Robotics & Automation. Lecture 25. Dynamics of Constrained Systems, Dynamic Control. John T. Wen. April 26, 2007
|
|
- Darcy Gilbert
- 5 years ago
- Views:
Transcription
1 Robotics & Automation Lecture 25 Dynamics of Constrained Systems, Dynamic Control John T. Wen April 26, 2007
2 Last Time Order N Forward Dynamics (3-sweep algorithm) Factorization perspective: causal-anticausal (LDU decomposition) + state space (group) structure leads to order N inverse dynamics (matrix-vector multiplication). Inverse dynamics: Kalman filter interpretation, mass matrix inverse can be expressed in terms of Kalman filter (written as whitening filter). Today: Control of serial robot, dynamics of constrained mechanism (single constrained arm, parallel platform type of mechanism), dynamics of rolling wheel, dynamics of general mechanisms, joint flexibility. April 26, 2007Copyrighted by John T. Wen Page 1
3 Dynamics of Closed Kinematic Chain Consider the slider crank system: q 2 q 1 q 3 Kinematic constraints, φ(θ) = 0: θ T = θ 1 + θ 2 + θ 3 = 0 y T = l 1 s 1 +l 2 s 12 +l 3 s 123 = 0. April 26, 2007Copyrighted by John T. Wen Page 2
4 Differentiate to obtain the velocity constraints: θ T v T = ẋ T ẏ T = J(θ) θ = }{{} A ẋ T. }{{} v C From the force perspective, if we regard the load as part of the arm, then f Tx = 0, or A T f T = 0 where f T = If we regard the load as separate, then f Tx causes motion of the load: m c ẍ T = f Tx = A T f T. τ T f Tx f Ty. April 26, 2007Copyrighted by John T. Wen Page 3
5 Complete Dynamics The arm dynamics can be determined from either Lagrange-Euler or Newton-Euler formulation. In addition to kinematic parameters (in this case just lengths of the links), we need to furnish dynamic parameters: mass, location of center of mass, and inertia about center of mass, for each link. The complete dynamics of the constrained mechanism is then M(θ) θ+c(θ, θ) θ+g(θ) = A T f T = 0 τ J T (θ) f T J(θ) θ = Av c, J(θ) θ+ J(θ) θ = Aα c + Ȧv c. If the load is considered separately, A T f T = 0 equation should be replaced by the load dynamics equation: Note that in this case, Ȧ = 0, b c = G c = 0. m c α c + b c v c + G c = A T f T. April 26, 2007Copyrighted by John T. Wen Page 4
6 For simulation, solve for α c and f T simultaneously: 0 AT α c 0 = A JM 1 J T f T JM 1 ( C θ G) Ȧv c + J θ+jm 1 τ. The leading matrix is invertible if and only if A T JM 1 JA is invertible. For simulation, the states are x c and ẋ c, which may be combined with the constraints to solve for θ and θ using the inverse kinematics. For a given τ, the right hand side of the differential equation can then be evaluated for ẍ c. The constraint force f T can also be solved based on θ, θ, and τ. Order N computation can also be extended to this case. If the load is considered separately, the equation of motion becomes: M c A T α c b = c v c G c A JM 1 J T f T JM 1 ( C θ G) Ȧv c + J θ+jm 1 τ. Again, the right hand side only depends on θ, θ, and τ. Therefore, the equation of motion can be used to propagate x c and v c, and solve for f T. April 26, 2007Copyrighted by John T. Wen Page 5
7 Dynamics of Multifinger grasp Multiple arms rigidly holding an object and platform type of parallel mechanisms are special cases. Strategy: Write down the dynamics of each finger and the payload, then add all the kinematic constraints. Dynamics of ith finger: M i (θ i ) θ i +C i (θ i, θ i ) θ i + G i (θ i ) = τ i J T i f i, where f i is the spatial force that ith finger applies to the load. Constraints: A i v c = J i θ i + H i W i, A i = φ ie = I 0 p ci I. H T i f i = 0 April 26, 2007Copyrighted by John T. Wen Page 6
8 Dynamics of payload: M c α c + b c v c + G c = A T f = m A T i f i. i=1 Putting everything together and stacked up variables and form block diagonal matrices, we have the complete dynamics: Substitute f = H T η, we have M c A T A HJM 1 J T H T M(θ) θ+c(θ, θ) θ+g(θ) = H T f T = 0 τ J T (θ) f T J(θ) θ+hw = Av c J(θ) θ+ J(θ) θ+hẇ + ḢW = Aα c + Ȧv c M c α c + b c v c + G c = A T f. α c η = 0 HJM 1 τ+ which can be used to propagate α c and solve for η (and hence f )... }{{} depends on θ, θ April 26, 2007Copyrighted by John T. Wen Page 7,
9 Summary Dynamics of constrained mechanisms (single constrained robot, parallel platform type of mechanism). General strategy: Express kinematic constraints in terms of unconstrained variables: Single constrained arm and multiple arm with rigid grasp: J θ = Av c Multi-finger grasp: J θ+hw = Av c. Differentiate the velocity constraint to obtain acceleration constraint: Single constrained arm and multiple arm with rigid grasp: J θ+ J θ = Aα c + Ȧv c Multi-finger grasp: J θ+ J θ+hẇ+ḣẇ = Aα c + Ȧv c. Corresponding force constraint: Single constrained arm and multiple arm with rigid grasp: A T f = 0 or M c α c + b c + g c = A T f Multi-finger grasp: H T f = 0 and M c α c + b c + g c = A T f. April 26, 2007Copyrighted by John T. Wen Page 8
10 Eliminate θ from acceleration constraint by using the arm dynamics: θ = M 1 (τ C θ G J T f). Obtain the equation of motion to solve for unconstrained acceleration and the constrained force: Single constrained arm and multiple arm with rigid grasp: M c A T α c b = c + g c. A JM 1 J T f JM 1 (τ C θ G)+ J θ Ȧv c Multi-finger grasp ( f = Hη): M c HA = A T H T HJM 1 J T H T α c η b c + g c H[JM 1 (τ C θ G)+ J θ Ȧv c ḢW]. April 26, 2007Copyrighted by John T. Wen Page 9
11 Additional Topics on Dynamics Dynamics of a wheel Dynamics of general complex multibody systems Robots with flexibility April 26, 2007Copyrighted by John T. Wen Page 10
12 Unicycle Model z y x h p r Recall kinematics model: Planar: p T z = p z = r No tilt: z T Ry = 0 No slip: r ωz+ ṗ = 0 April 26, 2007Copyrighted by John T. Wen Page 11
13 Write as a velocity constraint (note planar constraint follows from no-slip constraint): I 0 ω = z eẑφ y φ. rẑ I v 0 0 θ or JV = Av c. Differentiate to obtain acceleration constraint: Jα = Aα c + Ȧv c. Force constraint: A T f = 0 or z T τ c = 0,y T e ẑφ τ c = 0. Dynamics of wheel (Newton-Euler Equation): Mα+b = τ J T f, τ = 0 eẑφ τ y τ z, where τ y is the driving torque and τ z is the steering torque. April 26, 2007Copyrighted by John T. Wen Page 12
14 Overall dynamics: 0 A AT JM 1 J T α c f = 0 JM 1 (τ+b) Ȧv c. Kinematics must still be included: ẋ ẏ φ = rc φ 0 rs φ θ. φ }{{} v c April 26, 2007Copyrighted by John T. Wen Page 13
15 General Multibody Systems How do we obtain the dynamics of a general multibody system with many constraints? Newton-Euler (also called body-coordinate) approach: Write down Newton-Euler equation for each body: M i α i + b i = φ T ji f j. j April 26, 2007Copyrighted by John T. Wen Page 14
16 Kinematic constraints: Force constraint: φ i j (V i + H i θ i ) = 0 j H T i f i = 0 (or applied force/torque τ i ). Acceleration constraint: j Stack all variables together: Eliminate α to solve for θ and f together. φ i j (α i + H i θ i )+ ( φ i j (V i + H i θ i )+φ i j Ḣ i θ i ) = 0. j Mα+b = A T f Aα+Ȧ(V + H θ)+ah θ+ AḢ θ = 0. General, but computationally cumbersome. April 26, 2007Copyrighted by John T. Wen Page 15
17 Alternative Approach Constrained coordinate approach: First remove constraints to make the system a tree structure. The dynamics of which can be obtained using Lagrange-Euler method: M θ+c θ+ G = τ J T f April 26, 2007Copyrighted by John T. Wen Page 16
18 where f contains all the constrained forces. Then impose kinematic constraints: J θ = Av c, J θ+ J θ = Aα c + α c, A T f = 0, we can solve α c and f simultaneously. There are also order N methods for general constrained multibody systems. Prof. Anderson in mechanical engineering is a world authority on this topic. April 26, 2007Copyrighted by John T. Wen Page 17
19 Joint Flexibility Suppose each joint also has a torsional spring (e.g., flexure joints, popular in micromechanisms). To obtain the dynamics, define the zero configuration to correspond to zero deformation of all the springs. The Lagrangian now needs to include an additional term in the potential energy (assume linear springs): P = K i 2 (θ i θ i 1 ) 2. Lagrangian equation of motion then becomes M θ+c θ+ G+Kθ = τ J T f, where K = K 1 + K 2 K 2... K 2 K 2 + K 3 K 3 K 3 K 3 + K If the links are also flexible and can be modeled based on finite element method (FEM), April 26, 2007Copyrighted by John T. Wen Page 18
20 the dynamic model can also be constructed. April 26, 2007Copyrighted by John T. Wen Page 19
21 Impedance Control Make the robot end effector appear as a specified mechanical impedance. Desired behavior ( f is the spatial force robot applies to the environment): M des α des + D des v des + K des (x des x re f ) = B( f f re f ). f ref f M des K des D des x des x ref If only K des is present, it s called a generalized spring. If only D des is present, it s called a generalized damper. April 26, 2007Copyrighted by John T. Wen Page 20
22 To achieve the desired behavior, any motion controller can be applied to track x des, v des, α des. Joint space control: f desired impedance x des v des a des inverse kinematics joint controller τ robot θ Task space control: forward kinematics f desired impedance x des v des a des task space controller τ robot θ April 26, 2007Copyrighted by John T. Wen Page 21
23 Feedback linearizing control: f desired impedance x des v des a des feedback linearization τ robot θ April 26, 2007Copyrighted by John T. Wen Page 22
24 How to Exert a Desired Amount of Force? Position Accommodation: ψ r p c For position control, choose R des = I, p des = p c + eĥψ r, where ψ = vt. How do we maintain a specified downward z force? Let s consider a generalized damper: D z ż des = ( f z f zd ). April 26, 2007Copyrighted by John T. Wen Page 23
25 If the pen is not contacting the table, f z = 0. Since f zd < 0, z des will continue to decrease until the pen hits the table. Then f z will become more and more negative until at the steady state, f z = f zd and ż des = 0. Stability analysis: suppose the surface is infinitely rigid. Then upon contact, f z = F z where F z is the applied force. Suppose PD position controller is used: F z = k p (z z des ) k v ż. Since the surface is rigid, z and ż are zero, then F z = k p z des = k p D z Closed loop system: f z = k p ( f z f zd )dt. ( f z f zd )dt, which is stable and robust with respect to the force measurement time delay. What happens if a generalized spring is used? i.e., K z (z des z re f ) = ( f z f zd ). April 26, 2007Copyrighted by John T. Wen Page 24
26 Now consider direct torque control: First consider the motion control portion. Dynamics of open serial chain: Motion Control τ = τ m + τ f. M θ+c θ+ G = τ. Most industrial controllers are independent joint level PID: τ i = k Pi (θ i θ ides ) k Ii (θ i θ ides )dt k Di θ i. There is sometimes friction compensation (for Coulomb friction), integral anti-windup, backlash compensation, in addition to the standard PID. Trajectory generation is used for θ des to ensure smooth motion. Linearized system about (θ d,0): M(θ d ) θ+ G θ (θ θ d ) = τ. θ=θd April 26, 2007Copyrighted by John T. Wen Page 25
27 Performance under constant PID will vary depending on configuration and speed. Gains are typically tuned for the outstretched configuration, and response is faster (but may overshoot) in other configurations (smaller inertia and gravity load). Feedback Linearization (cancelling all the nonlinearity): τ = C θ+g(θ)+m(θ)( θ des K p (θ θ des ) K d θ). This can be computed by using the recursive Newton-Euler algorithm. Feedback linearization can be performed in the task space also (useful for impedance control): τ = C θ+g(θ)+j T (JM 1 J T ) 1 ( J θ α des K p (x x des ) K d (v v des )). Passivity based controller (noting the open loop system is conservative, we can just add damping and reshape the potential energy): τ = K p (θ θ des ) K d θ+g(θ). April 26, 2007Copyrighted by John T. Wen Page 26
28 Now consider a closed kinematic chain: Let τ = τ m + τ f, τ f = J T F. Effect of F on α c : Direct Force Control θ = M 1 (τ J T f)+..., Aα c = J θ+..., A T f = 0. Aα c = JM 1 J T F +..., therefore A T (JM 1 J T ) 1 Aα c = A T F +... If we choose F N (A T ) (A T F = 0), then F does not affect α c, i.e., force control does not affect motion (e.g., for the project, force control only affects how hard the arm presses down against the table.) Effect of motion on force: Reparameterize force: f = à T η and F = à T ξ. Then Aα c = JM 1 (τ J T f)+... = JM 1 J T (F f)+jm 1 τ m +... Multiply à on both sides: ÃJM 1 J T à T (ξ η) = terms dependent on θ, θ, and τ m. April 26, 2007Copyrighted by John T. Wen Page 27
29 Force dynamics: η = ξ+d where d is the motion induced force in the constrained direction. Since constrained force and its control do not affect motion, d can be considered as a disturbance as far as the force control is concerned. Direct force control Open loop: ξ = η des (no disturbance rejection) Proportional force feedback: ξ = k p (η η des ) (not robust w.r.t. force measurement delay) Integral force feedback: ξ = k i (η ηdes ) (robust w.r.t. force measurement delay) April 26, 2007Copyrighted by John T. Wen Page 28
Robotics. Dynamics. Marc Toussaint U Stuttgart
Robotics Dynamics 1D point mass, damping & oscillation, PID, dynamics of mechanical systems, Euler-Lagrange equation, Newton-Euler recursion, general robot dynamics, joint space control, reference trajectory
More informationIn this section of notes, we look at the calculation of forces and torques for a manipulator in two settings:
Introduction Up to this point we have considered only the kinematics of a manipulator. That is, only the specification of motion without regard to the forces and torques required to cause motion In this
More informationTrajectory-tracking control of a planar 3-RRR parallel manipulator
Trajectory-tracking control of a planar 3-RRR parallel manipulator Chaman Nasa and Sandipan Bandyopadhyay Department of Engineering Design Indian Institute of Technology Madras Chennai, India Abstract
More informationRobotics. Dynamics. University of Stuttgart Winter 2018/19
Robotics Dynamics 1D point mass, damping & oscillation, PID, dynamics of mechanical systems, Euler-Lagrange equation, Newton-Euler, joint space control, reference trajectory following, optimal operational
More informationMultibody simulation
Multibody simulation Dynamics of a multibody system (Newton-Euler formulation) Dimitar Dimitrov Örebro University June 8, 2012 Main points covered Newton-Euler formulation forward dynamics inverse dynamics
More informationAdaptive Robust Tracking Control of Robot Manipulators in the Task-space under Uncertainties
Australian Journal of Basic and Applied Sciences, 3(1): 308-322, 2009 ISSN 1991-8178 Adaptive Robust Tracking Control of Robot Manipulators in the Task-space under Uncertainties M.R.Soltanpour, M.M.Fateh
More informationLecture Note 12: Dynamics of Open Chains: Lagrangian Formulation
ECE5463: Introduction to Robotics Lecture Note 12: Dynamics of Open Chains: Lagrangian Formulation Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio,
More informationMCE493/593 and EEC492/592 Prosthesis Design and Control
MCE493/593 and EEC492/592 Prosthesis Design and Control Control Systems Part 3 Hanz Richter Department of Mechanical Engineering 2014 1 / 25 Electrical Impedance Electrical impedance: generalization of
More informationIntroduction to Robotics
J. Zhang, L. Einig 277 / 307 MIN Faculty Department of Informatics Lecture 8 Jianwei Zhang, Lasse Einig [zhang, einig]@informatik.uni-hamburg.de University of Hamburg Faculty of Mathematics, Informatics
More information1. Consider the 1-DOF system described by the equation of motion, 4ẍ+20ẋ+25x = f.
Introduction to Robotics (CS3A) Homework #6 Solution (Winter 7/8). Consider the -DOF system described by the equation of motion, ẍ+ẋ+5x = f. (a) Find the natural frequency ω n and the natural damping ratio
More informationLecture Schedule Week Date Lecture (M: 2:05p-3:50, 50-N202)
J = x θ τ = J T F 2018 School of Information Technology and Electrical Engineering at the University of Queensland Lecture Schedule Week Date Lecture (M: 2:05p-3:50, 50-N202) 1 23-Jul Introduction + Representing
More informationLecture «Robot Dynamics»: Dynamics 2
Lecture «Robot Dynamics»: Dynamics 2 151-0851-00 V lecture: CAB G11 Tuesday 10:15 12:00, every week exercise: HG E1.2 Wednesday 8:15 10:00, according to schedule (about every 2nd week) office hour: LEE
More information(W: 12:05-1:50, 50-N202)
2016 School of Information Technology and Electrical Engineering at the University of Queensland Schedule of Events Week Date Lecture (W: 12:05-1:50, 50-N202) 1 27-Jul Introduction 2 Representing Position
More informationMultibody simulation
Multibody simulation Dynamics of a multibody system (Euler-Lagrange formulation) Dimitar Dimitrov Örebro University June 16, 2012 Main points covered Euler-Lagrange formulation manipulator inertia matrix
More informationAdvanced Robotic Manipulation
Advanced Robotic Manipulation Handout CS37A (Spring 017 Solution Set # Problem 1 - Redundant robot control The goal of this problem is to familiarize you with the control of a robot that is redundant with
More informationVirtual Passive Controller for Robot Systems Using Joint Torque Sensors
NASA Technical Memorandum 110316 Virtual Passive Controller for Robot Systems Using Joint Torque Sensors Hal A. Aldridge and Jer-Nan Juang Langley Research Center, Hampton, Virginia January 1997 National
More informationq 1 F m d p q 2 Figure 1: An automated crane with the relevant kinematic and dynamic definitions.
Robotics II March 7, 018 Exercise 1 An automated crane can be seen as a mechanical system with two degrees of freedom that moves along a horizontal rail subject to the actuation force F, and that transports
More informationApproach based on Cartesian coordinates
GraSMech course 2005-2006 Computer-aided analysis of rigid and flexible multibody systems Approach based on Cartesian coordinates Prof. O. Verlinden Faculté polytechnique de Mons Olivier.Verlinden@fpms.ac.be
More informationDynamics. describe the relationship between the joint actuator torques and the motion of the structure important role for
Dynamics describe the relationship between the joint actuator torques and the motion of the structure important role for simulation of motion (test control strategies) analysis of manipulator structures
More informationLecture «Robot Dynamics»: Dynamics and Control
Lecture «Robot Dynamics»: Dynamics and Control 151-0851-00 V lecture: CAB G11 Tuesday 10:15 12:00, every week exercise: HG E1.2 Wednesday 8:15 10:00, according to schedule (about every 2nd week) Marco
More informationGAIN SCHEDULING CONTROL WITH MULTI-LOOP PID FOR 2- DOF ARM ROBOT TRAJECTORY CONTROL
GAIN SCHEDULING CONTROL WITH MULTI-LOOP PID FOR 2- DOF ARM ROBOT TRAJECTORY CONTROL 1 KHALED M. HELAL, 2 MOSTAFA R.A. ATIA, 3 MOHAMED I. ABU EL-SEBAH 1, 2 Mechanical Engineering Department ARAB ACADEMY
More informationNumerical Methods for Rigid Multibody Dynamics
Numerical Methods for Rigid Multibody Dynamics Claus Führer Centre for Mathematical Sciences Lund University Lappenranta 2012 Unit 0: Preface These notes serve as a skeleton for the compact course. They
More informationDynamics. 1 Copyright c 2015 Roderic Grupen
Dynamics The branch of physics that treats the action of force on bodies in motion or at rest; kinetics, kinematics, and statics, collectively. Websters dictionary Outline Conservation of Momentum Inertia
More informationDYNAMICS OF SERIAL ROBOTIC MANIPULATORS
DYNAMICS OF SERIAL ROBOTIC MANIPULATORS NOMENCLATURE AND BASIC DEFINITION We consider here a mechanical system composed of r rigid bodies and denote: M i 6x6 inertia dyads of the ith body. Wi 6 x 6 angular-velocity
More informationLecture 9 Nonlinear Control Design. Course Outline. Exact linearization: example [one-link robot] Exact Feedback Linearization
Lecture 9 Nonlinear Control Design Course Outline Eact-linearization Lyapunov-based design Lab Adaptive control Sliding modes control Literature: [Khalil, ch.s 13, 14.1,14.] and [Glad-Ljung,ch.17] Lecture
More informationIntroduction to centralized control
ROBOTICS 01PEEQW Basilio Bona DAUIN Politecnico di Torino Control Part 2 Introduction to centralized control Independent joint decentralized control may prove inadequate when the user requires high task
More informationRobust Control of Cooperative Underactuated Manipulators
Robust Control of Cooperative Underactuated Manipulators Marcel Bergerman * Yangsheng Xu +,** Yun-Hui Liu ** * Automation Institute Informatics Technology Center Campinas SP Brazil + The Robotics Institute
More informationIntroduction to centralized control
Industrial Robots Control Part 2 Introduction to centralized control Independent joint decentralized control may prove inadequate when the user requires high task velocities structured disturbance torques
More informationDynamics. Basilio Bona. Semester 1, DAUIN Politecnico di Torino. B. Bona (DAUIN) Dynamics Semester 1, / 18
Dynamics Basilio Bona DAUIN Politecnico di Torino Semester 1, 2016-17 B. Bona (DAUIN) Dynamics Semester 1, 2016-17 1 / 18 Dynamics Dynamics studies the relations between the 3D space generalized forces
More informationRobot Dynamics II: Trajectories & Motion
Robot Dynamics II: Trajectories & Motion Are We There Yet? METR 4202: Advanced Control & Robotics Dr Surya Singh Lecture # 5 August 23, 2013 metr4202@itee.uq.edu.au http://itee.uq.edu.au/~metr4202/ 2013
More informationHW3 Physics 311 Mechanics
HW3 Physics 311 Mechanics FA L L 2 0 1 5 P H Y S I C S D E PA R T M E N T U N I V E R S I T Y O F W I S C O N S I N, M A D I S O N I N S T R U C T O R : P R O F E S S O R S T E F A N W E S T E R H O F
More informationGain Scheduling Control with Multi-loop PID for 2-DOF Arm Robot Trajectory Control
Gain Scheduling Control with Multi-loop PID for 2-DOF Arm Robot Trajectory Control Khaled M. Helal, 2 Mostafa R.A. Atia, 3 Mohamed I. Abu El-Sebah, 2 Mechanical Engineering Department ARAB ACADEMY FOR
More informationNonholonomic Constraints Examples
Nonholonomic Constraints Examples Basilio Bona DAUIN Politecnico di Torino July 2009 B. Bona (DAUIN) Examples July 2009 1 / 34 Example 1 Given q T = [ x y ] T check that the constraint φ(q) = (2x + siny
More informationSeul Jung, T. C. Hsia and R. G. Bonitz y. Robotics Research Laboratory. University of California, Davis. Davis, CA 95616
On Robust Impedance Force Control of Robot Manipulators Seul Jung, T C Hsia and R G Bonitz y Robotics Research Laboratory Department of Electrical and Computer Engineering University of California, Davis
More information1 Trajectory Generation
CS 685 notes, J. Košecká 1 Trajectory Generation The material for these notes has been adopted from: John J. Craig: Robotics: Mechanics and Control. This example assumes that we have a starting position
More informationAutomatic Control Systems. -Lecture Note 15-
-Lecture Note 15- Modeling of Physical Systems 5 1/52 AC Motors AC Motors Classification i) Induction Motor (Asynchronous Motor) ii) Synchronous Motor 2/52 Advantages of AC Motors i) Cost-effective ii)
More informationLaboratory Exercise 1 DC servo
Laboratory Exercise DC servo Per-Olof Källén ø 0,8 POWER SAT. OVL.RESET POS.RESET Moment Reference ø 0,5 ø 0,5 ø 0,5 ø 0,65 ø 0,65 Int ø 0,8 ø 0,8 Σ k Js + d ø 0,8 s ø 0 8 Off Off ø 0,8 Ext. Int. + x0,
More informationGeneral procedure for formulation of robot dynamics STEP 1 STEP 3. Module 9 : Robot Dynamics & controls
Module 9 : Robot Dynamics & controls Lecture 32 : General procedure for dynamics equation forming and introduction to control Objectives In this course you will learn the following Lagrangian Formulation
More informationELEC4631 s Lecture 2: Dynamic Control Systems 7 March Overview of dynamic control systems
ELEC4631 s Lecture 2: Dynamic Control Systems 7 March 2011 Overview of dynamic control systems Goals of Controller design Autonomous dynamic systems Linear Multi-input multi-output (MIMO) systems Bat flight
More informationManipulator Dynamics 2. Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Manipulator Dynamics 2 Forward Dynamics Problem Given: Joint torques and links geometry, mass, inertia, friction Compute: Angular acceleration of the links (solve differential equations) Solution Dynamic
More informationDr Ian R. Manchester
Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the s-plane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus 7 Root Locus 2 Assign
More informationECEn 483 / ME 431 Case Studies. Randal W. Beard Brigham Young University
ECEn 483 / ME 431 Case Studies Randal W. Beard Brigham Young University Updated: December 2, 2014 ii Contents 1 Single Link Robot Arm 1 2 Pendulum on a Cart 9 3 Satellite Attitude Control 17 4 UUV Roll
More informationAdaptive Tracking and Parameter Estimation with Unknown High-Frequency Control Gains: A Case Study in Strictification
Adaptive Tracking and Parameter Estimation with Unknown High-Frequency Control Gains: A Case Study in Strictification Michael Malisoff, Louisiana State University Joint with Frédéric Mazenc and Marcio
More informationRobotics & Automation. Lecture 06. Serial Kinematic Chain, Forward Kinematics. John T. Wen. September 11, 2008
Robotics & Automation Lecture 06 Serial Kinematic Chain, Forward Kinematics John T. Wen September 11, 2008 So Far... We have covered rigid body rotational kinematics: representations of SO(3), change of
More informationDesign and Control of Variable Stiffness Actuation Systems
Design and Control of Variable Stiffness Actuation Systems Gianluca Palli, Claudio Melchiorri, Giovanni Berselli and Gabriele Vassura DEIS - DIEM - Università di Bologna LAR - Laboratory of Automation
More informationContents. Dynamics and control of mechanical systems. Focus on
Dynamics and control of mechanical systems Date Day 1 (01/08) Day 2 (03/08) Day 3 (05/08) Day 4 (07/08) Day 5 (09/08) Day 6 (11/08) Content Review of the basics of mechanics. Kinematics of rigid bodies
More informationRigid Manipulator Control
Rigid Manipulator Control The control problem consists in the design of control algorithms for the robot motors, such that the TCP motion follows a specified task in the cartesian space Two types of task
More informationNonlinear PD Controllers with Gravity Compensation for Robot Manipulators
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 4, No Sofia 04 Print ISSN: 3-970; Online ISSN: 34-408 DOI: 0.478/cait-04-00 Nonlinear PD Controllers with Gravity Compensation
More informationControl of Mobile Robots Prof. Luca Bascetta
Control of Mobile Robots Prof. Luca Bascetta EXERCISE 1 1. Consider a wheel rolling without slipping on the horizontal plane, keeping the sagittal plane in the vertical direction. Write the expression
More informationDynamics Algorithms for Multibody Systems
International Conference on Multi Body Dynamics 2011 Vijayawada, India. pp. 351 365 Dynamics Algorithms for Multibody Systems S. V. Shah, S. K. Saha and J. K. Dutt Department of Mechanical Engineering,
More informationROBOTICS: ADVANCED CONCEPTS & ANALYSIS
ROBOTICS: ADVANCED CONCEPTS & ANALYSIS MODULE 5 VELOCITY AND STATIC ANALYSIS OF MANIPULATORS Ashitava Ghosal 1 1 Department of Mechanical Engineering & Centre for Product Design and Manufacture Indian
More informationArtificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik. Robot Dynamics. Dr.-Ing. John Nassour J.
Artificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik Robot Dynamics Dr.-Ing. John Nassour 25.1.218 J.Nassour 1 Introduction Dynamics concerns the motion of bodies Includes Kinematics
More informationChapter 3 Numerical Methods
Chapter 3 Numerical Methods Part 3 3.4 Differential Algebraic Systems 3.5 Integration of Differential Equations 1 Outline 3.4 Differential Algebraic Systems 3.4.1 Constrained Dynamics 3.4.2 First and Second
More informationLecture Note 12: Dynamics of Open Chains: Lagrangian Formulation
ECE5463: Introduction to Robotics Lecture Note 12: Dynamics of Open Chains: Lagrangian Formulation Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio,
More informationRobot Manipulator Control. Hesheng Wang Dept. of Automation
Robot Manipulator Control Hesheng Wang Dept. of Automation Introduction Industrial robots work based on the teaching/playback scheme Operators teach the task procedure to a robot he robot plays back eecute
More informationEE Homework 3 Due Date: 03 / 30 / Spring 2015
EE 476 - Homework 3 Due Date: 03 / 30 / 2015 Spring 2015 Exercise 1 (10 points). Consider the problem of two pulleys and a mass discussed in class. We solved a version of the problem where the mass was
More informationExponential Controller for Robot Manipulators
Exponential Controller for Robot Manipulators Fernando Reyes Benemérita Universidad Autónoma de Puebla Grupo de Robótica de la Facultad de Ciencias de la Electrónica Apartado Postal 542, Puebla 7200, México
More information557. Radial correction controllers of gyroscopic stabilizer
557. Radial correction controllers of gyroscopic stabilizer M. Sivčák 1, J. Škoda, Technical University in Liberec, Studentská, Liberec, Czech Republic e-mail: 1 michal.sivcak@tul.cz; jan.skoda@pevnosti.cz
More informationDecentralized PD Control for Non-uniform Motion of a Hamiltonian Hybrid System
International Journal of Automation and Computing 05(2), April 2008, 9-24 DOI: 0.007/s633-008-09-7 Decentralized PD Control for Non-uniform Motion of a Hamiltonian Hybrid System Mingcong Deng, Hongnian
More informationRobotics 2 Robot Interaction with the Environment
Robotics 2 Robot Interaction with the Environment Prof. Alessandro De Luca Robot-environment interaction a robot (end-effector) may interact with the environment! modifying the state of the environment
More informationLecture 9 Nonlinear Control Design
Lecture 9 Nonlinear Control Design Exact-linearization Lyapunov-based design Lab 2 Adaptive control Sliding modes control Literature: [Khalil, ch.s 13, 14.1,14.2] and [Glad-Ljung,ch.17] Course Outline
More informationDynamics of Open Chains
Chapter 9 Dynamics of Open Chains According to Newton s second law of motion, any change in the velocity of a rigid body is caused by external forces and torques In this chapter we study once again the
More informationTrajectory Planning, Setpoint Generation and Feedforward for Motion Systems
2 Trajectory Planning, Setpoint Generation and Feedforward for Motion Systems Paul Lambrechts Digital Motion Control (4K4), 23 Faculty of Mechanical Engineering, Control Systems Technology Group /42 2
More informationEN Nonlinear Control and Planning in Robotics Lecture 2: System Models January 28, 2015
EN53.678 Nonlinear Control and Planning in Robotics Lecture 2: System Models January 28, 25 Prof: Marin Kobilarov. Constraints The configuration space of a mechanical sysetm is denoted by Q and is assumed
More informationDesign and Control of Compliant Humanoids. Alin Albu-Schäffer. DLR German Aerospace Center Institute of Robotics and Mechatronics
Design and Control of Compliant Humanoids Alin Albu-Schäffer DLR German Aerospace Center Institute of Robotics and Mechatronics Torque Controlled Light-weight Robots Torque sensing in each joint Mature
More informationDecoupling Identification for Serial Two-link Robot Arm with Elastic Joints
Preprints of the 1th IFAC Symposium on System Identification Saint-Malo, France, July 6-8, 9 Decoupling Identification for Serial Two-link Robot Arm with Elastic Joints Junji Oaki, Shuichi Adachi Corporate
More information, respectively to the inverse and the inverse differential problem. Check the correctness of the obtained results. Exercise 2 y P 2 P 1.
Robotics I July 8 Exercise Define the orientation of a rigid body in the 3D space through three rotations by the angles α β and γ around three fixed axes in the sequence Y X and Z and determine the associated
More informationIntegrator Backstepping using Barrier Functions for Systems with Multiple State Constraints
Integrator Backstepping using Barrier Functions for Systems with Multiple State Constraints Khoi Ngo Dep. Engineering, Australian National University, Australia Robert Mahony Dep. Engineering, Australian
More informationNatural and artificial constraints
FORCE CONTROL Manipulator interaction with environment Compliance control Impedance control Force control Constrained motion Natural and artificial constraints Hybrid force/motion control MANIPULATOR INTERACTION
More informationIntroduction to Haptic Systems
Introduction to Haptic Systems Félix Monasterio-Huelin & Álvaro Gutiérrez & Blanca Larraga October 8, 2018 Contents Contents 1 List of Figures 1 1 Introduction 2 2 DC Motor 3 3 1 DOF DC motor model with
More informationFigure 5.28 (a) Spring-restrained cylinder, (b) Kinematic variables, (c) Free-body diagram
Lecture 30. MORE GENERAL-MOTION/ROLLING- WITHOUT-SLIPPING EXAMPLES A Cylinder, Restrained by a Spring and Rolling on a Plane Figure 5.28 (a) Spring-restrained cylinder, (b) Kinematic variables, (c) Free-body
More informationCh. 5: Jacobian. 5.1 Introduction
5.1 Introduction relationship between the end effector velocity and the joint rates differentiate the kinematic relationships to obtain the velocity relationship Jacobian matrix closely related to the
More informationDOUBLE ARM JUGGLING SYSTEM Progress Presentation ECSE-4962 Control Systems Design
DOUBLE ARM JUGGLING SYSTEM Progress Presentation ECSE-4962 Control Systems Design Group Members: John Kua Trinell Ball Linda Rivera Introduction Where are we? Bulk of Design and Build Complete Testing
More informationAdvanced Dynamics. - Lecture 4 Lagrange Equations. Paolo Tiso Spring Semester 2017 ETH Zürich
Advanced Dynamics - Lecture 4 Lagrange Equations Paolo Tiso Spring Semester 2017 ETH Zürich LECTURE OBJECTIVES 1. Derive the Lagrange equations of a system of particles; 2. Show that the equation of motion
More informationLecture Note 8: Inverse Kinematics
ECE5463: Introduction to Robotics Lecture Note 8: Inverse Kinematics Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio, USA Spring 2018 Lecture 8 (ECE5463
More informationProblem 1: Ship Path-Following Control System (35%)
Problem 1: Ship Path-Following Control System (35%) Consider the kinematic equations: Figure 1: NTNU s research vessel, R/V Gunnerus, and Nomoto model: T ṙ + r = Kδ (1) with T = 22.0 s and K = 0.1 s 1.
More informationReduced-order Forward Dynamics of Multi-Closed-Loop Systems
Noname manuscript No. (will be inserted by the editor) Reduced-order Forward Dynamics of Multi-Closed-Loop Systems Majid Koul Suril V Shah S K Saha M Manivannan the date of receipt and acceptance should
More informationPhysics 106a, Caltech 4 December, Lecture 18: Examples on Rigid Body Dynamics. Rotating rectangle. Heavy symmetric top
Physics 106a, Caltech 4 December, 2018 Lecture 18: Examples on Rigid Body Dynamics I go through a number of examples illustrating the methods of solving rigid body dynamics. In most cases, the problem
More informationThe Jacobian. Jesse van den Kieboom
The Jacobian Jesse van den Kieboom jesse.vandenkieboom@epfl.ch 1 Introduction 1 1 Introduction The Jacobian is an important concept in robotics. Although the general concept of the Jacobian in robotics
More informationGeometric Mechanics and Global Nonlinear Control for Multi-Body Dynamics
Geometric Mechanics and Global Nonlinear Control for Multi-Body Dynamics Harris McClamroch Aerospace Engineering, University of Michigan Joint work with Taeyoung Lee (George Washington University) Melvin
More informationCase Study: The Pelican Prototype Robot
5 Case Study: The Pelican Prototype Robot The purpose of this chapter is twofold: first, to present in detail the model of the experimental robot arm of the Robotics lab. from the CICESE Research Center,
More informationInverse differential kinematics Statics and force transformations
Robotics 1 Inverse differential kinematics Statics and force transformations Prof Alessandro De Luca Robotics 1 1 Inversion of differential kinematics! find the joint velocity vector that realizes a desired
More informationControl of Robot. Ioannis Manganas MCE Master Thesis. Aalborg University Department of Energy Technology
Control of Robot Master Thesis Ioannis Manganas MCE4-3 Aalborg University Department of Energy Technology Copyright c Aalborg University 8 LATEXhas been used for typesetting this document, using the TeXstudio
More information(r i F i ) F i = 0. C O = i=1
Notes on Side #3 ThemomentaboutapointObyaforceF that acts at a point P is defined by M O (r P r O F, where r P r O is the vector pointing from point O to point P. If forces F, F, F 3,..., F N act on particles
More informationNONLINEAR MECHANICAL SYSTEMS (MECHANISMS)
NONLINEAR MECHANICAL SYSTEMS (MECHANISMS) The analogy between dynamic behavior in different energy domains can be useful. Closer inspection reveals that the analogy is not complete. One key distinction
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.04A Systems and Controls Spring 2013
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.04A Systems and Controls Spring 2013 Problem Set #4 Posted: Thursday, Mar. 7, 13 Due: Thursday, Mar. 14, 13 1. Sketch the Root
More informationControl of Manufacturing Processes
Control of Manufacturing Processes Subject 2.830 Spring 2004 Lecture #18 Basic Control Loop Analysis" April 15, 2004 Revisit Temperature Control Problem τ dy dt + y = u τ = time constant = gain y ss =
More informationRobotics I. April 1, the motion starts and ends with zero Cartesian velocity and acceleration;
Robotics I April, 6 Consider a planar R robot with links of length l = and l =.5. he end-effector should move smoothly from an initial point p in to a final point p fin in the robot workspace so that the
More informationModeling and Simulation of the Nonlinear Computed Torque Control in Simulink/MATLAB for an Industrial Robot
Copyright 2013 Tech Science Press SL, vol.10, no.2, pp.95-106, 2013 Modeling and Simulation of the Nonlinear Computed Torque Control in Simulink/MATLAB for an Industrial Robot Dǎnuţ Receanu 1 Abstract:
More informationPassivity-based Control of Euler-Lagrange Systems
Romeo Ortega, Antonio Loria, Per Johan Nicklasson and Hebertt Sira-Ramfrez Passivity-based Control of Euler-Lagrange Systems Mechanical, Electrical and Electromechanical Applications Springer Contents
More informationPlanar Multi-body Dynamics of a Tracked Vehicle using Imaginary Wheel Model for Tracks
Defence Science Journal, Vol. 67, No. 4, July 2017, pp. 460-464, DOI : 10.14429/dsj.67.11548 2017, DESIDOC Planar Multi-body Dynamics of a Tracked Vehicle using Imaginary Wheel Model for Tracks Ilango
More informationTrajectory Planning from Multibody System Dynamics
Trajectory Planning from Multibody System Dynamics Pierangelo Masarati Politecnico di Milano Dipartimento di Ingegneria Aerospaziale Manipulators 2 Manipulator: chain of
More informationOn-line Learning of Robot Arm Impedance Using Neural Networks
On-line Learning of Robot Arm Impedance Using Neural Networks Yoshiyuki Tanaka Graduate School of Engineering, Hiroshima University, Higashi-hiroshima, 739-857, JAPAN Email: ytanaka@bsys.hiroshima-u.ac.jp
More informationarxiv: v1 [cs.ro] 3 Jul 2017
A Projected Inverse Dynamics Approach for Dual-arm Cartesian Impedance Control Hsiu-Chin Lin, Joshua Smith, Keyhan Kouhkiloui Babarahmati, Niels Dehio, and Michael Mistry arxiv:77.484v [cs.ro] 3 Jul 7
More informationChapter 4 Statics and dynamics of rigid bodies
Chapter 4 Statics and dynamics of rigid bodies Bachelor Program in AUTOMATION ENGINEERING Prof. Rong-yong Zhao (zhaorongyong@tongji.edu.cn) First Semester,2014-2015 Content of chapter 4 4.1 Static equilibrium
More informationA DAE approach to Feedforward Control of Flexible Manipulators
27 IEEE International Conference on Robotics and Automation Roma, Italy, 1-14 April 27 FrA1.3 A DAE approach to Feedforward Control of Flexible Manipulators Stig Moberg and Sven Hanssen Abstract This work
More informationSolving high order nonholonomic systems using Gibbs-Appell method
Solving high order nonholonomic systems using Gibbs-Appell method Mohsen Emami, Hassan Zohoor and Saeed Sohrabpour Abstract. In this paper we present a new formulation, based on Gibbs- Appell method, for
More informationChapter 7 Control. Part Classical Control. Mobile Robotics - Prof Alonzo Kelly, CMU RI
Chapter 7 Control 7.1 Classical Control Part 1 1 7.1 Classical Control Outline 7.1.1 Introduction 7.1.2 Virtual Spring Damper 7.1.3 Feedback Control 7.1.4 Model Referenced and Feedforward Control Summary
More informationRobotics, Geometry and Control - A Preview
Robotics, Geometry and Control - A Preview Ravi Banavar 1 1 Systems and Control Engineering IIT Bombay HYCON-EECI Graduate School - Spring 2008 Broad areas Types of manipulators - articulated mechanisms,
More informationModelling and Control of Variable Stiffness Actuated Robots
Modelling and Control of Variable Stiffness Actuated Robots Sabira Jamaludheen 1, Roshin R 2 P.G. Student, Department of Electrical and Electronics Engineering, MES College of Engineering, Kuttippuram,
More information