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Modeling and Control of Robotic Systems Course: from Fundamentals to Applications⁎
12th IFAC Symposium on Advances in Control Education 12th IFAC Symposium on Advances in Control Education July 7-9, Philadelphia, PA, USAin Control Education 12th IFAC2019. Symposium on Advances July 7-9, 2019. Philadelphia, PA, USA Available online at www.sciencedirect.com 12th IFAC2019. Symposium on Advances July 7-9, Philadelphia, PA, USAin Control Education July 7-9, 2019. Philadelphia, PA, USA
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IFAC PapersOnLine 52-9 (2019) 224–229
Modeling Modeling and and Control Control Modeling and of Robotic Course: ModelingSystems and Control Control of Robotic Systems Course: of Robotic Systems Course: from Fundamentals to Applications Robotic Systems Course: from of Fundamentals to Applications from Fundamentals to Applications from Fundamentals∗,∗∗to Applications Vladislav S. Gromov ∗,∗∗ Oleg I. Borisov ∗,∗∗ ∗,∗∗
Vladislav S. Gromov ∗,∗∗ Oleg I. Borisov ∗,∗∗ ∗,∗∗ ∗ ∗ Sergey S. Pyrkin ∗,∗∗ ∗ Anton ∗ Vladislav S. Shavetov Gromov ∗,∗∗ Oleg A. I. Borisov Sergey S. Shavetov Anton A. ∗ ∗ ∗Pyrkin ∗,∗∗ Vladislav S. Gromov Oleg I. Borisov ∗ Fatimat B. Karashaeva ∗Pyrkin ∗ Sergey S. Shavetov Anton A. Fatimat B. Karashaeva ∗ ∗Pyrkin ∗ Sergey S. Shavetov Anton A. ∗ Fatimat B. Karashaeva ∗ Fatimat B. Karashaeva ∗ ∗ Faculty of Control Systems and Robotics, ITMO University, Faculty of Control Systems and Robotics, ITMO University, ∗ ∗ Faculty of Control Systems and Robotics, ITMO University, Saint Petersburg, Russia [email protected]) Saint Petersburg, Russia (e-mail: (e-mail: [email protected]) ∗∗ ∗ Faculty of Control Systems and Robotics, ITMO University, for Technologies in Robotics and Mechatronics ∗∗ Center Saint Petersburg, Russia (e-mail: [email protected]) forPetersburg, Technologies in Robotics and Mechatronics Components, Components, ∗∗ Center Saint Russia (e-mail: [email protected]) ∗∗ Innopolis University, Russia in RoboticsInnopolis, and Mechatronics Components, Innopolis University, Innopolis, Russia ∗∗ Center for Technologies Center for Technologies in RoboticsInnopolis, and Mechatronics Innopolis University, Russia Components, Innopolis University, Innopolis, Russia Abstract: to build build the the Abstract: In In this this paper paper the the Modeling Modeling and and Control Control of of Robotic Robotic Systems Systems course course to Abstract: In this paper the is Modeling andThis Control of is Robotic Systems course to build the set of the professional skills presented. course included in the curriculum of set of the professional skills is presented. This course is included in the curriculum of the Abstract: In this paper theprovided and Control ofControl Robotic Systems to build master’s degree program and at the Faculty of Systems Robotics of ITMO set of the professional skills isModeling presented. This course is included in and thecourse curriculum of the the master’s degree program and provided at the Faculty of Control Systems and Robotics of ITMO set of the professional skills istracks presented. This course is included inpresented. the Robotics curriculum of the University. The selection of the to build the professional skills are An important master’s degree program and provided at the Faculty of Control Systems and of ITMO University. The selection of theprovided tracks toatbuild the professional skills are presented. An important master’s degree program and the Faculty of Control Systems andand Robotics of ITMO University. The selection of the trackstechnical to build the professional skills areskills presented. An important part of the tracks is a using various equipment to build the experience. part of the tracks is a using various technical equipment to build the skills and experience. University. selection of the trackstechnical to build the professional skillsthe areskills presented. An important part of the The tracks is a using various equipment to build and experience. © 2019, IFAC (International Federation Automatic Control) Hosting bythe Elsevier rights reserved. part of the tracks is a using various of technical equipment to build skillsLtd. andAllexperience. Keywords: Keywords: Control Control education, education, Control Control application, application, Robotics, Robotics, Ship Ship control, control, Industrial Industrial robots robots Keywords: Control education, Control application, Robotics, Ship control, Industrial robots Keywords: Control education, Control application, Robotics, Ship control, Industrial robots 1. INTRODUCTION INTRODUCTION achieving achieving learning learning results. results. Learning Learning results results have have to to include include 1. 1. INTRODUCTION knowledge, abilities and skills. All this results are presented achieving learning results. Learning results have to include knowledge, abilities and skills. All this results are presented 1. task INTRODUCTION achieving learning results. Learning results have to include knowledge, abilities and skills. All this results are presented for each educational track of the course in the paper. Each It is an important to develop both theoretical and each educational track of the course in theare paper. Each It is an important task to develop both theoretical and for knowledge, abilities and skills. All this results presented for each educational track of the course in the paper. Each track supplied with the computer model for completing It is an important task to applied develop science both theoretical and track supplied with the computer model for completing practical experience in any including conpractical experience in any applied science including confor each educational track of the course in the paper. Each It istheory. an important task toofapplied develop both material theoretical and home tasks and with experimental setup for the work track supplied with the computer model for completing practical experience in any science including control Understanding fundamental will be home tasks andwith withthe experimental setup for the work at at trol theory. Understanding of fundamental material will be track supplied computer model for completing practical experience in any sciencematerial including conthe lab. Each scenario the experience robotic tasks and with provides experimental setup for with the work at trol theory. Understanding fundamental will be much easier for students students inofapplied case of having having opportunity to home the lab. Each scenario provides the experience with robotic much easier for in case of opportunity to home tasks and with provides experimental setup for with the work at trol theory. Understanding fundamental material will be lab. Each scenario the experience robotic systems. much easier for studentsstudies inofcase of having opportunity to the carry out experimental testing various theoretical systems. carry out experimental studies testing various theoretical the lab. Each scenario provides the experience with robotic much easier for students in case of having opportunity to systems. carry out experimental studies testing various theoretical results. Various professional skills should be built built during The paper is organized as follows. The brief overview of the results. Various professional skills should be during carry out experimental studies testing various theoretical The paper is organized as follows. The brief overview of the results. Various professional skills should be fundamental built during systems. the master’s degree program so the different The paper organized follows. The overview of the and the tracks is in Section 2. the master’s degree program skills so theshould different results. Various professional be fundamental built during course course andis tracks as is presented presented in brief Section 2. Sections Sections the master’s degree program so thehere. different fundamental and applied exercises are welcome The paper isthe organized as follows. The brief overview of the 3-5 is devoted to description of the applied exercises on and applied exercises are welcome here. course and the tracks is presented in Section 2. Sections the master’s degree program so thehere. different fundamental 3-5 is devoted description of theinapplied exercises on and applied exercises are welcome course and the to tracks is presented Section 2.boat, Sections control of robotic systems includes the robotic the 3-5 is devoted to description of the applied exercises on This is quite common for teachers and professors across and applied of robotic systems includes the robotic boat, the This is quiteexercises commonare forwelcome teachershere. and professors across control is devoted to description of the applied exercises on manipulator and the quadrotor. The conclusion of control of robotic systems includes the robotic boat, the This is quite common for teachers and professors across 3-5 the world to include the usage of various applied exercises manipulator and the quadrotor. The conclusion of the the the world to include thefor usage of various applied exercises control of robotic systems includes the robotic boat, This is quite common teachers and professors across manipulator and the quadrotor. The conclusion of the paper is presented in Section 6. the world tostudents, include the usage control of various applied to motivate motivate promote science andexercises improve paper is presented in Section 6. to promote control science and improve and the quadrotor. themotivate world experience. tostudents, include the usage ofLEGO various applied paper is presented in Section 6. The conclusion of the to students, promote control science andexercises improve practical Using the Mindstorms is the the manipulator practical experience. Using the LEGO Mindstorms is paper is presented in Section 6. 2. COURSES OVERVIEW to motivate students, Using promote and improve practical experience. thecontrol LEGOscience Mindstorms is the common solution provide experience of 2. COURSES OVERVIEW AND AND FUNDAMENTAL FUNDAMENTAL common solution to to Using provide experience of the the designdesignTRACKS practical experience. the LEGO Mindstorms is the 2. COURSES OVERVIEW AND FUNDAMENTAL common solution experience of the designing control control systemstofor forprovide the robots robots (see Kapitonov Kapitonov et al. al. TRACKS 2. COURSES OVERVIEW AND ing systems the (see et common solution tofor provide experience of the designTRACKS FUNDAMENTAL ing control systems the robots (seeinKapitonov et al. (2019)). The remote control study kit Shavetov TRACKS (2019)). The remote control study kit in Shavetov et al. ing control systems the robots (see al. Building (2019)). remote for control study kit inKapitonov Shavetov ettestal. (2016) isThe considered as configurable application for et Building skills skills of of using using real real technical technical plants plants is is an an imim(2016) is considered as configurable application for testBuilding skills of using real technical plantsof is Robotic an important part of the Modeling and Control (2019)). The remote as control study kit in remote Shavetov ettestal. portant part of the Modeling (2016) is considered configurable application for ing various algorithms for a trajectory or control. and Control of Robotic Building skills of using real technical plants is an iming various algorithms for a trajectory or remote control. Systems course Faculty of and part of at thethe ControlSystems of Robotic (2016) is considered asfor configurable application test- portant ing various algorithms a the trajectory or remote control. In Belyavskyi et al. al. (2017) (2017) description of the thefor 2DOF Systems course at theModeling Faculty and of Control Control Systems and portant part of the Modeling and Control of Robotic In Belyavskyi et the description of 2DOF Robotics of University. educational course at the Faculty So of all Control Systemstracks and ingBelyavskyi various algorithms for a the trajectory or remote In et laboratory al. (2017) description of thecontrol. 2DOF Systems quadrotor-based bench for engineering engineering educaRobotics of ITMO ITMO University. So educational tracks Systems course at different the Faculty of all Control Systems and quadrotor-based laboratory bench for educaRobotics of ITMO University. So all educational tracks considered in the programs are related to the In Belyavskyi et al. (2017) the description of the 2DOF quadrotor-based laboratory bench for engineering education. considered inITMO the different programs are related to the Robotics of University. So all educational tracks tion. considered in the different programs are related to the robotics plants. There are three master’s degree programs quadrotor-based laboratory bench for engineering educa- robotics plants. There are three master’s degree programs tion. considered in the different programs are related to the In this paper the Modeling and Control of Robotic Syswhich have the similar kernel is about control systems, robotics plants. There are three master’s degree programs tion. In this paper the Modeling and Control of Robotic Sys- which have the There similararekernel ismaster’s about control systems, robotics plants. three degree programs In this paper the Modeling and Control of Robotic Systems course course to to build build the the set set of of the the professional professional skills skills which but different specializations. “Robotics” is have the similar kernel is about program control systems, tems different specializations. “Robotics” program is about about In presented. thiscourse paper to thebuild Modeling andof Control of Robotic Sys- but which have the similar kernel is about“Control control ofsystems, tems the set the professional skills is Our set of skills is based on professional but different specializations. “Robotics” program isCyberabout industrial robots modeling and control, is presented. Our set of skills is based on professional industrial robots modeling and control, “Control of Cybertems course to build the set of the professional skills but different specializations. “Robotics” program isCyberabout is presented. Our set of skills is based on professional standards and foresight of the Faculty of Control Systems industrial robots modeling and control, “Control of Physical Systems” program is about control systems synstandards andOur foresight of skills the Faculty of Control Systems Physical Systems” programand is about control systems synis presented. set University. of isThese based on professional industrial robots modeling control, “Control of Cyberstandards andofforesight of the Faculty ofprofessional Control Systems and Robotics ITMO skills thesis and “Digital control systems” program is about Physical Systems” program is about control systems synand Robotics of ITMO University. These professional skills thesis and “Digital control systems” program is about standards andofforesight of the Faculty ofprofessional Control Systems Systems” program isto about control systems synand Robotics ITMOskills University. These skills Physical rely on fundamental and are achieved by student’s control systems integration various applications. The thesis and “Digital control systems” program is about rely on fundamental skills and areThese achieved by student’s control systems integration to various program applications. The and Robotics of ITMO University. professional skills thesis and “Digital control systems” is second about rely onand fundamental skills and are achieved by student’s home class work. The educational trajectories are control systems integration to various applications. The professional course takes one academic year since home and class work. The educational trajectories are professional course takes one academic year since second rely on fundamental skills and are achieved by student’s control systems integration to various applications. The home andforclass work. The educational trajectories are prepared each educational track and should be used for semester and is devoted to mastering fundamental and professional course takes one academic year since second prepared each work. educational track and should be used are for semester andcourse is devoted to mastering fundamental and home andfor The educational trajectories takes one academic year since second prepared forclass each educational track and should be used for professional professional skills. semester and is devoted to mastering fundamental and professional skills. prepared each educational track and should beofused for semester andskills. is devoted to mastering fundamental and workfor was written with the support of the Ministry Science The professional The work was written with the support of the Ministry of Science We general professional and Higher Education thethe Russian Federation, project unique The work was writtenof with support of the Ministry of Science We consider considerskills. general fundamental fundamental and and professional professional skills skills and Higher Education of the Russian Federation, project unique The for the educational programs that have similar conWe consider general fundamental and professional skills work was written with the support of the Ministry of Science identifier RFMEFI57818X0271 “Adaptive Sensorless Control for for the educational programs that have similar conand Higher Education of the Russian Federation, project unique identifier RFMEFI57818X0271 “Adaptive Sensorless Control for We consider general fundamental and professional skills nected tracks (the one professional skill model). In an our and Higher Education of theinRussian Federation, project unique Synchronous Electric Drives Intelligent Robotics andControl Transport for the educational programs that have similar conidentifier RFMEFI57818X0271 “Adaptive Sensorless for nected tracks (the one professional skill model). In an our Synchronous Electric Drives in Intelligent Robotics and Transport for thetracks educational that similar conidentifier RFMEFI57818X0271 “Adaptive Robotics Sensorless for nected (the approach oneprograms professional skillhave model). In an our Systems”. practical-oriented for mastering skills the series Synchronous Electric Drives in Intelligent andControl Transport Systems”. practical-oriented approach for mastering skills the series nected tracks (the one professional skill model). In an our Synchronous Electric Drives in Intelligent Robotics and Transport Systems”. practical-oriented approach for mastering skills the series Systems”. practical-oriented approach for mastering skills the series 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2019 IFAC 237 Copyright 2019 IFAC 237 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2019 IFAC 237 10.1016/j.ifacol.2019.08.204 Copyright © 2019 IFAC 237
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Lb
Px
O Pe
225
αe Mz Ps
Py
Pb
Fig. 1. The robotic boat actuators configuration
Fig. 2. 3D model of the robotic boat with actuators forces projections (red lines)
of the tracks is included in the curriculum: Modern Control Theory, Technical Systems Modeling, Control Systems Programming, Cyber-Physical Systems and Technologies, Modeling and Control of Robotic Systems, Dynamic of Robotic Systems, Modeling and Motion Control of Surface Vessels, Sensorless Control and some others. They are includes the content about identification and control fundamentals as well as modeling and control of the robotic systems. Practical tracks may includes home tasks to work with 3D models of the robots and the experimental exercises with robotic equipment at the laboratory. Three types of robots are used in these exercises: • Robotic boat; • Robotic manipulator KUKA youBot; • Unmanned quadrotor.
use existing software packages (and, if necessary, to develop a new software) for the data processing and control in mechatronic and robotic systems” with the learning results: Knowledge — modern approaches for designing robotic systems mathematical models and its program realization; Ability — motion trajectories planning of robotic systems at given reference points; Skill — robotic systems programming for given motions performing. This is a fundamental track for the Control Systems and Robotics students. It is a basis for designing control algorithms for a robotic boat, manipulator or unmanned quadrotor. 3. ROBOTIC BOAT CONTROL TRACK
Each professional skill can be mastered in several courses but one prevails for each course. We provide control and robotic fundamental skills before studying courses mastering professional skills. Fundamental skills are used in all tracks of the master’s degree programs.
For the robotic boat control track of the course the 3D model of the plant is considered. During the track students are able to prepare the transformation algorithm from local to global coordinate systems for the model, the identification and control algorithms for dynamic positioning.
The first fundamental track is “Technical Systems Modeling” with fundamental skill “Able to apply modern theoretical and experimental methods for developing mathematical models of cyber-physical systems, including plants parametric identification” which includes the learning results: Knowledge — principles of mathematical describing systems, principles of synthesis algorithms and numerical solutions for models; Ability — digital control and information modules synthesis and analysis; Skill — calculation and design of individual units, devices and systems using mathematical modeling.
The model is based of the real robotic boat and contains parameters which have been identified from the real plant. The actuators configuration of the robotic boat are presented in the Fig. 1. Pb , Ps , Pe are the propelling forces produced by the bow, stern and end actuators respectively which can be controlled by control system. The bow and stern actuators are the tunnel thrusters. Px and Py are the generalized forces along x and y axis, Mz is the generalized rotational moment.
The track “Modern Control Theory” with the skill “Able to develop and research control systems functioning algorithms” can be considered both as fundamental and as professional skill with the following learning results: Knowledge — modern approaches of planning and motion control for robotic systems; Ability — Control algorithms realizations for various software environments; Skill — controllers synthesis for solving various tasks of the robotic plants with parametric and signal uncertainties, constraints, functioning specifics. Parts of the track are used for designing controllers for a robotic boat, manipulator or unmanned quadrotor. Connected with the previous one track “Control Systems Programming” contains professional skill “Able to 238
For education purposes the mathematical model of the each plant actuators considered like b W (s) = 2 (1) s − as where a & b are unknown parameters of the plant. For the identification task the regression model has to be obtained by students. The model (1) can be rewritten like y¨ = ay˙ + bu
(2)
where u is the control signal, y is the output coordinate. Both signals are measurable but not their derivatives. By λ2 introducing the linear filter of the form (s+λ 2 )2 the new measurable signals can be obtained ξy (s) =
λ2 λ2 y(s), ξ (s) = u(s). u (s + λ2 )2 (s + λ2 )2
(3)
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for simplification of the computer vision. The current coordinates of the robotic boat are provided by the computer vision system from the digital camera instead of satellite navigation systems due to the scale of the educational setup. The camera is attached to the tripod above the basin. The captured image is transmitted as a RGB signal to the computer, where it is processed in order to get the coordinates of the boat. Students are able to complete the experimental trials with a designed control system during the classes. The experience with the real robot helps to understand identification and control principles and form debugging skills for a future work.
Fig. 3. The robotic boat setup for education and research By substituting (3) into (2) and performing the Laplace transform obtain (4) ξ¨y (t) = −aξ˙y (t) + bξu (t) + (t) where is the exponentially decreasing term which can be neglected. The regression model of the plant can be written from (4) in the next form (5) Y = ωT θ T T ˙ ¨ where (Y ) = ξy , ω = [ξy ξu ], θ = [a b]. Students are able to apply the identification algorithm using the regression model (5) and the control approaches to the plant model (1). For ease of developing algorithms and debugging by students at home the 3D model has been prepared in SIMULINK. The screenshot of the modeling process presented on the Fig. 2. Red lines represents the forces produced by actuators, yellow triangle is the target position which has to be achieved. The identification or control system should be implemented by students in SIMULINK like the scheme or the code. Due to the fact that the 3D model uses parameters which has been identified from the real boat it is easy to transfer the algorithm from the model to the experimental setup (see Borisov et al. (2016); Wang et al. (2015) for examples of research projects). The robotic boat setup was designed for experimental study at the laboratory during the robotic boat control track of the course. The main parts of the setup are the robotic boat itself and experimental basin. The robotic boat is a scale model of a real trawler ship represented at a scale of 1 : 32. The actuators configuration is the same like on the Figs. 1 so the developed control system for the mathematical model can be transferred to the real robotic boat. The experimental basin represents the workspace of the boat. Its volume is about 150 L of water. The internal surface is covered by sealant and painted with dark color 239
The professional skill “Able to develop and research control systems functioning algorithms” of the track “Modeling and Motion Control of Surface Vessels” includes the learning results: Knowledge — dynamic models of the surface vessels motion; Ability — modern approaches for vessel’s position determination in different coordinate systems; Skill — synthesis of vessels’ dynamic positioning systems and stabilizing algorithms for given velocities. In this track the considered plant is the Robotic boat. 4. ROBOTIC MANIPULATOR CONTROL TRACKS For the robotic manipulator control tracks the model of the 5-DOF robot is considered (see Fig. 4). The parameters of the model are the same like on the robot KUKA youBot. Students are able to design algorithms for solving forward and inverse kinematics tasks as well as identification and control systems. According to Spong et al. (2006) the dynamical model of the robotic manipulator can be calculated from the EulerLagrange equations dL d dL − = τi i = 1, . . . , n (6) dt dq˙i dqi z5
x5
x3 , z4 z3
x4 x2
z2
x1
z1
z0 x0
Fig. 4. The kinematic scheme of the robotic manipulator KUKA youBot
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Fig. 5. 3D model of the robotic manipulator KUKA youBot and desired trajectory of the end-effector
Fig. 6. The robotic manipulator KUKA youBot for education and research in our laboratory
where L = K − P is the Lagrangian of the system, K is the kinetic energy of the system, P is the potential energy of the system, qi are the generalized coordinates, τi are the generalized load torques, applied to the actuators, n is the number of link (order of the system). By calculating kinetic and potential energy and by substituting it to the (6) it is possible to represent to dynamic model of the joints without external load in the matrix form M (q)¨ q + C(q, q) ˙ + G(q) = τ (7) where M (q) is inertial matrix sized (n × n), C(q, q) ˙ ∈ Rn n is Coriolis forces vector, G(q) ∈ R is gravitational forces vector, τ ∈ Rn is load torques vector, which compensated by actuators.
During the course students are able to estimate the parameters of the actuators dynamic model for the 3D model which was prepared in SIMULINK (see Fig. 5). The inertial and gravitational parameters of the model are acquired earlier during the procedure of the identification of the real robot parameters. After estimating the parameters the designing control system for compensating gravitational forces is required for students. The next step is solving the forward kinematics task and designing the control system to follow the desired trajectory by the end-effector of the robot. All this tasks can be completed by students at home using the designed 3D model with control approaches provided during the fundamental track. Also the algorithms for disturbance and time delay compensation can be modeled too (Pyrkin and Bobtsov (2011); Bobtsov et al. (2011)).
The model (7) can be simplified by students for the static case when q˙ = 0 & q¨ = 0 G(q) = τ (8) where G(q) is gravitational matrix calculated by T ∂P ∂P ∂P ∂P ∂P ∂P = . (9) G(q) = q1 q2 q3 q4 q5 ∂q The potential energy of the robot KUKA youBot calculated from the potential energy of the each link P=
5
= mi ghi (q) = m1 gh1 (q) + m2 gh2 (q)
(10)
i=1
+ m3 gh3 (q) + m4 gh4 (q) + m5 gh5 (q)
where mi is the mass of the i’s link, hi (q) is the zcoordinate from the center mass vector of the i’s link in the base coordinate system. To identify the gravitational component, regression models were formed for each joint τi := Gi (q) = φTi (q, τ )θi , where φi (q) is regressor made up of measured signals depending on the configuration q, θi is vector of unknown parameters (centres of mass of links and values of masses), estimated using the identification algorithm. 240
After completing the home tasks with 3D model students are able to perform an experimental exercises at the class with the real robot KUKA youBot (see Gromov et al. (2016) for detailed description of the experimental setup). Due to the fact that 3D model contains parameters which are close to the real ones it is not difficult to transfer the identification and control algorithms designed by students to the real robot. Classroom activities helps to build skills and experience with multi-link robots and mechanical systems, solving kinematic and dynamic tasks, developing and debugging control systems. The track “Modeling and Control of Robotic Systems” is characterized by professional skill “Able to develop experimental setups of control, information and actuator modules for mechatronic and robotic systems”. This skill can be represented in a form of learning results: Knowledge — robotic systems kinematic analysis methods; Ability — developing various scenarios of industrial operations using robotic systems; Skill — using Denavit-Hartenberg approach for robotic systems kinematic analysis. In the course robotic manipulator KUKA youBot is used as an object for kinematic analysis. Connected with the previous one the track “Dynamic of Robotic Systems” is characterized by professional skill
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“Able to develop automated process control systems” that includes the learning results: Knowledge — methods for constructing robotic systems dynamic models; Ability — developing conceptions of automation control for robotic systems; Skill — constructing robotic systems dynamic models using the Euler-Lagrange and Newton-Euler methods. The robotic manipulator KUKA youBot also is used in the course for dynamic analysis. 5. UNMANNED QUADROTOR CONTROL TRACK Another application example, which can be used to conduct illustrative basic as well as advanced studies on modeling and control of robotic systems, is the station-keeping of a quadrotor. Consider its dynamical model assuming the drag coefficient equal to zero at low speed Altuˇ g et al. (2002): 4 m¨ x= ui (cφ sθ cψ +sφ sψ ), Jθ θ¨= (−u1 −u2 +u3 +u4 ), m¨ y=
i=1 4 i=1
m¨ z=
4 i=1
Consider several structures of robust controllers under assumption that plant parameters are unknown, derivatives of the output variables are not available for measurements and the input signals are bounded. Once the control law generating Ui is defined, a last step is the inverse transformation of (11) in order to allocate the controlbetween the actuators u1 1 −1 −1 1 U1 u 2 1 −1 1 −1 U2 u = 0, 25 1 1 1 1 U . 3 3 1 1 −1 −1 U4 u4
5.1 Robust Output Controller
The robust output controller based on the consecutive compensator approach applied to the quadrotor model (13)–(15) is of the form υi = κi (αi,2 ξ˙i + αi,1 ξi ), (17) ξ˙i = σi (−ξi + ei ),
ui (sφ sθ cψ −cφ sψ ),
Jψ ψ¨ = (−u1 +u2 +u3 −u4 ),
in which κi > 0, σi > 0, αi,j > 0, j = {1, 2} are the design parameters. Remark 2. Note that the gravitational force mg can be added to for U1 in order to compensate its effect.
ui (cθ cψ )−mg,
Jφ φ¨ = C(u1 −u2 +u3 −u4 ),
5.2 Static Disturbance Compensation
where x, y, z, θ, ψ, φ are the linear and angular coordinates, ui , i = {1, 2, 3, 4} are the control signals, m, g, , Jθ , Jψ , Jφ , C are the physical parameters, cφ ≡ cos φ, sφ ≡ sin φ.
Decompose the quadrotor model choosing the set of quasicontrol signals 1 1 1 1 u1 U1 U2 −1 −1 1 1 u2 (11) U = −1 1 1 −1 u , 3 3 1 −1 1 −1 u4 U4 where the quasi-control signals Ui , i = {1, 2, 3, 4} satisfy the saturation condition Ui,max , if υi ≥ Ui,max , υi , if Ui,min < υi < Ui,max , (12) Ui = sat(υi ) = Ui,min , if υi ≤ Ui,min , Ui,min and Ui,max are the input saturation limits satisfying, υi are the nominal control signals generated by the independed SISO linear regulators. Following transformations similar to Tomashevich et al. (2017) use the linearized quadrotor model ¨ = −U5 , mx ¯ Jθ θ¨ = U2 , (13) ¨ = −U6 , Jψ ψ¨ = U3 , (14) my¯
Jφ φ¨ = CU4 , (15) m¨ z = U1 − mg, where the desired values of the roll and pitch angles are calculated as U5 U6 θ∗ = , ψ ∗ = − , U5 = U1 θ, U6 = −U1 ψ. (16) U1 U1 Remark 1. Note that in the final engineering implementation the calculation algorithm should be modified in order to avoid singularities in (16). Zero values of θ∗ and ψ ∗ should be assigned when U1 is approaching to zero. Values of θ∗ and ψ ∗ should be bounded to avoid critical reference peaks within the transient processes. 241
Static disturbances (such as the gravitational force and constant wind acting on the quadrotor) leading to steadystate error of the output variables can be cancelled by the integral term. Rewrite the control law (17) as t ˙ ξi (τ )dτ , υi = κi βi,3 ξi + βi,2 ξi + βi,1 (18) 0 ξ˙i = σi (−ξi + ei ), in which βi,j > 0, j = {1, 2, 3} are the design parameters. 5.3 Anti-Windup Compensation However the integral term needed to cancel the steadystate error might result in the integral windup in case of bounded input signals. To get rid of this effect anti-windup technique can be implemented as t υi =κi βi,3 ξ˙i +βi,2 ξi +βi,1 (ξi (τ )−νi κi (υi ))dτ ,
(19) 0 ξ˙i =σi (−ξi + ei ), where νi > 0 are the design parameter, κi (υi ) = υi − sat(υi ) are the anti-windup signal. 5.4 Adaptive Tunning Controller Parameters The controller (19) can further be updated by implementing the direct adaptation laws. The design parameters κ, σ and ν being replaced with functions κ(t), σ(t) and ν(t), respectively, can be adjusted as κ˙ i (t) = ζκ ξi2 − ςκ (κi (t) − κi (0)), σ˙ i (t) = ζσ (ri − ei − ξi )2 − ςσ (σi (t) − σi (0)), ν˙ i (t) = ζν κi2 (υi ) − ςν (νi (t) − νi (0)), in which ri are the given references, κi (0), σi (0), νi (0) are the initial conditions, ζj , ςj , j = {κ, σ, ν} are the design parameters.
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u1
u4
u2 u3 (a) Quadrotor scheme
(b) Modeling the quadrotor
Fig. 7. Quadrotor 5.5 Robust Output Controller in State-Space The controller can be redesigned in state-space as υi = κi (cTq,i ξi + ei ) − γi ηi , ξ˙i = Aq,i ξi + bq,i ei ,
(20)
η˙ i = −κi (cTq,i ξi
(22)
+ ei ) + νi κi (υi ),
(21)
where the matrices Aq,i and vectors bq,i , cq,i are of the form −qi,2 σi 1 qi,2 σi qi,1 , b , c , = Aq,i = = q,i q,i qi,2 −qi,1 σi2 σi2 0 qi,1
> 0, qi,j > 0, j = {1, 2} are the and γi > 0, σi > 0, qi,j design parameters.
The unmanned quadrotor is used like a practical examples of the course “Sensorless Control”, which is about modern trend of sensorless control allows to reduce some sensors in robotic systems, e.g. in electric drives and replace it with some mathematical algorithm. The professional skill is similar to the ones from the previous track but includes the learning results: Knowledge — concepts of nonlinear electromechanical systems, coordinate transformations for AC motors, methods for synthesizing nonlinear observers based on identification approaches, vector motor control method; Ability — using algorithms for estimating parameters and state variables of AC motors in the angular position and / or rotor speed sensors absence; Skill — synthesis sensorless control algorithms for electromechanical systems. This approach allows to increase systems reliability. 6. CONCLUSION The educational program as a set of tracks could be filled in different ways to achieve proper professional skills. We use practical-oriented approach with the connected tracks that allows us to join acquired fundamental knowledge with applications to build skills. REFERENCES Altuˇg, E., Ostrowski, J., and Mahony, R. (2002). Control of a quadrotor helicopter using visual feed242
back. Proceedings - IEEE International Conference on Robotics and Automation, 1, 72–77. doi: 10.1109/ROBOT.2002.1013341. Belyavskyi, A., Tomashevich, S., and Andrievsky, B. (2017). Application of 2dof quadrotor-based laboratory testbed for engineering education. 939–944. doi: 10.1109/MED.2017.7984240. Bobtsov, A., Kremlev, A., and Pyrkin, A. (2011). Compensation of harmonic disturbances in nonlinear plants with parametric and functional uncertainty. Automation and Remote Control, 72(1), 111–118. doi: 10.1134/S0005117911010097. Borisov, O., Gromov, V., Pyrkin, A., Bobtsov, A., Petranevsky, I., and Klyunin, A. (2016). Output robust control with anti-windup compensation for robotic boat. 13–18. doi:10.1109/MMAR.2016.7575080. Gromov, V., Borisov, O., Vedyakov, A., Pyrkin, A., Shavetov, S., Bobtsov, A., Salikhov, V., and Aranovskiy, S. (2016). Adaptive multisinusoidal signal tracking system with input delay. IFAC-PapersOnLine, 49(13), 105–110. doi:10.1016/j.ifacol.2016.07.935. Kapitonov, A., Antonov, E., Artemov, K., Dobriborsci, D., Zamotaev, E., Karavaev, A., Al-Naim, R., and Souzdalev, O. (2019). Lego mindstorms ev3 for teaching the basics of trajectory control problems. volume 2018October. doi:10.1109/FIE.2018.8659322. Pyrkin, A. and Bobtsov, A. (2011). Output control for nonlinear system with time-varying delay and stability analysis. 7392–7397. doi:10.1109/CDC.2011.6160224. Shavetov, S., Vedyakov, A., Pyrkin, A., Bobtsov, A., and Borisov, O. (2016). Advanced educational tool for remote control study. IFAC-PapersOnLine, 49(6), 303– 308. doi:10.1016/j.ifacol.2016.07.194. Spong, M.W., Hutchinson, S., Vidyasagar, M., et al. (2006). Robot modeling and control. Tomashevich, S., Borisov, O., Gromov, V., Pyrkin, A., and Bobtsov, A. (2017). Experimental study on robust output control for quadcopters. 1029–1034. doi: 10.1109/MED.2017.7984253. Wang, J., Borisov, O., Gromov, V., Pyrkin, A., and Bobtsov, A. (2015). Adaptive controller implementation for surface robotic vessel. volume 2015-September, 3230–3235. doi:10.1109/ChiCC.2015.7260138.
ScienceDirect
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Modeling Modeling and and Control Control Modeling and of Robotic Course: ModelingSystems and Control Control of Robotic Systems Course: of Robotic Systems Course: from Fundamentals to Applications Robotic Systems Course: from of Fundamentals to Applications from Fundamentals to Applications from Fundamentals∗,∗∗to Applications Vladislav S. Gromov ∗,∗∗ Oleg I. Borisov ∗,∗∗ ∗,∗∗
Vladislav S. Gromov ∗,∗∗ Oleg I. Borisov ∗,∗∗ ∗,∗∗ ∗ ∗ Sergey S. Pyrkin ∗,∗∗ ∗ Anton ∗ Vladislav S. Shavetov Gromov ∗,∗∗ Oleg A. I. Borisov Sergey S. Shavetov Anton A. ∗ ∗ ∗Pyrkin ∗,∗∗ Vladislav S. Gromov Oleg I. Borisov ∗ Fatimat B. Karashaeva ∗Pyrkin ∗ Sergey S. Shavetov Anton A. Fatimat B. Karashaeva ∗ ∗Pyrkin ∗ Sergey S. Shavetov Anton A. ∗ Fatimat B. Karashaeva ∗ Fatimat B. Karashaeva ∗ ∗ Faculty of Control Systems and Robotics, ITMO University, Faculty of Control Systems and Robotics, ITMO University, ∗ ∗ Faculty of Control Systems and Robotics, ITMO University, Saint Petersburg, Russia [email protected]) Saint Petersburg, Russia (e-mail: (e-mail: [email protected]) ∗∗ ∗ Faculty of Control Systems and Robotics, ITMO University, for Technologies in Robotics and Mechatronics ∗∗ Center Saint Petersburg, Russia (e-mail: [email protected]) forPetersburg, Technologies in Robotics and Mechatronics Components, Components, ∗∗ Center Saint Russia (e-mail: [email protected]) ∗∗ Innopolis University, Russia in RoboticsInnopolis, and Mechatronics Components, Innopolis University, Innopolis, Russia ∗∗ Center for Technologies Center for Technologies in RoboticsInnopolis, and Mechatronics Innopolis University, Russia Components, Innopolis University, Innopolis, Russia Abstract: to build build the the Abstract: In In this this paper paper the the Modeling Modeling and and Control Control of of Robotic Robotic Systems Systems course course to Abstract: In this paper the is Modeling andThis Control of is Robotic Systems course to build the set of the professional skills presented. course included in the curriculum of set of the professional skills is presented. This course is included in the curriculum of the Abstract: In this paper theprovided and Control ofControl Robotic Systems to build master’s degree program and at the Faculty of Systems Robotics of ITMO set of the professional skills isModeling presented. This course is included in and thecourse curriculum of the the master’s degree program and provided at the Faculty of Control Systems and Robotics of ITMO set of the professional skills istracks presented. This course is included inpresented. the Robotics curriculum of the University. The selection of the to build the professional skills are An important master’s degree program and provided at the Faculty of Control Systems and of ITMO University. The selection of theprovided tracks toatbuild the professional skills are presented. An important master’s degree program and the Faculty of Control Systems andand Robotics of ITMO University. The selection of the trackstechnical to build the professional skills areskills presented. An important part of the tracks is a using various equipment to build the experience. part of the tracks is a using various technical equipment to build the skills and experience. University. selection of the trackstechnical to build the professional skillsthe areskills presented. An important part of the The tracks is a using various equipment to build and experience. © 2019, IFAC (International Federation Automatic Control) Hosting bythe Elsevier rights reserved. part of the tracks is a using various of technical equipment to build skillsLtd. andAllexperience. Keywords: Keywords: Control Control education, education, Control Control application, application, Robotics, Robotics, Ship Ship control, control, Industrial Industrial robots robots Keywords: Control education, Control application, Robotics, Ship control, Industrial robots Keywords: Control education, Control application, Robotics, Ship control, Industrial robots 1. INTRODUCTION INTRODUCTION achieving achieving learning learning results. results. Learning Learning results results have have to to include include 1. 1. INTRODUCTION knowledge, abilities and skills. All this results are presented achieving learning results. Learning results have to include knowledge, abilities and skills. All this results are presented 1. task INTRODUCTION achieving learning results. Learning results have to include knowledge, abilities and skills. All this results are presented for each educational track of the course in the paper. Each It is an important to develop both theoretical and each educational track of the course in theare paper. Each It is an important task to develop both theoretical and for knowledge, abilities and skills. All this results presented for each educational track of the course in the paper. Each track supplied with the computer model for completing It is an important task to applied develop science both theoretical and track supplied with the computer model for completing practical experience in any including conpractical experience in any applied science including confor each educational track of the course in the paper. Each It istheory. an important task toofapplied develop both material theoretical and home tasks and with experimental setup for the work track supplied with the computer model for completing practical experience in any science including control Understanding fundamental will be home tasks andwith withthe experimental setup for the work at at trol theory. Understanding of fundamental material will be track supplied computer model for completing practical experience in any sciencematerial including conthe lab. Each scenario the experience robotic tasks and with provides experimental setup for with the work at trol theory. Understanding fundamental will be much easier for students students inofapplied case of having having opportunity to home the lab. Each scenario provides the experience with robotic much easier for in case of opportunity to home tasks and with provides experimental setup for with the work at trol theory. Understanding fundamental material will be lab. Each scenario the experience robotic systems. much easier for studentsstudies inofcase of having opportunity to the carry out experimental testing various theoretical systems. carry out experimental studies testing various theoretical the lab. Each scenario provides the experience with robotic much easier for students in case of having opportunity to systems. carry out experimental studies testing various theoretical results. Various professional skills should be built built during The paper is organized as follows. The brief overview of the results. Various professional skills should be during carry out experimental studies testing various theoretical The paper is organized as follows. The brief overview of the results. Various professional skills should be fundamental built during systems. the master’s degree program so the different The paper organized follows. The overview of the and the tracks is in Section 2. the master’s degree program skills so theshould different results. Various professional be fundamental built during course course andis tracks as is presented presented in brief Section 2. Sections Sections the master’s degree program so thehere. different fundamental and applied exercises are welcome The paper isthe organized as follows. The brief overview of the 3-5 is devoted to description of the applied exercises on and applied exercises are welcome here. course and the tracks is presented in Section 2. Sections the master’s degree program so thehere. different fundamental 3-5 is devoted description of theinapplied exercises on and applied exercises are welcome course and the to tracks is presented Section 2.boat, Sections control of robotic systems includes the robotic the 3-5 is devoted to description of the applied exercises on This is quite common for teachers and professors across and applied of robotic systems includes the robotic boat, the This is quiteexercises commonare forwelcome teachershere. and professors across control is devoted to description of the applied exercises on manipulator and the quadrotor. The conclusion of control of robotic systems includes the robotic boat, the This is quite common for teachers and professors across 3-5 the world to include the usage of various applied exercises manipulator and the quadrotor. The conclusion of the the the world to include thefor usage of various applied exercises control of robotic systems includes the robotic boat, This is quite common teachers and professors across manipulator and the quadrotor. The conclusion of the paper is presented in Section 6. the world tostudents, include the usage control of various applied to motivate motivate promote science andexercises improve paper is presented in Section 6. to promote control science and improve and the quadrotor. themotivate world experience. tostudents, include the usage ofLEGO various applied paper is presented in Section 6. The conclusion of the to students, promote control science andexercises improve practical Using the Mindstorms is the the manipulator practical experience. Using the LEGO Mindstorms is paper is presented in Section 6. 2. COURSES OVERVIEW to motivate students, Using promote and improve practical experience. thecontrol LEGOscience Mindstorms is the common solution provide experience of 2. COURSES OVERVIEW AND AND FUNDAMENTAL FUNDAMENTAL common solution to to Using provide experience of the the designdesignTRACKS practical experience. the LEGO Mindstorms is the 2. COURSES OVERVIEW AND FUNDAMENTAL common solution experience of the designing control control systemstofor forprovide the robots robots (see Kapitonov Kapitonov et al. al. TRACKS 2. COURSES OVERVIEW AND ing systems the (see et common solution tofor provide experience of the designTRACKS FUNDAMENTAL ing control systems the robots (seeinKapitonov et al. (2019)). The remote control study kit Shavetov TRACKS (2019)). The remote control study kit in Shavetov et al. ing control systems the robots (see al. Building (2019)). remote for control study kit inKapitonov Shavetov ettestal. (2016) isThe considered as configurable application for et Building skills skills of of using using real real technical technical plants plants is is an an imim(2016) is considered as configurable application for testBuilding skills of using real technical plantsof is Robotic an important part of the Modeling and Control (2019)). The remote as control study kit in remote Shavetov ettestal. portant part of the Modeling (2016) is considered configurable application for ing various algorithms for a trajectory or control. and Control of Robotic Building skills of using real technical plants is an iming various algorithms for a trajectory or remote control. Systems course Faculty of and part of at thethe ControlSystems of Robotic (2016) is considered asfor configurable application test- portant ing various algorithms a the trajectory or remote control. In Belyavskyi et al. al. (2017) (2017) description of the thefor 2DOF Systems course at theModeling Faculty and of Control Control Systems and portant part of the Modeling and Control of Robotic In Belyavskyi et the description of 2DOF Robotics of University. educational course at the Faculty So of all Control Systemstracks and ingBelyavskyi various algorithms for a the trajectory or remote In et laboratory al. (2017) description of thecontrol. 2DOF Systems quadrotor-based bench for engineering engineering educaRobotics of ITMO ITMO University. So educational tracks Systems course at different the Faculty of all Control Systems and quadrotor-based laboratory bench for educaRobotics of ITMO University. So all educational tracks considered in the programs are related to the In Belyavskyi et al. (2017) the description of the 2DOF quadrotor-based laboratory bench for engineering education. considered inITMO the different programs are related to the Robotics of University. So all educational tracks tion. considered in the different programs are related to the robotics plants. There are three master’s degree programs quadrotor-based laboratory bench for engineering educa- robotics plants. There are three master’s degree programs tion. considered in the different programs are related to the In this paper the Modeling and Control of Robotic Syswhich have the similar kernel is about control systems, robotics plants. There are three master’s degree programs tion. In this paper the Modeling and Control of Robotic Sys- which have the There similararekernel ismaster’s about control systems, robotics plants. three degree programs In this paper the Modeling and Control of Robotic Systems course course to to build build the the set set of of the the professional professional skills skills which but different specializations. “Robotics” is have the similar kernel is about program control systems, tems different specializations. “Robotics” program is about about In presented. thiscourse paper to thebuild Modeling andof Control of Robotic Sys- but which have the similar kernel is about“Control control ofsystems, tems the set the professional skills is Our set of skills is based on professional but different specializations. “Robotics” program isCyberabout industrial robots modeling and control, is presented. Our set of skills is based on professional industrial robots modeling and control, “Control of Cybertems course to build the set of the professional skills but different specializations. “Robotics” program isCyberabout is presented. Our set of skills is based on professional standards and foresight of the Faculty of Control Systems industrial robots modeling and control, “Control of Physical Systems” program is about control systems synstandards andOur foresight of skills the Faculty of Control Systems Physical Systems” programand is about control systems synis presented. set University. of isThese based on professional industrial robots modeling control, “Control of Cyberstandards andofforesight of the Faculty ofprofessional Control Systems and Robotics ITMO skills thesis and “Digital control systems” program is about Physical Systems” program is about control systems synand Robotics of ITMO University. These professional skills thesis and “Digital control systems” program is about standards andofforesight of the Faculty ofprofessional Control Systems Systems” program isto about control systems synand Robotics ITMOskills University. These skills Physical rely on fundamental and are achieved by student’s control systems integration various applications. The thesis and “Digital control systems” program is about rely on fundamental skills and areThese achieved by student’s control systems integration to various program applications. The and Robotics of ITMO University. professional skills thesis and “Digital control systems” is second about rely onand fundamental skills and are achieved by student’s home class work. The educational trajectories are control systems integration to various applications. The professional course takes one academic year since home and class work. The educational trajectories are professional course takes one academic year since second rely on fundamental skills and are achieved by student’s control systems integration to various applications. The home andforclass work. The educational trajectories are prepared each educational track and should be used for semester and is devoted to mastering fundamental and professional course takes one academic year since second prepared each work. educational track and should be used are for semester andcourse is devoted to mastering fundamental and home andfor The educational trajectories takes one academic year since second prepared forclass each educational track and should be used for professional professional skills. semester and is devoted to mastering fundamental and professional skills. prepared each educational track and should beofused for semester andskills. is devoted to mastering fundamental and workfor was written with the support of the Ministry Science The professional The work was written with the support of the Ministry of Science We general professional and Higher Education thethe Russian Federation, project unique The work was writtenof with support of the Ministry of Science We consider considerskills. general fundamental fundamental and and professional professional skills skills and Higher Education of the Russian Federation, project unique The for the educational programs that have similar conWe consider general fundamental and professional skills work was written with the support of the Ministry of Science identifier RFMEFI57818X0271 “Adaptive Sensorless Control for for the educational programs that have similar conand Higher Education of the Russian Federation, project unique identifier RFMEFI57818X0271 “Adaptive Sensorless Control for We consider general fundamental and professional skills nected tracks (the one professional skill model). In an our and Higher Education of theinRussian Federation, project unique Synchronous Electric Drives Intelligent Robotics andControl Transport for the educational programs that have similar conidentifier RFMEFI57818X0271 “Adaptive Sensorless for nected tracks (the one professional skill model). In an our Synchronous Electric Drives in Intelligent Robotics and Transport for thetracks educational that similar conidentifier RFMEFI57818X0271 “Adaptive Robotics Sensorless for nected (the approach oneprograms professional skillhave model). In an our Systems”. practical-oriented for mastering skills the series Synchronous Electric Drives in Intelligent andControl Transport Systems”. practical-oriented approach for mastering skills the series nected tracks (the one professional skill model). In an our Synchronous Electric Drives in Intelligent Robotics and Transport Systems”. practical-oriented approach for mastering skills the series Systems”. practical-oriented approach for mastering skills the series 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2019 IFAC 237 Copyright 2019 IFAC 237 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2019 IFAC 237 10.1016/j.ifacol.2019.08.204 Copyright © 2019 IFAC 237
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Lb
Px
O Pe
225
αe Mz Ps
Py
Pb
Fig. 1. The robotic boat actuators configuration
Fig. 2. 3D model of the robotic boat with actuators forces projections (red lines)
of the tracks is included in the curriculum: Modern Control Theory, Technical Systems Modeling, Control Systems Programming, Cyber-Physical Systems and Technologies, Modeling and Control of Robotic Systems, Dynamic of Robotic Systems, Modeling and Motion Control of Surface Vessels, Sensorless Control and some others. They are includes the content about identification and control fundamentals as well as modeling and control of the robotic systems. Practical tracks may includes home tasks to work with 3D models of the robots and the experimental exercises with robotic equipment at the laboratory. Three types of robots are used in these exercises: • Robotic boat; • Robotic manipulator KUKA youBot; • Unmanned quadrotor.
use existing software packages (and, if necessary, to develop a new software) for the data processing and control in mechatronic and robotic systems” with the learning results: Knowledge — modern approaches for designing robotic systems mathematical models and its program realization; Ability — motion trajectories planning of robotic systems at given reference points; Skill — robotic systems programming for given motions performing. This is a fundamental track for the Control Systems and Robotics students. It is a basis for designing control algorithms for a robotic boat, manipulator or unmanned quadrotor. 3. ROBOTIC BOAT CONTROL TRACK
Each professional skill can be mastered in several courses but one prevails for each course. We provide control and robotic fundamental skills before studying courses mastering professional skills. Fundamental skills are used in all tracks of the master’s degree programs.
For the robotic boat control track of the course the 3D model of the plant is considered. During the track students are able to prepare the transformation algorithm from local to global coordinate systems for the model, the identification and control algorithms for dynamic positioning.
The first fundamental track is “Technical Systems Modeling” with fundamental skill “Able to apply modern theoretical and experimental methods for developing mathematical models of cyber-physical systems, including plants parametric identification” which includes the learning results: Knowledge — principles of mathematical describing systems, principles of synthesis algorithms and numerical solutions for models; Ability — digital control and information modules synthesis and analysis; Skill — calculation and design of individual units, devices and systems using mathematical modeling.
The model is based of the real robotic boat and contains parameters which have been identified from the real plant. The actuators configuration of the robotic boat are presented in the Fig. 1. Pb , Ps , Pe are the propelling forces produced by the bow, stern and end actuators respectively which can be controlled by control system. The bow and stern actuators are the tunnel thrusters. Px and Py are the generalized forces along x and y axis, Mz is the generalized rotational moment.
The track “Modern Control Theory” with the skill “Able to develop and research control systems functioning algorithms” can be considered both as fundamental and as professional skill with the following learning results: Knowledge — modern approaches of planning and motion control for robotic systems; Ability — Control algorithms realizations for various software environments; Skill — controllers synthesis for solving various tasks of the robotic plants with parametric and signal uncertainties, constraints, functioning specifics. Parts of the track are used for designing controllers for a robotic boat, manipulator or unmanned quadrotor. Connected with the previous one track “Control Systems Programming” contains professional skill “Able to 238
For education purposes the mathematical model of the each plant actuators considered like b W (s) = 2 (1) s − as where a & b are unknown parameters of the plant. For the identification task the regression model has to be obtained by students. The model (1) can be rewritten like y¨ = ay˙ + bu
(2)
where u is the control signal, y is the output coordinate. Both signals are measurable but not their derivatives. By λ2 introducing the linear filter of the form (s+λ 2 )2 the new measurable signals can be obtained ξy (s) =
λ2 λ2 y(s), ξ (s) = u(s). u (s + λ2 )2 (s + λ2 )2
(3)
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for simplification of the computer vision. The current coordinates of the robotic boat are provided by the computer vision system from the digital camera instead of satellite navigation systems due to the scale of the educational setup. The camera is attached to the tripod above the basin. The captured image is transmitted as a RGB signal to the computer, where it is processed in order to get the coordinates of the boat. Students are able to complete the experimental trials with a designed control system during the classes. The experience with the real robot helps to understand identification and control principles and form debugging skills for a future work.
Fig. 3. The robotic boat setup for education and research By substituting (3) into (2) and performing the Laplace transform obtain (4) ξ¨y (t) = −aξ˙y (t) + bξu (t) + (t) where is the exponentially decreasing term which can be neglected. The regression model of the plant can be written from (4) in the next form (5) Y = ωT θ T T ˙ ¨ where (Y ) = ξy , ω = [ξy ξu ], θ = [a b]. Students are able to apply the identification algorithm using the regression model (5) and the control approaches to the plant model (1). For ease of developing algorithms and debugging by students at home the 3D model has been prepared in SIMULINK. The screenshot of the modeling process presented on the Fig. 2. Red lines represents the forces produced by actuators, yellow triangle is the target position which has to be achieved. The identification or control system should be implemented by students in SIMULINK like the scheme or the code. Due to the fact that the 3D model uses parameters which has been identified from the real boat it is easy to transfer the algorithm from the model to the experimental setup (see Borisov et al. (2016); Wang et al. (2015) for examples of research projects). The robotic boat setup was designed for experimental study at the laboratory during the robotic boat control track of the course. The main parts of the setup are the robotic boat itself and experimental basin. The robotic boat is a scale model of a real trawler ship represented at a scale of 1 : 32. The actuators configuration is the same like on the Figs. 1 so the developed control system for the mathematical model can be transferred to the real robotic boat. The experimental basin represents the workspace of the boat. Its volume is about 150 L of water. The internal surface is covered by sealant and painted with dark color 239
The professional skill “Able to develop and research control systems functioning algorithms” of the track “Modeling and Motion Control of Surface Vessels” includes the learning results: Knowledge — dynamic models of the surface vessels motion; Ability — modern approaches for vessel’s position determination in different coordinate systems; Skill — synthesis of vessels’ dynamic positioning systems and stabilizing algorithms for given velocities. In this track the considered plant is the Robotic boat. 4. ROBOTIC MANIPULATOR CONTROL TRACKS For the robotic manipulator control tracks the model of the 5-DOF robot is considered (see Fig. 4). The parameters of the model are the same like on the robot KUKA youBot. Students are able to design algorithms for solving forward and inverse kinematics tasks as well as identification and control systems. According to Spong et al. (2006) the dynamical model of the robotic manipulator can be calculated from the EulerLagrange equations dL d dL − = τi i = 1, . . . , n (6) dt dq˙i dqi z5
x5
x3 , z4 z3
x4 x2
z2
x1
z1
z0 x0
Fig. 4. The kinematic scheme of the robotic manipulator KUKA youBot
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Fig. 5. 3D model of the robotic manipulator KUKA youBot and desired trajectory of the end-effector
Fig. 6. The robotic manipulator KUKA youBot for education and research in our laboratory
where L = K − P is the Lagrangian of the system, K is the kinetic energy of the system, P is the potential energy of the system, qi are the generalized coordinates, τi are the generalized load torques, applied to the actuators, n is the number of link (order of the system). By calculating kinetic and potential energy and by substituting it to the (6) it is possible to represent to dynamic model of the joints without external load in the matrix form M (q)¨ q + C(q, q) ˙ + G(q) = τ (7) where M (q) is inertial matrix sized (n × n), C(q, q) ˙ ∈ Rn n is Coriolis forces vector, G(q) ∈ R is gravitational forces vector, τ ∈ Rn is load torques vector, which compensated by actuators.
During the course students are able to estimate the parameters of the actuators dynamic model for the 3D model which was prepared in SIMULINK (see Fig. 5). The inertial and gravitational parameters of the model are acquired earlier during the procedure of the identification of the real robot parameters. After estimating the parameters the designing control system for compensating gravitational forces is required for students. The next step is solving the forward kinematics task and designing the control system to follow the desired trajectory by the end-effector of the robot. All this tasks can be completed by students at home using the designed 3D model with control approaches provided during the fundamental track. Also the algorithms for disturbance and time delay compensation can be modeled too (Pyrkin and Bobtsov (2011); Bobtsov et al. (2011)).
The model (7) can be simplified by students for the static case when q˙ = 0 & q¨ = 0 G(q) = τ (8) where G(q) is gravitational matrix calculated by T ∂P ∂P ∂P ∂P ∂P ∂P = . (9) G(q) = q1 q2 q3 q4 q5 ∂q The potential energy of the robot KUKA youBot calculated from the potential energy of the each link P=
5
= mi ghi (q) = m1 gh1 (q) + m2 gh2 (q)
(10)
i=1
+ m3 gh3 (q) + m4 gh4 (q) + m5 gh5 (q)
where mi is the mass of the i’s link, hi (q) is the zcoordinate from the center mass vector of the i’s link in the base coordinate system. To identify the gravitational component, regression models were formed for each joint τi := Gi (q) = φTi (q, τ )θi , where φi (q) is regressor made up of measured signals depending on the configuration q, θi is vector of unknown parameters (centres of mass of links and values of masses), estimated using the identification algorithm. 240
After completing the home tasks with 3D model students are able to perform an experimental exercises at the class with the real robot KUKA youBot (see Gromov et al. (2016) for detailed description of the experimental setup). Due to the fact that 3D model contains parameters which are close to the real ones it is not difficult to transfer the identification and control algorithms designed by students to the real robot. Classroom activities helps to build skills and experience with multi-link robots and mechanical systems, solving kinematic and dynamic tasks, developing and debugging control systems. The track “Modeling and Control of Robotic Systems” is characterized by professional skill “Able to develop experimental setups of control, information and actuator modules for mechatronic and robotic systems”. This skill can be represented in a form of learning results: Knowledge — robotic systems kinematic analysis methods; Ability — developing various scenarios of industrial operations using robotic systems; Skill — using Denavit-Hartenberg approach for robotic systems kinematic analysis. In the course robotic manipulator KUKA youBot is used as an object for kinematic analysis. Connected with the previous one the track “Dynamic of Robotic Systems” is characterized by professional skill
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“Able to develop automated process control systems” that includes the learning results: Knowledge — methods for constructing robotic systems dynamic models; Ability — developing conceptions of automation control for robotic systems; Skill — constructing robotic systems dynamic models using the Euler-Lagrange and Newton-Euler methods. The robotic manipulator KUKA youBot also is used in the course for dynamic analysis. 5. UNMANNED QUADROTOR CONTROL TRACK Another application example, which can be used to conduct illustrative basic as well as advanced studies on modeling and control of robotic systems, is the station-keeping of a quadrotor. Consider its dynamical model assuming the drag coefficient equal to zero at low speed Altuˇ g et al. (2002): 4 m¨ x= ui (cφ sθ cψ +sφ sψ ), Jθ θ¨= (−u1 −u2 +u3 +u4 ), m¨ y=
i=1 4 i=1
m¨ z=
4 i=1
Consider several structures of robust controllers under assumption that plant parameters are unknown, derivatives of the output variables are not available for measurements and the input signals are bounded. Once the control law generating Ui is defined, a last step is the inverse transformation of (11) in order to allocate the controlbetween the actuators u1 1 −1 −1 1 U1 u 2 1 −1 1 −1 U2 u = 0, 25 1 1 1 1 U . 3 3 1 1 −1 −1 U4 u4
5.1 Robust Output Controller
The robust output controller based on the consecutive compensator approach applied to the quadrotor model (13)–(15) is of the form υi = κi (αi,2 ξ˙i + αi,1 ξi ), (17) ξ˙i = σi (−ξi + ei ),
ui (sφ sθ cψ −cφ sψ ),
Jψ ψ¨ = (−u1 +u2 +u3 −u4 ),
in which κi > 0, σi > 0, αi,j > 0, j = {1, 2} are the design parameters. Remark 2. Note that the gravitational force mg can be added to for U1 in order to compensate its effect.
ui (cθ cψ )−mg,
Jφ φ¨ = C(u1 −u2 +u3 −u4 ),
5.2 Static Disturbance Compensation
where x, y, z, θ, ψ, φ are the linear and angular coordinates, ui , i = {1, 2, 3, 4} are the control signals, m, g, , Jθ , Jψ , Jφ , C are the physical parameters, cφ ≡ cos φ, sφ ≡ sin φ.
Decompose the quadrotor model choosing the set of quasicontrol signals 1 1 1 1 u1 U1 U2 −1 −1 1 1 u2 (11) U = −1 1 1 −1 u , 3 3 1 −1 1 −1 u4 U4 where the quasi-control signals Ui , i = {1, 2, 3, 4} satisfy the saturation condition Ui,max , if υi ≥ Ui,max , υi , if Ui,min < υi < Ui,max , (12) Ui = sat(υi ) = Ui,min , if υi ≤ Ui,min , Ui,min and Ui,max are the input saturation limits satisfying, υi are the nominal control signals generated by the independed SISO linear regulators. Following transformations similar to Tomashevich et al. (2017) use the linearized quadrotor model ¨ = −U5 , mx ¯ Jθ θ¨ = U2 , (13) ¨ = −U6 , Jψ ψ¨ = U3 , (14) my¯
Jφ φ¨ = CU4 , (15) m¨ z = U1 − mg, where the desired values of the roll and pitch angles are calculated as U5 U6 θ∗ = , ψ ∗ = − , U5 = U1 θ, U6 = −U1 ψ. (16) U1 U1 Remark 1. Note that in the final engineering implementation the calculation algorithm should be modified in order to avoid singularities in (16). Zero values of θ∗ and ψ ∗ should be assigned when U1 is approaching to zero. Values of θ∗ and ψ ∗ should be bounded to avoid critical reference peaks within the transient processes. 241
Static disturbances (such as the gravitational force and constant wind acting on the quadrotor) leading to steadystate error of the output variables can be cancelled by the integral term. Rewrite the control law (17) as t ˙ ξi (τ )dτ , υi = κi βi,3 ξi + βi,2 ξi + βi,1 (18) 0 ξ˙i = σi (−ξi + ei ), in which βi,j > 0, j = {1, 2, 3} are the design parameters. 5.3 Anti-Windup Compensation However the integral term needed to cancel the steadystate error might result in the integral windup in case of bounded input signals. To get rid of this effect anti-windup technique can be implemented as t υi =κi βi,3 ξ˙i +βi,2 ξi +βi,1 (ξi (τ )−νi κi (υi ))dτ ,
(19) 0 ξ˙i =σi (−ξi + ei ), where νi > 0 are the design parameter, κi (υi ) = υi − sat(υi ) are the anti-windup signal. 5.4 Adaptive Tunning Controller Parameters The controller (19) can further be updated by implementing the direct adaptation laws. The design parameters κ, σ and ν being replaced with functions κ(t), σ(t) and ν(t), respectively, can be adjusted as κ˙ i (t) = ζκ ξi2 − ςκ (κi (t) − κi (0)), σ˙ i (t) = ζσ (ri − ei − ξi )2 − ςσ (σi (t) − σi (0)), ν˙ i (t) = ζν κi2 (υi ) − ςν (νi (t) − νi (0)), in which ri are the given references, κi (0), σi (0), νi (0) are the initial conditions, ζj , ςj , j = {κ, σ, ν} are the design parameters.
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u1
u4
u2 u3 (a) Quadrotor scheme
(b) Modeling the quadrotor
Fig. 7. Quadrotor 5.5 Robust Output Controller in State-Space The controller can be redesigned in state-space as υi = κi (cTq,i ξi + ei ) − γi ηi , ξ˙i = Aq,i ξi + bq,i ei ,
(20)
η˙ i = −κi (cTq,i ξi
(22)
+ ei ) + νi κi (υi ),
(21)
where the matrices Aq,i and vectors bq,i , cq,i are of the form −qi,2 σi 1 qi,2 σi qi,1 , b , c , = Aq,i = = q,i q,i qi,2 −qi,1 σi2 σi2 0 qi,1
> 0, qi,j > 0, j = {1, 2} are the and γi > 0, σi > 0, qi,j design parameters.
The unmanned quadrotor is used like a practical examples of the course “Sensorless Control”, which is about modern trend of sensorless control allows to reduce some sensors in robotic systems, e.g. in electric drives and replace it with some mathematical algorithm. The professional skill is similar to the ones from the previous track but includes the learning results: Knowledge — concepts of nonlinear electromechanical systems, coordinate transformations for AC motors, methods for synthesizing nonlinear observers based on identification approaches, vector motor control method; Ability — using algorithms for estimating parameters and state variables of AC motors in the angular position and / or rotor speed sensors absence; Skill — synthesis sensorless control algorithms for electromechanical systems. This approach allows to increase systems reliability. 6. CONCLUSION The educational program as a set of tracks could be filled in different ways to achieve proper professional skills. We use practical-oriented approach with the connected tracks that allows us to join acquired fundamental knowledge with applications to build skills. REFERENCES Altuˇg, E., Ostrowski, J., and Mahony, R. (2002). Control of a quadrotor helicopter using visual feed242
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