Robot Dynamics I - UFMGtorres/wp-content/uploads/2018/02/MCR... · 2020. 3. 9. · Leonardo Torres Robot Dynamics I { PPGEE/UFMG. Introduction to Robotics Robot Dynamics Spaces in

Robot Dynamics OverviewSpaces in Robotics

Kinematic and Dynamic ControlReference Frames

Robot Dynamics I

Leonardo Torres

Electrical Engineering Graduate Program – UFMG

July 4, 2020

Leonardo Torres Robot Dynamics I – PPGEE/UFMG



1 Robot Dynamics Overview

2 Spaces in Robotics

3 Kinematic and Dynamic Control

4 Reference Frames




What is a Robot?

According to RIA – Robot Institute ofAmerica

A Robot is a reprogrammablemultifunctional manipulator designed tomove material, tools, or specialized devicesthrough variable programmed motions forthe performance of a variety of tasks.

COMAU Smart Six Robot.




What is a Robot?

Our definition of Robot

A Robot is an integrated system comprised bymechanisms, sensors and processors, that isreprogrammable and multifunctional, anddesigned to move itself and possibly othermaterials, tools or specialized devices in orderto perform different tasks.

http://www.verlab.dcc.ufmg.br/projetos/

roomba/index


http://www.verlab.dcc.ufmg.br/projetos/roomba/index

http://www.verlab.dcc.ufmg.br/projetos/roomba/index



Kinematics × Dynamics

In Robot Kinematics we are interested in describing the relationsbetween positions/poses and velocities of each part of the robot,without considering forces and torques. Important concepts are:

1 Configuration Space and Degrees of Freedom;2 Local Reference Frames;3 Forward Kinematic maps and the associated Jacobian matrices;

In Robot Dynamics our goal is to understand how forces and torquesdetermine the accelerations of each part of the robot.




Kinematics × Dynamics

In Robot Kinematics we are interested in describing the relationsbetween positions/poses and velocities of each part of the robot,without considering forces and torques. Important concepts are:

1 Configuration Space and Degrees of Freedom;2 Local Reference Frames;3 Forward Kinematic maps and the associated Jacobian matrices;

In Robot Dynamics our goal is to understand how forces and torquesdetermine the accelerations of each part of the robot.




Robot Dynamics: Motivation

Robotic Manipulators and other robotic mechanisms are oftendesigned specifically to position tools and materials very accurately,such that only their kinematic description is necessary to programtheir movement.

However, in accurate computer simulations as well in the designof low-level controllers, one must take into consideration therequired forces and torques that make the robotic mechanisms moveas desired.

Finally, by studying the dynamics of robotic mechanisms we actuallyunderstand how to model mechanical systems, and this knowledge isvital in the design of controllers for a large and interesting class ofsystems: automobiles, submarines, boats and ships, airplanes,helicopters, legged and wheeled robots, etc.


















Course Objectives

The main topics to be discussed:

Dynamic Modelling: the Euler-Lagrange approach,

Dynamic Modelling: the Newton-Euler approach,

Modelling of Aerial Robots,

How to incorporate non-holonomic constraints,

Introduction to Robot Control Strategies.

Recommended books:

Mark W. Spong, Seth Hutchinson and M. Vidyasagar. RobotModeling and Control. John Wiley & Sons, Inc. 2006.

Bruno Siciliano, Lorenzo Sciavicco, Luigi Villani and GiuseppeOriolo. Robotics: Modelling, planning and control. Springer-VerlagLondon Limited. 2009.

Richard M. Murray, Zexiang Li and S. Shankar Sastry. AMathematical Introduction to Robotic Manipulation. 1994.




Course Objectives







Recommended books:







Course Objectives







Recommended books:







Course Objectives







Recommended books:







Course Objectives







Recommended books:







Course Objectives







Recommended books:







Classification of Robots

Robots can be grouped together according to different criteria:

Type of Movement:1 Fixed basis: robotic manipulators;2 Mobile basis: mobile robots.

Power source: electrical, hydraulic, pneumatic.

Method to Control/Specify the Movement:1 Point-to-point: only discrete target points in the robot trajectory are

specified;2 Continuous path: not only the target points, but the way to move

from one point to another is also specified.

Geometry: articulated (RRR), spherical (RRP), SCARA (RRP),cylindrical (RPP), Cartesians (PPP).





































Types of Joints

Revolute:

Source:http://www.mathworks.com/help/toolbox/physmod/mech/ref/revolute.html


http://www.mathworks.com/help/toolbox/physmod/mech/ref/revolute.html



Types of Joints

Prismatic:

Source:http://www.mathworks.com/help/toolbox/physmod/mech/ref/prismatic.html


http://www.mathworks.com/help/toolbox/physmod/mech/ref/prismatic.html



Types of Joints

Spherical:

Source:http://www.mathworks.com/help/toolbox/physmod/mech/ref/spherical.html

It is possible to have the same effect by combining three revolutejoints whose axes of rotation intersect at one point: spherical wrist.


http://www.mathworks.com/help/toolbox/physmod/mech/ref/spherical.html



Common Kinematic Arrangements

Articulated (RRR):

http://www6.district125.k12.il.us/teched/courses/tdresources/RobotTypes.html





Cylindrical (RPP):






Cartesian (PPP):






Spherical (RRP):





SCARA – Selective Compliant Assembly Robot Arm (RRP):




The World

World

The formal definition of the primary space of points where the robotmovements will take place. This is also the space where obstacles usuallywill be defined as forbidden sets of points.

Some examples:

The unlimited Euclidean 3D space;

The unlimited Euclidean 2D space (e.g. planar robots);

The limited 3D space of an office, considered as a subset of theEuclidean space;

The surface of a sphere (non-Euclidean space, e.g. robots for theinspection of spherical gas tanks in an oil refinery).




The World

World


Some examples:








The World

World


Some examples:








The World

World


Some examples:








The Workspace

Workspace

The total volume swept out by the end effector as the robot executes allpossible motions. The emphasis is on the allowable positions for a pointin the robot structure where a tool could be attached.

The Workspace is further divided in two sets:

The Reachable Workspace ΩR: the set of points reachable by therobot;

The Dexterous Workspace ΩD: the set of points reachable by therobot from which it is possible to have arbitrary orientations andvelocities of the end effector.

ΩD ⊆ ΩR




The Workspace

Workspace

The total volume swept out by the end effector as the robot executes allpossible motions. The emphasis is on the allowable positions for a pointin the robot structure where a tool could be attached.

The Workspace is further divided in two sets:

The Reachable Workspace ΩR: the set of points reachable by therobot;

The Dexterous Workspace ΩD: the set of points reachable by therobot from which it is possible to have arbitrary orientations andvelocities of the end effector.

ΩD ⊆ ΩR




Workspace – Examples

Source: [2]

Cylindrical Robotic Manipulator.





Source: [2]

Spherical Robotic Manipulator.





Source: [2]

SCARA Robotic Manipulator.1

1There is a small error in this figure, which was corrected in the second edition of[2]: the edges of the workspace at the limits of the angular displacement are actuallycurves instead of straight lines.





Source: [2]

Cartesian Robotic Manipulator.




The Configuration Space

Robot Configuration

A minimal set of variables necessary to determine the world position ofany material point of the robot, departing from the knowledge of itsgeometry (lengths, diameters, etc). This minimal set will be representedby ~q.

Configuration Space QThe set of all possible values for ~q. It will be denoted by Q.

Degrees of Freedom

The dimension of the Configuration Space Q.




Configuration Space – Examples

1 2 degrees of freedom – DoF planar robotic arm:

~q =

[q1q2

]=

[θ1θ2

].

Notice that, if one knows ~q and therobot geometry, the position of anymaterial point of the robot in theworld can be determined.




Configuration Space – Examples

1 2 degrees of freedom – DoF planar robotic arm:

http://en.wikipedia.org/wiki/File:Torus_cycles.png

The Configuration Space for the 2DoF manipulator is non-Euclidean.Indeed it is a differentiable manifoldsuch that q1 ∈ [0; 2π], e q2 ∈ [0; 2π].In other words Q ≡ S1 × S1, withS1 the set of points on a circle.Since it is not homeomorphic to R2,minimum distances between pointsin Q are not straight lines.


http://en.wikipedia.org/wiki/File:Torus_cycles.png



The State Space

State

A minimum set of variables ~x(t) whose values at each time t ≥ t0 can beuniquely determined from: (i) the knowledge of ~x(t0), (ii) the differentialequations describing the system dynamics and (iii) the external signals(inputs) acting on the system.

The State can be considered as a representation of the internal memoryof the system, and the state variables are the elements of the StateVector, or simply State, denoted by ~x. Sometimes the state variables arecalled internal or auxiliary variables.

State Space X

The set of all posible States, i.e. ~x ∈ X.




State Variables

How to tell if you have a valid set of state variables?

Answer: Try to write down the tendency to change of each candidatestate variable. If you succeed in showing that each variable tendency tochange is a function of the current values of the candidate state variablesand the inputs to the system1, then you have a valid set of states.

The goal is to be able to write that:

~x = f(~x,~u)⇔

x1x2...xn

=

f1(x1, x2, . . . , xn, u1, u2, . . . , um)f2(x1, x2, . . . , xn, u1, u2, . . . , um)

...fn(x1, x2, . . . , xn, u1, u2, . . . , um)

.

1And there is no way to reduce the number of variables and still be able to explainthe time evolution of all the variables of interest.




State Variables




~x = f(~x,~u)⇔

x1x2...xn

=


...fn(x1, x2, . . . , xn, u1, u2, . . . , um)

.





State Variables




~x = f(~x,~u)⇔

x1x2...xn

=


...fn(x1, x2, . . . , xn, u1, u2, . . . , um)

.





State Space Representations I

A set of differential equations, sometimes called dynamic equations,

~x = f(~x,~u) ⇔

x1

x2

...xn

=


...fn(x1, x2, . . . , xn, u1, u2, . . . , um)

,

together with a set of algebraic equations, called output equations,

~y = h(~x,~u) ⇔

y1y2...yp

=

h1(x1, x2, . . . , xn, u1, u2, . . . , um)h2(x1, x2, . . . , xn, u1, u2, . . . , um)

...hp(x1, x2, . . . , xn, u1, u2, . . . , um)

,

forms a State Space Representation of the system dynamics.




State Space Representations II

For a given system there is an infinite number of equivalent statespace representations since there are infinite possible choices for thestate variables.

To see this, notice that, for example, it is easy to obtain anotherState Space Representation just by using a new set of variablesgiven by ~x ′ = M~x, where M ∈ Rn×n is a non-singular matrix.




Robot Control: Two different views I

In Robotics, usually one has as state variables the configurationvariables and the corresponding time derivatives, i.e.

~x =

~q~q

,such that the dimension of the State Space X is twice thedimension of the Configuration Space Q. ~q is usually related to thePotential Energy of the Robot, and ~q is usually related to itsKinectic Energy. This is the common framework when discussinghow applied forces and torques drive the robot’s movements, i.e. aDynamic Description.




Robot Control: Two different views II

Another possibility, which is also quite common, is to represent theRobot’s movements as a consequence of defining, directly, thevelocities that will be exhibited by the mechanism, i.e. a KinematicDescription. In this case,

~x = ~q,

and the Configuration Space Q is the same as the State Space X.




Reference Frames and Robotics I

It is essential to know how to describe the robot movements consideringappropriately chosen reference frames.

http://www.quarcservice.com/ReleaseNotes/files/quarc_using_devices_standard_axes.html

As an example, for robotic manipulators it is usual to associate to eachjoint a reference frame whose z-axis coincides with the axis of rotation(revolute joints) or translation (prismatic joints).


http://www.quarcservice.com/ReleaseNotes/files/quarc_using_devices_standard_axes.html



Reference Frames and Robotics II

Source: [1].Source: [3].

Another example is the control of aerial robots. The movementdescription commonly depends on the definition of an inertial referenceframe, and at least another reference frame attached to the vehicle (bodyframe).Therefore, it is very important to know how to relate therepresentations of a point or free-vector in different reference frames.




Some Interesting Software

Peter Corke’s Robotics Toolbox for MATLAB. Functions torepresent the kinematic chain of robotic manipulators, including theuse of Denavit-Hartenberg parameters and trajectory generation:

https://petercorke.com/toolboxes/robotics-toolbox/

RoKiSim – Robotics Kinematics Simulator. Easy way tovisualize movements of some industrial robotic manipulators:

http://www.parallemic.org/RoKiSim.html

Coppelia Robotics – CoppeliaSim. Formerly known as VirtualRobot Experimentation Platform (V-REP). A virtualenvironment to create robots with collision detection and 3Danimation:

http://coppeliarobotics.com/


https://petercorke.com/toolboxes/robotics-toolbox/

http://www.parallemic.org/RoKiSim.html

http://coppeliarobotics.com/



Tarek Hamel and Robert Mahony.Image based visual servo control for a class of aerial robotic systems.Automatica, 43(11):1975 – 1983, 2007.

Mark W. Spong, Seth Hutchinson, and M. Vidyasagar.Robot Modeling and Control.Draft, first edition, 2005.

Haitao Xiang and Lei Tian.Development of a low-cost agricultural remote sensing: system basedon an autonomous unmanned aerial vehicle (uav).Biosystems Engineering, 108(2):174 – 190, 2011.


Documents

Robot Dynamics I - UFMGtorres/wp-content/uploads/2018/02/MCR... · 2020. 3. 9. · Leonardo Torres Robot Dynamics I { PPGEE/UFMG. Introduction to Robotics Robot Dynamics Spaces in