Robotics Chapter 3 – Forward Kinematics Dr. Amit Goradia

Topics Introduction – 2 hrs Coordinate transformations – 6 hrs Forward Kinematics - 6 hrs Inverse Kinematics - 6 hrs Velocity Kinematics - 2 hrs Trajectory Planning - 6 hrs Robot Dynamics (Introduction) - 2 hrs Force Control (Introduction) - 1 hrs Task Planning - 6 hrs

Robot – Mechanical Structure

Degrees Of Freedom The degrees of freedom of a rigid body is defined as the number of independent movements it has. The number of : Independent position variables needed to locate all parts of the mechanism, Different ways in which a robot arm can move, Joints

Forward Kinematics Given the various joint and link parameters, compute the end-effector (tool) location in task space. Given joint variables End-effector position and orientation, -Formula? or Joint Frame World Forward Inverse

Kinematic Chains A robot is a set of rigid links connected at joints All joints have a single degree of freedom Joints can be Revolute – rotation about one axis Variable: angle of rotation Prismatic – linear motion along an axis Variable: amount of linear displacement. More complex joints which are a combination of one or more revolute or prismatic joints can be thought of as a succession of 1DOF revolute or prismatic joints with zero link length between them.

Kinematic Chains Rigid bodies Kinematic Pairs of Rigid Bodies Robot Manipulators Kinematic Chains : Link + Joint Rigid bodies Kinematic Pairs of Rigid Bodies Basic Topologies of Kinematic Chain Necklace Open Chain Tree

Kinematic Chain Representation n+1 link manipulator Base of robot is link 0 Joints are numbered 1 to n joint variable denoted by Attach a coordinate frame to each link (total n+1 coordinate frame) is the homogenous transformation from coordinate frame to

Kinematic Chain Representation is a variable matrix with 1 variable Transformation Matrix Forward Kinematics: Represent a point on end effector w.r.t. base as a function of the joint variables Attach a coordinate frame to each link and represent the end effector using the transformation between various coordinate frames

Link Parameters Link Length Link twist Is the shortest distance measured along the common normal between joint axes The common normal always exists and is unique except when both axes are parallel. Link twist The angle between the projection of axis of joint i and axis of joint i+1 on the plane perpendicular to the common normal between the two joints

Joint Parameters A joint axis is established at the connection of two links. This joint will have two normals connected to it: one for each of the links. The relative position of two links is called link offset dn which is the distance between the links (the displacement, along the joint axes between the links). The joint angle between the normals is measured in a plane normal to the joint axis.

Link and Joint Parameters

Link and Joint Parameters ai = link length = the distance from zi to zi+1.measured along xi. ai = link twist = the angle between zi and zi+1.measured about xi. di = link offset = the distance from xi-1 to xi. measured along z i. qi = joint angle = the angle between xi-1 to xi.measured about z i Note: Since the joint has only 1 DOF, 3 out of the 4 parameters are constant and 1 represents the state of the link

Links and Joints For Revolute Joints: a, , and d. are all fixed, then “i” is the Joint Variable. For Prismatic Joints: a, , and . are all fixed, then “di” is the

Denavitt-Hartenberg Convention Systematic method to choose coordinate axes attached to links and represent the homogenous transformation between successive coordinate frames Analysis is possible without D-H, but it helps to be systematic Transformation is a composition of 4 basic transformations:

D-H Transformation In D-H convention, a general transformation between two bodies is defined as the product of four basic transformations: A translation along the initial z axis by d, A rotation about the initial z axis by q, A translation along the new x axis by a, and. A rotation about the new x axis by a.

Important Note Homogenous transformation between coordinate systems have 6 parameters How come Ai has only 4 parameters??? This is done by eliminating 2 or more parameters by choice of location of coordinate frame Note that Ai is only comprised of a rotation and translation about only the X and Z axes. We place the coordinate frames such that there is no movement about the Y axes. This is codified by using DH conventions.

Existence and Uniqueness There exists a unique transform between adjacent frames Conditions to follow: DH1: axis xn perpendicular to zn-1 DH2: axis xn intersects zn-1 Show that given we can, in all cases, uniquely derive the link parameters and vice versa

Existence and Uniquness Satisfying DH1: Column 1 of is an expression of in frame 0 Now show that there exist unique angles and such that Row and column vectors of R are orthonormal So and have a unique solution Given and we can easily show that remaining elements of R have the said form

Existence and Uniqueness Satisfying DH2: Displacement between o0 and o1 can be expressed as a linear combination of vectors and Combining results from DH1 and DH2, we can show that the link parameters can be uniquely determined from Homogenous transformation matrix

The Arm Matrix The arm matrix is the composite transformation from the base frame to tool frame n, s, a, p are respectively called the Normal, Sliding, Approach and Position Vectors

Link Numbering 2 3 1 A 3-DOF Manipulator Arm Base of the arm: Link-0 1st moving link: Link-1 . . Last moving link Link-n A 3-DOF Manipulator Arm 1 2 3

Assigning Coordinate Frames Assign Zi axes Identify all revolute joints and assign Zi-1 axis as axis of rotation Identify all prismatic joints and assign Zi-1 axis as axis of translation Xi is chosen perpendicular to both Zi and Zi-1 If zi and zi-1 donot intersect, xi is the common normal of Zi and Zi-1. Its positive direction is from zi-1 to zi If zi and zi-1 intersect, assignment of Xi can have 2 solutions If zi-1 and zi are parallel, then there are infinitely many solutions Yi is choosen to complete the right handed set

Assignment of X axis

DH Algorithm – Pass 1 Assign Coordinate Frames

DH Algorithm – Pass 2 Make Kinematic Parameter Table

Link Coordinate Diagram Example 1 4 DOF SCARA type robot

Kinematic Parameter Table Example 1

Kinematic Parameter Table Example 1 Location of sliding axis Z2 is arbitrary. For simplicity, make it coincident with Z3 . Thus a2 and d2 (d2 = 0) are arbitrarily set. Placement of O3 and X3 along Z3 is arbitrary, since Z2 and Z3 are coincident. Once we choose O3, the joint displacement d3 is defined. We have also placed the end link frame in a convenient manner, with the Z4 axis coincident with the Z3 axis and the origin O4 displaced down into the gripper by d4.

Link Coordinate Diagram Example 2 5 DOF Articulated

Kinematic Parameter Table Example 2

Example 3 – 3DOF Anthropomorphic Arm

Example 4 – Alpha II

Example 4 – Micro Alpha II z0 x0 x1 z1 x2 z2 x3 z3 x4 z4 x5 z5 shoulder elbow Tool pitch Tool roll base Tool 1 5 2 3 4

Example 4 – Micro Alpha II Link Var  d  a 1 1 215 -/2 2 2 177.8 3 3 4 4 4-/2 5 5 129.5

Example 4 – Micro Alpha II

Example 5 a0 a1 d2 Joint 3 Joint 1 Joint 2 Z0 X0 Y0 Z3 X2 Y1 X1 Y2 Z1

Example 5

Tool Configuration Vector Another method to express position and orientation Approach vector (a) represents the tool pitch and yaw angles. Also, a is a unit vector. So, append tool roll info into approach vector and we have another representation for position Note: existence and uniqueness of inverse transformation needs to be established.

TCV Scaling function for roll must be invertible TCV