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Robot Vision SS 2005 Matthias Rüther 1 ROBOT VISION Lesson 9: Robot Kinematics Matthias Rüther

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Robot Vision SS 2005 Matthias Rüther 2 Contents Introduction –Overview –Basic Terms –Typical Configurations Geometry –Homogeneous Transformations –Representing Orientation Direct Kinematics –Open Chain –Denavit-Hartenberg –Typical Configurations Inverse Kinematics –Problem –Numerical Algorithms

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Robot Vision SS 2005 Matthias Rüther 3 Overview An industrial robot consists of –A manipulator that consists of a sequence of rigid bodies (links) connected by means of articulation (joints). It is characterized by an arm (for mobility), a wrist (for dexterity) and an end effector (to perform a task). –Actuators that set the manipulator in motion by actuating the joints (electric, hydraulic or pneumatic) –Sensors that measure the manipulators status (proprioceptive sensors) or the status of the environment (exteroceptive sensors). –A control system that controls and supervises manipulator motion.

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Robot Vision SS 2005 Matthias Rüther 4 Overview An industrial robot has three fundamental capacities –Material handling: objects are transported from one location to another, their physical characteristics are not altered (e.g. pallettizing, warehouse loading/unloading, sorting, packaging) –Material Manipulation: physical characteristics of objects are changed or their physical identity is lost (e.g. welding, painting, gluing, cutting, drilling, screwing, assembly, …) –Measurement: object properties are measured by sensors on the end effector, so they are able to explore a 3D environment (e.g. quality control by image processing, tactile sensors, …)

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Robot Vision SS 2005 Matthias Rüther 5 Basic Terms Kinematic chain: sequence of links and joints connecting the end effector to the base. Can be open (single chain) or closed (loop within the chain). Joints: –Revolute: rotational joint –Prismatic: translational joint Degree of mobility: each joint typ. provides an additional DOM (e.g. 6DOM necessary for 3D motion) Accuracy: „How close does the robot get to a desired point?“, distance between desired point and actual point. (off line) Repeatability: „How close does the robot get to a position it has reached before?“ (on line)

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Robot Vision SS 2005 Matthias Rüther 6 Repeatability and Accuracy Where :P t is the target position, P a the average achieved position, R is the repeatability, and A is the accuracy. Repeatability: the measurement of how closely a robot returns to the same position n number of times. Accuracy: the measurement of how closely the robot moves to a given target coordinate.

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Robot Vision SS 2005 Matthias Rüther 7 Many factors can effect the accuracy of a robot such as: –environmental factors ( eg temperature, humidity and electrical noise ); –kinematic parameters ( eg robot link lengths etc ); –dynamic parameters ( eg structural compliance, friction ); –measurement factors ( eg resolution and non-linearity of encoders and resolvers ); –computational factors ( eg digital round off, steady-state control errors ); and –application factors ( eg installation errors and errors in defining work-piece coordinate frames ).

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Robot Vision SS 2005 Matthias Rüther 8 Why? Each of these can effect the robot kinematics –depending on the robot model design and manufacturing process the kinematics can be the most significant source of error. –if the internal kinematic model of the robot does not match the actual model we can not accurately predict where the end-effector is. Make them match by either –tighter manufacturing tolerances; or –by measuring actual kinematics and incorporating this information into the kinematic model.

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Robot Vision SS 2005 Matthias Rüther 9 Basic Terms Joint space: end effector position and orientation given in joint coordinates. Operational space: end effector position and orientation given in world coordinates (e.g. cartesian coordinate system) Workspace: –Reachable WS: set of positions the EE can reach –Dexterous WS: set of positions the EE can reach with a given orientation Redundancy: more DOM than needed are available

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Robot Vision SS 2005 Matthias Rüther 10 Revolute Joint 1 DOF ( Variable - ) Prismatic Joint 1 DOF (linear) (Variables - d) Basic Joints

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Robot Vision SS 2005 Matthias Rüther 11 The PUMA 560 has SIX revolute joints A revolute joint has ONE degree of freedom ( 1 DOF) that is defined by its angle There are two more joints on the end effector (the gripper) An Example - The PUMA 560

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Robot Vision SS 2005 Matthias Rüther 12 Manipulator Structures

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Robot Vision SS 2005 Matthias Rüther 13 Manipulator Structures

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Robot Vision SS 2005 Matthias Rüther 14 Forward and Inverse Kinematics xwxw Joint 1 Joint 2 Joint 3 Link 1 z1z1 World (Base) Coordinate Frame Tool Coordinate Frame Joint 1 Joint 2 Joint 3 Link 1 z1z1 World (Base) Coordinate Frame Tool Coordinate Frame 11 22 zwzw ytyt ztzt Link Space n variables ( 1 … n ) Tool Space 6 variables ( x,y,z, x, y, z ) Forward K Inverse K

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Robot Vision SS 2005 Matthias Rüther 15 Co-ordinate Frames right handed coordinate frames used as reference frames x z y xwxw World (Base) Coordinate Frame Link frame zwzw x 2x 2 z 2z 2 Tool Coordinate Frame ytyt ztzt

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Robot Vision SS 2005 Matthias Rüther 16 F o r w a r d K i n e m a t i c s

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Robot Vision SS 2005 Matthias Rüther 17 The Situation: You have a robotic arm that starts out aligned with the x o -axis. You tell the first link to move by 1 and the second link to move by 2. The Quest: What is the position of the end of the robotic arm? Solution: 1. Geometric Approach This might be the easiest solution for the simple situation. However, notice that the angles are measured relative to the direction of the previous link. (The first link is the exception. The angle is measured relative to it’s initial position.) For robots with more links and whose arm extends into 3 dimensions the geometry gets much more tedious. 2. Algebraic Approach Involves coordinate transformations.

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Robot Vision SS 2005 Matthias Rüther 18 X2X2 X3X3 Y2Y2 Y3Y3 11 22 3 Example Problem: You are have a three link arm that starts out aligned in the x- axis. Each link has lengths l 1, l 2, l 3, respectively. You tell the first one to move by 1, and so on as the diagram suggests. Find the Homogeneous matrix to get the position of the yellow dot in the X 0 Y 0 frame. H = R z ( 1 ) * T x1 (l 1 ) * R z ( 2 ) * T x2 (l 2 ) * R z ( 3 ) i.e. Rotating by 1 will put you in the X 1 Y 1 frame. Translate in the along the X 1 axis by l 1. Rotating by 2 will put you in the X 2 Y 2 frame. and so on until you are in the X 3 Y 3 frame. The position of the yellow dot relative to the X 3 Y 3 frame is (l 1, 0). Multiplying H by that position vector will give you the coordinates of the yellow point relative the the X 0 Y 0 frame. X1X1 Y1Y1 X0X0 Y0Y0

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Robot Vision SS 2005 Matthias Rüther 19 Slight variation on the last solution: Make the yellow dot the origin of a new coordinate X 4 Y 4 frame X2X2 X3X3 Y2Y2 Y3Y3 11 22 3 X1X1 Y1Y1 X0X0 Y0Y0 X4X4 Y4Y4 H = R z ( 1 ) * T x1 (l 1 ) * R z ( 2 ) * T x2 (l 2 ) * R z ( 3 ) * T x3 (l 3 ) This takes you from the X 0 Y 0 frame to the X 4 Y 4 frame. The position of the yellow dot relative to the X 4 Y 4 frame is (0,0). Notice that multiplying by the (0,0,0,1) vector will equal the last column of the H matrix.

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Robot Vision SS 2005 Matthias Rüther 20 More on Forward Kinematics… Denavit - Hartenberg Parameters

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Robot Vision SS 2005 Matthias Rüther 21 Z (i - 1) X (i -1) Y (i -1) ( i - 1) a (i - 1 ) Z i Y i X i a i d i i IDEA: Each joint is assigned a coordinate frame. Using the Denavit- Hartenberg notation, you need 4 parameters to describe how a frame (i) relates to a previous frame ( i -1 ). THE PARAMETERS/VARIABLES: , a, d, Denavit-Hartenberg Notation

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Robot Vision SS 2005 Matthias Rüther 22 Z (i - 1) X (i -1) Y (i -1) ( i - 1) a (i - 1 ) Z i Y i X i a i d id i i You can align the two axis just using the 4 parameters 1) a (i-1) Technical Definition: a (i-1) is the length of the perpendicular between the joint axes. The joint axes is the axes around which revolution takes place which are the Z (i-1) and Z (i) axes. These two axes can be viewed as lines in space. The common perpendicular is the shortest line between the two axis-lines and is perpendicular to both axis-lines. The Parameters

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Robot Vision SS 2005 Matthias Rüther 23 Z (i - 1) X (i -1) Y (i -1) ( i - 1) a (i - 1 ) Z i Y i X i a i d id i i a (i-1) cont... Visual Approach - “A way to visualize the link parameter a (i-1) is to imagine an expanding cylinder whose axis is the Z (i-1) axis - when the cylinder just touches the joint axis i the radius of the cylinder is equal to a (i-1). ” (Manipulator Kinematics) It’s Usually on the Diagram Approach - If the diagram already specifies the various coordinate frames, then the common perpendicular is usually the X (i-1) axis. So a (i-1) is just the displacement along the X (i-1) to move from the (i-1) frame to the i frame. If the link is prismatic, then a (i-1) is a variable, not a parameter.

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Robot Vision SS 2005 Matthias Rüther 24 2) (i-1) Technical Definition: Amount of rotation around the common perpendicular so that the joint axes are parallel. i.e. How much you have to rotate around the X (i-1) axis so that the Z (i-1) is pointing in the same direction as the Z i axis. Positive rotation follows the right hand rule. 3) d (i-1) Technical Definition: The displacement along the Z i axis needed to align the a (i-1) common perpendicular to the a i common perpendicular. In other words, displacement along the Z i to align the X (i-1) and X i axes. 4) i Amount of rotation around the Z i axis needed to align the X (i-1) axis with the X i axis. Z (i - 1) X (i -1) Y (i -1) ( i - 1) a (i - 1 ) Z i Y i X i a i d id i i

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Robot Vision SS 2005 Matthias Rüther 25 Just like the Homogeneous Matrix, the Denavit-Hartenberg Matrix is a transformation matrix from one coordinate frame to the next. Using a series of D-H Matrix multiplications and the D-H Parameter table, the final result is a transformation matrix from some frame to your initial frame. Z (i - 1) X (i -1) Y (i -1) ( i - 1) a (i - 1 ) Z i Y i X i a i d id i i The Denavit-Hartenberg Matrix

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Robot Vision SS 2005 Matthias Rüther 26 Z0Z0 X0X0 Y0Y0 Z1Z1 X2X2 Y1Y1 Z2Z2 X1X1 Y2Y2 d2d2 a0a0 a1a1 Denavit-Hartenberg Link Parameter Table Notice that the table has two uses: 1) To describe the robot with its variables and parameters. 2) To describe some state of the robot by having a numerical values for the variables. 3 Revolute Joints

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Robot Vision SS 2005 Matthias Rüther 27 Z0Z0 X0X0 Y0Y0 Z1Z1 X2X2 Y1Y1 Z2Z2 X1X1 Y2Y2 d2d2 a0a0 a1a1 Note: T is the D-H matrix with (i-1) = 0 and i = 1.

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Robot Vision SS 2005 Matthias Rüther 28 This is just a rotation around the Z 0 axis This is a translation by a 0 followed by a rotation around the Z 1 axis This is a translation by a 1 and then d 2 followed by a rotation around the X 2 and Z 2 axis

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Robot Vision SS 2005 Matthias Rüther 29 Review of Direct Kinematics What we have: –Geometric layout of robot (constant under robot motion) –Joint positions q nx1 (variable under robot motion) What we want: –End Effector (EE) position and orientation wrt base coordinate frame (variable under robot motion) A coordinate transformation T from EE coordinates to base coordinates, where T is a function of q: –E.g. position of EE coordinate origin is:

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Robot Vision SS 2005 Matthias Rüther 30 Review of Direct Kinematics How we get there: –Place local coordinate frame at each joint i, i=1..n: Determine common normal (CN) between joint i and i-1 (i.e. line connecting both joint axes in the shortest possible way) Place coordinate origin o i-1 at intersection of CN and joint axis i-1 –Attention Sciavicco/Siciliano: o i-1 at intersection of CN and joint axis i! Set z-axis in direction of joint axis Set x-axis in direction of CN –Model robot arm using Denavit-Hartenberg (DH) parameters: a: shortest distance between consecutive joint axes (length of common normal CN). : angle between consecutive joint axes around CN. d: distance between consecutive coordinate origins along i-th joint axis (distance between i-th coordinate origin and intersection of CN with i-th joint axis. y or : angle between consecutive CN‘s around joint axis. Depending on joint type, either d or y are variables.

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Robot Vision SS 2005 Matthias Rüther 31 Review of Direct Kinematics –From DH-table, calculate T recursively:

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Robot Vision SS 2005 Matthias Rüther 32 Inverse Kinematics ( q 1 … q n ) ( x,y,z, x, y, z ) K -1 From Position to Angles

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Robot Vision SS 2005 Matthias Rüther 33 Inverse Kinematics What we have: –Geometric layout of robot (constant under robot motion) –End Effector (EE) position and orientation wrt base coordinate frame (variable under robot motion) What we want: –Joint coordinates q nx1 (variable under robot motion). A mapping f from EE position and orientation to joint coordinates: Problematic, because: –EE position may be out of reachable workspace (no solution) –EE position and orientation may be out of dexterous workspace (no solution) –EE position may be reachable in several ways (multiple solutions, redundancy) –EE position may be reachable in infinitely many ways (infinite number of solutions, redundancy) –f is highly nonlinear, no closed form solution for complicated canfigurations.

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Robot Vision SS 2005 Matthias Rüther 34 Inverse Kinematics How we get there: –Calculate closed-form solution, if possible fast accurate unique for every arm configuration (no general concept) –Use Iterative Method based on differential kinematics slower gives approximate solution converges to a single solution (even if multiple solutions exist) general algorithm, applicable to all arm configurations

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Robot Vision SS 2005 Matthias Rüther 35 Multiple Solutions Because we are dealing with transcendental functions there is often more then one solution. eg: cos -1 (0.5) = 60º and 300º

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Robot Vision SS 2005 Matthias Rüther 36 Insoluble

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Robot Vision SS 2005 Matthias Rüther 37 Inverse Kinematics Solution easier if the three joint axes intersect at one point => a closed form or analytical solution possible (Pieper’s solution). Otherwise use a numerical (iterative) solution where: speed is poor; and we may not be able to calculate all solutions. x5x5 y5y5 x6x6 z6z6 x3x3 y3y3 x4x4 z4z4

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Robot Vision SS 2005 Matthias Rüther 38 11 X Y S Revolute and Prismatic Joints Combined (x, y) Finding : More Specifically : arctan2() specifies that it’s in the first quadrant Finding S : A Simple Example

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Robot Vision SS 2005 Matthias Rüther 39 22 11 (x, y) l2l2 l1l1 Given: l 1, l 2, x, y Find: 1, 2 Redundancy: A unique solution to this problem does not exist. Sometimes no solution is possible. (x, y) l2l2 l1l1 l2l2 l1l1 Inverse Kinematics of a Two Link Manipulator

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Robot Vision SS 2005 Matthias Rüther 40 l1l1 l2l2 22 11 (x, y) Using the Law of Cosines: Redundant since 2 could be in the first or fourth quadrant. Redundancy caused since 2 has two possible values The Geometric Solution

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Robot Vision SS 2005 Matthias Rüther 41 l1l1 l2l2 22 11 (x, y) Only Unknown The Algebraic Solution

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Robot Vision SS 2005 Matthias Rüther 42 We know what 2 is from the previous slide. We need to solve for 1. Now we have two equations and two unknowns (sin 1 and cos 1 ) Substituting for c 1 and simplifying many times Notice this is the law of cosines and can be replaced by x 2 + y 2

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Robot Vision SS 2005 Matthias Rüther 43 A General Solution A general solution to the IK problem is obtained using differential kinematics. There exists a linear mapping between EE velocity and Joint velocity. Joint positions can be computed by integrating velocities over time.

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Robot Vision SS 2005 Matthias Rüther 44 Differential Kinematics What we have: –Geometric layout of robot (constant under robot motion) –Joint velocities What we want: –Linear velocity –Angular velocity A mapping J from joint velocities to EE velocities:

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Robot Vision SS 2005 Matthias Rüther 45 The Jacobian Jacobian matrix J(q) gives linear and angular velocity (p‘, ) as a function of joint velocity (q‘) Depends on the location in joint space, hence J(q) Can be computed via closed formula.

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Robot Vision SS 2005 Matthias Rüther 46 Derivative of a Rotation Matrix Consider rotation matrix R(t), what is its derivative? Interpretation of S: S is the cross-matrix of angular velocities

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Robot Vision SS 2005 Matthias Rüther 47 Example: rotation about z-axis We want to compute the time-derivative of R z (t)

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Robot Vision SS 2005 Matthias Rüther 48 Velocity composition rule Consider the coordinate transformation of a point P from coordinate frame 1 to frame 0: point wrt 0 point wrt 1origin of 1 wrt 0rotation of 1 wrt 0 If p 1 is fixed:

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Robot Vision SS 2005 Matthias Rüther 49 Link Velocity Consider link i connecting joints i and i+1 with their coordinate frames Let p i and p i+1 be the coordinate frame origins wrt base coordinate frame Let r i i,i+1 be the coordinate frame origin of i+1 wrt coordinate frame i, expressed in coordinate frame i Lin. velocity of i+1 Lin. velocity of iLin. velocity of i+1 wrt i Angular velocity of i … angular velocity (without derivation) Angular velocity of i+1 Angular velocity of i Angular velocity of i+1 wrt i

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Robot Vision SS 2005 Matthias Rüther 50 Link Velocity For prismatic joint: For revolute joint:

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Robot Vision SS 2005 Matthias Rüther 51 Computation of Jacobian Let the Jacobian J be partitioned into the following 3x1 vectors: Prismatic joint: Revolute joint: … contribution of joint i to EE linear velocity … contribution of joint i to EE angular velocity

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Robot Vision SS 2005 Matthias Rüther 52 Computation of Jacobian … if prismatic … if revolute

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Robot Vision SS 2005 Matthias Rüther 53 Example: three link planar arm s i…j, c i…j denote sin(q i +…+q j ), cos(q i +…+q j )

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Robot Vision SS 2005 Matthias Rüther 54 Iterative Inverse Kinematics 1.Specify target transformation T you want to reach. 2.Make an initial guess for joint coords at time t k : q(t k ). 3.Compute T(t k ) using forward kinematics. 4.Compute differential motion required to move from T(t k ) to T: 5.Compute pseudo-inverse of Jacobian J -1 at the position of q(t k ). 6.Add Joint motion to initial guess, make result new initial guess. 7.Repeat 2-6 until differential motion is satisfactorily small.

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Robot Vision SS 2005 Matthias Rüther 55 Example: Robotics Toolbox

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