Robotics: Mechanics & Control Chapter 5: Dynamic Analysis
Transcript of Robotics: Mechanics & Control Chapter 5: Dynamic Analysis
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 June 1, 2021
Chapter 5: Dynamic AnalysisIn this chapter we review the dynamics analysis for serial robots. First the definition to angular and linear accelerations are given, then mass properties, linear and angular momentums, and kinetic energy of a rigid body in space is defined. Lagrange formulation is given in general form, and dynamics mass matrix, gravity and Coriolis and centrifugal vectors are defined and derived for several case studies. Dynamic formulation properties is given next, then actuator dynamics is elaborated for electrically driven robots with gearbox. Finally, linear regression method is used for dynamics calibration, and model verification methods are elaborated by introducing consistency measure.
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 June 1, 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 June 1, 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
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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 June 1, 2021
Contents
In this chapter we review the dynamics analysis for serial robots. First the definition to angular and linear accelerations are given, then mass properties, linear and angular momentums, and kinetic energy of a rigid body in space is defined. Lagrange formulation is given in general form, and dynamics mass matrix, gravity and Coriolis and centrifugal vectors are defined and derived for several case studies. Dynamic formulation properties is given next, then actuator dynamics is elaborated for electrically driven robots with gearbox. Finally, linear regression method is used for dynamics calibration, and model verification methods are elaborated by introducing consistency measure.
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Actuator DynamicsElectrical actuators, permanent magnet DC motors, servo amplifiers, gearbox, motor-gearbox-load dynamics, motor-gearbox-multiple joint robot,
4
Dynamics CalibrationLinear Regression, linear model with constant gravity, house holder reflection, varying gravity term, filtered velocity, model verification, consistency measure.
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PreliminariesAngular acceleration, linear acceleration of a point, mass properties, center of mass, moments of inertia, inertia matrix transformations, linear and angular momentum, kinetic energy.
1
Lagrange FormulationMotivating example, generalized coordinates and forces, kinetic energy, mass matrix, potential energy, gravity vector, Coriolis and centrifugal vector, case studies.
2
Dynamic Formulation PropertiesMass matrix properties, linearity in parameters, Christoffel Matrix, skew-symmetric property, general dynamic formulation, passivity,
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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 June 1, 2021
Introduction
โข Preliminaries
Angular Acceleration of a Rigid Body
Attribute of the whole rigid body
Importance of screw representation
๐ ร เท๐ = ๐.
Angular acceleration is also along เท๐.
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(๐ ร เท๐)
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 June 1, 2021
Preliminaries
Linear Acceleration of a Point
Start with the velocity vector of point ๐ท
Differentiate with respect to time and manipulate
Where the last two terms are centrifugal and
Coriolis acceleration terms.
If ๐ท is embedded in the rigid body, the relative
velocity and acceleration are equal to zero. Then
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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 June 1, 2021
Preliminaries
โข Mass Properties:
Consider body consists of differential material
bodies with density ๐ and volume ๐๐ then ๐๐
= ๐๐๐ is the differential mass while the mass of the
body is defined by
The center of mass ๐๐ is then defined by
In which, ๐ denote the position of ๐๐
The body frame attached to the center of mass is
denoted by {๐ถ}.
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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 June 1, 2021
Preliminaries
โข Mass Properties:
Moments of Inertia
As opposed to the mass, moment of inertia introduces inertia
to angular acceleration.
Moment of inertia is a tensor (matrix) defined by the second
moment of mass w.r.t a frame of reference
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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 June 1, 2021
Preliminaries
โข Mass Properties:
Moments of Inertia
Principal Axes:
A specific direction in space where the matrix is diagonal
Principal Axes are the eigenvectors of the inertia matrix
Physically, they are the axes of symmetry of the rigid body.
We never calculate the Moment of inertia by hand
Use tables:
Appendix D: J. L. Meriam et. Al. : Dynamics book (see next slide )
Or Cad Softwares to calculate them.
Autocad, Solidworks, etc
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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 June 1, 2021
Preliminaries
โข Mass Properties:
Moments of Inertia Table:
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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 June 1, 2021
Preliminaries
โข Mass Properties:
Inertia Matrix Transformation
Parallel Axis Theorem
Where ๐๐ denotes the position vector of ๐ถ w.r.t {๐ด}.
Pure Rotation:
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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 June 1, 2021
Preliminaries
โข Momentum and Energy
Linear Momentum
Angular Momentum
Kinetic Energy
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โ
โ
โ
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 June 1, 2021
Contents
In this chapter we review the dynamics analysis for serial robots. First the definition to angular and linear accelerations are given, then mass properties, linear and angular momentums, and kinetic energy of a rigid body in space is defined. Lagrange formulation is given in general form, and dynamics mass matrix, gravity and Coriolis and centrifugal vectors are defined and derived for several case studies. Dynamic formulation properties is given next, then actuator dynamics is elaborated for electrically driven robots with gearbox. Finally, linear regression method is used for dynamics calibration, and model verification methods are elaborated by introducing consistency measure.
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Actuator DynamicsElectrical actuators, permanent magnet DC motors, servo amplifiers, gearbox, motor-gearbox-load dynamics, motor-gearbox-multiple joint robot,
4
Dynamics CalibrationLinear Regression, linear model with constant gravity, house holder reflection, varying gravity term, filtered velocity, model verification, consistency measure.
5
PreliminariesAngular acceleration, linear acceleration of a point, mass properties, center of mass, moments of inertia, inertia matrix transformations, linear and angular momentum, kinetic energy.
1
Lagrange FormulationMotivating example, generalized coordinates and forces, kinetic energy, mass matrix, potential energy, gravity vector, Coriolis and centrifugal vector, case studies.
2
Dynamic Formulation PropertiesMass matrix properties, linearity in parameters, Christoffel Matrix, skew-symmetric property, general dynamic formulation, passivity,
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 June 1, 2021
Lagrange Formulation
โข Motivating Example
Consider a sprung mass pendulum as in figure
A pendulum with mass ๐, Length ๐ฟ, and moment of Inertia ๐ถ๐ผ๐ง๐ง = ๐ผ.
A sprung mass with mass ๐, and spring stiffness ๐พ
A motor torque ๐ to control the motion with no friction
Generalized Coordinate
The system consists of two masses ๐ and ๐ โ two generalized coordinate:
๐ =๐๐
RotationTranslation
Generalized Forces
Corresponding to each generalized coordinate, consider the external
force/torques:
๐ธ =๐0
TorqueForce
Lagrangian โ = ๐พ โ ๐
In which ๐พ is the total kinetic energy of all rigid bodies, ๐พ =1
2๐๐๐
๐ +1
2๐ถ๐ผ๐๐
while, ๐ is the total potential energy w.r.t a reference level. ๐ = ๐๐โ +1
2๐๐ฅ2
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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 June 1, 2021
Lagrange Formulation
โข Motivating Example
Governing Dynamic Equations:
Where, ๐๐ denotes the generalized coordinate, ๐๐ denotes the generalized force,
and m denotes the number of independent generalized coordinate
The key point is
To derive the angular and center of mass velocities correctly.
๐ฃ๐12 = โ แถ๐
2๐1 = แถ๐ and ๐ฃ๐2
2 = แถ๐2 + ๐ แถ๐2๐2 = แถ๐
Hence the total kinetic energy is
๐พ =1
2๐ โ แถ๐
2+
1
2๐ผ แถ๐2 +
1
2๐ แถ๐2 + ๐ แถ๐
2
Notice that the mass m is considered as a point mass with no moment of inertia.
The total potential energy is (๐๐ denotes the position where the spring is at equilibrium):
๐ = โ๐๐โ cos๐ โ ๐๐๐ cos ๐ +1
2๐ ๐ โ ๐๐
๐
Hence:
โ =1
2๐โ2 +๐๐2 + ๐ผ แถ๐2 +
1
2๐ แถ๐2 + ๐โ +๐๐ ๐ cos ๐ +
1
2๐ ๐ โ ๐0
2
๐
๐ฆ
๐ฅ
(๐1)
(๐2)
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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 June 1, 2021
Lagrange Formulation
โข Motivating Example
Governing Dynamic Equations:
For ๐1 = ๐
๐
๐๐ก
๐โ
๐ แถ๐โ
๐โ
๐๐= ๐1
This yields to:
๐
๐๐ก๐โ2 +๐๐2 + ๐ผ แถ๐ + ๐โ +๐๐ ๐ sin๐ = ๐
๐โ2 +๐๐2 + ๐ผ แท๐ + 2๐๐ แถ๐ แถ๐ + ๐โ +๐๐ ๐ sin๐ = ๐
For ๐2 = ๐
๐
๐๐ก
๐โ
๐ แถ๐โ
๐โ
๐๐= ๐2
This yields to:
๐ แท๐ โ ๐๐ แถ๐ โ ๐๐๐๐๐๐ ๐๐ ๐ โ ๐0 = 0
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โ
=1
2๐โ2 +๐๐2 + ๐ผ แถ๐2 +
1
2๐ แถ๐2
+ ๐โ +๐๐ ๐ cos ๐ +1
2๐ ๐ โ ๐0
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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 June 1, 2021
Lagrange Formulation
โข Generalized Coordinates and Forces
Serial robot with primary joints
Each joint introduces a generalized coordinate:
๐ = ๐1, ๐2, โฆ , ๐๐๐ in which ๐๐ = แ
๐๐ for a revolute joint๐๐ for a prismatic joint
Each joint actuator introduces a generalized force:
๐ธ = ๐1, ๐2, โฆ , ๐๐๐ in which ๐๐ = แ
๐๐ for a revolute joint๐๐ for a prismatic joint
If the robot is applying a wrench ๐๐ to the environment, Then
๐ธ = ๐ + ๐ฑ๐ป๐๐
If the joints have friction:
๐๐ = เต๐๐ โ ๐๐ for a revolute joint
๐๐ โ ๐๐ for a prismatic joint
In which, ๐๐/๐๐ denotes the joint friction torque/force.
They are called generalized: To include both rotational/translational motion and torque/force actuation.
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๐ญ๐ธ
๐๐ธ
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 June 1, 2021
Lagrange Formulation
โข Kinetic Energy
Serial robot with primary joints
For joint ๐: ๐พ๐ =1
2๐๐๐๐๐
๐ ๐๐๐ +1
2๐๐๐ ๐ถ๐ผ๐๐๐
Total Kinetic Energy: ๐พ = ฯ๐=1๐ ๐พ๐
The velocity vectors and the moment of inertia could be expressed in any frame.
Inertia about center of mass is simply found in the moving frame ๐๐ผ๐ (and it is constant)
To express it in the base frame: ๐ช๐ฐ๐ =๐๐น๐
๐๐ฐ๐๐๐น๐
๐ป.
The velocity ๐๐๐ can be found by conventional or screw-based Jacobian mapping
For conventional Jacobian: แถ๐๐๐ = ๐ฑ๐ แถ๐
Where แถ๐๐๐ =๐๐๐๐๐
and ๐ฑ๐ =๐ฑ๐๐๐ฑ๐๐
.
In which ๐ฝ๐ denotes the link Jacobian
and ๐ฝ๐ฃ๐ and ๐ฝ๐๐ denote the translational/rotational Jacobian submatrices
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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 June 1, 2021
Lagrange Formulation
โข Kinetic Energy
To derive the link Jacobians ๐ฑ๐ =๐ฑ๐๐๐ฑ๐๐
Use the general derivation method for Jacobian column vectors ๐ฝ๐ฃ๐j
and ๐ฝ๐๐
jfor ๐ โค ๐.
๐ฝ๐ฃ๐j= เต
๐๐โ1 ร๐โ1๐๐๐
โ revolute
๐๐โ1 prismaticand ๐ฝ๐๐
j= แ
๐๐โ1 revolute
๐ prismatic
Where ๐โ1๐๐๐โ is the position vector from the
origin of ๐ โ 1 link frame to the center of mass
of link ๐ and expressed in the base frame.
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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 June 1, 2021
โข Kinetic Energy
The procedure is like what on regular Jacobians, but the 0๐๐โ is replaced with the position vector
of the center of mass 0๐๐๐โ . (Chapter 04 โ Slide 21)
Note that the velocity ๐ฃ๐๐ depends only to the joint velocities 1 to ๐, and therefore, ๐ฝ๐ฃ๐j= ๐ฝ๐๐
j
= 0 for ๐ > ๐.
๐ฑ๐๐ = ๐ฑ๐๐๐ , ๐ฑ๐๐
๐ , โฆ , ๐ฑ๐๐๐, ๐, ๐, โฆ , ๐ , ๐ฑ๐๐ = ๐ฑ๐๐
๐ , ๐ฑ๐๐๐ , โฆ , ๐ฑ๐๐
๐ , ๐, ๐, โฆ , ๐
Lagrange Formulation20
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 June 1, 2021
Lagrange Formulation
โข Kinetic Energy Therefore, ๐พ =
1
2ฯ๐=1๐ (๐๐๐๐๐
๐ ๐๐๐ +๐๐๐ ๐ถ๐ผ๐๐๐)
Use Jacobians: =1
2แถ๐๐ ฯ๐=1
๐ (๐๐ ๐ฑ๐๐๐ป ๐ฑ๐๐+ ๐ฑ๐
๐ป๐
๐ถ๐ผ๐ ๐ฑ๐๐) แถ๐
Define Mass Matrix:
By this means: ๐พ =1
2แถ๐๐๐ด ๐ แถ๐
The Mass Matrix is a symmetric, and positive definite matrix.
The kinetic energy is also always positive unless the system is at rest.
โข Potential Energy
The work required to displace link ๐ to position ๐๐๐ is given by โ๐๐๐๐๐๐๐, in which ๐ is the vector of gravity
acceleration, Therefore,
๐ = โฯ๐=1๐ ๐๐๐
๐ ๐๐๐๐
If there is any spring (flexibility) in joints, then
In which ๐๐ denote the spring stiffness while ฮ๐ฅ denote its deflection
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๐ด(๐) =
๐=1
๐
(๐๐ ๐ฑ๐๐๐ป ๐ฑ๐๐+ ๐ฑ๐
๐ป๐
๐ถ๐ผ๐ ๐ฑ๐๐)
๐ = โ
๐=1
๐
๐๐๐๐ ๐๐๐๐ +
1
2
๐=1
๐
๐๐ ฮ๐ฅ2
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 June 1, 2021
Lagrange Formulation
โข General Formulation
Consider ๐DoF serial manipulator with primary joints
Denote ๐ = ๐1, ๐2, โฆ , ๐๐๐ป as the generalized coordinates and
Denote ๐ธ = ๐1, ๐2, โฆ , ๐๐๐ป as the generalized Forces.
Form Lagrangian by โ = ๐พ โ ๐
Derive the governing equation of motion in vector form by:
๐
๐๐ก
๐โ
๐ แถ๐โ๐โ
๐๐= ๐ธ
in which, โ = ๐พ(๐, แถ๐) โ ๐(๐)
hence, ๐
๐๐ก
๐โ
๐ แถ๐=
๐
๐๐ก
๐๐พ
๐ แถ๐=
๐
๐๐ก๐ ๐ แถ๐ = ๐ด ๐ แท๐ + แถ๐ด ๐ แถ๐
Furthermore, define the gravity vector as g ๐ =๐๐
๐๐= โฯ๐=1
๐ ๐๐
๐๐๐๐๐
๐ ๐๐๐๐ = โฯ๐=1๐ ๐๐๐ฑ๐๐
๐ป ๐.
In which, ๐ฑ๐๐๐ป denotes the linear velocity Jacobian of each center of motion. Hence the dynamics is written as:
๐ด ๐ แท๐ + แถ๐ด ๐ แถ๐ โ๐๐พ
๐๐+ g ๐ = ๐ธ
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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 June 1, 2021
Lagrange Formulation
โข General Formulation
This can be written in general form of
๐ด ๐ แท๐ + v ๐, แถ๐ + g ๐ = ๐ธ
In which v(๐, แถ๐) is called the Coriolis and centrifugal vector
v ๐, แถ๐ = แถ๐ด ๐ แถ๐ โ๐๐พ
๐๐= แถ๐ด ๐ แถ๐ โ
1
2
๐
๐๐แถ๐๐๐ด ๐ แถ๐
This vector may be written in a Christoffel matrix form v ๐, แถ๐ = ๐ช ๐, แถ๐ แถ๐
๐ด ๐ แท๐ + ๐ช ๐, แถ๐ แถ๐ + g ๐ = ๐ธ
The derivation and properties of dynamic matrices are elaborated next.
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 June 1, 2021
๐ฅ
๐ฆ
๐1
๐2
Lagrange Formulation
โข Examples:
Example 1: Planar Elbow Manipulator
Consider
slender bar links with mass ๐๐ @ their half length,
viscous friction ๐๐๐ = ๐๐ แถ๐๐ at the joints
No joint flexibility, and no applied wrench.
Denote ๐ = ๐1, ๐2๐ and ๐ธ = ๐1 โ ๐1 แถ๐1, ๐2 โ ๐2 แถ๐2
๐
The inertia in ๐ง direction is found by
๐๐ผ๐ง๐ง๐ =1
12๐๐๐๐
2 which is the same in base frame: ๐ถ๐ผ๐ง๐ง๐ =1
12๐๐๐๐
2
The position vector of the center of mass in base frame:
0๐๐1โ =
1
2๐1
๐1๐ 10
, 1๐๐2โ =
1
2๐2
๐12๐ 120
โ 0๐๐2โ =
๐1๐1 +1
2๐2๐12
๐1๐ 1 +1
2๐2๐ 12
0
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0๐๐1โ
0๐๐2โ 1๐๐2
โ
Matlab Code: Lagrange_2R.m
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 June 1, 2021
Lagrange Formulation
Example 1: Planar Elbow Manipulator
The link Jacobians are derived easily as:
๐ฑ๐๐ =1
2๐1
โ๐ 1๐10
000
, ๐ฑ๐๐=
001
000
๐ฑ๐๐ =
โ๐1๐ 1 โ1
2๐2๐ 12
๐1๐1 +1
2๐2๐12
0
โ1
2๐2๐ 12
1
2๐2๐12
0
, ๐ฑ๐๐=
001
001
The robot Mass Matrix is derived as:
๐ด ๐ =
๐=1
2
๐๐ ๐ฑ๐๐๐ป ๐ฑ๐๐+ ๐ฑ๐
๐ป๐
๐ถ๐ผ๐ ๐ฑ๐๐ =
1
3๐1๐1
2 +๐2 ๐12 + ๐1๐2๐2 +
1
3๐22 ๐2
1
2๐1๐2๐2 +
1
3๐22
๐2
1
2๐1๐2๐2 +
1
3๐22
1
3๐2๐2
2
25
๐ฅ
๐ฆ
๐1
๐2
Matlab Code: Lagrange_2R.m
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 June 1, 2021
Lagrange Formulation
Example 1: Planar Elbow Manipulator
The Gravity vector is derived as:
g ๐ = โ
๐=1
2
๐๐๐ฑ๐๐๐ป ๐ =
1
2๐1 +๐2 ๐๐1๐1 +
1
2๐2๐๐2๐12
1
2๐2๐๐2๐12
The Coriolis and centrifugal vector is derived as:
v ๐, แถ๐ = แถ๐ด ๐ แถ๐ โ๐๐พ
๐๐=
โ๐2๐1๐2๐ 2 แถ๐1 แถ๐2 +1
2แถ๐22
1
2๐2๐1๐2๐ 2 แถ๐1
2
The Lagrange equations is formulates by:
๐ =๐1 โ ๐1 แถ๐2๐2 โ ๐2 แถ๐2
= ๐ด ๐ แท๐ + v ๐, แถ๐ + g ๐ .
26
๐ฅ
๐ฆ
๐1
๐2
Matlab Code: Lagrange_2R.m
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 June 1, 2021
Lagrange Formulation
Example 2: Planar RRR Manipulator
Like example 1 consider:
slender bar links with mass ๐๐ @ their half length with no friction,
no joint flexibility, and no applied wrench.
The position vector of the center of mass in base frame:
0๐๐1โ =
1
2๐1
๐1๐ 10
, 1๐๐2โ =
1
2๐2
๐12๐ 120
โ 0๐๐2โ =
๐1๐1 +1
2๐2๐12
๐1๐ 1 +1
2๐2๐ 12
0
2๐๐3โ =
1
2๐3
๐123๐ 1230
โ 1๐๐3โ =
๐2๐12 +1
2๐3๐123
๐2๐ 12 +1
2๐3๐ 123
0
and 0๐๐3โ =
๐1๐1 + ๐2๐12 +1
2๐3๐123
๐1๐ 1 + ๐2๐ 12 +1
2๐3๐ 123
0
27
Matlab Code: Lagrange_3R.m
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 June 1, 2021
Lagrange Formulation
Example 2: Planar RRR Manipulator
The link Jacobians are derived easily as:
๐ฑ๐๐ =1
2๐1
โ๐ 1๐10
000
000
, ๐ฑ๐๐=
001
000
000
๐ฑ๐๐ =
โ๐1๐ 1 โ1
2๐2๐ 12 โ
1
2๐2๐ 12 0
๐1๐1 +1
2๐2๐12
1
2๐2๐12 0
0 0 0
, ๐ฑ๐๐=
001
001
000
๐ฑ๐๐ =
โ๐1๐ 1 โ ๐2๐ 12 โ1
2๐3๐ 123
๐1๐1 + ๐2๐12 +1
2๐3๐๐ 123
0
โ๐2๐ 12 โ1
2๐3๐ 123
๐2๐12 +1
2๐3๐123
0
โ1
2๐3๐ 123
1
2๐3๐123
0
, ๐ฑ๐๐=
001
001
001
28
Matlab Code: Lagrange_3R.m
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 June 1, 2021
Lagrange Formulation
Example 2: Planar RRR Manipulator
The robot Mass Matrix is derived as:
๐ด๐ร๐ ๐ =
๐=1
2
๐๐ ๐ฑ๐๐๐ป ๐ฑ๐๐+ ๐ฑ๐
๐ป๐
๐ถ๐ผ๐ ๐ฑ๐๐
In which,
๐11 =1
3๐1 +๐2 +๐3 ๐1
2 +1
3๐2 +๐3 ๐2
2 +1
3๐3๐3
2 + ๐2 + 2๐3 ๐1๐2๐2 + ๐3๐3(๐1๐23 + ๐2๐3)
๐22 =1
3๐2 +๐3 ๐2
2 +1
3๐3๐3
2 +๐3๐2๐3๐3, ๐33 =1
3๐3๐3
2
๐12 = ๐21 =1
3๐2 +๐3 ๐2
2 +๐3๐31
3๐3 +
1
2๐1๐23 +
1
2๐2๐1๐2๐2 +๐3๐2(๐1๐2 + ๐3๐3)
๐23 = ๐32 = ๐3๐31
3๐3 +
1
2๐2๐3 , ๐13 = ๐31 = ๐3๐3
1
3๐3 +
1
2๐1๐23 +
1
2๐2๐3
The Gravity vector is derived as:
g ๐ =
1
2๐1๐1๐ ๐1 +
1
2๐2๐ ๐1๐1 + ๐2๐12 +๐3๐ ๐2๐12 +
1
2๐3๐123
1
2๐2๐2๐ ๐12 +๐3๐ ๐2๐12 +
1
2๐3๐123
1
2๐3๐3๐ ๐123
29
Matlab Code: Lagrange_3R.m
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 June 1, 2021
Lagrange Formulation
Example 2: Planar RRR Manipulator
The Coriolis and centrifugal vector is derived as:
๐ฝ ๐, แถ๐ = แถ๐ด ๐ แถ๐ โ1
2
๐
๐๐แถ๐๐๐ด ๐ แถ๐ =
๐1๐2๐3
In which,
๐1 = โ แถ๐1 แถ๐2 ๐1๐3๐3๐ 23 + ๐1๐2๐2๐ 2 + 2๐1๐2๐3๐ 2 + แถ๐3 ๐1๐3๐3๐ 23 + ๐2๐3๐3๐ 3
โ แถ๐2 แถ๐21
2๐1๐3๐3๐ 23 +
1
2๐1๐2๐2๐ 2 + ๐1๐2๐3๐ 2 + แถ๐3
1
2๐1๐3๐3๐ 23 + ๐2๐3๐3๐ 3
โ แถ๐3 ๐3๐3แถ๐3
1
2๐1๐ 23 +
1
2๐2๐ 3 +
1
2๐1๐3๐3๐ 23 แถ๐2
๐2 =1
2๐1๐3๐3๐ 23 + ๐1๐2๐2๐ 2 + ๐1๐2๐3๐ 2 แถ๐1
2 โ1
2๐2๐3๐3๐ 3 แถ๐3
2 โ ๐2๐3๐3๐ 3 แถ๐1 แถ๐3 + แถ๐2 แถ๐3
๐3 =1
2๐3๐3((๐1๐ 23 + ๐2๐ 3) แถ๐1
2 + ๐2๐ 3 แถ๐22 + 2๐2๐ 3 แถ๐1 แถ๐2
The Lagrange equations is formulates by:
๐ =
๐1๐2๐3
= ๐ด ๐ แท๐ + ๐ฝ ๐, แถ๐ + ๐ฎ ๐ .
30
Matlab Code: Lagrange_3R.m
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 June 1, 2021
Lagrange Formulation
Example 3: SCARA Arm
Like example 1 consider:
slender bar links with mass ๐๐ @ their half length with no friction,
no joint flexibility, and no applied wrench.
The inertia matrix is found by
1๐ผ๐ฆ๐ฆ1 =1 ๐ผ๐ง๐ง1 =
1
12๐1๐1
2 ; 2๐ผ๐ฆ๐ฆ2 =2 ๐ผ๐ง๐ง2 =
1
12๐2๐2
2 ,
3๐ผ๐ฅ๐ฅ3 =3 ๐ผ๐ฆ๐ฆ3 =
1
12๐3โ
2 .
The position vector of the center of mass in base frame:
0๐๐1โ =
1
2๐1๐1
1
2๐1๐ 1
๐1
, 1๐๐2โ =
1
2๐2๐12
1
2๐2๐ 12
0
โ 0๐๐2โ =
๐1๐1 +1
2๐2๐12
๐1๐ 1 +1
2๐2๐ 12
๐1
2๐๐3โ =
00
๐3 โ1
2โ
โ 1๐๐3โ =
๐2๐12๐2๐ 12
โ๐3 +1
2โ
โ 0๐๐3โ =
๐1๐1 + ๐2๐12๐1๐ 1 + ๐2๐ 12
๐1 โ ๐3 +1
2โ
31
Matlab Code: Lagrange_SCARA.m
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 June 1, 2021
Lagrange Formulation
Example 3: SCARA Arm
The link Jacobians are derived easily as:
๐ฑ๐๐ =1
2๐1
โ๐ 1๐10
000
000
, ๐ฑ๐๐=
0 0 00 0 01 0 0
๐ฑ๐๐ =
โ๐1๐ 1 โ1
2๐2๐ 12 โ
1
2๐2๐ 12 0
๐1๐1 +1
2๐2๐12
1
2๐2๐12 0
0 0 0
, ๐ฑ๐๐=
0 0 00 0 01 1 0
๐ฑ๐๐ =โ๐1๐ 1 โ ๐2๐ 12 โ๐2๐ 12 0๐1๐1 + ๐2๐12 ๐2๐12 0
0 0 1, ๐ฑ๐๐
=001
001
000
32
Matlab Code: Lagrange_SCARA.m
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 June 1, 2021
Lagrange Formulation
Example 3: SCARA Arm
The robot Mass Matrix is derived as:
๐ด ๐ =
๐=1
2
๐๐ ๐ฑ๐๐๐ป ๐ฑ๐๐+ ๐ฑ๐
๐ป๐
๐ถ๐ผ๐ ๐ฑ๐๐
= ๐1
1
3๐12 0 0
0 0 00 0 0
+ ๐2
๐12 + ๐1๐2๐2 +
1
3๐22 1
2๐1๐2๐2 +
1
3๐22 0
1
2๐1๐2๐2 +
1
3๐22 1
3๐22 0
0 0 0
+
๐3
๐12 + 2๐1๐2๐2 + ๐2
2 ๐1๐2๐2 + ๐22 0
๐1๐2๐2 + ๐22 ๐2
2 00 0 1
33
๐
=
1
3๐1๐1
2 +๐2 ๐12 +
1
3๐22 + ๐1๐2๐2 +๐3 ๐1
2 + ๐22 +
1
12๐32 + 2๐1๐2๐2 โ๐2๐2
1
3๐2 +
1
2๐1๐2 โ๐3 ๐2
2 +1
12๐32 + ๐1๐2๐2 0
โ๐2๐21
3๐2 +
1
2๐1๐2 โ๐3 ๐2
2 +1
12๐32 + ๐1๐2๐2
1
3๐2๐2
2 +๐3 ๐22 +
1
12๐32 0
0 0 ๐3
Matlab Code: Lagrange_SCARA.m
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 June 1, 2021
Lagrange Formulation
Example 3: SCARA Arm
The Gravity vector is derived as:
g ๐ =00
โ๐3๐
The Coriolis and centrifugal vector is derived as:
v ๐, แถ๐ =
โ(๐2+2๐3)๐1๐2๐ 2 แถ๐1 แถ๐2 +1
2แถ๐22
1
2๐2 +๐3 ๐1๐2๐ 2 แถ๐1
2
0
The Lagrange equations is formulated by:
๐ =
๐1๐2๐3
= ๐ด ๐ แท๐ + v ๐, แถ๐ + g ๐ .
34
Matlab Code: Lagrange_SCARA.m
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 June 1, 2021
Lagrange Formulation
Example 3: SCARA Arm
The Lagrange equations is formulated by:
๐ด ๐ แท๐ + v ๐, แถ๐ + g ๐ = ๐ธ
35
๐3Matlab Code: Lagrange_SCARA.m
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 June 1, 2021
Scientist Bio36
Joseph-Louis Lagrange(25 January 1736 โ 10 April 1813)
Was an Italian mathematician and astronomer, later naturalized French. He made significant contributions to the
fields of analysis, number theory, and both classical and celestial mechanics.
In 1766, on the recommendation of Swiss Leonhard Euler and French d'Alembert, Lagrange succeeded Euler as
the director of mathematics at the Prussian Academy of Sciences in Berlin, Prussia, where he stayed for over
twenty years, producing volumes of work and winning several prizes of the French Academy of Sciences.
Lagrange's treatise on analytical mechanics written in Berlin and first published in 1788, offered the most
comprehensive treatment of classical mechanics since Newton and formed a basis for the development of
mathematical physics in the nineteenth century. In 1787, at age 51, he moved from Berlin to Paris and became a
member of the French Academy of Sciences. He remained in France until the end of his life. He was instrumental
in the decimalisation in Revolutionary France, became the first professor of analysis at the รcole
Polytechnique upon its opening in 1794, was a founding member of the Bureau des Longitudes, and
became Senator in 1799.
Lagrange was one of the creators of the calculus of variations, deriving the EulerโLagrange equations for
extrema of functionals. He extended the method to include possible constraints, arriving at the method
of Lagrange multipliers. Lagrange invented the method of solving differential equations known as variation of
parameters, applied differential calculus to the theory of probabilities and worked on solutions for algebraic
equations. He proved that every natural number is a sum of four squares. His treatise Theorie des fonctions analytiques laid some of the foundations of group theory, anticipating Galois. In calculus, Lagrange developed a
novel approach to interpolation and Taylor series. He studied the three-body problem for the Earth, Sun and
Moon (1764) and the movement of Jupiter's satellites (1766), and in 1772 found the special-case solutions to this
problem that yield what are now known as Lagrangian points. Lagrange is best known for
transforming Newtonian mechanics into a branch of analysis, Lagrangian mechanics, and presented the
mechanical "principles" as simple results of the variational calculus.
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 June 1, 2021
Contents
In this chapter we review the dynamics analysis for serial robots. First the definition to angular and linear accelerations are given, then mass properties, linear and angular momentums, and kinetic energy of a rigid body in space is defined. Lagrange formulation is given in general form, and dynamics mass matrix, gravity and Coriolis and centrifugal vectors are defined and derived for several case studies. Dynamic formulation properties is given next, then actuator dynamics is elaborated for electrically driven robots with gearbox. Finally, linear regression method is used for dynamics calibration, and model verification methods are elaborated by introducing consistency measure.
37
Actuator DynamicsElectrical actuators, permanent magnet DC motors, servo amplifiers, gearbox, motor-gearbox-load dynamics, motor-gearbox-multiple joint robot,
4
Dynamics CalibrationLinear Regression, linear model with constant gravity, house holder reflection, varying gravity term, filtered velocity, model verification, consistency measure.
5
PreliminariesAngular acceleration, linear acceleration of a point, mass properties, center of mass, moments of inertia, inertia matrix transformations, linear and angular momentum, kinetic energy.
1
Lagrange FormulationMotivating example, generalized coordinates and forces, kinetic energy, mass matrix, potential energy, gravity vector, Coriolis and centrifugal vector, case studies.
2
Dynamic Formulation PropertiesMass matrix properties, linearity in parameters, Christoffel Matrix, skew-symmetric property, general dynamic formulation, passivity,
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 June 1, 2021
Dynamic Formulation Properties
โข Dynamics Formulation Representation
๐ด ๐ แท๐ + v ๐, แถ๐ + g ๐ = ๐ธ
Mass Matrix Properties
Since the mass matrix is defined from kinetic energy: ๐พ =1
2แถ๐๐๐ด ๐ แถ๐
It is always symmetric and positive definite (โ Configurations ๐)
It is always invertible
It has upper and lower bound
๐ ๐ฐ๐ร๐ โค ๐ด ๐ โค าง๐ ๐ฐ๐ร๐
Furthermore,
1
เดฅ๐๐ฐ๐ร๐ โค ๐ดโ๐ ๐ โค
1
๐๐ฐ๐ร๐
The bounds may be represented by matrix norm
๐ โค ๐ด ๐ โค เดฅ๐
In which ๐ and เดฅ๐ are positive constants.
38
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 June 1, 2021
Dynamic Formulation Properties
โข Linearity in Parameters
๐ด ๐ แท๐ + v ๐, แถ๐ + g ๐ = ๐ธ
This formulation is nonlinear and multivariable w.r.t ๐
But It could be written in linear regression form w.r.t kinematic and dynamic
parameters
๐ด ๐ แท๐ + v ๐, แถ๐ + ๐ ๐ = ๐จ ๐, แถ๐, แท๐ ๐ฝ
In which, ๐จ denote the linear regressor form
While ๐ฝ denote the kinematic and dynamic parameter vector
This regression form is not unique
But could be found for any serial robot
The minimum number of parameters may found by inspection
Or calibration process and singular values decomposition
39
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 June 1, 2021
Dynamic Formulation Properties
โข Linearity in Parameters
Example: Planar Elbow Manipulator
Dynamic formulation:
๐1
=1
3๐1๐1
2 +๐2 ๐12 + ๐1๐2๐2 +
1
3๐22 แท๐1 + ๐2
1
2๐1๐2๐2 +
1
3๐22 แท๐2
โ๐2๐1๐2๐ 2 แถ๐1 แถ๐2 +1
2แถ๐22 +
1
2๐1 +๐2 ๐๐1๐1 +
1
2๐2๐๐2๐12 + ๐1 แถ๐1
๐2 = ๐2
1
2๐1๐2๐2 +
1
3๐22 แท๐1 +
1
3๐2๐2
2 แท๐2 +1
2๐2๐1๐2๐ 2 แถ๐1
2 +1
2๐2๐๐2๐12 + ๐2 แถ๐2
Choose the parameters as:
๐1 =1
3๐1๐1
2 +๐2 ๐12 +
1
3๐22 ๐11 = ๐1 + ๐2๐2, ๐22 = ๐3
๐2 = ๐2๐1๐2, ๐3 =1
3๐2๐2
2 ๐12 = ๐21 =1
2๐2๐2 + ๐3
๐4 =1
2๐1 +๐2 , ๐5 =
1
2๐2๐2 ๐1 = ๐4๐๐1 + ๐5๐๐12, ๐2 = ๐5๐๐12
๐6 = ๐1, ๐7 = ๐2 ๐ฃ1 = โ๐2๐ 2 แถ๐1 แถ๐2 +1
2แถ๐22 , ๐ฃ2 =
1
2๐2๐ 2 แถ๐1
2
40
โ
๐ฅ
๐ฆ
๐1
๐2
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 June 1, 2021
Dynamic Formulation Properties
โข Linearity in Parameters
Example: Planar Elbow Manipulator
Then
๐ด ๐ แท๐ + v ๐, แถ๐ + g ๐ = ๐จ ๐, แถ๐, แท๐ ๐ฝ
In which
๐จ =แท๐10
๐2 แท๐1 +1
2แท๐2 โ ๐ 2 แถ๐1 แถ๐2 +
1
2แถ๐22
1
2๐2 แท๐1 + ๐ 2 แถ๐1
2
แท๐2แท๐1 + แท๐2
๐๐10
๐๐12๐๐12
แถ๐10
0แถ๐2
,
and ๐ฝ =
๐1๐2๐3๐4๐5๐6๐7
=
1
3๐1๐1
2 +๐2 ๐12 +
1
3๐22
๐2๐1๐21
3๐2๐2
2
1
2๐1 +๐2
1
2๐2๐2
๐1๐2
.
41
๐ฅ
๐ฆ
๐1
๐2
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 June 1, 2021
Dynamic Formulation Properties
โข Christoffel Matrix
The Coriolis and centrifugal vector may be written in Matrix form:
v ๐, แถ๐ = ๐ช ๐, แถ๐ แถ๐
To derive this Matrix, recall Kronecker product:
For two matrices ๐จ๐ร๐, and ๐ฉ๐ร๐ โ ๐จโจ๐ฉ = ๐๐๐๐ฉ ๐๐ร๐๐
The matrix has ๐ ร ๐ blocks, each determined by term-by-term multiplication of ๐๐๐๐ฉ
Note that ๐ฐ๐ร๐โจ๐ โ ๐โจ๐ฐ๐ร๐ but ๐ฐ๐ร๐โจ๐ ๐ = ๐โจ๐ฐ๐ร๐ ๐
For any arbitrary ๐ โ โ๐
Finally,
๐๐จ
๐๐=
๐๐จ
๐๐๐
โฎ๐๐จ
๐๐๐
โ๐
๐๐๐จ ๐ ๐ฉ(๐) = ๐ฐ๐ร๐โจ๐จ
๐๐ฉ
๐๐+
๐๐จ
๐๐๐ฉ.
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 June 1, 2021
Dynamic Formulation Properties
โข Christoffel Matrix
The Coriolis and centrifugal vector is defined by
v ๐, แถ๐ = แถ๐ด ๐ แถ๐ โ1
2
๐
๐๐แถ๐๐๐ด ๐ แถ๐
Use Kronecker product, and note that ๐ แถ๐๐
๐๐= 0, then
v ๐, แถ๐ = แถ๐ด ๐ โ1
2๐ฐ๐ร๐โจ แถ๐๐ป
๐๐ด
๐๐แถ๐.
Hence,
v ๐, แถ๐ = ๐ช1 ๐, แถ๐ แถ๐
In which,
๐ช1 ๐, แถ๐ = แถ๐ด ๐ โ1
2๐ผ ๐, แถ๐
where, ๐ผ ๐, แถ๐ = ๐ฐ๐ร๐โจ แถ๐๐ป๐๐ด
๐๐=
แถ๐1 00 แถ๐2
โฏ 0โฏ 0
โฎ โฎ0 0
โฑ โฎโฏ แถ๐๐
๐๐ด
๐๐1
๐๐ด
๐๐2โฏ
๐๐ด
๐๐๐.
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 June 1, 2021
Dynamic Formulation Properties
โข Christoffel Matrix
This representation is not unique! Find another one.
๐ช2 ๐, แถ๐ = ๐ผ๐ป ๐, แถ๐ โ1
2๐ผ ๐, แถ๐
Notice แถ๐ด โ ๐ผ๐ป but แถ๐ด แถ๐ = ๐ผ๐ป แถ๐.
The most celebrated Christofell Matrix is derived by:
๐ช ๐, แถ๐ =1
2แถ๐ด ๐ + ๐ผ๐ป ๐, แถ๐ โ ๐ผ ๐, แถ๐
โข Skew-Symmetric Property
For all ๐ถ๐ โs this relation holds
แถ๐๐ป แถ๐ด ๐ โ ๐๐ช๐ ๐, แถ๐ แถ๐ = 0
But for ๐ช ๐, แถ๐ the own matrix แถ๐ด ๐ โ ๐๐ช ๐, แถ๐ is skew symmetric.
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 June 1, 2021
Dynamic Formulation Properties
โข Christoffel Matrix
Proof:
1) แถ๐๐ป แถ๐ด โ ๐๐ช๐ แถ๐ = แถ๐๐ป แถ๐ด โ ๐ แถ๐ด + ๐ผ แถ๐ = แถ๐๐ป๐ผโ แถ๐๐ป แถ๐ด แถ๐ = ๐ since แถ๐ด แถ๐ = ๐ผ๐ป แถ๐.
2) แถ๐๐ป แถ๐ด โ ๐๐ช๐ แถ๐ = แถ๐๐ป แถ๐ด โ ๐๐ผ๐ป + ๐ผ แถ๐ = แถ๐๐ป แถ๐ด แถ๐ โ (๐๐ผ๐ป โ ๐ผ) แถ๐ = แถ๐๐ปเตซโ๐ผ๐ป
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 June 1, 2021
Dynamic Formulation Properties
โข Passivity
Claim: the amount of energy dissipated by the robot has a negative lower
bound โ๐ฝ
เถฑ0
๐
แถ๐๐ ๐ ๐ ๐ ๐๐ โฅ โ๐ฝ
Note แถ๐๐๐ has the units of power, and the integral denotes the energy produced
(or dissipated) by the system in the interval [0, ๐]
How this is related to the rate of total energy แถ๐ป:
In robotic manipulators:
Total Energy ๐ป = ๐พ + ๐ =1
2แถ๐๐๐ด ๐ แถ๐ + ๐ ๐
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 June 1, 2021
Dynamic Formulation Properties
โข Passivity Rate of total energy is:
แถ๐ป = แถ๐๐๐ด ๐ แท๐ +1
2แถ๐๐ แถ๐ด ๐ แถ๐ + แถ๐๐ป
๐๐ ๐
๐๐= แถ๐๐ ๐ โ ๐ช ๐, แถ๐ แถ๐ โ g(๐) +
1
2แถ๐๐ แถ๐ด ๐ แถ๐ + แถ๐๐ป
๐๐ ๐
๐๐
= แถ๐๐๐ +1
2แถ๐๐ แถ๐ด ๐ โ ๐๐ช ๐, แถ๐ แถ๐ + แถ๐๐
๐๐ ๐
๐๐โ g(๐) = แถ๐๐๐
The last two terms are both zero by skew-symmetric property and definition of gravity vector.
Hence, because แถ๐ป = แถ๐๐๐
เถฑ0
๐
แถ๐ป ๐ ๐๐ = เถฑ0
๐
แถ๐๐ ๐ ๐ ๐ ๐๐ = ๐ป ๐ โ ๐ป 0 โฅ โ๐ป(0)
Since the total energy of the system is always positive, for any robotic system the passivity property is
held with ๐ฝ = ๐ป(0).
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 June 1, 2021
Dynamic Simulation
โข Forward Dynamics
In forward dynamics given the actuator forces
applied to the robot, the resulting output
trajectory of the robot is found.
General Dynamics:
๐ + ๐๐ = ๐ด ๐ แท๐ + ๐ช ๐, แถ๐ แถ๐ + g ๐
The Dynamics equations shall be integrated to find the trajectory
The mass matrix is positive definite, hence it is invertible in all configurations
แท๐ = ๐ดโ๐ ๐ ๐ + ๐๐ โ ๐ช ๐, แถ๐ แถ๐ โ g ๐
Use numerical integration like Runge-Kutta method (ode45 in Matlab or Simulink)
Given the torques ๐, ๐๐ and the initial conditions for the augmented states ๐ = ๐๐, ๐๐๐ป = ๐, แถ๐ ๐ป, use
numerical integration to solve for the trajectory.
แถ๐1 = แถ๐ = ๐2
แถ๐2 = แท๐ = ๐ดโ๐ ๐1 ๐ + ๐๐ โ ๐ช ๐๐, ๐๐ ๐๐ โ g ๐๐
48
Forward Dynamics๐๐
๐๐
Actuator Torques
Output Trajectory
Disturbance torque
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 June 1, 2021
Dynamic Simulation
โข Forward Dynamics in Feedback loop
By using, trajectory planner, and controller design, a desired trajectory is traversed.
(Chapter 6)
โข Inverse Dynamics
In inverse dynamics given the trajectory of the robot ๐ ๐ก , แถ๐ ๐ก , แท๐(๐ก) , the required
actuator forced to traverse this trajectory ๐ is found.
Use general dynamics: ๐ = ๐ด ๐ แท๐ + ๐ช ๐, แถ๐ แถ๐ + g ๐
No integration is needed, direct calculation is possible.
49
๐๐ Forward DynamicsController
๐๐๐ ๐
+ โ
Trajectory Planner
Inverse Dynamics๐ ๐ , แถ๐, แท๐ ๐
Trajectory Required actuator torque
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 June 1, 2021
Contents
In this chapter we review the dynamics analysis for serial robots. First the definition to angular and linear accelerations are given, then mass properties, linear and angular momentums, and kinetic energy of a rigid body in space is defined. Lagrange formulation is given in general form, and dynamics mass matrix, gravity and Coriolis and centrifugal vectors are defined and derived for several case studies. Dynamic formulation properties is given next, then actuator dynamics is elaborated for electrically driven robots with gearbox. Finally, linear regression method is used for dynamics calibration, and model verification methods are elaborated by introducing consistency measure.
50
Actuator DynamicsElectrical actuators, permanent magnet DC motors, servo amplifiers, gearbox, motor-gearbox-load dynamics, motor-gearbox-multiple joint robot,
4
Dynamics CalibrationLinear Regression, linear model with constant gravity, house holder reflection, varying gravity term, filtered velocity, model verification, consistency measure.
5
PreliminariesAngular acceleration, linear acceleration of a point, mass properties, center of mass, moments of inertia, inertia matrix transformations, linear and angular momentum, kinetic energy.
1
Lagrange FormulationMotivating example, generalized coordinates and forces, kinetic energy, mass matrix, potential energy, gravity vector, Coriolis and centrifugal vector, case studies.
2
Dynamic Formulation PropertiesMass matrix properties, linearity in parameters, Christoffel Matrix, skew-symmetric property, general dynamic formulation, passivity,
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 June 1, 2021
Actuator Dynamics
โข Robot Electrical Actuators
51
Permanent MagnetDC Motors
Brushless DC Motors
AC Induction Motors
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 June 1, 2021
Actuator Dynamics
โข DC Motors
Principle of operation
A current carrying conductor in a magnetic field
experiences a Force
๐น = ๐ ร ๐
๐: the current in the conductor, ๐: the magnetic field flux
DC motor consists of
A fixed Stator (Permanent magnet)
A rotating rotor (Armature)
A commutator: to switch the direction of current
The torque generated in the motor
๐๐ = ๐พ1๐๐๐๐๐: the armature current, ๐: the magnetic field flux
Lenzโs Law: back-emf voltage
๐๐ = ๐พ2๐๐๐๐๐: the angular velocity of the rotor, ๐: the magnetic field flux
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 June 1, 2021
Actuator Dynamics
โข DC Motors
Principle of operation
For a permanent magnet DC motor ๐ is constant
If SI unit is used:
๐๐ = ๐พ๐๐๐ and ๐๐ = ๐พ๐๐๐
๐พ๐: the DC motor torque or velocity constant
Electrical model of the PMDC motor
๐ฟ๐๐๐
๐๐ก+ ๐ ๐๐ + ๐๐ = ๐ ๐ก or ๐ฟ๐ + ๐ ๐๐ = ๐ โ ๐๐
๐๐ =๐ โ ๐๐๐ฟ๐ + ๐
Electro-mechanical model of the PMDC motor
๐๐ = ๐พ๐๐๐ = ๐๐ = ๐พ๐๐ โ ๐พ๐๐๐๐ฟ๐ + ๐
53
๐๐
๐๐
๐๐
๐๐๐(๐ก)
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 June 1, 2021
Actuator Dynamics
โข DC Motors
Principle of operation
Usually ๐ฟ
๐ โช 1 โ 10โ2 โ 10โ3
Then up to 100 โ 1000 ๐ป๐ง bandwidth neglect induction
๐๐ =๐พ๐
๐ ๐ โ ๐พ๐๐๐
The Motor characteristic torque-velocity curve is linear
To avoid overheating ๐๐ is clamped to ๐๐๐๐ฅ by current limiters
The stall torque ๐๐ is the maximum output torque of the motor,
when the velocity of the motor is stalled.
๐๐ =๐ ๐๐ ๐พ๐
The maximum speed of the motor ๐๐๐๐ฅ is obtained when no load
torque is applied to the motor
Back-emf acts like a disturbance on the output torque, the more
the velocity the less the output torque.
How to compensate?!!
54
๐๐
๐1
๐2
๐๐๐๐ฅ
๐๐๐๐๐๐ฅ
๐๐ ๐๐๐๐ฅ = ๐พ๐๐๐๐๐ฅ
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 June 1, 2021
Actuator Dynamics
โข Servo Amplifier
Principle of operation
Voltage is generated in PWM (pulse-width modulation)
If the duty cycle is = 50% โ ๐ฃ๐๐ข๐ก = 0
If the duty cycle is > 50% โ ๐ฃ๐๐ข๐ก > 0
If the duty cycle is < 50% โ ๐ฃ๐๐ข๐ก < 0
The PWM frequency is usually > 25๐ป๐ง
55
๐๐๐๐ฅ
๐๐๐๐
time
Duty cycle
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 June 1, 2021
Actuator Dynamics
โข Servo Amplifier
Principle of operation
Current (Torque) mode
An internal current feedback with a tuned PI controller
Ideally considered as a current (torque) source
Back-emf effect limits the performance up to 100 ๐ป๐ง bandwidth
Command signal ๐๐ is tracked within the bandwidth
Velocity (Voltage) Mode
An external tacho (or velocity) feedback with a tuned lead controller
Command signal ๐๐ is tracked within the bandwidth when switched to this mode
Current limiter with current foldback
Usually ยฑ๐๐๐๐ฅ at continuous operation while permitting higher current intermittently
FWD and REV current clamp
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 June 1, 2021
Actuator Dynamics
โข Servo Amplifier (Function Diagram)
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 June 1, 2021
Actuator Dynamics
โข Servo Amplifier (Current Mode Frequency Response)
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 June 1, 2021
Actuator Dynamics
โข Gearbox
Principle of operation
Electric motors provide low torque but have high velocity
Use Gear box to increase the torque while reducing the speed
Ideal Gearbox with ratio ๐
๐๐ = ๐ ๐ and ๐๐ = ๐ ๐ while ๐๐ =1
๐๐
By use principle of virtual work: ๐๐ โ ๐๐ = ๐ โ ๐
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 June 1, 2021
Actuator Dynamics
Motor โ Gearbox โ Load Dynamics
Notation: subscript ๐ for motor side, no subscript for load side
Suppose inertia ๐ผ๐, ๐ผ and viscous friction ๐๐, ๐
Motor side dynamics: Load side dynamics:
๐๐ = ๐ผ๐ แท๐๐ + ๐๐ แถ๐๐ +1
๐๐,
๐ = ๐ผ แท๐ + ๐ แถ๐
Overall dynamics:
๐๐ = ๐ผ๐ แท๐๐ + ๐๐ แถ๐๐ +1
๐๐ผ แท๐ + ๐ แถ๐
Substitute: แถ๐๐ = ๐ แถ๐ and แท๐๐ = ๐ แท๐ and consider ๐๐ =1
๐๐
@ motor side: ๐๐ = ๐ผ๐ +1
๐2๐ผ แท๐๐ + ๐๐ +
1
๐2๐ แถ๐๐
@ Load side: ๐ = ๐2๐ผ๐ + ๐ผ แท๐ + ๐2๐๐ + ๐ แถ๐
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 June 1, 2021
The effect of motor inertia and damping is highlighted
by a factor of ๐2
Actuator Dynamics
Motor โ Gearbox โ Multiple (n) DOFs Robot
Shorthand Dynamics:
๐ = ๐ผ๐ แท๐ + ๐๐ แถ๐
In which the effective inertia and viscous friction are:
๐ผ๐ = ๐2๐ผ๐ + ๐ผ and ๐๐ = ๐2๐๐ + ๐
Consider the general model of the robot
๐ โ ๐ แถ๐ = ๐ด ๐ แท๐ + ๐ช ๐, แถ๐ แถ๐ + g ๐
Add motor โ gearbox dynamics
๐ โ ๐๐ แถ๐ = ๐ด๐ แท๐ + ๐ช ๐, แถ๐ แถ๐ + g ๐
In which:
๐ด๐ ๐ = ๐ด ๐ +
๐12๐ผ๐1
โฆ 0
โฎ โฑ 00 โฆ ๐๐
2๐ผ๐๐
and ๐๐ =
๐1 + ๐2๐๐1
โฎ๐๐ + ๐2๐๐๐
61
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 June 1, 2021
Actuator Dynamics
Motor โ Gearbox โ Multiple (n) DOFs Robot
If ๐๐ โซ 1 โ ๐๐2 dominates and dynamic equations will be decoupled
๐๐ โ ๐๐๐ แถ๐๐ = ๐๐๐ ๐ + ๐๐2๐ผ๐๐
แท๐๐ + ๐ฃ๐ ๐, แถ๐ + ๐๐(๐)
If ๐๐ โ 1 โ ๐๐2 dominates and the dynamics becomes decoupled and linear
๐ผ๐๐ = ๐๐๐ + ๐๐2๐ผ๐๐
โ ๐๐2๐ผ๐๐
๐๐๐ = ๐๐ + ๐๐2๐๐๐
โ ๐๐2๐๐๐
Then ๐๐ ๐, แถ๐ + ๐บ๐ ๐ may become negligible compared to ๐ผ๐๐ แท๐๐ + ๐๐๐ แถ๐๐
We usually keep the gravity term ๐๐ ๐ but may neglect ๐ฃ๐ ๐, แถ๐ then
๐๐ = ๐ผ๐๐ แท๐๐ + ๐๐๐ แถ๐๐ + ๐๐(๐)
62
1
๐ผ๐๐๐ 2 + ๐๐๐๐
๐๐ ๐
+
๐๐(๐ )๐๐ โ
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 June 1, 2021
Contents
In this chapter we review the dynamics analysis for serial robots. First the definition to angular and linear accelerations are given, then mass properties, linear and angular momentums, and kinetic energy of a rigid body in space is defined. Lagrange formulation is given in general form, and dynamics mass matrix, gravity and Coriolis and centrifugal vectors are defined and derived for several case studies. Dynamic formulation properties is given next, then actuator dynamics is elaborated for electrically driven robots with gearbox. Finally, linear regression method is used for dynamics calibration, and model verification methods are elaborated by introducing consistency measure.
63
Actuator DynamicsElectrical actuators, permanent magnet DC motors, servo amplifiers, gearbox, motor-gearbox-load dynamics, motor-gearbox-multiple joint robot,
4
Dynamics CalibrationLinear Regression, linear model with constant gravity, house holder reflection, varying gravity term, filtered velocity, model verification, consistency measure.
5
PreliminariesAngular acceleration, linear acceleration of a point, mass properties, center of mass, moments of inertia, inertia matrix transformations, linear and angular momentum, kinetic energy.
1
Lagrange FormulationMotivating example, generalized coordinates and forces, kinetic energy, mass matrix, potential energy, gravity vector, Coriolis and centrifugal vector, case studies.
2
Dynamic Formulation PropertiesMass matrix properties, linearity in parameters, Christoffel Matrix, skew-symmetric property, general dynamic formulation, passivity,
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 June 1, 2021
Dynamics Calibration
โข Linear Regression
Given the structure of the linear model for each joint
๐๐ = ๐ผ๐๐ แท๐๐ + ๐๐๐ แถ๐๐ + ๐๐(๐)
The dynamics parameters ๐ผ๐๐, ๐๐๐ and ๐บ๐ may be identified by linear regression
Case (1)
Assume a set of experiments is done, and input torques ๐, and output motions ๐, แถ๐, แท๐ for each joint
are logged for a large number of samples
๐ =
๐1๐2โฎ๐๐
, ๐ =
๐1๐2โฎ๐๐
, แถ๐ =
แถ๐1แถ๐2โฎแถ๐๐
, แท๐ =
แท๐1แท๐2โฎแท๐๐
for ๐ = 1,2, โฆ ,๐ samples
Assume the gravity term is set as a constant parameter ๐๐ ๐ = ๐๐.
The dynamic formulation for each link may be reformulated as a linear regression:
๐ = ๐ผ๐๐ แท๐ + ๐๐๐ แถ๐ + ๐๐ = ๐จ ๐, แถ๐, แท๐ ๐ฝ In which ๐จ ๐, แถ๐, แท๐ =
แท๐1แท๐2โฎแท๐๐
แถ๐1แถ๐2โฎแถ๐๐
11โฎ1
,๐ฝ =
๐ผ๐๐๐๐๐๐๐
, ๐ =
๐1๐2โฎ๐๐
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 June 1, 2021
Dynamics Calibration
โข Linear Regression
The linear Regression ๐ = ๐จ ๐, แถ๐, แท๐ ๐ฝ is an overdetermined set of equations
Find least-squares solution for the parameters by left-pseudo-inverse
๐ฝ = ๐จโ โ ๐ in which ๐จโ = ๐จ๐๐จ โ1๐จ๐ป
Direct evaluation of ๐จโ might be intractable, use house holder reflection
Factorize ๐จ into ๐จ๐ท = ๐ธ๐น
In which ๐ท๐ร๐ is a permutation matrix, ๐ธ๐ร๐ is an orthogonal matrix, and ๐น๐ร๐ is upper triangular.
Then
๐ฝ = ๐ท๐นโ1 ๐ธ๐๐
And ๐นโ1 is calculated by back substitution.
Note: Matlab pinv(Y)calculates the right-pseudo inverse by householder reflection.
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 June 1, 2021
Dynamics Calibration
โข Linear Regression
Given the structure of the linear model for each joint
๐๐ = ๐ผ๐๐ แท๐๐ + ๐๐๐ แถ๐๐ + ๐๐(๐)
Case (2):
Complete input-output logging and general gravity term ๐๐ ๐ .
In many examples ๐๐ ๐ = ๐๐ cos ๐ .
๐๐1 =1
2๐1๐1๐ cos(๐1) , ๐๐2 =
1
2๐2๐2๐ cos(๐1 + ๐2)
Then
๐ = ๐ผ๐๐ แท๐ + ๐๐๐ แถ๐ + ๐๐ cos ๐ = ๐จ ๐, แถ๐, แท๐ ๐ฝ
In which
๐จ ๐, แถ๐, แท๐ =
แท๐1แท๐2โฎแท๐๐
แถ๐1แถ๐2โฎแถ๐๐
cos ๐1cos ๐2โฎ
cos ๐๐
, ๐ฝ =
๐ผ๐๐๐๐๐๐๐
, ๐ =
๐1๐2โฎ๐๐
And therefore,
๐ฝ = ๐จโ โ ๐ or ๐ฝ =pinv(๐จ)* ๐
66
๐ฅ
๐ฆ
๐1
๐2
๐1๐
๐2๐๐๐1
๐๐2
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 June 1, 2021
Dynamics Calibration
โข Linear Regression
Given the structure of the linear model for each joint
๐๐ = ๐ผ๐๐ แท๐๐ + ๐๐๐ แถ๐๐ + ๐๐(๐)
Case (3): Only ๐ and ๐ are measured
Use filtered differentiation
In which
๐๐ ๐ป๐ง =1
๐๐is selected such that 10 ๐๐ต๐ < ๐๐ < 0.1 ๐๐๐๐๐ ๐
Where ๐๐ต๐ denotes the system bandwidth frequency while ๐๐๐๐๐ ๐ denotes the major noise
frequency content
Practically 0.001 < ๐๐ < 0.05 or equivalently 20 < ๐๐ < 103 would be a good choice.
Use Matlab command filtfilt to remove the delay in the filtered signal
Use the filter twice to find the angular acceleration แท๐๐.
Check the filtered differentiation output and tune ๐๐ to reduce the output noise.
67
๐
๐๐๐ + 1
แถ๐๐๐
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 June 1, 2021
Dynamics Calibration
โข Model Verification
Note that the simplified linear model is good
For linear controller design
Shall be valid for different operating regimes
If ๐ > 100 the linear term dominates the nonlinear vector v(๐, แถ๐)
Verify the calibrated dynamic parameters
For different experiments by statistical analysis
68
Consistency Measure
AVGExp 10. . .Exp 2Exp 1
STDEV / ๐ผ๐๐๐ฃ๐๐ผ๐๐๐ฃ๐10๐ผ๐. . .2๐ผ๐
1๐ผ๐๐ผ๐
STDEV / ๐๐๐๐ฃ๐๐๐๐๐ฃ๐10๐๐. . .2๐๐
1๐๐๐๐
STDEV / ๐บ๐๐ฃ๐๐๐๐ฃ๐10๐. . .2๐1๐๐
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 June 1, 2021
Dynamics Calibration
โข Model Verification
Verify the calibrated dynamic parameters
For different experiments by statistical analysis
If C.M. < 30% The average parameter is suitable for controller design
If C.M. < 80% The averaged parameters could be used for controller design using
robust linear controllers
If C.M. > 80% Then your model is incomplete and you need to add terms in your
model
If for different experiments the obtained parameters are physically inconsistent
For example you get negative moment of inertia or damping parameters
This means your model is still not good enough for calibration
You may use constrained optimization to add bound on the parameters
Matlab fmincon command may be used in this case.
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 June 1, 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 June 1, 2021
Chapter 5: Dynamic 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