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An Introduction to Robot Kinematics

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An Introduction to Robot Kinematics Carnegie Mellon An Introduction to Robot Kinematics Howie Choset Hannah Lyness © Howie Choset, 2019 Carnegie Mellon Robot kinematics refers to the geometry and movement of robotic mechanisms Prof Michael Kaess 16-665: Robot Mobility In Air, Land, and Sea. Underwater Robotics Lecture 2 Slide 31. Used with permission. Carnegie Mellon A select history of robotics Elmer, 1948 Unimate, Wabot 2, Shakey, 1959 1980 1966 CyberKnife, 1991 Asimo, Baxter, Kuka KR AGILUS, 2000 Big Dog, 2011 2014 2005 https://en.wikipedia.org/wiki/Shakey_the_robot#/media/File:SRI_Shakey_with_callouts.jpg, http://roamerrobot.tumblr.com/post/23079345849/the-history-of- turtle-robots, https://www.robotics.org/joseph-engelberger/unimate.cfm, https://en.wikipedia.org/wiki/BigDog, https://www.techexplorist.com/wp- content/uploads/2016/03/Baxter_Robot-696x470.jpg Carnegie Mellon Goals - Use robotics kinematics terms to explain real world situations. - Express a point in one coordinate frame in a different coordinate frame. - Represent complex translations and rotations using a homogenous transformation matrix. - Determine the position and orientation of an end effector given link and joint information. Carnegie Mellon What does degrees of freedom mean? Carnegie Mellon Degrees of Freedom (DOF): the number of independent parameters that can fully define the configuration Carnegie Mellon How many degrees of freedom does this have? https://www.chamonix.net/english/leisure/sightseeing/mer-de-glace Carnegie Mellon 1 DOF https://www.chamonix.net/english/leisure/sightseeing/mer-de-glace Carnegie Mellon How many degrees of freedom does this have? http://www.andrew.cmu.edu/user/kbrennan/TeamZ-Lab9.html Carnegie Mellon 2 DOF http://www.andrew.cmu.edu/user/kbrennan/TeamZ-Lab9.html Carnegie Mellon How many degrees of freedom does this have? https://pureadvantage.org/news/2016/11/15/underwater-robots/ Carnegie Mellon 6 DOF https://pureadvantage.org/news/2016/11/15/underwater-robots/ Carnegie Mellon How many degrees of freedom does this have? https://www.shopecobambino.com/maple-landmark-starter-train-set.html Carnegie Mellon 1 DOF https://www.shopecobambino.com/maple-landmark-starter-train-set.html Carnegie Mellon How many degrees of freedom does this have? http://hades.mech.northwestern.edu/index.php/File:Human-arm.png Carnegie Mellon 7 DOF http://hades.mech.northwestern.edu/index.php/File:Human-arm.png Carnegie Mellon Definitions Reference Frame: Static coordinate system from which translations and rotations are based Link: Single rigid body Joint: Connection between links Constraints: Limitations on movement http://www.value-design-consulting.co.uk/co-ordinate-systems.html http://www.andrew.cmu.edu/user/kbrennan/TeamZ-Lab9.html Carnegie Mellon Grübler’s Formula to find degrees of freedom Basic Idea: DOF of mechanism = Link DOFs – Joint Constraints Carnegie Mellon Grübler’s Formula to find degrees of freedom = (6 ) 6 − � − =1 M is the degrees of freedom n is the number of moving links j is the number of joints f_i is the degrees of freedom of the ith joint Carnegie Mellon Grübler’s Formula – Simple Open Chain = � =1 M is the degrees of freedom n is the number of moving links j is the number of joints f_i is the degrees of freedom of the ith joint http://hades.mech.northwestern.edu/index.php/File:Human-arm.png Carnegie Mellon Grübler’s Formula – Simple Closed Chain = � − =1 M is the degrees of freedom n is the number of moving links j is the number of joints f_i is the degrees of freedom of the ith joint d is the dimension, 3 for planar, 6 for spatial http://ocw.upm.es/ingenieria-mecanica/mechanical-devices-for- industry/contenidos/lectura-obligatoria/lesson-1/four-bar-linkages Carnegie Mellon Types of Joints – Lower Pairs Revolute Joint 1 DOF ( Variable - ) Prismatic Joint 1 DOF (linear) (Variables - d) Spherical Joint 3 DOF ( Variables - 1, 2, 3) Carnegie Mellon Types of Joints – Higher Pairs Gears 1 DOF ( Variable - ) Cam and Follower 1 DOF (linear) (Variables - d) https://nptel.ac.in/courses/112103174/module4/lec3/1.html Carnegie Mellon Grübler’s Formula to find degrees of freedom Carnegie Mellon We are interested in two kinematics topics Forward Kinematics Inverse Kinematics (angles to position) (position to angles) What you are given: What you are given: The length of each link The length of each link The angle of each joint The position of some point on the robot What you can find: What you can find: The position of any point The angles of each joint needed to (i.e. it’s (x, y, z) coordinates) obtain that position Carnegie Mellon Forward Kinematics (angles to position) tf x,y l2 l2 l1 l1 t1 Given l1, l2, t1, t2 Find x, y, tf Carnegie Mellon Inverse Kinematics (angles to position) tf x,y t2 l l2 2 l1 l1 t1 Given l1, l2, x, y, tf Find t1, t2 Carnegie Mellon Quick Math Review Vector: A geometric object with magnitude and direction Carnegie Mellon Quick Math Review Vector: A geometric object with magnitude and direction Examples of vector quantities: Velocity, displacement, acceleration, force Carnegie Mellon Quick Math Review Vector Magnitude: Just the vector quantity without direction Examples: Magnitude of velocity is speed, magnitude of displacement is distance, etc. Carnegie Mellon Quick Math Review Unit Vector: Vector with magnitude of 1 Used to indicate direction Carnegie Mellon Quick Math Review Vector: A geometric object with magnitude and direction y x Can be written in matrix form as a column vector Carnegie Mellon Quick Math Review Vector Addition - Sum each component of the vector y x Yields a new vector Commutative Carnegie Mellon Quick Math Review y x Yields a scalar Commutative Carnegie Mellon Quick Math Review Yields a vector perpendicular to both original vectors Not commutative https://en.wikipedia.org/wiki/Cross_product Carnegie Mellon Quick Math Review Matrix Addition - Sum matching elements a b e f (a + e) (b + f ) + = c d g h (c + g) (d + h) Matrices must be of same size Yields a new matrix of the same size Commutative Carnegie Mellon Quick Math Review Matrix Multiplication - Multiply rows and columns and sum products a b e f (ae + bg) (af + bh) ∗ = c d g h (ce + dg) (cf + dh) Matrices must have the same inner dimension Yields a new matrix of the same size Not commutative Carnegie Mellon We can use vectors to succinctly represent a point with respect to a certain reference frame Y X Carnegie Mellon We will use superscripts to indicate our reference frame Y O X M L N Carnegie Mellon Basic Transformations Representing a point in a different frame: Translation along the x-axis Y O N X Carnegie Mellon Basic Transformations Representing a point in a different frame: Translation along the x-axis Y O N X Carnegie Mellon Basic Transformations Representing a point in a different frame: Translation along the x-axis Y O N X Px Px = distance between the XY and NO coordinate planes Notation: Carnegie Mellon XY NO Writing V in terms of V Y O N X Px Carnegie Mellon XY NO Writing V in terms of V Y O N X PX Carnegie Mellon Basic Transformations Representing a point in a different frame: Translation along the x- and y-axes O Y N X Carnegie Mellon Basic Transformations Representing a point in a different frame: Translation along the x- and y-axes O Y N X Carnegie Mellon Using Basis Vectors Basis vectors are unit vectors that point along a coordinate axis O n Unit vector along the N-Axis o Unit vector along the N-Axis V NO Magnitude of the VNO vector o n N NO NO V N V cosθ V cosθ V NO • n NO = = = = V O NO NO NO V V sinθ V cos(90 − θ) V • o Carnegie Mellon Basic Transformations Representing a point in a different frame: Rotation about z-axis (out of the board) Y V VY X VX = Angle of rotation between the XY and NO coordinate axis Carnegie Mellon Y V VY X VX (Substituting for VNO using the N and O components of the vector) Carnegie Mellon Y V VY X VX (Substituting for VNO using the N and O components of the vector) Carnegie Mellon Basic Transformations Representing a point in a different frame: Rotation about z-axis (out of the board) Y V VY X VX Carnegie Mellon Compound Transformations Representing a point in a different frame: Translation along the x- and y-axes and rotation Y1 (VN,VO) Y0 VNO VXY 1 P X Translation along P followed by rotation by θ X0 X N V Px cosθ − sinθV XY = = + V Y O V Py sinθ cosθ V Carnegie Mellon Compound Transformations Representing a point in a different frame: Translation along the x- and y-axes and rotation Y1 (VN,VO) Y0 VNO VXY 1 P X Translation along P followed by rotation by θ X0 X N V Px cosθ − sinθV XY = = + V Y O V Py sinθ cosθ V (Note : Px, Py are relative to the original coordinate frame. Translation followed by rotation is different than rotation followed by translation.) Carnegie Mellon Relative versus absolute translation Relative: - Can be composed to create Y0 homogenous transformation X1 matrix. P - Translations are with respect to a frame fixed to the robot or X0 point. Y0 Absolute: - Translations are with respect to a X1 fixed world frame. P X0 Carnegie Mellon The Homogeneous Matrix can represent both translation and rotation X N V Px cosθ − sinθV What we found by doing a XY = = + V Y O translation and a rotation V Py sinθ cosθ V V X P cosθ − sinθ 0V N x = Y = + O Padding with 0’s and 1’s V Py sinθ cosθ 0V 1 0 0 0 1 1 X N V cosθ − sinθ Px V Y O = = Simplifying into a matrix form V sinθ cosθ Py V 1 0 0 1 1 cosθ − sinθ P x Homogenous Matrix for a Translation in XY = H sinθ cosθ Py plane, followed by a Rotation around the z-axis 0 0 1 Carnegie Mellon Rotation Matrices in 3D cosθ − sinθ 0 Rotation around the Z-Axis = R z sinθ cosθ 0 0 0 1 cosθ 0 sinθ = Rotation around the Y-Axis R y 0 1 0 − sinθ 0 cosθ 1 0 0 = − Rotation around the X-Axis R x 0 cosθ sinθ 0 sinθ cosθ Carnegie Mellon Homogeneous Matrices in 3D H is a 4x4 matrix that can describe a translation, rotation, or both in one matrix O 1 0 0 Px Y 0 1 0 Py N P H = 0 0 1 Pz A X 0 0 0 1 Translation without rotation Z nx ox ax 0 Y O n o a 0 H = y y y N n o a 0 z z z X 0 0 0 1 Could be rotation around Z Rotation without translation z-axis, x-axis, y-axis or a A combination of the three.
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