forward kinematics - tu chemnitz
TRANSCRIPT
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Forward Kinematics
Dr.-Ing. John Nassour
Artificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik
Serial link manipulators
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Suggested literature
• Robot Modeling and Control• 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
𝒙
𝒚
𝒙 × 𝒚
𝒛
𝒚
𝒛
𝒙
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
𝝎Wrap 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 𝜽 for a revolute joint and 𝒅 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
𝟔 𝒑𝒂𝒓𝒂𝒎𝒆𝒕𝒆𝒓𝒔/𝒍𝒊𝒏𝒌 𝟑 𝒑𝒐𝒔𝒊𝒕𝒊𝒐𝒏𝒔𝟑 𝒐𝒓𝒊𝒆𝒏𝒕𝒂𝒕𝒊𝒐𝒏𝒔
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Generalized Coordinates
A set of independent configuration parameters
𝟔 𝒑𝒂𝒓𝒂𝒎𝒆𝒕𝒆𝒓𝒔/𝒍𝒊𝒏𝒌 𝟑 𝒑𝒐𝒔𝒊𝒕𝒊𝒐𝒏𝒔𝟑 𝒐𝒓𝒊𝒆𝒏𝒕𝒂𝒕𝒊𝒐𝒏𝒔
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𝐿𝑖𝑛𝑘 1
Base
𝐿𝑖𝑛𝑘 2
𝐿𝑖𝑛𝑘 𝑛
6n parameters for n moving links
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Generalized Coordinates
A set of independent configuration parameters
𝟔 𝒑𝒂𝒓𝒂𝒎𝒆𝒕𝒆𝒓𝒔/𝒍𝒊𝒏𝒌 𝟑 𝒑𝒐𝒔𝒊𝒕𝒊𝒐𝒏𝒔𝟑 𝒐𝒓𝒊𝒆𝒏𝒕𝒂𝒕𝒊𝒐𝒏𝒔
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𝐿𝑖𝑛𝑘 1
Base
𝐿𝑖𝑛𝑘 2
𝐿𝑖𝑛𝑘 𝑛𝟓 𝑪𝒐𝒏𝒔𝒕𝒓𝒂𝒊𝒏𝒕
6n parameters for n moving links
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Generalized Coordinates
A set of independent configuration parameters
𝟔 𝒑𝒂𝒓𝒂𝒎𝒆𝒕𝒆𝒓𝒔/𝒍𝒊𝒏𝒌 𝟑 𝒑𝒐𝒔𝒊𝒕𝒊𝒐𝒏𝒔𝟑 𝒐𝒓𝒊𝒆𝒏𝒕𝒂𝒕𝒊𝒐𝒏𝒔
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𝐿𝑖𝑛𝑘 1
Base
𝐿𝑖𝑛𝑘 2
𝐿𝑖𝑛𝑘 𝑛𝟓 𝑪𝒐𝒏𝒔𝒕𝒓𝒂𝒊𝒏𝒕
6n parameters for n moving links5n constraints for n joints
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Generalized Coordinates
A set of independent configuration parameters
𝟔 𝒑𝒂𝒓𝒂𝒎𝒆𝒕𝒆𝒓𝒔/𝒍𝒊𝒏𝒌 𝟑 𝒑𝒐𝒔𝒊𝒕𝒊𝒐𝒏𝒔𝟑 𝒐𝒓𝒊𝒆𝒏𝒕𝒂𝒕𝒊𝒐𝒏𝒔
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𝐿𝑖𝑛𝑘 1
Base
𝐿𝑖𝑛𝑘 2
𝐿𝑖𝑛𝑘 𝑛𝟓 𝑪𝒐𝒏𝒔𝒕𝒓𝒂𝒊𝒏𝒕
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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𝒐𝟎 𝒙𝟎 𝒚𝟎 𝒛𝟎
𝒐𝒏+𝟏 𝒙𝒏+𝟏 𝒚𝒏+𝟏 𝒛𝒏+𝟏
End-effector configuration parameters are a set of 𝒎 parameters (𝒙𝟏, 𝒙𝟐, 𝒙𝟑, . . , 𝒙𝒎) that completely specify the end-effector position and orientation with respect to the frame 𝒐𝟎 𝒙𝟎 𝒚𝟎 𝒛𝟎.
𝒐𝒏+𝟏 is the operational point.
A set (𝒙𝟏, 𝒙𝟐, 𝒙𝟑, . . , 𝒙𝒎𝟎) of
independent configuration Parameters 𝒎𝟎: number of degree of freedom of the end-effector.
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Operational Coordinates
Base
𝒙𝒚
𝒙
𝜶
𝒚
𝜶
Is also called Operational Space
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Joint Coordinates
Base
𝜽1
𝜽2𝜽3
𝜽1
𝜽2
𝜽3
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
𝜽1
𝜽2
𝜽3
𝒙
𝜶
𝒚
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Redundancy
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A robot is said to be redundant if 𝒏 > 𝒎𝟎. Degree of redundancy: 𝒏 −𝒎𝟎
Base
𝒎𝟎 = 𝟑𝒏 = 𝟒
how many solutions exist?
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Redundancy
14.11.2017
A robot is said to be redundant if 𝒏 > 𝒎𝟎. Degree of redundancy: 𝒏 −𝒎𝟎
Base
𝒎𝟎 = 𝟑𝒏 = 𝟒
how many solutions exist?
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Redundancy
14.11.2017
A robot is said to be redundant if 𝒏 > 𝒎𝟎. Degree of redundancy: 𝒏 −𝒎𝟎
Base
𝒎𝟎 = 𝟑𝒏 = 𝟒
how many solutions exist?
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Redundancy
14.11.2017
A robot is said to be redundant if 𝒏 > 𝒎𝟎. Degree of redundancy: 𝒏 −𝒎𝟎
Base
𝒎𝟎 = 𝟑𝒏 = 𝟑
how many solutions exist?
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Redundancy
14.11.2017
A robot is said to be redundant if 𝒏 > 𝒎𝟎. Degree of redundancy: 𝒏 −𝒎𝟎
Base
𝒎𝟎 = 𝟑𝒏 = 𝟑
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 𝒊 is actuated, link 𝒊 moves.
𝒒𝒊 is the joint variable
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Kinematic Arrangements
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Spherical wrist 3 D.O.F
spherical wrist: RRR Links’ 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 𝒐𝒊𝒙𝒊 𝒚𝒊 𝒛𝒊 to 𝒍𝒊𝒏𝒌 𝒊. This means that, whatever motion the robot executes, the coordinates of any point 𝒑 on
link 𝒊 are constant when expressed in the 𝒊𝒕𝒉 coordinate frame 𝒑𝒊 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕.When 𝒋𝒐𝒊𝒏𝒕 𝒊 is actuated, 𝒍𝒊𝒏𝒌 𝒊 and its attached frame, 𝒐𝒊𝒙𝒊 𝒚𝒊 𝒛𝒊, experience a resulting motion.
The frame 𝒐𝟎𝒙𝟎 𝒚𝟎 𝒛𝟎, which is attached to the robot base, is referred to as the reference frame.
𝒐𝒊𝒙𝒊 𝒚𝒊 𝒛𝒊
𝒐𝟎𝒙𝟎 𝒚𝟎 𝒛𝟎
𝒍𝒊𝒏𝒌 𝒊𝒑
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Kinematic Arrangements
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Base
𝒐𝟎𝒙𝟎 𝒚𝟎 𝒛𝟎
𝒐𝒊𝒙𝒊 𝒚𝒊 𝒛𝒊
𝒍𝒊𝒏𝒌 𝒊
The frame 𝒐𝟎𝒙𝟎 𝒚𝟎 𝒛𝟎, 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 𝒐𝒊𝒙𝒊 𝒚𝒊 𝒛𝒊 to 𝒍𝒊𝒏𝒌 𝒊. This means that, whatever motion the robot executes, the coordinates of any point 𝒑 on
link 𝒊 are constant when expressed in the 𝒊𝒕𝒉 coordinate frame 𝒑𝒊 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕.When 𝒋𝒐𝒊𝒏𝒕 𝒊 is actuated, 𝒍𝒊𝒏𝒌 𝒊 and its attached frame, 𝒐𝒊𝒙𝒊 𝒚𝒊 𝒛𝒊, experience a resulting motion.
𝒑
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Kinematic Arrangements
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Base
𝒐𝟎𝒙𝟎 𝒚𝟎 𝒛𝟎
𝒐𝒊𝒙𝒊 𝒚𝒊 𝒛𝒊
𝒍𝒊𝒏𝒌 𝒊
The frame 𝒐𝟎𝒙𝟎 𝒚𝟎 𝒛𝟎, 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 𝒐𝒊𝒙𝒊 𝒚𝒊 𝒛𝒊 to 𝒍𝒊𝒏𝒌 𝒊. This means that, whatever motion the robot executes, the coordinates of any point 𝒑 on
link 𝒊 are constant when expressed in the 𝒊𝒕𝒉 coordinate frame 𝒑𝒊 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕.When 𝒋𝒐𝒊𝒏𝒕 𝒊 is actuated, 𝒍𝒊𝒏𝒌 𝒊 and its attached frame, 𝒐𝒊𝒙𝒊 𝒚𝒊 𝒛𝒊, experience a resulting motion.
𝒑
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Joint And Link Labelling
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Joint And Link Labelling
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𝒐𝟎𝒙𝟎 𝒚𝟎 𝒛𝟎Link 0 (fixed)Base frame
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Joint And Link Labelling
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𝒐𝟎𝒙𝟎 𝒚𝟎 𝒛𝟎Link 0 (fixed)
Joint 1
Link 1
Joint variable 𝜽1
Base frame
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Joint And Link Labelling
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𝒐𝟎𝒙𝟎 𝒚𝟎 𝒛𝟎Link 0 (fixed)
Joint 1
Link 1
Joint variable 𝜽1
𝒐𝟏𝒙𝟏 𝒚𝟏 𝒛𝟏
Joint 2 Link 2
Joint variable 𝜽2
Base frame
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Joint And Link Labelling
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𝒐𝟎𝒙𝟎 𝒚𝟎 𝒛𝟎Link 0 (fixed)
Joint 1
Link 1
Joint variable 𝜽1
𝒐𝟏𝒙𝟏 𝒚𝟏 𝒛𝟏
Joint 2 Link 2
Joint variable 𝜽2
Link 3
𝒐𝟐𝒙𝟐 𝒚𝟐 𝒛𝟐
Joint 3
Joint variable 𝜽3
Base frame
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Joint And Link Labelling
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𝒐𝟎𝒙𝟎 𝒚𝟎 𝒛𝟎Link 0 (fixed)
Joint 1
Link 1
Joint variable 𝜽1
𝒐𝟏𝒙𝟏 𝒚𝟏 𝒛𝟏
Joint 2 Link 2
Joint variable 𝜽2
Link 3
𝒐𝟐𝒙𝟐 𝒚𝟐 𝒛𝟐
Joint 3
Joint variable 𝜽3
𝒐𝟑𝒙𝟑 𝒚𝟑 𝒛𝟑
Base frame
Do we need a specific way to orientate the axes?
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Transformation Matrix
𝒐𝟎𝒙𝟎 𝒚𝟎 𝒛𝟎Link 0 (fixed)
Joint 1
Link 1
Joint variable 𝜽1
𝒐𝟏𝒙𝟏 𝒚𝟏 𝒛𝟏
Joint 2 Link 2
Joint variable 𝜽2
Link 3
𝒐𝟐𝒙𝟐 𝒚𝟐 𝒛𝟐
Joint 3
Joint variable 𝜽3
𝒐𝟑𝒙𝟑 𝒚𝟑 𝒛𝟑
Base frame
Suppose 𝑨𝒊 is the homogeneous transformation matrix that describe the position and the orientation of 𝒐𝒊𝒙𝒊 𝒚𝒊 𝒛𝒊 with respect to 𝒐𝒊−𝟏𝒙𝒊−𝟏 𝒚𝒊−𝟏 𝒛𝒊−𝟏.𝑨𝒊 is derived from joint and link 𝑖. 𝑨𝒊 is a function of only a single joint variable.
𝑨𝒊 = 𝑨𝒊(𝒒𝒊)
𝑨𝒊(𝒒𝒊) =𝑹 𝒊
𝒊−𝟏 𝒐 𝒊𝒊−𝟏
𝟎 𝟏
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Transformation Matrix
The position and the orientation of the end effector (reference frame 𝒐𝒏𝒙𝒏 𝒚𝒏 𝒛𝒏) with respect to the base (reference frame 𝒐𝟎𝒙𝟎 𝒚𝟎 𝒛𝟎) can be expressed by the transformation matrix:
𝐇 = 𝑻𝒏𝟎 = 𝑨𝟏 𝒒𝟏 …𝑨𝒏(𝒒𝒏) =
𝑹𝒏𝟎 𝒐𝒏
𝟎
𝟎 𝟏
The position and the orientation of a reference frame 𝒐𝒋𝒙𝒋 𝒚𝒋 𝒛𝒋) with
respect to a reference frame 𝒐𝒊𝒙𝒊 𝒚𝒊 𝒛𝒊 can be expressed by the transformation matrix:
𝑻𝒋𝒊 =
𝑨𝒊+𝟏𝑨𝒊+𝟐…𝑨𝒋−𝟏 𝑨𝒋
𝑰
(𝑻𝒊𝒋)−𝟏
𝑖𝑓 𝒊 < 𝒋𝑖𝑓 𝒊 = 𝒋𝑖𝑓 𝒊 > 𝒋
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Transformation Matrix
𝑻𝒋𝒊 =
𝑨𝒊+𝟏𝑨𝒊+𝟐…𝑨𝒋−𝟏 𝑨𝒋
𝑰
(𝑻𝒊𝒋)−𝟏
𝑖𝑓 𝒊 < 𝒋𝑖𝑓 𝒊 = 𝒋𝑖𝑓 𝒊 > 𝒋
if 𝒊 < 𝒋 then
𝑻𝒋𝒊 = 𝑨𝒊+𝟏𝑨𝒊+𝟐…𝑨𝒋−𝟏 𝑨𝒋 =
𝑹𝒋𝒊 𝒐𝒋
𝒊
𝟎 𝟏
The orientation part: 𝑹𝒋𝒊 = 𝑹𝒊+𝟏
𝒊 …𝑹 𝒋𝒋−𝟏
The translation part: 𝒐𝒋𝒊 = 𝒐𝒋−𝟏
𝒊+𝑹𝒋−𝟏𝒊 𝒐 𝒋
𝒋−𝟏
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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:• the link length a• the link twist a
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Link Description
Axis(i-1) Axis(i)Link(i-1)
𝒂𝒊−𝟏 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.
𝒂𝒊−𝟏
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Link Description
Axis(i-1) Axis(i)Link(i-1)
𝜶𝒊−𝟏 Link Twist
Project both axes 𝑖 − 1 and 𝑖onto the plane whose normal isthe mutually perpendicular line.
Measured in the right-handsense about 𝒂𝒊−𝟏.
𝒂𝒊−𝟏
𝜶𝒊−𝟏
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Link Description
Intersecting joint axis !
𝒂𝒊−𝟏 Link length ?
𝜶𝒊−𝟏 Link Twist ?The sense of 𝛼𝑖−1 is free.
Axis(i-1)Axis(i)
𝜶𝒊−𝟏
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Joint Parameters
Axis(i-1) Axis(i)Link(i-1)
𝒂𝒊−𝟏
𝜶𝒊−𝟏
A joint axis is established atthe connection of two links.
This joint will have twonormals connected to itone for each of the links.
𝒂𝒊
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Joint Parameters
Axis(i-1) Axis(i)Link(i-1)
𝒂𝒊−𝟏
𝜶𝒊−𝟏
𝒂𝒊
𝒅𝒊 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).
𝒅𝒊
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Joint Parameters
Axis(i-1) Axis(i)Link(i-1)
𝒂𝒊−𝟏
𝜶𝒊−𝟏
𝒂𝒊
𝒅𝒊 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).
𝜽𝒊 Joint AngleVariable if joint is revolute.
The joint angle between the normals is measured in a plane normal to the joint axis.
𝒅𝒊
𝜽𝒊
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Link Description
Axis(i-1) Axis(i)Link(i-1)
𝒂𝒊−𝟏 Link Length
and
𝜶𝒊−𝟏 Link Twist
depend on joint axes𝑖 − 1 and 𝑖.
𝒂𝒊−𝟏
𝜶𝒊−𝟏
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Joint Parameters
Axis(i-1) Axis(i)Link(i-1)
𝒂𝒊−𝟏
𝜶𝒊−𝟏
𝒂𝒊
𝒅𝒊 Link Offsetand
𝜽𝒊 Joint Angle
depend on links 𝑖 − 1and 𝑖.
𝒅𝒊
𝜽𝒊
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Denavit-Hartenberg Convention
Each 𝐴 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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14.11.2017
Denavit-Hartenberg Convention
Each 𝐴 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.
𝒂𝒊 is link length of like i (constant unless you reconfigure the robot)
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14.11.2017
Denavit-Hartenberg Convention
Each 𝐴 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.
𝒂𝒊 is link length of like i (constant unless you reconfigure the robot)𝜶𝒊 is link twist of link i (constant unless you reconfigure the robot)
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14.11.2017
Denavit-Hartenberg Convention
Each 𝐴 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.
𝒂𝒊 is link length of like i (constant unless you reconfigure the robot)𝜶𝒊 is link twist of link i (constant unless you reconfigure the robot)𝒅𝒊 is link offset of link i (prismatic variable)
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14.11.2017
Denavit-Hartenberg Convention
Each 𝐴 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.
𝒂𝒊 is link length of like i (constant unless you reconfigure the robot)𝜶𝒊 is link twist of link i (constant unless you reconfigure the robot)𝒅𝒊 is link offset of link i (prismatic variable)𝜽𝒊 is joint angle of link i (revolute variable)
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Denavit-Hartenberg Matrix
Each homogeneous transformation 𝐴𝑖 is represented as a product of four basic transformations:
where the four quantities are parameters associated with 𝑙𝑖𝑛𝑘 𝑖 and 𝑗𝑜𝑖𝑛𝑡 𝑖.
𝒂𝒊 is link length 𝜶𝒊 is link twist𝒅𝒊 is link offset𝜽𝒊 is joint angle
Reminder:
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Denavit-Hartenberg Matrix
Axis(i-1) Axis(i)Link(i-1)
𝒂𝒊−𝟏
𝜶𝒊−𝟏
𝒂𝒊
𝒂𝒊 is link length 𝜶𝒊 is link twist𝒅𝒊 is link offset𝜽𝒊 is joint angle
𝒅𝒊
𝜽𝒊
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Denavit-Hartenberg Matrix
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it is not necessary that the origin of 𝑓𝑟𝑎𝑚𝑒 𝑖 be placed at the physical end of 𝑙𝑖𝑛𝑘 𝑖.
it is not necessary that frame 𝑖 be placed within the physical link; 𝑓𝑟𝑎𝑚𝑒 𝑖 could lie in free space — so long as 𝑓𝑟𝑎𝑚𝑒 𝑖 is rigidly attached to 𝑙𝑖𝑛𝑘 𝑖.
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 𝑥1is perpendicular to the axis 𝑧0. (DH2) The axis 𝑥1 intersects the axis 𝑧0.
Under these conditions, there existunique numbers a, d, 𝜽, 𝜶 such that:
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Denavit-Hartenberg Convention
Positive sense for 𝜃 and 𝛼
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Rules For Assigning Frames
Rule 1: 𝑧𝑖−1 is axis of actuation of joint 𝑖.Axis of revolution of revolute jointAxis of translation of prismatic joint
Rule 2: Axis 𝑥𝑖 is set so it is perpendicular to and intersects 𝑧𝑖−1.
Rule 3: Derive 𝑦𝑖 from 𝑥𝑖 and 𝑧𝑖.
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Rules For Assigning Frames
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Rules For Assigning Frames
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𝒚𝟎Link 0 (fixed)Base frame
𝒛𝟎
𝒙𝟎
Rule 1: 𝒛𝒊−𝟏 is axis of actuation of joint 𝒊
Base frame
𝒛𝟎 is axis of actuation of joint 𝟏.
𝒙𝟎 and 𝒚𝟎 are set according tothe right hand rule.
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Rules For Assigning Frames
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Rule 1: 𝒛𝒊−𝟏 is axis of actuation of joint 𝒊
Tool frame
𝑧𝑛 is the approach direction of the tool.𝑦𝑛 is the slide direction of the gripper.𝑥𝑛 is the normal direction to other axes.
𝒚𝒏
𝒙𝒏
𝒛𝒏
J.Nassour 81
![Page 82: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/82.jpg)
Rules For Assigning Frames
14.11.2017
Rule 2: Axis 𝒙𝒊 is set so it is perpendicular to and intersects 𝒛𝒊−𝟏
Case 1: 𝒛𝒊−𝟏 and 𝒛𝒊 are not coplanar.
• There is only one line possible for 𝑥𝑖 , which is the shortest line from 𝑧𝑖−1 to 𝑧𝑖 . • 𝑜𝑖 is at intersection of 𝑥𝑖 and 𝑧𝑖 .
𝒚𝒊−𝟏
𝒛𝒊−𝟏
𝒙𝒊−𝟏
𝒛𝒊
𝒙𝒊
J.Nassour 82
![Page 83: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/83.jpg)
Rules For Assigning Frames
14.11.2017
Rule 2: Axis 𝒙𝒊 is set so it is perpendicular to and intersects 𝒛𝒊−𝟏
Case 2: 𝒛𝒊−𝟏 and 𝒛𝒊 are parallel.
• There are an infinite number of possibilities for 𝑥𝑖 from 𝑧𝑖−1 to 𝑧𝑖 . • Usually easiest to choose an 𝑥𝑖 that passes through 𝑜𝑖−1(so that 𝑑𝑖 = 0). • 𝑜𝑖 is at intersection of 𝑥𝑖 and 𝑧𝑖 . • 𝛼𝑖 = 0 always for this case.
𝒚𝒊−𝟏
𝒛𝒊−𝟏
𝒙𝒊−𝟏
𝒛𝒊
𝒙𝒊
J.Nassour 83
![Page 84: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/84.jpg)
Rules For Assigning Frames
14.11.2017
Rule 2: Axis 𝒙𝒊 is set so it is perpendicular to and intersects 𝒛𝒊−𝟏
Case 3: 𝒛𝒊−𝟏 intersects 𝒛𝒊.
• 𝑥𝑖 is normal to the plane of 𝑧𝑖−1 and 𝑧𝑖 . • Positive direction of 𝑥𝑖 is arbitrary. • 𝑜𝑖 naturally sits at intersection of 𝑧𝑖−1 and 𝑧𝑖 but can be anywhere on 𝑧𝑖 . • 𝑎𝑖 = 0 always for this case.
𝒚𝒊−𝟏
𝒛𝒊−𝟏
𝒙𝒊−𝟏
𝒛𝒊
𝒙𝒊
J.Nassour 84
![Page 85: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/85.jpg)
Rules For Assigning Frames
14.11.2017
Rule 2: Axis 𝒙𝒊 is set so it is perpendicular to and intersects 𝒛𝒊−𝟏
Case 3: 𝒛𝒊−𝟏 intersects 𝒛𝒊.
• 𝑥𝑖 is normal to the plane of 𝑧𝑖−1 and 𝑧𝑖 . • Positive direction of 𝑥𝑖 is arbitrary. • 𝑜𝑖 naturally sits at intersection of 𝑧𝑖−1 and 𝑧𝑖 but can be anywhere on 𝑧𝑖 . • 𝑎𝑖 = 0 always for this case.
𝒚𝒊−𝟏
𝒛𝒊−𝟏
𝒙𝒊−𝟏
𝒛𝒊
𝒙𝒊
J.Nassour 85
![Page 86: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/86.jpg)
Rules For Assigning Frames
14.11.2017
Rule 2: Axis 𝒙𝒊 is set so it is perpendicular to and intersects 𝒛𝒊−𝟏
Case 3: 𝒛𝒊−𝟏 intersects 𝒛𝒊.
• 𝑥𝑖 is normal to the plane of 𝑧𝑖−1 and 𝑧𝑖 . • Positive direction of 𝑥𝑖 is arbitrary. • 𝑜𝑖 naturally sits at intersection of 𝑧𝑖−1 and 𝑧𝑖 but can be anywhere on 𝑧𝑖 . • 𝑎𝑖 = 0 always for this case.
𝒚𝒊−𝟏
𝒛𝒊−𝟏
𝒙𝒊−𝟏
𝒛𝒊
𝒙𝒊
J.Nassour 86
![Page 87: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/87.jpg)
D-H Parameters
14.11.2017
𝒂𝒊 is distance from 𝒛𝒊−𝟏 to 𝒛𝒊 measured along 𝒙𝒊.𝜶𝒊 is angle from 𝒛𝒊−𝟏 to 𝒛𝒊 measured about 𝒙𝒊. 𝒅𝒊 is distance from 𝒙𝒊−𝟏 to 𝒙𝒊 measured along 𝒛𝒊−𝟏.
𝜽𝒊 is angle from 𝒙𝒊−𝟏 to 𝒙𝒊 measured about 𝒛𝒊−𝟏.
𝒛𝒊−𝟏
𝒙𝒊−𝟏
𝒛𝒊
𝒙𝒊
𝒛𝒊−𝟏
𝒂𝒊
𝒅𝒊
𝜽𝒊
𝜶𝒊
J.Nassour 87
![Page 88: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/88.jpg)
D-H Parameters
14.11.2017
𝒂𝒊 is distance from 𝒛𝒊−𝟏 to 𝒛𝒊 measured along 𝒙𝒊.𝜶𝒊 is angle from 𝒛𝒊−𝟏 to 𝒛𝒊 measured about 𝒙𝒊. 𝒅𝒊 is distance from 𝒙𝒊−𝟏 to 𝒙𝒊 measured along 𝒛𝒊−𝟏.
𝜽𝒊 is angle from 𝒙𝒊−𝟏 to 𝒙𝒊 measured about 𝒛𝒊−𝟏.
𝒛𝒊−𝟏
𝒙𝒊−𝟏
𝒛𝒊
𝒙𝒊
𝒛𝒊−𝟏
𝒂𝒊
𝒅𝒊
𝜽𝒊
𝜶𝒊
J.Nassour 88
![Page 89: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/89.jpg)
D-H Parameters
14.11.2017
𝒂𝒊 is distance from 𝒛𝒊−𝟏 to 𝒛𝒊 measured along 𝒙𝒊.𝜶𝒊 is angle from 𝒛𝒊−𝟏 to 𝒛𝒊 measured about 𝒙𝒊. 𝒅𝒊 is distance from 𝒙𝒊−𝟏 to 𝒙𝒊 measured along 𝒛𝒊−𝟏.
𝜽𝒊 is angle from 𝒙𝒊−𝟏 to 𝒙𝒊 measured about 𝒛𝒊−𝟏.
𝒛𝒊−𝟏
𝒙𝒊−𝟏
𝒛𝒊
𝒙𝒊
𝒛𝒊−𝟏
𝒂𝒊
𝒅𝒊
𝜽𝒊
𝜶𝒊
J.Nassour 89
![Page 90: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/90.jpg)
D-H Parameters
14.11.2017
𝒂𝒊 is distance from 𝒛𝒊−𝟏 to 𝒛𝒊 measured along 𝒙𝒊.𝜶𝒊 is angle from 𝒛𝒊−𝟏 to 𝒛𝒊 measured about 𝒙𝒊. 𝒅𝒊 is distance from 𝒙𝒊−𝟏 to 𝒙𝒊 measured along 𝒛𝒊−𝟏.
𝜽𝒊 is angle from 𝒙𝒊−𝟏 to 𝒙𝒊 measured about 𝒛𝒊−𝟏.
𝒛𝒊−𝟏
𝒙𝒊−𝟏
𝒛𝒊
𝒙𝒊
𝒛𝒊−𝟏
𝒂𝒊
𝒅𝒊
𝜽𝒊
𝜶𝒊
J.Nassour 90
![Page 91: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/91.jpg)
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
𝜽1
𝜽2
𝒅3
J.Nassour 91
![Page 92: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/92.jpg)
Example: RRP Robot
14.11.2017
Joint 3
Joint 1
Joint 2
Tool 𝒛𝟎
𝒙𝟎
𝒚𝟎
3𝑚
Assign coordinate frames so that wecan find DH parameters for this robot.
J.Nassour 92
![Page 93: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/93.jpg)
Example: RRP Robot
14.11.2017
Joint 3
Joint 1
Joint 2
Tool 𝒛𝟎
𝒙𝟎
𝒚𝟎
𝒛𝟏𝒙𝟏𝒚𝟏
3𝑚
Assign coordinate frames so that wecan find DH parameters for this robot.
J.Nassour 93
![Page 94: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/94.jpg)
Example: RRP Robot
14.11.2017
Joint 3
Joint 1
Joint 2
Tool 𝒛𝟎
𝒙𝟎
𝒚𝟎
𝒛𝟏𝒙𝟏𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒛𝟐
3𝑚
Assign coordinate frames so that wecan find DH parameters for this robot.
J.Nassour 94
![Page 95: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/95.jpg)
Example: RRP Robot
14.11.2017
Joint 3
Joint 1
Joint 2
Tool 𝒛𝟎
𝒙𝟎
𝒚𝟎
𝒛𝟏𝒙𝟏𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒛𝟐
𝒛𝟑
𝒚𝟑
𝒙𝟑
𝑳3
3𝑚
Assign coordinate frames so that wecan find DH parameters for this robot.
J.Nassour 95
![Page 96: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/96.jpg)
Example: RRP Robot
14.11.2017
Find DH parameters for thisrobot. Identify the jointvariables.
Joint 3
Joint 1
Joint 2
Tool 𝒛𝟎
𝒙𝟎
𝒚𝟎
𝒛𝟏𝒙𝟏𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒛𝟐
𝒛𝟑
𝒚𝟑
𝒙𝟑
𝑳3
3𝑚
J.Nassour 96
![Page 97: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/97.jpg)
Example: RRP Robot
14.11.2017
Find DH parameters for thisrobot. Identify the jointvariables.
𝒂𝒊 is distance from 𝒛𝒊−𝟏 to 𝒛𝒊 measured along 𝒙𝒊.𝜶𝒊 is angle from 𝒛𝒊−𝟏 to 𝒛𝒊 measured about 𝒙𝒊. 𝒅𝒊 is distance from 𝒙𝒊−𝟏 to 𝒙𝒊 measured along 𝒛𝒊−𝟏.
𝜽𝒊 is angle from 𝒙𝒊−𝟏 to 𝒙𝒊 measured about 𝒛𝒊−𝟏.
Joint 3
Joint 1
Joint 2
Tool 𝒛𝟎
𝒙𝟎
𝒚𝟎
𝒛𝟏𝒙𝟏𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒛𝟐
𝒛𝟑
𝒚𝟑
𝒙𝟑
𝑳3
3𝑚
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1
2
3
J.Nassour 97
![Page 98: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/98.jpg)
Example: RRP Robot
14.11.2017
Find DH parameters for thisrobot. Identify the jointvariables.
𝒂𝒊 is distance from 𝒛𝒊−𝟏 to 𝒛𝒊 measured along 𝒙𝒊.𝜶𝒊 is angle from 𝒛𝒊−𝟏 to 𝒛𝒊 measured about 𝒙𝒊. 𝒅𝒊 is distance from 𝒙𝒊−𝟏 to 𝒙𝒊 measured along 𝒛𝒊−𝟏.
𝜽𝒊 is angle from 𝒙𝒊−𝟏 to 𝒙𝒊 measured about 𝒛𝒊−𝟏.
Joint 3
Joint 1
Joint 2
Tool 𝒛𝟎
𝒙𝟎
𝒚𝟎
𝒛𝟏𝒙𝟏𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒛𝟐
𝒛𝟑
𝒚𝟑
𝒙𝟑
𝑳3
3𝑚
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 3m 𝜽𝟏 = 0 °
2
3
J.Nassour 98
![Page 99: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/99.jpg)
Example: RRP Robot
14.11.2017
Find DH parameters for thisrobot. Identify the jointvariables.
𝒂𝒊 is distance from 𝒛𝒊−𝟏 to 𝒛𝒊 measured along 𝒙𝒊.𝜶𝒊 is angle from 𝒛𝒊−𝟏 to 𝒛𝒊 measured about 𝒙𝒊. 𝒅𝒊 is distance from 𝒙𝒊−𝟏 to 𝒙𝒊 measured along 𝒛𝒊−𝟏.
𝜽𝒊 is angle from 𝒙𝒊−𝟏 to 𝒙𝒊 measured about 𝒛𝒊−𝟏.
Joint 3
Joint 1
Joint 2
Tool 𝒛𝟎
𝒙𝟎
𝒚𝟎
𝒛𝟏𝒙𝟏𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒛𝟐
𝒛𝟑
𝒚𝟑
𝒙𝟑
𝑳3
3𝑚
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 3m 𝜽𝟏 = 0 °
2 0 −90 ° 0 𝜽𝟐 = −90 °
3
J.Nassour 99
![Page 100: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/100.jpg)
Example: RRP Robot
14.11.2017
Find DH parameters for thisrobot. Identify the jointvariables.
𝒂𝒊 is distance from 𝒛𝒊−𝟏 to 𝒛𝒊 measured along 𝒙𝒊.𝜶𝒊 is angle from 𝒛𝒊−𝟏 to 𝒛𝒊 measured about 𝒙𝒊. 𝒅𝒊 is distance from 𝒙𝒊−𝟏 to 𝒙𝒊 measured along 𝒛𝒊−𝟏.
𝜽𝒊 is angle from 𝒙𝒊−𝟏 to 𝒙𝒊 measured about 𝒛𝒊−𝟏.
Joint 3
Joint 1
Joint 2
Tool 𝒛𝟎
𝒙𝟎
𝒚𝟎
𝒛𝟏𝒙𝟏𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒛𝟐
𝒛𝟑
𝒚𝟑
𝒙𝟑
𝑳3
3𝑚
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 3m 𝜽𝟏 = 0 °
2 0 −90 ° 0 𝜽𝟐 = −90 °
3 0 0 ° 𝒅𝟑 = 𝑳𝟑 0 °
J.Nassour 100
![Page 101: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/101.jpg)
Example: RRP Robot
14.11.2017
Find DH parameters for thisrobot. Identify the jointvariables.
𝒂𝒊 is distance from 𝒛𝒊−𝟏 to 𝒛𝒊 measured along 𝒙𝒊.𝜶𝒊 is angle from 𝒛𝒊−𝟏 to 𝒛𝒊 measured about 𝒙𝒊. 𝒅𝒊 is distance from 𝒙𝒊−𝟏 to 𝒙𝒊 measured along 𝒛𝒊−𝟏.
𝜽𝒊 is angle from 𝒙𝒊−𝟏 to 𝒙𝒊 measured about 𝒛𝒊−𝟏.
Joint 3
Joint 1
Joint 2
Tool 𝒛𝟎
𝒙𝟎
𝒚𝟎
𝒛𝟏𝒙𝟏𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒛𝟐
𝒛𝟑
𝒚𝟑
𝒙𝟑
𝑳3
3𝑚
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 3m 𝜽𝟏 = 0 °
2 0 −90 ° 0 𝜽𝟐 = −90 °
3 0 0 ° 𝒅𝟑 = 𝑳𝟑 0 °
Find the A matrices
J.Nassour 101
![Page 102: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/102.jpg)
Example: RRP Robot
14.11.2017
Find the A matrices Reminder: 𝑨𝒊
𝐴 1 =
𝑐1 0𝑠1 0
−𝑠1 0𝑐1 0
0 −10 0
0 30 1
𝐴 2 =
𝑐2 0𝑠2 0
−𝑠2 0𝑐2 0
0 −10 0
0 00 1
𝐴 3 =
1 00 1
0 00 0
0 00 0
1 𝑳𝟑0 1
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 3m 𝜽𝟏 = 0 °
2 0 −90 ° 0 𝜽𝟐 = −90 °
3 0 0 ° 𝒅𝟑 = 𝑳𝟑 0 °
J.Nassour 102
![Page 103: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/103.jpg)
Example: RRP Robot
14.11.2017
Find the A matrices Reminder: 𝑨𝒊
𝐴 1 =
𝑐1 0𝑠1 0
−𝑠1 0𝑐1 0
0 −10 0
0 30 1
𝐴 2 =
𝑐2 0𝑠2 0
−𝑠2 0𝑐2 0
0 −10 0
0 00 1
𝐴 3 =
1 00 1
0 00 0
0 00 0
1 𝑳𝟑0 1
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 3m 𝜽𝟏 = 0 °
2 0 −90 ° 0 𝜽𝟐 = −90 °
3 0 0 ° 𝒅𝟑 = 𝑳𝟑 0 °
J.Nassour 103
![Page 104: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/104.jpg)
Example: RRP Robot
14.11.2017
Find the A matrices Reminder: 𝑨𝒊
𝐴 1 =
𝑐1 0𝑠1 0
−𝑠1 0𝑐1 0
0 −10 0
0 30 1
𝐴 2 =
𝑐2 0𝑠2 0
−𝑠2 0𝑐2 0
0 −10 0
0 00 1
𝐴 3 =
1 00 1
0 00 0
0 00 0
1 𝑳𝟑0 1
𝑇 10= 𝐴 1 𝑇 2
0= 𝐴 1 𝐴 2 =
𝑐1𝑐2 𝑠1𝑠1𝑐2 −𝑐1
−𝑐1𝑠2 0−𝑠1𝑠2 0
−𝑠2 00 0
−𝑐2 30 1
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 3m 𝜽𝟏 = 0 °
2 0 −90 ° 0 𝜽𝟐 = −90 °
3 0 0 ° 𝒅𝟑 = 𝑳𝟑 0 °
J.Nassour 104
![Page 105: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/105.jpg)
Example: RRP Robot
14.11.2017
Find the A matrices
𝑇 10= 𝐴 1 𝑇 2
0= 𝐴 1 𝐴 2 =
𝑐1𝑐2 𝑠1𝑠1𝑐2 −𝑐1
−𝑐1𝑠2 0−𝑠1𝑠2 0
−𝑠2 00 0
−𝑐2 30 1
=
? ?? ?
? 0? 0
? 00 0
? 30 1
In the current configuration
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 3m 𝜽𝟏 = 0 °
2 0 −90 ° 0 𝜽𝟐 = −90 °
3 0 0 ° 𝒅𝟑 = 𝑳𝟑 0 °
J.Nassour 105
Joint 3
Joint 1
Joint 2
Tool 𝒛𝟎
𝒙𝟎
𝒚𝟎
𝒛𝟏𝒙𝟏𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒛𝟐
𝒛𝟑
𝒚𝟑
𝒙𝟑
𝑳3
3𝑚
![Page 106: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/106.jpg)
Example: RRP Robot
14.11.2017
Find the A matrices
𝑇 10= 𝐴 1
In the current configuration
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 3m 𝜽𝟏 = 0 °
2 0 −90 ° 0 𝜽𝟐 = −90 °
3 0 0 ° 𝒅𝟑 = 𝑳𝟑 0 °
J.Nassour 106
Joint 3
Joint 1
Joint 2
Tool 𝒛𝟎
𝒙𝟎
𝒚𝟎
𝒛𝟏𝒙𝟏𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒛𝟐
𝒛𝟑
𝒚𝟑
𝒙𝟑
𝑳3
3𝑚
𝑇 20= 𝐴 1 𝐴 2 =
𝑐1𝑐2 𝑠1𝑠1𝑐2 −𝑐1
−𝑐1𝑠2 0−𝑠1𝑠2 0
−𝑠2 00 0
−𝑐2 30 1
=
0 00 −1
1 00 0
1 00 0
0 30 1
![Page 107: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/107.jpg)
Example: RRP Robot
14.11.2017
Find the A matrices Reminder: 𝑨𝒊
𝐴 1 =
𝑐1 0𝑠1 0
−𝑠1 0𝑐1 0
0 −10 0
0 30 1
𝐴 2 =
𝑐2 0𝑠2 0
−𝑠2 0𝑐2 0
0 −10 0
0 00 1
𝐴 3 =
1 00 1
0 00 0
0 00 0
1 𝑳𝟑0 1
𝑇 30= 𝑇2
0 𝐴 3 =
𝑐1𝑐2 𝑠1𝑠1𝑐2 −𝑐1
−𝑐1𝑠2 0−𝑠1𝑠2 0
−𝑠2 00 0
−𝑐2 30 1
1 00 1
0 00 0
0 00 0
1 𝐿30 1
=
𝑐1𝑐2 𝑠1𝑠1𝑐2 −𝑐1
−𝑐1𝑠2 −𝐿𝟑𝑐1𝑠2−𝑠1𝑠2 −𝐿𝟑𝑠1𝑠2
−𝑠2 00 0
−𝑐2 3 −𝐿𝟑𝑐20 1
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 3m 𝜽𝟏 = 0 °
2 0 −90 ° 0 𝜽𝟐 = −90 °
3 0 0 ° 𝒅𝟑 = 𝑳𝟑 0 °
J.Nassour 107
![Page 108: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/108.jpg)
Example: RRP Robot
14.11.2017
Find the A matrices
𝑇30 =
𝑐1𝑐2 𝑠1𝑠1𝑐2 −𝑐1
−𝑐1𝑠2 −𝐿𝟑𝑐1𝑠2−𝑠1𝑠2 −𝐿𝟑𝑠1𝑠2
−𝑠2 00 0
−𝑐2 3 −𝐿𝟑𝑐20 1
=
? ?? ?
? ?? ?
? 00 0
? ?0 1
In the current configuration
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 3m 𝜽𝟏 = 0 °
2 0 −90 ° 0 𝜽𝟐 = −90 °
3 0 0 ° 𝒅𝟑 = 𝑳𝟑 0 °
Joint 3
Joint 1
Joint 2
Tool 𝒛𝟎
𝒙𝟎
𝒚𝟎
𝒛𝟏𝒙𝟏𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒛𝟐
𝒛𝟑
𝒚𝟑
𝒙𝟑
𝑳3
3𝑚
J.Nassour 108
![Page 109: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/109.jpg)
Example: RRP Robot
14.11.2017
Find the A matrices
𝑇30 =
𝑐1𝑐2 𝑠1𝑠1𝑐2 −𝑐1
−𝑐1𝑠2 −𝐿𝟑𝑐1𝑠2−𝑠1𝑠2 −𝐿𝟑𝑠1𝑠2
−𝑠2 00 0
−𝑐2 3 −𝐿𝟑𝑐20 1
=
0 00 −1
1 𝐿30 0
1 00 0
0 30 1
In the current configuration
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 3m 𝜽𝟏 = 0 °
2 0 −90 ° 0 𝜽𝟐 = −90 °
3 0 0 ° 𝒅𝟑 = 𝑳𝟑 0 °
Joint 3
Joint 1
Joint 2
Tool 𝒛𝟎
𝒙𝟎
𝒚𝟎
𝒛𝟏𝒙𝟏𝒚𝟏
𝒙𝟐
𝒚𝟐
𝒛𝟐
𝒛𝟑
𝒚𝟑
𝒙𝟑
𝑳3
3𝑚
J.Nassour 109
![Page 110: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/110.jpg)
Example: Two-Link Planar Robot
14.11.2017
Assign coordinate frames so that wecan find DH parameters for this robot.
J.Nassour 110
![Page 111: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/111.jpg)
Example: Two-Link Planar Robot
14.11.2017
Find DH parameters for this robot. Identify the joint variables.
𝒂𝒊 is distance from 𝒛𝒊−𝟏 to 𝒛𝒊 measured along 𝒙𝒊.𝜶𝒊 is angle from 𝒛𝒊−𝟏 to 𝒛𝒊 measured about 𝒙𝒊. 𝒅𝒊 is distance from 𝒙𝒊−𝟏 to 𝒙𝒊 measured along 𝒛𝒊−𝟏.
𝜽𝒊 is angle from 𝒙𝒊−𝟏 to 𝒙𝒊 measured about 𝒛𝒊−𝟏.
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 𝑎1 0 ° 0 𝜽𝟏
2 𝑎2 0 ° 0 𝜽𝟐
J.Nassour 111
![Page 112: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/112.jpg)
Example: Two-Link Planar Robot
14.11.2017
Find DH parameters for this robot. Identify the joint variables.
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 𝑎1 0 ° 0 𝜽𝟏
2 𝑎2 0 ° 0 𝜽𝟐
J.Nassour 112
![Page 113: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/113.jpg)
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.
• Assign coordinate frames so that we can find DH parameters for this robot.• Find DH parameters for this robot. Identify the joint variables.
J.Nassour 113
![Page 114: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/114.jpg)
Stanford Arm
14.11.2017 J.Nassour 114
𝜽𝟏
![Page 115: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/115.jpg)
Stanford Arm
14.11.2017 J.Nassour 115
𝜽𝟏
𝜽𝟐
![Page 116: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/116.jpg)
Stanford Arm
14.11.2017 J.Nassour 116
𝒅𝟑
𝜽𝟏
𝜽𝟐
![Page 117: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/117.jpg)
Stanford Arm
14.11.2017 J.Nassour 117
𝒅𝟑
𝜽𝟏
𝜽𝟐
𝜽𝟒
![Page 118: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/118.jpg)
Stanford Arm
14.11.2017 J.Nassour 118
𝒅𝟑
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
![Page 119: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/119.jpg)
Stanford Arm
14.11.2017 J.Nassour 119
𝒅𝟑
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
![Page 120: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/120.jpg)
Stanford Arm
14.11.2017 J.Nassour 120
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝑑2
![Page 121: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/121.jpg)
Stanford Arm
14.11.2017 J.Nassour 121
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝑑2
![Page 122: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/122.jpg)
Stanford Arm
14.11.2017 J.Nassour 122
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝑑2
![Page 123: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/123.jpg)
Stanford Arm
14.11.2017 J.Nassour 123
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝑑2
𝒙𝟏
![Page 124: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/124.jpg)
Stanford Arm
14.11.2017 J.Nassour 124
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝑑2
𝒙𝟏
𝒛𝟐
![Page 125: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/125.jpg)
Stanford Arm
14.11.2017 J.Nassour 125
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝑑2
𝒙𝟏
𝒛𝟐
𝒙𝟐
![Page 126: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/126.jpg)
Stanford Arm
14.11.2017 J.Nassour 126
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝑑2
𝒙𝟏
𝒛𝟐
𝒙𝟐
𝒛𝟑
![Page 127: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/127.jpg)
Stanford Arm
14.11.2017 J.Nassour 127
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝑑2
𝒙𝟏
𝒛𝟐
𝒙𝟐
𝒛𝟑
𝒙
![Page 128: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/128.jpg)
Stanford Arm
14.11.2017 J.Nassour 128
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝑑2
𝒙𝟏
𝒛𝟐
𝒙𝟐
𝒛𝟑
𝒛𝟒𝒙
![Page 129: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/129.jpg)
Stanford Arm
14.11.2017 J.Nassour 129
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝑑2
𝒙𝟏
𝒛𝟐
𝒙𝟐
𝒛𝟒𝒙
𝒛𝟑 𝒛𝟓
![Page 130: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/130.jpg)
Stanford Arm
14.11.2017 J.Nassour 130
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝑑2
𝒙𝟏
𝒛𝟐
𝒙𝟐
𝒛𝟒𝒙
𝒛𝟑 𝒛𝟓
𝒛𝟔
![Page 131: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/131.jpg)
Stanford Arm
14.11.2017 J.Nassour 131
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝒛𝟐
𝒛𝟒
𝒛𝟑 𝒛𝟓
𝑑2
𝒛𝟔
𝒙𝟏 𝒙𝟐
𝒙
𝒚𝟔
![Page 132: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/132.jpg)
Stanford Arm
14.11.2017 J.Nassour 132
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝒛𝟐
𝒛𝟒
𝒛𝟑 𝒛𝟓
𝑑2
𝒛𝟔
𝒙𝟏 𝒙𝟐
𝒙
𝒚𝟔
𝒙𝟔
![Page 133: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/133.jpg)
14.11.2017 J.Nassour 133
Stanford Arm
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 0 𝜽𝟏∗
2 0 +90 ° 𝑑2 𝜽𝟐∗
3 0 0 °𝒅𝟑
∗ 0
4 0 −90 ° 0 𝜽𝟒∗
5 0 +90 ° 0 𝜽𝟓∗
6 0 0 ° 𝑑6 𝜽𝟔∗
𝒂𝒊 is distance from 𝒛𝒊−𝟏 to 𝒛𝒊 measured along 𝒙𝒊.𝜶𝒊 is angle from 𝒛𝒊−𝟏 to 𝒛𝒊 measured about 𝒙𝒊. 𝒅𝒊 is distance from 𝒙𝒊−𝟏 to 𝒙𝒊 measured along 𝒛𝒊−𝟏.
𝜽𝒊 is angle from 𝒙𝒊−𝟏 to 𝒙𝒊 measured about 𝒛𝒊−𝟏.
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝒛𝟐
𝒛𝟒
𝒛𝟑 𝒛𝟓
𝑑2
𝒛𝟔
𝒙𝟏 𝒙𝟐
𝒙
𝒚𝟔
𝒙𝟔
![Page 134: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/134.jpg)
14.11.2017 J.Nassour 134
Stanford Arm
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 0 𝜽𝟏∗
2 0 +90 ° 𝑑2 𝜽𝟐∗
3 0 0 °𝒅𝟑
∗ 0
4 0 −90 ° 0 𝜽𝟒∗
5 0 +90 ° 0 𝜽𝟓∗
6 0 0 ° 𝑑6 𝜽𝟔∗
𝑇60 = 𝐴1𝐴2𝐴3𝐴4𝐴5𝐴6 =
𝑟11 𝑟12𝑟21 𝑟22
𝑟13 𝑑𝑥𝑟23 𝑑𝑦
𝑟31 𝑟320 0
𝑟33 𝑑𝑧0 1
𝒂𝒊 is distance from 𝒛𝒊−𝟏 to 𝒛𝒊 measured along 𝒙𝒊.𝜶𝒊 is angle from 𝒛𝒊−𝟏 to 𝒛𝒊 measured about 𝒙𝒊. 𝒅𝒊 is distance from 𝒙𝒊−𝟏 to 𝒙𝒊 measured along 𝒛𝒊−𝟏.
𝜽𝒊 is angle from 𝒙𝒊−𝟏 to 𝒙𝒊 measured about 𝒛𝒊−𝟏.
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝒛𝟐
𝒛𝟒
𝒛𝟑 𝒛𝟓
𝑑2
𝒛𝟔
𝒙𝟏 𝒙𝟐
𝒙
𝒚𝟔
𝒙𝟔
![Page 135: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/135.jpg)
14.11.2017 J.Nassour 135
Stanford Arm
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 0 𝜽𝟏∗
2 0 +90 ° 𝑑2 𝜽𝟐∗
3 0 0 °𝒅𝟑
∗ 0
4 0 −90 ° 0 𝜽𝟒∗
5 0 +90 ° 0 𝜽𝟓∗
6 0 0 ° 𝑑6 𝜽𝟔∗
𝑇60 = 𝐴1𝐴2𝐴3𝐴4𝐴5𝐴6 =
𝑟11 𝑟12𝑟21 𝑟22
𝑟13 𝑑𝑥𝑟23 𝑑𝑦
𝑟31 𝑟320 0
𝑟33 𝑑𝑧0 1
𝒂𝒊 is distance from 𝒛𝒊−𝟏 to 𝒛𝒊 measured along 𝒙𝒊.𝜶𝒊 is angle from 𝒛𝒊−𝟏 to 𝒛𝒊 measured about 𝒙𝒊. 𝒅𝒊 is distance from 𝒙𝒊−𝟏 to 𝒙𝒊 measured along 𝒛𝒊−𝟏.
𝜽𝒊 is angle from 𝒙𝒊−𝟏 to 𝒙𝒊 measured about 𝒛𝒊−𝟏.
𝑷𝟎 = 𝑻𝟔𝟎 𝑷𝟔
𝒑𝟎 =?
𝒑𝟔
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝒛𝟐
𝒛𝟒
𝒛𝟑 𝒛𝟓
𝑑2
𝒛𝟔
𝒙𝟏 𝒙𝟐
𝒙
𝒚𝟔
𝒙𝟔
![Page 136: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/136.jpg)
14.11.2017 J.Nassour 136
Stanford Arm
𝐴1 =
𝑐1 0𝑠1 0
−𝑠1 0𝑐1 0
0 −10 0
0 00 1
Reminder: 𝑨𝒊
𝐴2 =
𝑐2 0𝑠2 0
𝑠2 0−𝑐2 0
0 10 0
0 𝑑20 1
𝐴3 =
1 00 1
0 00 0
0 00 0
1 𝑑30 1
𝐴4 =
𝑐4 0𝑠4 0
−𝑠4 0𝑐4 0
0 −10 0
0 00 1
𝐴5 =
𝑐5 0𝑠5 0
𝑠5 0−𝑐5 0
0 −10 0
0 00 1
𝐴6 =
𝑐6 −𝑠6𝑠6 𝑐6
0 00 0
0 00 0
1 𝑑60 1
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 0 𝜽𝟏∗
2 0 +90 ° 𝑑2 𝜽𝟐∗
3 0 0 °𝒅𝟑
∗ 0
4 0 −90 ° 0 𝜽𝟒∗
5 0 +90 ° 0 𝜽𝟓∗
6 0 0 ° 𝑑6 𝜽𝟔∗
![Page 137: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/137.jpg)
14.11.2017 J.Nassour 137
Stanford Arm
![Page 138: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/138.jpg)
14.11.2017 J.Nassour 138
Stanford Arm[ s6*(c4*s1 + c1*c2*s4) – c6*(c5*(s1*s4 – c1*c2*c4) + c1*s2*s5), s6*(c5*(s1*s4 – c1*c2*c4) + c1*s2*s5) + c5*(c4*s1 + c1*c2*s4), c1*c5*s2 – s5*(s1*s4 – c1*c2*c4), d3*c1*s2 - d6*(s5*(s1*s4 – c1*c2*c4) – c1*c5*s2) - d2*s1 ][ c6*(c5*(c1*s4 + c2*c4*s1) – s1*s2*s5) – s6*(c1*c4 – c2*s1*s4), - s6*(c5*(c1*s4 + c2*c4*s1) – s1*s2*s5) – c5*(c1*c4 – 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) – s2*s4*s6, s6*(c2*s5 + c4*c5*s2) – c5*s2*s4, c2*c5 – c4*s2*s5, d6*(c2*c5 – c4*s2*s5) + d3*c2][ 0, 0, 0, 1]
𝑟11 = 𝑠6. (𝑐4. 𝑠1 + 𝑐1. 𝑐2. 𝑠4)– 𝑐6. (𝑐5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) + 𝑐1. 𝑠2. 𝑠5)𝑟21 = 𝑐6. (𝑐5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1)– 𝑠1. 𝑠2. 𝑠5)– 𝑠6. (𝑐1. 𝑐4 – 𝑐2. 𝑠1. 𝑠4)𝑟31 = −𝑐6. (𝑐2. 𝑠5 + 𝑐4. 𝑐5. 𝑠2)– 𝑠2. 𝑠4. 𝑠6
𝑟12 = 𝑠6. (𝑐5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) + 𝑐1. 𝑠2. 𝑠5) + 𝑐5. (𝑐4. 𝑠1 + 𝑐1. 𝑐2. 𝑠4)𝑟22 = − 𝑠6. (𝑐5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) – 𝑠1. 𝑠2. 𝑠5) – 𝑐5. (𝑐1. 𝑐4 – 𝑐2. 𝑠1. 𝑠4)𝑟32 = 𝑠6. (𝑐2. 𝑠5 + 𝑐4. 𝑐5. 𝑠2) – 𝑐5. 𝑠2. 𝑠4
𝑟31 = 𝑐1. 𝑐5. 𝑠2 – 𝑠5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4)𝑟32 = 𝑠5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) + 𝑐5. 𝑠1. 𝑠2𝑟33 = 𝑐2. 𝑐5 – 𝑐4. 𝑠2. 𝑠5
𝑑𝑥 = 𝑑3. 𝑐1. 𝑠2 − 𝑑6. (𝑠5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) – 𝑐1. 𝑐5. 𝑠2) − 𝑑2. 𝑠1𝑑𝑦 = 𝑑2. 𝑐1 + 𝑑6. (𝑠5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) + 𝑐5. 𝑠1. 𝑠2) + 𝑑3. 𝑠1. 𝑠2𝑑𝑧 = 𝑑6. (𝑐2. 𝑐5 – 𝑐4. 𝑠2. 𝑠5) + 𝑑3. 𝑐2
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝒛𝟐
𝒛𝟒
𝒛𝟑 𝒛𝟓
𝑑2
𝒛𝟔
𝒙𝟏 𝒙𝟐
𝒙
𝒚𝟔
𝒙𝟔
![Page 139: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/139.jpg)
14.11.2017 J.Nassour 139
Stanford Arm
𝑟11 = 𝑠6. (𝑐4. 𝑠1 + 𝑐1. 𝑐2. 𝑠4)– 𝑐6. (𝑐5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) + 𝑐1. 𝑠2. 𝑠5)𝑟21 = 𝑐6. (𝑐5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1)– 𝑠1. 𝑠2. 𝑠5)– 𝑠6. (𝑐1. 𝑐4 – 𝑐2. 𝑠1. 𝑠4)𝑟31 = −𝑐6. (𝑐2. 𝑠5 + 𝑐4. 𝑐5. 𝑠2)– 𝑠2. 𝑠4. 𝑠6
𝑟12 = 𝑠6. (𝑐5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) + 𝑐1. 𝑠2. 𝑠5) + 𝑐5. (𝑐4. 𝑠1 + 𝑐1. 𝑐2. 𝑠4)𝑟22 = − 𝑠6. (𝑐5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) – 𝑠1. 𝑠2. 𝑠5) – 𝑐5. (𝑐1. 𝑐4 – 𝑐2. 𝑠1. 𝑠4)𝑟32 = 𝑠6. (𝑐2. 𝑠5 + 𝑐4. 𝑐5. 𝑠2) – 𝑐5. 𝑠2. 𝑠4
𝑟31 = 𝑐1. 𝑐5. 𝑠2 – 𝑠5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4)𝑟32 = 𝑠5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) + 𝑐5. 𝑠1. 𝑠2𝑟33 = 𝑐2. 𝑐5 – 𝑐4. 𝑠2. 𝑠5
𝑑𝑥 = 𝑑3. 𝑐1. 𝑠2 − 𝑑6. (𝑠5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) – 𝑐1. 𝑐5. 𝑠2) − 𝑑2. 𝑠1𝑑𝑦 = 𝑑2. 𝑐1 + 𝑑6. (𝑠5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) + 𝑐5. 𝑠1. 𝑠2) + 𝑑3. 𝑠1. 𝑠2𝑑𝑧 = 𝑑6. (𝑐2. 𝑐5 – 𝑐4. 𝑠2. 𝑠5) + 𝑑3. 𝑐2
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝒛𝟐
𝒛𝟒
𝒛𝟑 𝒛𝟓
𝑑2
𝒛𝟔
𝒙𝟏 𝒙𝟐
𝒙
𝒚𝟔
𝒙𝟔
𝑇60 = 𝐴1𝐴2𝐴3𝐴4𝐴5𝐴6 =
𝑟11 𝑟12𝑟21 𝑟22
𝑟13 𝑑𝑥𝑟23 𝑑𝑦
𝑟31 𝑟320 0
𝑟33 𝑑𝑧0 1
![Page 140: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/140.jpg)
14.11.2017 J.Nassour 140
Stanford Arm
𝑟11 = 𝑠6. (𝑐4. 𝑠1 + 𝑐1. 𝑐2. 𝑠4)– 𝑐6. (𝑐5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) + 𝑐1. 𝑠2. 𝑠5)𝑟21 = 𝑐6. (𝑐5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1)– 𝑠1. 𝑠2. 𝑠5)– 𝑠6. (𝑐1. 𝑐4 – 𝑐2. 𝑠1. 𝑠4)𝑟31 = −𝑐6. (𝑐2. 𝑠5 + 𝑐4. 𝑐5. 𝑠2)– 𝑠2. 𝑠4. 𝑠6
𝑟12 = 𝑠6. (𝑐5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) + 𝑐1. 𝑠2. 𝑠5) + 𝑐5. (𝑐4. 𝑠1 + 𝑐1. 𝑐2. 𝑠4)𝑟22 = − 𝑠6. (𝑐5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) – 𝑠1. 𝑠2. 𝑠5) – 𝑐5. (𝑐1. 𝑐4 – 𝑐2. 𝑠1. 𝑠4)𝑟32 = 𝑠6. (𝑐2. 𝑠5 + 𝑐4. 𝑐5. 𝑠2) – 𝑐5. 𝑠2. 𝑠4
𝑟31 = 𝑐1. 𝑐5. 𝑠2 – 𝑠5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4)𝑟32 = 𝑠5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) + 𝑐5. 𝑠1. 𝑠2𝑟33 = 𝑐2. 𝑐5 – 𝑐4. 𝑠2. 𝑠5
𝑑𝑥 = 𝑑3. 𝑐1. 𝑠2 − 𝑑6. (𝑠5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) – 𝑐1. 𝑐5. 𝑠2) − 𝑑2. 𝑠1𝑑𝑦 = 𝑑2. 𝑐1 + 𝑑6. (𝑠5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) + 𝑐5. 𝑠1. 𝑠2) + 𝑑3. 𝑠1. 𝑠2𝑑𝑧 = 𝑑6. (𝑐2. 𝑐5 – 𝑐4. 𝑠2. 𝑠5) + 𝑑3. 𝑐2
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝒛𝟐
𝒛𝟒
𝒛𝟑 𝒛𝟓
𝑑2
𝒛𝟔
𝒙𝟏 𝒙𝟐
𝒙
𝒚𝟔
𝒙𝟔
𝑇60 =
𝑟11 𝑟12𝑟21 𝑟22
𝑟13 𝑑𝑥𝑟23 𝑑𝑦
𝑟31 𝑟320 0
𝑟33 𝑑𝑧0 1
In the configuration shown, find:
![Page 141: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/141.jpg)
14.11.2017 J.Nassour 141
Stanford Arm
𝑟11 = 𝑠6. (𝑐4. 𝑠1 + 𝑐1. 𝑐2. 𝑠4)– 𝑐6. (𝑐5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) + 𝑐1. 𝑠2. 𝑠5)𝑟21 = 𝑐6. (𝑐5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1)– 𝑠1. 𝑠2. 𝑠5)– 𝑠6. (𝑐1. 𝑐4 – 𝑐2. 𝑠1. 𝑠4)𝑟31 = −𝑐6. (𝑐2. 𝑠5 + 𝑐4. 𝑐5. 𝑠2)– 𝑠2. 𝑠4. 𝑠6
𝑟12 = 𝑠6. (𝑐5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) + 𝑐1. 𝑠2. 𝑠5) + 𝑐5. (𝑐4. 𝑠1 + 𝑐1. 𝑐2. 𝑠4)𝑟22 = − 𝑠6. (𝑐5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) – 𝑠1. 𝑠2. 𝑠5) – 𝑐5. (𝑐1. 𝑐4 – 𝑐2. 𝑠1. 𝑠4)𝑟32 = 𝑠6. (𝑐2. 𝑠5 + 𝑐4. 𝑐5. 𝑠2) – 𝑐5. 𝑠2. 𝑠4
𝑟31 = 𝑐1. 𝑐5. 𝑠2 – 𝑠5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4)𝑟32 = 𝑠5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) + 𝑐5. 𝑠1. 𝑠2𝑟33 = 𝑐2. 𝑐5 – 𝑐4. 𝑠2. 𝑠5
𝑑𝑥 = 𝑑3. 𝑐1. 𝑠2 − 𝑑6. (𝑠5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) – 𝑐1. 𝑐5. 𝑠2) − 𝑑2. 𝑠1𝑑𝑦 = 𝑑2. 𝑐1 + 𝑑6. (𝑠5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) + 𝑐5. 𝑠1. 𝑠2) + 𝑑3. 𝑠1. 𝑠2𝑑𝑧 = 𝑑6. (𝑐2. 𝑐5 – 𝑐4. 𝑠2. 𝑠5) + 𝑑3. 𝑐2
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝒛𝟐
𝒛𝟒
𝒛𝟑 𝒛𝟓
𝑑2
𝒛𝟔
𝒙𝟏 𝒙𝟐
𝒙
𝒚𝟔
𝒙𝟔
𝑇60 =
𝑟11 𝑟12𝑟21 𝑟22
𝑟13 𝑑𝑥𝑟23 𝑑𝑦
𝑟31 𝑟320 0
𝑟33 𝑑𝑧0 1
In the configuration shown, find:
![Page 142: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/142.jpg)
14.11.2017 J.Nassour 142
Stanford Arm
𝑟11 = 𝑠6. (𝑐4. 𝑠1 + 𝑐1. 𝑐2. 𝑠4)– 𝑐6. (𝑐5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) + 𝑐1. 𝑠2. 𝑠5)𝑟21 = 𝑐6. (𝑐5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1)– 𝑠1. 𝑠2. 𝑠5)– 𝑠6. (𝑐1. 𝑐4 – 𝑐2. 𝑠1. 𝑠4)𝑟31 = −𝑐6. (𝑐2. 𝑠5 + 𝑐4. 𝑐5. 𝑠2)– 𝑠2. 𝑠4. 𝑠6
𝑟12 = 𝑠6. (𝑐5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) + 𝑐1. 𝑠2. 𝑠5) + 𝑐5. (𝑐4. 𝑠1 + 𝑐1. 𝑐2. 𝑠4)𝑟22 = − 𝑠6. (𝑐5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) – 𝑠1. 𝑠2. 𝑠5) – 𝑐5. (𝑐1. 𝑐4 – 𝑐2. 𝑠1. 𝑠4)𝑟32 = 𝑠6. (𝑐2. 𝑠5 + 𝑐4. 𝑐5. 𝑠2) – 𝑐5. 𝑠2. 𝑠4
𝑟31 = 𝑐1. 𝑐5. 𝑠2 – 𝑠5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4)𝑟32 = 𝑠5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) + 𝑐5. 𝑠1. 𝑠2𝑟33 = 𝑐2. 𝑐5 – 𝑐4. 𝑠2. 𝑠5
𝑑𝑥 = 𝑑3. 𝑐1. 𝑠2 − 𝑑6. (𝑠5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) – 𝑐1. 𝑐5. 𝑠2) − 𝑑2. 𝑠1𝑑𝑦 = 𝑑2. 𝑐1 + 𝑑6. (𝑠5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) + 𝑐5. 𝑠1. 𝑠2) + 𝑑3. 𝑠1. 𝑠2𝑑𝑧 = 𝑑6. (𝑐2. 𝑐5 – 𝑐4. 𝑠2. 𝑠5) + 𝑑3. 𝑐2
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝒛𝟐
𝒛𝟒
𝒛𝟑 𝒛𝟓
𝑑2
𝒛𝟔
𝒙𝟏 𝒙𝟐
𝒙
𝒚𝟔
𝒙𝟔
𝑇60 =
𝑟11 𝑟12𝑟21 𝑟22
𝑟13 𝑑𝑥𝑟23 𝑑𝑦
𝑟31 𝑟320 0
𝑟33 𝑑𝑧0 1
In the configuration shown, find:
![Page 143: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/143.jpg)
14.11.2017 J.Nassour 143
Stanford Arm
𝑟11 = 𝑠6. (𝑐4. 𝑠1 + 𝑐1. 𝑐2. 𝑠4)– 𝑐6. (𝑐5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) + 𝑐1. 𝑠2. 𝑠5)𝑟21 = 𝑐6. (𝑐5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1)– 𝑠1. 𝑠2. 𝑠5)– 𝑠6. (𝑐1. 𝑐4 – 𝑐2. 𝑠1. 𝑠4)𝑟31 = −𝑐6. (𝑐2. 𝑠5 + 𝑐4. 𝑐5. 𝑠2)– 𝑠2. 𝑠4. 𝑠6
𝑟12 = 𝑠6. (𝑐5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) + 𝑐1. 𝑠2. 𝑠5) + 𝑐5. (𝑐4. 𝑠1 + 𝑐1. 𝑐2. 𝑠4)𝑟22 = − 𝑠6. (𝑐5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) – 𝑠1. 𝑠2. 𝑠5) – 𝑐5. (𝑐1. 𝑐4 – 𝑐2. 𝑠1. 𝑠4)𝑟32 = 𝑠6. (𝑐2. 𝑠5 + 𝑐4. 𝑐5. 𝑠2) – 𝑐5. 𝑠2. 𝑠4
𝑟31 = 𝑐1. 𝑐5. 𝑠2 – 𝑠5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4)𝑟32 = 𝑠5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) + 𝑐5. 𝑠1. 𝑠2𝑟33 = 𝑐2. 𝑐5 – 𝑐4. 𝑠2. 𝑠5
𝑑𝑥 = 𝑑3. 𝑐1. 𝑠2 − 𝑑6. (𝑠5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) – 𝑐1. 𝑐5. 𝑠2) − 𝑑2. 𝑠1𝑑𝑦 = 𝑑2. 𝑐1 + 𝑑6. (𝑠5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) + 𝑐5. 𝑠1. 𝑠2) + 𝑑3. 𝑠1. 𝑠2𝑑𝑧 = 𝑑6. (𝑐2. 𝑐5 – 𝑐4. 𝑠2. 𝑠5) + 𝑑3. 𝑐2
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝒛𝟐
𝒛𝟒
𝒛𝟑 𝒛𝟓
𝑑2
𝒛𝟔
𝒙𝟏 𝒙𝟐
𝒙
𝒚𝟔
𝒙𝟔
𝑇60 =
𝑟11 𝑟12𝑟21 𝑟22
𝑟13 𝑑𝑥𝑟23 𝑑𝑦
𝑟31 𝑟320 0
𝑟33 𝑑𝑧0 1
In the configuration shown, find:
![Page 144: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/144.jpg)
14.11.2017 J.Nassour 144
Stanford Arm
𝑟11 = 𝑠6. (𝑐4. 𝑠1 + 𝑐1. 𝑐2. 𝑠4)– 𝑐6. (𝑐5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) + 𝑐1. 𝑠2. 𝑠5)𝑟21 = 𝑐6. (𝑐5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1)– 𝑠1. 𝑠2. 𝑠5)– 𝑠6. (𝑐1. 𝑐4 – 𝑐2. 𝑠1. 𝑠4)𝑟31 = −𝑐6. (𝑐2. 𝑠5 + 𝑐4. 𝑐5. 𝑠2)– 𝑠2. 𝑠4. 𝑠6
𝑟12 = 𝑠6. (𝑐5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) + 𝑐1. 𝑠2. 𝑠5) + 𝑐5. (𝑐4. 𝑠1 + 𝑐1. 𝑐2. 𝑠4)𝑟22 = − 𝑠6. (𝑐5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) – 𝑠1. 𝑠2. 𝑠5) – 𝑐5. (𝑐1. 𝑐4 – 𝑐2. 𝑠1. 𝑠4)𝑟32 = 𝑠6. (𝑐2. 𝑠5 + 𝑐4. 𝑐5. 𝑠2) – 𝑐5. 𝑠2. 𝑠4
𝑟31 = 𝑐1. 𝑐5. 𝑠2 – 𝑠5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4)𝑟32 = 𝑠5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) + 𝑐5. 𝑠1. 𝑠2𝑟33 = 𝑐2. 𝑐5 – 𝑐4. 𝑠2. 𝑠5
𝑑𝑥 = 𝑑3. 𝑐1. 𝑠2 − 𝑑6. (𝑠5. (𝑠1. 𝑠4 – 𝑐1. 𝑐2. 𝑐4) – 𝑐1. 𝑐5. 𝑠2) − 𝑑2. 𝑠1𝑑𝑦 = 𝑑2. 𝑐1 + 𝑑6. (𝑠5. (𝑐1. 𝑠4 + 𝑐2. 𝑐4. 𝑠1) + 𝑐5. 𝑠1. 𝑠2) + 𝑑3. 𝑠1. 𝑠2𝑑𝑧 = 𝑑6. (𝑐2. 𝑐5 – 𝑐4. 𝑠2. 𝑠5) + 𝑑3. 𝑐2
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝒛𝟐
𝒛𝟒
𝒛𝟑 𝒛𝟓
𝑑2
𝒛𝟔
𝒙𝟏 𝒙𝟐
𝒙
𝒚𝟔
𝒙𝟔
𝑇60 =
𝑟11 𝑟12𝑟21 𝑟22
𝑟13 𝑑𝑥𝑟23 𝑑𝑦
𝑟31 𝑟320 0
𝑟33 𝑑𝑧0 1
In the configuration shown, find:
![Page 145: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/145.jpg)
14.11.2017 J.Nassour 145
Stanford Arm
𝐴1 =
𝑐1 0𝑠1 0
−𝑠1 0𝑐1 0
0 −10 0
0 00 1
Reminder: 𝑨𝒊
𝐴2 =
𝑐2 0𝑠2 0
𝑠2 0−𝑐2 0
0 10 0
0 𝑑20 1
𝐴3 =
1 00 1
0 00 0
0 00 0
1 𝑑30 1
𝐴4 =
𝑐4 0𝑠4 0
−𝑠4 0𝑐4 0
0 −10 0
0 00 1
𝐴5 =
𝑐5 0𝑠5 0
𝑠5 0−𝑐5 0
0 −10 0
0 00 1
𝐴6 =
𝑐6 −𝑠6𝑠6 𝑐6
0 00 0
0 00 0
1 𝑑60 1
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 0 𝜽𝟏∗
2 0 +90 ° 𝑑2 𝜽𝟐∗
3 0 0 °𝒅𝟑
∗ 0
4 0 −90 ° 0 𝜽𝟒∗
5 0 +90 ° 0 𝜽𝟓∗
6 0 0 ° 𝑑6 𝜽𝟔∗
![Page 146: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/146.jpg)
14.11.2017 J.Nassour 146
Stanford Arm
![Page 147: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/147.jpg)
14.11.2017 J.Nassour 147
Stanford Arm
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝒛𝟐
𝒛𝟒
𝒛𝟑 𝒛𝟓
𝑑2
𝒛𝟔
𝒙𝟏 𝒙𝟐
𝒙
𝒚𝟔
𝒙𝟔
𝑟11 = − 𝑐6 𝑐5 𝑠1𝑠4 − 𝑐1𝑐2𝑐4 + 𝑐1𝑠2𝑠5 − 𝑠6 𝑐4𝑠1 + 𝑐1𝑐2𝑠4𝑟21 = 𝑐6 𝑐5 𝑐1𝑠4 + 𝑐2𝑐4𝑠1 − 𝑠1𝑠2𝑠5 + 𝑠6 𝑐1𝑐4 − 𝑐2𝑠1𝑠4𝑟31 = 𝑠2𝑠4𝑠6 − 𝑐6 𝑐2𝑠5 + 𝑐4𝑐5𝑠2
𝑟12 = 𝑠6 𝑐5 𝑠1𝑠4 − 𝑐1𝑐2𝑐4 + 𝑐1𝑠2𝑠5 − 𝑐6 𝑐4𝑠1 + 𝑐1𝑐2𝑠4𝑟22 = 𝑐6 𝑐1𝑐4 − 𝑐2𝑠1𝑠4 − 𝑠6 𝑐5 𝑐1𝑠4 + 𝑐2𝑐4𝑠1 − 𝑠1𝑠2𝑠5𝑟32 = 𝑠6 𝑐2𝑠5 + 𝑐4𝑐5𝑠2 + 𝑐6𝑠2𝑠4
𝑟13 = 𝑐1𝑐5𝑠2 − 𝑠5 𝑠1𝑠4 − 𝑐1𝑐2𝑐4𝑟23 = 𝑠5 𝑐1𝑠4 + 𝑐2𝑐4𝑠1 + 𝑐5𝑠1𝑠2𝑟33 = 𝑐2𝑐5 − 𝑐4𝑠2𝑠5
𝑑𝑥 = 𝑑3𝑐1𝑠2 − 𝑑6 𝑠5 𝑠1𝑠4 − 𝑐1𝑐2𝑐4 − 𝑐1𝑐5𝑠2 − 𝑑2𝑠1𝑑𝑦 = 𝑑2𝑐1 + 𝑑6 𝑠5 𝑐1𝑠4 + 𝑐2𝑐4𝑠1 + 𝑐5𝑠1𝑠2 + 𝑑3𝑠1𝑠2𝑑𝑧 = 𝑑6 𝑐2𝑐5 − 𝑐4𝑠2𝑠5 + 𝑑3𝑐2
![Page 148: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/148.jpg)
𝑟11 = − 𝑐6 𝑐5 𝑠1𝑠4 − 𝑐1𝑐2𝑐4 + 𝑐1𝑠2𝑠5 − 𝑠6 𝑐4𝑠1 + 𝑐1𝑐2𝑠4𝑟21 = 𝑐6 𝑐5 𝑐1𝑠4 + 𝑐2𝑐4𝑠1 − 𝑠1𝑠2𝑠5 + 𝑠6 𝑐1𝑐4 − 𝑐2𝑠1𝑠4𝑟31 = 𝑠2𝑠4𝑠6 − 𝑐6 𝑐2𝑠5 + 𝑐4𝑐5𝑠2
𝑟12 = 𝑠6 𝑐5 𝑠1𝑠4 − 𝑐1𝑐2𝑐4 + 𝑐1𝑠2𝑠5 − 𝑐6 𝑐4𝑠1 + 𝑐1𝑐2𝑠4𝑟22 = 𝑐6 𝑐1𝑐4 − 𝑐2𝑠1𝑠4 − 𝑠6 𝑐5 𝑐1𝑠4 + 𝑐2𝑐4𝑠1 − 𝑠1𝑠2𝑠5𝑟32 = 𝑠6 𝑐2𝑠5 + 𝑐4𝑐5𝑠2 + 𝑐6𝑠2𝑠4
𝑟13 = 𝑐1𝑐5𝑠2 − 𝑠5 𝑠1𝑠4 − 𝑐1𝑐2𝑐4𝑟23 = 𝑠5 𝑐1𝑠4 + 𝑐2𝑐4𝑠1 + 𝑐5𝑠1𝑠2𝑟33 = 𝑐2𝑐5 − 𝑐4𝑠2𝑠5
𝑑𝑥 = 𝑑3𝑐1𝑠2 − 𝑑6 𝑠5 𝑠1𝑠4 − 𝑐1𝑐2𝑐4 − 𝑐1𝑐5𝑠2 − 𝑑2𝑠1𝑑𝑦 = 𝑑2𝑐1 + 𝑑6 𝑠5 𝑐1𝑠4 + 𝑐2𝑐4𝑠1 + 𝑐5𝑠1𝑠2 + 𝑑3𝑠1𝑠2𝑑𝑧 = 𝑑6 𝑐2𝑐5 − 𝑐4𝑠2𝑠5 + 𝑑3𝑐2
14.11.2017 J.Nassour 148
Stanford Arm
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝒛𝟐
𝒛𝟒
𝒛𝟑 𝒛𝟓
𝑑2
𝒛𝟔
𝒙𝟏 𝒙𝟐
𝒙
𝒚𝟔
𝒙𝟔
![Page 149: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/149.jpg)
14.11.2017 J.Nassour 149
Stanford Arm
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝒛𝟐
𝒛𝟒
𝒛𝟑 𝒛𝟓
𝑑2
𝒛𝟔
𝒙𝟏 𝒙𝟐
𝒙
𝒚𝟔
𝒙𝟔
𝑟11 = − 𝑐6 𝑐5 𝑠1𝑠4 − 𝑐1𝑐2𝑐4 + 𝑐1𝑠2𝑠5 − 𝑠6 𝑐4𝑠1 + 𝑐1𝑐2𝑠4𝑟21 = 𝑐6 𝑐5 𝑐1𝑠4 + 𝑐2𝑐4𝑠1 − 𝑠1𝑠2𝑠5 + 𝑠6 𝑐1𝑐4 − 𝑐2𝑠1𝑠4𝑟31 = 𝑠2𝑠4𝑠6 − 𝑐6 𝑐2𝑠5 + 𝑐4𝑐5𝑠2
𝑟12 = 𝑠6 𝑐5 𝑠1𝑠4 − 𝑐1𝑐2𝑐4 + 𝑐1𝑠2𝑠5 − 𝑐6 𝑐4𝑠1 + 𝑐1𝑐2𝑠4𝑟22 = 𝑐6 𝑐1𝑐4 − 𝑐2𝑠1𝑠4 − 𝑠6 𝑐5 𝑐1𝑠4 + 𝑐2𝑐4𝑠1 − 𝑠1𝑠2𝑠5𝑟32 = 𝑠6 𝑐2𝑠5 + 𝑐4𝑐5𝑠2 + 𝑐6𝑠2𝑠4
𝑟13 = 𝑐1𝑐5𝑠2 − 𝑠5 𝑠1𝑠4 − 𝑐1𝑐2𝑐4𝑟23 = 𝑠5 𝑐1𝑠4 + 𝑐2𝑐4𝑠1 + 𝑐5𝑠1𝑠2𝑟33 = 𝑐2𝑐5 − 𝑐4𝑠2𝑠5
𝑑𝑥 = 𝑑3𝑐1𝑠2 − 𝑑6 𝑠5 𝑠1𝑠4 − 𝑐1𝑐2𝑐4 − 𝑐1𝑐5𝑠2 − 𝑑2𝑠1𝑑𝑦 = 𝑑2𝑐1 + 𝑑6 𝑠5 𝑐1𝑠4 + 𝑐2𝑐4𝑠1 + 𝑐5𝑠1𝑠2 + 𝑑3𝑠1𝑠2𝑑𝑧 = 𝑑6 𝑐2𝑐5 − 𝑐4𝑠2𝑠5 + 𝑑3𝑐2
![Page 150: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/150.jpg)
14.11.2017 J.Nassour 150
Stanford Arm
𝒅𝟑
𝑑6
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
𝒛𝟎
𝒛𝟏
𝒛𝟐
𝒛𝟒
𝒛𝟑 𝒛𝟓
𝑑2
𝒛𝟔
𝒙𝟏 𝒙𝟐
𝒙
𝒚𝟔
𝒙𝟔
𝑟11 = − 𝑐6 𝑐5 𝑠1𝑠4 − 𝑐1𝑐2𝑐4 + 𝑐1𝑠2𝑠5 − 𝑠6 𝑐4𝑠1 + 𝑐1𝑐2𝑠4𝑟21 = 𝑐6 𝑐5 𝑐1𝑠4 + 𝑐2𝑐4𝑠1 − 𝑠1𝑠2𝑠5 + 𝑠6 𝑐1𝑐4 − 𝑐2𝑠1𝑠4𝑟31 = 𝑠2𝑠4𝑠6 − 𝑐6 𝑐2𝑠5 + 𝑐4𝑐5𝑠2
𝑟12 = 𝑠6 𝑐5 𝑠1𝑠4 − 𝑐1𝑐2𝑐4 + 𝑐1𝑠2𝑠5 − 𝑐6 𝑐4𝑠1 + 𝑐1𝑐2𝑠4𝑟22 = 𝑐6 𝑐1𝑐4 − 𝑐2𝑠1𝑠4 − 𝑠6 𝑐5 𝑐1𝑠4 + 𝑐2𝑐4𝑠1 − 𝑠1𝑠2𝑠5𝑟32 = 𝑠6 𝑐2𝑠5 + 𝑐4𝑐5𝑠2 + 𝑐6𝑠2𝑠4
𝑟13 = 𝑐1𝑐5𝑠2 − 𝑠5 𝑠1𝑠4 − 𝑐1𝑐2𝑐4𝑟23 = 𝑠5 𝑐1𝑠4 + 𝑐2𝑐4𝑠1 + 𝑐5𝑠1𝑠2𝑟33 = 𝑐2𝑐5 − 𝑐4𝑠2𝑠5
𝑑𝑥 = 𝑑3𝑐1𝑠2 − 𝑑6 𝑠5 𝑠1𝑠4 − 𝑐1𝑐2𝑐4 − 𝑐1𝑐5𝑠2 − 𝑑2𝑠1𝑑𝑦 = 𝑑2𝑐1 + 𝑑6 𝑠5 𝑐1𝑠4 + 𝑐2𝑐4𝑠1 + 𝑐5𝑠1𝑠2 + 𝑑3𝑠1𝑠2𝑑𝑧 = 𝑑6 𝑐2𝑐5 − 𝑐4𝑠2𝑠5 + 𝑑3𝑐2
![Page 151: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/151.jpg)
PUMA 260
14.11.2017 J.Nassour 151
𝜽𝟏
𝜽𝟐
𝜽𝟒
𝜽𝟓
𝜽𝟔
![Page 152: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/152.jpg)
14.11.2017 J.Nassour 152
PUMA 260
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓𝜽𝟔
![Page 153: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/153.jpg)
14.11.2017 J.Nassour 153
PUMA 260
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓𝜽𝟔
𝒛𝟎
𝒙𝟎𝒚𝟎
![Page 154: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/154.jpg)
14.11.2017 J.Nassour 154
PUMA 260
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓𝜽𝟔
𝒛𝟎
𝒙𝟎𝒚𝟎
𝒙𝟏
𝒛𝟏
𝒚𝟏
![Page 155: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/155.jpg)
14.11.2017 J.Nassour 155
PUMA 260
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓𝜽𝟔
𝒛𝟎
𝒙𝟎𝒚𝟎
𝒙𝟏
𝒛𝟏
𝒚𝟏
𝒙𝟐
𝒛𝟐
𝒚𝟐
𝒅𝟐
𝒂𝟐
![Page 156: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/156.jpg)
14.11.2017 J.Nassour 156
PUMA 260
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓𝜽𝟔
𝒛𝟎
𝒙𝟎𝒚𝟎
𝒙𝟏
𝒛𝟏
𝒚𝟏
𝒙𝟐
𝒛𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
𝒛𝟑
𝒅𝟐
𝒂𝟐
𝒂𝟑
![Page 157: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/157.jpg)
14.11.2017 J.Nassour 157
PUMA 260
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓𝜽𝟔
𝒛𝟎
𝒙𝟎𝒚𝟎
𝒙𝟏
𝒛𝟏
𝒚𝟏
𝒙𝟐
𝒛𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
𝒛𝟑
𝒛𝟒𝒚𝟒
𝒙𝟒
𝒅𝟐
𝒂𝟐
𝒅𝟒
𝒂𝟑
![Page 158: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/158.jpg)
14.11.2017 J.Nassour 158
PUMA 260
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓𝜽𝟔
𝒛𝟎
𝒙𝟎𝒚𝟎
𝒙𝟏
𝒛𝟏
𝒚𝟏
𝒙𝟐
𝒛𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
𝒛𝟑
𝒛𝟒𝒚𝟒
𝒙𝟒
𝒙𝟓
𝒚𝟓
𝒛𝟓
𝒅𝟐
𝒂𝟐
𝒅𝟒
𝒂𝟑
![Page 159: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/159.jpg)
14.11.2017 J.Nassour 159
PUMA 260
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓𝜽𝟔
𝒛𝟎
𝒙𝟎𝒚𝟎
𝒙𝟏
𝒛𝟏
𝒚𝟏
𝒙𝟐
𝒛𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
𝒛𝟑
𝒛𝟒𝒚𝟒
𝒙𝟒
𝒙𝟓
𝒚𝟓
𝒛𝟓
𝒙𝟔
𝒚𝟔
𝒛𝟔
𝒅𝟐
𝒂𝟐
𝒅𝟒
𝒂𝟑
𝒅𝟔
![Page 160: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/160.jpg)
14.11.2017 J.Nassour 160
PUMA 260
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓𝜽𝟔
𝒛𝟎
𝒙𝟎𝒚𝟎
𝒙𝟏
𝒛𝟏
𝒚𝟏
𝒙𝟐
𝒛𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
𝒛𝟑
𝒛𝟒𝒚𝟒
𝒙𝟒
𝒙𝟓
𝒚𝟓
𝒛𝟓
𝒙𝟔
𝒚𝟔
𝒛𝟔
𝒅𝟐
𝒂𝟐
𝒅𝟒
𝒂𝟑
𝒅𝟔
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 0 𝜽𝟏∗
2 𝑎2 0 ° 𝑑2 𝜽𝟐∗
3 𝑎3 90 ° 0 𝜽𝟑∗
4 0 −90 ° 𝑑4 𝜽𝟒∗
5 0 +90 ° 0 𝜽𝟓∗
6 0 0 ° 𝑑6 𝜽𝟔∗
𝒂𝒊 is distance from 𝒛𝒊−𝟏 to 𝒛𝒊 measured along 𝒙𝒊.𝜶𝒊 is angle from 𝒛𝒊−𝟏 to 𝒛𝒊 measured about 𝒙𝒊. 𝒅𝒊 is distance from 𝒙𝒊−𝟏 to 𝒙𝒊 measured along 𝒛𝒊−𝟏.
𝜽𝒊 is angle from 𝒙𝒊−𝟏 to 𝒙𝒊 measured about 𝒛𝒊−𝟏.
![Page 161: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/161.jpg)
14.11.2017 J.Nassour 161
PUMA 260
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓𝜽𝟔
𝒛𝟎
𝒙𝟎𝒚𝟎
𝒙𝟏
𝒛𝟏
𝒚𝟏
𝒙𝟐
𝒛𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
𝒛𝟑
𝒛𝟒𝒚𝟒
𝒙𝟒
𝒙𝟓
𝒚𝟓
𝒛𝟓
𝒙𝟔
𝒚𝟔
𝒛𝟔
𝒅𝟐
𝒂𝟐
𝒅𝟒
𝒂𝟑
𝒅𝟔
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 0 𝟗𝟎
2 𝑎2 0 ° 𝑑2 𝟎
3 𝑎3 90 ° 0 𝟗𝟎
4 0 −90 ° 𝑑4 𝟎
5 0 +90 ° 0 𝟎
6 0 0 ° 𝑑6 𝟎
𝒂𝒊 is distance from 𝒛𝒊−𝟏 to 𝒛𝒊 measured along 𝒙𝒊.𝜶𝒊 is angle from 𝒛𝒊−𝟏 to 𝒛𝒊 measured about 𝒙𝒊. 𝒅𝒊 is distance from 𝒙𝒊−𝟏 to 𝒙𝒊 measured along 𝒛𝒊−𝟏.
𝜽𝒊 is angle from 𝒙𝒊−𝟏 to 𝒙𝒊 measured about 𝒛𝒊−𝟏.
In the configuration shown, find 𝜽𝒊?
![Page 162: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/162.jpg)
14.11.2017 J.Nassour 162
PUMA 260𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 0 𝜽𝟏∗
2 𝑎2 0 ° 𝑑2 𝜽𝟐∗
3 𝑎3 90 ° 0 𝜽𝟑∗
4 0 −90 ° 𝑑4 𝜽𝟒∗
5 0 +90 ° 0 𝜽𝟓∗
6 0 0 ° 𝑑6 𝜽𝟔∗
Reminder: 𝑨𝒊
![Page 163: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/163.jpg)
14.11.2017 J.Nassour 163
r11 = − s6 c4s1 − s4 c1s2s3 − c1c2c3 − c6 c5 s1s4 + c4 c1s2s3 − c1c2c3 + s5 c1c2s3 + c1c3s2
𝑟12 = 𝑠6 𝑐5 𝑠1𝑠4 + 𝑐4 𝑐1𝑠2𝑠3 − 𝑐1𝑐2𝑐3 + 𝑠5 𝑐1𝑐2𝑠3 + 𝑐1𝑐3𝑠2 − 𝑐6 𝑐4𝑠1 − 𝑠4 𝑐1𝑠2𝑠3 − 𝑐1𝑐2𝑐3
r13 = c5 c1c2s3 + c1c3s2 − s5 s1s4 + c4 c1s2s3 − c1c2c3
𝑑𝑥 = 𝑑4 𝑐1𝑐2𝑠3 + 𝑐1𝑐3𝑠2 − 𝑑2𝑠1 − 𝑑6 𝑠5 𝑠1𝑠4 + 𝑐4 𝑐1𝑠2𝑠3 − 𝑐1𝑐2𝑐3 − 𝑐5 𝑐1𝑐2𝑠3 + 𝑐1𝑐3𝑠2 + 𝑎2𝑐1𝑐2 + 𝑎3𝑐1𝑐2𝑐3 − 𝑎3𝑐1𝑠2𝑠3
𝑟21 = 𝑠6 𝑐1𝑐4 + 𝑠4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 + 𝑐6 𝑐5 𝑐1𝑠4 − 𝑐4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 − 𝑠5 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2
𝑟22 = 𝑐6 𝑐1𝑐4 + 𝑠4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 − 𝑠6 𝑐5 𝑐1𝑠4 − 𝑐4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 − 𝑠5 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2
𝑟23 = 𝑠5 𝑐1𝑠4 − 𝑐4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 + 𝑐5 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2
𝑑𝑦 = 𝑑6 𝑠5 𝑐1𝑠4 − 𝑐4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 + 𝑐5 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2 + 𝑑4 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2 + 𝑑2𝑐1 + 𝑎2𝑐2𝑠1 + 𝑎3𝑐2𝑐3𝑠1 − 𝑎3𝑠1𝑠2𝑠3
𝑟31 = 𝑠4𝑠6 𝑐2𝑠3 + 𝑐3𝑠2 − 𝑐6 𝑠5 𝑐2𝑐3 − 𝑠2𝑠3 + 𝑐4𝑐5 𝑐2𝑠3 + 𝑐3𝑠2𝑟32 = 𝑠6 𝑠5 𝑐2𝑐3 − 𝑠2𝑠3 + 𝑐4𝑐5 𝑐2𝑠3 + 𝑐3𝑠2 + 𝑐6𝑠4 𝑐2𝑠3 + 𝑐3𝑠2𝑟33 = 𝑐5 𝑐2𝑐3 − 𝑠2𝑠3 − 𝑐4𝑠5 𝑐2𝑠3 + 𝑐3𝑠2𝑑𝑧 = 𝑑4 𝑐2𝑐3 − 𝑠2𝑠3 − 𝑎2𝑠2 + 𝑑6 𝑐5 𝑐2𝑐3 − 𝑠2𝑠3 − 𝑐4𝑠5 𝑐2𝑠3 + 𝑐3𝑠2 − 𝑎3𝑐2𝑠3 − 𝑎3𝑐3𝑠2
PUMA 260
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14.11.2017 J.Nassour 164
r11 = − s6 c4s1 − s4 c1s2s3 − c1c2c3 − c6 c5 s1s4 + c4 c1s2s3 − c1c2c3 + s5 c1c2s3 + c1c3s2
𝑟12 = 𝑠6 𝑐5 𝑠1𝑠4 + 𝑐4 𝑐1𝑠2𝑠3 − 𝑐1𝑐2𝑐3 + 𝑠5 𝑐1𝑐2𝑠3 + 𝑐1𝑐3𝑠2 − 𝑐6 𝑐4𝑠1 − 𝑠4 𝑐1𝑠2𝑠3 − 𝑐1𝑐2𝑐3
r13 = c5 c1c2s3 + c1c3s2 − s5 s1s4 + c4 c1s2s3 − c1c2c3
𝑑𝑥 = 𝑑4 𝑐1𝑐2𝑠3 + 𝑐1𝑐3𝑠2 − 𝑑2𝑠1 − 𝑑6 𝑠5 𝑠1𝑠4 + 𝑐4 𝑐1𝑠2𝑠3 − 𝑐1𝑐2𝑐3 − 𝑐5 𝑐1𝑐2𝑠3 + 𝑐1𝑐3𝑠2 + 𝑎2𝑐1𝑐2 + 𝑎3𝑐1𝑐2𝑐3 − 𝑎3𝑐1𝑠2𝑠3
𝑟21 = 𝑠6 𝑐1𝑐4 + 𝑠4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 + 𝑐6 𝑐5 𝑐1𝑠4 − 𝑐4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 − 𝑠5 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2
𝑟22 = 𝑐6 𝑐1𝑐4 + 𝑠4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 − 𝑠6 𝑐5 𝑐1𝑠4 − 𝑐4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 − 𝑠5 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2
𝑟23 = 𝑠5 𝑐1𝑠4 − 𝑐4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 + 𝑐5 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2
𝑑𝑦 = 𝑑6 𝑠5 𝑐1𝑠4 − 𝑐4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 + 𝑐5 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2 + 𝑑4 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2 + 𝑑2𝑐1 + 𝑎2𝑐2𝑠1 + 𝑎3𝑐2𝑐3𝑠1 − 𝑎3𝑠1𝑠2𝑠3
𝑟31 = 𝑠4𝑠6 𝑐2𝑠3 + 𝑐3𝑠2 − 𝑐6 𝑠5 𝑐2𝑐3 − 𝑠2𝑠3 + 𝑐4𝑐5 𝑐2𝑠3 + 𝑐3𝑠2𝑟32 = 𝑠6 𝑠5 𝑐2𝑐3 − 𝑠2𝑠3 + 𝑐4𝑐5 𝑐2𝑠3 + 𝑐3𝑠2 + 𝑐6𝑠4 𝑐2𝑠3 + 𝑐3𝑠2𝑟33 = 𝑐5 𝑐2𝑐3 − 𝑠2𝑠3 − 𝑐4𝑠5 𝑐2𝑠3 + 𝑐3𝑠2𝑑𝑧 = 𝑑4 𝑐2𝑐3 − 𝑠2𝑠3 − 𝑎2𝑠2 + 𝑑6 𝑐5 𝑐2𝑐3 − 𝑠2𝑠3 − 𝑐4𝑠5 𝑐2𝑠3 + 𝑐3𝑠2 − 𝑎3𝑐2𝑠3 − 𝑎3𝑐3𝑠2
PUMA 260
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓𝜽𝟔
𝒛𝟎
𝒙𝟎𝒚𝟎
𝒙𝟔
𝒚𝟔
𝒛𝟔
![Page 165: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/165.jpg)
14.11.2017 J.Nassour 165
r11 = − s6 c4s1 − s4 c1s2s3 − c1c2c3 − c6 c5 s1s4 + c4 c1s2s3 − c1c2c3 + s5 c1c2s3 + c1c3s2
𝑟12 = 𝑠6 𝑐5 𝑠1𝑠4 + 𝑐4 𝑐1𝑠2𝑠3 − 𝑐1𝑐2𝑐3 + 𝑠5 𝑐1𝑐2𝑠3 + 𝑐1𝑐3𝑠2 − 𝑐6 𝑐4𝑠1 − 𝑠4 𝑐1𝑠2𝑠3 − 𝑐1𝑐2𝑐3
r13 = c5 c1c2s3 + c1c3s2 − s5 s1s4 + c4 c1s2s3 − c1c2c3
𝑑𝑥 = 𝑑4 𝑐1𝑐2𝑠3 + 𝑐1𝑐3𝑠2 − 𝑑2𝑠1 − 𝑑6 𝑠5 𝑠1𝑠4 + 𝑐4 𝑐1𝑠2𝑠3 − 𝑐1𝑐2𝑐3 − 𝑐5 𝑐1𝑐2𝑠3 + 𝑐1𝑐3𝑠2 + 𝑎2𝑐1𝑐2 + 𝑎3𝑐1𝑐2𝑐3 − 𝑎3𝑐1𝑠2𝑠3
𝑟21 = 𝑠6 𝑐1𝑐4 + 𝑠4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 + 𝑐6 𝑐5 𝑐1𝑠4 − 𝑐4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 − 𝑠5 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2
𝑟22 = 𝑐6 𝑐1𝑐4 + 𝑠4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 − 𝑠6 𝑐5 𝑐1𝑠4 − 𝑐4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 − 𝑠5 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2
𝑟23 = 𝑠5 𝑐1𝑠4 − 𝑐4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 + 𝑐5 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2
𝑑𝑦 = 𝑑6 𝑠5 𝑐1𝑠4 − 𝑐4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 + 𝑐5 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2 + 𝑑4 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2 + 𝑑2𝑐1 + 𝑎2𝑐2𝑠1 + 𝑎3𝑐2𝑐3𝑠1 − 𝑎3𝑠1𝑠2𝑠3
𝑟31 = 𝑠4𝑠6 𝑐2𝑠3 + 𝑐3𝑠2 − 𝑐6 𝑠5 𝑐2𝑐3 − 𝑠2𝑠3 + 𝑐4𝑐5 𝑐2𝑠3 + 𝑐3𝑠2𝑟32 = 𝑠6 𝑠5 𝑐2𝑐3 − 𝑠2𝑠3 + 𝑐4𝑐5 𝑐2𝑠3 + 𝑐3𝑠2 + 𝑐6𝑠4 𝑐2𝑠3 + 𝑐3𝑠2𝑟33 = 𝑐5 𝑐2𝑐3 − 𝑠2𝑠3 − 𝑐4𝑠5 𝑐2𝑠3 + 𝑐3𝑠2𝑑𝑧 = 𝑑4 𝑐2𝑐3 − 𝑠2𝑠3 − 𝑎2𝑠2 + 𝑑6 𝑐5 𝑐2𝑐3 − 𝑠2𝑠3 − 𝑐4𝑠5 𝑐2𝑠3 + 𝑐3𝑠2 − 𝑎3𝑐2𝑠3 − 𝑎3𝑐3𝑠2
PUMA 260
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓𝜽𝟔
𝒛𝟎
𝒙𝟎𝒚𝟎
𝒙𝟔
𝒚𝟔
𝒛𝟔
![Page 166: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/166.jpg)
14.11.2017 J.Nassour 166
PUMA 260
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓𝜽𝟔
𝒛𝟎
𝒙𝟎𝒚𝟎
𝒙𝟏
𝒛𝟏
𝒚𝟏
𝒙𝟐
𝒛𝟐
𝒚𝟐
𝒙𝟑
𝒚𝟑
𝒛𝟑
𝒛𝟒𝒚𝟒
𝒙𝟒
𝒙𝟓
𝒚𝟓
𝒛𝟓
𝒙𝟔
𝒚𝟔
𝒛𝟔
𝒅𝟐
𝒂𝟐
𝒅𝟒
𝒂𝟑
𝒅𝟔
𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
1 0 −90 ° 0 𝟗𝟎
2 𝑎2 0 ° 𝑑2 𝟎
3 𝑎3 90 ° 0 𝟗𝟎
4 0 −90 ° 𝑑4 𝟎
5 0 +90 ° 0 𝟎
6 0 0 ° 𝑑6 𝟎
𝒂𝒊 is distance from 𝒛𝒊−𝟏 to 𝒛𝒊 measured along 𝒙𝒊.𝜶𝒊 is angle from 𝒛𝒊−𝟏 to 𝒛𝒊 measured about 𝒙𝒊. 𝒅𝒊 is distance from 𝒙𝒊−𝟏 to 𝒙𝒊 measured along 𝒛𝒊−𝟏.
𝜽𝒊 is angle from 𝒙𝒊−𝟏 to 𝒙𝒊 measured about 𝒛𝒊−𝟏.
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14.11.2017 J.Nassour 167
r11 = − s6 c4s1 − s4 c1s2s3 − c1c2c3 − c6 c5 s1s4 + c4 c1s2s3 − c1c2c3 + s5 c1c2s3 + c1c3s2
𝑟12 = 𝑠6 𝑐5 𝑠1𝑠4 + 𝑐4 𝑐1𝑠2𝑠3 − 𝑐1𝑐2𝑐3 + 𝑠5 𝑐1𝑐2𝑠3 + 𝑐1𝑐3𝑠2 − 𝑐6 𝑐4𝑠1 − 𝑠4 𝑐1𝑠2𝑠3 − 𝑐1𝑐2𝑐3
r13 = c5 c1c2s3 + c1c3s2 − s5 s1s4 + c4 c1s2s3 − c1c2c3
𝑑𝑥 = 𝑑4 𝑐1𝑐2𝑠3 + 𝑐1𝑐3𝑠2 − 𝑑2𝑠1 − 𝑑6 𝑠5 𝑠1𝑠4 + 𝑐4 𝑐1𝑠2𝑠3 − 𝑐1𝑐2𝑐3 − 𝑐5 𝑐1𝑐2𝑠3 + 𝑐1𝑐3𝑠2 + 𝑎2𝑐1𝑐2 + 𝑎3𝑐1𝑐2𝑐3 − 𝑎3𝑐1𝑠2𝑠3
𝑟21 = 𝑠6 𝑐1𝑐4 + 𝑠4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 + 𝑐6 𝑐5 𝑐1𝑠4 − 𝑐4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 − 𝑠5 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2
𝑟22 = 𝑐6 𝑐1𝑐4 + 𝑠4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 − 𝑠6 𝑐5 𝑐1𝑠4 − 𝑐4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 − 𝑠5 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2
𝑟23 = 𝑠5 𝑐1𝑠4 − 𝑐4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 + 𝑐5 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2
𝑑𝑦 = 𝑑6 𝑠5 𝑐1𝑠4 − 𝑐4 𝑠1𝑠2𝑠3 − 𝑐2𝑐3𝑠1 + 𝑐5 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2 + 𝑑4 𝑐2𝑠1𝑠3 + 𝑐3𝑠1𝑠2 + 𝑑2𝑐1 + 𝑎2𝑐2𝑠1 + 𝑎3𝑐2𝑐3𝑠1 − 𝑎3𝑠1𝑠2𝑠3
𝑟31 = 𝑠4𝑠6 𝑐2𝑠3 + 𝑐3𝑠2 − 𝑐6 𝑠5 𝑐2𝑐3 − 𝑠2𝑠3 + 𝑐4𝑐5 𝑐2𝑠3 + 𝑐3𝑠2𝑟32 = 𝑠6 𝑠5 𝑐2𝑐3 − 𝑠2𝑠3 + 𝑐4𝑐5 𝑐2𝑠3 + 𝑐3𝑠2 + 𝑐6𝑠4 𝑐2𝑠3 + 𝑐3𝑠2𝑟33 = 𝑐5 𝑐2𝑐3 − 𝑠2𝑠3 − 𝑐4𝑠5 𝑐2𝑠3 + 𝑐3𝑠2𝑑𝑧 = 𝑑4 𝑐2𝑐3 − 𝑠2𝑠3 − 𝑎2𝑠2 + 𝑑6 𝑐5 𝑐2𝑐3 − 𝑠2𝑠3 − 𝑐4𝑠5 𝑐2𝑠3 + 𝑐3𝑠2 − 𝑎3𝑐2𝑠3 − 𝑎3𝑐3𝑠2
PUMA 260
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓𝜽𝟔
𝒛𝟎
𝒙𝟎𝒚𝟎
𝒙𝟔
𝒚𝟔
𝒛𝟔
![Page 168: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/168.jpg)
14.11.2017 J.Nassour 168
NAO Left Arm
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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.
![Page 170: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/170.jpg)
14.11.2017 J.Nassour 170
NAO Zero Position
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14.11.2017 J.Nassour 171
NAO Left Arm
𝜽𝟏
𝜽𝟐
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14.11.2017 J.Nassour 172
NAO Left Arm
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
![Page 173: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/173.jpg)
14.11.2017 J.Nassour 173
NAO Left Arm
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓
![Page 174: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/174.jpg)
14.11.2017 J.Nassour 174
NAO Left Arm
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓
𝒛𝑻
𝒙𝑻𝒚𝑻
![Page 175: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/175.jpg)
14.11.2017 J.Nassour 175
NAO Left Arm
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓
𝒛𝑻
𝒙𝑻𝒚𝑻
𝒛𝟎𝒙𝟎
𝒚𝟎
![Page 176: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/176.jpg)
14.11.2017 J.Nassour 176
NAO Left Arm
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓
𝒛𝑻
𝒙𝑻𝒚𝑻
𝒛𝟎𝒙𝟎
𝒚𝟎
𝒛𝟏
𝒚𝟏
𝒙𝟏
![Page 177: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/177.jpg)
14.11.2017 J.Nassour 177
NAO Left Arm
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓
𝒛𝑻
𝒙𝑻𝒚𝑻
𝒛𝟎𝒙𝟎
𝒚𝟎
𝒛𝟏
𝒚𝟏
𝒙𝟏
𝒛𝟐
𝒚𝟐
𝒙𝟐
![Page 178: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/178.jpg)
14.11.2017 J.Nassour 178
NAO Left Arm
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓
𝒛𝑻
𝒙𝑻𝒚𝑻
𝒛𝟎𝒙𝟎
𝒚𝟎
𝒛𝟏
𝒚𝟏
𝒙𝟏
𝒛𝟐
𝒚𝟐
𝒙𝟐
𝒚𝟑
𝒙𝟑
𝒛𝟑
![Page 179: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/179.jpg)
14.11.2017 J.Nassour 179
NAO Left Arm
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓
𝒛𝑻
𝒙𝑻𝒚𝑻
𝒛𝟎𝒙𝟎
𝒚𝟎
𝒛𝟏
𝒚𝟏
𝒙𝟏
𝒛𝟐
𝒚𝟐
𝒙𝟐
𝒚𝟑
𝒙𝟑
𝒛𝟑
𝒛𝟒
𝒙𝟒
𝒚𝟒
![Page 180: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/180.jpg)
14.11.2017 J.Nassour 180
NAO Left Arm
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓
𝒛𝑻
𝒙𝑻𝒚𝑻
𝒛𝟎𝒙𝟎
𝒚𝟎
𝒛𝟏
𝒚𝟏
𝒙𝟏
𝒛𝟐
𝒚𝟐
𝒙𝟐
𝒚𝟑
𝒙𝟑
𝒛𝟑
𝒛𝟒
𝒙𝟒
𝒚𝟒
𝒛𝟓
𝒙𝟓𝒚𝟓
![Page 181: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/181.jpg)
14.11.2017 J.Nassour 181
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓
𝒛𝑻
𝒙𝑻𝒚𝑻
𝒛𝟎𝒙𝟎
𝒚𝟎
𝒛𝟏
𝒚𝟏
𝒙𝟏
𝒛𝟐
𝒚𝟐
𝒙𝟐
𝒚𝟑
𝒙𝟑
𝒛𝟑
𝒛𝟒
𝒙𝟒
𝒚𝟒
NAO Left Arm𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
0 𝑇 0𝐵𝐴𝑆𝐸 (0, 𝑆ℎ𝑜𝑢𝑙𝑑𝑒𝑟𝑂𝑓𝑓𝑠𝑒𝑡𝑌, 𝑆ℎ𝑜𝑢𝑙𝑑𝑒𝑟𝑂𝑓𝑓𝑠𝑒𝑡𝑍)
1 0 90 ° 0 𝜽𝟏∗
2 𝑎2 90 ° 0 𝝅
𝟐+ 𝜽𝟐
∗
3 0 −90 ° 𝑑3 𝜽𝟑∗
4 0 +90 ° 0 𝜽𝟒∗
5 𝑎5 0 ° 𝑑5𝝅
𝟐+ 𝜽𝟓
∗
𝒂𝒊 is distance from 𝒛𝒊−𝟏 to 𝒛𝒊 measured along 𝒙𝒊.𝜶𝒊 is angle from 𝒛𝒊−𝟏 to 𝒛𝒊 measured about 𝒙𝒊. 𝒅𝒊 is distance from 𝒙𝒊−𝟏 to 𝒙𝒊 measured along 𝒛𝒊−𝟏.
𝜽𝒊 is angle from 𝒙𝒊−𝟏 to 𝒙𝒊 measured about 𝒛𝒊−𝟏.
𝒛𝟓
𝒙𝟓𝒚𝟓
![Page 182: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/182.jpg)
14.11.2017 J.Nassour 182
𝜽𝟏
𝜽𝟐
𝜽𝟑
𝜽𝟒
𝜽𝟓
𝒛𝑻
𝒙𝑻𝒚𝑻
𝒛𝟎𝒙𝟎
𝒚𝟎
𝒛𝟏
𝒚𝟏
𝒙𝟏
𝒛𝟐
𝒚𝟐
𝒙𝟐
𝒚𝟑
𝒙𝟑
𝒛𝟑
𝒛𝟒
𝒙𝟒
𝒚𝟒
NAO Left Arm𝒊 𝒂𝒊 𝜶𝒊 𝒅𝒊 𝜽𝒊
0 𝑇 0𝐵𝐴𝑆𝐸 (0, 𝑆ℎ𝑜𝑢𝑙𝑑𝑒𝑟𝑂𝑓𝑓𝑠𝑒𝑡𝑌, 𝑆ℎ𝑜𝑢𝑙𝑑𝑒𝑟𝑂𝑓𝑓𝑠𝑒𝑡𝑍)
1 0 90 ° 0 𝜽𝟏∗
2 𝑎2 90 ° 0 𝝅
𝟐+ 𝜽𝟐
∗
3 0 −90 ° 𝑑3 𝜽𝟑∗
4 0 +90 ° 0 𝜽𝟒∗
5 𝑎5 0 ° 𝑑5𝝅
𝟐+ 𝜽𝟓
∗
𝒂𝒊 is distance from 𝒛𝒊−𝟏 to 𝒛𝒊 measured along 𝒙𝒊.𝜶𝒊 is angle from 𝒛𝒊−𝟏 to 𝒛𝒊 measured about 𝒙𝒊. 𝒅𝒊 is distance from 𝒙𝒊−𝟏 to 𝒙𝒊 measured along 𝒛𝒊−𝟏.
𝜽𝒊 is angle from 𝒙𝒊−𝟏 to 𝒙𝒊 measured about 𝒛𝒊−𝟏.
𝑻 𝟓𝑩𝑨𝑺𝑬 = ?
𝒛𝟓
𝒙𝟓𝒚𝟓
![Page 183: Forward Kinematics - TU Chemnitz](https://reader030.vdocuments.site/reader030/viewer/2022012717/61af4ef36212e30e733f8bd9/html5/thumbnails/183.jpg)
14.11.2017 J.Nassour 183