Lecture «Robot Dynamics» : Kinematics 3
|
|
- Gabriella Marshall
- 5 years ago
- Views:
Transcription
1 Lecture «Robot Dynamics» : Kinematics V lecture: CAB G11 Tuesday 10:15-12:00, every week exercise: HG G1 Wednesday 8:15-10:00, according to schedule (about every 2nd week) office hour: LEE H303 Friday Marco Hutter, Roland Siegwart and Thomas Stastny Robot Dynamics - Kinematics
2 Outline Kinematic control methods Inverse kinematics Singularities, redundancy Multi-task control Iterative inverse differential kinematics Kinematic trajectory control 2
3 What we saw so far Relative pose between frame coordinate frames 3
4 What we saw so far Relative pose between frame coordinate frames [ ] CAB T AB = A r AB 0 T 1 4
5 What we saw so far Relative pose between frame coordinate frames T AB = [ CAB ] A r AB 0 T 1 5
6 What we saw so far Forward kinematics Description of end-effector configuration (position & orientation) as a function of joint coordinates 6
7 What we saw so far Forward kinematics Description of end-effector configuration (position & orientation) as a function of joint coordinates Use the homogeneous transformation matrix 7
8 What we saw so far Forward kinematics Description of end-effector configuration (position & orientation) as a function of joint coordinates {E} Use the homogeneous transformation matrix T IE (q) = [ CIE (q) Ir IE (q) 0 T 1 ] R 4 4 {I} Forward* Kinematics Joint*angles 8
9 What we saw so far Forward kinematics Description of end-effector configuration (position & orientation) as a function of joint coordinates {E} Use the homogeneous transformation matrix T IE (q) = [ CIE (q) Ir IE (q) 0 T 1 Parametrized description [ ] rie (q) x E = = f(q) ϕ IE (q) ] R 4 4 {I} Forward* Kinematics Joint*angles 9
10 What we saw so far Inverse kinematics Description of joint angles as a function of the end-effector configuration Use the homogeneous transformation matrix [ ] CIE (q) T IE (q) = Ir IE (q) 0 T R Parametrized description [ ] rie (q) x E = = f(q) q = f 1 (x ϕ IE (q) E ) {I} {E} Configuration end7effector Inverse* Kinematics Joint*angles 10
11 Closed form solutions Geometric or Algebra Analytic solutions exist for a large class of mechanisms 3 intersecting neighbouring axes (most industrial robots) 11
12 Closed form solutions Geometric or Algebraic Analytic solutions exist for a large class of mechanisms Geometric 3 intersecting neighbouring axes (most industrial robots) Decompose spatial geometry of manipulator into several plane problems and apply geometric laws 12
13 Closed form solutions Geometric or Algebraic Analytic solutions exist for a large class of mechanisms Geometric 3 intersecting neighbouring axes (most industrial robots) Decompose spatial geometry of manipulator into several plane problems and apply geometric laws Algebraic Manipulate transformation matrix equation to get the joint angles TIE = T01 (! 1) T12 (! 2 ) T23 (! 3 ) T34 (! 4 ) T45 (! 5 ) T56 (! 6 ) " (! ) 1 = (! ) (! ) (! ) (! ) (! ) T T T T T T T 01 1 IE " 1 ( T01 (! 1) T12 (! 2 )) TIE = T23 (! 3 ) T34 (! 4 ) T45 (! 5 ) T56 (! 6 ) " 1 ( T01 (! 1) T12 (! 2 ) T23 (! 3 )) TIE = T34 (! 4 ) T45 (! 5 ) T56 (! 6 ) 13
14 Inverse Differential Kinematics We have seen how Jacobians map velocities from joint space to task-space w e = J e0 q In general, we are interested in the inverse problem Simple method: use the pseudoinverse q = J + e0 w e however, the Jacobian might be singular! 14
15 Singularities A singurality is a joint-space configuration q s such that J e0 (q s ) is column-rank deficient the Jacobian becomes badly conditioned w e small desired velocities produce high joint velocities q Singularities can be classified into: boundary (e.g. a stretched out manipulator) easy to avoid during motion planning internal harder to prevent, requires careful motion planning 15
16 Singularities A singurality is a joint-space configuration q s such that J e0 (q s ) is column-rank deficient the Jacobian becomes badly conditioned w e small desired velocities produce high joint velocities q Use a damped version of the Moore-Penrose pseudo inverse q = J T e0(j e0 J T e0 + λ 2 I) 1 w e λ>0,λ R 16
17 Singularities A singurality is a joint-space configuration q s such that J e0 (q s ) is column-rank deficient the Jacobian becomes badly conditioned w e small desired velocities produce high joint velocities q Use a damped version of the Moore-Penrose pseudo inverse q = J T e0(j e0 J T e0 + λ 2 I) 1 w e min w e J e0 q 2 + λ 2 q 2 λ>0,λ R 17
18 Redundancy A kinematic structure is redundant if the dimension of the task-space is smaller than the dimension of the joint-space E.g. the human arm has 7DoF (three in the shoulder, one in the elbow, and three in the wrist) 18
19 Redundancy A kinematic structure is redundant if the dimension of the task-space is smaller than the dimension of the joint-space E.g. the human arm has 7DoF (three in the shoulder, one in the elbow, and three in the wrist) q R 7 19
20 Redundancy A kinematic structure is redundant if the dimension of the task-space is smaller than the dimension of the joint-space E.g. the human arm has 7DoF (three in the shoulder, one in the elbow, and three in the wrist) q R 7 w R 6 20
21 Redundancy A kinematic structure is redundant if the dimension of the task-space is smaller than the dimension of the joint-space E.g. the human arm has 7DoF (three in the shoulder, one in the elbow, and three in the wrist) q R 7 w R 6 J e0 R
22 Redundancy A kinematic structure is redundant if the dimension of the task-space is smaller than the dimension of the joint-space E.g. the human arm has 7DoF (three in the shoulder, one in the elbow, and three in the wrist) q R 7 w R 6 J e0 R 6 7 Redundancy implies infinite solutions q = J + e0 w e 22
23 Redundancy A kinematic structure is redundant if the dimension of the task-space is smaller than the dimension of the joint-space E.g. the human arm has 7DoF (three in the shoulder, one in the elbow, and three in the wrist) q R 7 w R 6 J e0 R 6 7 Redundancy implies infinite solutions q = J + e0 w e + N q 0 N = N (J e0 ) J e0 N = 0 23
24 Redundancy A kinematic structure is redundant if the dimension of the task-space is smaller than the dimension of the joint-space E.g. the human arm has 7DoF (three in the shoulder, one in the elbow, and three in the wrist) q R 7 w R 6 J e0 R 6 7 Redundancy implies infinite solutions q = J + e0 w e + N q 0 N = N (J e0 ) J e0 N = 0 J e0 (J + e0 w e + N q 0 )=w e 24
25 Redundancy A kinematic structure is redundant if the dimension of the task-space is smaller than the dimension of the joint-space E.g. the human arm has 7DoF (three in the shoulder, one in the elbow, and three in the wrist) q R 7 w R 6 J e0 R 6 7 Redundancy implies infinite solutions q = J + e0 w e + N q 0 N = N (J e0 ) J e0 N = 0 One way to compute the nullspace projection matrix N = I J + e0 J e0 J e0 (J + e0 w e + N q 0 )=w e 25
26 Multi-task control Manipulation (as well as locomotion! ) is a complex combination of high level tasks track a desired position ensure kinematic constraints reach a desired end-effector orientation Break down the complexity into smaller tasks Two methods Multi-task with equal priority task i := {J i, wi } Multi-task with Prioritization 26
27 Multi-task control Equal priority Assume that t tasks have been defined The generalised velocity is given by q = J 1.. J nt + } {{ } J w 1.. w n t } {{ } w The pseudo inversion will try to solve all tasks at the same time in an optimal way It is possible to weigh some tasks higher than others J +W = ( JT W J ) 1 JT W W = diag(w 1,...,w m ) 27
28 Multi-task control Prioritization Instead of solving all tasks at once, we can use consecutive nullspace projection to ensure a strict priority We already saw that q = J + 1 w 1 + N 1 q 0 The solution for task 2 should not violate the one found for task 1 ( w 2 = J 2 q = J 2 J + 1 w1 ) + N 1 q 0 This can be solved for Back substituting yields The iterative solution for T tasks is then given by q = n T i=1 q 0 N i q i, with q i =(J i N i ) + (w i J q 0 =(J 2 N 1 ) + ( w2 J 2 J + ) 1 w q = J + 1 w 1 + N 1 (J 2 N 1 ) + ( 1 w2 J 2 J + ) 1 w 1 i 1 k=1 N k q k ) 28
29 Multi-task control Example - single task 3DoF planar robot arm with unitary link lengths Find the generalised velocities, given q t =(π/6,π/3,π/3) T 0ṙ E,t =(1, 1) T l 1 c 1 + l 2 c 12 + l 3 c 123 l 2 c 12 + l 3 c 123 l 3 c 123 IJ e0p = l 1 s 1 l 2 s 12 l 3 c 123 l 2 s 12 l 3 s 123 l 3 s 123 ϕ 1 l 1 e 1 z ϕ 3 l 3 e 3 e l E z 2 z ϕ 2 e 2 z e 3 e E v e x x e 2 x e 1 x l 0 e 0 z = e I z e 0 x = e I x 29
30 Multi-task control Example - single task 3DoF planar robot arm with unitary link lengths Find the generalised velocities, given q t =(π/6,π/3,π/3) T 0ṙ E,t =(1, 1) T l 1 c 1 + l 2 c 12 + l 3 c 123 l 2 c 12 + l 3 c 123 l 3 c 123 IJ e0p = l 1 s 1 l 2 s 12 l 3 c 123 l 2 s 12 l 3 s 123 l 3 s 123 ϕ 1 l 1 e 1 z ϕ 3 l 3 e 3 e l E z 2 z ϕ 2 e 2 z e 3 e E v e x x e 2 x w E = 0 r E = [ ] 1 1 q t = J + E (q t)w E,t = ( [ + [ ] = ]) l 0 e 1 x e 0 z = e I z e 0 x = e I x 30
31 Multi-task control Example - two tasks 3DoF planar robot arm with unitary link lengths Find the generalised velocities, given q t =(π/6,π/3,π/3) T 0ṙ E,t =(1, 1) T IJ e0p = l 1 c 1 + l 2 c 12 + l 3 c 123 l 2 c 12 + l 3 c 123 l 3 c l 1 s 1 l 2 s 12 l 3 c 123 l 2 s 12 l 3 s 123 l 3 s 123 Additionally, we want to fulfill a second task with the same priority as the first, namely that the first and third joint velocities are zero ϕ 1 l 1 e 1 z ϕ 3 l 3 e 3 e l E z 2 z ϕ 2 e 2 z e 3 e E v e x x e 2 x e 1 x l 0 e 0 z = e I z e 0 x = e I x 31
32 Multi-task control Example - two tasks 3DoF planar robot arm with unitary link lengths Find the generalised velocities, given q t =(π/6,π/3,π/3) T 0ṙ E,t =(1, 1) T IJ e0p = Additionally, we want to fulfill a second task with the same priority as the first, namely that the first and third joint velocities are zero w j = [ 0 q = [ JE J j 0 ] T ] + [ ] w E wj J j = [ 1 0 ] l 1 c 1 + l 2 c 12 + l 3 c 123 l 2 c 12 + l 3 c 123 l 3 c l 1 s 1 l 2 s 12 l 3 c 123 l 2 s 12 l 3 s 123 l 3 s 123 l 0 ϕ 3 l 3 e 3 e l E z 2 z ϕ 2 e 2 z e 3 e E v e x x l 1 ϕ 1 e 2 e 1 x z e 1 x e 0 z = e I z e 0 x = e I x 32
33 Prioritized multi-task control Quadrupedal locomotion Main body pose Kinematic limits Foot motion tracking Friction cone constraints Constraint points are not allowed to move 33
34 Prioritized multi-task control Quadrupedal locomotion 34
35 Mapping associated with the Jacobian 35
36 Numerical solutions Inverse differential kinematics Jacobians map joint-space velocities to end-effector velocities χ e = J ea (q) q 36
37 Numerical solutions Inverse differential kinematics Jacobians map joint-space velocities to end-effector velocities χ e = J ea (q) q We can use this to iteratively solve the inverse kinematics problem target configuration χ e, initial joint space guess q 0 1. q q 0 start configuration 2. while χ e χ e (q) tol do while the solution is not reached 3. J ea J ea (q) = χ e q (q) evaluate Jacobian 4. J + ea (J ea(q)) + compute the pseudo inverse 5. χ e χ e χ e (q) find the end-effector configuration error vector 6. q q + J + ea χ e updated the generalized coordinates 37
38 Numerical solutions Inverse differential kinematics Jacobians map joint-space velocities to end-effector velocities χ e = J ea (q) q We can use this to iteratively solve the inverse kinematics problem target configuration χ e, initial joint space guess q 0 1. q q 0 start configuration 2. while χ e χ e (q) tol do while the solution is not reached 3. J ea J ea (q) = χ e q (q) evaluate Jacobian 4. J + ea (J ea(q)) + compute the pseudo inverse 5. χ e χ e χ e (q) find the end-effector configuration error vector 6. q q + J + ea χ e updated the generalized coordinates 38
39 Inverse kinematics Three-link arm example Determine end-effector Jacobian 39
40 Inverse kinematics Three-link arm example Determine end-effector Jacobian 1. Introduce coordinate frames 40
41 Inverse kinematics Three-link arm example Determine end-effector Jacobian 1. Introduce coordinate frames 2. Introduce generalized coordinates 41
42 Inverse kinematics Three-link arm example Determine end-effector Jacobian 1. Introduce coordinate frames 2. Introduce generalized coordinates 3. Determine end-effector position l 0 + l 1 cos(q 1 )+l 2 cos(q 1 + q 2 )+l 3 cos(q 1 + q 2 + q 3 ) 0r 0E (q) = l 1 sin(q 1 )+l 2 sin(q 1 + q 2 )+l 3 sin(q 1 + q 2 + q 3 ) 0 42
43 Inverse kinematics Three-link arm example Determine end-effector Jacobian 1. Introduce coordinate frames 2. Introduce generalized coordinates 3. Determine end-effector position l 0 + l 1 cos(q 1 )+l 2 cos(q 1 + q 2 )+l 3 cos(q 1 + q 2 + q 3 ) 0r 0E (q) = l 1 sin(q 1 )+l 2 sin(q 1 + q 2 )+l 3 sin(q 1 + q 2 + q 3 ) 0 4. Compute the Jacobian 0J ep = q 0 r 0E (q) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )!# l1sin q1 # l2sin q1+ q2 # l3sin q1+ q2 + q3 # l2sin q1+ q2 # l3sin q1+ q2 + q3 # l3sin q1+ q2 + q3 " $ % = $ l1cos q1 + l2cos q1+ q2 + l3cos q1+ q2 + q3 l2cos q1+ q2 + l3cos q1+ q2 + q3 l3cos q1+ q2 + q3 % $ & % ' 43
44 Inverse kinematics Three-link arm example Iterative inverse kinematics to find desired configuration q i+1 = q i + J + ep (r goal r i ) 44
45 Inverse kinematics Three-link arm example Iterative inverse kinematics to find desired configuration q i+1 = q i + J + ep (r goal r i ) start value 0 q 0 =
46 Inverse kinematics Three-link arm example Iterative inverse kinematics to find desired configuration q i+1 = q i + J + ep (r goal r i ) start value π/2 q 0 =
47 Inverse kinematics Three-link arm example Iterative inverse kinematics to find desired configuration q i+1 = q i + J + ep (r goal r i ) start value π/2 q 0 = π/2 0 Same goal position, multiple solutions joint-space bigger than task-space, redundant system 47
48 Inverse kinematics Iterative methods Let s have a closer look at the joint update rule q i+1 = q i + J + ea χ 48
49 Inverse kinematics Iterative methods Let s have a closer look at the joint update rule q i+1 = q i + J + ea χ Two main issues Scaling if the current error is too large, the error linearization implemented by the jacobian is not accurate enough use a scaling factor 0 <k<1 q i+1 = q i + kj + ea χ unfortunately, this will lead to slower convergence 49
50 Inverse kinematics Iterative methods Let s have a closer look at the joint update rule q i+1 = q i + J + ea χ Two main issues Singular configurations When the Jacobian is rank-deficient, the inversion becomes a badly conditioned problem Use the damped pseudoinverse (Levenberg-Marquardt) q i+1 = q i + J T ea(j ea J T ea + λ 2 I) 1 χ Use the transpose of the Jacobian q i+1 = q i + αj T ea χ For a detailed explanation, check Introduction to Inverse Kinematics with Jacobian Transpose, Pseudoinverse and Damped Least Squares methods, Samuel Buss,
51 Inverse differential kinematics Orientation error 3D rotations are defined in the Special Orthogonal group SO(3) The parametrization affects convergence from start to goal orientation Rotate along shortest path in SO(3): use rotational vectors which parametrize rotation from start to goal χ rotvec = ϕ = C GS ( ϕ) =C GI (ϕ )C T SI(ϕ t ) The update law for rotations will then be q q + k PR J + e0 R ϕ This is NOT the difference between rotation vectors, but the rotation vector extracted from the relative rotation between start and goal 51
52 Trajectory control Position Consider a planned desired motion of the end effector r e(t) ṙ e(t) Let s see how to kinematically control the end-effector position Feedback term r t e = r e(t) r e (q t ) We can design a nonlinear stabilizing controller law q = J + e0 P (ṙ + k PP r t e) If we substitute this into the differential kinematics equation, we get w e = J ep q = ṙ + k PP r t e = ṙ t e + k PP r t e = 0 Stable error dynamics for positive k PP 52
53 Trajectory control Orientation Derivation more involved Final control law similar to the position case q = J + e0 R (ω(t) e + k PR ϕ) Note that we are not using the analytical Jacobian since we are dealing with angular velocities and rotational vectors 53
Robot Dynamics Instantaneous Kinematiccs and Jacobians
Robot Dynamics Instantaneous Kinematiccs and Jacobians 151-0851-00 V Lecture: Tuesday 10:15 12:00 CAB G11 Exercise: Tuesday 14:15 16:00 every 2nd week Marco Hutter, Michael Blösch, Roland Siegwart, Konrad
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 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 informationLecture «Robot Dynamics»: Kinematics 2
Lecture «Robot Dynamics»: Kinematics 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) Marco Hutter,
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 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 informationDifferential Kinematics
Differential Kinematics Relations between motion (velocity) in joint space and motion (linear/angular velocity) in task space (e.g., Cartesian space) Instantaneous velocity mappings can be obtained through
More informationExercise 1b: Differential Kinematics of the ABB IRB 120
Exercise 1b: Differential Kinematics of the ABB IRB 120 Marco Hutter, Michael Blösch, Dario Bellicoso, Samuel Bachmann October 5, 2016 Abstract The aim of this exercise is to calculate the differential
More informationGiven U, V, x and θ perform the following steps: a) Find the rotation angle, φ, by which u 1 is rotated in relation to x 1
1 The Jacobian can be expressed in an arbitrary frame, such as the base frame located at the first joint, the hand frame located at the end-effector, or the global frame located somewhere else. The SVD
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 informationRobotics & Automation. Lecture 17. Manipulability Ellipsoid, Singularities of Serial Arm. John T. Wen. October 14, 2008
Robotics & Automation Lecture 17 Manipulability Ellipsoid, Singularities of Serial Arm John T. Wen October 14, 2008 Jacobian Singularity rank(j) = dimension of manipulability ellipsoid = # of independent
More informationRobotics 1 Inverse kinematics
Robotics 1 Inverse kinematics Prof. Alessandro De Luca Robotics 1 1 Inverse kinematics what are we looking for? direct kinematics is always unique; how about inverse kinematics for this 6R robot? Robotics
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 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 informationRobotics 1 Inverse kinematics
Robotics 1 Inverse kinematics Prof. Alessandro De Luca Robotics 1 1 Inverse kinematics what are we looking for? direct kinematics is always unique; how about inverse kinematics for this 6R robot? Robotics
More informationMEAM 520. More Velocity Kinematics
MEAM 520 More Velocity Kinematics Katherine J. Kuchenbecker, Ph.D. General Robotics, Automation, Sensing, and Perception Lab (GRASP) MEAM Department, SEAS, University of Pennsylvania Lecture 12: October
More informationRobotics 1 Inverse kinematics
Robotics 1 Inverse kinematics Prof. Alessandro De Luca Robotics 1 1 Inverse kinematics what are we looking for? direct kinematics is always unique; how about inverse kinematics for this 6R robot? Robotics
More informationPosition and orientation of rigid bodies
Robotics 1 Position and orientation of rigid bodies Prof. Alessandro De Luca Robotics 1 1 Position and orientation right-handed orthogonal Reference Frames RF A A p AB B RF B rigid body position: A p AB
More informationDIFFERENTIAL KINEMATICS. Geometric Jacobian. Analytical Jacobian. Kinematic singularities. Kinematic redundancy. Inverse differential kinematics
DIFFERENTIAL KINEMATICS relationship between joint velocities and end-effector velocities Geometric Jacobian Analytical Jacobian Kinematic singularities Kinematic redundancy Inverse differential kinematics
More informationRobotics: Tutorial 3
Robotics: Tutorial 3 Mechatronics Engineering Dr. Islam Khalil, MSc. Omar Mahmoud, Eng. Lobna Tarek and Eng. Abdelrahman Ezz German University in Cairo Faculty of Engineering and Material Science October
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 informationROBOTICS 01PEEQW. Basilio Bona DAUIN Politecnico di Torino
ROBOTICS 01PEEQW Basilio Bona DAUIN Politecnico di Torino Kinematic Functions Kinematic functions Kinematics deals with the study of four functions(called kinematic functions or KFs) that mathematically
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 information8 Velocity Kinematics
8 Velocity Kinematics Velocity analysis of a robot is divided into forward and inverse velocity kinematics. Having the time rate of joint variables and determination of the Cartesian velocity of end-effector
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 informationNumerical Methods for Inverse Kinematics
Numerical Methods for Inverse Kinematics Niels Joubert, UC Berkeley, CS184 2008-11-25 Inverse Kinematics is used to pose models by specifying endpoints of segments rather than individual joint angles.
More informationRobotics I. February 6, 2014
Robotics I February 6, 214 Exercise 1 A pan-tilt 1 camera sensor, such as the commercial webcams in Fig. 1, is mounted on the fixed base of a robot manipulator and is used for pointing at a (point-wise)
More informationMaths for Signals and Systems Linear Algebra in Engineering
Maths for Signals and Systems Linear Algebra in Engineering Lectures 13 15, Tuesday 8 th and Friday 11 th November 016 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE
More informationRobotics I. June 6, 2017
Robotics I June 6, 217 Exercise 1 Consider the planar PRPR manipulator in Fig. 1. The joint variables defined therein are those used by the manufacturer and do not correspond necessarily to a Denavit-Hartenberg
More informationOperational Space Control of Constrained and Underactuated Systems
Robotics: Science and Systems 2 Los Angeles, CA, USA, June 27-3, 2 Operational Space Control of Constrained and Underactuated Systems Michael Mistry Disney Research Pittsburgh 472 Forbes Ave., Suite Pittsburgh,
More informationRobot Dynamics Lecture Notes. Robotic Systems Lab, ETH Zurich
Robot Dynamics Lecture Notes Robotic Systems Lab, ETH Zurich HS 217 Contents 1 Introduction 1 1.1 Nomenclature.............................. 2 1.2 Operators................................ 3 2 Kinematics
More informationRobotics I. Classroom Test November 21, 2014
Robotics I Classroom Test November 21, 2014 Exercise 1 [6 points] In the Unimation Puma 560 robot, the DC motor that drives joint 2 is mounted in the body of link 2 upper arm and is connected to the joint
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 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 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 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 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 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 informationKinematic representation! Iterative methods! Optimization methods
Human Kinematics Kinematic representation! Iterative methods! Optimization methods Kinematics Forward kinematics! given a joint configuration, what is the position of an end point on the structure?! Inverse
More informationRobotics I Kinematics, Dynamics and Control of Robotic Manipulators. Velocity Kinematics
Robotics I Kinematics, Dynamics and Control of Robotic Manipulators Velocity Kinematics Dr. Christopher Kitts Director Robotic Systems Laboratory Santa Clara University Velocity Kinematics So far, we ve
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 informationLarge Scale Data Analysis Using Deep Learning
Large Scale Data Analysis Using Deep Learning Linear Algebra U Kang Seoul National University U Kang 1 In This Lecture Overview of linear algebra (but, not a comprehensive survey) Focused on the subset
More informationMath, Numerics, & Programming. (for Mechanical Engineers)
DRAFT V1.2 From Math, Numerics, & Programming (for Mechanical Engineers) Masayuki Yano James Douglass Penn George Konidaris Anthony T Patera September 212 The Authors. License: Creative Commons Attribution-Noncommercial-Share
More informationExample: RR Robot. Illustrate the column vector of the Jacobian in the space at the end-effector point.
Forward kinematics: X e = c 1 + c 12 Y e = s 1 + s 12 = s 1 s 12 c 1 + c 12, = s 12 c 12 Illustrate the column vector of the Jacobian in the space at the end-effector point. points in the direction perpendicular
More informationRobotics I June 11, 2018
Exercise 1 Robotics I June 11 018 Consider the planar R robot in Fig. 1 having a L-shaped second link. A frame RF e is attached to the gripper mounted on the robot end effector. A B y e C x e Figure 1:
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 informationCLASS NOTES Computational Methods for Engineering Applications I Spring 2015
CLASS NOTES Computational Methods for Engineering Applications I Spring 2015 Petros Koumoutsakos Gerardo Tauriello (Last update: July 27, 2015) IMPORTANT DISCLAIMERS 1. REFERENCES: Much of the material
More informationRobotics. Kinematics. Marc Toussaint University of Stuttgart Winter 2017/18
Robotics Kinematics 3D geometry, homogeneous transformations, kinematic map, Jacobian, inverse kinematics as optimization problem, motion profiles, trajectory interpolation, multiple simultaneous tasks,
More informationLecture Notes: (Stochastic) Optimal Control
Lecture Notes: (Stochastic) Optimal ontrol Marc Toussaint Machine Learning & Robotics group, TU erlin Franklinstr. 28/29, FR 6-9, 587 erlin, Germany July, 2 Disclaimer: These notes are not meant to be
More informationMultiple-priority impedance control
Multiple-priority impedance control Robert Platt Jr, Muhammad Abdallah, and Charles Wampler Abstract Impedance control is well-suited to robot manipulation applications because it gives the designer a
More informationMinimal representations of orientation
Robotics 1 Minimal representations of orientation (Euler and roll-pitch-yaw angles) Homogeneous transformations Prof. lessandro De Luca Robotics 1 1 Minimal representations rotation matrices: 9 elements
More informationWeek 3: Wheeled Kinematics AMR - Autonomous Mobile Robots
Week 3: Wheeled Kinematics AMR - Paul Furgale Margarita Chli, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Wheeled Kinematics 1 AMRx Flipped Classroom A Matlab exercise is coming later
More informationA new large projection operator for the redundancy framework
21 IEEE International Conference on Robotics and Automation Anchorage Convention District May 3-8, 21, Anchorage, Alaska, USA A new large projection operator for the redundancy framework Mohammed Marey
More informationLecture 7: Kinematics: Velocity Kinematics - the Jacobian
Lecture 7: Kinematics: Velocity Kinematics - the Jacobian Manipulator Jacobian c Anton Shiriaev. 5EL158: Lecture 7 p. 1/?? Lecture 7: Kinematics: Velocity Kinematics - the Jacobian Manipulator Jacobian
More informationPose Tracking II! Gordon Wetzstein! Stanford University! EE 267 Virtual Reality! Lecture 12! stanford.edu/class/ee267/!
Pose Tracking II! Gordon Wetzstein! Stanford University! EE 267 Virtual Reality! Lecture 12! stanford.edu/class/ee267/!! WARNING! this class will be dense! will learn how to use nonlinear optimization
More informationThe Acrobot and Cart-Pole
C H A P T E R 3 The Acrobot and Cart-Pole 3.1 INTRODUCTION A great deal of work in the control of underactuated systems has been done in the context of low-dimensional model systems. These model systems
More informationReconstructing Null-space Policies Subject to Dynamic Task Constraints in Redundant Manipulators
Reconstructing Null-space Policies Subject to Dynamic Task Constraints in Redundant Manipulators Matthew Howard Sethu Vijayakumar September, 7 Abstract We consider the problem of direct policy learning
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 informationMulti-Priority Cartesian Impedance Control
Multi-Priority Cartesian Impedance Control Robert Platt Jr. Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology rplatt@csail.mit.edu Muhammad Abdallah, Charles
More informationLinear Algebra and Robot Modeling
Linear Algebra and Robot Modeling Nathan Ratliff Abstract Linear algebra is fundamental to robot modeling, control, and optimization. This document reviews some of the basic kinematic equations and uses
More informationThe Virtual Linkage: A Model for Internal Forces in Multi-Grasp Manipulation
The Virtual Linkage: A Model for Internal Forces in Multi-Grasp Manipulation David Williams Oussama Khatib Robotics Laboratory Department of Computer Science Stanford University Stanford, California 94305
More informationPose estimation from point and line correspondences
Pose estimation from point and line correspondences Giorgio Panin October 17, 008 1 Problem formulation Estimate (in a LSE sense) the pose of an object from N correspondences between known object points
More informationKinematics of a UR5. Rasmus Skovgaard Andersen Aalborg University
Kinematics of a UR5 May 3, 28 Rasmus Skovgaard Andersen Aalborg University Contents Introduction.................................... Notation.................................. 2 Forward Kinematics for
More informationRobotics I. Test November 29, 2013
Exercise 1 [6 points] Robotics I Test November 9, 013 A DC motor is used to actuate a single robot link that rotates in the horizontal plane around a joint axis passing through its base. The motor is connected
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 I Midterm classroom test November 24, 2017
xercise [8 points] Robotics I Midterm classroom test November, 7 Consider the -dof (RPR) planar robot in ig., where the joint coordinates r = ( r r r have been defined in a free, arbitrary way, with reference
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 informationAdvanced Robotic Manipulation
Advanced Robotic Manipulation Handout CS37A (Spring 017) Solution Set #3 Problem 1 - Inertial properties In this problem, you will explore the inertial properties of a manipulator at its end-effector.
More informationSingle Exponential Motion and Its Kinematic Generators
PDFaid.Com #1 Pdf Solutions Single Exponential Motion and Its Kinematic Generators Guanfeng Liu, Yuanqin Wu, and Xin Chen Abstract Both constant velocity (CV) joints and zero-torsion parallel kinematic
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 informationMathematics for Intelligent Systems Lecture 5 Homework Solutions
Mathematics for Intelligent Systems Lecture 5 Homework Solutions Advanced Calculus I: Derivatives and local geometry) Nathan Ratliff Nov 25, 204 Problem : Gradient and Hessian Calculations We ve seen that
More informationDYNAMICS OF PARALLEL MANIPULATOR
DYNAMICS OF PARALLEL MANIPULATOR PARALLEL MANIPULATORS 6-degree of Freedom Flight Simulator BACKGROUND Platform-type parallel mechanisms 6-DOF MANIPULATORS INTRODUCTION Under alternative robotic mechanical
More informationLinear vector spaces and subspaces.
Math 2051 W2008 Margo Kondratieva Week 1 Linear vector spaces and subspaces. Section 1.1 The notion of a linear vector space. For the purpose of these notes we regard (m 1)-matrices as m-dimensional vectors,
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 informationCLASS NOTES Computational Methods for Engineering Applications I Spring 2015
CLASS NOTES Computational Methods for Engineering Applications I Spring 2015 Petros Koumoutsakos Gerardo Tauriello (Last update: July 2, 2015) IMPORTANT DISCLAIMERS 1. REFERENCES: Much of the material
More informationScrew Theory and its Applications in Robotics
Screw Theory and its Applications in Robotics Marco Carricato Group of Robotics, Automation and Biomechanics University of Bologna Italy IFAC 2017 World Congress, Toulouse, France Table of Contents 1.
More informationAn experimental robot load identification method for industrial application
An experimental robot load identification method for industrial application Jan Swevers 1, Birgit Naumer 2, Stefan Pieters 2, Erika Biber 2, Walter Verdonck 1, and Joris De Schutter 1 1 Katholieke Universiteit
More informationMulti-body Singularity Equations IRI Technical Report
IRI-TR-10-02 Multi-body Singularity Equations IRI Technical Report On how to obtain the equations for computation with CUIK O. Bohigas-Nadal L. Ros Abstract This technical report explains how to obtain
More informationGetting Started with Communications Engineering
1 Linear algebra is the algebra of linear equations: the term linear being used in the same sense as in linear functions, such as: which is the equation of a straight line. y ax c (0.1) Of course, if we
More information13. Nonlinear least squares
L. Vandenberghe ECE133A (Fall 2018) 13. Nonlinear least squares definition and examples derivatives and optimality condition Gauss Newton method Levenberg Marquardt method 13.1 Nonlinear least squares
More informationMATH 612 Computational methods for equation solving and function minimization Week # 2
MATH 612 Computational methods for equation solving and function minimization Week # 2 Instructor: Francisco-Javier Pancho Sayas Spring 2014 University of Delaware FJS MATH 612 1 / 38 Plan for this week
More informationAnalysis of the Acceleration Characteristics of Non-Redundant Manipulators
Analysis of the Acceleration Characteristics of Non-Redundant Manipulators Alan Bowling and Oussama Khatib Robotics Laboratory Computer Science Depart men t Stanford University Stanford, CA, USA 94305
More informationJim Lambers MAT 419/519 Summer Session Lecture 11 Notes
Jim Lambers MAT 49/59 Summer Session 20-2 Lecture Notes These notes correspond to Section 34 in the text Broyden s Method One of the drawbacks of using Newton s Method to solve a system of nonlinear equations
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 informationBindel, Fall 2009 Matrix Computations (CS 6210) Week 8: Friday, Oct 17
Logistics Week 8: Friday, Oct 17 1. HW 3 errata: in Problem 1, I meant to say p i < i, not that p i is strictly ascending my apologies. You would want p i > i if you were simply forming the matrices and
More informationReconfiguration Manipulability Analyses for Redundant Robots in View of Strucuture and Shape
SCIS & ISIS 2010, Dec. 8-12, 2010, Okayama Convention Center, Okayama, Japan Reconfiguration Manipulability Analyses for Redundant Robots in View of Strucuture and Shape Mamoru Minami, Tongxiao Zhang,
More informationNumerical Methods I Solving Nonlinear Equations
Numerical Methods I Solving Nonlinear Equations Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fall 2014 October 16th, 2014 A. Donev (Courant Institute)
More information13 Path Planning Cubic Path P 2 P 1. θ 2
13 Path Planning Path planning includes three tasks: 1 Defining a geometric curve for the end-effector between two points. 2 Defining a rotational motion between two orientations. 3 Defining a time function
More informationCLASS NOTES Models, Algorithms and Data: Introduction to computing 2018
CLASS NOTES Models, Algorithms and Data: Introduction to computing 2018 Petros Koumoutsakos, Jens Honore Walther (Last update: April 16, 2018) IMPORTANT DISCLAIMERS 1. REFERENCES: Much of the material
More informationInverse Theory. COST WaVaCS Winterschool Venice, February Stefan Buehler Luleå University of Technology Kiruna
Inverse Theory COST WaVaCS Winterschool Venice, February 2011 Stefan Buehler Luleå University of Technology Kiruna Overview Inversion 1 The Inverse Problem 2 Simple Minded Approach (Matrix Inversion) 3
More informationControl of a Handwriting Robot with DOF-Redundancy based on Feedback in Task-Coordinates
Control of a Handwriting Robot with DOF-Redundancy based on Feedback in Task-Coordinates Hiroe HASHIGUCHI, Suguru ARIMOTO, and Ryuta OZAWA Dept. of Robotics, Ritsumeikan Univ., Kusatsu, Shiga 525-8577,
More informationMA201: Further Mathematical Methods (Linear Algebra) 2002
MA201: Further Mathematical Methods (Linear Algebra) 2002 General Information Teaching This course involves two types of teaching session that you should be attending: Lectures This is a half unit course
More informationPartially Observable Markov Decision Processes (POMDPs)
Partially Observable Markov Decision Processes (POMDPs) Sachin Patil Guest Lecture: CS287 Advanced Robotics Slides adapted from Pieter Abbeel, Alex Lee Outline Introduction to POMDPs Locally Optimal Solutions
More informationChapter 1. Rigid Body Kinematics. 1.1 Introduction
Chapter 1 Rigid Body Kinematics 1.1 Introduction This chapter builds up the basic language and tools to describe the motion of a rigid body this is called rigid body kinematics. This material will be the
More informationSegment Prismatic 1 DoF Physics
edx Robo4 Mini MS Locomotion Engineering Block 1 Week 2 Unit 1 A Linear Time Invariant Mechanical System Video 2.1 Prismatic 1 DoF Physics Daniel E. Koditschek with Wei-Hsi Chen, T. Turner Topping and
More informationROBOTICS Laboratory Problem 02
ROBOTICS 2015-2016 Laboratory Problem 02 Basilio Bona DAUIN PoliTo Problem formulation The planar system illustrated in Figure 1 consists of a cart C sliding with or without friction along the horizontal
More information3 Systems of Equations
14102, Math for Economists Fall 2004 Lecture Notes, 9/16/2004 These notes are primarily based on those written by George Marios Angeletos for the Harvard Math Camp in 1999 and 2000, and updated by Stavros
More informationME 115(b): Homework #2 Solution. Part (b): Using the Product of Exponentials approach, the structure equations take the form:
ME 5(b): Homework #2 Solution Problem : Problem 2 (d,e), Chapter 3 of MLS. Let s review some material from the parts (b) of this problem: Part (b): Using the Product of Exponentials approach, the structure
More informationRobotics. 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 informationCOMP 558 lecture 18 Nov. 15, 2010
Least squares We have seen several least squares problems thus far, and we will see more in the upcoming lectures. For this reason it is good to have a more general picture of these problems and how to
More information