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## Kinematics

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**Amirkabir University of TechnologyComputer Engineering &**Information Technology Department Kinematics Time to Derive Kinematics Model of the Robotic Arm**Direct Kinematics**Where is my hand? Direct Kinematics: HERE!**Kinematics of Manipulators**Objective: • To drive a method to compute the position and orientation of the manipulator’s end-effector relative to the base of the manipulator as a function of the joint variables.**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**DOF of a Rigid Body**In a plane In space**Degrees of Freedom**3 position As DOF 3D Space = 6 DOF 3 orientation In robotics: DOF = number of independently driven joints positioning accuracy computational complexity cost flexibility power transmission is more difficult**Robot Links and Joints**A manipulator may be thought of as a set of bodies (links) connected in a chain by joints. • In open kinematics chains (i.e. Industrial Manipulators): {No of D.O.F. = No of Joints}**Lower Pair**• The connection between a pair of bodies when the relative motion is characterized by two surfaces sliding over one another**Higher Pair**• A higher pair joint is one which contact occurs only at isolated points or along a line segments**Robot Joints**Revolute Joint 1 DOF ( Variable - q) Spherical Joint 3 DOF ( Variables - q1, q2, q3) Prismatic Joint 1 DOF (linear) (Variables - d)**Robot Specifications**Number of axes • Major axes, (1-3) => position the wrist • Minor axes, (4-6) => orient the tool • Redundant, (7-n) => reaching around obstacles, avoiding undesirable configuration**An Example - The PUMA 560**2 3 1 4 The PUMA 560 hasSIXrevolute 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)**Note on Joints**Without loss of generality, we will consider only manipulators which have joints with a single degree of freedom. A joint having n degrees of freedom can be modeled as n joints of one degree of freedom connected with n-1 links of zero length.**zn+1**zn zn qn+1 an xn+1 qn xn xn Link n Joint n+1 Joint n Link • A link is considered as a rigid body which defines the relationship between two neighboring joint axes of a manipulator.**The Kinematics Function of a Link**The kinematics function of a link is to maintain a fixed relationship between the two joint axes it supports. This relationship can be described with two parameters: the link length a, the link twist a**Link Length**Is measured along a line which is mutually perpendicular to both axes. The mutually perpendicular always exists and is unique except when both axes are parallel.**Link twist**Project both axes i-1 and i onto the plane whose normal is the mutually perpendicular line, and measure the angle between them Right-hand sense**Axis i**Axis i-1 ai-1 i-1 Link Length and Twist**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 offsetdn whish is the distance between the links (the displacement, along the joint axes between the links). • The joint angleqn between the normals is measured in a plane normal to the joint axis.**Axis i**Axis i-1 di ai-1 i-1 Link and Joint Parameters ai-1 i**Link and Joint Parameters**4 parameters are associated with each link. You can align the two axis using these parameters. • Link parameters: a0 the length of the link. anthe twist angle between the joint axes. • Joint parameters: qn the angle between the links. dn the distance between the links**Link Connection Description:**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. Joint Variable. These four parameters: (Link-Length ai-1), (Link-Twist i-1(, (Link-Offset di), (Joint-Angle i)are known as theDenavit-HartenbergLink Parameters.**2**3 1 0 A 3-DOF Manipulator Arm Links Numbering Convention Base of the arm: Link-0 1st moving link: Link-1 . . . . . . Last moving link Link-n Link 2 Link 3 Link 1 Link 0**First and Last Links in the Chain**• a0= an=0.0 • a0= an=0.0 • If joint 1 is revolute: d0= 0 and q1 is arbitrary • If joint 1 is prismatic: d0= arbitraryand q1 = 0**Affixing Frames to Links**In order to describe the location of each link relative to its neighbors we define a frame attached to each link. • The Z axis is coincident with the joint axis i. • The origin of frame is located where ai perpendicular intersects the joint i axis. • The X axis points along ai( from i to i+1). • If ai = 0 (i.E. The axes intersect) then Xiis perpendicular to axes i and i+1. • The Y axis is formed by right hand rule.**Affixing Frames to Links**First and last links • Base frame (0) is arbitrary • Make life easy • Coincides with frame {1} when joint parameter is 0 • Frame {n} (last link) • Revolute joint n: • Xn= Xn-1 when qn = 0 • Origin {n} such that dn=0 • Prismatic joint n: • Xn such that qn = 0 • Origin {n} at intersection of joint axis n and Xnwhen dn=0**Joint n**Joint n-1 Link n Joint n+1 Link n-1 zn zn+1 xn zn-1 an yn-1 dn an xn+1 an-1 yn yn+1 xn-1 Affixing Frames to Links**Affixing Frames to Links**Note: assign link frames so as to cause as many link parameters as possible to become zero! The reference vector z of a link-frame is always on a joint axis. The parameter di is algebraic and may be negative. It is constant if joint i is revolute and variable when joint i is prismatic. The parameter ai is always constant and positive. a i is always chosen positive with the smallest possible magnitude.**The Kinematics Model**The robot can now be kinematically modeled by using the link transforms ie: Where 0nT is the pose of the end-effector relative to base; Tiis the link transform for the ith joint; and nis the number of links.**The Denavit-Hartenberg (D-H) Representation**• In the robotics literature, the Denavit-Hartenberg (D-H) representation has been used, almost universally, to derive the kinematic description of robotic manipulators.**The Denavit-Hartenberg (D-H) Representation**• The appeal of the D-H representation lies in its algorithmic approach. • The method begins with a systematic approach to assigning and labeling an orthonormal (x,y,z) coordinate system to each robot joint. It is then possible to relate one joint to the next and ultimately to assemble a complete representation of a robot's geometry.**Axis i-1**ai-1 i-1 Denavit-Hartenberg Parameters Axis i i Link i di**The Link Parameters**ai = the distance from zi to zi+1. measured along xi. ai = the angle between zi and zi+1. measured about xi. di = the distance from xi-1 to xi. measured along zi. qi = the angle between xi-1 to xi. measured about zi**General Transformation Between Two Bodies**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.**A General Transformation in D-H Convention**D-H transformation for adjacent coordinate frames:**Denavit-Hartenberg Convention**• D1. Establish the base coordinate system. Establish a right-handed orthonormal coordinate system at the supporting base with axis lying along the axis of motion of joint 1. • D2. Initialize and loop Steps D3 to D6 for I=1,2,….n-1 • D3. Establish joint axis. Align the Zi with the axis of motion (rotary or sliding) of joint i+1. • D4. Establish the origin of the ith coordinate system. Locate the origin of the ith coordinate at the intersection of the Zi & Zi-1 or at the intersection of common normal between the Zi & Zi-1 axes and the Zi axis. • D5. Establish Xi axis. Establish or along the common normal between the Zi-1 & Zi axes when they are parallel. • D6. Establish Yi axis. Assign to complete the right-handed coordinate system.**Denavit-Hartenberg Convention**• D7. Establish the hand coordinate system • D8. Find the link and joint parameters : d,a,a,q D-H transformation for adjacent coordinate frames:**Z3**Z1 Z0 Joint 3 X3 Y0 Y1 Z2 d2 Joint 1 X0 X1 X2 Joint 2 Y2 a0 a1 Example**Example (3.3):**Link Frame Assignments**Example:SCARA Robot**• The location of the sliding axis Z2is arbitrary, since it is a free vector. For simplicity, we make it coincident with Z3 . thus a2and d2are arbitrarily set. • The placement of O3and X3along Z3is arbitrary, since Z2and Z3are coincident. Once we choose O3, however, then the joint displacement d3is defined. • We have also placed the end link frame in a convenient manner, with the Z4axis coincident with the Z3axis and the origin O4displaced down into the gripper by d4.**d4**a3 x3 x4 y3 z4 x6 x5 z6 y5 Forearm of a PUMA Spherical joint**Example: Puma 560**Different Configuration