Forward Kinematics - TU Chemnitz
Transcript of Forward Kinematics - TU Chemnitz
Forward Kinematics
Dr.-Ing. John Nassour
Artificial Intelligence & Neuro Cognitive Systems FakultΓ€t fΓΌr Informatik
Serial link manipulators
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Suggested literature
β’ Robot Modeling and Controlβ’ Robotics: Modelling, Planning and Control
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Reminder: Right Hand Rules
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Cross product
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Reminder: Right Hand Rules
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A right-handed coordinate frame
π
π
π Γ π
π
π
π
π
The first three fingers of your right hand which indicate the relative directions of the x-, y- and z-axes respectively.
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Reminder: Right Hand Rules
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Rotation about a vector
πWrap your right hand around the vector withyour thumb (your x-finger) in the direction ofthe arrow. The curl of your fingers indicatesthe direction of increasing angle.
+
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Kinematics
The problem of kinematics is to describe the motion ofthe manipulator without consideration of the forces andtorques causing that motion.
The kinematic description is therefore a geometric one.
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Forward Kinematics
Determine the position and orientation of theend-effector given the values for the jointvariables of the robot.
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End-EffectorLink 1
Base
Link 2
Link n-1
Joint 1
Joint 2
Joint 3
Joint n
Joint n-1
Robot Manipulators are composed oflinks connected by joints to form akinematic chain.
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Robot Manipulators
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Base
Link i
Prismatic joint
Revolute joint
Revolute joint (R): allows a relative rotation about a single axis. Prismatic joint (P): allows a linear motion along a single axis (extension or retraction).
Spherical wrist: A three degree of freedom rotational joint with all three axes of rotation crossing at a point is typically called a spherical wrist.
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The Workspace Of A Robot
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Base
Link i
Prismatic joint
Revolute joint
The total volume its end - effector could sweep as the robotexecutes all possible motions. It is constrained by the geometry ofthe manipulator as well as mechanical limits imposed on thejoints.
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Robot Manipulators
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Symbolic representation of robot joints
e.g. A three-link arm with three revolute joints was denoted by RRR.
Joint variables, denoted by π½ for a revolute joint and π for the prismatic joint,represent the relative displacement between adjacent links.
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Articulated Manipulators (RRR)
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Three joints of the rotational type (RRR). It resembles the human arm. The second joint axis is perpendicular to the first one. The third joint axis is parallel to the second one.The workspace of the anthropomorphic robot arm, encompassing all the points that can be reached by the robot end point.
Also called: Anthropomorphic Manipulators
Articulated Manipulators (RRR)
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Elbow Manipulator (RRR)
Workspace
Structure
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Spherical Manipulator
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The Stanford Arm
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Spherical Manipulator RRP
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Two rotation and one translation (RRP). The second joint axis is perpendicular to the first one and the third axis is perpendicular to the second one.
Workspace
Structure
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Spherical Manipulator RRP
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Two rotation and one translation (RRP). The second joint axis is perpendicular to the first one and the third axis is perpendicular to the second one. The workspace of the robot arm has a spherical shape as in the case of the anthropomorphic robot arm.
Workspace
Structure
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Spherical Manipulator RRR
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Workspace?
Structure
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SCARA Manipulator
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Two joints are rotational and one is translational (RRP).The axes of all three joints are parallel.
Workspace
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SCARA Manipulator
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Two joints are rotational and one is translational (RRP).The axes of all three joints are parallel.The workspace of SCARA robot arm is of cylindrical shape.
Workspace
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Cylindrical Manipulator
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One rotational and two translational (RPP).The axis of the second joint is parallel to the first axis. The third joint axis is perpendicular to the second one.
Workspace
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Cylindrical Manipulator
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One rotational and two translational (RPP).The axis of the second joint is parallel to the first axis. The third joint axis is perpendicular to the second one.
Workspace
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The Cartesian Manipulators
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Three joints of the translational type (PPP).The joint axes are perpendicular one to another.
Workspace
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The Cartesian Manipulators
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Three joints of the translational type (PPP).The joint axes are perpendicular one to another.
Workspace
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A set of position parameters that describes the full configuration of the system.
Base
Configuration Parameters
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A set of position parameters that describes the full configuration of the system.
Base
9 parameters/link
Configuration Parameters
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Generalized Coordinates
A set of independent configuration parameters
π ππππππππππ/ππππ π ππππππππππ ππππππππππππ
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Generalized Coordinates
A set of independent configuration parameters
π ππππππππππ/ππππ π ππππππππππ ππππππππππππ
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πΏπππ 1
Base
πΏπππ 2
πΏπππ π
6n parameters for n moving links
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Generalized Coordinates
A set of independent configuration parameters
π ππππππππππ/ππππ π ππππππππππ ππππππππππππ
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πΏπππ 1
Base
πΏπππ 2
πΏπππ ππ πͺπππππππππ
6n parameters for n moving links
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Generalized Coordinates
A set of independent configuration parameters
π ππππππππππ/ππππ π ππππππππππ ππππππππππππ
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πΏπππ 1
Base
πΏπππ 2
πΏπππ ππ πͺπππππππππ
6n parameters for n moving links5n constraints for n joints
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Generalized Coordinates
A set of independent configuration parameters
π ππππππππππ/ππππ π ππππππππππ ππππππππππππ
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πΏπππ 1
Base
πΏπππ 2
πΏπππ ππ πͺπππππππππ
6n parameters for n moving links5n constraints for n jointsD.O.F: 6n - 5n = n
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Generalized Coordinates
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D.O.F: n joints + ?
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Generalized Coordinates
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The robot is free to move forward/backward, up/down, left/right (translation in three perpendicular axes) combined with rotation about three perpendicular axes, often termed pitch, yaw, and roll.
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Generalized Coordinates
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The robot is free to move forward/backward, up/down, left/right (translation in three perpendicular axes) combined with rotation about three perpendicular axes, often termed pitch, yaw, and roll.
D.O.F: n joints + 6
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Operational Coordinates
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ππ ππ ππ ππ
ππ+π ππ+π ππ+π ππ+π
End-effector configuration parameters are a set of π parameters (ππ, ππ, ππ, . . , ππ) that completely specify the end-effector position and orientation with respect to the frame ππ ππ ππ ππ.
ππ+π is the operational point.
A set (ππ, ππ, ππ, . . , πππ) of
independent configuration Parameters ππ: number of degree of freedom of the end-effector.
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Operational Coordinates
Base
ππ
π
πΆ
π
πΆ
Is also called Operational Space
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Joint Coordinates
Base
π½1
π½2π½3
π½1
π½2
π½3
Is also called Joint Space
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Joint Space -> Operational Space
Determine the position and orientation of the end-effector given the values for the joint variables of the robot.
Base
π½1
π½2
π½3
π
πΆ
π
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Redundancy
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A robot is said to be redundant if π > ππ. Degree of redundancy: π βππ
Base
ππ = ππ = π
how many solutions exist?
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Redundancy
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A robot is said to be redundant if π > ππ. Degree of redundancy: π βππ
Base
ππ = ππ = π
how many solutions exist?
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Redundancy
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A robot is said to be redundant if π > ππ. Degree of redundancy: π βππ
Base
ππ = ππ = π
how many solutions exist?
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Redundancy
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A robot is said to be redundant if π > ππ. Degree of redundancy: π βππ
Base
ππ = ππ = π
how many solutions exist?
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Redundancy
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A robot is said to be redundant if π > ππ. Degree of redundancy: π βππ
Base
ππ = ππ = π
how many solutions exist?
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Kinematic Arrangements
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The objective of forward kinematic analysis is to determine the cumulative effect of the entire set of joint variables, that is, to determine the position and orientation of the end effector given the values of these joint variables.
We assume that each joint has one D.O.F
The action of each joint can be described by one real number: the angle of rotation in the case of a revolute joint or the displacement in the case of a prismatic joint.
When joint π is actuated, link π moves.
ππ is the joint variable
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Kinematic Arrangements
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Spherical wrist 3 D.O.F
spherical wrist: RRR Linksβ lengths = 0
The objective of forward kinematic analysis is to determine the cumulative effect of the entire set of joint variables, that is, to determine the position and orientation of the end effector given the values of these joint variables.
We assume that each joint has one D.O.F
The action of each joint can be described by one real number: the angle of rotation in the case of a revolute joint or the displacement in the case of a prismatic joint.
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Kinematic Arrangements
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Base
To perform the kinematic analysis, we attach a coordinate frame rigidly to each link. In particular, we attach ππππ ππ ππ to ππππ π. This means that, whatever motion the robot executes, the coordinates of any point π on
link π are constant when expressed in the πππ coordinate frame ππ = ππππππππ.When πππππ π is actuated, ππππ π and its attached frame, ππππ ππ ππ, experience a resulting motion.
The frame ππππ ππ ππ, which is attached to the robot base, is referred to as the reference frame.
ππππ ππ ππ
ππππ ππ ππ
ππππ ππ
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Kinematic Arrangements
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Base
ππππ ππ ππ
ππππ ππ ππ
ππππ π
The frame ππππ ππ ππ, which is attached to the robot base, is referred to as the reference frame.
To perform the kinematic analysis, we attach a coordinate frame rigidly to each link. In particular, we attach ππππ ππ ππ to ππππ π. This means that, whatever motion the robot executes, the coordinates of any point π on
link π are constant when expressed in the πππ coordinate frame ππ = ππππππππ.When πππππ π is actuated, ππππ π and its attached frame, ππππ ππ ππ, experience a resulting motion.
π
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Kinematic Arrangements
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Base
ππππ ππ ππ
ππππ ππ ππ
ππππ π
The frame ππππ ππ ππ, which is attached to the robot base, is referred to as the reference frame.
To perform the kinematic analysis, we attach a coordinate frame rigidly to each link. In particular, we attach ππππ ππ ππ to ππππ π. This means that, whatever motion the robot executes, the coordinates of any point π on
link π are constant when expressed in the πππ coordinate frame ππ = ππππππππ.When πππππ π is actuated, ππππ π and its attached frame, ππππ ππ ππ, experience a resulting motion.
π
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Joint And Link Labelling
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Joint And Link Labelling
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ππππ ππ ππLink 0 (fixed)Base frame
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Joint And Link Labelling
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ππππ ππ ππLink 0 (fixed)
Joint 1
Link 1
Joint variable π½1
Base frame
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Joint And Link Labelling
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ππππ ππ ππLink 0 (fixed)
Joint 1
Link 1
Joint variable π½1
ππππ ππ ππ
Joint 2 Link 2
Joint variable π½2
Base frame
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Joint And Link Labelling
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ππππ ππ ππLink 0 (fixed)
Joint 1
Link 1
Joint variable π½1
ππππ ππ ππ
Joint 2 Link 2
Joint variable π½2
Link 3
ππππ ππ ππ
Joint 3
Joint variable π½3
Base frame
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Joint And Link Labelling
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ππππ ππ ππLink 0 (fixed)
Joint 1
Link 1
Joint variable π½1
ππππ ππ ππ
Joint 2 Link 2
Joint variable π½2
Link 3
ππππ ππ ππ
Joint 3
Joint variable π½3
ππππ ππ ππ
Base frame
Do we need a specific way to orientate the axes?
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Transformation Matrix
ππππ ππ ππLink 0 (fixed)
Joint 1
Link 1
Joint variable π½1
ππππ ππ ππ
Joint 2 Link 2
Joint variable π½2
Link 3
ππππ ππ ππ
Joint 3
Joint variable π½3
ππππ ππ ππ
Base frame
Suppose π¨π is the homogeneous transformation matrix that describe the position and the orientation of ππππ ππ ππ with respect to ππβπππβπ ππβπ ππβπ.π¨π is derived from joint and link π. π¨π is a function of only a single joint variable.
π¨π = π¨π(ππ)
π¨π(ππ) =πΉ π
πβπ π ππβπ
π π
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Transformation Matrix
The position and the orientation of the end effector (reference frame ππππ ππ ππ) with respect to the base (reference frame ππππ ππ ππ) can be expressed by the transformation matrix:
π = π»ππ = π¨π ππ β¦π¨π(ππ) =
πΉππ ππ
π
π π
The position and the orientation of a reference frame ππππ ππ ππ) with
respect to a reference frame ππππ ππ ππ can be expressed by the transformation matrix:
π»ππ =
π¨π+ππ¨π+πβ¦π¨πβπ π¨π
π°
(π»ππ)βπ
ππ π < πππ π = πππ π > π
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Transformation Matrix
π»ππ =
π¨π+ππ¨π+πβ¦π¨πβπ π¨π
π°
(π»ππ)βπ
ππ π < πππ π = πππ π > π
if π < π then
π»ππ = π¨π+ππ¨π+πβ¦π¨πβπ π¨π =
πΉππ ππ
π
π π
The orientation part: πΉππ = πΉπ+π
π β¦πΉ ππβπ
The translation part: πππ = ππβπ
π+πΉπβππ π π
πβπ
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Link Description
Axis(i-1) Axis(i)Link(i-1)
A link is considered as a rigid body whichdefines the relationship between twoneighboring joint axes of a manipulator.
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Link Description
Axis(i-1) Axis(i)Link(i-1)
The kinematics function of a link is tomaintain a fixed relationship betweenthe two joint axes it supports.
This relationship can be describedwith two parameters:β’ the link length aβ’ the link twist a
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Link Description
Axis(i-1) Axis(i)Link(i-1)
ππβπ Link Lengthmutual perpendicular
Is measured along a line which ismutually perpendicular to bothaxes.The mutually perpendicular alwaysexists and is unique except whenboth axes are parallel.
ππβπ
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Link Description
Axis(i-1) Axis(i)Link(i-1)
πΆπβπ Link Twist
Project both axes π β 1 and πonto the plane whose normal isthe mutually perpendicular line.
Measured in the right-handsense about ππβπ.
ππβπ
πΆπβπ
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Link Description
Intersecting joint axis !
ππβπ Link length ?
πΆπβπ Link Twist ?The sense of πΌπβ1 is free.
Axis(i-1)Axis(i)
πΆπβπ
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Joint Parameters
Axis(i-1) Axis(i)Link(i-1)
ππβπ
πΆπβπ
A joint axis is established atthe connection of two links.
This joint will have twonormals connected to itone for each of the links.
ππ
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Joint Parameters
Axis(i-1) Axis(i)Link(i-1)
ππβπ
πΆπβπ
ππ
π π Link OffsetVariable if joint is prismatic.
The relative position of two links is called link offset whish is the distance between the links (the displacement, along the joint axes between the links).
π π
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Joint Parameters
Axis(i-1) Axis(i)Link(i-1)
ππβπ
πΆπβπ
ππ
π π Link OffsetVariable if joint is prismatic.
The relative position of two links is called link offset whish is the distance between the links (the displacement, along the joint axes between the links).
π½π Joint AngleVariable if joint is revolute.
The joint angle between the normals is measured in a plane normal to the joint axis.
π π
π½π
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Link Description
Axis(i-1) Axis(i)Link(i-1)
ππβπ Link Length
and
πΆπβπ Link Twist
depend on joint axesπ β 1 and π.
ππβπ
πΆπβπ
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Joint Parameters
Axis(i-1) Axis(i)Link(i-1)
ππβπ
πΆπβπ
ππ
π π Link Offsetand
π½π Joint Angle
depend on links π β 1and π.
π π
π½π
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Denavit-Hartenberg Convention
Each π΄ matrix has 6 variables- 3 in the rotation matrix and 3 in the position vector.
DH parameters collapse 6 variables to 4 link and joint parameters if we follow a certain procedure for setting coordinate frames.
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Denavit-Hartenberg Convention
Each π΄ matrix has 6 variables- 3 in the rotation matrix and 3 in the position vector.
DH parameters collapse 6 variables to 4 link and joint parameters if we follow a certain procedure for setting coordinate frames.
ππ is link length of like i (constant unless you reconfigure the robot)
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Denavit-Hartenberg Convention
Each π΄ matrix has 6 variables- 3 in the rotation matrix and 3 in the position vector.
DH parameters collapse 6 variables to 4 link and joint parameters if we follow a certain procedure for setting coordinate frames.
ππ is link length of like i (constant unless you reconfigure the robot)πΆπ is link twist of link i (constant unless you reconfigure the robot)
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Denavit-Hartenberg Convention
Each π΄ matrix has 6 variables- 3 in the rotation matrix and 3 in the position vector.
DH parameters collapse 6 variables to 4 link and joint parameters if we follow a certain procedure for setting coordinate frames.
ππ is link length of like i (constant unless you reconfigure the robot)πΆπ is link twist of link i (constant unless you reconfigure the robot)π π is link offset of link i (prismatic variable)
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Denavit-Hartenberg Convention
Each π΄ matrix has 6 variables- 3 in the rotation matrix and 3 in the position vector.
DH parameters collapse 6 variables to 4 link and joint parameters if we follow a certain procedure for setting coordinate frames.
ππ is link length of like i (constant unless you reconfigure the robot)πΆπ is link twist of link i (constant unless you reconfigure the robot)π π is link offset of link i (prismatic variable)π½π is joint angle of link i (revolute variable)
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Denavit-Hartenberg Matrix
Each homogeneous transformation π΄π is represented as a product of four basic transformations:
where the four quantities are parameters associated with ππππ π and πππππ‘ π.
ππ is link length πΆπ is link twistπ π is link offsetπ½π is joint angle
Reminder:
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Denavit-Hartenberg Matrix
Axis(i-1) Axis(i)Link(i-1)
ππβπ
πΆπβπ
ππ
ππ is link length πΆπ is link twistπ π is link offsetπ½π is joint angle
π π
π½π
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Denavit-Hartenberg Matrix
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it is not necessary that the origin of πππππ π be placed at the physical end of ππππ π.
it is not necessary that frame π be placed within the physical link; πππππ π could lie in free space β so long as πππππ π is rigidly attached to ππππ π.
By a clever choice of the origin and the coordinate axes, it is possible to cut down the number of parameters needed from six to four (or even fewer in some cases).
Denavit-Hartenberg Convention
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Denavit-Hartenberg Convention
DH Coordinate Frame Assumptions
(DH1) The axis π₯1is perpendicular to the axis π§0. (DH2) The axis π₯1 intersects the axis π§0.
Under these conditions, there existunique numbers a, d, π½, πΆ such that:
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Denavit-Hartenberg Convention
Positive sense for π and πΌ
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Rules For Assigning Frames
Rule 1: π§πβ1 is axis of actuation of joint π.Axis of revolution of revolute jointAxis of translation of prismatic joint
Rule 2: Axis π₯π is set so it is perpendicular to and intersects π§πβ1.
Rule 3: Derive π¦π from π₯π and π§π.
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Rules For Assigning Frames
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Rules For Assigning Frames
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ππLink 0 (fixed)Base frame
ππ
ππ
Rule 1: ππβπ is axis of actuation of joint π
Base frame
ππ is axis of actuation of joint π.
ππ and ππ are set according tothe right hand rule.
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Rules For Assigning Frames
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Rule 1: ππβπ is axis of actuation of joint π
Tool frame
π§π is the approach direction of the tool.π¦π is the slide direction of the gripper.π₯π is the normal direction to other axes.
ππ
ππ
ππ
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Rules For Assigning Frames
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Rule 2: Axis ππ is set so it is perpendicular to and intersects ππβπ
Case 1: ππβπ and ππ are not coplanar.
β’ There is only one line possible for π₯π , which is the shortest line from π§πβ1 to π§π . β’ ππ is at intersection of π₯π and π§π .
ππβπ
ππβπ
ππβπ
ππ
ππ
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Rules For Assigning Frames
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Rule 2: Axis ππ is set so it is perpendicular to and intersects ππβπ
Case 2: ππβπ and ππ are parallel.
β’ There are an infinite number of possibilities for π₯π from π§πβ1 to π§π . β’ Usually easiest to choose an π₯π that passes through ππβ1(so that ππ = 0). β’ ππ is at intersection of π₯π and π§π . β’ πΌπ = 0 always for this case.
ππβπ
ππβπ
ππβπ
ππ
ππ
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Rules For Assigning Frames
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Rule 2: Axis ππ is set so it is perpendicular to and intersects ππβπ
Case 3: ππβπ intersects ππ.
β’ π₯π is normal to the plane of π§πβ1 and π§π . β’ Positive direction of π₯π is arbitrary. β’ ππ naturally sits at intersection of π§πβ1 and π§π but can be anywhere on π§π . β’ ππ = 0 always for this case.
ππβπ
ππβπ
ππβπ
ππ
ππ
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Rules For Assigning Frames
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Rule 2: Axis ππ is set so it is perpendicular to and intersects ππβπ
Case 3: ππβπ intersects ππ.
β’ π₯π is normal to the plane of π§πβ1 and π§π . β’ Positive direction of π₯π is arbitrary. β’ ππ naturally sits at intersection of π§πβ1 and π§π but can be anywhere on π§π . β’ ππ = 0 always for this case.
ππβπ
ππβπ
ππβπ
ππ
ππ
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Rules For Assigning Frames
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Rule 2: Axis ππ is set so it is perpendicular to and intersects ππβπ
Case 3: ππβπ intersects ππ.
β’ π₯π is normal to the plane of π§πβ1 and π§π . β’ Positive direction of π₯π is arbitrary. β’ ππ naturally sits at intersection of π§πβ1 and π§π but can be anywhere on π§π . β’ ππ = 0 always for this case.
ππβπ
ππβπ
ππβπ
ππ
ππ
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D-H Parameters
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ππ is distance from ππβπ to ππ measured along ππ.πΆπ is angle from ππβπ to ππ measured about ππ. π π is distance from ππβπ to ππ measured along ππβπ.
π½π is angle from ππβπ to ππ measured about ππβπ.
ππβπ
ππβπ
ππ
ππ
ππβπ
ππ
π π
π½π
πΆπ
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D-H Parameters
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ππ is distance from ππβπ to ππ measured along ππ.πΆπ is angle from ππβπ to ππ measured about ππ. π π is distance from ππβπ to ππ measured along ππβπ.
π½π is angle from ππβπ to ππ measured about ππβπ.
ππβπ
ππβπ
ππ
ππ
ππβπ
ππ
π π
π½π
πΆπ
J.Nassour 88
D-H Parameters
14.11.2017
ππ is distance from ππβπ to ππ measured along ππ.πΆπ is angle from ππβπ to ππ measured about ππ. π π is distance from ππβπ to ππ measured along ππβπ.
π½π is angle from ππβπ to ππ measured about ππβπ.
ππβπ
ππβπ
ππ
ππ
ππβπ
ππ
π π
π½π
πΆπ
J.Nassour 89
D-H Parameters
14.11.2017
ππ is distance from ππβπ to ππ measured along ππ.πΆπ is angle from ππβπ to ππ measured about ππ. π π is distance from ππβπ to ππ measured along ππβπ.
π½π is angle from ππβπ to ππ measured about ππβπ.
ππβπ
ππβπ
ππ
ππ
ππβπ
ππ
π π
π½π
πΆπ
J.Nassour 90
Example: RRP Robot
14.11.2017
Assign coordinate frames so that wecan find DH parameters for this robot.
Joint 3
Joint 1
Joint 2
Tool
π½1
π½2
π 3
J.Nassour 91
Example: RRP Robot
14.11.2017
Joint 3
Joint 1
Joint 2
Tool ππ
ππ
ππ
3π
Assign coordinate frames so that wecan find DH parameters for this robot.
J.Nassour 92
Example: RRP Robot
14.11.2017
Joint 3
Joint 1
Joint 2
Tool ππ
ππ
ππ
ππππππ
3π
Assign coordinate frames so that wecan find DH parameters for this robot.
J.Nassour 93
Example: RRP Robot
14.11.2017
Joint 3
Joint 1
Joint 2
Tool ππ
ππ
ππ
ππππππ
ππ
ππ
ππ
3π
Assign coordinate frames so that wecan find DH parameters for this robot.
J.Nassour 94
Example: RRP Robot
14.11.2017
Joint 3
Joint 1
Joint 2
Tool ππ
ππ
ππ
ππππππ
ππ
ππ
ππ
ππ
ππ
ππ
π³3
3π
Assign coordinate frames so that wecan find DH parameters for this robot.
J.Nassour 95
Example: RRP Robot
14.11.2017
Find DH parameters for thisrobot. Identify the jointvariables.
Joint 3
Joint 1
Joint 2
Tool ππ
ππ
ππ
ππππππ
ππ
ππ
ππ
ππ
ππ
ππ
π³3
3π
J.Nassour 96
Example: RRP Robot
14.11.2017
Find DH parameters for thisrobot. Identify the jointvariables.
ππ is distance from ππβπ to ππ measured along ππ.πΆπ is angle from ππβπ to ππ measured about ππ. π π is distance from ππβπ to ππ measured along ππβπ.
π½π is angle from ππβπ to ππ measured about ππβπ.
Joint 3
Joint 1
Joint 2
Tool ππ
ππ
ππ
ππππππ
ππ
ππ
ππ
ππ
ππ
ππ
π³3
3π
π ππ πΆπ π π π½π
1
2
3
J.Nassour 97
Example: RRP Robot
14.11.2017
Find DH parameters for thisrobot. Identify the jointvariables.
ππ is distance from ππβπ to ππ measured along ππ.πΆπ is angle from ππβπ to ππ measured about ππ. π π is distance from ππβπ to ππ measured along ππβπ.
π½π is angle from ππβπ to ππ measured about ππβπ.
Joint 3
Joint 1
Joint 2
Tool ππ
ππ
ππ
ππππππ
ππ
ππ
ππ
ππ
ππ
ππ
π³3
3π
π ππ πΆπ π π π½π
1 0 β90 Β° 3m π½π = 0 Β°
2
3
J.Nassour 98
Example: RRP Robot
14.11.2017
Find DH parameters for thisrobot. Identify the jointvariables.
ππ is distance from ππβπ to ππ measured along ππ.πΆπ is angle from ππβπ to ππ measured about ππ. π π is distance from ππβπ to ππ measured along ππβπ.
π½π is angle from ππβπ to ππ measured about ππβπ.
Joint 3
Joint 1
Joint 2
Tool ππ
ππ
ππ
ππππππ
ππ
ππ
ππ
ππ
ππ
ππ
π³3
3π
π ππ πΆπ π π π½π
1 0 β90 Β° 3m π½π = 0 Β°
2 0 β90 Β° 0 π½π = β90 Β°
3
J.Nassour 99
Example: RRP Robot
14.11.2017
Find DH parameters for thisrobot. Identify the jointvariables.
ππ is distance from ππβπ to ππ measured along ππ.πΆπ is angle from ππβπ to ππ measured about ππ. π π is distance from ππβπ to ππ measured along ππβπ.
π½π is angle from ππβπ to ππ measured about ππβπ.
Joint 3
Joint 1
Joint 2
Tool ππ
ππ
ππ
ππππππ
ππ
ππ
ππ
ππ
ππ
ππ
π³3
3π
π ππ πΆπ π π π½π
1 0 β90 Β° 3m π½π = 0 Β°
2 0 β90 Β° 0 π½π = β90 Β°
3 0 0 Β° π π = π³π 0 Β°
J.Nassour 100
Example: RRP Robot
14.11.2017
Find DH parameters for thisrobot. Identify the jointvariables.
ππ is distance from ππβπ to ππ measured along ππ.πΆπ is angle from ππβπ to ππ measured about ππ. π π is distance from ππβπ to ππ measured along ππβπ.
π½π is angle from ππβπ to ππ measured about ππβπ.
Joint 3
Joint 1
Joint 2
Tool ππ
ππ
ππ
ππππππ
ππ
ππ
ππ
ππ
ππ
ππ
π³3
3π
π ππ πΆπ π π π½π
1 0 β90 Β° 3m π½π = 0 Β°
2 0 β90 Β° 0 π½π = β90 Β°
3 0 0 Β° π π = π³π 0 Β°
Find the A matrices
J.Nassour 101
Example: RRP Robot
14.11.2017
Find the A matrices Reminder: π¨π
π΄ 1 =
π1 0π 1 0
βπ 1 0π1 0
0 β10 0
0 30 1
π΄ 2 =
π2 0π 2 0
βπ 2 0π2 0
0 β10 0
0 00 1
π΄ 3 =
1 00 1
0 00 0
0 00 0
1 π³π0 1
π ππ πΆπ π π π½π
1 0 β90 Β° 3m π½π = 0 Β°
2 0 β90 Β° 0 π½π = β90 Β°
3 0 0 Β° π π = π³π 0 Β°
J.Nassour 102
Example: RRP Robot
14.11.2017
Find the A matrices Reminder: π¨π
π΄ 1 =
π1 0π 1 0
βπ 1 0π1 0
0 β10 0
0 30 1
π΄ 2 =
π2 0π 2 0
βπ 2 0π2 0
0 β10 0
0 00 1
π΄ 3 =
1 00 1
0 00 0
0 00 0
1 π³π0 1
π ππ πΆπ π π π½π
1 0 β90 Β° 3m π½π = 0 Β°
2 0 β90 Β° 0 π½π = β90 Β°
3 0 0 Β° π π = π³π 0 Β°
J.Nassour 103
Example: RRP Robot
14.11.2017
Find the A matrices Reminder: π¨π
π΄ 1 =
π1 0π 1 0
βπ 1 0π1 0
0 β10 0
0 30 1
π΄ 2 =
π2 0π 2 0
βπ 2 0π2 0
0 β10 0
0 00 1
π΄ 3 =
1 00 1
0 00 0
0 00 0
1 π³π0 1
π 10= π΄ 1 π 2
0= π΄ 1 π΄ 2 =
π1π2 π 1π 1π2 βπ1
βπ1π 2 0βπ 1π 2 0
βπ 2 00 0
βπ2 30 1
π ππ πΆπ π π π½π
1 0 β90 Β° 3m π½π = 0 Β°
2 0 β90 Β° 0 π½π = β90 Β°
3 0 0 Β° π π = π³π 0 Β°
J.Nassour 104
Example: RRP Robot
14.11.2017
Find the A matrices
π 10= π΄ 1 π 2
0= π΄ 1 π΄ 2 =
π1π2 π 1π 1π2 βπ1
βπ1π 2 0βπ 1π 2 0
βπ 2 00 0
βπ2 30 1
=
? ?? ?
? 0? 0
? 00 0
? 30 1
In the current configuration
π ππ πΆπ π π π½π
1 0 β90 Β° 3m π½π = 0 Β°
2 0 β90 Β° 0 π½π = β90 Β°
3 0 0 Β° π π = π³π 0 Β°
J.Nassour 105
Joint 3
Joint 1
Joint 2
Tool ππ
ππ
ππ
ππππππ
ππ
ππ
ππ
ππ
ππ
ππ
π³3
3π
Example: RRP Robot
14.11.2017
Find the A matrices
π 10= π΄ 1
In the current configuration
π ππ πΆπ π π π½π
1 0 β90 Β° 3m π½π = 0 Β°
2 0 β90 Β° 0 π½π = β90 Β°
3 0 0 Β° π π = π³π 0 Β°
J.Nassour 106
Joint 3
Joint 1
Joint 2
Tool ππ
ππ
ππ
ππππππ
ππ
ππ
ππ
ππ
ππ
ππ
π³3
3π
π 20= π΄ 1 π΄ 2 =
π1π2 π 1π 1π2 βπ1
βπ1π 2 0βπ 1π 2 0
βπ 2 00 0
βπ2 30 1
=
0 00 β1
1 00 0
1 00 0
0 30 1
Example: RRP Robot
14.11.2017
Find the A matrices Reminder: π¨π
π΄ 1 =
π1 0π 1 0
βπ 1 0π1 0
0 β10 0
0 30 1
π΄ 2 =
π2 0π 2 0
βπ 2 0π2 0
0 β10 0
0 00 1
π΄ 3 =
1 00 1
0 00 0
0 00 0
1 π³π0 1
π 30= π2
0 π΄ 3 =
π1π2 π 1π 1π2 βπ1
βπ1π 2 0βπ 1π 2 0
βπ 2 00 0
βπ2 30 1
1 00 1
0 00 0
0 00 0
1 πΏ30 1
=
π1π2 π 1π 1π2 βπ1
βπ1π 2 βπΏππ1π 2βπ 1π 2 βπΏππ 1π 2
βπ 2 00 0
βπ2 3 βπΏππ20 1
π ππ πΆπ π π π½π
1 0 β90 Β° 3m π½π = 0 Β°
2 0 β90 Β° 0 π½π = β90 Β°
3 0 0 Β° π π = π³π 0 Β°
J.Nassour 107
Example: RRP Robot
14.11.2017
Find the A matrices
π30 =
π1π2 π 1π 1π2 βπ1
βπ1π 2 βπΏππ1π 2βπ 1π 2 βπΏππ 1π 2
βπ 2 00 0
βπ2 3 βπΏππ20 1
=
? ?? ?
? ?? ?
? 00 0
? ?0 1
In the current configuration
π ππ πΆπ π π π½π
1 0 β90 Β° 3m π½π = 0 Β°
2 0 β90 Β° 0 π½π = β90 Β°
3 0 0 Β° π π = π³π 0 Β°
Joint 3
Joint 1
Joint 2
Tool ππ
ππ
ππ
ππππππ
ππ
ππ
ππ
ππ
ππ
ππ
π³3
3π
J.Nassour 108
Example: RRP Robot
14.11.2017
Find the A matrices
π30 =
π1π2 π 1π 1π2 βπ1
βπ1π 2 βπΏππ1π 2βπ 1π 2 βπΏππ 1π 2
βπ 2 00 0
βπ2 3 βπΏππ20 1
=
0 00 β1
1 πΏ30 0
1 00 0
0 30 1
In the current configuration
π ππ πΆπ π π π½π
1 0 β90 Β° 3m π½π = 0 Β°
2 0 β90 Β° 0 π½π = β90 Β°
3 0 0 Β° π π = π³π 0 Β°
Joint 3
Joint 1
Joint 2
Tool ππ
ππ
ππ
ππππππ
ππ
ππ
ππ
ππ
ππ
ππ
π³3
3π
J.Nassour 109
Example: Two-Link Planar Robot
14.11.2017
Assign coordinate frames so that wecan find DH parameters for this robot.
J.Nassour 110
Example: Two-Link Planar Robot
14.11.2017
Find DH parameters for this robot. Identify the joint variables.
ππ is distance from ππβπ to ππ measured along ππ.πΆπ is angle from ππβπ to ππ measured about ππ. π π is distance from ππβπ to ππ measured along ππβπ.
π½π is angle from ππβπ to ππ measured about ππβπ.
π ππ πΆπ π π π½π
1 π1 0 Β° 0 π½π
2 π2 0 Β° 0 π½π
J.Nassour 111
Example: Two-Link Planar Robot
14.11.2017
Find DH parameters for this robot. Identify the joint variables.
π ππ πΆπ π π π½π
1 π1 0 Β° 0 π½π
2 π2 0 Β° 0 π½π
J.Nassour 112
F.K. For Cylindrical Manipulator
14.11.2017
One rotational and two translational (RPP).The axis of the second joint is parallel to the first axis. The third joint axis is perpendicular to the second one.
β’ Assign coordinate frames so that we can find DH parameters for this robot.β’ Find DH parameters for this robot. Identify the joint variables.
J.Nassour 113
Stanford Arm
14.11.2017 J.Nassour 114
π½π
Stanford Arm
14.11.2017 J.Nassour 115
π½π
π½π
Stanford Arm
14.11.2017 J.Nassour 116
π π
π½π
π½π
Stanford Arm
14.11.2017 J.Nassour 117
π π
π½π
π½π
π½π
Stanford Arm
14.11.2017 J.Nassour 118
π π
π½π
π½π
π½π
π½π
Stanford Arm
14.11.2017 J.Nassour 119
π π
π½π
π½π
π½π
π½π
π½π
Stanford Arm
14.11.2017 J.Nassour 120
π π
π6
π½π
π½π
π½π
π½π
π½π
π2
Stanford Arm
14.11.2017 J.Nassour 121
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
π2
Stanford Arm
14.11.2017 J.Nassour 122
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
π2
Stanford Arm
14.11.2017 J.Nassour 123
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
π2
ππ
Stanford Arm
14.11.2017 J.Nassour 124
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
π2
ππ
ππ
Stanford Arm
14.11.2017 J.Nassour 125
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
π2
ππ
ππ
ππ
Stanford Arm
14.11.2017 J.Nassour 126
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
π2
ππ
ππ
ππ
ππ
Stanford Arm
14.11.2017 J.Nassour 127
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
π2
ππ
ππ
ππ
ππ
π
Stanford Arm
14.11.2017 J.Nassour 128
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
π2
ππ
ππ
ππ
ππ
πππ
Stanford Arm
14.11.2017 J.Nassour 129
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
π2
ππ
ππ
ππ
πππ
ππ ππ
Stanford Arm
14.11.2017 J.Nassour 130
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
π2
ππ
ππ
ππ
πππ
ππ ππ
ππ
Stanford Arm
14.11.2017 J.Nassour 131
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
ππ
ππ
ππ ππ
π2
ππ
ππ ππ
π
ππ
Stanford Arm
14.11.2017 J.Nassour 132
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
ππ
ππ
ππ ππ
π2
ππ
ππ ππ
π
ππ
ππ
14.11.2017 J.Nassour 133
Stanford Arm
π ππ πΆπ π π π½π
1 0 β90 Β° 0 π½πβ
2 0 +90 Β° π2 π½πβ
3 0 0 Β°π π
β 0
4 0 β90 Β° 0 π½πβ
5 0 +90 Β° 0 π½πβ
6 0 0 Β° π6 π½πβ
ππ is distance from ππβπ to ππ measured along ππ.πΆπ is angle from ππβπ to ππ measured about ππ. π π is distance from ππβπ to ππ measured along ππβπ.
π½π is angle from ππβπ to ππ measured about ππβπ.
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
ππ
ππ
ππ ππ
π2
ππ
ππ ππ
π
ππ
ππ
14.11.2017 J.Nassour 134
Stanford Arm
π ππ πΆπ π π π½π
1 0 β90 Β° 0 π½πβ
2 0 +90 Β° π2 π½πβ
3 0 0 Β°π π
β 0
4 0 β90 Β° 0 π½πβ
5 0 +90 Β° 0 π½πβ
6 0 0 Β° π6 π½πβ
π60 = π΄1π΄2π΄3π΄4π΄5π΄6 =
π11 π12π21 π22
π13 ππ₯π23 ππ¦
π31 π320 0
π33 ππ§0 1
ππ is distance from ππβπ to ππ measured along ππ.πΆπ is angle from ππβπ to ππ measured about ππ. π π is distance from ππβπ to ππ measured along ππβπ.
π½π is angle from ππβπ to ππ measured about ππβπ.
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
ππ
ππ
ππ ππ
π2
ππ
ππ ππ
π
ππ
ππ
14.11.2017 J.Nassour 135
Stanford Arm
π ππ πΆπ π π π½π
1 0 β90 Β° 0 π½πβ
2 0 +90 Β° π2 π½πβ
3 0 0 Β°π π
β 0
4 0 β90 Β° 0 π½πβ
5 0 +90 Β° 0 π½πβ
6 0 0 Β° π6 π½πβ
π60 = π΄1π΄2π΄3π΄4π΄5π΄6 =
π11 π12π21 π22
π13 ππ₯π23 ππ¦
π31 π320 0
π33 ππ§0 1
ππ is distance from ππβπ to ππ measured along ππ.πΆπ is angle from ππβπ to ππ measured about ππ. π π is distance from ππβπ to ππ measured along ππβπ.
π½π is angle from ππβπ to ππ measured about ππβπ.
π·π = π»ππ π·π
ππ =?
ππ
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
ππ
ππ
ππ ππ
π2
ππ
ππ ππ
π
ππ
ππ
14.11.2017 J.Nassour 136
Stanford Arm
π΄1 =
π1 0π 1 0
βπ 1 0π1 0
0 β10 0
0 00 1
Reminder: π¨π
π΄2 =
π2 0π 2 0
π 2 0βπ2 0
0 10 0
0 π20 1
π΄3 =
1 00 1
0 00 0
0 00 0
1 π30 1
π΄4 =
π4 0π 4 0
βπ 4 0π4 0
0 β10 0
0 00 1
π΄5 =
π5 0π 5 0
π 5 0βπ5 0
0 β10 0
0 00 1
π΄6 =
π6 βπ 6π 6 π6
0 00 0
0 00 0
1 π60 1
π ππ πΆπ π π π½π
1 0 β90 Β° 0 π½πβ
2 0 +90 Β° π2 π½πβ
3 0 0 Β°π π
β 0
4 0 β90 Β° 0 π½πβ
5 0 +90 Β° 0 π½πβ
6 0 0 Β° π6 π½πβ
14.11.2017 J.Nassour 137
Stanford Arm
14.11.2017 J.Nassour 138
Stanford Arm[ s6*(c4*s1 + c1*c2*s4) β c6*(c5*(s1*s4 β c1*c2*c4) + c1*s2*s5), s6*(c5*(s1*s4 β c1*c2*c4) + c1*s2*s5) + c5*(c4*s1 + c1*c2*s4), c1*c5*s2 β s5*(s1*s4 β c1*c2*c4), d3*c1*s2 - d6*(s5*(s1*s4 β c1*c2*c4) β c1*c5*s2) - d2*s1 ][ c6*(c5*(c1*s4 + c2*c4*s1) β s1*s2*s5) β s6*(c1*c4 β c2*s1*s4), - s6*(c5*(c1*s4 + c2*c4*s1) β s1*s2*s5) β c5*(c1*c4 β c2*s1*s4), s5*(c1*s4 + c2*c4*s1) + c5*s1*s2, d2*c1 + d6*(s5*(c1*s4 + c2*c4*s1) + c5*s1*s2) + d3*s1*s2][ -c6*(c2*s5 + c4*c5*s2) β s2*s4*s6, s6*(c2*s5 + c4*c5*s2) β c5*s2*s4, c2*c5 β c4*s2*s5, d6*(c2*c5 β c4*s2*s5) + d3*c2][ 0, 0, 0, 1]
π11 = π 6. (π4. π 1 + π1. π2. π 4)β π6. (π5. (π 1. π 4 β π1. π2. π4) + π1. π 2. π 5)π21 = π6. (π5. (π1. π 4 + π2. π4. π 1)β π 1. π 2. π 5)β π 6. (π1. π4 β π2. π 1. π 4)π31 = βπ6. (π2. π 5 + π4. π5. π 2)β π 2. π 4. π 6
π12 = π 6. (π5. (π 1. π 4 β π1. π2. π4) + π1. π 2. π 5) + π5. (π4. π 1 + π1. π2. π 4)π22 = β π 6. (π5. (π1. π 4 + π2. π4. π 1) β π 1. π 2. π 5) β π5. (π1. π4 β π2. π 1. π 4)π32 = π 6. (π2. π 5 + π4. π5. π 2) β π5. π 2. π 4
π31 = π1. π5. π 2 β π 5. (π 1. π 4 β π1. π2. π4)π32 = π 5. (π1. π 4 + π2. π4. π 1) + π5. π 1. π 2π33 = π2. π5 β π4. π 2. π 5
ππ₯ = π3. π1. π 2 β π6. (π 5. (π 1. π 4 β π1. π2. π4) β π1. π5. π 2) β π2. π 1ππ¦ = π2. π1 + π6. (π 5. (π1. π 4 + π2. π4. π 1) + π5. π 1. π 2) + π3. π 1. π 2ππ§ = π6. (π2. π5 β π4. π 2. π 5) + π3. π2
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
ππ
ππ
ππ ππ
π2
ππ
ππ ππ
π
ππ
ππ
14.11.2017 J.Nassour 139
Stanford Arm
π11 = π 6. (π4. π 1 + π1. π2. π 4)β π6. (π5. (π 1. π 4 β π1. π2. π4) + π1. π 2. π 5)π21 = π6. (π5. (π1. π 4 + π2. π4. π 1)β π 1. π 2. π 5)β π 6. (π1. π4 β π2. π 1. π 4)π31 = βπ6. (π2. π 5 + π4. π5. π 2)β π 2. π 4. π 6
π12 = π 6. (π5. (π 1. π 4 β π1. π2. π4) + π1. π 2. π 5) + π5. (π4. π 1 + π1. π2. π 4)π22 = β π 6. (π5. (π1. π 4 + π2. π4. π 1) β π 1. π 2. π 5) β π5. (π1. π4 β π2. π 1. π 4)π32 = π 6. (π2. π 5 + π4. π5. π 2) β π5. π 2. π 4
π31 = π1. π5. π 2 β π 5. (π 1. π 4 β π1. π2. π4)π32 = π 5. (π1. π 4 + π2. π4. π 1) + π5. π 1. π 2π33 = π2. π5 β π4. π 2. π 5
ππ₯ = π3. π1. π 2 β π6. (π 5. (π 1. π 4 β π1. π2. π4) β π1. π5. π 2) β π2. π 1ππ¦ = π2. π1 + π6. (π 5. (π1. π 4 + π2. π4. π 1) + π5. π 1. π 2) + π3. π 1. π 2ππ§ = π6. (π2. π5 β π4. π 2. π 5) + π3. π2
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
ππ
ππ
ππ ππ
π2
ππ
ππ ππ
π
ππ
ππ
π60 = π΄1π΄2π΄3π΄4π΄5π΄6 =
π11 π12π21 π22
π13 ππ₯π23 ππ¦
π31 π320 0
π33 ππ§0 1
14.11.2017 J.Nassour 140
Stanford Arm
π11 = π 6. (π4. π 1 + π1. π2. π 4)β π6. (π5. (π 1. π 4 β π1. π2. π4) + π1. π 2. π 5)π21 = π6. (π5. (π1. π 4 + π2. π4. π 1)β π 1. π 2. π 5)β π 6. (π1. π4 β π2. π 1. π 4)π31 = βπ6. (π2. π 5 + π4. π5. π 2)β π 2. π 4. π 6
π12 = π 6. (π5. (π 1. π 4 β π1. π2. π4) + π1. π 2. π 5) + π5. (π4. π 1 + π1. π2. π 4)π22 = β π 6. (π5. (π1. π 4 + π2. π4. π 1) β π 1. π 2. π 5) β π5. (π1. π4 β π2. π 1. π 4)π32 = π 6. (π2. π 5 + π4. π5. π 2) β π5. π 2. π 4
π31 = π1. π5. π 2 β π 5. (π 1. π 4 β π1. π2. π4)π32 = π 5. (π1. π 4 + π2. π4. π 1) + π5. π 1. π 2π33 = π2. π5 β π4. π 2. π 5
ππ₯ = π3. π1. π 2 β π6. (π 5. (π 1. π 4 β π1. π2. π4) β π1. π5. π 2) β π2. π 1ππ¦ = π2. π1 + π6. (π 5. (π1. π 4 + π2. π4. π 1) + π5. π 1. π 2) + π3. π 1. π 2ππ§ = π6. (π2. π5 β π4. π 2. π 5) + π3. π2
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
ππ
ππ
ππ ππ
π2
ππ
ππ ππ
π
ππ
ππ
π60 =
π11 π12π21 π22
π13 ππ₯π23 ππ¦
π31 π320 0
π33 ππ§0 1
In the configuration shown, find:
14.11.2017 J.Nassour 141
Stanford Arm
π11 = π 6. (π4. π 1 + π1. π2. π 4)β π6. (π5. (π 1. π 4 β π1. π2. π4) + π1. π 2. π 5)π21 = π6. (π5. (π1. π 4 + π2. π4. π 1)β π 1. π 2. π 5)β π 6. (π1. π4 β π2. π 1. π 4)π31 = βπ6. (π2. π 5 + π4. π5. π 2)β π 2. π 4. π 6
π12 = π 6. (π5. (π 1. π 4 β π1. π2. π4) + π1. π 2. π 5) + π5. (π4. π 1 + π1. π2. π 4)π22 = β π 6. (π5. (π1. π 4 + π2. π4. π 1) β π 1. π 2. π 5) β π5. (π1. π4 β π2. π 1. π 4)π32 = π 6. (π2. π 5 + π4. π5. π 2) β π5. π 2. π 4
π31 = π1. π5. π 2 β π 5. (π 1. π 4 β π1. π2. π4)π32 = π 5. (π1. π 4 + π2. π4. π 1) + π5. π 1. π 2π33 = π2. π5 β π4. π 2. π 5
ππ₯ = π3. π1. π 2 β π6. (π 5. (π 1. π 4 β π1. π2. π4) β π1. π5. π 2) β π2. π 1ππ¦ = π2. π1 + π6. (π 5. (π1. π 4 + π2. π4. π 1) + π5. π 1. π 2) + π3. π 1. π 2ππ§ = π6. (π2. π5 β π4. π 2. π 5) + π3. π2
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
ππ
ππ
ππ ππ
π2
ππ
ππ ππ
π
ππ
ππ
π60 =
π11 π12π21 π22
π13 ππ₯π23 ππ¦
π31 π320 0
π33 ππ§0 1
In the configuration shown, find:
14.11.2017 J.Nassour 142
Stanford Arm
π11 = π 6. (π4. π 1 + π1. π2. π 4)β π6. (π5. (π 1. π 4 β π1. π2. π4) + π1. π 2. π 5)π21 = π6. (π5. (π1. π 4 + π2. π4. π 1)β π 1. π 2. π 5)β π 6. (π1. π4 β π2. π 1. π 4)π31 = βπ6. (π2. π 5 + π4. π5. π 2)β π 2. π 4. π 6
π12 = π 6. (π5. (π 1. π 4 β π1. π2. π4) + π1. π 2. π 5) + π5. (π4. π 1 + π1. π2. π 4)π22 = β π 6. (π5. (π1. π 4 + π2. π4. π 1) β π 1. π 2. π 5) β π5. (π1. π4 β π2. π 1. π 4)π32 = π 6. (π2. π 5 + π4. π5. π 2) β π5. π 2. π 4
π31 = π1. π5. π 2 β π 5. (π 1. π 4 β π1. π2. π4)π32 = π 5. (π1. π 4 + π2. π4. π 1) + π5. π 1. π 2π33 = π2. π5 β π4. π 2. π 5
ππ₯ = π3. π1. π 2 β π6. (π 5. (π 1. π 4 β π1. π2. π4) β π1. π5. π 2) β π2. π 1ππ¦ = π2. π1 + π6. (π 5. (π1. π 4 + π2. π4. π 1) + π5. π 1. π 2) + π3. π 1. π 2ππ§ = π6. (π2. π5 β π4. π 2. π 5) + π3. π2
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
ππ
ππ
ππ ππ
π2
ππ
ππ ππ
π
ππ
ππ
π60 =
π11 π12π21 π22
π13 ππ₯π23 ππ¦
π31 π320 0
π33 ππ§0 1
In the configuration shown, find:
14.11.2017 J.Nassour 143
Stanford Arm
π11 = π 6. (π4. π 1 + π1. π2. π 4)β π6. (π5. (π 1. π 4 β π1. π2. π4) + π1. π 2. π 5)π21 = π6. (π5. (π1. π 4 + π2. π4. π 1)β π 1. π 2. π 5)β π 6. (π1. π4 β π2. π 1. π 4)π31 = βπ6. (π2. π 5 + π4. π5. π 2)β π 2. π 4. π 6
π12 = π 6. (π5. (π 1. π 4 β π1. π2. π4) + π1. π 2. π 5) + π5. (π4. π 1 + π1. π2. π 4)π22 = β π 6. (π5. (π1. π 4 + π2. π4. π 1) β π 1. π 2. π 5) β π5. (π1. π4 β π2. π 1. π 4)π32 = π 6. (π2. π 5 + π4. π5. π 2) β π5. π 2. π 4
π31 = π1. π5. π 2 β π 5. (π 1. π 4 β π1. π2. π4)π32 = π 5. (π1. π 4 + π2. π4. π 1) + π5. π 1. π 2π33 = π2. π5 β π4. π 2. π 5
ππ₯ = π3. π1. π 2 β π6. (π 5. (π 1. π 4 β π1. π2. π4) β π1. π5. π 2) β π2. π 1ππ¦ = π2. π1 + π6. (π 5. (π1. π 4 + π2. π4. π 1) + π5. π 1. π 2) + π3. π 1. π 2ππ§ = π6. (π2. π5 β π4. π 2. π 5) + π3. π2
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
ππ
ππ
ππ ππ
π2
ππ
ππ ππ
π
ππ
ππ
π60 =
π11 π12π21 π22
π13 ππ₯π23 ππ¦
π31 π320 0
π33 ππ§0 1
In the configuration shown, find:
14.11.2017 J.Nassour 144
Stanford Arm
π11 = π 6. (π4. π 1 + π1. π2. π 4)β π6. (π5. (π 1. π 4 β π1. π2. π4) + π1. π 2. π 5)π21 = π6. (π5. (π1. π 4 + π2. π4. π 1)β π 1. π 2. π 5)β π 6. (π1. π4 β π2. π 1. π 4)π31 = βπ6. (π2. π 5 + π4. π5. π 2)β π 2. π 4. π 6
π12 = π 6. (π5. (π 1. π 4 β π1. π2. π4) + π1. π 2. π 5) + π5. (π4. π 1 + π1. π2. π 4)π22 = β π 6. (π5. (π1. π 4 + π2. π4. π 1) β π 1. π 2. π 5) β π5. (π1. π4 β π2. π 1. π 4)π32 = π 6. (π2. π 5 + π4. π5. π 2) β π5. π 2. π 4
π31 = π1. π5. π 2 β π 5. (π 1. π 4 β π1. π2. π4)π32 = π 5. (π1. π 4 + π2. π4. π 1) + π5. π 1. π 2π33 = π2. π5 β π4. π 2. π 5
ππ₯ = π3. π1. π 2 β π6. (π 5. (π 1. π 4 β π1. π2. π4) β π1. π5. π 2) β π2. π 1ππ¦ = π2. π1 + π6. (π 5. (π1. π 4 + π2. π4. π 1) + π5. π 1. π 2) + π3. π 1. π 2ππ§ = π6. (π2. π5 β π4. π 2. π 5) + π3. π2
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
ππ
ππ
ππ ππ
π2
ππ
ππ ππ
π
ππ
ππ
π60 =
π11 π12π21 π22
π13 ππ₯π23 ππ¦
π31 π320 0
π33 ππ§0 1
In the configuration shown, find:
14.11.2017 J.Nassour 145
Stanford Arm
π΄1 =
π1 0π 1 0
βπ 1 0π1 0
0 β10 0
0 00 1
Reminder: π¨π
π΄2 =
π2 0π 2 0
π 2 0βπ2 0
0 10 0
0 π20 1
π΄3 =
1 00 1
0 00 0
0 00 0
1 π30 1
π΄4 =
π4 0π 4 0
βπ 4 0π4 0
0 β10 0
0 00 1
π΄5 =
π5 0π 5 0
π 5 0βπ5 0
0 β10 0
0 00 1
π΄6 =
π6 βπ 6π 6 π6
0 00 0
0 00 0
1 π60 1
π ππ πΆπ π π π½π
1 0 β90 Β° 0 π½πβ
2 0 +90 Β° π2 π½πβ
3 0 0 Β°π π
β 0
4 0 β90 Β° 0 π½πβ
5 0 +90 Β° 0 π½πβ
6 0 0 Β° π6 π½πβ
14.11.2017 J.Nassour 146
Stanford Arm
14.11.2017 J.Nassour 147
Stanford Arm
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
ππ
ππ
ππ ππ
π2
ππ
ππ ππ
π
ππ
ππ
π11 = β π6 π5 π 1π 4 β π1π2π4 + π1π 2π 5 β π 6 π4π 1 + π1π2π 4π21 = π6 π5 π1π 4 + π2π4π 1 β π 1π 2π 5 + π 6 π1π4 β π2π 1π 4π31 = π 2π 4π 6 β π6 π2π 5 + π4π5π 2
π12 = π 6 π5 π 1π 4 β π1π2π4 + π1π 2π 5 β π6 π4π 1 + π1π2π 4π22 = π6 π1π4 β π2π 1π 4 β π 6 π5 π1π 4 + π2π4π 1 β π 1π 2π 5π32 = π 6 π2π 5 + π4π5π 2 + π6π 2π 4
π13 = π1π5π 2 β π 5 π 1π 4 β π1π2π4π23 = π 5 π1π 4 + π2π4π 1 + π5π 1π 2π33 = π2π5 β π4π 2π 5
ππ₯ = π3π1π 2 β π6 π 5 π 1π 4 β π1π2π4 β π1π5π 2 β π2π 1ππ¦ = π2π1 + π6 π 5 π1π 4 + π2π4π 1 + π5π 1π 2 + π3π 1π 2ππ§ = π6 π2π5 β π4π 2π 5 + π3π2
π11 = β π6 π5 π 1π 4 β π1π2π4 + π1π 2π 5 β π 6 π4π 1 + π1π2π 4π21 = π6 π5 π1π 4 + π2π4π 1 β π 1π 2π 5 + π 6 π1π4 β π2π 1π 4π31 = π 2π 4π 6 β π6 π2π 5 + π4π5π 2
π12 = π 6 π5 π 1π 4 β π1π2π4 + π1π 2π 5 β π6 π4π 1 + π1π2π 4π22 = π6 π1π4 β π2π 1π 4 β π 6 π5 π1π 4 + π2π4π 1 β π 1π 2π 5π32 = π 6 π2π 5 + π4π5π 2 + π6π 2π 4
π13 = π1π5π 2 β π 5 π 1π 4 β π1π2π4π23 = π 5 π1π 4 + π2π4π 1 + π5π 1π 2π33 = π2π5 β π4π 2π 5
ππ₯ = π3π1π 2 β π6 π 5 π 1π 4 β π1π2π4 β π1π5π 2 β π2π 1ππ¦ = π2π1 + π6 π 5 π1π 4 + π2π4π 1 + π5π 1π 2 + π3π 1π 2ππ§ = π6 π2π5 β π4π 2π 5 + π3π2
14.11.2017 J.Nassour 148
Stanford Arm
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
ππ
ππ
ππ ππ
π2
ππ
ππ ππ
π
ππ
ππ
14.11.2017 J.Nassour 149
Stanford Arm
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
ππ
ππ
ππ ππ
π2
ππ
ππ ππ
π
ππ
ππ
π11 = β π6 π5 π 1π 4 β π1π2π4 + π1π 2π 5 β π 6 π4π 1 + π1π2π 4π21 = π6 π5 π1π 4 + π2π4π 1 β π 1π 2π 5 + π 6 π1π4 β π2π 1π 4π31 = π 2π 4π 6 β π6 π2π 5 + π4π5π 2
π12 = π 6 π5 π 1π 4 β π1π2π4 + π1π 2π 5 β π6 π4π 1 + π1π2π 4π22 = π6 π1π4 β π2π 1π 4 β π 6 π5 π1π 4 + π2π4π 1 β π 1π 2π 5π32 = π 6 π2π 5 + π4π5π 2 + π6π 2π 4
π13 = π1π5π 2 β π 5 π 1π 4 β π1π2π4π23 = π 5 π1π 4 + π2π4π 1 + π5π 1π 2π33 = π2π5 β π4π 2π 5
ππ₯ = π3π1π 2 β π6 π 5 π 1π 4 β π1π2π4 β π1π5π 2 β π2π 1ππ¦ = π2π1 + π6 π 5 π1π 4 + π2π4π 1 + π5π 1π 2 + π3π 1π 2ππ§ = π6 π2π5 β π4π 2π 5 + π3π2
14.11.2017 J.Nassour 150
Stanford Arm
π π
π6
π½π
π½π
π½π
π½π
π½π
ππ
ππ
ππ
ππ
ππ ππ
π2
ππ
ππ ππ
π
ππ
ππ
π11 = β π6 π5 π 1π 4 β π1π2π4 + π1π 2π 5 β π 6 π4π 1 + π1π2π 4π21 = π6 π5 π1π 4 + π2π4π 1 β π 1π 2π 5 + π 6 π1π4 β π2π 1π 4π31 = π 2π 4π 6 β π6 π2π 5 + π4π5π 2
π12 = π 6 π5 π 1π 4 β π1π2π4 + π1π 2π 5 β π6 π4π 1 + π1π2π 4π22 = π6 π1π4 β π2π 1π 4 β π 6 π5 π1π 4 + π2π4π 1 β π 1π 2π 5π32 = π 6 π2π 5 + π4π5π 2 + π6π 2π 4
π13 = π1π5π 2 β π 5 π 1π 4 β π1π2π4π23 = π 5 π1π 4 + π2π4π 1 + π5π 1π 2π33 = π2π5 β π4π 2π 5
ππ₯ = π3π1π 2 β π6 π 5 π 1π 4 β π1π2π4 β π1π5π 2 β π2π 1ππ¦ = π2π1 + π6 π 5 π1π 4 + π2π4π 1 + π5π 1π 2 + π3π 1π 2ππ§ = π6 π2π5 β π4π 2π 5 + π3π2
PUMA 260
14.11.2017 J.Nassour 151
π½π
π½π
π½π
π½π
π½π
14.11.2017 J.Nassour 152
PUMA 260
π½π
π½π
π½π
π½π
π½ππ½π
14.11.2017 J.Nassour 153
PUMA 260
π½π
π½π
π½π
π½π
π½ππ½π
ππ
ππππ
14.11.2017 J.Nassour 154
PUMA 260
π½π
π½π
π½π
π½π
π½ππ½π
ππ
ππππ
ππ
ππ
ππ
14.11.2017 J.Nassour 155
PUMA 260
π½π
π½π
π½π
π½π
π½ππ½π
ππ
ππππ
ππ
ππ
ππ
ππ
ππ
ππ
π π
ππ
14.11.2017 J.Nassour 156
PUMA 260
π½π
π½π
π½π
π½π
π½ππ½π
ππ
ππππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
π π
ππ
ππ
14.11.2017 J.Nassour 157
PUMA 260
π½π
π½π
π½π
π½π
π½ππ½π
ππ
ππππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππππ
ππ
π π
ππ
π π
ππ
14.11.2017 J.Nassour 158
PUMA 260
π½π
π½π
π½π
π½π
π½ππ½π
ππ
ππππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππππ
ππ
ππ
ππ
ππ
π π
ππ
π π
ππ
14.11.2017 J.Nassour 159
PUMA 260
π½π
π½π
π½π
π½π
π½ππ½π
ππ
ππππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
π π
ππ
π π
ππ
π π
14.11.2017 J.Nassour 160
PUMA 260
π½π
π½π
π½π
π½π
π½ππ½π
ππ
ππππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
π π
ππ
π π
ππ
π π
π ππ πΆπ π π π½π
1 0 β90 Β° 0 π½πβ
2 π2 0 Β° π2 π½πβ
3 π3 90 Β° 0 π½πβ
4 0 β90 Β° π4 π½πβ
5 0 +90 Β° 0 π½πβ
6 0 0 Β° π6 π½πβ
ππ is distance from ππβπ to ππ measured along ππ.πΆπ is angle from ππβπ to ππ measured about ππ. π π is distance from ππβπ to ππ measured along ππβπ.
π½π is angle from ππβπ to ππ measured about ππβπ.
14.11.2017 J.Nassour 161
PUMA 260
π½π
π½π
π½π
π½π
π½ππ½π
ππ
ππππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
π π
ππ
π π
ππ
π π
π ππ πΆπ π π π½π
1 0 β90 Β° 0 ππ
2 π2 0 Β° π2 π
3 π3 90 Β° 0 ππ
4 0 β90 Β° π4 π
5 0 +90 Β° 0 π
6 0 0 Β° π6 π
ππ is distance from ππβπ to ππ measured along ππ.πΆπ is angle from ππβπ to ππ measured about ππ. π π is distance from ππβπ to ππ measured along ππβπ.
π½π is angle from ππβπ to ππ measured about ππβπ.
In the configuration shown, find π½π?
14.11.2017 J.Nassour 162
PUMA 260π ππ πΆπ π π π½π
1 0 β90 Β° 0 π½πβ
2 π2 0 Β° π2 π½πβ
3 π3 90 Β° 0 π½πβ
4 0 β90 Β° π4 π½πβ
5 0 +90 Β° 0 π½πβ
6 0 0 Β° π6 π½πβ
Reminder: π¨π
14.11.2017 J.Nassour 163
r11 = β s6 c4s1 β s4 c1s2s3 β c1c2c3 β c6 c5 s1s4 + c4 c1s2s3 β c1c2c3 + s5 c1c2s3 + c1c3s2
π12 = π 6 π5 π 1π 4 + π4 π1π 2π 3 β π1π2π3 + π 5 π1π2π 3 + π1π3π 2 β π6 π4π 1 β π 4 π1π 2π 3 β π1π2π3
r13 = c5 c1c2s3 + c1c3s2 β s5 s1s4 + c4 c1s2s3 β c1c2c3
ππ₯ = π4 π1π2π 3 + π1π3π 2 β π2π 1 β π6 π 5 π 1π 4 + π4 π1π 2π 3 β π1π2π3 β π5 π1π2π 3 + π1π3π 2 + π2π1π2 + π3π1π2π3 β π3π1π 2π 3
π21 = π 6 π1π4 + π 4 π 1π 2π 3 β π2π3π 1 + π6 π5 π1π 4 β π4 π 1π 2π 3 β π2π3π 1 β π 5 π2π 1π 3 + π3π 1π 2
π22 = π6 π1π4 + π 4 π 1π 2π 3 β π2π3π 1 β π 6 π5 π1π 4 β π4 π 1π 2π 3 β π2π3π 1 β π 5 π2π 1π 3 + π3π 1π 2
π23 = π 5 π1π 4 β π4 π 1π 2π 3 β π2π3π 1 + π5 π2π 1π 3 + π3π 1π 2
ππ¦ = π6 π 5 π1π 4 β π4 π 1π 2π 3 β π2π3π 1 + π5 π2π 1π 3 + π3π 1π 2 + π4 π2π 1π 3 + π3π 1π 2 + π2π1 + π2π2π 1 + π3π2π3π 1 β π3π 1π 2π 3
π31 = π 4π 6 π2π 3 + π3π 2 β π6 π 5 π2π3 β π 2π 3 + π4π5 π2π 3 + π3π 2π32 = π 6 π 5 π2π3 β π 2π 3 + π4π5 π2π 3 + π3π 2 + π6π 4 π2π 3 + π3π 2π33 = π5 π2π3 β π 2π 3 β π4π 5 π2π 3 + π3π 2ππ§ = π4 π2π3 β π 2π 3 β π2π 2 + π6 π5 π2π3 β π 2π 3 β π4π 5 π2π 3 + π3π 2 β π3π2π 3 β π3π3π 2
PUMA 260
14.11.2017 J.Nassour 164
r11 = β s6 c4s1 β s4 c1s2s3 β c1c2c3 β c6 c5 s1s4 + c4 c1s2s3 β c1c2c3 + s5 c1c2s3 + c1c3s2
π12 = π 6 π5 π 1π 4 + π4 π1π 2π 3 β π1π2π3 + π 5 π1π2π 3 + π1π3π 2 β π6 π4π 1 β π 4 π1π 2π 3 β π1π2π3
r13 = c5 c1c2s3 + c1c3s2 β s5 s1s4 + c4 c1s2s3 β c1c2c3
ππ₯ = π4 π1π2π 3 + π1π3π 2 β π2π 1 β π6 π 5 π 1π 4 + π4 π1π 2π 3 β π1π2π3 β π5 π1π2π 3 + π1π3π 2 + π2π1π2 + π3π1π2π3 β π3π1π 2π 3
π21 = π 6 π1π4 + π 4 π 1π 2π 3 β π2π3π 1 + π6 π5 π1π 4 β π4 π 1π 2π 3 β π2π3π 1 β π 5 π2π 1π 3 + π3π 1π 2
π22 = π6 π1π4 + π 4 π 1π 2π 3 β π2π3π 1 β π 6 π5 π1π 4 β π4 π 1π 2π 3 β π2π3π 1 β π 5 π2π 1π 3 + π3π 1π 2
π23 = π 5 π1π 4 β π4 π 1π 2π 3 β π2π3π 1 + π5 π2π 1π 3 + π3π 1π 2
ππ¦ = π6 π 5 π1π 4 β π4 π 1π 2π 3 β π2π3π 1 + π5 π2π 1π 3 + π3π 1π 2 + π4 π2π 1π 3 + π3π 1π 2 + π2π1 + π2π2π 1 + π3π2π3π 1 β π3π 1π 2π 3
π31 = π 4π 6 π2π 3 + π3π 2 β π6 π 5 π2π3 β π 2π 3 + π4π5 π2π 3 + π3π 2π32 = π 6 π 5 π2π3 β π 2π 3 + π4π5 π2π 3 + π3π 2 + π6π 4 π2π 3 + π3π 2π33 = π5 π2π3 β π 2π 3 β π4π 5 π2π 3 + π3π 2ππ§ = π4 π2π3 β π 2π 3 β π2π 2 + π6 π5 π2π3 β π 2π 3 β π4π 5 π2π 3 + π3π 2 β π3π2π 3 β π3π3π 2
PUMA 260
π½π
π½π
π½π
π½π
π½ππ½π
ππ
ππππ
ππ
ππ
ππ
14.11.2017 J.Nassour 165
r11 = β s6 c4s1 β s4 c1s2s3 β c1c2c3 β c6 c5 s1s4 + c4 c1s2s3 β c1c2c3 + s5 c1c2s3 + c1c3s2
π12 = π 6 π5 π 1π 4 + π4 π1π 2π 3 β π1π2π3 + π 5 π1π2π 3 + π1π3π 2 β π6 π4π 1 β π 4 π1π 2π 3 β π1π2π3
r13 = c5 c1c2s3 + c1c3s2 β s5 s1s4 + c4 c1s2s3 β c1c2c3
ππ₯ = π4 π1π2π 3 + π1π3π 2 β π2π 1 β π6 π 5 π 1π 4 + π4 π1π 2π 3 β π1π2π3 β π5 π1π2π 3 + π1π3π 2 + π2π1π2 + π3π1π2π3 β π3π1π 2π 3
π21 = π 6 π1π4 + π 4 π 1π 2π 3 β π2π3π 1 + π6 π5 π1π 4 β π4 π 1π 2π 3 β π2π3π 1 β π 5 π2π 1π 3 + π3π 1π 2
π22 = π6 π1π4 + π 4 π 1π 2π 3 β π2π3π 1 β π 6 π5 π1π 4 β π4 π 1π 2π 3 β π2π3π 1 β π 5 π2π 1π 3 + π3π 1π 2
π23 = π 5 π1π 4 β π4 π 1π 2π 3 β π2π3π 1 + π5 π2π 1π 3 + π3π 1π 2
ππ¦ = π6 π 5 π1π 4 β π4 π 1π 2π 3 β π2π3π 1 + π5 π2π 1π 3 + π3π 1π 2 + π4 π2π 1π 3 + π3π 1π 2 + π2π1 + π2π2π 1 + π3π2π3π 1 β π3π 1π 2π 3
π31 = π 4π 6 π2π 3 + π3π 2 β π6 π 5 π2π3 β π 2π 3 + π4π5 π2π 3 + π3π 2π32 = π 6 π 5 π2π3 β π 2π 3 + π4π5 π2π 3 + π3π 2 + π6π 4 π2π 3 + π3π 2π33 = π5 π2π3 β π 2π 3 β π4π 5 π2π 3 + π3π 2ππ§ = π4 π2π3 β π 2π 3 β π2π 2 + π6 π5 π2π3 β π 2π 3 β π4π 5 π2π 3 + π3π 2 β π3π2π 3 β π3π3π 2
PUMA 260
π½π
π½π
π½π
π½π
π½ππ½π
ππ
ππππ
ππ
ππ
ππ
14.11.2017 J.Nassour 166
PUMA 260
π½π
π½π
π½π
π½π
π½ππ½π
ππ
ππππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
π π
ππ
π π
ππ
π π
π ππ πΆπ π π π½π
1 0 β90 Β° 0 ππ
2 π2 0 Β° π2 π
3 π3 90 Β° 0 ππ
4 0 β90 Β° π4 π
5 0 +90 Β° 0 π
6 0 0 Β° π6 π
ππ is distance from ππβπ to ππ measured along ππ.πΆπ is angle from ππβπ to ππ measured about ππ. π π is distance from ππβπ to ππ measured along ππβπ.
π½π is angle from ππβπ to ππ measured about ππβπ.
14.11.2017 J.Nassour 167
r11 = β s6 c4s1 β s4 c1s2s3 β c1c2c3 β c6 c5 s1s4 + c4 c1s2s3 β c1c2c3 + s5 c1c2s3 + c1c3s2
π12 = π 6 π5 π 1π 4 + π4 π1π 2π 3 β π1π2π3 + π 5 π1π2π 3 + π1π3π 2 β π6 π4π 1 β π 4 π1π 2π 3 β π1π2π3
r13 = c5 c1c2s3 + c1c3s2 β s5 s1s4 + c4 c1s2s3 β c1c2c3
ππ₯ = π4 π1π2π 3 + π1π3π 2 β π2π 1 β π6 π 5 π 1π 4 + π4 π1π 2π 3 β π1π2π3 β π5 π1π2π 3 + π1π3π 2 + π2π1π2 + π3π1π2π3 β π3π1π 2π 3
π21 = π 6 π1π4 + π 4 π 1π 2π 3 β π2π3π 1 + π6 π5 π1π 4 β π4 π 1π 2π 3 β π2π3π 1 β π 5 π2π 1π 3 + π3π 1π 2
π22 = π6 π1π4 + π 4 π 1π 2π 3 β π2π3π 1 β π 6 π5 π1π 4 β π4 π 1π 2π 3 β π2π3π 1 β π 5 π2π 1π 3 + π3π 1π 2
π23 = π 5 π1π 4 β π4 π 1π 2π 3 β π2π3π 1 + π5 π2π 1π 3 + π3π 1π 2
ππ¦ = π6 π 5 π1π 4 β π4 π 1π 2π 3 β π2π3π 1 + π5 π2π 1π 3 + π3π 1π 2 + π4 π2π 1π 3 + π3π 1π 2 + π2π1 + π2π2π 1 + π3π2π3π 1 β π3π 1π 2π 3
π31 = π 4π 6 π2π 3 + π3π 2 β π6 π 5 π2π3 β π 2π 3 + π4π5 π2π 3 + π3π 2π32 = π 6 π 5 π2π3 β π 2π 3 + π4π5 π2π 3 + π3π 2 + π6π 4 π2π 3 + π3π 2π33 = π5 π2π3 β π 2π 3 β π4π 5 π2π 3 + π3π 2ππ§ = π4 π2π3 β π 2π 3 β π2π 2 + π6 π5 π2π3 β π 2π 3 β π4π 5 π2π 3 + π3π 2 β π3π2π 3 β π3π3π 2
PUMA 260
π½π
π½π
π½π
π½π
π½ππ½π
ππ
ππππ
ππ
ππ
ππ
14.11.2017 J.Nassour 168
NAO Left Arm
14.11.2017 J.Nassour 169
NAO Zero Position
Provided by Aldebaran Robotics
The torso is the point where all the kinematic chains begin and is located at the center of the NAO body.
14.11.2017 J.Nassour 170
NAO Zero Position
14.11.2017 J.Nassour 171
NAO Left Arm
π½π
π½π
14.11.2017 J.Nassour 172
NAO Left Arm
π½π
π½π
π½π
π½π
14.11.2017 J.Nassour 173
NAO Left Arm
π½π
π½π
π½π
π½π
π½π
14.11.2017 J.Nassour 174
NAO Left Arm
π½π
π½π
π½π
π½π
π½π
ππ»
ππ»ππ»
14.11.2017 J.Nassour 175
NAO Left Arm
π½π
π½π
π½π
π½π
π½π
ππ»
ππ»ππ»
ππππ
ππ
14.11.2017 J.Nassour 176
NAO Left Arm
π½π
π½π
π½π
π½π
π½π
ππ»
ππ»ππ»
ππππ
ππ
ππ
ππ
ππ
14.11.2017 J.Nassour 177
NAO Left Arm
π½π
π½π
π½π
π½π
π½π
ππ»
ππ»ππ»
ππππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
14.11.2017 J.Nassour 178
NAO Left Arm
π½π
π½π
π½π
π½π
π½π
ππ»
ππ»ππ»
ππππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
14.11.2017 J.Nassour 179
NAO Left Arm
π½π
π½π
π½π
π½π
π½π
ππ»
ππ»ππ»
ππππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
14.11.2017 J.Nassour 180
NAO Left Arm
π½π
π½π
π½π
π½π
π½π
ππ»
ππ»ππ»
ππππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππππ
14.11.2017 J.Nassour 181
π½π
π½π
π½π
π½π
π½π
ππ»
ππ»ππ»
ππππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
NAO Left Armπ ππ πΆπ π π π½π
0 π 0π΅π΄ππΈ (0, πβππ’ππππππππ ππ‘π, πβππ’ππππππππ ππ‘π)
1 0 90 Β° 0 π½πβ
2 π2 90 Β° 0 π
π+ π½π
β
3 0 β90 Β° π3 π½πβ
4 0 +90 Β° 0 π½πβ
5 π5 0 Β° π5π
π+ π½π
β
ππ is distance from ππβπ to ππ measured along ππ.πΆπ is angle from ππβπ to ππ measured about ππ. π π is distance from ππβπ to ππ measured along ππβπ.
π½π is angle from ππβπ to ππ measured about ππβπ.
ππ
ππππ
14.11.2017 J.Nassour 182
π½π
π½π
π½π
π½π
π½π
ππ»
ππ»ππ»
ππππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
ππ
NAO Left Armπ ππ πΆπ π π π½π
0 π 0π΅π΄ππΈ (0, πβππ’ππππππππ ππ‘π, πβππ’ππππππππ ππ‘π)
1 0 90 Β° 0 π½πβ
2 π2 90 Β° 0 π
π+ π½π
β
3 0 β90 Β° π3 π½πβ
4 0 +90 Β° 0 π½πβ
5 π5 0 Β° π5π
π+ π½π
β
ππ is distance from ππβπ to ππ measured along ππ.πΆπ is angle from ππβπ to ππ measured about ππ. π π is distance from ππβπ to ππ measured along ππβπ.
π½π is angle from ππβπ to ππ measured about ππβπ.
π» ππ©π¨πΊπ¬ = ?
ππ
ππππ
14.11.2017 J.Nassour 183