robotics: mechanics & control chapter 3: kinematic analysis
TRANSCRIPT
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Chapter 3: Kinematic AnalysisIn this chapter we define the forward and inverse kinematic of serial manipulators. Different geometric, algorithmic, and screw-based solution methods will be examined, and Denavit-Hartenberg, and homogeneous transformation is introduced. The loop closure method in forward and inverse problem is solved for a number of case studies.
Robotics: Mechanics & Control
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
WelcomeTo Your Prospect Skills
On Robotics :
Mechanics and Control
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
About ARAS
ARAS Research group originated in 1997 and is proud of its 22+ years of brilliant background, and its contributions to
the advancement of academic education and research in the field of Dynamical System Analysis and Control in the
robotics application.ARAS are well represented by the industrial engineers, researchers, and scientific figures graduated
from this group, and numerous industrial and R&D projects being conducted in this group. The main asset of our
research group is its human resources devoted all their time and effort to the advancement of science and technology.
One of our main objectives is to use these potentials to extend our educational and industrial collaborations at both
national and international levels. In order to accomplish that, our mission is to enhance the breadth and enrich the
quality of our education and research in a dynamic environment.
01
Get more global exposure Self confidence is the first key
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Contents
In this chapter we define the forward and inverse kinematic of serial manipulators. Different geometric, algorithmic, and screw-based solution methods will be examined, and Denavit- Hartenberg, and homogeneous transformation is introduced. The loop closure method in forward and inverse problem is solved for a number of case studies.
4
IntroductionDefinitions, kinematic loop closure, forward and inverse kinematics, joint and task space variables, 1
Forward Kinematics
Motivating example, geometric and algorithmic approach, frame assignment, DH parameters, Craigβs and Paulβs conventions, DH homogeneous transformations, case studies.Successive screw method; Screw-based transformations, Case studies, frame terminology.
2
Inverse KinematicsInverse problem, solvability, existence of solutions, reachable and dexterous workspace. Methods of solution, Algebraic, trigonometric, geometric solutions, reduction to polynomials, Pieperβs solution, method of successive screws, Case studies.
3
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Kinematic Analysis
β’ Definitions
The study of the geometry of motion in a robot, without considering the
forces and torques that cause the motion.
A serial robot consist of
A single kinematic loop
A number of links and joints
The joints might be primary (P or R) or compound (U, C, S)
Kinematic loop closure
A loop consists of the consecutive links and joint to the end-effector
Rigid links with primary joints
Compound joints are reduced to a number of primary joints
The loop is written in a vector form
The joint motion variables form the Joint Space
The end-effector final motion DoFβs form the Task Space
5
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Kinematic Analysis
Example: Elbow Manipulator
3DoF spatial manipulator (RRR)
Joint variables: π = π1 π2 π3π
Task variables: π = π₯π π¦π π§π π
The position and orientation variables of end-effector
Forward kinematics Inverse kinematics
Given π find π Given π find π
6
Joint Spaceπ
Task Spaceπ
FK
IK
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Contents
In this chapter we define the forward and inverse kinematic of serial manipulators. Different geometric, algorithmic, and screw-based solution methods will be examined, and Denavit- Hartenberg, and homogeneous transformation is introduced. The loop closure method in forward and inverse problem is solved for a number of case studies.
7
IntroductionDefinitions, kinematic loop closure, forward and inverse kinematics, joint and task space variables, 1
Forward Kinematics
Motivating example, geometric and algorithmic approach, frame assignment, DH parameters, Craigβs and Paulβs conventions, DH homogeneous transformations, case studies.Successive screw method., Screw-based transformations, Case studies, frame terminology.
2
Inverse KinematicsInverse problem, solvability, existence of solutions, reachable and dexterous workspace. Methods of solution, Algebraic, trigonometric, geometric solutions, reduction to polynomials, Pieperβs solution, method of successive screws, Case studies.
3
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Motivating Example
Kinematic loop closure
Assign base coordinate frame {0}
Denote π = π1, π2π and π = π₯π , π¦π
π
Denote the link vectors
and the end-effector vector
Write the loop closure vector equation:
π1 + π2 = π
π1 cos π1 + π2 cos(π1 + π2) = π₯ππ1 sin π1 + π2 sin(π1 + π2) = π¦π
Shorthand notation (FK)
π1π1 + π2π12 = π₯ππ1π 1 + π2π 12 = π¦π
In which π1 = cos π1 , π 1 = sin π1 , π12 = cos(π1 + π2) , π 12 = sin(π1 + π2).
Given joint variables π = π1, π2π, the task space variables π = π₯π , π¦π
π is found from FK formulation.
Inverse problem (IK) may be found by algebraic calculations.
8
π = π₯π, π¦ππ
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Algorithmic Approach
General Link Parameters
Link length ππ and link twist πΌπStart from zero frame based on the joint axes
Link length ππβ1 common normal line lengths
Link twist πΌπβ1 relative angle of two joint axes
Link offset ππ and joint angle ππNeighboring link distance ππ and angle ππFor rotary joints ππ = ππ is the joint variable
For prismatic joints ππ = ππ is the joint variable
First and Last link in the chain
Consider base of the robot the 0 link and frame
For the last frame on the end-effector coplanar to the
previous frame
9
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematicsβ’ Algorithmic Approach
Frame Assignment (Craigβs Convention)
β’ The αππ axis of rotation of {π} joint (R)
β’ OR the axis of translation of {π} joint (P)
β’ The origin of frame {π} is at the intersection of
perpendicular line to the axis π.
β’ The ππ axis points along ππ along the common
normal. In case ππ is zero ππ is normal to the
plane of αππ and αππ+1
Denavit-Hartenberg (DH) Parameters
ππ = The distance from αππ to αππ+1 measured along ππ
πΌπ = The angle from αππ to αππ+1 measured about ππ
ππ = The distance from ππβ1 to ππ measured along αππ
ππ = The angle from ππβ1 to ππ measured about αππ.
10
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematicsβ’ Algorithmic Approach
Frame Assignment (Paulβs Convention)
β’ The αππβ1 axis of rotation of {π} joint (R)
β’ OR the axis of translation of {π} joint (P)
β’ The origin of frame {π} is at the intersection of
perpendicular line to the axis π.
β’ The ππ axis points along ππ along the common
normal. In case ππ is zero ππ is normal to the
plane of αππβ1 and αππ
Denavit-Hartenberg (DH) Parameters
ππ = The distance from αππβ1 to αππ measured along ππ
πΌπ = The angle from αππβ1 to αππ measured about ππ
ππ = The distance from ππβ1 to ππ measured along αππβ1
ππ = The angle from ππβ1 to ππ measured about αππβ1.
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(π§π+1 in Craig)(π§π in Craig)
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Algorithmic Approach
Denavit-Hartenberg (DH) Parameters
12
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Algorithmic Approach
DH Homogeneous Transformations (Craigβs Convention)
Consider three intermediate Frames
π β 1 , π , π , π , {π}
The general transformation will be found by:
Considering four transformation about fixed
Frames (post-multiplication)
In which,
Or
13
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Algorithmic Approach
DH Homogeneous Transformations (Craigβs Convention)
ππβ1π =
1 00 ππΌπβ1
0 ππβ1βπ πΌπβ1 0
0 π πΌπβ10 0
ππΌπβ1 00 1
πππ βπ πππ ππ πππ
0 00 0
0 00 0
1 ππ0 1
ππβ1π =
πππ βπ πππ ππππΌπβ1 πππππΌπβ1
0 ππβ1βπ πΌπβ1 βπ πΌπβ1ππ
π πππ πΌπβ1 ππππ πΌπβ10 0
ππΌπβ1 ππΌπβ1ππ0 1
And the inverse is:
πβ1ππ =
πππ π ππππΌπβ1βπ ππ πππππΌπβ1
π πππ πΌπβ1 βππβ1πππππππ πΌπβ1 βππβ1π ππ
0 βπ πΌπβ10 0
ππΌπβ1 βππ0 1
.
14
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Algorithmic Approach
DH Homogeneous Transformations (Paulβs Convention)
Consider three intermediate Frames
π β 1 , π , π , π , {π}
The general transformation will be found by
Considering four transformation about moving
frames (pre-multiplication)
ππβ1π = π
πβ1π πππ π
ππ π
π π
In which,
ππβ1π = π·π§ ππ π π§ ππ π·π₯ ππ π π₯ πΌπ
Or
ππβ1π = Screwπ§ ππ , ππ Screwπ₯ ππ , πΌπ
15
ππ·, ππΈ
πΏπ·
πΏπΈ πΏπΉ
ππΉ ππ
ππβπ
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Algorithmic Approach
DH Homogeneous Transformations (Paulβs Convention)
ππβ1π =
πππ βπ πππ ππ πππ
0 00 0
0 00 0
1 ππ0 1
1 00 ππΌπ
0 ππβπ πΌπ 0
0 π πΌπ0 0
ππΌπ 00 1
ππβ1π =
πππ βπ ππππΌππ ππ πππππΌπ
π πππ πΌπ πππππβππππ πΌπ πππ ππ
0 π πΌπ0 0
ππΌπ ππ0 1
And the inverse is
πβ1ππ =
πππ π ππβπ ππππΌπ πππππΌπ
0 βπππ πΌπ βπππ πΌπ
π πππ πΌπ βππππ πΌπ0 0
ππΌπ βππππΌπ0 1
16
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Examples:
Example 1: Planar RRR Manipulator
Geometric Approach (3DoFs)
Denote π = π1, π2, π3π and π = π₯πΈ , π¦πΈ , ΞπΈ
π
Denote the link, and the end-effector vectors:
Write the loop closure vector equation:
π1 + π2 + π3 = π ; ΞπΈ = π1 + π2 + π3.
π1 cos π1 + π2 cos(π1 + π2) + π3 cos(π1 + π2 + π3) = π₯πΈπ1 sin π1 + π2 sin(π1 + π2) + π2 sin(π1 + π2 + π3) = π¦πΈ
ΞπΈ = π1 + π2 + π3
Shorthand notation (FK)
π1π1 + π2π12 + π3π123 = π₯πΈπ1π 1 + π2π 12 + π3π 123 = π¦πΈ
ΞπΈ = π1 + π2 + π3
In which π1 = cos π1 , π 1 = sin π1 , π12 = cos(π1 + π2) , π 12 = sin(π1 + π2) , π123= cos(π1 + π2 + π3) , π 123 = sin(π1 + π2 + π3) .
17
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Examples:
Example 1: Planar RRR Manipulator
Algorithmic Approach (Craigβs Convention)
Denote π = π1, π2, π3π and π = π₯πΈ , π¦πΈ , ΞπΈ
π
Affix the frames and find DH-parameters.
Find the homogeneous transformations:
10π =
π1 βπ 1π 1 π1
0 00 0
0 00 0
1 00 1
, 21π =
π2 βπ 2π 2 π2
0 π10 0
0 00 0
1 00 1
,
32π =
π3 βπ 3π 3 π3
0 π20 0
0 00 0
1 00 1
, 43π =
1 00 1
0 π30 0
0 00 0
1 00 1
.
Find the loop closure equation in matrix form:
πΈ0π = 4
0π = 10π 2
1π 32π 4
3π
18
ππ
π1
π3
π1
π2π2
π3
π½ππ πππβππΆπβππ
π10001
π20π102
π30π203
00π304
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Examples:
Example 1: (Cont.)
Algorithmic Approach (Craigβs Convention)
Calculate the loop closure matrix equation:
πΈ0π = 4
0π = 10π 2
1π 32π 4
3π
πΈ0π =
πΞπΈ βπ ΞπΈπ ΞπΈ πΞπΈ
0 π₯πΈ0 π¦πΈ
0 00 0
1 00 1
40π = 1
0π 21π 3
2π 43π =
π123 βπ 123π 123 π123
0 π1π1 + π2π12 + π3π1230 π1π 1 + π2π 12 + π3π 123
0 00 0
1 00 1
.
Shorthand notation (FK)
π1π1 + π2π12 + π3π123 = π₯πΈπ1π 1 + π2π 12 + π3π 123 = π¦πΈ
ΞπΈ = π1 + π2 + π3
19
ππ
π1
π3
π1
π2π2
π3
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Examples:
Example 1: Planar RRR Manipulator
Algorithmic Approach (Paulβs Convention)
Denote π = π1, π2, π3π and π = π₯πΈ , π¦πΈ , ΞπΈ
π
Affix the frames and find DH-parameters.
Find the homogeneous transformations:
10π =
π1 βπ 1π 1 π1
0 π1π10 π1π 1
0 00 0
1 00 1
, 21π =
π2 βπ 2π 2 π2
0 π2π20 π2π 2
0 00 0
1 00 1
,
32π =
π3 βπ 3π 3 π3
0 π3π30 π3π 3
0 00 0
1 00 1
.
Find the loop closure equation in matrix form:
πΈ0π = 3
0π = 10π 2
1π 32π
20
π½ππ ππππΆππ
π10π101
π20π202
π30π303
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Examples:
Example 1: (Cont.)
Algorithmic Approach (Paulβs Convention)
Calculate the loop closure matrix equation:
πΈ0π = 3
0π = 10π 2
1π 32π
πΈ0π =
πΞπΈ βπ ΞπΈπ ΞπΈ πΞπΈ
0 π₯πΈ0 π¦πΈ
0 00 0
1 00 1
30π = 1
0π 21π 3
2π =
π123 βπ 123π 123 π123
0 π1π1 + π2π12 + π3π1230 π1π 1 + π2π 12 + π3π 123
0 00 0
1 00 1
.
Shorthand notation (FK)
π1π1 + π2π12 + π3π123 = π₯πΈπ1π 1 + π2π 12 + π3π 123 = π¦πΈ
ΞπΈ = π1 + π2 + π3
21
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Examples:
Example 2: SCARA Manipulator
Algorithmic Approach (Paulβs Convention)
Denote π = π1, π2, π3, π4π and π = π₯πΈ , π¦πΈ , π§πΈ , ΞπΈ
π
Affix the frames and find DH-parameters.
Find the homogeneous transformations:
10π =
π1 βπ 1π 1 π1
0 π1π10 π1π 1
0 00 0
1 π10 1
, 21π =
π2 π 2π 2 βπ2
0 π2π20 π2π 2
0 00 0
β1 00 1
,
32π =
1 00 1
0 00 0
0 00 0
1 π30 1
, 43π =
π4 βπ 4π 4 π4
0 00 0
0 00 0
1 π40 1
.
Find the loop closure equation in matrix form:
πΈ0π = 3
0π = 10π 2
1π 32π 4
3π
22
π½ππ ππππΆππ
π1π1π101
π20π2π2
0π3003
π4π4004
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Examples:
Example 2: (Cont.)
Algorithmic Approach (Paulβs Convention)
Calculate the loop closure matrix equation:
πΈ0π = 4
0π = 10π 2
1π 32π 4
3π
πΈ0π =
πΞπΈ βπ ΞπΈπ ΞπΈ πΞπΈ
0 π₯πΈ0 π¦πΈ
0 00 0
1 π§πΈ0 1
40π = 1
0π 21π 3
2π 43π =
π12π4 + π 12π 4 βπ12π 4 + π 12π4π 12π4 β π12π 4 βπ 12π 4 β π12π4
0 π1π1 + π2π120 π1π 1 + π2π 12
0 00 0
β1 π1 β π3 β π40 1
.
Shorthand notation (FK)
π1π1 + π2π12 = π₯πΈ, π§πΈ = π1 β π3 β π4,
π1π 1 + π2π 12 = π¦πΈ, βΞπΈ = βπ1 β π2 + π4.
23
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Examples:
Example 3: Cylindrical Robot (RPP)
Algorithmic Approach (Paulβs Convention)
Denote π = π1, π2, π3π and π = π₯πΈ , π§πΈ , ΞπΈ
π
Affix the frames and find DH-parameters.
Find the homogeneous transformations:
10π =
π1 βπ 1π 1 π1
0 00 0
0 00 0
1 π10 1
, 21π =
1 00 0
0 01 0
0 β10 0
0 π20 1
,
32π =
1 00 1
0 00 0
0 00 0
1 π30 1
.
Find the loop closure equation in matrix form:
πΈ0π = 3
0π = 10π 2
1π 32π
24
π½ππ ππππΆππ
π1π1001
0π20βπ/22
0π3003
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Examples:
Example 3: (Cont.)
Algorithmic Approach (Paulβs Convention)
Calculate the loop closure matrix equation:
πΈ0π = 3
0π = 10π 2
1π 32π
πΈ0π =
πΞπΈ βπ ΞπΈπ ΞπΈ πΞπΈ
0 π₯πΈ0 π¦πΈ
0 00 0
1 π§πΈ0 1
40π = 1
0π 21π 3
2π =
π1 0π 1 0
βπ 1 βπ 1π3π1 π1π3
0 β10 0
0 π1 + π20 1
.
Shorthand notation (FK)
βπ 1π3 = π₯πΈ , π§πΈ = π1 + π2,
π1π3 = π¦πΈ , ΞπΈ= π(π1).
25
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Examples:
Example 4: Spherical Wrist (RRR)
Algorithmic Approach (Paulβs Convention)
Denote π = π4, π5, π5π and π = π, π, π π
Affix the frames and find DH-parameters.
Find the homogeneous transformations:
43π =
π4 0π 4 0
βπ 4 0π4 0
0 β10 0
0 00 1
, 54π =
π5 0π 5 0
π 5 0βπ5 0
0 10 0
0 00 1
,
65π =
π6 0π 6 0
βπ 6 0π6 0
0 00 0
1 π60 1
.
Find the loop closure equation in matrix form:
πΈ3π = 6
3π = 43π 5
4π 65π
26
π½ππ ππππΆππ
π400βπ/24
π500π/25
π6π6006
π§6
π₯6
π6
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Examples:
Example 4: (Cont.)
Algorithmic Approach (Paulβs Convention)
Calculate the loop closure matrix equation:
πΈ3π = 6
3π = 43π 5
4π 65π
πΈ3π = π (π, π, π) ππΈ
3
0 1
63π = 4
3π 54π 6
5π =
27
π§6
π₯6
π6
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Examples:
Example 5: Scorbot 5R Manipulator
Algorithmic Approach (Paulβs Convention)
Denote π = π1, π2, π3, π4, π5π and π = ππΈ , ΞπΈ
π
Affix the frames and find DH-parameters.
28
π½ππ ππππΆππ
π1π1π1βπ/21
π20π202
π30π303
π400βπ/24
π5π5005
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Examples:
Example 5: (Cont.)
Algorithmic Approach (Paulβs Convention)
Calculate the loop closure matrix equation:
πΈ0π = 5
0π = π΄1 π΄2 π΄3 π΄4 π΄5
πΈ0π = π (π―π¬) ππ¬
0
0 1
29
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics Example 5: (Cont.)
30
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Examples:
Example 6: Stanford Manipulator (2RP3R)
Algorithmic Approach (Paulβs Convention)
Denote π = π1, π2, π3, π4, π5, π6π and π = ππ¬, π―π¬
π
Affix the frames and find DH-parameters.
31
π½ππ ππππΆππ
π100βπ/21
π2π20π/22
0π3003
π400βπ/24
π500π/25
π6π6006
π2
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Examples:
Example 6: (Cont.)
Algorithmic Approach (Paulβs Convention)
Calculate the loop closure matrix equation:
πΈ0π = 6
0π = π΄1 π΄2 π΄3 π΄4 π΄5 π΄6
πΈ0π = π (π―π¬) ππ¬
0
0 1
32
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics Example 6: (Cont.)
33
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Successive Screw Method
Use Screw Displacement Representation
Screw axis ΰ·π along
the rotation axis in π joints
the translation axis in π joints
The screw displacement ππ In base frame.
Consider the screw representation $π in home configuration
Find the target screw axis by consecutive screw displacement
Find the Homogeneous transformations π΄πfrom screw
representation $πThe joint variables ππ and ππ are used in the homogeneous
transformations
Apply the loop closure equation by π΄πΈ = π΄1 π΄2β―π΄πβ1 π΄πNo Frame assignment; just the base and the end effector frames are
needed
34
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Successive Screw Method
Review Screw Displacement Formulation
General Motion =
Rotation about ΰ·π + Translation along ΰ·π
ΰ·π, π + {ππ, π}
where
Given Screw Parameters Find π΄π by:
while,
35
π΄π
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematicsβ’ Successive Screw Method
Recipe
Consider the manipulator in Reference Position
Where the joint variables are all zero ππ = ππ = 0
Determine the end effector position ππ¬π
and orientation πΉπΈ0 = ΰ·π0 ΰ·π0 ΰ·π0
And the Screw axis representations for n joints
$π = {ΰ·ππ, πππ} for π = 1,2, β¦ , π.
Consider the manipulator in target Position
Determine the end effector position ππ¬ and orientation πΉπ¬ = ΰ·π ΰ·π ΰ·π
This is found based on task space variables π.
Apply the loop closure equation
Calculate the homogeneous transformation of $πDetermine the target screw axis by consecutive screw displacement
π΄πΈ = π΄1 π΄2β―π΄πβ1 π΄π β ππ¬ = π΄1 π΄2β―π΄πβ1 π΄π ππ¬π
In forward kinematics given π the task space variable π is found.
In inverse kinematics given π the joint space variable π is found.
36
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Successive Screw Method
Example 1
Consider Planar RRR Manipulator
Reference Pose:
Consider all the joint variables ππ = 0.
Consider the base frame π₯ β π¦ β π§ and
The end effector frame π’ β π£ β π€
Derive the reference pose by:
π0 = 1, 0, 0 π , π0 = 0, 1, 0 π , π0 = 0, 0, 1 π
ππΈ0 = ππ = π1 + π2 + π3, 0, 0
π
And for the wrist point π·
ππ = π1 + π2, 0, 0π.
37
π₯
π¦
π1 π2 π3
$1 $2 $3
π·πΈ
π’
π£
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Successive Screw Method
Example 1: (Cont.)
Reference Pose:
Denote the screws $π as on the diagram
Find the screw parameters ΰ·ππ and πππ as
given in the table
Target Pose:
Let the target position of the wrist be:
And for the end effector π = ππ₯, ππ¦, ππ§π.
38
π₯
π¦
π1 π2 π3
$1 $2 $3
π·πΈ
π’
π£
πππππJoin i
(0, 0, 0)(0, 0, 1)1
(π1, 0, 0)(0, 0, 1)2
(π1 + π2, 0, 0)(0, 0, 1)3
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Successive Screw Method
Example 1: (Cont.)
Screw Transformation Matrices:
π΄1 =
ππ1π π100
βπ π1ππ100
0010
0001
; π΄2 =
ππ2π π200
βπ π2ππ200
0010
π1π£π2βπ1π π2
01
; π΄2 =
ππ3π π300
βπ π3ππ300
0010
(π1 + π2)π£π3β(π1 + π2)π π3
01
;
π΄1π΄2π΄3 =
ππ123π π12300
βπ π123ππ12300
0010
π1ππ1 + π2ππ12 β π1 + π2 ππ123π1π π1 + π2π π12 β π1 + π2 π π123
01
;
Loop closure equation: π = π΄1π΄2π΄3ππ ;
ππ₯ = π1ππ1 + π2ππ12 + π3ππ123
ππ¦ = π1π π1 + π2π π12 + π3π π123
ππ§ = 0
The results is the same as before.
39
Orientation Equivalence:
πΈ0π =
πΞπΈπ ΞπΈ0
βπ ΞπΈπΞπΈ0
001
=ππ123π π1230
βπ π123ππ1230
001
OR β ΞπΈ= π1 + π2 + π3
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Successive Screw Method
Example 2
Consider 6DoF Elbow Manipulator (6R)
Reference Pose:
Consider all the joint variables ππ = 0.
Consider the base frame π₯ β π¦ β π§ and
The end effector frame π’ β π£ β π€
Derive the reference pose by:
π0 = 0,0,1 π , π0 = 0,β1,0 π , π0 = 1,0,0 π
ππΈ0 = ππ = π2 + π3 + π4 + π6, 0,0
π
And for the wrist point π·
ππ = π2 + π3 + π4, 0,0π.
40
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Successive Screw Method
Example 2: (Cont.)
Reference Pose:
Denote the screws $π as on the diagram
Find the screw parameters ΰ·ππ and πππ as
given in the table
Target Pose:
Let the target position of the wrist be:
And for the end effector π = ππ₯, ππ¦, ππ§π.
41
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Successive Screw Method
Example 2: (Cont.)
Screw Transformation Matrices:
42
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Successive Screw Method
Example 3
Consider 6DoF Stanford Arm (2RP3R)
Reference Pose:
Consider all the joint variables ππ = ππ = 0.
Consider the base frame π₯ β π¦ β π§ and
The end effector frame π’ β π£ β π€
Derive the reference pose by:
π0 = 1, 0, 0 π , π0 = 0,0,β1 π , π0 = 0, 1, 0 π
ππΈ0 = ππ = π, β, 0 π
And for the wrist point π·
ππ = π, 0,0 π.
43
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Successive Screw Method
Example 3: (Cont.)
Reference Pose:
Denote the screws $π as on the diagram
Find the screw parameters ΰ·ππ and πππ as
given in the table
Target Pose:
Let the target position of the wrist be:
And for the end effector π = ππ₯, ππ¦, ππ§π,
Where π = π β βπ.
44
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Successive Screw Method
Example 3: (Cont.)
Screw Transformation Matrices:
Wrist Position: π = π΄1π΄2π΄3 ππ
45
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Successive Screw Method
Example 3: (Cont.)
Screw Transformation Matrices:
End Effector Orientation:
46
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Forward Kinematics
β’ Frame Terminology
The Base Frame, π΅
The Station Frame, π
The Wrist Frame, π
The Tool Frame, π
The Goal Frame, πΊ
Where is The Tool?
Where means the position and orientation
πππ indicates where the tool is with respect to
the station frame π
To reach the tool to the goal one may solve
the kinematics problem of:
πππ=πΊ
ππ.
47
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Contents
In this chapter we define the forward and inverse kinematic of serial manipulators. Different geometric, algorithmic, and screw-based solution methods will be examined, and Denavit- Hartenberg, and homogeneous transformation is introduced. The loop closure method in forward and inverse problem is solved for a number of case studies.
48
IntroductionDefinitions, kinematic loop closure, forward and inverse kinematics, joint and task space variables, 1
Forward Kinematics
Motivating example, geometric and algorithmic approach, frame assignment, DH parameters, Craigβs and Paulβs conventions, DH homogeneous transformations, case studies.Successive screw method., Screw-based transformations, Case studies, frame terminology.
2
Inverse KinematicsInverse problem, solvability, existence of solutions, reachable and dexterous workspace. Methods of solution, Algebraic, trigonometric, geometric solutions, reduction to polynomials, Pieperβs solution, method of successive screws, Case studies.
3
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Kinematic Analysis
β’ Inverse Kinematics
Solve the inverse problem
Forward kinematics Inverse kinematics
Given π find π Given π find π
Solve the loop closure equation for: Given π find π
πΈ0π = n
0π = 10π 2
1πβ― nβ1nβ2π n
nβ1π
32π β1
21π β1
10π β1
πΈ0π = 4
3πβ― nβ1nβ2π n
nβ1π
Direct inversion is not practical, and usually using the wrist frame {π€} for
separation of position and orientation equation is very effective
49
Joint Spaceπ
Task Spaceπ
FK
IK
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematicβ’ Solvability
The vector loop closure equations are nonlinear and trigonometric
For 6DoF manipulator, the number of variables are six
The number of equations are twelve (9 for πΈ0π, 3 for π₯πΈ
0)
Only three out of 9 equations are independent
Finding the solution is difficult
IK might have no solution (out of workspace) or multiple solutions
Existence of Solutions
Solution exists if the end-effector is within the reachable workspace of the robot (might have multiple solutions)
On the border of the reachable workspace the solution is unique.
Reachable Workspace (RW)
The Volume of Space (6DoF space in general) that the end-effector of the robot can reach
Dexterous Workspace (DW)
The Volume of space that the end-effector of the robot can reach with all configuration
Dextereous Workspace is a subset of Reachable Workspace
Constant Orientation Workspace (COW)
The Volume of Space (6DoF Space) that the end-effector of the robot can reach with constant configuration
This is used to better visualization (6DoF workspace)
50
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematicβ’ Workspace
Reachable & Dexterous Workspace
Consider Planar RR Manipulator
IF πΏ1 = πΏ2 (with no joint limit)
RW: A disc with radius 2πΏ1 DW: Only one point; the origin
IF πΏ1 > πΏ2 (with no joint limit)
RW: A Ring with outer radius πΏ1 + πΏ2, and the inner radius πΏ1 β πΏ2;
DW: Empty Set; β
51
On the boundaries of the workspace:One double solution for
Fully extended/folded Arm
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematic
β’ Methods of Solution
Closed-Form Solution & Numerical Solution
Restrict ourselves to closed-form solution
IK of a 6DoF manipulator with R and P joints are solvable
Algebraic (Trigonometric) Solution
Analytical Geometric Solution
Reduction to Polynomial
Pieperβs Solution (When three axes intersects)
Method of Successive Screws
52
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematicsβ’ Algebraic (Trigonometric) Solution
Kinematic loop closure
Consider Planar RRR Manipulator
Shorthand notation (FK)
π1π1 + π2π12 + π3π123 = π₯πΈπ1π 1 + π2π 12 + π3π 123 = π¦πΈπ = ΞπΈ = π1 + π2 + π3
Consider the wrist point π vs the end effector point π
ππ₯ = π₯πΈ β π3ππ; ππ¦ = π¦πΈ β π3π π
On the other side,
By this means π3 disappears. Note that the distance between π to π is independent of π1.
Hence eliminate π1 by summing the squares of above equations:
Solving forπ2:
where
53
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematic
β’ Algebraic Approach
Kinematic loop closure
Consider Planar RRR Manipulator
Inverse cosine equation yields to two real roots if π < 1.
If π2 = πβ is the solution, then π2 = βπβ is as well.
(Elbow up and Elbow down config)
One double root if π = 1 ( Border of the workspace).
No solution if π > 1 (Out of workspace).
Solve for π1 , by expanding the loop closure equations
Solve for π1
In which
We might use Atan2.
54
Two solutions for π < 1One double root for fully extended Arm
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematics
β’ Geometric Approach
Decompose the spatial geometry to a number of plane-geometry
When πΌπ = 0 or Β±π/2 it is plausible.
Consider Planar RRR Manipulator
Consider the solid triangle apply the law of cosines
In which, cos 180 β π2 = βπ2, therefore
This has a solution if the radius of the goal point P is less than π1 + π2.
To solve for π1 follow the procedure given in the previous slide.
Note that the geometric approach gives much faster solution.
55
π2β
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematic
β’ Reduction to Polynomial
Trigonometric relation to polynomial
Use half-angle tangent relation
Example: Convert the following equation to algebraic polynomials
Substitute the above relation and manipulate:
Collecting the powers:
This is solved by quadratic polynomial solution
Hence:
56
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematicβ’ Pieperβs Solution (When three axes intersects)
For 6DoF robots with three consecutive joint axis intersection
Consider the three last joint intersecting (most of commercial Robots)
Origin of frames 4 , {5}, and 6 are at the point of intersection
ππ40 = 1
0π 21π 3
2π ππ43
Use DH-convention
ππ40 = 1
0π 21π 3
2π
π3βπ4π πΌ3π4ππΌ31
Determine this point by calculation of general DH-transformation 10π 2
1π 32π
ππ40 =
π1π1 β π 1π2π 1π1 + π1π2
π31
In which while,
57
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematic
β’ Pieperβs Solution (When three axes intersects)
Find magnitude of ππ40 as π
therefore
Write this equation along with the z-component of ππ40
where
In this equation π1 is eliminated and is of simple form of π2. Now Solve for π3:
58
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematic
β’ Pieperβs Solution (When three axes intersects)Furthermore,
Having solved for π3 , one may solve for π2, and then for π1.
Complete the solution:
We need to solve for π4, π5, π6, since these axis intersect:
These joint angles determines the orientation of the end effector
This can be computed by the orientation of the goal 60π .
First determine the orientation of link frame {4} relative to the base frame when π4 = 0, denote
this with 40π Θπ4=0. This found by
This part of the orientation may be found by suitable Euler angles, usually π β π β π one.
59
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematic
β’ Examples:
See IK solution examples in the references.
Example 1: Consider the 6DoF Fanuc S-900w robot
Algorithmic Approach (Paulβs Convention)
Denote π = π1, π2, π3, π4, π5, π6π and π = ππΈ , ΞπΈ
π
Affix the frames and find DH-parameters as given in the next slide.
Find the homogeneous transformations:
60
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematic
Example 1: Fanuc S-900w robot
Algorithmic Approach (Paulβs Convention)
Approach
61
π½ππ ππππΆππ
π10π1π/21
π20π202
π30π3π/23
π4π40βπ/24
π500π/25
π6π6006
Denavit-Hartenberg parameters
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematic Example 1: Fanuc S-900w robot
Note that the last three joints intersect @ π·
While: and:
By inspection it is easy to see
Transform it to the base frame by:
Now manipulate
Use homogeneous transformations to find β¦
Where ππ₯ , ππ¦ , ππ§ is found from ππ₯ , ππ¦, ππ§.
A solution for π1is found by the last equation:
There will be two solutions for this joint angle.
62
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematic Example 1: Fanuc S-900w robot
By observation the distance between π¨ and π· is independent to π1 and
π2. Therefore these two variables can be eliminated simultaneously
Sum the squares of (2.81) β (2.83)
Where
Convert (2.85) into polynomial by half angle relations
to obtain
Hence,
This yields to two real roots (Elbow up and down solution)
63
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematic Example 1: Fanuc S-900w robot
Once π1 and π3 are known, π2 is found by back substitution
Expand (2.81) and (2.82)
Where
Solve (2.88) and (2.89) for ππ2 and ππ2, a unique solution is found for π2
Four solution is found for wrist position, but only two is physically possible.
64
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematic Example 1: Fanuc S-900w robot
Loop closure equation for end effector orientation:
In which 30π΄ β1 can be found, knowing π1, π2 and π3
Using homogeneous transformations
From the rotation matrices π5 can be found
Two real roots are found.
65
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematic
Example 1: Fanuc S-900w robot
Assuming π π1 β 0 we can solve for π4 and π6
Use (1,3) and (2,3) components of the rotation matrix:
Hence, a unique solution can be found by
Similarly, se (3,1) and (3,2) components of the rotation matrix:
A unique solution of π6 is found
66
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematic
β’ Method of Successive Screws
Loop closure for wrist position π·
π· = π΄1 π΄2β¦π΄π π·π
Matrix manipulation: for given π , find the joint space variable π.
π΄1β1π· = π΄2β¦π΄π π·π , β¦
Loop closure end effector orientation.
use π = π 1 π 2β¦π 6 ππ
or π = π 1 π 2β¦π 6 ππor π = π 1 π 2β¦π 6 ππ
Where π π is the rotation matrix corresponding to π΄π Matrix manipulation: for given π , find the joint space variable π.
π 3ππ 2
ππ 1ππ = π 4 π 5π 6 ππ, β¦;
67
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematics
β’ Successive Screw Method
Example 1
Consider 6DoF Elbow Manipulator (6R)
Screw Transformation Matrices:
68
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematics
β’ Successive Screw Method
Example 1: (Cont.)
Loop closure for wrist position π·
π· = π΄1π΄2π΄3π΄4 π·π
Manipulate
A1β1
ππ₯ππ¦ππ§1
= π΄2π΄3π΄4
π2 + π3 + π4001
,
69
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematics
β’ Successive Screw Method
Example 1: (Cont.)
This leads to:
From (2.145) two solutions are found by
For this manipulator position and orientation is not decoupled
Write the orientation loop closure
This results in:
70
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematics
β’ Successive Screw Method
Example 1: (Cont.)
From (2.150) two solutions are found by
Equations (2.149) and (2.151) may be used to solve for π234
Next use (2.144) and (2.146) to solve for π2 and π3. Lets rewrite them as
Where
Summing the squares
This results in:
71
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematics
β’ Successive Screw Method
Example 1: (Cont.)
To solve for π6, write the orientation loop closure for π.
This results in:
Solve (2.159) and (2.160) for π π6
And use (2.161)
By this means the inverse kinematics is completed.
72
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematics
β’ Successive Screw Method
Example 2: Stanford Arm (2RP3R)
Screw Transformation Matrices:
Wrist Position: π = π΄1π΄2π΄3 ππ
73
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematics
β’ Successive Screw Method
Example 2: (Cont.)Find π3 by summing the squares of the equations:
Hence,
This equation yield to two real roots if the end-effector is in the reachable workspace, but only the
positive π3 is acceptable.
Now use (2.169) to solve for π2:
Here two solutions could be found, π2 = π2β, π β π2
β
Now solve for π1:
hence
74
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematicsβ’ Successive Screw Method
Example 2: (Cont.)
End effector orientation loop closure equation for π
Manipulate
In which
Let us define
where
Note that π is independent of π6. Solve (2.180) for π5:
This equation yields to two real roots in the reachable workspace.
75
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Inverse Kinematics
β’ Successive Screw Method
Example 2: (Cont.)Assume that π π5 β 0 use (2.179) and (2.181) solve for π5:
Now solve for π6 by using loop closure equation for π.
Manipulate
Define This yields to:
where
Multiply (2.186) by π π4 and (2.188) by ππ4
This yields to
And completes the IK solution.
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Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Hamid D. Taghirad has received his B.Sc. degree in mechanical engineering
from Sharif University of Technology, Tehran, Iran, in 1989, his M.Sc. in mechanical
engineering in 1993, and his Ph.D. in electrical engineering in 1997, both
from McGill University, Montreal, Canada. He is currently the University Vice-
Chancellor for Global strategies and International Affairs, Professor and the Director
of the Advanced Robotics and Automated System (ARAS), Department of Systems
and Control, Faculty of Electrical Engineering, K. N. Toosi University of Technology,
Tehran, Iran. He is a senior member of IEEE, and Editorial board of International
Journal of Robotics: Theory and Application, and International Journal of Advanced
Robotic Systems. His research interest is robust and nonlinear control applied to
robotic systems. His publications include five books, and more than 250 papers in
international Journals and conference proceedings.
About Hamid D. Taghirad
Hamid D. TaghiradProfessor
Robotics: Mechanics and Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems March 2, 2021
Chapter 3: Kinematic Analysis
To read more and see the course videos visit our course website:
http://aras.kntu.ac.ir/arascourses/robotics/
Thank You
Robotics: Mechanics & Control