 4. 5.

Minimise journey time. Minimise energy consumption. Maximise safety. Maximise passenger comfort. Minimise forces on the equipment.

Joint space planning and trajectory planning are terms used in robotic manipulator arms to describe the process of planning the displacement, velocity and acceleration against time. Velocity-time profiling is the term used in elevators. This paper looks at the two different approaches used for kinematic-time profiling, as well as the use of cascade control as a control structure of implementing the motion control system. 2. THE SINGLE POLYNOMIAL APPROACH IN ROBOTIC MANIPULATOR ARMS One of the simple methods of planning joint angles against time is the use of a single polynomial ([1], [2]). The polynomial for displacement against time can be a cubic polynomial (i.e., 3rd order in time), a quintic polynomial (i.e., 5th order in time) or a heptic/septic (i.e., 7th oder in time). Page 1 of 11

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Using the same notation as in [1], the following is the development of the quintic displacement polynomial. 𝜃 𝑡

𝐴𝑡

𝐵𝑡

𝐶𝑡

𝐷𝑡

𝐸𝑡

𝐹

……….(1)

It will be assumed that this polynomial will cover the period from 0 to T seconds, which is the required duration of the journey. Differentiating with respect to time gives the following polynomial for the angular velocity: 𝜃 𝑡

5𝐴𝑡

4𝐵𝑡

3𝐶𝑡

2𝐷𝑡

𝐸

……….(2)

Differentiating with respect to time gives the following polynomial for the angular acceleration: 𝜃 𝑡

20𝐴𝑡

12𝐵𝑡

6𝐶𝑡

2𝐷

……….(3)

Substituting 0 for t gives the initial value of the angular displacement gives: 𝜃 0

𝐹

……….(4)

Substituting T seconds for t gives the final value of the angular displacement gives: 𝜃 𝑇

𝐴𝑇

𝐵𝑇

𝐶𝑇

𝐷𝑇

𝐸𝑇

𝐹

……….(5)

Substituting 0 for t gives the initial value of the angular velocity gives: 𝜃 0

𝐸

……….(6)

Substituting T seconds for t gives the final value of the angular velocity gives: 𝜃 𝑇

4𝐵𝑇

5𝐴𝑇

3𝐶𝑇

2𝐷𝑇

𝐸

……….(7)

Substituting 0 for t gives the initial value of the angular acceleration gives: 𝜃 0

2𝐷

……….(8)

Substituting T seconds for t gives the final value of the angular acceleration gives: 𝜃 𝑇

20𝐴𝑇

12𝐵𝑇

6𝐶𝑇

2𝐷

……….(9)

Equations (4) to (9) can be combined in a very concise and compact format using matrices as shown in equation (10) below:

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𝜃 ⎡ 𝜃 ⎢ ⎢𝜃 ⎢𝜃 ⎢ ⎢𝜃 ⎣𝜃

0 𝑇 0 𝑇 0 𝑇

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0 ⎡ 𝑇 ⎢ ⎢ 0 ⎢ 5𝑇 ⎢ 0 ⎣20𝑇

0 𝑇 0 4𝑇 0 12𝑇

0 𝑇 0 3𝑇 0 6𝑇

0 𝑇 0 2𝑇 2 2

0 𝑇 1 1 0 0

1 𝐴 1⎤ ⎡ 𝐵 ⎤ ⎥ ⎢ ⎥ 0⎥ ⎢ 𝐶 ⎥ ∙ 0 ⎥ ⎢𝐷 ⎥ 0⎥ ⎢ 𝐸 ⎥ 0⎦ ⎣ 𝐹 ⎦

……….(10)

The left-hand side of equation (10) is known (which is the initial and final conditions for the angular displacement, velocity and acceleration. The coefficients A to F are unknown. This can be solved by finding the inverse of the 6x6 square matrix. 𝐴 ⎡𝐵 ⎤ ⎢ ⎥ ⎢𝐶 ⎥ ⎢𝐷 ⎥ ⎢𝐸 ⎥ ⎣𝐹 ⎦

0 ⎡ 𝑇 ⎢ ⎢ 0 ⎢ 5𝑇 ⎢ 0 ⎣20𝑇

0 𝑇 0 4𝑇 0 12𝑇

0 𝑇 0 3𝑇 0 6𝑇

0 𝑇 0 2𝑇 2 2

0 𝑇 1 1 0 0

1 1⎤ ⎥ 0⎥ 0⎥ 0⎥ 0⎦

𝜃 ⎡ 𝜃 ⎢ ⎢𝜃 ∙⎢ 𝜃 ⎢ ⎢𝜃 ⎣𝜃

0 𝑇 0 𝑇 0 𝑇

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

……….(10)

An example on this method is given below. A revolute joint will be displaced by 60 degrees. The initial and final angular velocity should be zero. The initial and final angular acceleration should also be zero. It will be assumed that the initial displacement is zero and the final displacement is π/3 radians (i.e., 60°). It is necessary to develop a quintic polynomial (quintic: to the fifth degree) for the angular displacement of the revolute joint. The total time will be 1 second (i.e., T=1 s). Starting with the initial angular displacement gives the value of F as zero: 𝜃 0

𝐹

0

……….(11)

Using the initial value of the angular velocity gives the value of E as zero: 𝜃 0

𝐸

0

……….(12)

Using the initial value for the angular acceleration gives the value of D as zero: 𝜃 0

2𝐷

0

……….(13)

Thus, three of the coefficients are zero (D, E and F). Using these values and the final value of the angular displacement gives: 𝜃 1

𝐴𝑇 𝐴

𝐵𝑇 𝐵

𝐶

𝐶𝑇 𝜋 3

……….(12)

Using the final value of the angular displacement (i.e., 0) gives: 𝜃 1

4𝐵𝑇 5𝐴𝑇 5𝐴 4𝐵 3𝐶

3𝐶𝑇 0

Using the final value of the angular acceleration (i.e., 0) gives: Page 3 of 11

……….(13)

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𝜃 1

12𝐵𝑇 20𝐴𝑇 20𝐴 12 𝐵 6 𝐶

6𝐶𝑇 0

……….(14)

Solving for the three simultaneous equations (12), (13) and (14) gives the following values for A, B and C: 𝐴 2𝜋 𝑟𝑎𝑑/𝑠 𝐵

5𝜋 𝑟𝑎𝑑/𝑠

𝐶

10 𝜋 𝑟𝑎𝑑/𝑠 3

Thus, the final polynomials for angular displacement, angular velocity and angular acceleration are: 𝜃 𝑡

2𝜋𝑡

𝜃 𝑡 𝜃 𝑡

10𝜋𝑡 40𝜋𝑡

10𝜋 𝑡 3 20𝜋𝑡 10𝜋𝑡 60𝜋𝑡 20𝜋𝑡

5𝜋𝑡

……….(15) ……….(16) ……….(17)

The polynomials for angular displacement, velocity and acceleration have been plotted against time in Figure 1, Figure 2 and Figure 3 respectively.

Figure 1: Angular displacement against time throughout the journey.

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Figure 2: Angular velocity against time throughout the journey.

Figure 3: Angular acceleration against time throughout the journey.

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3. VELOCITY TIME PROFILING APPROACH IN ELEVATOR SYSTEMS USING A BLENDED POLYNOMIAL In robotic manipulator arm systems, one of the approaches to developing speed-time profiles is to use a quartic polynomial (t4) for the velocity time profiles, which is equivalent to using a quintic polynomial (t5) for the displacement time profile. The advantage of the qunitic polynomial approach for displacement against time is that one single polynomial is needed for the full duration of the journey, the journey time can be specified as a requirement and it is possible to specify zero values for the initial and final velocity and acceleration, which leads to smoother motion. However, the main disadvantage of this approach is the system spends a very short period running at the rated speed. This leads to excessive journey times. It is too expensive to change the motor/drive/mechanical drive to one with a higher rated speed. In order to overcome this problem in elevator systems, a different approach is adopted. This approach uses a multi-segment (also referred to as blended polynomial) journey with different order polynomials (t3, t2, t3, t, t3, t2, t3) for the displacement against time polynomial ([3], [4], [5], [6], [7], [8]). This offers the advantage that the elevator will travel for a significant period of time at the rated speed. This ensures minimum journey time for a specific rated speed. Figure 4 below shows an example of a velocity-time profile commonly employed in elevator systems. It can be seen how it is composed of 7 segments. Of particular importance is the middle segment (duration of t4 covering a distance of d4). As the displacement increases, the duration of t4 is simply increased in order to accommodate this increase.

Figure 4: Blended polynomial approach adopted in elevator speed-time profiling.

There are three advantages to this approach, listed below: 1. The jerk is finite contributing to the passenger comfort. 2. The initial and final value of the velocity and acceleration are zeros, leading to Page 6 of 11

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smooth operation. 3. The elevator spends a significant amount of time at the rated speed, leading to a reduction in journey time. This blended polynomial approach is also currently used in modern drive systems. An example of the use of speed-time profiling in a Siemens drive system is shown in Figure 5. the values of the acceleration and deceleration can be independently set, as well as being able to set four independent values for the jerk (four values for the four transition points).

Figure 5: Setting of the Blended Polynomial Velocity Time Profile inside a Drive.

The velocity time profiling has several important applications in elevators systems, such as evaluating the round trip time ([9], [10]), calculating the energy consumption ([11], [12]) and safety distance control in elevator system with multiple cars in the same shaft [13]. 4. COMPARISON BETWEEN THE BLENDED POLYNOMIAL (MULTI-SEGMENT) TO THE SINGLE POLYNOMIAL APPROACH A comparison has been made between the single polynomial approach and the blended polynomial approach for a 600 m journey. In the blended polynomial approach, the following parameters are assumed: Displacement: 600 m. Rated velocity: 24 m/s. Rated acceleration: 1 m/s2. Rated jerk: 1 m/s3. Page 7 of 11

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Based on these values, the total journey time is: 𝑡

𝑑 𝑣

𝑣 𝑎

𝑎 𝑗

600 24

24 1

1 1

50 𝑠

……….(18)

Thus, the total journey time is 50 s. Using the two values of the total displacement of 600 m and the total journey time of 50 s, a single polynomial can be developed assuming zero values for the initial and final velocity and acceleration. 𝑣 𝑡 𝑣 𝑡

5 ∗ 1.15𝑥10

∗𝑡

4 ∗ 0.00144 ∗ 𝑡

3 ∗ 0.048 ∗ 𝑡

……….(19)

5.75𝑥10

∗𝑡

5.75𝑥10

0.144 ∗ 𝑡

……….(20)

∗𝑡

For this journey, the two speed time profiles (the single polynomial and the blended polynomial) have been plotted in Figure 6.

Figure 6: Comparison between the single polynomial and blended polynomial for a journey of 600 m and a duration of 50 s.

Using the blended polynomial approach, the rated speed, rated acceleration and rated jerk are specified as inputs and the duration of the journey is an output. Using the single polynomial approach, the duration of the journey is an input and the maximum velocity, maximum acceleration and maximum jerk are the outputs. 5. THE USE OF CASCADE CONTROL SYSTEMS In many position and speed control systems, cascade control is used. Cascade control is Page 8 of 11

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a structure for a control system whereby two or more nested loops are used, where the error from the outer loop becomes the set value for the inner loop. It is possible to think of cascade control as an alternative method to state space feedback control: “Full state information may be unavailable, or difficult to use at once. In cascade control, successive control loops are used, each using a single measurement and a single actuated variable. The output of the primary controller is the input to the secondary and so on. A judicious choice of how to pair variables and the ordering of the multiple loops can lead to a very efficient control implementation without the need for a full state feedback control.” “… Thus, the main concept behind the cascade control is: feedback any information in the plant as soon as it is available.” [14], (Albertos & Mareels, 2010, pp 216-217) This section illustrates the importance of generating the kinematic-time profile as a reference value for the cascade control system. In the two-loop cascade control structure, a velocity-time profile is the reference value; in the three-loop cascade control structure, a displacement-time profile is the reference value. 5.1 Implementation a Two-Loop Cascade Setup In a velocity control system, the reference value is the velocity and thus two nested loops are required (one for torque and one for velocity). Torque is inferred by measuring the armature current of a dc motor (as torque is proportional to the armature current at constant magnetic flux excitation). The block diagram of this setup is shown in Figure 7. It is worth noting that the inner loop is the faster of the two loops and the outer loop is the slower of the two loops. Although this setup will result in negligible error in the velocity, it might accumulate error in displacement. This accumulated error in displacement would show as an error in stopping level of the elevator, for example.

Figure 7: Velocity control system employing two cascaded loops.

5.2 Implementation a Three-Loop Cascade Setup An improvement on the two-loop setup is to use three loops in a position control system. In a position control system, the reference value is the displacement and thus three nested loops are required (one for torque, one for velocity and one for displacement). Such a setup is shown in Figure 8. It is worth noting that the outer loop that is the slowest loop and it is the position loop. The intermediate loop which is faster than the position loop and is denoted as the velocity loop. And finally, the inner loop which is the torque or current loop, and this is the fastest of the three loops.

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Figure 8: Displacement control system employing three cascaded loops.

The speed is measured using an incremental shaft encoder or using a tacho-generator. Integrating the output from the incremental shaft encoder gives the value of the position. A current sensor that measures the dc motor armature current gives a signal that is representative of the torque. The reference value must be the displacement vs time profile for the full journey. The error in distance is applied to a PI or PID controller. This becomes the set value for the speed. The error in the speed is then also applied to the PI or PID controller and this becomes the set value for the torque loop. The measured armature current is used as representative of the actual torque. The error in the current/torque is then applied to the PI or PID controller. This is then used to drive the power electronic DC drive system (e.g., PWM H bridge) by feeding it with the value of the duty cycle (dc%). As the error changes it changes the duty cycle of the output. The output from the DC motor is coupled to the mechanical load and to the incremental or absolute shaft encoder. 6. CONCLUSIONS In mechatronic motion control systems, kinematic time profiling is very critical. Before motion can take place, the full journey must be planned. This planning is denoted in this paper as “kinematic-time profiling”. Kinematic time profiling can involve either velocity-time profiling or displacementtime profiling. Velocity-time profiling is defined as the planning of the system velocity at every point in the journey such that a parameter or more of interest are optimised. For example, the velocity time profile can be planned in order to minimise the energy consumption or to minimise the journey time. There are two broad approaches to planning the velocity against time or displacement against time: the single polynomial and the blended polynomial. The single polynomial is used in robotic manipulator arms, where the displacement versus time polynomial is quintic (i.e., 5th order against time). The blended polynomial is used in elevator systems and allows a constant speed to be used to traverse the journey. The kinematic time profiling is inextricably linked to the control system that will be used in the mechatronic motion control system. Cascade control can be used whereby two loops or three loops are nested. In a two-loop cascade system, the reference is the velocity and hence a velocity-time profile is required. In a three-loop cascade control system, the displacement-time profile is required as the reference. REFERENCES [1] Corke P. Robotics, Vision and Control: Fundamental algorithms in MATLAB. Springer, 2011. pp 43-44 [2] Craig J J. Introduction to Robotics: Mechanics and Control. Pearson Prentice Hall, 3rd Edition, 2005. (All of chapter 7 is dedicated to trajectory generation). Page 10 of 11

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[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

Peters R D. Vertical Transportation Planning in Buildings. Ph.D. Thesis, Brunel University, February 1998. Al-Sharif L. Intermediate Elevator Kinematics and Preferred Numbers (METE III). Lift Report 2014; 40(6): 20-31. Peters R D. Ideal lift kinematics: derivation of formulae for the equations of motion of a lift. International Journal of Elevator Engineers, 1996; 1(1): 60-71. Peters R D. Ideal Lift Kinematics: Complete Equations for Plotting Optimum Motion. Elevator Technology 6, 1995; 6: 175-184. March 1995, Hong Kong. Molz H D. On the ideal kinematics of lifts, Part I. Elevatori, 1991; 20: 41-46. January/February 1991. Molz H D. On the ideal kinematics of lifts, Part II. Elevatori, 1991; 20: 39-43. March/April 1991. Al-Sharif L. Introduction and Assessing Demand in Elevator Traffic Systems (METE I). Lift Report 2014; 40(4): 16-24. July/August 2014. Al-Sharif L. Calculating the Elevator Round Trip Time for the Most Basic of Cases (METE II). Lift Report 2014; 40(5): 18-30. Sep/Oct 2014. Al-Sharif L. Lift energy consumption: general overview (1974-2001). Elevator World 2004; 52(10): 61-67. Smith R S, Peters R D and Al-Sharif L. Elevator system to minimize entrapment of passengers during a power failure. US Patent 7,967,113, 18 October 2005. Gerstenmeyer S and Peters R. Safety distance control for multi-car lifts. Building Services Engineering Research and Technology 2016; 37(6): 730 - 754. Albertos P and Mareels I. Feedback and Control for Everyone. Springer, 2010.

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