1 ROBOT VISION Lesson 9: Robot Kinematics Matthias Rüther Kawada Industries Inc. has introduced the HRP-2P for Robodex 2002. This humanoid appears to be very impressive. It is 154 cm (60") tall, weighs 58kg (127 lbs) and has 30 DOF. Here is a news release. Notice the LACK of a battery pack. Here is a new story about HRP2.
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.
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, …)
5 Basic TermsKinematic 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 jointPrismatic: translational jointDegree 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)
6 Repeatability and 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.Where: Pt is the target position,Pa the average achieved position,R is the repeatability, andA is the accuracy.
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); andapplication factors (eg installation errors and errors in defining work-piece coordinate frames).
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 eithertighter manufacturing tolerances; orby measuring actual kinematics and incorporating this information into the kinematic model.
9 Basic TermsJoint 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 reachDexterous WS: set of positions the EE can reach with a given orientationRedundancy: more DOM than needed are available
11 An Example - The PUMA 560 2 3 4 1 The PUMA 560 has SIX revolute joints There are two more joints on the end effector (the gripper)The PUMA 560 has SIX revolute jointsA revolute joint has ONE degree of freedom ( 1 DOF) that is defined by its angle
14 Forward and Inverse Kinematics xwJoint 1Joint 2Joint 3Link 1z1World (Base) Coordinate FrameTool Coordinate Frameq1q2zwytztLink Spacen variables (q1 … qn)Tool Space6 variables (x,y,z,qx,qy,qz)Forward KAnother way to think of this is a transform between Link space (i.e. variables such as angles and distances) to Tool space (i.e. the x,z coordinates and orientation angles of the tool frame), and back again.Inverse K
15 Co-ordinate Frames z y x z 2 x 2 right handed coordinate frames used as reference framesxzyxwWorld (Base) Coordinate FrameLink framezwx 2z 2Tool Coordinate FrameytztThe coordinate frames are a set of three orthogonal unit vectors.We use them as reference frames and attach them to each link on the robot, so that each link has its own reference point.
16 F o r w a r d K i n e m a t i c sF o r w a r d K i n e m a t i c s
17 The Situation:You have a robotic arm that starts out aligned with the xo-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 ApproachThis 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 ApproachInvolves coordinate transformations.
18 Example Problem:You are have a three link arm that starts out aligned in the x-axis. Each link has lengths l1, l2, l3, 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 X0Y0 frame.Y3323Y2X32X2H = Rz(1 ) * Tx1(l1) * Rz(2 ) * Tx2(l2) * Rz(3 )i.e. Rotating by 1 will put you in the X1Y1 frame.Translate in the along the X1 axis by l1.Rotating by 2 will put you in the X2Y2 frame.and so on until you are in the X3Y3 frame.The position of the yellow dot relative to the X3Y3 frame is(l1, 0). Multiplying H by that position vector will give you thecoordinates of the yellow point relative the the X0Y0 frame.Y01X11Y1X0
19 Slight variation on the last solution: Make the yellow dot the origin of a new coordinate X4Y4 frameY3Y4323Y2X32X2X4H = Rz(1 ) * Tx1(l1) * Rz(2 ) * Tx2(l2) * Rz(3 ) * Tx3(l3)This takes you from the X0Y0 frame to the X4Y4 frame.The position of the yellow dot relative to the X4Y4 frame is (0,0).Y01X11Y1X0Notice that multiplying by the (0,0,0,1) vector will equal the last column of the H matrix.
20 Denavit - Hartenberg Parameters More on Forward Kinematics…Denavit - Hartenberg Parameters
21 Denavit-Hartenberg Notation Z(i - 1)Y(i -1)Y iZ iX ia ia(i - 1 )d iX(i -1) i( i - 1)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,
22 The ParametersYou can align the two axis just using the 4 parametersZ(i - 1)Y(i -1)Y iZ iX ia ia(i - 1 )d iX(i -1) i( i - 1)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.
23 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.Z(i - 1)X(i -1)Y(i -1)( i - 1)a(i - 1 )Z iY iX ia id i i
24 Technical Definition: The displacement 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 Zi axis. Positive rotation follows the right hand rule.3) d(i-1)Technical Definition: The displacementalong the Zi axis needed to align the a(i-1)common perpendicular to the ai commonperpendicular.In other words, displacement along theZi to align the X(i-1) and Xi axes.4) iAmount of rotation around the Zi axis needed to align the X(i-1) axis with the Xi axis.Z(i - 1)X(i -1)Y(i -1)( i - 1)a(i - 1 )Z iY iX ia id i i
25 The Denavit-Hartenberg Matrix 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)Y(i -1)Y iZ iX ia ia(i - 1 )d iX(i -1) i( i - 1)
26 Denavit-Hartenberg Link Parameter Table 3 Revolute JointsZ0X0Y0Z1X2Y1Z2X1Y2d2a0a1Denavit-Hartenberg Link Parameter TableNotice 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.
27 d2 a0 a1 Note: T is the D-H matrix with (i-1) = 0 and i = 1. Z0 X0 Y0
28 This is just a rotation around the Z0 axis This is a translation by a0 followed by a rotation around the Z1 axisThis is a translation by a1 and then d2 followed by a rotation around the X2 and Z2 axis
29 Review of Direct Kinematics What we have:Geometric layout of robot (constant under robot motion)Joint positions qnx1 (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:
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 oi-1 at intersection of CN and joint axis i-1Attention Sciavicco/Siciliano: oi-1 at intersection of CN and joint axis i!Set z-axis in direction of joint axisSet x-axis in direction of CNModel 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.
31 Review of Direct Kinematics From DH-table, calculate T recursively:
32 From Position to Angles Inverse KinematicsFrom Position to AnglesK-1(q1 … qn)(x,y,z,qx,qy,qz)We do inverse kinematics unwittingly, our eyes can determine where an object is in 3D space, and our sub-sub-conscious can figure out the variables required to move our hand to that position (or the tool in our hand to that position…gosh we are clever!)But for robots we have to think a bit harder and mathematically, not intuitively!
33 Inverse Kinematics What we have: What we want: Problematic, because: 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 qnx1 (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.
34 Inverse Kinematics How we get there: Calculate closed-form solution, if possiblefastaccurateunique for every arm configuration (no general concept)Use Iterative Method based on differential kinematicsslowergives approximate solutionconverges to a single solution (even if multiple solutions exist)general algorithm, applicable to all arm configurations
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
37 Inverse Kinematics x4 x3 z4 y3 x6 x5 z6 y5 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; andwe may not be able to calculate all solutions.x4z4x3y3x5y5x6z6
38 A Simple Example Finding : Revolute and Prismatic Joints Combined More Specifically:(x , y)arctan2() specifies that it’s in the first quadrantYFinding S:S1X
39 Inverse Kinematics of a Two Link Manipulator Given: l1, l2 , x , yFind: 1, 2Redundancy:A unique solution to this problem does not exist.Sometimes no solution is possible.(x , y)2l2l11l2l1(x , y)l2l1
40 The Geometric Solution Using the Law of Cosines:2(x , y)l11Using the Law of Cosines:Redundant since 2 could be in the first or fourth quadrant.Redundancy caused since 2 has two possible values
41 The Algebraic Solution (x , y)l22l11Only Unknown
42 Substituting for c1 and simplifying many times 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 c1 and simplifying many timesNotice this is the law of cosines and can be replaced by x2+ y2
43 A General SolutionA 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.
44 Differential Kinematics What we have:Geometric layout of robot (constant under robot motion)Joint velocitiesWhat we want:Linear velocityAngular velocityA mapping J from joint velocities to EE velocities:
45 The JacobianJacobian 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.
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
47 Example: rotation about z-axis We want to compute the time-derivative of Rz(t)
48 Velocity composition rule Consider the coordinate transformation of a point P from coordinate frame 1 to frame 0:point wrt 0origin of 1 wrt 0rotation of 1 wrt 0point wrt 1If p1 is fixed:
49 Link VelocityConsider link i connecting joints i and i+1 with their coordinate framesLet pi and pi+1 be the coordinate frame origins wrt base coordinate frameLet rii,i+1 be the coordinate frame origin of i+1 wrt coordinate frame i, expressed in coordinate frame iLin. velocity of iLin. velocity of i+1 wrt iAngular velocity of iLin. velocity of i+1… angular velocity (without derivation)Angular velocity of i+1Angular velocity of iAngular velocity of i+1 wrt i
50 Link VelocityFor prismatic joint:For revolute joint:
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
52 Computation of Jacobian … if prismatic… if revolute
53 Example: three link planar arm si…j, ci…j denote sin(qi+…+qj), cos(qi+…+qj)
54 Iterative Inverse Kinematics Specify target transformation T you want to reach.Make an initial guess for joint coords at time tk: q(tk).Compute T(tk) using forward kinematics.Compute differential motion required to move from T(tk) to T:Compute pseudo-inverse of Jacobian J-1 at the position of q(tk).Add Joint motion to initial guess, make result new initial guess.Repeat 2-6 until differential motion is satisfactorily small.