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2 Abstract Impedance and admittance control are complementary in terms of stability and performance properties. In general, impedance controlled system have stable dynamic interaction with stiff environments but poor accuracy in free-space, while for admittance control is the other way around. It was also performed some simulations to verify and for better understanding that the slave coupled with an impedance control indeed behaves better when there is contact with stiff environment and that in free flight, admittance control is the best option. The aim of this assignment is to compare impedance and admittance control in the teleoperation of unmanned aerial vehicles. More precisely, the two controllers will be implemented on master device, which commands the flying vehicle controlled by an admittance control, these are the two frameworks used. These frameworks were considered in order to decide which one is better during only free flight considering its transparency. It is simulated in Matlab/Simulink. The simulations shows that the framework using admittance control as both master s and slave s controller is the best option for free flight, due to the fact that the human receives a good haptic cue at the same range of frequency that the slave is able to track the reference better. i

3 Contents I. Introduction... 1 II. Model... 1 A. Impedance Control... 1 B. Admittance control... 2 III. Complete teleoperation framework... 3 IV. Simulations and analysis... 5 A. Slave... 5 B. Complete teleoperation... 6 V. Conclusions... 9 VI. Acknowledgment References Appendix A. Evaluation of impedance and admittance ii

4 I. Introduction Unmanned Aerial Vehicles (UAVs) have the potential to support human beings in a wide range of applications that are dull or inflict a risk to human life, such as inspecting infrastructures and repairing it, at lower operational costs. [4] As a result, the human is able to stay at a safe distance to perform such activity. On bilateral teleoperation, there is a remote coupling between a master and a slave. The human interacts with the haptic device, the master in this case, to send a signal to the UAV that interacts with the environment. The force generated from the interaction between the slave and the environment is fedback to the human. [4] Ideally, with human-in-the-loop control strategies, the UAVs can work as a flying hand and perform high-level tasks. The master and the slave have their respective controllers, which can be impedance or admittance control. In the literature, there are papers considering either impedance or admittance as the controllers for both master and slave [1], [2]. The aim of this assignment is to compare impedance and admittance control in the teleoperation of unmanned aerial vehicles. More precisely, the two controllers will be implemented on master device, which commands the flying vehicle, and compared during a free flight and in contact with a virtual wall. In order to decide which one is better depending on the task, the following metrics will be used: position error, time of execution and transparency. The remainder of this paper is organized as follows: in Section II describes the formulation of the slave model with either impedance or admittance control. In Section III, it is described both frameworks with the complete teleoperation that were used for the simulations. In Section IV, the analysis for the slave simulations and the complete framework simulations. A short summary is provided in Section V. II. Model Initially, the system considered the vehicle, i.e., the slave, to be a free single mass with one degree of freedom, and modelled as a simplified second order system (1) where is the generalized inertia, is the position of the mass, the control force applied to the vehicle and is the force resulted from the interaction between the mass and the environment. The environment is considered to be a virtual wall modelled by a spring ( ) (2) where is the stiffness of the environment. When there is no interaction, is considered to be zero, which means that there is no feedback force in free flight for the slave s controller. Impedance and admittance control are complementary in terms of stability and performance properties. In general, impedance controlled systems have stable dynamic interaction with stiff environments but poor accuracy in freespace, while admittance controlled systems have high accuracy in noncontact tasks, but they can become unstable when interacting with stiff environments. [3] A. Impedance Control In impedance control, the controller is a mechanical impedance, i.e., the system receives a reference position as input and yields a force as output (see Figure 1), and the plant behaves as an admittance. Figure 1 The block illustrates an impedance 1

5 The controller was modelled as a proportional-derivative controller (PD controller) which is equivalent to a virtual spring and damper. The Ideal Physical Model (IPM) of the system can be seen in Figure 2. The block diagram in Figure 3 shows a schematic of the slave system, the inputs and outputs of each block characterizes them as impedance and admittance. A detailed block diagram can be seen in Section IV. As most of plants usually behave as an admittance, to emulate an impedance behaviour it is required to consider part of the controller as being part of the plant, see Figure 6. This way we can have an admittance control for the plant. The admittance control is modelled as a set of spring, damper and mass in order to have a position as output and the position control is a set of spring and damper, Figure 5 shows the IPM of the system. Figure 2 IPM of the Impedance control + vehicle Figure 3 Schematics of Impedance control + Plant Dynamics The system dynamics can be described by combining the equations (1), (2) and (3). ( ) ( ) (3) where is the stiffness of the spring, is the damping coefficient of the damper and is the position reference provided. B. Admittance control In admittance control is the other way around, the controller is a mechanical admittance, i.e., the system receives a reference force and yields a position (see Figure 4), and the plant behaves as an impedance. Figure 5 - IPM of the Admittance control + Plant The parameters are different from each control. The second set of spring and damper are stiffer than the first one, what makes the second mass-springdamper system behaves like a rigid mass tracking the position that is the output of the mass Md. The schematic of the slave system is presented in Figure 6, the inputs and outputs of each block are the one used to mimic the desired characteristic. A detailed block diagram can be seen in Section IV. Figure 6 Schematic of Admittance control + Plant dynamics To describe the system dynamics it was used equations (1) and a modified version of equation (3) that is now equation (4), for the plant, equation (2) for the interaction with the environment and (5) for the controller. ( ) ( ) (4) Figure 4 The block illustrates an admittance 2

6 ( ) ( ) ( ) (5) where, and are the parameters of the first spring-damper-mass system, i.e., the controller, and is the output position of the admittance control. III. Complete teleoperation framework The complete teleoperation schematic is shown in Figure 7, the human interacts with the master that is coupled with the controller and the command given by the human is sent the slave with no delays. After the slave has performed the action, there is a feedback to the human that depends on the master s control, if it is an impedance, then the feedback is a virtual force. On the other hand, if it is an admittance, the feedback is a position. Figure 7 - Complete teleoperation Virtual Force For the impedance control in the master side, ideally, when the human interacts with the haptic device, they are imposing a position that will be sent to the vehicle regardless of the necessary force needed to deliver it, then it is computed the difference from the vehicle s position to the reference signal sent by the master and a virtual force is sent a feedback to the human, as a result the human is able to feel what is happening in the slave. For the admittance control in the master side, the human imposes a force to the master regardless of the final position, which is decided by the control. Then the position of the master is sent to the slave without delays and the slave sends a feedback of the output position that is computed with the slave s reference input, so the human is able to feel what is happening in the slave. It will be considered two combinations of controllers to evaluate in which conditions one works better than the other in free flight. The two combinations are: 1. the slave controller is considered to be an admittance control and the master controller is an impedance control, Figure 9; 2. the slave controller is an admittance and the master controller is also an admittance, Figure 10, the difference between both controllers is that the master s one was modelled as an ideal admittance and it has reversed causality. Since the slave s control is set to be an admittance in both combinations, for now on, when it is referred to impedance or admittance control, it is actually referring to the master s control, unless told otherwise. With these block diagrams it is not possible to perform simulations based on the system tracking response because now the difference between both frameworks is how the reference position that is sent to the slave is generated, then the slave will behave the same. In order to analyse it, then, it was necessary to derive the transfer function from each block diagram. For the impedance, the transfer function used was based on the reference position sent to the slave,, as being the input and the force feedback, ( ), as output, both indicated in the block diagram. ( ) ( ) ( ) (6) For the admittance is the other way around, what is taken as input was the force feedback, ( ), and the output is the reference position sent to the slave,. ( ) ( ) ( ) (7) 3

7 Figure 8 - Complete teleoperation with impedance control in the master side Figure 9 - Complete teleoperation with admittance control in the master side 4

8 Figure 10 - Admittance control + plant Figure 11 Impedance Control + Plant IV. Simulations and analysis A. Slave It is well known in the literature that impedance and admittance control behave differently in different tasks, however it was performed some simulations with the systems presented in Section II by a way of understanding why one is considered better than the other in free flight and in contact with a virtual wall and to see the effect of varying the parameters. The analysis follows bellow and the complete simulation can be found in Appendix Part B. Figures 11 and 12 show the block diagrams of both systems used for the simulations in Matlab/Simulink. 1. Free flight During free flight is zero. The response from both systems behaved like it was expected, i.e., if the damping coefficients were high enough or the stiffness of the springs were low enough the response were less oscillatory and even critically damped. On the other hand, when the damping coefficients were low and stiffness of the springs were high, the response showed higher overshoot and to be more oscillatory. The responses from the system with admittance control did not become oscillatory as the parameters were varied, they were more robust; the overshoot was not greater than 30% and the stead state errors were always zero. The responses from the system with impedance control became oscillatory, the highest overshoot was 100% and the steady state error was not always zero. Based on the maximum overshoot, how oscillatory the response was, the settling time and the steady state error, admittance control behaved better than impedance control. 5

9 2. Interaction with environment The fact that the system did not become unstable as the parameters were varied when impedance control was being used is the main argument on why it behaves better than admittance when there is interaction with the environment, even though the responses using admittance control were less oscillatory than the ones using impedance control, stability is important. 3. Admittance control in frequency domain As it was already stated in Section III that both complete teleoperation frameworks use admittance control as the slave controller in this part is presented the frequency response of the system to realise what is the bandwidth of the system that the output position and the reference position are the same, i.e., the vehicle can track the position input. ( ) ( ) ( ) Figure 12 - Admittance frequency response (8) It is possible to see that for low frequencies, the response has a unit gain, which means that the vehicle is able to track the reference input. For high frequencies the response is attenuated and no longer can track the reference accurately. B. Complete teleoperation For the complete teleoperation rather than analysing the response of the system based on the reference input and the position output, it is analysed the transparency, i.e., how much the human can through the haptic device what is going on with the vehicle. The initial parameters are: Slave: Master -Impedance control -Admittance The simulations were performed on Matlab/Simulink using the block diagrams in Figures 12 and 17 using a Linear Analysis Bode Plot to indicate the respective inputs and output in order to get the bode plot of the transfer function presented in equations (6) and (7). The simulations will show the effect of the following parameters and for the impedance, and and for the admittance. 1. Impedance control Figure 13 show the response of the system, for low frequencies the system response is attenuated, which means that the feedback does not provide proper feeling of what is happening in the slave side. For high frequencies, there is a unit gain, which means that the human now receives the force feedback without information loss. 6

10 Figure 15 - Complete teleoperation with impedance control in the master side Figure 13 - Transfer function of the impedance framework with the initial parameters By tuning and it is possible to move the bandwidth, the constant gain and the pick. Figures 15 and 16 show the system responses when and were varied, respectively. m_h=1.9 m_h=100 m_h=1e-4 Figure 14 Comparison between different values of the haptic mass It is possible to notice that has an indirect relation to the bandwidth, i.e., if is increases, the bandwidth is smaller, if is decreased, the bandwidth gets wider. 7

11 k_m=1e3 k_m=1 k_m=1e-3 As the frequency gets higher, above the bandwidth frequency the response attenuates, that means that the human is no longer able to receive an accurate feedback. By tuning the and it is possible to move the bandwidth and the constant gain. Figures 18 and 19 show the system responses when and were varied. Figure 16 - Comparison between different values of the virtual force constant k_m s The constant gain is directly related to, since is the scaling factor that gives the virtual force feedback, it increases if is increased and decreases if is decreased. 2. Admittance control Using the initial parameters, the bode plot of the admittance control is in Figure 18. m_a=1e-4 m_a=1.9 m_a=100 Figure 18 - Comparison between different values of the haptic mass m_a It is possible to notice that, just like for the impedance framework, has an indirect relation to the bandwidth, i.e., if is increases, the bandwidth is smaller, if is decreased, the bandwidth gets more wide. k_a=1e-3 k_a=1 Figure 17 - Transfer function of the admittance framework with the initial parameters For low frequencies the system response has a unit gain, the position feedback allow the human to feel what is happening with the slave. k_a=1e3 Figure 19 Comparison between different scaling factors 8

12 Figure 20 - Complete teleoperation with admittance control in the master side is analog to, is the scaling factor that gives the reference position to the slave. The constant gain is directly related to, it increases if is increased and decreases if is decreased. Admittance Impedance Figure 21 Response from both frameworks using the initial parameters Admittance control is the best choice for the master when the slave s controller is also an admittance control and the vehicle is in free flight. The reason for that is that the human will have a better feel for low frequencies, which is also the region where the slave can track the reference better. V. Conclusions After characterizing admittance and impedance control, they were implemented to both the slave and master in order to analyse their responses. It was possible to realise that for the slave alone, impedance is preferable when there is interaction with the environment, as opposed to admittance that works better in free flight. For the complete teleoperation of the flying vehicle, during free flight, the framework using admittance control for both master and slave is the ideal due to the fact that the human receives a good haptic cue of what is happening at the slave at a range of frequency that the 9

13 slave is presenting a good response to the input, i.e., the feeling that the human will have at the best will be when the vehicle is tracking the reference without attenuation. VI. Acknowledgment I would like to thank Abeje Mersha and Raffaella Carloni for all the patience and help. Also, I want to thank CNPq- Brasil for the opportunity of studying abroad by granting me a scholarship. References [1] X. Hou, R. Mahony, F. Schill. Representation of Vehicle Dynamics in Haptic Teleoperation of Aerial Robots. [2] A. Mersha, S. Stramigioli, R. Carloni. On Bilateral Teleoperation of Aerial Robots. [3] C. Ott, R. Mukherjee, Y. Nakamura. Unified Impedance and Admittance Control. [4] Airobots. [5]Matlab/Simulink [6]20-sim Camila Dalécio Soares Student Bachelor s Assignment Committee Dr. R. Carloni Chairperson A. Y. Mersha, MSc Daily supervisor Dr. H.K. Hemmes External member 10

14 Appendix A. Evaluation of impedance and admittance It is known that admittance control behaves better than impedance control when there is no interaction with the environment, i.e., in free flight. While impedance control behaves better than admittance control when there is contact with a (virtual) wall. In this part, I show the simulations that were made using the block diagrams presented in Section II. They all show the output position of the plant and the reference position that was given. Firstly, Impedance and Admittance will be compared when there is no external force and it will be chosen the one that behaves better in this case. Next, the same comparison will be made for the case that there is external force to determine which one works better when there is interaction. Only the parameters considered to be part of the controller were varied, one at a time; the caption under the figure specifies which parameter was changed and its new value. The parameters were varied without concern regarding the amount of force that the actuator can actually deliver. The initial parameters are: 1. No external force IMPEDANCE Figure 22 - Impedance - initial parameters Figure 22 depicts the response using the impedance controller with the initial parameters, we can see it has a fast response and the overshoot is close to zero. 11

15 Figure 23 - kp=1e10 Figure 24 - dp=14 When the kp was greatly increased or dp decreased, Figure 23 and Figure 24 respectively, the response was very oscillatory, the overshoot was around 100% and the steady state error was not zero, the steady state response oscillated around 5% of the reference. As kp was decreased, it became less oscillatory until there was no overshoot it was critically damped and went from rising rapidly, Figure 25, to very slowly, Figure 26. Figure 25 - kp=8e5 Figure 26 - kp=1e4 As dp was increased, Figure 27 and Figure 28, the overshoot was lowered and the steady state error was zero. 12

16 ADMITTANCE Figure 27 - dp=140 Figure 28 - dp=1.4e4 Figure 29 Admittance - Initial parameters Figure 29 shows the response using the admittance controller with the initial parameters; it shows a fast response with an overshoot of 20% of the reference. When Kd was increased, Figure 30, the overshoot was never higher than 30% and the steady state error zero, and it did not need much oscillations to arrive at the steady state response. When it was decreased, the overshoot would decrease as well until it became critically damped and increasing very slowly, Figure 31-Figure

17 Figure 30 - Kd=1e8 Figure 31 - Kd=10 Figure 32 - Kd=5 Figure 33 - Kd=0,1 When Dd was decreased, the maximum overshoot was 30% and the error went to zero within a small number of oscillations, Figure 34. When Dd was increased, Figure 35, it became critically damped with the error going to zero. Figure 34 - Dd=0,001 Figure 35 - Dd=1 14

18 2. With external force In this part, the stiffness of the virtual wall, one of the initial parameters. IMPEDANCE will be considered to be Figure 36 Impedance - Initial Parameters - kw=10 Figure 36 depicts the response using the impedance controller with the initial parameters interacting with a virtual wall; we can see that the response is not different from the one with no external force. When the stiffness of the virtual wall, kw, was increased, the response, at first showed an overshoot of 1% and the steady state error was zero, Figure 37. As kw increased, so did the error, there was no overshoot when compared to the final value, however there was a constant error, but it was still stable, Figure 38. The stiffness one order higher made the system become unstable. The changes in kw did not influenced the overshoot, the response was always fast and it quickly arrived to the steady state. Figure 37 - kw=100 Figure 38 - kw=1e5 When kp was lowered, Figure 39, it became critically damped and rising slowly, but the error went to zero. When kp was increased, it became more oscillatory with an higher overshoot, but small settling time. 15

19 Figure 39 - kp=1e4 Figure 40 - kp=1e8 As dp was lowered, the response became very oscillatory with an overshoot of almost 100% and the steady state error was very oscillatory and different than zero. When dp was increased, it became critically damped and the error went, very slowly, to zero. Figure 41 - dp=14 Figure 42 - dp=14e4 16

20 ADMITTANCE Figure 43 - Admittance - Initial parameters - kw=10 Figure 43 shows the response of the admittance control with the initial parameters interacting with a virtual wall; we can see that the shape of the response is not different from the one with no external force, however the final value is not the reference, there is a constant steady state error now of 15% of the reference. When kw was increased to 1, Figure 44, the overshoot was about 25% and there was a constant steady state error of 1% of the reference value. For kw one order higher, it became unstable. Figure 44 - kw=1 Figure 45 - kw=100 17

21 When Kd was lowered, it became unstable, Figure 46, but when Kd was increased, Figure 47, there was an overshoot and the steady state error was constant. At Kd=1e4, the steady state error was (virtually) zero. Figure 46 - Kd=10 Figure 47 - Kd=1e5 When Dd was increased, the response was critically damped shape-like, but with a constant steady state error above the reference value. When Dd was decreased, there was an high overshoot and a constant steady state error. Figure 48 - Dd=1 Figure 49 - Dd=