Robotics 1 Inverse kinematics
|
|
- Maryann Henderson
- 5 years ago
- Views:
Transcription
1 Robotics 1 Inverse kinematics Prof. Alessandro De Luca Robotics 1 1
2 Inverse kinematics what are we looking for? direct kinematics is always unique; how about inverse kinematics for this 6R robot? Robotics 1
3 Inverse kinematics problem! given a desired end-effector pose (position + orientation), find the values of the joint variables that will realize it! a synthesis problem, with input data in the form! T = R p classical formulation: inverse kinematics for a given end-effector pose! a typical nonlinear problem! existence of a solution (workspace definition)! uniqueness/multiplicity of solutions! solution methods p " r =, or any other task vector! generalized formulation: inverse kinematics for a given value of task variables Robotics 1 3
4 Solvability and robot workspace! primary workspace WS 1 : set of all positions p that can be reached with at least one orientation (! or R)! out of WS 1 there is no solution to the problem! for p " WS 1 and a suitable! (or R) there is at least one solution! secondary (or dexterous) workspace WS : set of positions p that can be reached with any orientation (among those feasible for the robot direct kinematics)! for p " WS there is at least one solution for any feasible! (or R)! WS # WS 1 Robotics 1 4
5 Workspace of Fanuc R-000i/165F section for a constant angle $ 1 WS 1 (! WS for spherical wrist without joint limits) rotating the base joint angle $ 1 Robotics 1 5
6 Workspace of planar R arm y p orientations l 1 +l l q l 1 -l l 1! if l 1 % l q 1! WS 1 = {p " R : l 1 -l &!p!& l 1 +l }! WS = '! if l 1 = l = "! WS 1 = {p " R :!p!& "} x (WS 1 inner and outer boundaries 1 orientation! WS = {p = 0} (infinite number of feasible orientations at the origin) Robotics 1 6
7 Wrist position and E-E pose inverse solutions for an articulated 6R robot LEFT DOWN Unimation PUMA 560 RIGHT DOWN 4 inverse solutions out of singularities (for the position of the wrist center only) LEFT UP 8 inverse solutions considering the complete E-E pose (spherical wrist: alternative solutions for the last 3 joints) RIGHT UP Robotics 1 7
8 Counting and visualizing the 8 solutions to the inverse kinematics of a Unimation Puma 560 RIGHT UP RIGHT DOWN LEFT UP LEFT DOWN Robotics 1 8
9 Multiplicity of solutions some examples! E-E positioning of a planar R robot arm! regular solutions in WS 1! 1 solution on (WS 1! for l 1 = l : ) solutions in WS singular solutions! E-E positioning of an articulated elbow-type 3R robot arm! 4 regular solutions in WS 1! spatial 6R robot arms! & 16 distinct solutions, out of singularities: this upper bound of solutions was shown to be attained by a particular instance of orthogonal robot, i.e., with twist angles * i = 0 or ±+/ (,i)! analysis based on algebraic transformations of robot kinematics! transcendental equations are transformed into a single polynomial equation of one variable! seek for an equivalent polynomial equation of the least possible degree Robotics 1 9
10 A planar 3R arm workspace and number/type of inverse solutions y q 3 l1 = l = l 3 = " p l 1 l q l 3 WS 1 = {p " R :!p!& 3"} WS = {p " R :!p!& "} q 1 x any planar orientation is feasible in WS 1. in WS 1 : ) 1 regular solutions (except for. and 3.), at which the E-E can take a continuum of ) orientations (but not all orientations in the plane!). if!p!= 3" : only 1 solution, singular 3. if!p!= " : ) 1 solutions, 3 of which singular 4. if!p!< " : ) 1 regular solutions (never singular) Robotics 1 10
11 ! if m = n!! solutions Multiplicity of solutions summary of the general cases! a finite number of solutions (regular/generic case)! degenerate solutions: infinite or finite set, but anyway different in number from the generic case (singularity)! if m < n (robot is redundant for the kinematic task)!! solutions! ) n-m solutions (regular/generic case)! a finite or infinite number of singular solutions! use of the term singularity will become clearer when dealing with differential kinematics! instantaneous velocity mapping from joint to task velocity! lack of full rank of the associated m"n Jacobian matrix J(q) Robotics 1 11
12 Dexter robot (8R arm)! m = 6 (position and orientation of E-E)! n = 8 (all revolute joints)! ) inverse kinematic solutions (redundancy degree = n-m = ) video exploring inverse kinematic solutions by a self-motion Robotics 1 1
13 Solution methods ANALYTICAL solution (in closed form) " preferred, if it can be found * " use ad-hoc geometric inspection " algebraic methods (solution of polynomial equations) " systematic ways for generating a reduced set of equations to be solved * sufficient conditions for 6-dof arms 3 consecutive rotational joint axes are incident (e.g., spherical wrist), or 3 consecutive rotational joint axes are parallel D. Pieper, PhD thesis, Stanford University, 1968 NUMERICAL solution (in iterative form) " certainly needed if n>m (redundant case), or at/close to singularities " slower, but easier to be set up " in its basic form, it uses the (analytical) Jacobian matrix of the direct kinematics map J r (q) = (f r (q) (q " Newton method, Gradient method, and so on Robotics 1 13
14 y p y Inverse kinematics of planar R arm l q p direct kinematics p x = l 1 c 1 + l c 1 l 1 p y = l 1 s 1 + l s 1 q 1 p x x data q 1, q unknowns squaring and summing the equations of the direct kinematics p x + p y - (l 1 + l ) = l 1 l (c 1 c 1 + s 1 s 1 ) = l 1 l c and from this in analytical form c = (p x + p y - l 1 - l )/ l 1 l, s = ±-1 - c q = ATAN {s, c } must be in [-1,1] (else, point p is outside robot workspace!) solutions Robotics 1 14
15 y p y. Inverse kinematics of R arm (cont d) q 1 * q p p x solutions (one for each value of s ) x by geometric inspection q 1 = * -. q 1 = ATAN {p y, p x } - ATAN {l s, l 1 + l c } q 1 q p note: difference of ATAN needs to be re-expressed in (-+, +]! {q 1,q } UP/LEFT {q q 1,q } DOWN/RIGHT q 1 q e q have same absolute value, but opposite signs Robotics 1 15
16 Algebraic solution for q 1 another solution method p x = l 1 c 1 + l c 1 = l 1 c 1 + l (c 1 c - s 1 s ) p y = l 1 s 1 + l s 1 = l 1 s 1 + l (s 1 c + c 1 s ) linear in s 1 and c 1 l 1 + l c - l s c 1 = p x l s l 1 + l c s 1 p y except for l 1 =l and c =-1 det = (l 1 + l + l 1 l c ) > 0 being then q 1 undefined (singular case: ) 1 solutions) q 1 = ATAN {s 1, c 1 } = ATAN {(p y (l 1 +l c )-p x l s )/det, (p x (l 1 +l c )+p y l s )/det} notes: a) this method provides directly the result in (-+, +] b) when evaluating ATAN, det > 0 can be eliminated from the expressions of s 1 and c 1 Robotics 1 16
17 Inverse kinematics of polar (RRP) arm q is NOT a DH variable! p z q 3 p x = q 3 c c 1 p y = q 3 c s 1 d 1 q p z = d 1 + q 3 s p x + p y + (p z - d 1 ) = q 3 p y q 3 = + -p x + p y + (p z - d 1 ) p x q 1 our choice: take here only the positive value... if q 3 = 0, then q 1 and q remain both undefined (stop); else q = ATAN{(p z - d 1 )/q 3, ± -(p x + p y )/q 3 } if p x +p y = 0, then q 1 remains undefined (stop); else q 1 = ATAN{p y /c, p x /c } ( regular solutions {q 1,q,q 3 }) (if it stops, a singular case: ) or ) 1 solutions) Robotics 1 we have eliminated q 3 >0 from both arguments! 17
18 Inverse kinematics for robots with spherical wrist j6 first 3 robot joints of any type (RRR, RRP, PPP, ) W z 4 z 3 z 5 y 6 x 6 O 6 = p z 6 = a z 0 j5 j4 d 6 x 0 j1 y 0 find q 1,, q 6 from the input data: p (origin O 6 ) R = [n s a] (orientation of RF 6 ) 1. W = p - d 6 a / q 1, q, q 3 (inverse position kinematics for main axes). R = 0 R 3 (q 1, q, q 3 ) 3 R 6 (q 4, q 5, q 6 ) / 3 R 6 (q 4, q 5, q 6 ) = 0 R T 3 R / q 4, q 5, q 6 (inverse orientation given known, Euler ZYZ or ZXZ kinematics for wrist) after 1. rotation matrix Robotics 1 18
19 6R example: Unimation PUMA 600 spherical wrist n = 0 x 6 (q) s = 0 y 6 (q) a function of q 1, q, q 3 only! a = 0 z 6 (q) p = 0 6 (q) 8 different inverse solutions that can be found in closed form (see Paul, Shimano, Mayer; 1981) here d 6 =0, Robotics 1 so that 0 6 =W directly 19
20 Numerical solution of inverse kinematics problems! use when a closed-form solution q to r d = f r (q) does not exist or is too hard to be found! J r (q) = (f r (analytical Jacobian) (q! Newton method (here for m=n)! r d = f r (q) = f r (q k ) + J r (q k ) (q - q k ) + o(!q - q k! ) q k+1 = q k + J r -1 (q k ) [r d - f r (q k )] 0 neglected! convergence if q 0 (initial guess) is close enough to some q * : f r (q * ) = r d! problems near singularities of the Jacobian matrix J r (q)! in case of robot redundancy (m<n), use the pseudo-inverse J r# (q)! has quadratic convergence rate when near to solution (fast!) Robotics 1 0
21 Operation of Newton method! in the scalar case, also known as method of the tangent! for a differentiable function f(x), find a root of f(x)=0 by iterating as x k+1 = x k " f(x k ) f'(x k ) an approximating sequence { x 1,x,x 3,x 4,x 5,...} animation from Robotics 1 1
22 Numerical solution of inverse kinematics problems (cont d)! Gradient method (max descent)! minimize the error function H(q) = #!r d - f r (q)! = # [r d - f r (q)] T [r d - f r (q)] q k+1 = q k - * 1 q H(q k ) from 1H(q) = - J rt (q) [r d - f r (q)], we get q k+1 = q k + * J rt (q k ) [r d - f r (q k )]! the scalar step size * > 0 should be chosen so as to guarantee a decrease of the error function at each iteration (too large values for * may lead the method to miss the minimum)! when the step size * is too small, convergence is extremely slow Robotics 1
23 Revisited as a feedback scheme r d + - r e J rt (q) f r (q). q q(0) 3 q r d = cost (* = 1) e = r d - f r (q) / 0 4 closed-loop equilibrium e=0 is asymptotically stable V = # e T e 5 0 Lyapunov candidate function... V = e T e = e T d ( q dt (r d - f r (q)) = - et J r = - e T J r J rt e & 0. V = 0 4 e " Ker(J rt ) in particular e = 0 asymptotic stability Robotics 1 3
24 Properties of Gradient method! computationally simpler: Jacobian transpose, rather than its (pseudo)-inverse! direct use also for robots that are redundant for the task! may not converge to a solution, but it never diverges! the discrete time evolution of the continuous scheme q k+1 = q k + 6T J rt (q k ) [r d - f(q k )] (* = 6T) is equivalent to an iteration of the Gradient method! scheme can be accelerated by using a gain matrix K>0. q = J rt (q) K e note: K can be used also to escape from being stuck in a stationary point, by rotating the error e out of the kernel of J r T (if a singularity is encountered) Robotics 1 4
25 A case study analytic expressions of Newton and gradient iterations! R robot with l 1 =l =1, desired end-effector position r d = (1,1)! direct kinematic function and error " f r (q) = c 1 + c 1 % " $ ' e = r # s 1 + s 1 & d - f r (q) = $ 1% ' 1 - f # & r (q)! Jacobian matrix J r (q) = " f r(q) " q = # -(s + s ) -s & % ( $ c 1 + c 1 c 1 '! Newton versus Gradient iteration det J r (q) q k+1 = q k + J -1 r (q k ) 1 " c 1 s 1 % $ ' s #-(c 1 + c 1 ) -(s 1 + s 1 )& # " -(s + s ) c + c & % ( $ -s 1 c 1 ' J r T (q k ) " 1- (c 1 + c 1 )% $ ' # 1- (s 1 + s 1 )& Robotics 1 5 q=q k q=q k e k q=q k
26 Error function! R robot with l1=l=1, desired end-effector position rd = (1,1) e = rd - fr(q) plot of!e! as a function of q = (q1,q) Robotics 1 two local minima (inverse kinematic solutions) 6
27 Error reduction by Gradient method!! flow of iterations along the negative (or anti-) gradient two possible cases: convergence or stuck (at zero gradient) start one solution. local maximum (stop if this is the initial guess). saddle point (stop after some iterations) another start......the other solution (q1,q) =(0,!/) (q1,q) =(!/,-!/) Robotics 1 (q1,q)max =(-3!/4,0) (q1,q)saddle =(!/4,0) e " Ker(JrT)! 7
28 ! initial guess q 0 Issues in implementation! only one inverse solution is generated for each guess! multiple initializations for obtaining other solutions! optimal step size * in Gradient method! a constant step may work good initially, but not close to the solution (or vice versa)! an adaptive one-dimensional line search (e.g., Armijo s rule) could be used to choose the best * at each iteration! stopping criteria Cartesian error (possibly, separate for position and orientation)!r d - f(q k )! $ #! understanding closeness to singularities algorithm increment!q k+1 -q k! $ # q $ min {J(q k )} % $ 0 numerical conditioning of Jacobian matrix (SVD) (or a simpler test on its determinant, for m=n) Robotics 1 8
29 Numerical tests on RRP robot! RRP/polar robot: desired E-E position r d = p d = (1, 1,1) see slide 17, d 1 =0.5! the two (known) analytic solutions, with q 3 % 0, are: q * = (0.7854, , 1.5) q ** = (q 1* +, + q *, q 3* ) = (-.356,.8018, 1.5)! norms # = 10-5 (max Cartesian error), # q =10-6 (min joint increment)! k max =15 (max iterations), det(j r ) $ 10-4 (closeness to singularity)! numerical performance of Gradient (with different *) vs. Newton! test 1: q 0 = (0, 0, 1) as initial guess! test : q 0 = (-+/4, +/, 1) singular start, since c =0 (see slide 17)! test 3: q 0 = (0, +/, 0) double singular start, since also q 3 =0! solution and plots with Matlab code Robotics 1 9
30 Numerical test! -1 test 1: q0 = (0, 0, 1) as initial guess; evolution of error norm Gradient: * = 0.5 Gradient: * = 1 too large, oscillates around solution slow, 15 (max) iterations good, converges in 11 iterations 0.57䏙10-5 Newton very fast, converges in 5 iterations Gradient: * = 0.7 Cartesian errors component-wise ex ey ez Robotics 䏙
31 Numerical test - 1! test 1: q 0 = (0, 0, 1) as initial guess; evolution of joint variables Gradient: * = 1 Gradient: * = 0.7 not converging to a solution converges in 11 iterations Newton converges in 5 iterations both to solution q * = (0.7854, , 1.5) Robotics 1 31
32 Numerical test -! test : q 0 = (-+/4, +/, 1): singular start error norms Gradient * = 0.7 Newton with check of singularity: blocked at start without check: it diverges! starts toward solution, but slowly stops (in singularity): when Cartesian error vector e Ker(J rt ) joint variables!! Robotics 1 3
33 Numerical test - 3! test 3: q 0 = (0, +/, 0): double singular start error norm Gradient (with * = 0.7) starts toward solution exits the double singularity slowly converges in 19 iterations to the solution q * =(0.7854, , 1.5) Newton is either blocked at start or (w/o check) explodes (NaN)!! Cartesian errors joint variables Robotics 1 33
34 Final remarks! an efficient iterative scheme can be devised by combining! initial iterations with Gradient method ( sure but slow, having linear convergence rate)! switch then to Newton method (quadratic terminal convergence rate)! joint range limits are considered only at the end! check if the found solution is feasible, as for analytical methods! if the problem has to be solved on-line! execute iterations and associate an actual robot motion: repeat steps at times t 0, t 1 =t 0 +T,..., t k =t k-1 +T (e.g., every T=40 ms)! the good choice for the initial q 0 at t k is the solution of the previous problem at t k-1 (gives continuity, needs only 1- Newton iterations)! crossing of singularities and handling of joint range limits need special care in this case! Jacobian-based inversion schemes are used also for kinematic control, along a continuous task trajectory r d (t) Robotics 1 34
Robotics 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 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 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 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 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 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 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 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 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 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 informationLecture «Robot Dynamics» : Kinematics 3
Lecture «Robot Dynamics» : Kinematics 3 151-0851-00 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
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 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 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 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 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 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 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 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 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 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 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 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 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 informationControl of Mobile Robots
Control of Mobile Robots Regulation and trajectory tracking Prof. Luca Bascetta (luca.bascetta@polimi.it) Politecnico di Milano Dipartimento di Elettronica, Informazione e Bioingegneria Organization and
More informationVideo 8.1 Vijay Kumar. Property of University of Pennsylvania, Vijay Kumar
Video 8.1 Vijay Kumar 1 Definitions State State equations Equilibrium 2 Stability Stable Unstable Neutrally (Critically) Stable 3 Stability Translate the origin to x e x(t) =0 is stable (Lyapunov stable)
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 z A p AB B RF B z B x B y A rigid body
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 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 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 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 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 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 informationDirectional Redundancy for Robot Control
ACCEPTED FOR PUBLICATION IN IEEE TRANSACTIONS ON AUTOMATIC CONTROL Directional Redundancy for Robot Control Nicolas Mansard, François Chaumette, Member, IEEE Abstract The paper presents a new approach
More informationAlgorithms for Sensor-Based Robotics: Potential Functions
Algorithms for Sensor-Based Robotics: Potential Functions Computer Science 336 http://www.cs.jhu.edu/~hager/teaching/cs336 Professor Hager http://www.cs.jhu.edu/~hager The Basic Idea A really simple idea:
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 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 informationMathematical optimization
Optimization Mathematical optimization Determine the best solutions to certain mathematically defined problems that are under constrained determine optimality criteria determine the convergence of the
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 informationMath 302 Outcome Statements Winter 2013
Math 302 Outcome Statements Winter 2013 1 Rectangular Space Coordinates; Vectors in the Three-Dimensional Space (a) Cartesian coordinates of a point (b) sphere (c) symmetry about a point, a line, and a
More informationGENG2140, S2, 2012 Week 7: Curve fitting
GENG2140, S2, 2012 Week 7: Curve fitting Curve fitting is the process of constructing a curve, or mathematical function, f(x) that has the best fit to a series of data points Involves fitting lines and
More informationPreliminary Examination in Numerical Analysis
Department of Applied Mathematics Preliminary Examination in Numerical Analysis August 7, 06, 0 am pm. Submit solutions to four (and no more) of the following six problems. Show all your work, and justify
More informationKinematic Analysis of the 6R Manipulator of General Geometry
Kinematic Analysis of the 6R Manipulator of General Geometry Madhusudan Raghavan Powertrain Systems Research Lab General Motors R&D Center Warren, MI 48090-9055 and Bernard Roth, Professor Design Division
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 informationChapter 3 Numerical Methods
Chapter 3 Numerical Methods Part 2 3.2 Systems of Equations 3.3 Nonlinear and Constrained Optimization 1 Outline 3.2 Systems of Equations 3.3 Nonlinear and Constrained Optimization Summary 2 Outline 3.2
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 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 informationMULTIVARIABLE CALCULUS, LINEAR ALGEBRA, AND DIFFERENTIAL EQUATIONS
T H I R D E D I T I O N MULTIVARIABLE CALCULUS, LINEAR ALGEBRA, AND DIFFERENTIAL EQUATIONS STANLEY I. GROSSMAN University of Montana and University College London SAUNDERS COLLEGE PUBLISHING HARCOURT BRACE
More informationIntroduction to Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz
Introduction to Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Vectors Arrays of numbers Vectors represent a point in a n dimensional space
More informationMinimization of Static! Cost Functions!
Minimization of Static Cost Functions Robert Stengel Optimal Control and Estimation, MAE 546, Princeton University, 2017 J = Static cost function with constant control parameter vector, u Conditions for
More informationConstrained optimization. Unconstrained optimization. One-dimensional. Multi-dimensional. Newton with equality constraints. Active-set method.
Optimization Unconstrained optimization One-dimensional Multi-dimensional Newton s method Basic Newton Gauss- Newton Quasi- Newton Descent methods Gradient descent Conjugate gradient Constrained optimization
More informationMath 411 Preliminaries
Math 411 Preliminaries Provide a list of preliminary vocabulary and concepts Preliminary Basic Netwon s method, Taylor series expansion (for single and multiple variables), Eigenvalue, Eigenvector, Vector
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 informationComputational Methods. Least Squares Approximation/Optimization
Computational Methods Least Squares Approximation/Optimization Manfred Huber 2011 1 Least Squares Least squares methods are aimed at finding approximate solutions when no precise solution exists Find the
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 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 informationNonlinear Optimization for Optimal Control
Nonlinear Optimization for Optimal Control Pieter Abbeel UC Berkeley EECS Many slides and figures adapted from Stephen Boyd [optional] Boyd and Vandenberghe, Convex Optimization, Chapters 9 11 [optional]
More informationECEN 615 Methods of Electric Power Systems Analysis Lecture 18: Least Squares, State Estimation
ECEN 615 Methods of Electric Power Systems Analysis Lecture 18: Least Squares, State Estimation Prof. om Overbye Dept. of Electrical and Computer Engineering exas A&M University overbye@tamu.edu Announcements
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 informationVector Derivatives and the Gradient
ECE 275AB Lecture 10 Fall 2008 V1.1 c K. Kreutz-Delgado, UC San Diego p. 1/1 Lecture 10 ECE 275A Vector Derivatives and the Gradient ECE 275AB Lecture 10 Fall 2008 V1.1 c K. Kreutz-Delgado, UC San Diego
More informationNonlinear Optimization
Nonlinear Optimization (Com S 477/577 Notes) Yan-Bin Jia Nov 7, 2017 1 Introduction Given a single function f that depends on one or more independent variable, we want to find the values of those variables
More informationMobile Robotics 1. A Compact Course on Linear Algebra. Giorgio Grisetti
Mobile Robotics 1 A Compact Course on Linear Algebra Giorgio Grisetti SA-1 Vectors Arrays of numbers They represent a point in a n dimensional space 2 Vectors: Scalar Product Scalar-Vector Product Changes
More informationCHAPTER 2 EXTRACTION OF THE QUADRATICS FROM REAL ALGEBRAIC POLYNOMIAL
24 CHAPTER 2 EXTRACTION OF THE QUADRATICS FROM REAL ALGEBRAIC POLYNOMIAL 2.1 INTRODUCTION Polynomial factorization is a mathematical problem, which is often encountered in applied sciences and many of
More information, b = 0. (2) 1 2 The eigenvectors of A corresponding to the eigenvalues λ 1 = 1, λ 2 = 3 are
Quadratic forms We consider the quadratic function f : R 2 R defined by f(x) = 2 xt Ax b T x with x = (x, x 2 ) T, () where A R 2 2 is symmetric and b R 2. We will see that, depending on the eigenvalues
More informationIntroduction to Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Maren Bennewitz, Diego Tipaldi, Luciano Spinello
Introduction to Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Cyrill Stachniss, Maren Bennewitz, Diego Tipaldi, Luciano Spinello Vectors Arrays of numbers Vectors represent a point
More informationCHAPTER 2: QUADRATIC PROGRAMMING
CHAPTER 2: QUADRATIC PROGRAMMING Overview Quadratic programming (QP) problems are characterized by objective functions that are quadratic in the design variables, and linear constraints. In this sense,
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 information5 Handling Constraints
5 Handling Constraints Engineering design optimization problems are very rarely unconstrained. Moreover, the constraints that appear in these problems are typically nonlinear. This motivates our interest
More informationStructural topology, singularity, and kinematic analysis. J-P. Merlet HEPHAISTOS project INRIA Sophia-Antipolis
Structural topology, singularity, and kinematic analysis J-P. Merlet HEPHAISTOS project INRIA Sophia-Antipolis 1 Parallel robots Definitions: a closed-loop mechanism whose end-effector is linked to the
More informationCS 542G: Robustifying Newton, Constraints, Nonlinear Least Squares
CS 542G: Robustifying Newton, Constraints, Nonlinear Least Squares Robert Bridson October 29, 2008 1 Hessian Problems in Newton Last time we fixed one of plain Newton s problems by introducing line search
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 information1. Method 1: bisection. The bisection methods starts from two points a 0 and b 0 such that
Chapter 4 Nonlinear equations 4.1 Root finding Consider the problem of solving any nonlinear relation g(x) = h(x) in the real variable x. We rephrase this problem as one of finding the zero (root) of a
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 informationAPPENDIX A. Background Mathematics. A.1 Linear Algebra. Vector algebra. Let x denote the n-dimensional column vector with components x 1 x 2.
APPENDIX A Background Mathematics A. Linear Algebra A.. Vector algebra Let x denote the n-dimensional column vector with components 0 x x 2 B C @. A x n Definition 6 (scalar product). The scalar product
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 informationIntroduction to Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Bastian Steder
Introduction to Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Bastian Steder Reference Book Thrun, Burgard, and Fox: Probabilistic Robotics Vectors Arrays of numbers Vectors represent
More informationMATHEMATICS COMPREHENSIVE EXAM: IN-CLASS COMPONENT
MATHEMATICS COMPREHENSIVE EXAM: IN-CLASS COMPONENT The following is the list of questions for the oral exam. At the same time, these questions represent all topics for the written exam. The procedure for
More informationUnconstrained optimization
Chapter 4 Unconstrained optimization An unconstrained optimization problem takes the form min x Rnf(x) (4.1) for a target functional (also called objective function) f : R n R. In this chapter and throughout
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 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 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 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 informationRidig Body Motion Homogeneous Transformations
Ridig Body Motion Homogeneous Transformations Claudio Melchiorri Dipartimento di Elettronica, Informatica e Sistemistica (DEIS) Università di Bologna email: claudio.melchiorri@unibo.it C. Melchiorri (DEIS)
More informationA nonlinear equation is any equation of the form. f(x) = 0. A nonlinear equation can have any number of solutions (finite, countable, uncountable)
Nonlinear equations Definition A nonlinear equation is any equation of the form where f is a nonlinear function. Nonlinear equations x 2 + x + 1 = 0 (f : R R) f(x) = 0 (x cos y, 2y sin x) = (0, 0) (f :
More informationChapter 3: The Derivative in Graphing and Applications
Chapter 3: The Derivative in Graphing and Applications Summary: The main purpose of this chapter is to use the derivative as a tool to assist in the graphing of functions and for solving optimization problems.
More informationME451 Kinematics and Dynamics of Machine Systems
ME451 Kinematics and Dynamics of Machine Systems Newton-Raphson Method 4.5 Closing remarks, Kinematics Analysis October 25, 2010 Dan Negrut, 2011 ME451, UW-Madison Success is going from failure to failure
More information1 Newton s Method. Suppose we want to solve: x R. At x = x, f (x) can be approximated by:
Newton s Method Suppose we want to solve: (P:) min f (x) At x = x, f (x) can be approximated by: n x R. f (x) h(x) := f ( x)+ f ( x) T (x x)+ (x x) t H ( x)(x x), 2 which is the quadratic Taylor expansion
More informationLecture Notes: Geometric Considerations in Unconstrained Optimization
Lecture Notes: Geometric Considerations in Unconstrained Optimization James T. Allison February 15, 2006 The primary objectives of this lecture on unconstrained optimization are to: Establish connections
More informationAdvanced Robotic Manipulation
Lecture Notes (CS327A) Advanced Robotic Manipulation Oussama Khatib Stanford University Spring 2005 ii c 2005 by Oussama Khatib Contents 1 Spatial Descriptions 1 1.1 Rigid Body Configuration.................
More informationLECTURE NOTES ELEMENTARY NUMERICAL METHODS. Eusebius Doedel
LECTURE NOTES on ELEMENTARY NUMERICAL METHODS Eusebius Doedel TABLE OF CONTENTS Vector and Matrix Norms 1 Banach Lemma 20 The Numerical Solution of Linear Systems 25 Gauss Elimination 25 Operation Count
More informationLinear Regression. Robot Image Credit: Viktoriya Sukhanova 123RF.com
Linear Regression These slides were assembled by Eric Eaton, with grateful acknowledgement of the many others who made their course materials freely available online. Feel free to reuse or adapt these
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 informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 5 Nonlinear Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction
More informationReview of some mathematical tools
MATHEMATICAL FOUNDATIONS OF SIGNAL PROCESSING Fall 2016 Benjamín Béjar Haro, Mihailo Kolundžija, Reza Parhizkar, Adam Scholefield Teaching assistants: Golnoosh Elhami, Hanjie Pan Review of some mathematical
More informationOptimal control problems with PDE constraints
Optimal control problems with PDE constraints Maya Neytcheva CIM, October 2017 General framework Unconstrained optimization problems min f (q) q x R n (real vector) and f : R n R is a smooth function.
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 informationIntroduction to gradient descent
6-1: Introduction to gradient descent Prof. J.C. Kao, UCLA Introduction to gradient descent Derivation and intuitions Hessian 6-2: Introduction to gradient descent Prof. J.C. Kao, UCLA Introduction Our
More informationNumerical Methods. Root Finding
Numerical Methods Solving Non Linear 1-Dimensional Equations Root Finding Given a real valued function f of one variable (say ), the idea is to find an such that: f() 0 1 Root Finding Eamples Find real
More information